Log-Gases and Random Matrices
London Mathematical Society Monographs Editors: Martin Bridson, Terry Lyons, and Peter Sarnak Editorial Advisers: Mikhail Gromov, Jean-Francois Le Gall, and Richard Taylor The London Mathematical Society Monographs Series was established in 1968. Since that time it has published outstanding volumes that have been critically acclaimed by the mathematics community. The aim of this series is to publish authoritative accounts of current research in mathematics and high-quality expository works bringing the reader to the frontiers of research. Of particular interest are topics that have developed rapidly in the last ten years but that have reached a certain level of maturity. Clarity of exposition is important and each book should be accessible to those commencing work in its field. The original series was founded in 1968 by the Society and Academic Press; the second series was launched by the Society and Oxford University Press in 1983. In January 2003, the Society and Princeton University Press united to expand the number of books published annually and to make the series more international in scope.
LMS-34. Log-Gases and Random Matrices, by P. J. Forrester LMS-33. Prime-Detecting Sieves, by Glyn Harman LMS-32. The Geometry and Topology of Coxeter Groups, by Michael W. Davis LMS-31. Analysis of Heat Equations on Domains, by El Maati Ouhabaz
Log-Gases and Random Matrices
P.J. Forrester
PRINCETON UNIVERSITY PRESS PRINCETON AND OXFORD
Copyright © 2010 by Princeton University Press Requests for permission to reproduce material from this work should be sent to Permissions, Princeton University Press Published by Princeton University Press, 41 William Street, Princeton, New Jersey 08540 In the United Kingdom: Princeton University Press, 6 Oxford Street, Woodstock, Oxfordshire OX20 1TW press.princeton.edu All Rights Reserved
Library of Congress Cataloging-in-Publication Data Forrester, Peter (Peter John) Log-gases and random matrices / P.J. Forrester. p. cm. -- (London Mathematical Society monographs) ISBN 978-0-691-12829-0 (hardcover : alk. paper) 1. Random matrices. 2. Jacobi polynomials. 3. Integral theorems. I. Title. QA188.F656 2010 519.2--dc22 2009053314
British Library Cataloging-in-Publication Data is available The publisher would like to acknowledge the author of this volume for providing the camera-ready copy from which this book was printed Printed on acid-free paper. f Printed in the United States of America 10 9 8 7 6 5 4 3 2 1
Preface
Often it is asked what makes a mathematical topic interesting. Some qualities which come to mind are usefulness, beauty, depth and fertility. Usefulness is usually measured by the utility of the topic outside mathematics. Beauty is an alluring quality of much of mathematics, with the caveat that it is often something only a trained eye can see. Depth comes via the linking together of multiple ideas and topics, often seemingly removed from the original context. And fertility means that with a reasonable effort there are new results, some useful, some with beauty, and a few maybe with depth, still awaiting to be found. More than fifteen years ago I embarked on a project to write in monograph form a development of the theory of solvable log-gas systems in statistical mechanics. As a researcher in the field, I had personally witnessed and experienced some of the interesting qualities of this topic, and I was keen that these be recorded in a form which could serve as a reference for researchers in related fields. Little did I realize that in the ensuing years these related fields would be the subject of intense research activity, requiring a revision of both the focus of the book, and my own research directions, to properly reflect these developments. Although my focus thus evolved away from the statistical mechanics of log-gas systems, this subject still proved to be a unifying theme in the presentation of the subject matter. And as a further give away as to my own research origins, there is a fairly strong flavor of the language of classical equilibrium statistical mechanics throughout, although a similar background of the reader can hardly be expected. More likely the motivation of the reader will come from the topics of random matrices, Painlev´e systems, stochastic growth processes, or Jack polynomials. These are some of the the related fields referred to above, which have been the subject of much recent activity, and which promise to remain interesting topics into the future. Of these it is random matrices which appears along side log-gases as the unifying theme of the book. This marriage of topics has a fine historical pedigree, with the log-gas picture of eigenvalues of random matrices being used to great advantage in the pioneering work of Dyson [147]. While providing a directed logical framework, a development of the intersection between log-gases and random matrices necessarily excludes substantial portions of each of the topics taken separately. However, the latter is necessary in order to achieve a mostly self-contained presentation. Seeking the common intersection of two topics can then be seen as a way of achieving this in a fairly democratic manner. In addition there is intersection with a third topic at work, keeping a further bound on the content, but also being responsible for much of the richness of the mathematics. This third topic is integrable systems. In general the exact calculation of correlations and probability distributions for interacting statistical mechanical systems is an intractable problem; however, underlying integrable structures make log-gases and random matrices an exception. The development of this topic leads to the study of determinantal and Pfaffian processes and the corresponding orthogonal polynomials, as well as Painlev´e systems and Jack polynomials. The quality of usefulness marked the beginning of the study of random matrices and log-gases in mathematical physics. As already mentioned, log-gases were introduced as a tool by Dyson to study random matrices, or as expressed in [201], to liberate the mathematics where none yet exists. Random matrices themselves were introduced by Wigner as a model for the statistical properties of the highly excited energy levels of heavy nuclei. Many of the early works on this theme (up to 1965) are conveniently collected together in the work of Porter [447], along with an introductory review. Long before their occurrence in physics, random matrices appeared in mathematics, especially in relation to the Haar measure on classical groups. Perhaps the first work of this type is due to Hurwitz, who computed the
vi
PREFACE
volume form of a general unitary matrix parametrized in terms of Euler angles [301]. The book of Weyl [540] contains the Haar volume form written in terms of eigenvalues and eigenvectors for the classical groups, and the book of Hua [300] inter-relates these forms to similar measures relating to spaces of Hermitian matrices. In mathematical statistics Wishart [547] gave the volume form of a rectangular matrix X in terms of the volume of the corresponding positive definite matrix XT X. Two other early mathematical works of lasting importance to the field are those of Dixon [137] and Selberg [483], both of which relate to multidimensional integrals with integrands which can be interpreted as probability measures associated with random matrices. The historical development of random matrices is well documented. Two recent informative accounts are [226], [75]. However, as already stated, the present work addresses only the intersection of the topics of log-gases, random matrices and integrable systems, and so a more extensive historical introduction beyond that already given does not serve as as an informative introduction to the content. Instead it is perhaps worth isolating some of our major topics, giving them some context and providing commentary on how they are to be developed. Jacobians All the works referenced above in relation to how random matrices appear in mathematics relate to Jacobians. To gain insight into the prevalance of Jacobians throughout random matrix theory, consider, for example, the problem of studying the eigenvalues of an N × N real symmetric random matrix, in the situation that the joint distribution on the space of the independent elements is given. The dimension of this space is N (N + 1)/2. The eigenvalue/eigenvector decomposition provides a change of variables from the independent elements of the matrix to its N eigenvalues and N (N − 1)/2 variables associated with its eigenvectors. A strategy then to study the eigenvalues is to perform this change of variables, and an essential ingredient for this task is the computation of the corresponding Jacobian. In the case of real symmetric matrices, complex Hermitian and quaternion real Hermitian matrices, these Jacobians are computed in Chapter 1. Chapter 1 also contains the computation of the Jacobian for a change of variables from the independent elements of an N ×N real symmetric tridiagonal matrix to its eigenvalues and a further N − 1 independent variables relating to its eigenvectors, and a Jacobian relating to the Householder transformation. In Chapter 2 Jacobians are computed in relation to spaces of unitary matrices, including orthogonal and symplectic unitary matrices, which have dimensions O(N 2 ). Jacobians are also computed for the change of variables from the elements to the eigenvalues and variables relating to the eigenvectors for certain unitary and real orthogonal Hessenberg matrices. In these latter circumstances the underlying spaces are of dimension O(N ). The singular value decomposition of rectangular matrices (or equivalently certain decompositions of positive definite matrices), the block decomposition of unitary matrices, and positive definite matrices formed from bidiagonal matrices are some of the settings which give rise to calculations of Jacobians undertaken in Chapter 3. Jacobians of a different sought appear in Chapter 4. Here rational functions with random coefficients in their partial fraction expansion are encountered, and we seek to change variables from a description in terms of these coefficients to one in terms of the zeros. For this purpose use is make of tools already known from the computation of Jacobians in Chapters 1–3, in particular the calculus of wedge products, and also the classical Vandermonde and Cauchy determinants. In Chapter 11 Jacobians are encountered in the change of variables of differential operators given in terms of the elements of parameter-dependent random matrices, to the differential operators given in terms of corresponding eigenvalues and variables relating to the eigenvectors. Finally, in Chapter 15, a task similar to that addressed in Chapter 1 is undertaken, namely the change of variables from the description of N × N real, complex, or quaternion real matrices in terms of the independent elements, to one in terms of the eigenvalues (which are typically complex) and an appropriate number of other variables. Also computed are some Jacobians relating to the change of variables of a random polynomial from its coefficients to its zeros. Determinantal point processes and orthogonal polynomials of one variable A determinantal point process is a statistical system of many particles (points) in which the k-point corre-
PREFACE
vii
lation function is a k × k determinant for each k. The study of eigenvalues of random matrices with complex entries, and also of log-gas systems at the special coupling β = 2 (in the cases considered in this work, the former are mostly special cases of the latter, due to our subject matter being typically restricted to the intersection of the two fields) gives rise to determinantal point processes. Furthermore, the corresponding determinants are determined by just one quantity, referred to as the correlation kernel. To exhibit this fact an essential role is played by orthogonal polynomials. It turns out that in the cases of interest it is the classical orthogonal polynomials which are required. Because full information on the asymptotic properties of these polynomials is known in the existing literature, it is possible to proceed and calculate scaling limits. A generalization of a determinant point process is a Pfaffian point process, in which the k-point correlation function is a 2k × 2k Pfaffian (or equivalently a k × k quaternion determinant) for each k. The eigenvalues of matrix ensembles studied in Chapters 1–3 in which the matrices are diagonalized by real orthogonal or symplectic unitary matrices are examples of Pfaffian point processes. These eigenvalues can be interpreted in terms of log-gas systems at the particular coupling β = 1 and β = 4 respectively. In the theory of Pfaffian processes skew orthogonal polynomials play a role analogous to that played by orthogonal polynomials in the theory of determinantal processes. For the particular skew inner products encountered from the random matrix problems of Chapters 1–3, the required skew orthogonal polynomials can be expressed in terms of classical orthogonal polynomials, and moreover the elements of the Pfaffian are determined by a single 2 × 2 block, the elements of which can be expressed in a summed form suitable for asymptotic analysis. In Chapter 15 non-Hermitian Gaussian random matrices are studied, with real, complex, and real quaternion entries. The eigenvalues in the complex case form a determinantal point process, while in the other two cases a Pfaffian point process results. The Selberg integral, Jack polynomials and generalized hypergeometric functions Familiar in the theory of the Gauss hypergeometric function is the Euler integral, which has the feature that it can be evaluated in terms of gamma functions. The Selberg integral can be considered as an N -dimensional generalization of the Euler integral. In a random matrix context, it appears as the normalization of various ensembles considered in Chapters 1–3. In a log-gas context, it gives the partition function for general β > 0. When written in a trigonometric form, extra parameters can be interpreted as providing the full distribution of certain linear statistics in the circular β-ensemble. In the case β = 2, and in the limit N → ∞, this ties in with the Fisher-Hartwig asymptotic formula from the theory of Toeplitz determinants, covered in Chapter 14. One of the structures underlying the Selberg integral is a further multidimensional integral referred to as the Dixon-Anderson integral. Like the Selberg integral, it can arrived at by the consideration of a problem in random matrix theory, and it too can be evaluated in terms of gamma functions. The many free parameters in the Dixon-Anderson integral allow for an interpretation giving an inter-relation between the distribution of every second eigenvalue in classical matrix ensembles at β = 1, and the joint distribution of the eigenvalues for a related classical matrix ensemble at β = 4. The integrand of the Selberg integral and its various limits is, up to normalization, the eigenvalue probability density function of the various classical β-ensembles given in Chapters 1–3. Theory linking the Selberg integral with the Dixon-Anderson integral can also be used to provide stochastic three-term recurrences (in the degree N ) for the corresponding characteristic polynomials. The integrand of the Euler integral is the weight function for the classical Jacobi polynomials (when defined on the interval [0, 1]). Likewise, the integrand of the Selberg integral, and its various limiting forms, can be used to define inner products which permit complete sets of orthogonal polynomials with special properties. The most fundamental are the Jack polynomials, which relate to the integrand of the Selberg integral in trigonometric form, specialized to correspond to the eigenvalue probability density function for the circular β-ensemble. Using the Jack polynomials as a basis, generalized classical Hermite, Laguerre and Jacobi polynomials, which are multivariable counterparts of the one-variable classical orthogonal polynomials of the same name, can be studied. Another viewpoint of the Jack polynomials is as the polynomial (in complex exponential variables) portion of the eigenfunctions for the Fokker-Planck operator of the Dyson Brownian motion model of the log-gas on a circle. This topic is developed in Chapter 11. A crucial feature is an alge-
viii
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braic theory of the Fokker-Planck operator, in which it is decomposed into fundamental commuting operators relating to the degenerate Hecke algebra of type A, and involving exchange operators. The presentation of the theory of Jack polynomials given in Chapter 12 begins from a study of the simultaneous nonsymmetric polynomial eigenfunctions of these commuting operators. The underlying degenerate Hecke algebra allows these polynomials to be constructed inductively using two fundamental operations (transposition and raising), and these operations allow for the explicit evaluation of associated scalar quantities such as various normalizations. The operations of symmetrization and antisymmetrization also play an important role. It is well known that the Euler integral can be extended to provide the solutions of the Gauss hypergeometric differential equation. Likewise, weighting the Selberg integrand by an appropriate factor gives rise to multidimensional integrals which relate to multidimensional hypergeometric functions based on Jack polynomials. These are studied in Chapter 13. With the parameters specialized, these integrals can be interpreted as correlations for log-gas systems. Duality formulas, in which multidimensional integrals of this type are expressed as other multidimensional integrals, this time of dimension independent of N , provide the basis for the asymptotic analysis of the corresponding correlations for all even β at least. Furthermore, Jack polynomial theory can be used to compute the bulk dynamical two-point density-density correlation for the Dyson Brownian motion model perturbed from its equilibrium state for all values of rational values of β. Painlev´e transcendents The Painlev´e differential equations are a distinguished family of second order nonlinear equations. In applied mathematics they are perhaps best known for their role in soliton theory, and thus the study of integrable partial differential equations. Certain solutions of the Painlev´e differential equations — the Painlev´e transcendents — appear in the calculation of gap probabilities for classical random matrix systems corresponding to log-gas systems with β = 1, 2 and 4 (although the latter two are restricted to those instances in which their is an inter-relation with a β = 2 log-gas system; one way the latter comes about is by superimposing two β = 1 ensembles, and integrating over every second eigenvalue, while in the bulk the β = 1 gap probability is transformed by making use of an evenness symmetry). The viewpoint taken in Chapter 8 on these calculations is an algebraic theory of Painlev´e systems based on a Hamiltonian formulation, due mainly to Okamoto, which has the feature of using the Toda lattice equation to inductively construct determinant solutions from a seed solution (the latter relating to an underlying linear second order equation). These determinants can be identified with the gap probabilities of certain log-gas systems at β = 2. Moreover (formal) scaling of the differential operators gives analogous characterization of the gap probabilities in various scaling limits. As a consequence of these characterizations, high precision calculation of the gap probabilities can be undertaken. In Chapter 9 additional viewpoints on these results are considered. One is the study of function theoretic properties of the gap probabilities expressed as Fredholm determinants. Indeed, starting with the Fredholm determinant form seems necessary to account for the scaled limit rigorously. Instead of using function theoretic properties, this starting point can also be developed from a Riemann-Hilbert viewpoint, which in turn is closely related to studying isomonodromic deformations of linear second order differential equations. The Fredholm determinant evaluations allow the high precision calculations of the gap probabilities initiated from the Painlev´e evaluations to be extended. Macroscopic electrostatics and asymptotic formulas Averaging a linear statistic against the eigenvalue spectrum of a random matrix gives a mean value proportional to N (the number of eigenvalues), but a variance of order unity. In applications, this effect shows itself in the study of the statistical properties of the conductance of a quantum wire, noted in Chapter 3. It can be anticipated, and a precise formula for the variance formulated, by hypothesizing that for large length scales the log-gas behaves like a macroscopic conductor, and then using linear response arguments based on the predictions of two-dimensional electrostatics. Moreover, this hypothesis leads to the prediction that the full distribution of the linear statistic will be a Gaussian. For some log-gas systems this has been rigorously established, one of these being that corresponding to the Szeg¨o asymptotic formula from the theory of Toeplitz determinants.
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One of the most basic predictions from macroscopic electrostatics is the leading form of the density profile for random matrix ensembles. It can be used too to predict the O(1/N ) correction to this form. Another application is to gap probabilities, for which the large gap size asymptotics can be predicted to the leading two orders, and at the soft edge the large deviation forms of the left and right tails can be computed. When available, the exact results agree with these predictions. Non-intersecting paths and models in statistical mechanics The generating function for non-intersecting paths on an acyclic directed graph is well known to be given in the form of a determinant. In a number of cases of interest this determinant can be evaluated, revealing that the joint probability density function for the paths possessing prescribed coordinates is of a log-gas form, with β = 1 or 2. Non-intersecting paths underly a number of statistical mechanical models, in particular the polynuclear growth model and the Hammersely model of directed percolation. To understand how this comes about requires a study of the Robinson-Schensted-Knuth correspondence from bijective combinatorics. This in turn leads naturally to the study of Schur polynomials, which are in fact examples of Jack polynomials. It is shown that fluctuations of the primary observable quantities in the polynuclear growth model and the Hammersely model of directed percolation (the height of the profile and length of the path, respectively) can be expressed as random matrix averages over the unitary group, and that these matrix averages can be rigorously analyzed in the appropriate scaling limits. Various symmetrizations of the Hammersely model of directed percolation are particularly natural. Examples of these relate to averages over random matrices from the orthogonal and symplectic groups. Transformations of these averages to relate to gap probabilities in Laguerre random matrix ensembles with β = 1 and 4 allows the the rigorous analysis of the scaling limits. Applications of random matrix theory All the random matrix ensembles introduced in Chapters 1–3, for β = 1, 2 and 4 at least, can be associated with problems in quantum physics. The work of Wigner and Dyson relates the Gaussian ensembles to quantum Hamiltonians; the circular ensembles relate to scattering from a disordered cavity; Verbaarschot has given an interpretation of chiral random matrices in terms of the Dirac equation as it relates to QCD; and quantum transport problems lead to the Jacobi ensemble. For general values of β > 0 the eigenvalue p.d.f.’s of the β-ensembles appear as the ground state wave function of a class of quantum many-body problems with the 1/r2 pair potential. The eigenvalue p.d.f. for the complex random matrices of Chapter 15 has the interpretation as the absolute value squared of the ground state wave function for spinless fermions confined to a plane in the presence of a perpendicular magnetic field. An application of the GOE to the statistics of high-dimensional random energy landscapes is given in Chapter 1. In Chapter 3 features of Wishart matrices and the Jacobi ensemble relating to multivariate statistics are discussed, as is the application of Wishart matrices to wireless communication, numerical analysis and quantum entanglement (the latter requires a further constraint on the trace). In Chapters 5 and 14 an account is given too of the application of both the GUE and CUE to the study of statistical properties of the zeros of the Riemann zeta function. The applications to statistical mechanics, as summarized under the previous heading, is given in Chapter 10. It is clear from the above descriptions that the chapters have not been organized according to these headings. Instead the ordering has been determined by the desire to first define and motivate the various classical random matrix ensembles and their generalizations (for example, β extensions, minor processes), to give the mathematics leading to the determination of the corresponding eigenvalue p.d.f.’s, and to relate the latter to log-gases. This accounts for Chapters 1–4. Chapters 5–7 are about the calculation of correlations for the p.d.f’s encountered in Chapters 1–4 when the former can be expressed in terms of determinants (Chapter 5) or Pfaffians (Chapter 6). Chapters 8 and 9 give the theory leading to the computation of gap probabilities and spacing distributions in some of the systems for which the correlations were computed in the previous three chapters. With knowledge of the evaluation of gap probabilities and related random matrix averages in terms
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of Painlev´e transcendents thus established, we proceed in Chapter 10 to show how this can be put to use in the analysis of certain models in statistical mechanics relating to non-intersecting paths. The generalization of the Gaussian ensembles to be parameter dependent, or equivalently to have Brownian-motion valued entries, is introduced in Chapter 11, leading to the Calogero-Sutherland quantum many-body system and families of commuting operators. The polynomial eigenfunctions of these commuting operators are studied in Chapters 12 and 13, culminating in the computation of correlation functions for general β. Theory from Chapter 4 on the Dixon-Anderson integral again appears in Chapter 13, for its relevance to the computation of correlations for general β (or more precisely, for the inter-relations it provides), while theory from Chapters 1–3 relating to the β-ensembles is developed to give characterizations of the general β bulk and edge states in terms of stochastic differential equations. Continuing the general β theme, the study of fluctuation formulas is taken up in Chapter 14. The topic of the log-gas in a two-dimensional domain (which is in fact where my own studies began), and the corresponding random matrix ensembles in which the eigenvalues are complex, is the theme of final chapter of the book. After this introduction to the content and organization, a few words about the presentation are appropriate. As already remarked it has been my desire to give enough detail so that the development is self-contained. A large portion of the necessary working is carried out in the body of the text, but use too has been made of an exercises format which is both more space efficient and less laborious. I have aimed to structure the exercises with sufficient intermediate results so that they can reasonably be worked through, without the need to consult the original references. Generally it has been my intention to keep the subject matter moving. Consequently, there are a small number of results requiring a technical working beyond the main stream of the book, which necessarily have been omitted. I have been most fortunate to have had research fellowships from the Australian Research Council for the duration of this project. This has freed up time and energy for me to follow, and to be part of, many of the developments which have taken place since I began writing. Both being an active researcher in the field and following the developments have been necessary for writing this monograph. While rewarding, studying the research literature is often difficult and inefficient. My own learning was most efficient when studying instead monographs, in particular those of Gupta and Nagar [279], Haake [284], Hua [300], Mehta [398], Macdonald [376] and Muirhead [410]. Similarly it is my hope that this work will prove itself to be an efficient learning resource in preparation for future researches. There are a number of individuals who have over the years lent their assistance to this project, both directly and indirectly. My wife Gail places value on the worth of such academic pursuits, and provided a home environment to make it possible. For getting me started in research, and teaching me some fundamentals, I thank R.J. Baxter, B. Jancovici and (the late) E.R. Smith. Collaborations with K. Aomoto, T.H. Baker, P. Desrosiers, N.E. Frankel, T. Nagao, E.M. Rains and N.S. Witte have been of great value. E. Due˜nez provided some critical comments on my earlier writing on the circular ensembles which were of much help, and P. Sarnak saw enough potential in these earlier writings to recommend the work to Princeton University Press. Most recently F. Bornemann has provided me with high precision numerical data calculated from Fredholm determinants for use in Chapters 8 and 9, and A. Mays provided some help in relation to the proofreading.
Peter Forrester Melbourne, Australia January 2010
Contents
Preface
v
Chapter 1. Gaussian matrix ensembles 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9
Random real symmetric matrices The eigenvalue p.d.f. for the GOE Random complex Hermitian and quaternion real Hermitian matrices Coulomb gas analogy High-dimensional random energy landscapes Matrix integrals and combinatorics Convergence The shifted mean Gaussian ensembles Gaussian β -ensemble
Chapter 2. Circular ensembles 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9
Scattering matrices and Floquet operators Definitions and basic properties The elements of a random unitary matrix Poisson kernel Cauchy ensemble Orthogonal and symplectic unitary random matrices Log-gas systems with periodic boundary conditions Circular β -ensemble Real orthogonal β -ensemble
Chapter 3. Laguerre and Jacobi ensembles 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 3.10 3.11 3.12
Chiral random matrices Wishart matrices Further examples of the Laguerre ensemble in quantum mechanics The eigenvalue density Correlated Wishart matrices Jacobi ensemble and Wishart matrices Jacobi ensemble and symmetric spaces Jacobi ensemble and quantum conductance A circular Jacobi ensemble Laguerre β -ensemble Jacobi β -ensemble Circular Jacobi β -ensemble
Chapter 4. The Selberg integral 4.1 4.2 4.3
Selberg’s derivation Anderson’s derivation Consequences for the β -ensembles
1 1 5 11 20 30 33 41 42 43
53 53 56 61 66 68 71 73 76 81
85 85 90 98 106 110 111 115 118 125 127 129 130
133 133 137 145
xii
4.4 4.5 4.6 4.7 4.8
CONTENTS
Generalization of the Dixon-Anderson integral Dotsenko and Fateev’s derivation Aomoto’s derivation Normalization of the eigenvalue p.d.f.’s Free energy
Chapter 5. Correlation functions at β = 2 5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8 5.9
Successive integrations Functional differentiation and integral equation approaches Ratios of characteristic polynomials The classical weights Circular ensembles and the classical groups Log-gas systems with periodic boundary conditions Partition function in the case of a general potential Biorthogonal structures Determinantal k-component systems
Chapter 6. Correlation functions at β = 1 and 4 6.1 6.2 6.3 6.4 6.5 6.6 6.7
Correlation functions at β = 4 Construction of the skew orthogonal polynomials at β = 4 Correlation functions at β = 1 Construction of the skew orthogonal polynomials and summation formulas Alternate correlations at β = 1 Superimposed β = 1 systems A two-component log-gas with charge ratio 1:2
Chapter 7. Scaled limits at β = 1, 2 and 4 7.1 7.2 7.3 7.4 7.5 7.6 7.7 7.8 7.9 7.10
Scaled limits at β = 2 — Gaussian ensembles Scaled limits at β = 2 — Laguerre and Jacobi ensembles Log-gas systems with periodic boundary conditions Asymptotic behavior of the one- and two-point functions at β = 2 Bulk scaling and the zeros of the Riemann zeta function Scaled limits at β = 4 — Gaussian ensemble Scaled limits at β = 4 — Laguerre and Jacobi ensembles Scaled limits at β = 1 — Gaussian ensemble Scaled limits at β = 1 — Laguerre and Jacobi ensembles Two-component log-gas with charge ratio 1:2
Chapter 8. Eigenvalue probabilities — Painleve´ systems approach 8.1 8.2 8.3 8.4 8.5 8.6
Definitions Hamiltonian formulation of the Painlev´e theory σ -form Painlev´e equation characterizations The cases β = 1 and 4 — circular ensembles and bulk Discrete Painlev´e equations Orthogonal polynomial approach
Chapter 9. Eigenvalue probabilities — Fredholm determinant approach 9.1 9.2 9.3 9.4 9.5
Fredholm determinants Numerical computations using Fredholm determinants The sine kernel The Airy kernel Bessel kernels
156 160 165 172 180
186 186 193 197 200 207 212 217 223 229
236 236 246 251 263 269 274 278
283 283 290 297 298 301 308 312 316 319 323
328 328 333 349 363 372 375
380 380 385 386 393 399
CONTENTS
9.6 9.7 9.8 9.9 9.10
Eigenvalue expansions for gap probabilities The probabilities Eβsoft (n; (s, ∞)) for β = 1, 4 The probabilities Eβhard (n; (0, s); a) for β = 1, 4 Riemann-Hilbert viewpoint Nonlinear equations from the Virasoro constraints
Chapter 10. Lattice paths and growth models 10.1 10.2 10.3 10.4 10.5 10.6 10.7 10.8 10.9
Counting formulas for directed nonintersecting paths Dimers and tilings Discrete polynuclear growth model Further interpretations and variants of the RSK correspondence Symmetrized growth models The Hammersley process Symmetrized permutation matrices Gap probabilities and scaled limits Hammersley process with sources on the boundary
Chapter 11. The Calogero–Sutherland model 11.1 11.2 11.3 11.4 11.5 11.6 11.7
Shifted mean parameter-dependent Gaussian random matrices Other parameter-dependent ensembles The Calogero-Sutherland quantum systems The Schr¨odinger operators with exchange terms The operators H (H,Ex) , H (L,Ex) and H (J,Ex) Dynamical correlations for β = 2 Scaled limits
Chapter 12. Jack polynomials 12.1 12.2 12.3 12.4 12.5 12.6 12.7 12.8
Nonsymmetric Jack polynomials Recurrence relations Application of the recurrences A generalized binomial theorem and an integration formula Interpolation nonsymmetric Jack polynomials The symmetric Jack polynomials Interpolation symmetric Jack polynomials Pieri formulas
Chapter 13. Correlations for general β 13.1 13.2 13.3 13.4 13.5 13.6 13.7
Hypergeometric functions and Selberg correlation integrals Correlations at even β Generalized classical polynomials Green functions and zonal polynomials Inter-relations for spacing distributions Stochastic differential equations Dynamical correlations in the circular β ensemble
Chapter 14. Fluctuation formulas and universal behavior of correlations 14.1 14.2 14.3 14.4 14.5 14.6
Perfect screening Macroscopic balance and density Variance of a linear statistic Gaussian fluctuations of a linear statistic Charge and potential fluctuations Asymptotic properties of Eβ (n; J) and Pβ (n; J)
xiii
403 416 421 426 435
440 440 456 463 471 480 487 492 495 500
505 505 512 516 521 524 530 540
543 543 550 553 555 558 564 579 583
592 592 601 613 627 633 634 640
658 658 663 665 672 680 688
xiv
14.7
CONTENTS
Dynamical correlations
Chapter 15. The two-dimensional one-component plasma 15.1 15.2 15.3 15.4 15.5 15.6 15.7 15.8 15.9 15.10 15.11
Complex random matrices and polynomials Quantum particles in a magnetic field Correlation functions General properties of the correlations and fluctuation formulas Spacing distributions The sphere The pseudosphere Metallic boundary conditions Antimetallic boundary conditions Eigenvalues of real random matrices Classification of non-Hermitian random matrices
698
701 701 706 711 718 725 729 738 744 747 752 760
Bibliography
765
Index
785
Chapter One Gaussian matrix ensembles The Gaussian ensembles are introduced as Hermitian matrices with independent elements distributed as Gaussians, and joint distribution of all independent elements invariant under conjugation by appropriate unitary matrices. The Hermitian matrices are divided into classes according to the elements being real, complex or real quaternion, and their invariance under conjugation by orthogonal, unitary, and unitary symplectic matrices, respectively. These invariances are intimately related to time reversal symmetry in quantum physics, and this in turn leads to the eigenvalues of the Gaussian ensembles being good models of the highly excited spectra of certain quantum systems. Calculation of the eigenvalue p.d.f.’s is essentially an exercise in change of variables, and to calculate the corresponding Jacobians both wedge products and metric forms are used. The p.d.f.’s coincide with the Boltzmann factor for a log-gas system at three special values of the inverse temperature β = 1, 2 and 4. Thus the eigenvalues behave as charged particles, all of like sign, which are in equilibrium. The Coulomb gas analogy, through the study of various integral equations, allows for the prediction of the leading asymptotic form of the eigenvalue density. After scaling, this leading asymptotic form is referred to as the Wigner semicircle law. The Wigner semicircle law is applied to the study of the statistics of critical points for a model of high-dimensional energy landscapes, and to relating matrix integrals to some combinatorial problems on the enumeration of maps. Conversely, the latter considerations also lead to the proof of the Wigner semicircle law in the case of the GUE. The shifted mean Gaussian ensembles are introduced, and it is shown how the Wigner semicircle law can be used to predict the condition for the separation of the largest eigenvalue. In the last section a family of random tridiagonal matrices, referred to as the Gaussian β-ensemble, are presented. These interpolate continuously between the eigenvalue p.d.f.’s of the Gaussian ensembles studied previously.
1.1 RANDOM REAL SYMMETRIC MATRICES Quantum mechanics singles out three classes of random Hermitian matrices. We will begin our study by specifying one of these—Hermitian matrices with all entries real, or equivalently real symmetric matrices. The independent elements are taken to be distributed as independent Gaussians, but with the variance different for the diagonal and off-diagonal entries. D EFINITION 1.1.1 A random real symmetric N × N matrix X is said to belong to the Gaussian orthogonal ensemble (GOE) if the diagonal and upper triangular elements are independently chosen with p.d.f.’s 2 1 √ e−xjj /2 2π
and
2 1 √ e−xjk , π
respectively. The p.d.f.’s of Definition 1.1.1 are examples of the normal (or Gaussian) distribution 2 2 1 √ e−(x−μ) /2σ , 2 2πσ
denoted N[μ, σ]. With this notation, note that an equivalent construction of GOE matrices is to let Y be an N × N random matrix of independent standard Gaussians N[0, 1] and to form X = 12 (Y + YT ).
2
CHAPTER 1
The joint p.d.f. of all the independent elements is P (X) :=
N j=1
N 2 1 −x2jj /2 1 −x2jk √ e √ e = AN e−xjk /2 π 2π 1≤j
= AN e−
PN
j,k=1
x2jk /2
= AN e−(1/2)TrX , 2
(1.1)
where AN is the normalization and Tr denotes the trace. This structure is behind the choices of the independent Gaussians in Definition 1.1.1. It provides the starting point to identify features of the GOE which make it relevant to quantum physics [447]. P ROPOSITION 1.1.2 Let X be a member of the GOE and let R be an N × N real orthogonal matrix. One has P (RT XR) = P (X). Furthermore, the most general p.d.f. satisfying this equation which has the factorization property P (X) = 1≤j≤k≤N fjk (xjk ) for fjk differentiable is P (X) = Ae−a
PN
j,k=1 (xjk )
2
−b
PN j=1
xjj
= Ae−aTr(X
2
)−bTrX
.
Proof. See Exercises 1.1 q.1. P ROPOSITION 1.1.3 Define the entropy S of the joint p.d.f. P of the independent elements of X by S[P ] := − P log P μ(dX) =: −log P P where μ(dX) := 1≤j≤k≤N dxjk . Then P as given by (1.1) maximizes S subject to the constraint TrX2 P = N 2 . Proof. Because of the constraint on the second moment, and the normalization constraint, we can write “ ” ” “ S[P ] = −log P P − λ Tr X2 P − N 2 + (log A + 1) 1P − 1 , where λ and −(log A + 1) are Lagrange multipliers. The condition for a maximum is δS = 0, where the variation is made with respect to P . This gives − log P − λTrX2 + log A = 0 2
and thus P = Ae−λTrX . The value of λ is determined to be
1 2
from the given constraint.
From these properties an understanding of the applicability of the GOE in the study of quantum energy spectra can be obtained. However as a further prerequisite some theory from quantum mechanics is required [401], [284]. 1.1.1 Time reversal in quantum systems First it is necessary to understand the relevance of an N × N matrix to quantum energy spectra. A basic axiom of quantum mechanics says the energy spectrum of a quantum system is given by the eigenvalues of its (Hermitian) Hamiltonian operator H, the latter being in general infinite dimensional. Now, to model the discrete portion of the spectrum of a complicated quantum system, a reasonable approximation is to replace H by a finite-dimensional N × N Hermitian matrix, which has a discrete spectrum only. Next we need to understand the significance of real symmetric matrices in quantum mechanics. This is related to the fact that in general the structure of a matrix modeling H is constrained by the symmetries of H. D EFINITION 1.1.4 A quantum Hamiltonian H is said to have a symmetry A if [H, A] = 0, where [·, ·] denotes the commutator. One basic symmetry of most quantum systems is time reversal.
3
GAUSSIAN MATRIX ENSEMBLES
D EFINITION 1.1.5 A general time reversal operator T is any antiunitary operator, which means T = U K where U is unitary and K is the complex conjugation operator. Hence we say a quantum system has a time reversal symmetry if the Hamiltonian commutes with an antiunitary operator. Study of time reversal operators in the context of physical systems further restricts their form. For systems with an even number or no spin 12 particles, it is required that T 2 = 1, while for a finite-dimensional system with an odd number of spin T 2 = −1
and
1 2
particles
T = Z2N K,
where Z2N is a 2N × 2N block diagonal matrix with each 2 × 2 diagonal block given by 0 −1 1 0
(1.2)
(a tensor product formula for Z2N is given in Exercises 1.1 q.2) which has the effect of reversing the spins. Real symmetric matrices arise in the former situation. P ROPOSITION 1.1.6 Let H be a quantum Hamiltonian which is invariant with respect to a time reversal symmetry T , where T has the additional property T 2 = 1. Then H can always be given a T -invariant orthogonal basis, and with respect to this basis the (in general infinite) matrix representation of H is real. Proof. See Exercises 1.1 q.3. The above result tells us that a matrix chosen to model the discrete energy spectra of a quantum system with a time reversal symmetry T such that T 2 = 1 must be real symmetric. A further general property in quantum mechanics is that two operators related by a similarity transformation of unitary operators are equally valid descriptions of the operator, in that all observables are the same for both operators. A requirement of (1.1) is therefore that any two real symmetric matrices related by a similarity transformation of unitary matrices must have the same p.d.f. for the elements. For the two real symmetric matrices to be so related the unitary matrix must be real orthogonal (or i times a real orthogonal matrix; see Exercises 1.1 q.4). Thus this requirement is guaranteed by Proposition 1.1.2. We are assuming no information on the Hamiltonian other than the time reversal symmetry. Proposition 1.1.3 says that the p.d.f. (1.1) is the most random subject to the given constraint, in that it maximizes the entropy. These considerations thus show the applicability of the GOE in the study of quantum spectra. Explicitly, it is hypothesized that the statistical properties of the highly excited states of a complex quantum system with a time reversal symmetry T 2 = 1 coincide with the statistical properties of the bulk eigenvalues from large GOE matrices (see Section 7.1.1 for the notion of bulk eigenvalues). Here it is assumed that both spectra have been scaled (technically referred to as unfolded) so that the mean spacing is unity. The meaning of a complex quantum system requires further explanation. Wigner first made this hypothesis for the spectra of heavy nuclei in the 1950’s. In 1984 Bohigas, Giannoni and Schmit made the same hypothesis for a single particle quantum billiard system, provided the underlying classical mechanics is chaotic and the system has a time reversal symmetry T 2 = 1. It is of interest to note that a GOE hypothesis also applies to eigenmodes of microwave cavities (this is not surprising as the Helmholtz equation is formally equivalent to the stationary Schr¨odinger equation), and also to the eigenmodes of systems governed by classical wave equations — vibrations of irregular shaped metal plates, electromechanical eigenmodes of aluminium and quartz blocks, among other examples. (For references to the original literature, and an extended discussion of GOE hypotheses, see [276].)
4
CHAPTER 1
E XERCISES 1.1
1. The objective of this exercise is to prove Proposition 1.1.2.
(i) Note that the invariance P (RT XR) = P (X) with R a permutation matrix requires that the distribution of all elements on the diagonal be equal, fjj = f , and similarly the distribution of all elements on the off diagonal be equal, fjk = g (j < k), for some f and g. (ii) Choose
2 6 6 6 R=6 6 4
1 − 0 .. . 0
1 0 .. . 0
0 0 1 .. . 0
... ... ... .. . ...
where || 1. Ignoring terms O(2 ), show that 2 x11 − 2x12 x12 + (x11 − x22 ) 6 ∗ x22 + 2x12 6 6 −1 ∗ ∗ R XR = 6 6 .. .. 4 . . ∗ ∗
0 0 0 .. . 1
3 7 7 7 7, 7 5
x13 − x23 x23 + x13 x33 .. . ∗
... ... ... .. . ...
x1N − x2N x2N + x1N x3N .. . xNN
3 7 7 7 7, 7 5
where the elements ∗ are such that the matrix is symmetric. (iii) Use the result of (ii) to show that at first order in the requirement N Y
Y
f (xjj )
j=1
g(xjk ) =
N Y j=1
1≤j
Y
f (˜ xjj )
g(˜ xjk ),
1≤j
where x ˜jk := [R−1 XR]jk implies (x11 − x22 )g (x12 ) x12 f (x11 ) x12 f (x22 ) X −2 +2 − g(x12 ) f (x11 ) f (x22 ) j=3 N
„
x2j g (x1j ) x1j g (x2j ) − g(x1j ) g(x2j )
« = 0,
which in turn, by separation of variables, implies −
f (x11 ) f (x12 ) (x11 − x22 )g (x12 ) + + =γ f (x11 ) f (x12 ) 2x12 g(x12 )
for some constant γ. (iv) By a further separation of variables in the last equation conclude g (x12 ) = −b x12 g(x12 ) for some constant b. Solve this differential equation. (v) Note that the invariance P (RT XR) = P (X) requires that P be a symmetric function of the eigenvalues, and thus a function of Tr(Xk ) k = 1, 2, . . . . Now combine the results of (i) and (iv) to deduce the result. 2. Let A = [aij ] be a p × q matrix and B = [bi j ] be an r × s matrix. The tensor product, denoted A ⊗ B, is the pr × qs matrix with elements (A ⊗ B)ii ,jj = ai,j bi ,j , and thus
2
a11 B 6 .. A⊗B=4 . ap1 B
a12 B .. . ap2 B
··· .. . ···
3 a1q B 7 .. 5. . apq B
5
GAUSSIAN MATRIX ENSEMBLES
With Z2N defined as above (1.2), show that » Z2N = 1N ⊗
0 1
−1 0
– .
(1.3)
1 and φ 1 are vectors, T is anti-unitary and T 2 = 1. Note 1 = α1 φ 1 + T (α1 φ 1 ), where α1 is a scalar, ψ 3. [401] Let ψ that T ψ1 = ψ1 . Here Proposition 1.1.6 will be established. (i) From the antiunitarity property it follows that in general u|T v = T u|v , where ·|· denotes the inner u|T v. product. Use this to show that u|v = T 2 + T (α2 φ 2 ) is orthogonal to ψ 1 . Use (i) to show that ψ 2 := α2 φ 1 , and note 2 is orthogonal to ψ (ii) Suppose φ how this construction can be used to create an orthogonal basis of vectors with the T -invariance property n = ψ n . Tψ (iii) Consider a Hamiltonian H which has symmetry T . Use the above properties of T to show that with respect m |H ψ n are real. n } the matrix elements ψ to the basis {ψ 4. Let X be an arbitrary real symmetric N × N matrix and suppose X = U−1 XU, where U is unitary and X is real symmetric. Assume that the only symmetry of X and X in general (other than some constant times the identity) is the time reversal operator T with T 2 = 1. (i) Deduce that T UT −1 U−1 commutes with X. (ii) Use (i) to show T U = cUT and take the inverse of this equation to conclude c = ±1. n }, ψ n |Uψ m = cψ n |Uψ m . (iii) Use (ii) and q.3(i) to show that with respect to the T invariant basis {ψ Hence conclude that U has either real elements (c = 1) or pure imaginary elements (c = −1) and is thus either a real orthogonal matrix or i times a real orthogonal matrix.
1.2 THE EIGENVALUE P.D.F. FOR THE GOE The p.d.f. for the elements of the matrices in the GOE is given by (1.1). We want to calculate the corresponding eigenvalue p.d.f. This was first accomplished as long ago as 1939 [299]. We will follow a more recent treatment [410]. The new variables and the final expression The p.d.f. (1.1) has N (N + 1)/2 independent variables, whereas there are only N eigenvalues, say, λ1 < · · · < λN . The remaining variables are linear combinations of the independent elements of the eigenvectors, denoted p1 , . . . , pN (N −1)/2 say. Our task is to change variables 1 exp − Tr(X2 ) 2 where the Jacobian is given by
1≤j≤k≤N
⎡
⎢ ⎢ J := det ⎢ ⎢ ⎣
N (N −1)/2 N N 1 dpj , dxjk = exp − λ2l |J| dλj 2 j=1 j=1 l=1
∂x11 ∂λ1 ∂x11 ∂λ2
∂x12 ∂λ1 ∂x12 ∂λ2
∂x11 ∂pN (N −1)/2
∂x12 ∂pN (N −1)/2
.. .
.. .
... ... .. . ...
∂xN N ∂λ1 ∂xN N ∂λ2
.. .
∂xN N ∂pN (N −1)/2
⎤ ⎥ ⎥ ⎥. ⎥ ⎦
Thus we must evaluate the Jacobian and then integrate over the variables p1 , . . . , pN (N −1)/2 to obtain the eigenvalue p.d.f.
6
CHAPTER 1
Below we will show that J factorizes, J=
(λk − λj ) f (p1 , . . . , pN (N −1)/2 )
1≤j
so the integration over the variables p1 , . . . , pN (N −1)/2 only alters the normalization constant. Hence the final expression for the eigenvalue p.d.f. of the GOE is 1 1 exp − λ2 CN 2 j=1 j
N
|λk − λj |,
(1.4)
1≤j
where CN is the normalization constant. From the viewpoint of application to quantum mechanics, the important feature is the product of differences due to the Jacobian. It can be proved that the correlations are determined entirely by the product of differences, 2 in the sense that the same so-called bulk correlations (see Section 7.1) result if the one body terms e−λ /2 −V (λ)/2 are replaced by some different functional forms e , provided the local density is constant [125]. This feature is referred to as universality and gives rise to the notion [53] that the essential feature of a random matrix hypothesis applying to a quantum system is that the spectral correlations are geometrical, meaning that they are due to this Jacobian. 1.2.1 Wedge products In the theory of multivariable calculus (see, e.g., [485]) the wedge product operation, which is linear and antisymmetric, is defined to give a signed volume element in the tangent space at a point in the manifold. However, for our purpose the latter concept plays no explicit role, and we can make do with the following definition. D EFINITION 1.2.1 With dui (j) := δi,j dui define du1 ∧ · · · ∧ duN =:
N
duj := det[dui (j)]i,j=1,...,N .
(1.5)
j=1
Note that it follows from (1.5) that f (u1 , . . . , uN )du1 ∧ · · · ∧ duN = f (u1 , . . . , uN )du1 · · · duN , Ω
Ω
since only the diagonal entries in the determinant are nonzero. When changing variables from {u1 , . . . , uN } to {v1 , . . . , vN } the fundamental formula dui =
N ∂ui
∂vl
l=1
dvl
applies. Substituting this in (1.5), and noting the factorization N ∂u l=1
i
∂vl
dvl (j)
i,j=1,...,N
=
∂u i
∂vj
i,j=1,...,N
dvi (j)
i,j=1,...,N
shows N j=1
duj = det
∂u
N
∂vj
j=1
i
i,j=1,...,N
dvj .
(1.6)
The determinant in (1.6) is precisely the Jacobian for the change of variables. The practical use of calculating
7
GAUSSIAN MATRIX ENSEMBLES
Jacobians from this formula relies on an alternative way of calculating the l.h.s. of (1.6) in terms of {vj }. For the problem at hand, this in turn is done by using the special feature that all the variables are connected by matrix relations. The following definitions are helpful. D EFINITION 1.2.2 For any N × N matrix X = [xjk ], the matrix of differentials is defined as ⎡ ⎤ dx11 dx12 . . . dx1N ⎢ dx21 dx22 . . . dx2N ⎥ ⎢ ⎥ dX = ⎢ ⎥. .. .. .. . . ⎣ ⎦ . . . . dxN 1 dxN 2 . . . dxN N With this definition the usual product rule for differentiation holds, d(XY) = dX Y + X dY. D EFINITION 1.2.3 The symbol (dX) denotes the wedge product of the independent elements of dX. In particular, if X is a real symmetric matrix, (dX) = dxjk , 1≤j≤k≤N
while if X = [xjk + iyjk ]j,k=1,...,N is Hermitian (xjk = xkj , yjk = −ykj ) (dX) =
N j=1
dxjj
dxjk dyjk .
1≤j
In integration formulas only the absolute value of the Jacobian occurring in the change of variables formula (1.6) is required, so consequently there is no need to strictly adhere to the ordering of wedge products specified in Definition 1.2.3 (according to the definition, reversing the order of two differentials changes the sign of the wedge product). Because of this, any overall factor of −1 will be ignored in subsequent formulas involving (dX). With this convention (dX) will be referred to as a volume form, or volume measure. In preparation for the calculation of J, we note a result for the wedge product (AT dMA), where A is a real N × N matrix and M is a real symmetric N × N matrix [410]. P ROPOSITION 1.2.4 Let A and M be real N ×N matrices, and suppose furthermore that M is symmetric. We have (AT dMA) = (det A)N +1 (dM). Proof. We note from Definition 1.2.3 that (AT dMA) = p(A)(dM),
(1.7)
where p is a polynomial in the elements of A. Furthermore, if B is also an N × N matrix, then (BT AT dMAB) = p(B)(AT dMA) = p(B)p(A)(dM), so we must have p(AB) = p(A)p(B), for arbitrary A and B. But it is known [377] that the only polynomial in the matrix elements satisfying such a factorization is p(A) = (det A)k ,
k ∈ Z≥0 .
The value of k can be determined by making the special choice A = diag(a, 1, . . . , 1) in (1.7). For an alternative proof of this result, see Exercises 1.3 q.2.
8
CHAPTER 1
1.2.2 Calculation of the Jacobian From Definition 1.2.3 and (1.6) we see J
N (N −1)/2
N
dλi
i=1
dpj = (dX).
j=1
To calculate (dX) in terms of the eigenvalues and eigenvectors we use the fact that all symmetric matrices are orthogonally diagonalizable [8] (see Exercises 1.9 q.3) to write X = RLRT .
(1.8)
Here L is a diagonal matrix consisting of the N eigenvalues of X and the columns of the real orthogonal matrix R consist of the corresponding normalized eigenvectors. Using the notation of Definition 1.2.2, the product rule for differentiation gives dX = dR LRT + RdL RT + RLdRT . Rather than take the wedge product of both sides of this equation, it is simpler to premultiply by RT and postmultiply by R to obtain RT dXR = RT dR L + LdRT R + dL = RT dR L − LRT dR + dL,
(1.9) T
where to obtain the last line the formula dR R = −R dR has been used (this follows from RR = 1). According to Proposition 1.2.4 T
T
(RT dXR) = (det R)N +1 (dX).
(1.10)
But R is an orthogonal matrix and so det R = ±1. As already noted, since only the modulus of J occurs in the change of variables formula, this sign factor can be ignored. The wedge product of the r.h.s. of (1.9) can be taken with the aid of the following result. P ROPOSITION 1.2.5 With the notation rk = (r1k , r2k , . . . , rN k )T for the kth column of R, we have RT dRL − LRT dR + dL ⎡ dλ1 ⎢ (λ2 − λ1 )r1 T · dr2 ⎢ =⎢ .. ⎣ .
(λ2 − λ1 )r1 T · dr2 dλ2 .. .
... ... .. .
(λN − λ1 )r1 T · drN (λN − λ2 )r2 T · drN .. .
(λN − λ1 )r1 T · drN
(λN − λ2 )r2 T · drN
...
dλN
⎤ ⎥ ⎥ ⎥. ⎦
Proof. This is obtained by explicitly forming the matrix products, and simplifying the resulting expression by noting from dRT dR = −RT dR that RT dR is antisymmetric. From Proposition 1.2.5 and Definition 1.2.3, the wedge product of the r.h.s. of (1.9) can be written down (note in particular that the matrix in Proposition 1.2.5 is symmetric), whereas (1.10) gives the wedge product of the l.h.s. of (1.9). Equating these expressions gives (dX) =
1≤j
(λk − λj )
N
dλj (RT dR).
(1.11)
j=1
The factorization property of the Jacobian between the eigenvalues and the variables involving the eigenvectors is evident and the expression (1.4) for the eigenvalue p.d.f. of the GOE follows. The p.d.f. for the components of the eigenvectors is calculated in Exercises 1.2 q.2.
9
GAUSSIAN MATRIX ENSEMBLES
1.2.3 Scaling of the Jacobian Here we will show how the eigenvalue factor in the Jacobian can be deduced by considering a simple scaling property of the wedge product. Since there are N (N + 1)/2 independent elements in X, (dX) consists of the product of N (N + 1)/2 independent differentials. Thus if we multiply X by a scalar a, we have that (d aX) = aN (N +1)/2 (dX). On the other hand, with X = RLRT , we know that (dX) is a polynomial in λ1 , . . . , λN . Since aX = RaLRT , the scaling property of (d aX) gives that in fact (dX) is a homogeneous polynomial of degree N (N − 1)/2 (here we have subtracted N from N (N + 1)/2 to account for the scaling of the measure dλ1 · · · dλN ). Furthermore, analysis of the 2 × 2 case reveals that the Jacobian must vanish linearly for λj → λk (see Exercises 1.2 q.3). Hence the polynomial factor is necessarily proportional to j
1.2.4 Metric forms Another approach to deriving (1.11) is through the use of a metric form defined on the space of symmetric matrices [300]. For an N × N real symmetric matrix X, the metric form of the line element ds is specified by (ds)2 = Tr(dXdXT ) =
N
(dxjj )2 + 2
j=1
(dxjk )2
(1.12)
j
(of course dXT = dX, but it is convenient to write as presented), and the volume measure is (dX) = dxjk . j≤k
If one now makes a change of variables, expressing the elements xjk in terms of some new variables yjk such that 2
(ds) =
N
(hjj dyjj )2 + 2
j=1
(hjk dyjk )2 ,
(1.13)
j
where the hjk typically depend on {yjk }, the corresponding volume measure is hjk (dY), (dX) =
(1.14)
j≤k
thus giving a change of variable formula for the volume measure. More generally the metric forms method gives that if (ds)2 is a symmetric quadratic form in some independent infinitesimals {dyμ }, so that gμ,ν dyμ dyν , gμ,ν = gν,μ , (1.15) (ds)2 = μ,ν
then the corresponding volume measure is
1/2 det[gμ,ν ] dyμ .
(1.16)
μ
Comparing (1.15) with (1.13), we see that there are only diagonal terms present in the formula for the line element. The determinant is then the product of the diagonal terms, which is consistent with (1.14).
10
CHAPTER 1
We can apply this formalism by noting from (1.9) and Proposition 1.2.5 that N 2 2 Tr(dXdX ) = 2 (λk − λj ) (rj · drk ) + (dλj )2 . T
j=1
j
Application of (1.14) then reclaims (1.11). 1. (i) Let R be a N × N real orthogonal matrix. Show that in general R has N 2 − N (N − 1)/2 − N independent elements.
E XERCISES 1.2
(ii) Use (i) to show that the number of independent elements on the two sides of the equation X = RLRT , where X is real symmetric and L is diagonal, are equal. (iii) For a, b, N × 1 real column vectors related by a = Ab for some N × N matrix A, show that (da) = | det A|(db).
(1.17)
2. [284] Here the distribution of the components of the eigenvectors in the GOE is calculated. (i) Note that for matrices in the GOE every eigenvector can be transformed by an arbitrary real orthogonal matrix, and still remain an eigenvector of a matrix in the GOE. Conclude from this that the only invariant of the eigenvectors is their norm, and so the joint distribution of the components (u1 , . . . , uN ) is given by N ” X 1 “ u2p , δ 1− C p=1
where C = 2π N/2 /Γ(N/2) represents the surface area of the unit (N − 1)-sphere. (ii) Show that the marginal joint distribution p(u1 , . . . , un ), obtained by integrating out the variables un+1 , . . . , uN , is given by n ”(N−n−2)/2 “ X Γ(N/2) u2p . p(u1 , . . . , un ) = π −n/2 1− Γ((N − n)/2) p=1 For this purpose write the delta function in (i) as a Fourier integral. (iii) From (ii) show that for large N , 1 N n/2
“ 2 ”n/2 1 Pn “ u 2 un ” 1 ∼ p √ ,..., √ e− 2 p=1 up . π N N
(1.18)
(iv) Show that forming a vector (u1 , . . . , uN ) in which each component has distribution xj /(x21 + · · · + x2N )1/2 , with the xj s standard normal random variables, implies that the vector is uniformly distributed on the unit (N − 1)-sphere and thus has joint density as in (i). Use this fact to rederive (1.18). 3.
(i) For a general 2 × 2 real symmetric matrix » A :=
a b
b c
– ,
show that the (unordered) eigenvalues are given by λ± =
”1/2 1“ 1 (a + c) ± (a − c)2 + 4b2 . 2 2
Note that the condition for a degenerate eigenvalue is b = 0 and a = c, and thus has codimension 2 in the space of matrix entries. (ii) For the matrix in (i) parametrize the matrix of eigenvectors as » – cos θ − sin θ R= sin θ cos θ
11
GAUSSIAN MATRIX ENSEMBLES
and from the diagonalization equation A = R diag[λ+ , λ1 ]RT , read off that a = λ+ cos2 θ + λ− sin2 θ, (iii) Deduce from (ii) that
b = (λ+ − λ− ) cos θ sin θ,
˛ ∂a ˛ ˛ ˛ ∂λ+ ˛ ∂a J := ˛˛ ˛ ∂λ− ˛ ∂a ˛ ∂θ
∂b ∂λ+ ∂b ∂λ− ∂b ∂θ
∂c ∂λ+ ∂c ∂λ− ∂c ∂θ
c = λ+ sin2 θ + λ− cos2 θ.
˛ ˛ ˛ ˛ ˛ ˛ = (λ+ − λ− ). ˛ ˛ ˛ ˛
1.3 RANDOM COMPLEX HERMITIAN AND QUATERNION REAL HERMITIAN MATRICES Since most physical systems possess a time reversal symmetry, the GOE correctly models statistical properties of the spectra of many quantum systems (recall the discussion at the end of Section 1.1). Nonetheless the considerations of time reversal symmetry of Section 1.1.1 indicate two further random matrix ensembles [149]. 1.3.1 The Gaussian unitary ensemble For a quantum system without time reversal symmetry the only constraint on the complex Hermitian matrix used to model the discrete portion of the energy spectrum is that two matrices related by a similarity transformation of unitary operators have the same joint p.d.f. for the elements. This requirement is satisfied by the following choice of matrix ensemble. D EFINITION 1.3.1 A random Hermitian N × N matrix X is said to belong to the Gaussian unitary ensemble (GUE) if the diagonal elements (which must be real) and the upper triangular elements xjk = ujk + ivjk are independently chosen with p.d.f.’s 2 1 √ e−xjj π
2 2 2 −2(u2jk +vjk 2 ) e = e−2|xjk | , π π √ respectively. Equivalently, the diagonal entries have distribution N[0, 1/ 2], while the upper triangular elements have distribution N[0, 12 ] + iN[0, 12 ], and X can be specified in terms of the complex random matrix Y √ √ with entries independently chosen from N[0, 1/ 2] + iN[0, 1/ 2], according to X = (Y + Y† )/2.
and
From this definition the joint p.d.f. of all the independent elements is P (X) :=
N 2 1 √ e−xjj π j=1
1≤j
N 2 2 2 −2|x2jk | e = AN e−|xjk | = AN e−Tr(X ) , π j,k=1
−1
where AN is the normalization. The invariance P (U immediately.
XU) = P (X) for any unitary matrix U follows
1.3.2 The Gaussian symplectic ensemble In Section 1.1.1 it was remarked that in quantum systems with a time reversal symmetry T , either T 2 = 1 or T 2 = −1 with T = Z2N K. Consideration of the former case leads to real symmetric matrices. Here the latter possibility will be discussed. Now, since T commutes with the 2N × 2N matrix X modeling the Hamiltonian, X must in addition to being Hermitian have the property −1 ¯ −1 X = T XT −1 = Z2N KXK −1 Z−1 2N = Z2N KXKZ2N = Z2N XZ2N .
(1.19)
12
CHAPTER 1
Since Z2N is block diagonal, with blocks (1.2), a 2N × 2N matrix X with this property can be viewed as an N × N matrix with elements consisting of 2 × 2 blocks of the form z w , (1.20) −w ¯ z¯ where z and w are complex numbers. A 2 × 2 matrix of this form is said to be real quaternion. From an abstract perspective the quaternions are an algebra with elements of the form i2 = j 2 = k 2 = −1, ijk = −1,
a0 + a1 i + a2 j + a3 k,
(1.21)
where a0 , . . . , a3 are scalars. The basis elements 1, i, j, k can be realized as 2 × 2 matrices with complex elements given by 1 0 i 0 0 1 0 i e1 := iσz = e2 := iσy = 1 := e3 := iσx = , (1.22) 0 1 0 −i −1 0 i 0 respectively. Forming a general linear combination, consisting of real scalar multiples of these basis elements, gives the structure (1.20). For future reference we note that with a real quaternion q written in the form q = c0 1+ c1 e1 + c2 e2 + c3 e3 ¯ or qD , is defined as the dual, denoted q ¯ = qD = c0 1 − c1 e1 − c2 e2 − c3 e3 . q With this definition the dual of (1.20) is
z¯ w ¯
−w z
(1.23)
.
(1.24)
¯ q = q¯ Furthermore, with |q|2 := q q, we have |q|2 = c20 + c21 + c22 + c23 , the relation |q1 q2 | = |q1 ||q2 | holds, ¯ /|q|2 . and each nonzero q has a unique inverse, q−1 = q An N × N matrix with real quaternion elements is said to be quaternion real. This structure underlies the definition of the third and final ensemble of Gaussian random matrices as motivated by quantum physics. D EFINITION 1.3.2 A random Hermitian N × N matrix X with real quaternion elements is said to belong to the Gaussian symplectic ensemble (GSE) if the elements zjj of each diagonal real quaternion (which must be real) are independently chosen with p.d.f. 2 2 −2zjj e π (or equivalently have distribution N[0, 1/2]) while the upper triangular off-diagonal elements zjk = ujk + ivjk and wjk = ujk + ivjk are independently chosen with p.d.f. 4 −4|wjk |2 4 −4|zjk |2 and e e π π √ √ (or equivalently have distribution N[0, 1/2 2]+iN[0, 1/2 2]). Thus X = (Y+Y† )/2, where Y is an N ×N random matrix of independent real quaternions with z and w in (1.20) having distribution N [0, 12 ]+iN [0, 12 ]. A fundamental property of quaternion real Hermitian matrices, which follows from the first equation in (1.19), is that their spectrum is doubly degenerate (see Exercises 1.3 q.1). It follows from Definition 1.3.2 that the joint p.d.f. of all the independent elements of the GSE is given by P (X) = AN e−2Tr(X ) , 2
where AN denotes the normalization and Tr denotes the trace with X2 regarded as a quaternion real matix (i.e., Tr(X2 ) equals the sum of the scalar multiples of 12 on the diagonal of X2 ). This satisfies the general
13
GAUSSIAN MATRIX ENSEMBLES
requirement of being invariant with respect to similarity transformations of appropriate unitary matrices. In fact the appropriate unitary matrices are those which under a similarity transformation map a quaternion real Hermitian matrix into another quaternion real Hermitian matrix. This subgroup of unitary matrices is specified by the following result. P ROPOSITION 1.3.3 (a) Let X be an arbitrary N × N Hermitian matrix with real quaternion elements, so that in general the only symmetry of X (other than some multiple of the identity) is the operator T = Z2N K. Then any unitary matrix U which under a similarity transformation maps X into another Hermitian matrix with real quaternion elements must commute or anticommute with T . (b) A unitary matrix U which commutes with T = Z2N K has the property UZ2N UT = Z2N ,
(1.25)
which implies U is equivalent to a symplectic matrix, while a unitary matrix U which anticommutes with T has the property −UZ2N UT = Z2N . Proof. (a) Let X be such that U−1 XU = X . Since both X and X are quaternion real, T commutes with both of these
matrices. This implies XT UT −1 = T UT −1 X . Comparing these two equations gives that T UT −1 U−1 commutes with X. But the only operators which commute with X are T and some multiple of the identity, so the above combination of operators must equal one of these operators. We see that the first choice leads to T = 1, which is a contradiction, while the second gives T U = ±UT (regarding the signs, recall Exercises 1.1 q.4(ii)) as required. (b) At the beginning of this subsection it was noted that any matrix, in this case U, which commutes with T has the ¯ −1 . Equation (1.25) follows after noting U ¯ = (U−1 )T and rearranging. To continue, recall that property U = Z2N UZ 2N by definition a matrix is symplectic if – » 1N 0N T . (1.26) J2N := X J2N X = J2N , −1N 0N If X is also unitary, this implies X has the block structure » Z X= ¯ −W
W ¯ Z
–
(cf. (1.20)). Now, the matrix J2N is related to Z2N by a similarity transformation J2N = Q−1 Z2N Q, where Q is a unitary matrix with elements ±1 (there must therefore be exactly one nonzero element in each row/column). We thus conclude from (1.25) that Q−1 UQ is symplectic. The only difference in the anticommuting case is a minus sign, which gives the second result.
1.3.3 The eigenvalue p.d.f.’s The calculation of the eigenvalue p.d.f.’s from the joint p.d.f.’s for the elements of the GUE and GSE can be done in a similar way to that presented in Section 1.2 for the GOE. The required working is sketched in Exercises 1.3 q.3 and q.4, and the final results are summarized in the following, which for completeness also contains the eigenvalue p.d.f. for the GOE. P ROPOSITION 1.3.4 Let H be a Hermitian matrix with real (β = 1), complex (β = 2), or real quaternion (β = 4) elements, and let H be decomposed in terms of its eigenvalues and eigenvectors via the formula H = ULU† , where L is a diagonal matrix consisting of the eigenvalues of H, and U is a unitary matrix with real (β = 1), complex (β = 2) or real quaternion (β = 4) elements consisting of the corresponding eigenvectors. We have (dH) =
1≤j
|λk − λj |β
N j=1
dλj (U† dU),
(1.27)
14
CHAPTER 1
and hence for an appropriate choice of the normalization Gβ,N , which is given explicitly by (1.163), 1 Gβ,N
N β exp − λ2j 2 j=1
|λk − λj |β ,
(1.28)
1≤j
with β = 1,2 and 4 is the eigenvalue p.d.f. for the GOE, GUE and GSE, respectively. We remark that for the decomposition H = ULU† to be unique the eigenvalues must be ordered and the first component of the eigenvectors must be real and positive. Because (1.28) is a symmetric function of the eigenvalues, the ordering constraint can conveniently be removed, and the normalization appropriately adjusted. In particular, Gβ,N is the normalization without the ordering constraint. 1.3.4 Relationship to Lie algebras The sets of matrices gl(N, R) := {all N × N real matrices}, gl(N, C) := {all N × N complex matrices}, u∗ (2N ) := {all N × N real quaternion matrices} are each closed under commutation and so form matrix Lie algebras. Now, in general a matrix can be decomposed as the sum of a Hermitian and an anti-Hermitian matrix. We see that the Hermitian component of the above Lie algebras corresponds to Hermitian matrices with real, complex and real quaternion elements, respectively. This is significant for a number of reasons. One is from a classification perspective. One can identify ten infinite families of matrix Lie algebras, in correspondence with the ten infinite families of symmetric spaces, as catalogued by Cartan [295]. We will see that each of the remaining seven cases also occurs in a basic quantum mechanics problem constrained by a global symmetry. Furthermore, the identification with matrix Lie algebras implies a one-to-one correspondence between the ten families of Hermitian matrices and the ten families of unitary matrices. This comes about because of the relationship between matrix Lie algebras and symmetric spaces. To each matrix Lie algebra there corresponds a noncompact and compact symmetric space, with the former being isomorphic to a certain set of Hermitian matrices, and the latter isomorphic to a certain set of unitary matrices. Some more details are given in Section 2.1.2. The isomorphism with symmetric spaces has the consequence that the eigenvalue-dependent portion of the Jacobian in (dH) can be written in the form | α, λ|mα , α ∈R+
where α , an N component Euclidean vector, is a so-called root of the root system corresponding to the symmetric space, ·, · is the dot product, R+ is the set of positive roots, and mα the multiplicity of α . This structure, in the case of the symmetric spaces corresponding to the classical groups, appears in the so called Weyl integration formula [540]. For the symmetric spaces corresponding to the Gaussian ensembles the positive roots are ej − ek (j < k) (root system of type A — see Section 4.7.2) with multiplicities mα = β, and this reclaims the eigenvalue-dependent portion of (1.27). However ,we will not pursue the derivation of these facts, which can be found in [295]. 1.3.5 Octonions and the N = 2, β = 8 Gaussian ensemble The p.d.f. (1.28) for N = 2, β = 8 can be realized as the eigenvalues of a random Hermitian matrix with real octonion elements. To see this, we must first revise aspects of the theory of real octonions [513]. The real octonions can be constructed out of the real quaternions. Let p1 , p2 , q1 , q2 be abstract real quaternions,
15
GAUSSIAN MATRIX ENSEMBLES
and thus linear combinations with real coefficients of {1, i, j, k} as specified by (1.21). Let q¯ denote the quaternionic dual defined by (1.23), and let l denote a quantity algebraically distinct from the real quaternions. The real octonion algebra then consists of elements of the form a = p1 + p2 l, b = q1 + q2 l, with addition and multiplication defined by ab = (p1 q1 − q¯2 p2 ) + (q2 p1 + p2 q¯1 )l,
a + b = (p1 + q1 ) + (p2 + q2 )l,
(1.29)
respectively. It follows that the real octonions are an eight-dimensional algebra with basis 1, e1 := i, e2 := j, e3 := k, e4 := l, e5 := il, e6 := jl, e7 := kl. In general a(bc) = (ab)c (for example with a = e5 , b = e6 , c = e7 we have a(bc) = −e4 and (ab)c = e4 ), so unlike the real quaternions the real octonions are not associative. On the other√hand, with ¯ := p¯1 − p2 l, p¯1 denoting the √ a quaternionic dual (1.23), we have ab = ¯b¯ a and thus with |a| := a¯ a= a ¯a, |ab| = |a||b|.
(1.30)
7
Furthermore, with a general real octonion written as a = a0 + j=1 aj ej , we have |a| = a20 + a21 + · · · + a27 ,
(1.31)
and it is also true that each a = 0 has a unique inverse specified by ¯/(¯ aa). a−1 = a
(1.32)
The properties (1.30)–(1.32) say that the real octonions are a normed division algebra. In fact a theorem of Hurwitz [301] says that up to isomorphisms, the only normed division algebras over the reals, with a unit element, are the reals, complex numbers, real quaternions and real octonions. Because the real octonions are not associative, they cannot be represented as a matrix algebra. Nonetheless, the actions of right and left multiplication by a given real octonion a on a general real octonion x can be represented as a matrix. To specify these matrices, we first require the corresponding result for the real quaternions, which follows immediately from the explicit form of the multiplication rule. P ROPOSITION 1.3.5 Let x = x0 + x1 i + x2 j + x3 k be a real quaternion and let x = (x0 , x1 , x2 , x3 )T denote the column vector formed from the coefficients. Then for a a real quaternion ax = φ(a)x, where
⎡
a0 ⎢ a1 φ(a) = ⎢ ⎣ a2 a3
−a1 a0 a3 −a2
−a2 −a3 a0 a1
⎤ −a3 a2 ⎥ ⎥, −a1 ⎦ a0
xa = τ (a)x ⎡
a0 ⎢ a1 T τ (a) = Kφ (a)K = ⎢ ⎣ a2 a3
−a1 a0 −a3 a2
−a2 a3 a0 −a1
⎤ −a3 −a2 ⎥ ⎥, a1 ⎦ a0
with K = diag[1, −1, −1, −1]. Using Proposition 1.3.5, the corresponding result for the real octonions follows from the multiplication rule (1.29). P ROPOSITION 1.3.6 Let x = x0 + 7j=1 xj ej be a real octonion, and let x = (x0 , x1 , . . . , x7 )T denote the column vector formed from the coefficients. Then with a = a(1) + a(2) l a real octonion, and thus a(1) , a(2)
16
CHAPTER 1
˜ := diag[K, 14 ] we have real quaternions, and K ax = ω(a)x, where
⎡
ω(a) =
φ(a(1) ) φ(a(2) )K
⎢ ⎢ ⎢ ⎢ (2) ⎢ −τ (a )K =⎢ (1) ⎢ τ (a ) ⎢ ⎢ ⎢ ⎣
a0 a1 a2 a3 a4 a5 a6 a7
xa = ν(a)x, −a1 a0 a3 −a2 a5 −a4 −a7 a6
−a2 −a3 a0 a1 a6 a7 −a4 −a5
−a3 a2 −a1 a0 a7 −a6 a5 −a4
−a4 −a5 −a6 −a7 a0 a1 a2 a3
−a5 a4 −a7 a6 −a1 a0 −a3 a2
−a6 a7 a4 −a5 −a2 a3 a0 −a1
−a7 −a6 a5 a4 −a3 −a2 a1 a0
⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥, ⎥ ⎥ ⎥ ⎥ ⎦
˜ T (a)K. ˜ ν(a) = Kω Consider now the 2 × 2 Hermitian matrix with real octonion entries a b . A= ¯ b c For A to be Hermitian, the elements a and c must in fact be real, and thus a18 ω(b) ω(A) = . ω T (b) c18
(1.33)
Adding together appropriate (octonion) multiples of rows and columns shows that this matrix is similar to the matrix ⎤ ⎡ b 0 a18 14 ⊗ ⎢ ⎥ 0 ¯b ⎢ ⎥ ¯b 0 ⎣ ⎦ 14 ⊗ c18 0 b and thus the characteristic polynomial is given by det(ω(A) − λ116 ) = ((a − λ)(c − λ) − b¯b)8 . This shows that each eigenvalue is eightfold degenerate. Regarding the eigenvectors, as the number of independent real elements in (1.33) is ten, and there are two distinct eigenvalues, there are a total of eight independent components. This implies that in the analogue of Proposition 1.2.5, exactly eight components are to be multiplied together in any one term, and consequently (dA) = (λ1 − λ2 )8 dλ1 dλ2 (U† dU).
(1.34)
Furthermore, choosing the elements a, c and the components bj of ω(b) in (1.33) to have the Gaussian distributions 2 −4a2 8 −8b2j 2 −4c2 √ e e , √ e , , π π π respectively, we have that the joint distribution of the independent elements is proportional to e−Tr((ω(A))
2
)/2
.
This together with (1.34) implies that the eigenvalue p.d.f. is given by (1.28) with N = 2, β = 8. 1. The aim of this exercise is to show that if a 2N ×2N Hermitian matrix X commutes with the time reversal operator T = Z2N K, then the eigenvalues of X are doubly degenerate (this is known as Kramer’s
E XERCISES 1.3
17
GAUSSIAN MATRIX ENSEMBLES
degeneracy). is an eigenvector of X with eigenvalue λ. State why T φ is also an eigenvector with eigenvalue (i) Suppose φ λ. φ = 0 and (ii) Use the facts that T satisfies the formula of Exercises 1.1 q.3(i) and T 2 = −1 to show that φ|T hence deduce the desired result. 2. [389, p. 32] Let A and M be N × N matrices, where A is nonsingular. In this exercise it will be shown that for A real (β = 1), complex (β = 2) and real quaternion (β = 4), and M real symmetric (β = 1), Hermitian (β = 2) and quaternion real Hermitian (β = 4), (A† dMA) =
“
det(A† A)
”β(N−1)/2+1
(dM),
(1.35)
up to a ± sign. In the case β = 1, this is the statement of Proposition 1.2.4. The idea is to decompose A in terms of elementary matrices A = Ep Ep−1 · · · E1 . Each elementary matrix is either a permutation matrix E(j↔k) (the identity matrix with rows j and k interchanged), a matrix E(j→αj) which multiplies row j by the constant α with α real (β = 1), complex (β = 2) or real quaternion (β = 4) (the identity matrix with row j multiplied by α), or the matrix E(j→j+k) which adds together two rows (the identity matrix with row k added to row j). (i) Show by explicit calculation that for any matrix X of the same type as M (E(j↔k) dXE(j↔k)† ) = (dX),
(E(j→j+k) dXE(j→j+k)† ) = (dX),
while, up to a ± sign, (E(j→αj) dXE(j→αj)† ) = |α|β(N−1)+2 (dX) = | det E(j→αj) |β(N−1)+2 (dX). (ii) Use the result of (i) to deduce the stated result. ¯ is used in the exercises below to denote the complex conjugate of For printing purposes, the symbol α∗ rather than α α. 3. The aim of this exercise is to calculate the change of variables from the independent elements of a Hermitian matrix X to the eigenvalues λ1 , . . . , λN and other independent variables. (i) From the diagonalization formula X = ULU−1 , where L := diag[λ1 , . . . , λN ] and U is a unitary matrix with columns given by the eigenvectors of X, show that U−1 dXU = U−1 dUL − LU−1 dU + dL and write down a formula for the Jacobian in terms of (dX). Use the result of q.2 to show that the wedge product of the independent elements on the l.h.s. is equal to (dX). (ii) Show that U−1 dUL − LU−1 dU + dL equals 2 6 6 6 4
dλ1 (λ2 − λ1 )( u†1 · d u2 )∗ ... u†1 · d uN )∗ (λN − λ1 )(
(λ2 − λ1 ) u†1 · d u2 dλ2 .. . (λN − λ2 )( u†2 · d uN )∗
... ... .. . ...
(λN − λ1 ) u†1 · d uN (λN − λ2 ) u†2 · d uN .. . dλN
3 7 7 7. 5
(iii) Use the facts that u†j · d uk has independent real and imaginary parts and that only the elements on and above the diagonal are independent to conclude that the wedge product of the independent elements of the matrix in (ii) equals N Y ^ (λk − λj )2 dλj (U† dU). 1≤j
j=1
18
CHAPTER 1
(iv) Show that the factor dependent on the λj ’s is consistent with the form required by the scaling X → aX (recall Section 1.2.3). 4. Here the objective is the same as in q.3 above, except X is now an N × N Hermitian matrix with real quaternion elements. (i) From the diagonalization formula X = ULU−1 where L = diag[λ1 12 , . . . , λN 12 ] and U is an N × N unitary matrix with real quaternion elements, write down the formulas analogous to those in q.3(i) and q.3(ii) above. To write down the analogue of (ii) use a matrix notation for the quaternion elements, uk = (u1k , . . . , uNk )T ,
u†j · uk =
N X
u†pj upk .
p=1
(ii) Use the facts that u†j · d uk has four independent terms, corresponding to the real and imaginary parts of the two independent terms in each real quaternion element, to deduce the formula analogous to q.3(iii) above. Also repeat the scaling analysis of q.3(iv) above. 5. A Hermitian matrix with zero real part is antisymmetric. (i) Show that the nonzero eigenvalues of antisymmetric Hermitian matrices come in ± pairs, λj and −λj , say, j and φ ∗j , and that for N odd λ = 0 is an eigenvalue. with corresponding eigenvectors φ j , λN/2+j = −λj , uN/2+j = u∗j (j = (ii) Use (i) to deduce that the equation of q.3(ii) holds with uj = φ ∗ j , λ(N−1)/2+j = −λj (j = 1, . . . , (N − uN = φ0 , λN = 0, uj = φ 1, . . . , N/2) N even, and uN = 1)/2) N odd. Use the fact that eigenvectors of a Hermitian matrix corresponding to distinct eigenvalues 0 † · dφ ∗j = 0 (j = 0), and note too that φ † · dφ k and φ ∗† · dφ ∗j are not are orthogonal to deduce that φ 0 0 independent. (iii) Use (ii) to show that for an antisymmetric Hermitian N × N matrix Hi , diagonalized by Hi = ULU−1 , (dHi ) =
Y
Y
Y
(N−1)/2
j=1
dλj (U† dU),
N even,
j=1
1≤j
(dHi ) =
^
N/2
(λ2j − λ2k )2
λ2j
^
(N−1)/2
(λ2j − λ2k )2
dλj (U† dU),
N odd.
j=1
1≤j
(iv) Conclude from the result of (iii) that for a random antisymmetric Hermitian N × N matrix with upper p 2 triangular elements ixjk chosen with p.d.f. 1/πe−xjk , the eigenvalue p.d.f. of the positive eigenvalues is equal to N/2 1 Y −λ2j e CN j=1
1 CN
Y
(N−1)/2
Y
N even,
1≤j
Y
2
λ2j e−λj
j=1
(λ2j − λ2k )2 ,
(λ2j − λ2k )2 ,
N odd,
1≤j
where the normalizations CN are given explicitly in (4.157) below. 6.
(i) Let Qr be a quaternion real Hermitian matrix in which all entries of each real quaternion are real, and let H be the Hermitian matrix formed by replacing each quaternion element (1.20) by the scalar z + iw. Show that Qr and H have the same distinct and that the eigenvalues of Qr are doubly degenerate with h r eigenvalues, i φk (1) (2) = Z2N ψ = [φrk + iφik ]k=1,...,N is an (1) where φ eigenvectors of the form ψ = φi and ψ k
k=1,...,N
eigenvector of H. Hence write down the eigenvalue p.d.f. of Qr .
19
GAUSSIAN MATRIX ENSEMBLES
(ii) Let Q be an N × N real quaternion Hermitian matrix. Proceed in a converse fashion to (i) to write down a 4N × 4N real symmetric matrix R such that Q and R have the same distinct eigenvalues, and thus the same eigenvalue p.d.f. Also, relate the corresponding eigenvectors. Similarly, for H an N × N complex Hermitian matrix, replace each entry x + iy by its 2 × 2 real matrix representation » – x y (1.36) −y x to obtain a doubly degenerate 2N × 2N matrix for which the distinct eigenvalues coincide with those of H. 7. Consider a quaternion real Hermitian matrix Qi in which all entries of each real quaternion are pure imaginary so that Qi is antisymmetric. j as in (i) With the pair of eigenvectors corresponding to the doubly degenerate eigenvalues λj denoted by u q.4, note from the theory of q.5(i) that u∗j is equal to the pair of eigenvectors corresponding to the doubly degenerate eigenvalue −λj . (ii) Use the fact that eigenvectors of a Hermitian matrix corresponding to distinct eigenvalues are orthogonal to deduce that – – » N N » X X wpj 0 −2Im (zpj wpj ) zpj , ujp := , u†j · u∗j := u†pj u∗pj = ∗ ∗ −2Im (zpj wpj ) 0 −wpj zpj p=1
p=1
†j · d †j · d and conclude from this that u u∗j has only one independent component. Note too that u uk and ∗† ∗ uj · d uk are not independent. (iii) With the analogue of the equation of q.3(ii) in the quaternion case modified as in the first sentence of q.5(ii) (N even case), show from (i) that for an antisymmetric N × N quaternion real Hermitian matrix Qi diagonalized by Qi = ULU−1 , Y
Y
N/2
(dQi ) =
λj
j=1
Y
Y
λ5j
j=1
dλj (U† dU),
N even,
j=1
1≤j
(N−1)/2
(dQi ) =
^
N/2
(λ2j − λ2k )4
^
(N−1)/2
(λ2j − λ2k )4
dλj (U† dU),
N odd.
j=1
1≤j
8. [146] Let Q be a quaternion real matrix with the property that iQ is Hermitian. (i) Note that Q must anticommute with the time reversal operator T = Z2N K, and use this to show that if |φ is an eigenvector with eigenvalue −λ. is an eigenvector of Q with eigenvalue λ, then T φ (ii) Proceed as in q.4 to show that with iQ diagonalized by iQ = UiLU† , where L = diag(λ1 , −λ1 , . . . , λN , −λN ), and U is unitary with real quaternion elements in which all elements are real, (dQ) =
N Y
(2λj )2
j=1
Y
(λ2k − λ2j )2
N ^
dλj (U† dU).
j=1
1≤j
9. Let Gβ,N be the normalization in (1.28), which has the evaluation (1.163) below, and let Aβ,N = Z
so that Aβ,N
“ β ”N/2 “ β ”βN(N−1)/4 2π π 2
e−(β/2)Tr(H ) (dH) = 1.
(1.37)
20
CHAPTER 1
Deduce from this last equation and (1.27) that for β = 1, 2 Z N! (U† dU) = , Aβ,N Gβ,N
(1.38)
where the first entry of each column of U is chosen to be real and positive. 10. Define the matrix
„ ˜ = H
c Tr H2
«1/2 H,
where H is a member of one of the Gaussian ensembles and c > 0 is a constant. By changing variables from ˜ j } of H ˜ have distribution ˜ and Tr H2 , show that the eigenvalues {λ the elements of H to the elements of H, proportional to N ” Y “ X ˜ 2j ˜k − λ ˜ j |β . λ δ c− |λ j=1
1≤j
1.4 COULOMB GAS ANALOGY The eigenvalue p.d.f. (1.28) can be identified with the Boltzmann factor of a particular log-gas, an observation which goes back to Dyson [146]. To appreciate this, we must revise some basic theory from statistical mechanics [390], and show how the Boltzmann factor of a log-gas, or more generally a one-component Coulomb system, is computed.
1.4.1 Boltzmann factors The canonical formalism of statistical mechanics applies to any mechanical system of N particles free to move in a fixed domain Ω, in equilibrium at absolute temperature T . A fundamental postulate gives the p.d.f. for the event that the particles are at positions r1 , . . . , rN as 1 −βU(r1 ,...,rN ) e . ZˆN Here U (r1 , . . . , rN ) denotes the total potential energy of the system, β := 1/(kB T ) (kB is Boltzmann’s constant), and the normalization ZˆN is given by dr1 · · · drN e−βU(r1 ,...,rN ) . (1.39) ZˆN = Ω
Ω
−βU( r1 ,..., rN )
is referred to as the Boltzmann factor and ZˆN /N ! =: ZN is called the (canonical) The term e partition function. For log-potential Coulomb systems the potential energy U is calculated according to the laws of twodimensional electrostatics, and Ω must be one- or two-dimensional. The particles can be thought of as infinitely long parallel charged lines, which are perpendicular to the confining domain. In a vacuum the electrostatic potential Φ at a point r = (x, y) due to a two-dimensional unit charge at r = (x , y ) is given by the solution of the Poisson equation ∇2r Φ(r, r ) = −2πδ(r − r ), where ∇2r :=
∂2 ∂2 + . ∂x2 ∂y 2
(1.40)
21
GAUSSIAN MATRIX ENSEMBLES
It is straightforward to verify that the solution of the Poisson equation is (see Exercises 1.4 q.1) Φ(r, r ) = − log(|r − r |/l),
(1.41)
where l is some arbitrary length scale which will henceforth be set to unity. A Coulomb system is said to consist of one component if all N particles are of like charge, q, say. To stop the particles from all repelling to the boundary, a neutralizing background charge density −qρb (r) is imposed, with the electroneutrality condition Ω ρb (r ) dr = N. The total potential energy U therefore consists of the sum of the electrostatic energy of the particle-particle interaction log |rk − rj |, U1 := −q 2 1≤j
the particle-background interaction U2 := q 2
N
V (rj )
where
log |r − rj | ρb (r) dr,
V (rj ) :=
(1.42)
Ω
j=1
and the background-background interaction q2 q2 dr ρb (r ) dr ρb (r) log |r − r| = − ρb (r )V (r ) dr . U3 := − 2 Ω 2 Ω Ω
(1.43)
The factor of 12 in U3 is included to compensate for the double counting of the potential energy implicit in the double integration. From this expression for U we conclude that the Boltzmann factor of a one-component log-potential Coulomb system (log-gas) is of the form e−βU3
N l=1
e−ΓV (rl )
|rk − rj |Γ ,
(1.44)
1≤j
where Γ := q 2 /kB T . Furthermore, for a given geometry and background density the potentials V (r) and U3 can readily be evaluated. As an illustration, we have the following result. P ROPOSITION 1.4.1 The Boltzmann factor of a one-component log-potential Coulomb system of N particles of charge q = 1, confined to a circle of radius R with a uniform neutralizing background, is given by R−N β/2 |eiθk − eiθj |β , 1≤j
where the position of each particle has been specified in polar coordinates. Proof. It is generally true that for two points r and r in the plane |r − r | = |z − z |, where z and z are the
corresponding points in the complex plane. Hence, if r and r are both on a circle of radius R with positions specified using polar coordinates, then |r − r | = R|eiθ − eiθ |. Use of this formula gives the required expression for the product over pairs in (1.44). It also allows the potential V (r) to be written as Z 2π Z 2π N N V (r) = log |Reiθ − Reiθ | Rdθ = N log R + log |eiθ − 1| dθ . 2πR 0 2π 0
But it is straightforward to show that the last integral vanishes (see Exercises 1.4 q.2), and so V (r) = N log R. Use of 2 this result gives U3 = − q2 N 2 log R. Substituting these evaluations in (1.44) and noting that since q = 1, Γ = β gives the desired expression for the Boltzmann factor.
The Boltzmann factor, being proportional to the p.d.f. for the location of the particles, occurs in the defi-
22
CHAPTER 1
nition of all statistical quantities associated with the equilibrium state. In particular the canonical average of any function f (r1 , . . . , rN ) is given by 1 dr1 · · · drN f (r1 , . . . , rN )e−βU(r1 ,...,rN ) . (1.45) f := ˆ ZN Ω Ω N With f = r − rj ) the canonical average is called the one-point correlation function, or particle j=1 δ( density N N ρ(1) (r) := δ(r − rj ) = dr2 · · · drN e−βU(r,r2 ,...,rN ) , (1.46) ˆN Ω Z Ω j=1 where the equality is valid for a system of identical particles, and thus when the Boltzmann factor is a symmetric function of the particle coordinates. 1.4.2 The potential and calculation of ρb (y) Comparison of (1.28) with (1.44) shows immediately that the eigenvalue p.d.f. is identical to the Boltzmann factor of a one-component log-potential Coulomb system confined to a line, with the position of the charged particles corresponding identically to the location of the eigenvalues. Furthermore, the background charge density −qρb (y) is such that ∞ x2 +C = ρb (y) log |x − y| dy, (1.47) 2 −∞ where C is a constant. Note that it is not possible to satisfy (1.47) for |x| → ∞, since in this limit the r.h.s. is to leading order N log |x| and is thus a different order from the l.h.s. Instead we seek to solve the integral equation for ρb (y) with support on the finite interval (−a, a) say, and x confined to the same interval. Then (1.47) reads a x2 +C = ρb (y) log |x − y| dy, x ∈ (−a, a). (1.48) 2 −a The solution of the equation can be computed exactly by the method of eigenfunction expansions (see, e.g., [448]). P ROPOSITION 1.4.2 Suppose all the eigenvalues {λn }n=0,1,... and corresponding normalized eigenfunctions {φn }n=0,1,... of a linear operator A are known, all the eigenvalues are nonzero, and the eigenfunctions form a complete set. Then the operator equation g = Af, where g is given, has the solution ∞ g|φn f= φn , λn n=0
where ·|· denotes the inner product. P P∞ Proof. Since the eigenfunctions form a complete set, g = ∞ n=0 g|φn φn Also f = n=0 f |φn φn and so Af = P ∞ n=0 f |φn λn φn . The result follows by equating the coefficients of φn in the operator equation.
To make use of this method, it is necessary to make the further change of variables y = a cos θ, so that (1.48) reads 1 − cos 2σ − 4
x = a cos σ,
1 N C − log a + 2 4 a2 a
sin θρb (a cos θ) =: aφ(θ)
=− 0
π
log | cos θ − cos σ| φ(θ) dθ.
(1.49)
23
GAUSSIAN MATRIX ENSEMBLES
Note that cos θ − cos σ = 2 sin(σ − θ)/2 sin(σ + θ)/2. Since 2| sin(σ − θ)/2| gives the chord length for two points on the unit circle with angles σ and θ, the r.h.s. of (1.49) can be interpreted as giving (up to an additive constant) the electrostatic potential at the angle σ due to a charge density φ(θ) and φ(2π − θ) between 0 and π and π and 2π, respectively, on the unit circle. The eigenvalues and eigenfunctions of the integral operator π A[φ](σ) := − log | cos θ − cos σ|φ(θ) dθ (1.50) 0
are known (see Exercises 1.4 q.4). They are λ0 = π log 2,
φ0 (θ) =
1 π 1/2
,
π λn = , n
1/2 2 φn (θ) = cos nθ π
(n = 1, 2, . . . ).
In terms of these eigenfunctions 1 π 12 C C 1 1 1 N 1 N − − log a + 2 = − φ2 (σ) − log a + 2 π 2 φ0 (σ), − cos 2σ − 4 4 a2 a 4 2 4 a2 a and so from the general formula of Proposition 1.9 the solution of the transformed integral equation is 1 1 N C 1 1 − 2 log a + 2 + log 2 , φ(θ) = − (cos 2θ − 1) − 2π π log 2 4 a a 2 where 1/(2π) has been added and subtracted for later convenience. Reverting back to the original variables we obtain the following result. P ROPOSITION 1.4.3 The solution of the integral equation a x2 +C = ρb (y) log |x − y| dy, 2 −a is a ρb (y) = π
1−
y 2 a
1 − π log 2
−a ≤ x ≤ a,
1 N C 1 − log a + 2 + log 2 4 a2 a 2
a . 1 − (y/a)2
We see that there are two drastically different classes of solution depending on the value of C. Unless we choose a2 a2 − log 2 (1.51) 4 2 the density profile ρb (y) has an inverse square root singularity at y = ±a. However, with C according to (1.51) the term proportional to (1 − (y/a)2 )−1/2 vanishes and a a physically sensible result is obtained. Making this choice of C and fixing a by the neutrality condition −a ρb (y) dy = N gives the desired analogy between the Boltzmann factor of a one-component log-potential Coulomb system and the eigenvalue p.d.f.’s of Proposition 1.3.4. C = N log a −
P ROPOSITION 1.4.4 The Boltzmann factor of the one-component Coulomb system with par√ log-potential √ ticles of charge q = 1 at x1 , . . . , xN , confined to the interval [− 2N, 2N ], with a neutralizing background charge density √ y2 2N , −ρb (y) = − 1− π 2N
24
CHAPTER 1
is N β x2j A exp − 2 j=1
|xk − xj |β ,
1≤j
3βN 2 βN 2 log(N/2) + . A = exp − 4 8
Proof. Apply the general formula (1.44) for the Boltzmann √ factor of a one-component log-potential Coulomb system with rk = xk . From Proposition 1.4.3 and (1.51) with a = 2N « „ p x2 1 1 , V (x) = + 4N log N/2 − 2 4 8 «– » 2Z 1 p „ p 1 N 1 U3 = −q 2 x2 1 − x2 dx + 2N 2 log N/2 − . π −1 4 8
A simple change of variables x = cos θ shows that the integral in the above equals π/8. The stated formula for the Boltzmann factor follows.
Proposition 1.4.4 can be used to predict the eigenvalue density profile for Gaussian β-ensembles with eigenvalue p.d.f. (1.28). Physically, we expect that to leading order in N Coulomb systems are locally charge neutral, which for a one-component system implies that to leading order the particle density will equal the background density. For the log-potential system of Proposition 1.4.4 this gives the particle density as √ 2N y2 ρ(1) (y) = . (1.52) 1− π 2N But the statistical properties of the log-potential system in Proposition 1.4.4 are identical to those of the eigenvalues of Gaussian random matrices, so we expect that the eigenvalue density profile will to leading order in N be given by this formula. Consequently we expect the so-called global density √ ρ˜(1) (Y ) := lim 2/Nρ(1) ( 2N Y ) (1.53) N →∞
to obey the limit formula
ρ˜(1) (x) =
2 π (1
0,
− x2 )1/2 , |x| < 1, |x| ≥ 1,
(1.54)
known as the Wigner semicircle law. The validity of this statement is known rigorously from [328]. Wigner’s derivation, which is applicable to GUE matrices, is given in Exercises 1.6 q.1. In Figure 1.1 we have plotted the empirical eigenvalue density for 1000 10 × 10 matrices from the GUE, √ using the variable Y = y/ 2N . The accuracy of the Wigner semicircle law is evident.
1.4.3 The complex electric field and calculation of ρb (y) The integral equation (1.48) is the special case V (x) = x2 /2 of the integral equation a ρb (y) log |x − y| dy, x ∈ (−a, a). V (x) + C =
(1.55)
−a
For the log-gas at β = 2 a rigorous derivation of this integral equation for the particle density is given in Exercises 14.4 q.4 below. We seek the solution such that ρb (y) is bounded at y = ±a and normalized so that a ρb (y) dy = N. (1.56) −a
25
GAUSSIAN MATRIX ENSEMBLES
density
0.6 0.5 0.4 0.3 0.2 0.1 -1
-0.5
1
0.5
Y
Figure 1.1 Empirical demonstration of the Wigner semicircle law for 10 × 10 matrices from the GUE.
If the primary concern is the calculation of ρb (y) and not C, an alternative to the method of eigenfunctions used above is to introduce the complex electric field a ρb (y) dy. (1.57) E(z) := − −a z − y Note that for z ∈ / [−a, a], E(z) is analytic and has the asymptotic behavior E(z)
∼
|z|→∞
−
N , z
(1.58)
while for z ∼ ±a, by the assumption that ρb (y) is bounded, E(z) ∼ O(log(z ∓ a)). z→±a
(1.59)
Furthermore, if ρb (y) can be analytically continued to a neighborhood of the interval y ∈ (−a, a), it follows by deforming the path of integration in the neighborhood of y = x and Cauchy’s theorem that E + (x) − E − (x) = 2πiρb (x),
x ∈ (−a, a),
(1.60)
where E ± (x) = lim E(x ± i). →0+
Differentiating (1.55) shows 1 + (E (x) + E − (x)) = −V (x), for x ∈ (−a, a). (1.61) 2 The properties (1.58), (1.59) and (1.61) can be used to characterize E(z), with ρb (x) then computed from (1.60). For the quantity W (z) := eE(z) the properties (1.58), (1.59), (1.60), where ρb (x) is given, specify a scalar Riemann-Hilbert problem. Restricting attention to potentials V (x) even in x, one can check that the function a 1 2 V (t) √ E(z) = − z − a2 dt, (1.62) 2 2 π −a (z − t) a − t Re E(x) =
26
CHAPTER 1
with a such that 1 π
a
−a
tV (t) √ dt = N a 2 − t2
(1.63)
has the properties (1.58), (1.59) and (1.61) and is thus the sought complex electric field. In particular, to verify (1.61), we note that (1.62) gives a V (t) i 2 ± 2 √ E (x) = ∓ a − x lim dt (1.64) π →0+ −a (x ± i − t) a2 − t2 and then make use of Cauchy’s theorem. Using this formula in (1.60) gives an explicit formula for ρb (x) in terms of the potential V [412]. P ROPOSITION 1.4.5 In the case V (x) even, the solution of the integral equation (1.55) with ρb (y) bounded at y = ±a and normalized as in (1.56) is a V (y) − V (t) 1 1 2 2 √ ρb (y) = 2 a − y dt, (1.65) 2 π y − t a − t2 −a where a is specified by (1.63). Proof. After substituting (1.64) in (1.60), we subtract an appropriate multiple of the identity “Z
a
lim
→0
−a
1 √ dt − (x + i − t) a2 − t2
Z
−a
from both sides to obtain 2πiρb (x) = −
a
2i p 2 a − x2 Re lim π →0+
Z
” 1 √ dt = 0 2 2 (x − i − t) a − t
a
−a
(1.66)
V (t) − V (x) √ dt. (x + i − t) a2 − t2
The limit can be taken inside the integrand because the numerator vanishes for x = t, giving (1.65).
In the special case V (y) = y 2 /2, (1.65) gives ρb (y) =
a 1 − (y/a)2 , π
√ while it follows from (1.63) that a = 2N , in agreement with (1.52). In this same special case the explicit form of the complex electric field can also be computed; this is done in Exercises 1.6 q.2. The generalization of (1.65) for V (x) not necessarily even is b V (y) − V (t) 1 1 ρb (y) = 2 (y − a)(b − y) dt, (1.67) π y − t (t − a)(b − t) a where a and b are such that b V (t) dt = 0, (t − a)(b − t) a
1 π
a
b
tV (t) dt = N, (t − a)(b − t)
(1.68)
as can be checked from a similar analysis. It can happen that for certain V (x) the solution (1.65) or (1.67) of (1.55) does not in fact correspond to the background density because it becomes negative within the interval (−a, a). An example is the potential V (x) = −cx2 + gx4 for c large enough. Formula (1.65) gives 1 (1.69) ρb (y) = (−2c + 2ga2 + 4gy 2 ) a2 − y 2 , π
27
GAUSSIAN MATRIX ENSEMBLES
where, according to (1.63), −ca2 +
3ga4 = N. 2
(1.70)
The solution (1.69) will take on negative values for some y whenever c > ga2 . According to (1.70), for this to happen it is sufficient that c2 > 2gN . In such a circumstance, the original assumption that the support is on a single interval breaks down, and one must seek a solution supported on a double interval (−a2 , −a1 ) ∪ (a1 , a2 ). E XERCISES 1.4
1.
“ ” (i) By explicit differentiation show that Φ(r, r ) = − log |r − r |/l , satisfies the two-
dimensional Laplace equation ∇ 2r Φ(r, r ) = 0 for r = r . (ii) Use the divergence theorem in the plane Z Z ∇2 V (r ) dr = n · ∇V (r ) dr D
C
with V (r) = Φ(r, r ), D a disk centered on r and C the circle which is the boundary of the disk, to conclude Z ∇ 2r Φ(r, r ) dr = −2π. D
Relate this result to the Poisson equation (1.40). 2. Use the power series expansion of log(1 − z) for |z| < 1 to show that for all |μ| < 1, Z 2π log |1 − μeiθ | dθ = 0. 0
Show that this integral is equal to 2π log |μ| for |μ| > 1 by using the result for |μ| < 1, and use the continuity of the integral as a function of μ to deduce its value for |μ| = 1. 3. Suppose there are N mobile particles of charge q in a disk filled with a uniform neutralizing background ρb = N/πR2 . This specifies the two-dimensional one-component plasma confined to a disk. (i) With the position of the particles specified in polar coordinates, use the integral evaluations of q.2 and the definition of V (r) (1.42) to show V (r) = πρb (r 2 /2 + R2 log R − R2 /2).
(1.71)
Write down the Poisson equation satisfied by V (r). (ii) Use this expression for V (r) to calculate U3 and thus show that the Boltzmann factor is equal to PN Y 2 2 e−ΓN ((1/2) log R−3/8) e−πΓρb j=1 | rj | /2 |rk − rj |Γ , Γ := q 2 β.
(1.72)
1≤j
4.
(i) Assuming the validity of the formula log |1 − aeix | = −
∞ X an cos nx , n n=1
0 ≤ a < 1,
x ∈ R,
for a = 1 provided x = 0 mod(2π), deduce that log |2 sin(x − t)/2| = −
∞ X cos n(x − t) n n=1
for x − t = 0 mod(2π), and write down a similar formula for log |2 sin(x + t)/2|. Hence derive the cosine
28
CHAPTER 1
expansion log(2| cos x − cos t|) = −
∞ X 2 cos nx cos nt. n n=1
(1.73)
(ii) Use the above cosine expansion to verify that the eigenvalues and normalized eigenfunctions of the integral operator Z π ˆ dθ ˆ log | cos θ − cos σ|φ(θ) A[φ](σ) := − 0
are as specified below (1.50). 5. The objective of this exercise is to compute the background density and the Boltzmann factor for the onecomponent log-potential system confined to the interval (−a, a) with unit charges and one-body potential V (x) =
x4 x2 +g + C. 2 N
(1.74)
This calculation is of interest in the graphical expansion of matrix integrals [98], [555], and will be used in this context in the next section. (i) With Y = cos θ verify that
cos 4θ − 1 = −8Y 2 (1 − Y 2 )1/2 . sin θ
(ii) Use the eigenfunction expansion method, the result of (i) and Proposition 1.4.3 to show that the solution of the integral equation Z a
V (x) = −a
ρb (y) log |x − y| dy,
−a ≤ x ≤ a,
which is bounded at y = ±a is ρb (y) = provided
„ C = −a
2
4g 2 ”p a“ 2ga2 + y 1 − (y/a)2 , 1+ π N N
« N 3ga2 3ga2 1 1 − 2 log a + log 2 + + log 2 . 4 a 2 8N 2N
(iii) Use the neutrality condition to show a2 3ga4 + = N, 2 2N and use this in the formula for C to obtain the simplification C=−
(1.75)
a N a2 + N log − . 8 2 4
(iv) Use the trigonometric Euler integral in Exercises 4.1 q.1(i) below and the neutrality condition to show U3 = −
CN a4 a2 N N2 + − − , 2 192 24 16
and thus (U2 + U3 ) − (U2 + U3 )|g=0 – » N √ √ ´ ´` g X 4 N2 1 ` √ = xj − (a/ 2N )2 − 1 9 − (a/ 2N )2 − log(a/ 2N ) , N j=1 2 24
(1.76)
where (U2 + U3 )|g=0 is as implicit in Proposition 1.4.4. 6.
(i) For a general potential u(x), use the eigenfunction expansion method to show that the solution ρb (y) of the
29
GAUSSIAN MATRIX ENSEMBLES
integral equation
Z
a
u(x) + C = −a
ρb (y) log |x − y| dy,
x ∈ [−a, a],
(1.77)
which is bounded at y = ±a can be written ρb (a cos θ) = −
∞ “Z π ”“ ” X 2 p u(a cos σ) cos pσ dσ cos pθ − 1 . 2 aπ sin θ p=1 0
(ii) For u(x) = x2n , n ∈ Z+ , use the integration formula Z π π “ 2n ” cos2n σ cos 2pσ dσ = 2n , 2 n+p 0 verified using complex exponentials, and the transformation identity n n “ X 2(n − l) ” 1 X “ 2n ” 1 − cos 2pθ p (2 cos θ)2(l−1) = 2 n−l 2n p=1 n + p 1 − cos θ l=1
to show that [99] ρb (x) =
„ n «r “ x ”2 4n “ a ”2n−1 X “ 2(n − l) ”“ 2x ”2(l−1) 1− . n−l π 2 a a l=1
Check that this is consistent with ρb (y) in q.2(ii). 7. [124] The task of this exercise is to solve the integral equation Z a x2 ρb (y) log |x − y| dy, +C = 2 −b √ with a = 2N s, s < 1, subject to the neutrality constraint Z a ρb (y) dy = N,
(1.78)
(1.79)
−b
and to the constraint that ρb (y) be bounded at y = −b. (i) Change variables according to y=Y +
a−b , 2
x=X+
a−b 2
and then according to Y =
a+b cos θ, 2
Y =
a+b cos σ, 2
sin θρb
“a + b 2
cos θ +
a − b” a+b = φ(θ), 2 2
to rewrite (1.78) to read −
“ 2 ”2 “ (a + b)2 a 2 − b2 (a − b)2 (a + b)2 a + b” cos 2σ − cos σ + + + C − N log a+b 16 2 8 16 2 Z π φ(θ) log | cos θ − cos φ| dφ. =− 0
(ii) Use the method of derivation of Proposition 1.4.4 to show that only for the value √ 9N l2 l N C = N log √ − + 2N ls − N s2 , 8 2
(1.80)
30
CHAPTER 1
√ where a + b = N l, does (1.78) permit a solution bounded at y = −b, and furthermore show that the latter has the explicit form √ √ √ √ l − 2s “ l − s + y/ 2N ”1/2 √ 2N √ . (1.81) ρb (y) = (s − y/ 2N )1/2 (l − s + y/ 2N )1/2 + N √ π 2π s − y/ 2N (iii) By making use of (4.2) below, show that the neutrality condition (1.79) gives l=
p 2 (s + s2 + 3). 3
(1.82)
8. In this exercise the location of the minimum of the function H(x1 , . . . , xN ) :=
N 1X 2 x − 2 j=1 j
X
log |xk − xj |
1≤j
will be determined by following a calculation of Stieltjes [503]. This gives the equilibrium points of the system of Proposition 1.4.4. P ∂2H (i) Show that H is convex by establishing that for tj = 0 (j = 1, . . . , N ), N j,k=1 tj tk ∂xj xk > 0, and conclude that H has a unique minimum. Q (0) (ii) Let g(x) = N l=1 (x − xl ). Show that the equations for the minimum ∂H/∂xj = 0 (j = 1, . . . , N ) can be written g (xj ) − 2xj g (xj ) = 0 (j = 1, . . . , N ). (iii) Observe that the l.h.s. of the above equation is a polynomial of degree N which vanishes at the zeros of g(x), and so must be proportional to g(x), to deduce the d.e. g (x) − 2xg (x) + 2N g(x) = 0. Hence show that the minimum of H(x1 , . . . , xN ) occurs at the zeros of the Hermite polynomial HN (x).
1.5 HIGH-DIMENSIONAL RANDOM ENERGY LANDSCAPES This and the next three sections all relate to the Wigner semicircle law for the eigenvalue density in the Gaussian ensembles. In this section it is the Wigner semicircle law in the case of the GOE which arises. As we have seen, the GOE was formulated as a model of the eigenvalues of classically chaotic quantum Hamiltonians with a time reversal symmetry. Later, the GOE received prominence for its relevance to the study of the so-called replica trick in the theory of disordered systems [161]. As applied to random matrix theory, the replica trick corresponds to the identity 1 ∂Zn () Tr ( − H)−1 , Zn () := det n ( − H)H∈GOE . = lim (1.83) n→0 n ∂ H∈GOE In practice the difficulty with the implementation of (1.83) is that orthogonal polynomial methods to evaluate Zn () (see Chapter 5) require n to be a positive integer, and one is then faced with the problem of analytic continuation off the positive integers in order to take the limit. More recently the GOE has been shown to be of relevance to another problem relating to the theory of disordered systems [248]. The problem is the computation of the distribution of the critical points for certain high-dimensional Gaussian random potentials (often referred to as landscapes). Specifically, consider the energy μ 2 xj + V (x1 , . . . , xN ), 2 N
H :=
j=1
(1.84)
31
GAUSSIAN MATRIX ENSEMBLES
where μ > 0 and V is Gaussian distributed with zero mean and covariance 1 (x1 − x2 )2 . V (x1 )V (x2 ) = N f (1.85) 2N A critical point of H is characterized by the simultaneous stationarity conditions ∂H/∂xj = 0 (j = 1, . . . , N ). Let ρ(1) (x) be the density of critical points, so that N (D) — the expected number of critical points in the region D — is given by N (D) = D ρ(1) (x) dx. With {xk }k=1,...,N ∗ denoting the critical points, one has the change of variables type formula ∗
N k=1
N ∂2H ∂H δ(x − xk ) = δ , det ∂x ∂x ∂x i,j=1,...,N i i j i=1
(1.86)
and hence ρ(1) (x) can be computed as the ensemble average of the r.h.s. This form of ρ(1) (x) is referred to as the generalized Kac-Rice formula (see also (15.56) below). P ROPOSITION 1.5.1 Let GOE# refer to the GOE with matrices X → N/2f (0)X. We have N ∞ −N t2 /2 N −N N (R ) = μ e dt. det (μ + f (0)t)1N − X) 2π −∞ X∈GOE#
(1.87)
Proof. We begin with the formula implied by the sentence including (1.86). Recalling (1.84) this gives x) = ρ(1) (
N DY
E δ(μxi + ∂i V )| det[μδj,k + ∂j ∂k V ]j,k=1,...,N |
i=1
=
N DY
ED δ(μxi + ∂i V )
E | det[μδj,k + ∂j ∂k V ]j,k=1,...,N | ,
(1.88)
i=1
where ∂i := ∂/∂xi and the second equality follows by noting that ∂i V and ∂j ∂k V are statistically independent. The Gaussian field formed by ∂i V has, according to (1.85), covariance ∂j V ∂k V = a2 δj,k , a2 := −f (0). Hence, after making use of the Fourier integral representation of the delta function, one has N DY
E 2 PN 2 2 1 δ(μxi + ∂i V ) = √ e−μ j=1 xj /2a 2 )N ( 2πa i=1
(see (1.93) below). The second average in (1.88) is independent of xk , and so we can integrate over D = RN to obtain ˛E D˛ ˛ ˛ N (RN ) = μ−N ˛ det[μδj,k + ∂j ∂k V ]j,k=1,...,N ˛ . Set Hjk := ∂j ∂k V . It follows from (1.85) that Hil Hjm =
” f (0) “ δij δlm + δim δlj + δil δjm . N
(1.89)
Now let the diagonal elements Hii and upper triangular elements Hjk (j < k) collectively be indexed Hμ , and form the = (Hμ ). Being Gaussian variables, for some matrix A of appropriate size they have distribution proportional to vector H H). Furthermore A is completely determined by Hμ Hν (see (1.95) below) with the task being to compute exp(− 12 HA the inverse of the matrix of these averages. The final result can be written in a structured form, showing that H := [Hjk ] is a real symmetric Gaussian random matrix with p.d.f. proportional to ”” “ 1 N “ Tr H2 − (Tr H)2 . (1.90) exp − 4f (0) N +2
32
CHAPTER 1
By completing the square in t we see that Z ∞ “ 2 e−Nt /2 exp − −∞
”2 ”” p N “ “ (0)t1 f Tr H − dt, N 4f (0)
is proportional to (1.90) and (1.87) follows. (The N -dependent proportionality follows by requiring that N (RN ) → 1 for μ → ∞.)
With J :=
f (0), changing variables X → J 2/N X in the average of (1.87) gives (J 2/N )N det N/2((μ/J) + t)1N − X = (J
X∈GOE
2/N )N eN ((μ/J)+t)
2
G1,N +1 /4 ρ(1),N +1 ( N/2((μ/J) + t)) (N + 1)G1,N
(1.91)
where ρ(1),N +1 refers to the density in the GOE with N + 1 eigenvalues, and G1,N is given by (1.163). The equality in (1.91) follows by writing the determinant as a product of eigenvalues, writing the average in terms of eigenvalues using (1.27), and recalling the formula for the density (one-particle correlation) (1.46). Substituting (1.91) in (1.87) shows ∞ J √2 N N + 1 2 N N N (μ/J)2 /2 N (R ) = √ e Γ e−N (t−(μ/J)) /4 ρ(1),N +1 ( N/2((μ/J) + t)) dt. 2 π −∞ μ N For large N , after making use of Stirling’s formula Γ(x + 1) ∼ (2πx)1/2 ex log x−x
as
x → ∞, Re(x) > 0,
(1.92)
and noting the delta function type behavior of the integral, we see that for the argument of ρ(1),N +1 inside its support, and thus μ < J, √ 2 N (RN ) ∼ 2(2π)1/2 (J/μ)N eN (μ/J) /2 e−N/2 ρ(1),N +1 ( 2N (μ/J)). Making use now of (1.52), one obtains that for μ < J 1 1 μ2 log N (RN ) = − 1 − log(μ/J). N →∞ N 2 J2 √ Note that Σ(J) = 0. In fact analysis of ρ(1),N +1 ( 2NX) for |X| > 1 undertaken in Exercises 14.4 q.5 below can be used to show that Σ(μ) = 0 for μ > J, and so the number of critical points undergoes a phase transition at μ = J. Σ(μ) := lim
1. (i) Let A be an n × n positive definite matrix. By changing variables y = A1/2 x and completing the square show Z ∞ Z ∞ “ 1 ” dx1 · · · dxn exp − x + b · x In [A, b] := x T A 2 −∞ −∞ ” “1 n/2 −1/2 T = (2π) (det A) (1.93) exp b A−1b . 2
E XERCISES 1.5
(ii) Let
Z ∞ Z ∞ “ 1 ” 1 x T A dx1 · · · dxn f exp − x . 2 In [A, 0] −∞ −∞ Use (1.93) and the method of derivation of (1.99) below to show that for l even X −1 A−1 xk1 xk2 · · · xkl A = kp kp · · · A kp kp , f A =
1
all possible pairings of (k1 ···kl )
2
l−1
l
(1.94)
33
GAUSSIAN MATRIX ENSEMBLES
while for l odd this average vanishes. (iii) By choosing l = 2, deduce from (1.94) that A−1 = [xj xk A ]j,k=1,...,n , which in words says that the covariance matrix associated with the average ·A is given by A−1 . (iv) Replace b by ib in (1.93), and integrate over bk+1 , . . . , bn (k ≤ n) to deduce that Z ∞ Z ∞ “ 1 ” ˜ −1b dbk+1 · · · dbn exp − b T A (2π)k/2 (det A)−1/2 2 −∞ −∞ “ 1 ”˛˛ ˜ −1/2 exp − b T A−1b ˛ , = (2π)k/2 (det A) ˛ 2 bk+1 =···=bn =0 ˜ is the k × k submatrix of A formed from the first k rows and columns. where A (v) With b replaced by ib, regard the r.h.s. of (1.93) as a p.d.f. for b (up to normalization), so that the covariance matrix is now A = [bj bk A−1 ]j,k=1,...,n . Show that under the linear change of variables b = Lc, the vector c has a Gaussian distribution “ 1 ” (2π)n/2 (det B)−1/2 exp − c T B−1c , B = cj ck A−1 . 2
(1.95)
1.6 MATRIX INTEGRALS AND COMBINATORICS 1.6.1 Combinatorics of Tr(X2k )GUE∗ In this section we put our knowledge of the asymptotic density for the GUE to use in the solution of a combinatorial problem. It has long been known [98] that the matrix integrals 2 f (X)e−Tr(X )/2 (dX), for X a particular class of random matrices and suitable f (X), have combinatorial significance in that they count certain diagrams embedded on surfaces according to their genus. Here, following [557], [239], we will detail such a combinatorial interpretation of the matrix integral 2 1 Tr(X2k )e−Tr(X )/2 (dX) =: Tr(X2k )GUE∗ , (1.96) C √ where C is the normalization, and GUE∗ is identical to the GUE except that X → X/ 2. By changing variables X = ULU−1 for the eigenvalues and eigenvectors we see from the result of Exercises 1.3 q.3 that ∞ N N 2 1 ∞ Tr(X2k )GUE∗ = dλ1 · · · dλN e−λl /2 λ2k |λk − λj |2 . (1.97) j C −∞ −∞ j=1 1≤j
l=1
√ Changing variables λl → 2λl we see from the definition (1.46) that in terms of the density ρ(1) (λ) for the GUE we have ∞ 2k k Tr(X )GUE∗ = 2 λ2k ρ(1) (λ) dλ. (1.98) −∞
However, it is not from (1.98) that the combinatorics arise; this comes from the evaluation of (1.96) as a Gaussian integral over the independent elements of the matrix X. The latter task can be achieved by using a particular matrix version of Wick’s theorem.
34
CHAPTER 1
P ROPOSITION 1.6.1 Let X = [zjk ]j,k=1,...,N , zjk = xjk + iyjk be Hermitian so that e−Tr(X
2
)/2
(dX) = e−Tr(X
2
)/2
N
dxjj
j=1
dxjk dyjk .
1≤j
Let I be a finite ordered set of pairs of indices (j, k), 1 ≤ j, k ≤ N , and let P denote a matching of the elements of I in pairs. Then we have zij = zij zkl GUE∗ . (1.99) ∗ GUE
(i,j)∈I
pairings P of I
(i,j), (k,l)
Proof. Introducing the Hermitian matrix Y = [wjk ]j,k=1,...,N we observe that D Y
E zij
(i,j)∈I
=
GUE∗
„ Y (i,j)∈I
«D ˛ E ∂ ˛ eTr(YX) . ˛ ∂wji GUE∗ Y=0
(1.100)
In the integrand, writing Tr X2 − 2Tr (YX) = Tr((X − Y)2 ) − Tr Y 2 , we see from the change of variables X → X + Y that D
eTr(YX)
Thus (1.100) gives
E GUE∗
D Y
= eTr(Y
2 )/2
=
N Y j=1
E zij
(i,j)∈I
GUE∗
=
Y
2
ewjj /2
ewjk wkj .
1≤j
X
Y
pairings P of I
(i,j), (k,l)
δi,l δj,k ,
which reduces to (1.99) after noting zij zkl GUE∗ = δi,l δj,k .
(1.101)
Our task is to compute Tr(X2k )GUE∗ :=
N
zi1 i2 zi2 i3 · · · zi2k−1 i2k zi2k i1
i1 ,...,i2k =1
GUE∗
.
(1.102)
According to (1.99) we have Tr(X ) 2k
GUE∗
=
N
i1 ,...,i2k =1
pairings P of {(i1 ,i2 ),(i2 ,i3 ),...,(i2k ,i1 )}
(j,j ), (l,l )
zij ij zil il GUE∗ ,
(1.103)
and (1.101) shows that various labels must coincide for a given term in this expression to be nonzero. For example, with k = 4 consider the particular term in (1.103) zi1 i2 zi3 i4 zi2 i3 zi8 i1 zi4 i5 zi6 i7 zi5 i6 zi7 i8 = (δi1 ,i4 δi2 ,i3 )(δi2 ,i1 δi3 ,i8 )(δi4 ,i7 δi5 ,i6 )(δi5 ,i8 δi6 ,i7 ). (1.104) For this to be nonzero we must have i1 = i2 = · · · = i8 , giving only one independent label. As another example, consider the term zi1 i2 zi4 i5 zi2 i3 zi3 i4 zi5 i6 zi8 i1 zi6 i7 zi7 i8 = (δi1 ,i5 δi2 ,i4 )(δi2 ,i4 )(δi5 ,i1 δi6 ,i8 )(δi6 ,i8 ), which is nonzero for i1 = i5 , i2 = i4 , i6 = i8 , giving five independent labels, i1 , i2 , i3 , i6 , i7 , say.
(1.105)
35
GAUSSIAN MATRIX ENSEMBLES
Figure 1.2 Graphical representation of the contributions (1.104) and (1.105). The heavy lines identify edges and the dot marks the location of the vertex labeled i1 , with the other vertices labeled clockwise.
11 00 00 11
11 00 00 11
Figure 1.3 The dual graphical representation of Figure 1.2 for the contributions of (1.104) and (1.105). The dot marks the location of the vertex labeled i1 , with the other vertices labeled clockwise.
In general the nonzero terms in (1.103) can be represented graphically in two related ways, both of which involve a regular 2k-gon, with the vertices labeled i1 , . . . , i2k , and the edges oriented clockwise. One method to carry out the pairing between consecutive vertices (ij , ik ) and consecutive vertices (il , im ) is to join the corresponding edges on the 2k-gon according to the rule that edges must be joined in opposite directions (see Figure 1.2 for this representation of (1.104) and (1.105)). Another approach to carrying out the pairing is to draw a straight line segment perpendicular to and outward from the ends of each edge of the 2k-gon. These segments are to be given the directions of out and in alternately around the 2k-gon. Then each nonzero contribution to (1.103) can be represented by joining different pairs (ij , ik ) and (il , im ) of parallel straight line segments to form roadways (see Figure 1.3 for this representation of (1.104) and (1.105)). Note that the joining is such that edges of the roadways have definite directions. The diagrams of Figure 1.2 can be catalogued according to the number ν of independent vertices after pairing. This of course is just the number of independent summation labels in (1.103) so we can write Tr(X2k )GUE∗ =
k+1
aν (k)N ν ,
(1.106)
ν=1
where aν (k) denotes the number of different pairings which have ν vertices. On the other hand, the diagrams of Figure 1.3 are topological duals of Figure 1.2 with the ν independent vertices now ν independent faces. These can be determined by following the edge of a roadway and its continuation according to its direction, until arriving back at the starting point. The formal meaning of the faces is obtained by shrinking the width of the roadways in Figure 1.3 to single lines, at the same time as shrinking the 2k-gon to a single vertex so the lines become loops, and then embedding the diagram on a closed surface as a map.
D EFINITION 1.6.2 A map is a graph (a collection of vertices and edges) drawn on a closed surface such that the edges do not intersect and, if we cut the surface along the edges, a disjoint union of sets topologically
36
CHAPTER 1
Figure 1.4 Embedding of the graphs of Figure 1.3 (after shrinking the k-gon to a single vertex, and the roadways to single lines) onto a closed surface to form a map with a single vertex.
equivalent to an open disk results. The number of such disks is by definition the number of faces of the map. It is similarly the case that the number of independent vertices in the diagrams of Figure 1.2 can be specified in terms of the corresponding map. The index ν in (1.106) determines the genus g (number of holes) of the closed surface. This follows from Euler’s relation 2 − 2g = V − E + F,
(1.107)
where V denotes the number of vertices, E the number of edges and F the number of faces. In the diagrams of Figure 1.2 V = ν, F = 1 and E = k, while in the diagrams of Figure 1.3 the roles of V and F are interchanged so that V = 1, F = ν and E = k. Either way (1.107) gives ν = k + 1 − 2g.
(1.108)
As shown in Figure 1.4, the diagrams of Figure 1.3 can be directly embedded on a surface of particular genus, thereby illustrating (1.108) (thus in the first case k = 4, ν = 1, g = 2 while in the second case k = 4, ν = 5, g = 0). Using (1.108) in (1.106) gives
[k/2]
Tr(X2k )GUE∗ = N k+1
ak+1−2g (k)N −2g .
(1.109)
g=0
In particular lim N −k−1 Tr(X2k )GUE∗ = ak+1 (k)
N →∞
(1.110)
where ak+1 (k) denotes the number of matchings of the 2k-gon which are planar (i.e. can be embedded on the surface of a sphere, which has g = 0). Substituting for Tr(X2k )GUE∗ using (1.97), and then substituting for ρ(1) (x) using (1.52), evaluating the integral using (4.3) below and simplifying the resulting gamma functions using the duplication formula 22z−1 Γ(z)Γ(z + 1/2) = π 1/2 Γ(2z), one finds
(1.111)
1 2k . (1.112) k+1 k This number is familiar in combinatorics and is called the kth Catalan number. In fact (1.112) can easily be derived without using (1.98), which has the significance of providing an alternative derivation of the Wigner semicircle law (1.52) for Hermitian matrices (see Exercises 1.6 q.1), one which applies to establishing the Wigner semicircle for a large class of symmetric random matrices with independent entries (see, e.g., [551]). ak+1 (k) =
37
GAUSSIAN MATRIX ENSEMBLES
However, this is not the case for the coefficients ak−1 (k), ak−3 (k), . . . for which the use of (1.98) is the most efficient. We will return to the evaluation of these numbers in Chapter 5 when the exact value of ρ(1) (λ) is available. We remark that Tr(X2k )GOE allows for a similar combinatorial description in terms of maps on surfaces, although now the surfaces may be nonorientable (corresponding to graphs with twisted ribbons) [348]. 1.6.2 Combinatorics of the β = 2 partition function with a general power series potential Closely related to the combinatorial interpretation of (1.96) is the combinatorial interpretation of ZN ({gj }) :=
N
e
P∞
j=1
gj xjl /jN j/2−1
(1.113)
GUE∗
l=1
when expanded in a power series in {gi }. For the latter, expanding the exponentials gives ∞
∞
ZN ({gj }) =
n1 ,n2 ,···=0 j=1
∞ N
n
gj j j nj nj !N nj (j/2−1)
j=1
xjl
nj GUE∗
l=1
,
(1.114)
while ∞ N j=1
xjl
nj GUE∗
l=1
=
∞ nj Tr Xj j=1
GUE∗
.
(1.115)
From the discussion of the previous subsection we know how to give a combinatorial interpretation of (1.115) in the special case nj = 1 (j = k), nj = 0 (j = k). A natural generalization of this interpretation extends to the general case [557], [239]. Each factor of Tr Xj is represented as a j-gon with vertices labeled i1 , i2 , . . . , ij clockwise, starting at a marked vertex. These labels on vertices extend to labels on pairs of oppositely directed roadway edges coming into and out of each vertex. Whereas in the case of a single factor of Tr Xj the combinatorial interpretation of computing (1.115) via Wick’s theorem involved connecting roadways within the single j-gon, the graphical representation of contributions to (1.115) is to connect roadways among or within any of the nj j-gons (j = 1, 2, . . . ). The resulting structure, referred to as a labeled fatgraph, has weight N ν , where ν is the number ∞ of faces (which in turn is equal to the number of unpaired labels). The number of edges is equal to required to be an integer, while the number of vertices — defined as the number of j=1 jnj /2, which is ∞ j-gons — is equal to j=1 nj . Recalling Euler’s relation (1.107) we see that (1.114) can thus be written ZN ({gj }) =
n ∞ gj j ag ({nj })N 2−2g , nj n ! j j g ,···=0 j=1
∞ n1 ,n2
where ag ({nj }) is the number of labeled graphs constructed out of nj j-gons (j = 1, 2, . . . ) which can be embedded on a surface of genus g. The various j-gons in the labeled fatgraph will not in general be connected. However, as ZN ({gj }) is an exponential generating function for these quantities, it is a well-known fact that taking the logarithm restricts to connected components. Thus, denoting this restriction by an asterisk, we have n ∞ ∞ g j j ∗ log ZN ({gj }) = ag ({nj })N 2−2g . (1.116) nj n ! j j g n ,n ,···=0 j=1 1
2
Fatgraphs which are topologically equivalent define a class of maps Γ. For each class the maximum value ∞ ∞ of ag ({nj }) is j=1 j nj nj ! and furthermore j=1 j nj nj !/ag ({nj }) is an integer written as |Aut Γ|. As the notation suggests, |Aut Γ| is in fact equal to the order of the group of automorphisms associated with Γ. This
38
CHAPTER 1
Figure 1.5 Three classes of fatgraphs can be constructed out of two 4-gons. An example from each class, together with the corresponding map is given. For the first class |Aut Γ| = 8, while for the second and third classes |Aut Γ| = 2.
can be specified as the number of equivalent labelings of the faces of Γ, which means the number of different labelings in the plane which result from topological transformations of the map on the closed surface. In terms of |Aut Γ| (1.116) reads log ZN ({gj }) =
connected Γ
∞ 1 V (Γ) N 2−2g(Γ) gj j , |Aut Γ| j=1
(1.117)
where nj in (1.116) has been written Vj (Γ) in (1.117) to emphasize that it counts the number of vertices with coordination number j in the corresponding map. In particular 1 log ZN ({gj }) = N →∞ N 2 lim
connected Γ g(Γ)=0
∞ 1 V (Γ) g j , |Aut Γ| j=1 j
(1.118)
and thus we obtain a generating function for maps weighted according to the coordination number of the vertices. To illustrate (1.118), in Figure 1.5 we display the contributions to the coefficient of g42 in terms of fatgraphs and the corresponding maps. Suppose gj = 0 for j = 4. From the definition (1.113) we have 1 {g log Z } lim N j N →∞ N 2 gj =0 (j=4) N N 1 1 g4 x4l /4N g4 x4l /N log e = lim log e . = lim N →∞ N 2 N →∞ N 2 GUE∗ GUE l=1
(1.119)
l=1
To evaluate this limit we make use of the log-gas interpretation of the average as a ratio of configuration
39
GAUSSIAN MATRIX ENSEMBLES
integrals relating to one-component log-potential systems with particular neutralizing background charge densities. The Boltzmann factor for these systems contains constant terms (i.e., terms independent of the particle coordinates) which are not present in (1.119). If these terms, AN say, were included, its logarithm would then be expected to be proportional to N as the difference between two free energies is being calculated (see (4.160) below). Thus one expects 1 1 1 lim {g = lim log Z } log . N j 2 N →∞ N N →∞ N 2 AN gj =0 (j=4) The value of 1/AN has been calculated in Exercises 1.4 q.5(iv). It is equal to the exponential of the xj independent terms in (1.76) with g in (1.75) replaced by −g4 . This gives 1 1 1 (u − 1)(9 − u) − log u , (1.120) lim {g = − log Z } N j N →∞ N 2 24 2 gj =0 (j=4) where u is defined in terms of g4 as the solution of u → 1 as g4 → 0.
u − 3g4 u2 = 1,
(1.121)
According to the result of Exercises 1.6 q.1(iii), u=−
∞ k=0
1 2k (3g4 )k . k+1 k
(1.122)
After substituting this in the r.h.s. of (1.120), and substituting (1.118) in the l.h.s, the following result is obtained [98]. P ROPOSITION 1.6.3 We have connected Γ g(Γ)=0
∞
(2k − 1)! 1 V (Γ) g4 4 = (3g4 )k . |Aut Γ| k!(k + 2)!
(1.123)
k=1
Proof. The remaining task is to expand the functions of u on the r.h.s. of (1.120) as power series in g4 . For the quadratic, this is immediate from (1.122) and (1.121). For log u this follows from the result of Exercises 1.6 q.3. Let us denote the coefficient of g4k in (1.123) by ak , which then represents the number of (weighted) planar fatgraphs that can be constructed out of k 4-gons. Making use of Stirling’s formula (1.92) shows ak ∼
12k . k 7/2 π
The particular value of the exponent of the algebraic term k −7/2 has meaning in the conformal field theory associated with the graphical expansion [239]. If we cut an edge in any of the planar maps giving rise to (1.123), we obtain a planar fatgraph constructed from 4-gons, but now with two external legs in the same face. The external legs, when distinguished by different labelings, break the symmetry of the maps, so for all classes Γ2 of such maps |Aut Γ2 | = 1. Because the legs have been distinguished, and because there are twice as many edges as vertices, one sees [241] ˜ = 1 + 4g4 ∂ G, G ∂g4 ˜ denotes the generating function for the maps with external legs, and G denotes the l.h.s. of (1.123). where G ˜ has positive integer coefficients, as it must. Substituting the r.h.s. of (1.123) we see that the power series of G 2 In particular, the coefficient of g4 is 9. One contribution results from the first map in Figure 1.5, while four result from each of the other two maps therein.
40
CHAPTER 1
E XERCISES 1.6
1. Here the number ck := ak+1 (k) of diagrams which can be constructed from a 2k-gon according to the prescription of Figure 1.2, and which contain no intersecting lines, will be computed directly. (i) Suppose the lines from edge 1 join the lines from edge 2j (j = 1, . . . , k). Argue that inside these lines there can be cj−1 configurations of the allowed type, while there are ck−j configurations of the allowed type joining the edges 2j + 1, . . . , 2k. Hence deduce that ck =
k−1 X
cj ck−1−j ,
c0 = 1.
(1.124)
j=0
(ii) Verify that the Catalan numbers (1.112) solve this recurrence. P k (iii) Introduce the generating function C(t) = ∞ k=0 ck t . Use the recurrence (1.124) to show that C(t) satisfies the quadratic equation C(t) = 1 + t(C(t))2 , (1.125) and consequently has the explicit form tC(t) =
1 (1 − (1 − 4t)1/2 ). 2
(1.126)
(iv) With the value of ak+1 (k) known independently of the average in (1.110) according to the result of (ii), use (1.98) to deduce that the scaled density (1.53) is such that Z ∞ 1 “ 2k ” x2k ρ˜(1) (x) dx = 22k (1.127) k+1 k −∞ while the odd moments vanish. (v) A sufficient condition for a density function to be determined by its moments {c0 , c1 , c2 , . . . } is that ∞ X ck tk k=0
2.
k!
(1.128)
converges for some t > 0 (this implies the Fourier transform of the density function is analytic in the neighbourhood of the origin). Verify that this is the case for the moments in (iv). Now use the fact that (1.54) reproduces these values to conclude from (iv) that the Wigner semicircle law is valid. √ (i) Consider the Gaussian β-ensemble p.d.f. in (1.160) below, scaled so that λl → 2βN /J, J > 0. Use the result of Proposition 1.4.4 to show that to leading order the density is then supported on the interval [−J, J] and is given by 2N p 1 − (y/J)2 . (1.129) πJ (ii) Use (1.55) with suitable V (x) and a, to show that for ρb (y) given by (1.129) and z ∈ (−J, J), Z J 2N z ρb (y) . dy = J2 −J z − y
(1.130)
Also, deduce from (1.127) that Z
J −J
y 2k ρb (y) dy =
2N (J/2)2k+1 ck , J
(1.131)
where ck denotes the kth Catalan number (1.112). From this and the result of q.1(iii) deduce that for |z| > J, Z J ” 2N z “ ρb (y) dy = 1 − (1 − J 2 /z 2 )1/2 . (1.132) 2 J −J z − y
41
GAUSSIAN MATRIX ENSEMBLES
3. [541] Let f (z) and φ(z) be analytic in a neighborhood Ω of z = a. According to the Lagrange inversion formula, for t small enough that |tφ(z)| < |z − a|, z ∈ Ω, the equation ζ = a + tφ(ζ) has one solution in Ω, and furthermore ∞ ” X tn dn−1 “ f (a)(φ(a))n . f (ζ) = f (a) + n−1 n! da n=1 Use this formula to show that for x defined as the solution of the equation x = 1 + yxp with the property x → 1 as y → 0, one has ∞ X (kp − 1)! k y . log x = k!(kp − k)! k=1
1.7 CONVERGENCE Consider fordefiniteness GUE matrices. As stated the Wigner semicircle law tells us the leading large N √ N √ form of N1 j=1 2N δ( 2Ny − λj )GUE . As the normalized empirical density integrated over an interval [a, b] is the proportion of eigenvalues in that interval, #[a, b] say, equivalently the Wigner semicircle law tells us the expected value of this quantity when averaged over GUE matrices. Indeed, this was how Figure 1.1 was produced, with the theoretical means in each bin of the bar graph substituted by their empirical averages. What if instead one considers #[a, b] for a sequence √ of single n × n matrices, n = 1, 2, . . . , each chosen from the GUE and with eigenvalues scaled λj → λj / 2N . Does the resulting sequence of values for #[a, b] converge to that predicted by the Wigner semicircle law? And what is the meaning of convergence in this setting? Regarding the latter point, two possibilities are convergence in probability, and almost sure convergence. Convergence in probability says that for a given > 0, and sequence of single n × n GUE matrices (n = 1, 2, . . . ), Pr(|μn −μ| > ) → 0, where μn is the empirical value of #[a, b] for each matrix, and μ is the limiting ensemble average (the value implied by the Wigner semicircle law). Almost sure convergence says that the measure of the sequence of matrices for which μn → μ is equal to 1. A well-known consequence of the Borel-Cantelli lemma in probability theory (see, e.g., [66]) is that almost sure convergence is equivalent ∞ to the statement that for a given > 0, n=1 Pr(|μn − μ| > ) < ∞. Note that a necessary condition for this is that Pr(|μn − μ| > ) → 0, and thus almost sure convergence implies convergence in probability. To estimate Pr(|μn − μ| > ), the Chebyshev inequality [66] Pr(|μn − μ| > ) ≤
(μn − μ)2 GUE 2
can be employed. Hence for convergence in probability, (μn − μ)2 GUE → 0 as n → ∞, it∞is sufficient that 2 while for almost sure convergence, it is sufficient that n=1 (μn − μ) GUE < ∞. In Section 1.6.1 and Exercises 1.6 q.1 the Wigner semicircle law has been studied through its moments. We have shown that N −k−1 Tr(X2k )GUE∗ → m2k where m2k is the corresponding moment of the Wigner semicircle law. To study convergence in probability and almost sure convergence one thus must study 2 Var(N −k−1 TrX2k ) := (N −k−1 TrX2k )2 GUE∗ − N −k−1 TrX2k GUE∗ . Now, analogous to (1.102) we have (TrX2k )2 GUE∗ =
i1 ,...,i2k =1 j1 ,...,j2k =1
zi1 i2 · · · zi2k i1 zj1 j2 · · · zj2k j1 .
42
CHAPTER 1
Regarding i1 , . . . , i2k as fixed, and taking into consideration (1.103), one sees [274] (TrX2k )2 GUE∗ = zi1 i2 · · · zi2k i1 z · · · z j j j j 1 2 1 2k ∗ GUE
i1 ,...,i2k =1
j1 ,...,j2k =1
GUE∗
1+O
1 N2
1 . = TrX2k 2GUE∗ 1 + O N2 ∞ It follows from this that N =1 Var(N −k−1 TrX2k ) < ∞, so we can conclude that almost sure convergence holds and so the Wigner semicircle law is the limiting density of all sequences of GUE matrices, up to a set of measure zero.
1.8 THE SHIFTED MEAN GAUSSIAN ENSEMBLES The Gaussian orthogonal, unitary and symplectic ensembles have joint p.d.f. for the elements proportional to exp(−(β/2)Tr H2 ). We know that this is equivalent to the independent entries in H having Gaussian distribution with mean zero and particular variance. It follows that for a fixed Hermitian matrix H0 , a p.d.f. proportional to exp(−(β/2)Tr (H − H0 )2 ) specifies a Gaussian ensemble in which the mean of each element is equal to the corresponding element in H0 . The simplest case is when all elements of H0 are constant, equal to c say. Then H = A + cxxT ,
(1.133)
where x is a column vector with all entries equal to 1, and A is a member of the corresponding zero mean Gaussian ensemble. Thus in this case the shifted mean Gaussian ensembles correspond to a rank 1 perturbation of the original ensembles. Consider for definiteness the GOE. Diagonalizing A, A = OLOT , L = diag (a1 , . . . , aN ), and writing OT x =: y shows that from the viewpoint of the eigenvalues, the r.h.s. of (1.133) can be replaced by L+cy yT . We seek the eigenvalues of this matrix. P ROPOSITION 1.8.1 The eigenvalues of the matrix ˜ := diag(a1 , . . . , aN ) + cyy T H are given by the solutions of the equation 0=1−c
N i=1
yi2 . λ − ai
(1.134)
Assuming the ordering a1 > · · · > aN , and that c > 0, a corollary is that the eigenvalues satisfy the interlacing λ1 > a1 > λ2 > a2 > · · · > λN > aN .
(1.135)
˜ = diag(a1 , . . . , aN ) we have Proof. With A ˜ = det(1N − A) ˜ det(1N λ − c ˜ −1 ). det(1N λ − H) y y T (1N λ − A)
(1.136)
But the matrix product in the second determinant has rank 1, and so ˜ −1 ) = 1 − cTr( ˜ −1 ) = 1 − c det(1N − c y y T (1N λ − A) y y T (1N λ − A)
N X i=1
yi2 . λ − ai
(1.137)
˜ due The characteristic polynomial (1.136) vanishes at the zeros of this determinant, but not at the zeros of det(1N λ − A) to the poles in (1.137), implying the first result. The interlacing condition can be seen by sketching a graph, and noting in
43
GAUSSIAN MATRIX ENSEMBLES
the process that cyi2 > 0.
(s) The GUE and GSE lead to the same equation (1.134) but with yi2 replaced by |yi |2 := βs=1 (yi )2 , where (s) the yi are the independent real parts of the complex and real quaternion entries, respectively. We remark too that in the case of the GSE the eigenvalues of A are doubly degenerate and the rank 1 perturbation leaves one copy of the eigenvalues unchanged; see the discussion about (4.20) below. ˜ and thus H, as a function of c and N , Of particular interest is the position of the largest eigenvalue of H, which unlike the other eigenvalues is not trapped by the eigenvalues of A. Following [335] this can analyzed by making use of the results of Exercises 1.6 q.2. P ROPOSITION 1.8.2 Consider the Gaussian ensembles scaled so that the distribution of the elements is proportional to exp(−(βN/J 2 )Tr (H − H0 )2 ), with H0 the constant matrix having all elements equal to c/N , c > 0. Suppose N is large. Then for 2c > J a single eigenvalue splits off from the bulk of the eigenvalues, these being supported on (−J, J), and is located at λ=c+ Proof. We seek a solution λ > J of
J2 . 4c
c D X |yj |2 E , N j=1 λ − λj N
1=
(1.138)
where each |yj |2 has mean unity, and {λj } are the eigenvalues of a member of the specified Gaussian ensemble but with H0 = 0. We know that the density of the {λj } is then given by the semicircle law (1.129). Hence for N large Z J N “ DX 2N λ “ J 2 ”1/2 ” |yj |2 E ρb (y) dy = ∼ 1− 1− 2 , 2 λ − λj J λ −J λ − y j=1 where the first relation follows from the fact that the eigenvalues and eigenvectors are independently distributed and |yj |2 = 1, while the equality, which requires that λ > J, follows from (1.132). Substituting this in (1.138) and solving for λ gives the stated result.
1.9 GAUSSIAN β-ENSEMBLE The p.d.f. (1.28) is realized by the eigenvalues of the GOE, GUE and GSE for the values of β equal to 1, 2 and 4 respectively. In this section a family of random tridiagonal matrices, referred to as the Gaussian βensemble, with (1.28) as their eigenvalue p.d.f. for general β > 0, will be studied. They can be motivated by the reduction of GOE or GUE matrices to tridiagonal form. 1.9.1 Householder transformations A familiar technique in numerical linear algebra is the similarity transformation of a real symmetric matrix to tridiagonal form using a sequence of reflection matrices, referred to as Householder transformations. Explicitly, let A be a real symmetric matrix [aij ]i,j=1,...,N . Then one can construct a sequence of symmetric real orthogonal matrices U(1) , U(2) , . . . , U(N −2) such that the transformed matrix U(N −2) U(N −3) · · · U(1) AU(1) U(2) · · · U(N −2) =: B(N −2) is a symmetric tridiagonal matrix. These matrices have the structure 0j×N −j 1j (j) (j) (j)T , = U = 1N − 2u u 0N −j×j VN −j×N −j
(1.139)
(1.140)
44
CHAPTER 1
where u(j)T u(j) = 1 and VN −j×N −j is symmetric real orthogonal. Geometrically U(j) corresponds to a reflection in the hyperplane orthogonal to u(j) . (1) Consider first the construction of U(1) . Choosing the components ul of u(1) as 1 a12 1/2 a1l (1) (1) (1) 1− u1 = 0, u2 = , ul = − (l ≥ 3), (1.141) (1) 2 α 2αu2 (1)
where α = (a212 + · · · + a21N )1/2 , we then have u(1)T [al1 ]l=1,...,N = (a12 − α)/2u2 . This in turn implies that B(1) := U(1) AU(1)
(1.142)
has b11 = a11 ,
b1k = bk1 = 0 (k ≥ 3)
b12 = b21 = α,
and is thus tridiagonal with respect to the first row and column. The matrices U(j) , j = 2, 3, . . . in order are (j) (j) (j) now defined by the formulas (1.141), but with u1 = u2 = · · · = uj = 0, and the analogue of the entries a1l replaced by the elements in the first row of the bottom right (N − j + 1) × (N − j + 1) submatrix of B(j−1) . A number of works (see [157] and references therein) posed the question as to the form of B(N −2) when A is a member of the GOE. It was found that like A itself, the elements of B(N −2) are all independent (apart from the requirement that B(N −2) be symmetric) with a distribution that can be calculated explicitly. P ROPOSITION 1.9.1 Let N[0, 1] refer to the standard normal distribution as defined below Definition 1.1.1, and let χ ˜k denote the square root of the gamma distribution Γ[k/2, 1], the latter being specified by the p.d.f. (1/Γ(k/2))uk/2−1 e−u , u > 0, and realized by the sum of the squares of k independent Gaussian √ 2 distributions N[0, 1/ 2]. (The p.d.f. of χ ˜k is thus equal to (2/Γ(k/2))uk−1 e−u , u > 0.) For A a member of the GOE, the tridiagonal matrix B(N −2) obtained by successive Householder transformations is given by ⎤ ⎡ N[0, 1] χ ˜N −1 ⎥ ⎢ χ ˜N −2 ⎥ ⎢ ˜N −1 N[0, 1] χ ⎥ ⎢ χ ˜ N[0, 1] χ ˜ N −2 N −3 ⎥ ⎢ ⎥. ⎢ . . . .. .. .. ⎥ ⎢ ⎥ ⎢ ⎣ χ ˜2 N[0, 1] χ ˜1 ⎦ χ ˜1 N[0, 1] Proof. Let GOEn denote the ensemble of n × n GOE matrices. From the Householder algorithm, the first row and column of B(N−2) are he same as those of B(1) in (1.142), and thus from (1.141) we have (N−2)
b11
= N[0, 1],
(N−2)
b12
=χ ˜N−1 ,
where use has been made of the assumption that A is a member of GOEN , and the definition of χ ˜2N−1 as a sum of squares of Gaussians. To proceed further we must compute the distribution of the bottom N −1×N −1 block of B(1) . In general, (1) denoting such a block of the matrix X by XN−1 , it follows from (1.140) that BN−1 = VN−1 AN−1 VN−1 . Since the elements of the real orthogonal matrix VN−1 are independent of the elements of AN−1 , which itself is a member of (1) GOEN−1 , it follows immediately from the general invariance of the GOE under orthogonal transformations that BN−1 (1) is also a member of GOEN−1 . Applying the Householder transformation to BN−1 , we thus get (N−2)
b22
Continuing inductively gives the stated result.
= N [0, 1],
(N−2)
b23
=χ ˜N−2 .
45
GAUSSIAN MATRIX ENSEMBLES
1.9.2 Tridiagonal matrices The result of Proposition 1.9.1 suggests investigating the Jacobian for the change of variables from a general real symmetric tridiagonal matrix ⎡ ⎤ an bn−1 ⎢ bn−1 an−1 bn−2 ⎥ ⎢ ⎥ ⎢ ⎥ bn−2 an−2 bn−3 ⎢ ⎥ (1.143) T=⎢ ⎥, .. .. .. ⎢ ⎥ . . . ⎢ ⎥ ⎣ b2 a2 b1 ⎦ b1 a1 to its eigenvalues and variables relating to its eigenvectors. First, for each eigenvalue λk and correspond(1) ing eigenvector vk , it is easy to see by direct substitution that once the first component vk =: qk of vk is specified, all other components can be expressed in terms of λk and the elements of T. To make the eigendecomposition unique we specify that qk > 0, and furthermore note that T, being symmetric, can be orthogonally diagonalized, and so doing this we have n
qk2 = 1.
(1.144)
k=1
The Jacobian for the change of variables from b := (bn−1 , . . . , b1 ),
a := (an , an−1 , . . . , a1 ),
(1.145)
to λ := (λ1 , . . . , λn ),
q := (q1 , . . . , qn−1 )
(1.146)
can be calculated using the method of wedge products. However, one must first establish some auxiliary results. P ROPOSITION 1.9.2 Let (X)11 denote the top-left hand entry of the matrix X. We have n
qj2 . λj − λ
(1.147)
n−1 2i bi (λi − λj ) = i=1 n 2 . i=1 qi
(1.148)
((T − λ1)−1 )11 =
j=1
Also
2
1≤i<j≤n
Proof. Now
((T − λ1)−1 )11 = e1 · (T − λ1)−1e1 ,
where e1 := (1, 0, . . . , 0) . Since {vj } is an orthonormal set, T
e1 =
n X
(e1 · vj )vj =
j=1
n X
qj vj ,
(1.149)
j=1
and substituting into the above equation gives (1.147). This derivation makes no explicit use of T being tridiagonal, rather just that (1.149) holds, for which it is sufficient T be real symmetric. To derive (1.148) [140] we begin by recalling that in general for X an n × n nonsingular matrix, (X−1 )11 =
det Xn−1 , det X
(1.150)
46
CHAPTER 1
where Xn−1 denotes the bottom right n − 1 × n − 1 submatrix of X. Hence we can rewrite (1.147) to read Qn−1 n (n−1) X qj2 ) i=1 (λ − λi Qn , = λ − λj i=1 (λ − λi ) j=1 (n−1)
where {λi
(1.151)
} denotes the eigenvalues of Xn−1 . It follows from this that qj2 =
Pn−1 (λj ) , Pn (λj )
Pk (λ) :=
k Y (k) (λ − λi ),
(1.152)
i=1 (k)
where Pk (λ) is the characteristic polynomial of the bottom right k × k submatrix of T, say Tk , and {λi } the corresponding eigenvalues. Hence Qn n Y |Pn−1 (λi )| qi2 = Q i=1 . (1.153) 2 1≤i<j≤n (λi − λj ) i=1 Next, by expanding along the first row of λ1k − Tk , one obtains the three-term recurrence Pk (λ) = (λ − ak )Pk−1 (λ) − b2k−1 Pk−2 (λ) and it follows from this that
k−1 Y
(k−1)
|Pk (λi
2(k−1)
)| = bk−1
i=1
Since
k−1 Y
k−1 Y k−2 Y
)| =
i=1
this can be rewritten as
(k−1)
|Pk−2 (λi
(k−1)
|λi
(k−2)
− λj
|=
i=1 j=1 k−1 Y
)|.
i=1
(k−1)
|Pk−2 (λi
k−1 Y
(1.154)
(k−1)
|Pk (λi
k−2 Y
(k−2)
|Pk−1 (λj
)|,
(1.155)
j=1
2(k−1)
)| = bk−1
i=1
k−2 Y
(k−2)
|Pk−1 (λj
)|,
j=1
and iteration shows
n−1 Y
(n−1)
|Pn (λi
)| =
i=1
n−1 Y
b2i i .
i=1
Use of (1.155) with k = n + 1 and substitution into (1.153) gives (1.148).
P ROPOSITION 1.9.3 The Jacobian for the change of variables (1.145) to (1.146) can be written as n−1 1 i=1 bi . (1.156) qn ni=1 qi
Proof. [223] Rewriting (1.147) in the form ((1 − λT)−1 )11 =
n X j=1
qj2 1 − λλj
(1.157)
47
GAUSSIAN MATRIX ENSEMBLES
and equating successive powers of λ on both sides gives 1=
n X
qj2 ,
n X
an =
j=1
qj2 λj ,
∗ + b2n−1 =
j=1
∗ + an−1 b2n−1 =
n X
n X j=1
qj2 λ3j ,
∗ + b2n−2 b2n−1 =
j=1
∗ + an−2 b2n−2 b2n−1 =
qj2 λ2j ,
n X
qj2 λ4j ,
j=1 n X
qj2 λ5j , . . . ,
∗ + a1 b21 · · · b2n−2 b2n−1 =
j=1
n X
qj2 λ2n−1 , j
j=1
where the ∗ denotes terms involving only variables already having appeared on the l.h.s. of preceding equations (thus the variables an , bn−1 , an−1 , bn−2 , . . . occur in a triangular structure). The first of these equations implies qn dqn = −
n−1 X
qj dqj .
(1.158)
j=1
Taking differentials of the remaining equations, substituting for qn dqn , and then taking wedge products of both sides (making use of the triangular structure on the l.h.s.) shows n−1 Y
b4j−1 da ∧ db = qn2 j
n−1 Y
j=1
i h q, qj3 det [λjk − λjn ] j=1,...,2n−1 [jλj−1 ] j=1,...,2n−1 dλ ∧ d k k=1,...,n−1
j=1
where da :=
n ^
daj ,
db :=
j=1
n−1 ^ j=1
dbj ,
dλ :=
n ^ j=1
dλj ,
k=1,...,n
d q :=
n−1 ^
dqj .
j=1
By definition the Jacobian J is positive and such that da ∧ db = ±Jdλ ∧ d q for some sign ±. Making use of the determinant evaluation (1.175) below we thus read off that Qn−1 Qn 2 1 j=1 bj “ j=1 qj ”2 Y Q (λk − λj )4 . J= Qn−1 2j n qn j=1 qj j=1 bj 1≤j
Using Proposition 1.9.3, the fact that the tridiagonal matrix of Proposition 1.9.1 has the same eigenvalue p.d.f. as GOE matrices can be reclaimed. Moreover, one can prescribe a tridiagonal matrix with eigenvalue p.d.f. (1.28) for general β > 0 [140]. P ROPOSITION 1.9.4 Let β > 0 be fixed. In the notation of Proposition 1.9.1 ensemble as the set of symmetric tridiagonal matrices ⎡ N[0, 1] χ ˜(N −1)β ⎢ χ ˜ N[0, 1] χ ˜(N −2)β (N −1)β ⎢ ⎢ χ ˜ N[0, 1] χ ˜(N −3)β (N −2)β ⎢ Tβ := ⎢ . . .. . . ⎢ . . . ⎢ ⎣ χ ˜2β N[0, 1] χ ˜β χ ˜β N[0, 1]
define the Gaussian β⎤ ⎥ ⎥ ⎥ ⎥ ⎥. ⎥ ⎥ ⎦
(1.159)
The eigenvalues and first component of the eigenvectors (which form the vector q) are independent, with the
48
CHAPTER 1
distribution of the former given by 1
N
˜ β,N G
l=1
e
−λ2l /2
|λk − λj | dλ,
˜ β,N = (2π)N/2 G
β
N −1 j=0
1≤j
Γ(1 + (j + 1)β/2) , Γ(1 + β/2)
(1.160)
and the distribution of the latter given by 1 cβ,N qN
N
qiβ−1 d q,
qi > 0,
i=1
N
qi2 = 1,
where
cβ,N =
i=1
ΓN (β/2) 2N −1 Γ(βN/2)
.
(1.161)
Proof. Denote the joint distribution of Tβ by P (Tβ ). We have N−1 βl−1 −b2 Y N 2 e l 2N−1 Y bl e−al /2 da ∧ db (2π)N/2 l=1 Γ(βl/2) l=1 QN−1 βl N−1 bl −Tr(T2β )/2 2N−1 Y 1 1 e dλ ∧ d q, = Ql=1 N (2π)N/2 l=1 Γ(βl/2) qN l=1 ql
P (Tβ )(dTβ ) =
where the second equality follows using (1.156). But 2
e−Tr(Tβ )/2 = e−
PN
j=1
λ2 j /2
,
N−1 Y
bβl l =
l=1
N Y l=1
qlβ
Y
|λj − λi |β ,
1≤i<j≤N
q factorizes into the functional forms where the latter formula follows from (1.148), so indeed the dependence on λ and specified in (1.160) and (1.161). The normalization for (1.161) follows from the Dirichlet integral [541] Z Pn+1 i=1
dρ1 · · · dρn ρi =1, ρi >0
n+1 Y
ρsi i −1 =
i=1
Γ(s1 ) · · · Γ(sn+1 ) Γ(s1 + · · · + sn+1 )
(1.162)
˜ β,N with n = N − 1, si = β/2 and the change of variables ρi = qi2 . With this normalization specified, the value of G follows (an extra factor of N ! is included to effectively remove the ordering on {λi } implicit in the above working; recall the remark below Proposition 1.3.4).
˜ β,N given in (1.160) implies, after a simple change of variables, that We remark that the evaluation of G the normalization constant in (1.28) has the evaluation Gβ,N = β −N/2−N β(N −1)/4 (2π)N/2
N −1 j=0
Γ(1 + (j + 1)β/2) . Γ(1 + β/2)
(1.163)
Another point of interest is that the recurrence (1.154) with ak ∈ N[0, 1],
b2k ∈ Γ[kβ/2, 1]
(1.164)
can be used to generate the characteristic polynomial for a member of the Gaussian β-ensemble, so the p.d.f. (1.160) can be sampled by simply computing the zeros of this polynomial. 1.9.3 Sturm sequences For tridiagonal matrices, the task of computing the cumulative microscopic eigenvalue density N (μ), that is, the number of eigenvalues less than μ, has a number of special features. This in turn follows from special features of the corresponding Sturm sequences [13]. D EFINITION 1.9.5 Let An be a general n × n matrix, and let An−k (k = 1, . . . , n − 1) denote the matrix
49
GAUSSIAN MATRIX ENSEMBLES
obtained by deleting the first k rows and columns. Let di := det Ai (i = 1, . . . , n) and set d0 := 1. The Sturm sequence refers to (d0 , d1 , . . . , dn ). P ROPOSITION 1.9.6 Let An be a real symmetric matrix with no repeated eigenvalues and no zero eigenvalues, and similarly An−k . The number of sign changes in the Sturm sequence (reading from right-to-left, say) is equal to the number of negative eigenvalues of An . Proof. For a given k = 2, . . . , n it is a fundamental result (see Exercises 4.2 q.2(iii) below) that the eigenvalues {ai } of Ak interlace the eigenvalues {αi } of Ak−1 , ak < αk−1 < ak−1 < · · · < α1 < a1 . We know too that the determinant is equal to the product of eigenvalues. Consequently, the number of negative eigenvalues of Ak equals the number of negative eigenvalues of Ak−1 , if dk /dk−1 is positive, while we must add one if dk /dk−1 is negative. Iteratively applying this for k = n, . . . , 1 gives the stated result.
Applying Proposition 1.9.6 to the matrix An − μ1n gives that N (μ) is equal to the number of sign changes in the Sturm sequence for An − μ1n . In the case that An is the tridiagonal matrix (1.143), one has that dk = (−1)k Pk (μ) as specified by (1.154), and using the recurrence (1.154) shows that ri := di /di−1 can be specified by the recursive formula i = 1, a1 − μ, ri = (1.165) (ai − μ) − b2i−1 /ri−1 , i = 2, . . . , n. As each sign change in the Sturm sequence {di } corresponds to a negative in the ratio sequence {ri }, we see that the number of negative values in {ri } equals N (μ). This latter result can be related to so-called shooting eigenvectors. D EFINITION 1.9.7 The vector x satisfying all but the first of the n linear equations implied by the matrix equation (An − μ1n )x = 0, with x = (xn , . . . , x1 )T and x1 given, is referred to as a shooting eigenvector. (Note that the first equation can only be satisfied as well if and only if μ is an eigenvalue.) For the tridiagonal matrix (1.143), and with xn+1 defined as the first component of (An − μ1n )x, a recurrence for the ratio si = xi /xi−1 , i = 2, . . . , n + 1 is readily obtained, and comparison with (1.165) shows si = −ri−1 /bi−1 (in the case i = n + 1 this requires setting bn := 1). Thus with each bi > 0, the number of positive values in {si } equals N (μ). This can equivalently be stated in terms of {xi }. P ROPOSITION 1.9.8 The number of sign changes in the shooting eigenvector x equals n − N (μ), which is the number of eigenvalues of A greater than μ. 1.9.4 Prufer ¨ phases There is a parametrization, in terms of Pr¨ ufer phases and amplitudes, of the shooting vectors well suited to analysis of the large n limit of the bulk eigenvalues (see Section 13.6). To introduce the parametrization, first observe that the three-term recurrence satisfied by the shooting vector bj xj+1 + aj xj + bj−1 xj−1 = μxj
(j = 1, . . . , n; b0 := 0, bn := −1)
is equivalent to the matrix equation uj uj+1 (μ − aj )/bj −1/bj = bj 0 vj vj+1 where
uj vj
=
1 0 0 bj−1
xj xj−1
(j = 1, . . . , n),
(1.166)
(1.167)
(1.168)
50
CHAPTER 1
(note that the matrix in (1.167) has unit determinant and so as a transformation is volume preserving). Choosing the initial condition u1 = 1, v1 = 0 we see that 1 uj , = Tj 0 vj where Tj := Vj−1 · · · V1 is referred to as a transfer matrix D EFINITION 1.9.9 The Pr¨ufer phases θjμ and amplitudes Rjμ > 0 are such that μ Rj cos θjμ uj = , vj Rjμ sin θjμ
(1.169)
μ where −π/2 < θj+1 − θjμ < 3π/2.
Note that it follows from (1.167) and (1.168) that {θjμ } satisfies the first order recurrence μ b2j cot θj+1 = − tan θjμ + (μ − aj ),
θ1μ = 0.
(1.170)
A consequence is an identity which tells us that θjμ is a decreasing function of μ (see also Exercises 1.9 q.5). P ROPOSITION 1.9.10 We have ∂ μ θj = − u2l . ∂μ j−1
(Rjμ )2
(1.171)
l=1
Proof. Differentiating (1.170) with respect to μ and making use of (1.168) and (1.169) gives the recurrence μ )2 (Rj+1
μ ∂θj+1 ∂θjμ = (Rjμ )2 − u2j . ∂μ ∂μ
This together with the initial condition ∂θ1μ /∂μ = 0 implies (1.171).
We are now in a position to relate θnμ to N (μ) for the tridiagonal matrix (1.143) [326] . First note from the recurrence (1.166) that for μ → ∞, xj is positive while xj−1 /xj → 0. Recalling (1.169), this implies limμ→∞ θjμ = 0. But it has just been shown that θjμ is a decreasing function of μ. The facts that xn+1 = μ μ un+1 = Rn+1 cos θn+1 and that xn+1 = 0 if and only if μ is an eigenvalue then imply the kth largest μ λk eigenvalue λk of T is such that θn+1 = (π/2) + π(k − 1), and moreover that θn+1 relates to the number of eigenvalues of T greater that μ, n − N (μ), according to 1 1 μ (1.172) θn+1 − (n − N (μ)) ≤ . π 2 E XERCISES 1.9 1. The objective of this exercise is to derive the Vandermonde determinant evaluation
]j,k=1,...,N det[xk−1 j
˛ ˛ ˛ ˛ ˛ := ˛ ˛ ˛ ˛
1 1 .. . 1
x1 x2 .. . xN
x21 x22 .. . x2N
··· ··· .. . ···
xN−1 1 xN−1 2 .. . xN−1 N
˛ ˛ ˛ ˛ Y ˛ (xk − xj ). ˛= ˛ ˛ 1≤j
(1.173)
(i) Verify that both the determinant and product of differences are antisymmetric polynomials which are homogeneous of degree 21 N (N − 1) and hence must be proportional. (ii) Show that the proportionality constant is unity by comparing the coefficients of the term x01 x12 . . . xN−1 on N both sides. 2.
(i) In the Vandermonde determinant identity (1.173), replace N by pN . Subtract row one from row two, divide this row by x2 − x1 and take the limit x2 → x1 by first differentiating the top and bottom lines with respect
51
GAUSSIAN MATRIX ENSEMBLES
to x2 . Next subtract the first and second row from the third, divide this row by (x3 − x1 )2 and take the limit x3 → x1 by differentiating top and bottom lines with respect to x3 twice. Proceed in this fashion by subtracting rows 1, 2, . . . , j − 1 from row j (j = 4, . . . , p), dividing by (xj − x1 )j−1 , and taking the limit xj → x1 by differentiating top and bottom lines with respect to xj j − 1 times. Repeat this procedure for successive blocks of p variables to deduce the confluent Vandermonde determinant identity 2 3 xk−1 j ` ´ 6 7 k−1 Y xk−2 6 7 2 j 1 7 det 6 = (xk − xj )p . (1.174) .. 6 7 4 5 . 1≤j
(ii) Consider the identity (1.174) in the case p = 2. Take the transpose of the determinant, and rearrange columns so that it reads h i Y ] j=1,...,2N [jλj−1 ] j=1,...,2N = (λk − λj )4 . (−1)N(N−1)/2 det [λj−1 k k k=1,...,N
k=1,...,N
1≤j
Subtract column N from columns 1, . . . , k − 1, then expand by the first row to deduce that i h Y (−1)(N−1)(N−2)/2 det [λjk − λjN ] j=1,...,2N −1 [jλj−1 ] j=1,...,2N −1 = (λk − λj )4 . (1.175) k k=1,...,N −1
k=1,...,N
1≤j
3. [546], [249] In this exercise the Householder transformation will be used to establish an identity of relevance to the Schur decomposition (15.3) below, and also to establish the diagonalization formula (1.8). (i) Let a and b be unit vectors, and form the unit vector v = (a + b)/|a + b|. From the fact that v bisects the angle of a and b, deduce from the geometrical interpretation of the Householder transformation UN = 1 − 2vv T as a reflection in the hyperplane orthogonal to v that UN a = −b, UNb = −a. Also derive these equations algebraically. (ii) Let AN be an N × N matrix and let λ be an eigenvalue of AN with corresponding normalized eigenvector and use the formulas w. Let e1 := (1, 0, . . . , 0)T be an N × 1 elementary vector. In (i) set a = e1 , b = w, therein to deduce that UN AN UN e1 = λe1 . »
Hence conclude UN A N UN =
λ 0N−1
α TN−1 AN−1
– (1.176)
for some 1 × (N − 1) vector α TN−1 and (N − 1) × (N − 1) matrix AN−1 . (iii) Let PN be a real orthogonal diagonal matrix (each diagonal entry ±1). With VN = PN UN note from (1.176) that – » T N−1 λ β T (1.177) VN AN VN = ˜ N−1 0N−1 A T ˜ N−1 . Now use a Householder transformation of the form N−1 ,A for some β » – 0TN−1 1 0N−1 UN−1
˜ N−1 to triangular form and proceed inductively to deduce that there exists a real orthogonal to reduce A matrix R such that RAN RT = T, (1.178)
52
CHAPTER 1
where T is upper triangular with diagonal entries equal to the eigenvalues of AN−1 , and note that R is unique up to an overall sign of each column. (iv) Show that (1.178) implies the diagonalization formula (1.8). 4. [156] In this exercise the change of variables implied by (1.176) will be used to derive a generalization of (1.11). N−1 = 0N−1 , deduce that (i) From the decomposition (1.176) in the case AN is symmetric so that α » – » – – » 0TN−1 0TN−1 0TN−1 λ1 λ1 dλ1 − UN dUN + , UN dAN UN = UN dUN 0N−1 AN−1 0N−1 AN−1 0N−1 dAN−1 where use is made of the fact that UN dUN is antisymmetric. Make further use of this latter fact to show that it is permissible to write – » 0 −dsTN−1 UN dUN := ˜ N−1 dsN−1 dU and so obtain UN dAN UN =
»
0 (λ1 − AN−1 )dsN−1
dsTN−1 (λ1 − AN−1 ) ˜ N−1 AN−1 − AN−1 dU ˜ N−1 dU
–
» +
dλ1 0N−1
0TN−1 dAN−1
– .
(ii) Using (1.10) and (1.17) read off from the final equation in (i) that (dAN ) = | det(λ1 − AN−1 )|dλ1 (dsN−1 )(dAN−1 ),
(1.179)
˜ N−1 is a function of the components of dsN−1 and thus where use has been made of the fact that dU ˜ N−1 = 0. dsN−1 ∧ dU (iii) Iterate (1.179) to obtain a result equivalent to (1.11). 5.
(i) Use the fact that ri = −Pi (μ)/Pi−1 (μ), to show that the Pr¨ufer phase for j = 2, . . . , n satisfies cot θjμ =
1 Pj−1 (μ) . (bj−1 )2 Pj−2 (μ)
(1.180)
(ii) Use the fact that {Pj (μ)}j=0,1,... are, as a consequence of their obeying the three-term recurrence (1.154), a set of orthogonal polynomials with respect to an inner product defined by its moment (the so called Favard (μ)Pi (μ) > 0. theorem, see, e.g., [384]), together with (5.13) below to show that Pi (μ)Pi−1 (μ) − Pi−1 μ After differentiating (1.180) with respect to μ, use this fact to deduce dθj /dμ < 0.
Chapter Two Circular ensembles Invariance of the probability measure on the space of matrices under conjugation by the appropriate unitary matrices does not uniquely determine the Gaussian ensembles. This fact prompted Dyson to develop a theory of random unitary matrices with the same invariances as the Hermitian matrices used to model quantum Hamiltonians. In quantum mechanics, scattering matrices and Floquet operators are quantities which can be modeled by random unitary matrices. In the case of no time reversal symmetry, the unitary matrices have no further constraints and form the group U (N ). It is well known that the Haar measure is the unique uniform measure on U (N ). In the case of time reversal symmetry, the unitary matrices must be invariant under an appropriate transpose, and the required invariance properties of the Haar measure must be appropriately modified. The corresponding eigenvalue p.d.f.’s can be computed by using either the method of wedge products or metric forms. Another method is to map the unitary matrices to Hermitian matrices through a Cayley transform, and to make use of the known Jacobians for Hermitian matrices. The p.d.f.’s correspond to a log-gas system on a circle, again at three special values of the inverse temperature β = 1, 2 and 4. Although the Haar measure does not make explicit the distribution of the elements of a random unitary matrix, a suitable decomposition into certain elementary unitary matrices allows for distributions to be specified. It is further possible to average over the elements of a random unitary matrix. Unitary matrices are one of the three compact classical groups, the other two being the real orthogonal matrices and the unitary symplectic matrices. The eigenvalue p.d.f.’s corresponding to the uniform distribution (Haar measure) are calculated. In the last two sections unitary and real orthogonal Hessenberg matrices are used to provide generalizations of the eigenvalue p.d.f.’s obtained for special β previously in the chapter.
2.1 SCATTERING MATRICES AND FLOQUET OPERATORS The three ensembles of random Hermitian matrices introduced at the beginning of Chapter 1 were motivated by their relevance as models of quantum Hamiltonians. Likewise, three ensembles of random unitary matrices can be isolated by considering settings in quantum physics, but now with the focus being on scattering matrices and evolution operators rather than Hamiltonians. The settings to be considered here are the scattering of plane waves within an irregular shaped cavity, and the evolution of periodic kicked quantum systems through their Floquet operators. 2.1.1 Random scattering matrices We will consider first the random unitary matrices relating to the scattering of plane waves within an irregular shaped domain (cavity), or a cavity of arbitrary shape containing random scattering impurities (see, e.g., [53], [276]). The plane waves enter and leave the cavity through a lead (wave guide) which is assumed to permit N distinct plane wave states (channels). With the amplitudes of the N incoming plane wave states denoted and the amplitudes of the N outgoing states denoted by O, the N × N scattering matrix is defined so by I, that SI = O.
(2.1)
54
CHAPTER 2
2 = |O| 2 , and from this it follows that SI|S I = I| I for arbitrary I. This Flux conservation requires |I| † in turn implies SS = 1, and thus S must be unitary. In fact, it is generally true (see, e.g., [401]) that S is a limiting form of the evolution operator S=
lim
lim U (t, t0 )
(2.2)
t0 →−∞ t→∞
where, with T denoting time ordering,
i t U (t, t0 ) := T exp − H(t ) dt t0
(2.3)
for some Hamiltonian H. Thus S is necessarily unitary. However, analogous to the theory of Section 1.1.1 for time-independent Hamiltonians, the structure of the N × N unitary matrix modeling S is constrained by the requirements of time reversal symmetry. In Section 1.1.1 a time reversal symmetry T was defined by the statement that T is anti-unitary and [H, T ] = 0. This definition assumed that the Hamiltonian was independent of time. For time-dependent Hamiltonians the following definition is used. D EFINITION 2.1.1 A time-dependent Hamiltonian H(t) is said to have a time reversal symmetry T if T is anti-unitary and T −1 H(t)T = H(−t). Notice that this reduces to the previous definition when H is independent of t. From this definition the action of T on the evolution operator is easily deduced. P ROPOSITION 2.1.2 If T is a time reversal symmetry of a time-dependent Hamiltonian H, then T −1 U (t, t0 )T = U (−t, −t0 ).
(2.4)
Proof. As is well known (see, e.g., [401]), and can easily be checked from its definition in terms of a time ordered exponential, U (t, t0 ) satisfies the integral equation U (t, t0 ) = 1 + Thus T −1 U (t, t0 )T = 1 −
1 i
Z
t
1 i
Z
t
H(t )U (t , t0 ) dt .
t0
T −1 H(t )U (t , t0 )T dt = 1 −
t0
1 i
Z
t
H(−t )T −1 U (t , t0 )T dt ,
t0
where to obtain the first equality we used the fact that T −1 is anti-unitary, and to obtain the second equality we used Definition 2.1.1. Hence Z −t Z t 1 1 H(−t )T −1 U (t , −t0 )T dt = 1 + H(t )T −1 U (−t , −t0 )T dt . T −1 U (−t, −t0 )T = 1 − i −t0 i t0 We therefore have that U (t, t0 ) and T −1 U (−t, −t0 )T satisfy the same integral equation and are thus equal.
We now assume that for an irregular shaped domain the statistical properties of the scattering matrix are determined solely by the global time reversal symmetry. To see what constraint a time reversal symmetry imposes on S, suppose T is a time reversal symmetry of a time-dependent Hamiltonian H. By taking the limits t0 → −∞, t → ∞ in (2.4) and using (2.2) we conclude T −1 ST = S† .
(2.5)
Consider now (2.5) with T 2 = 1. From Exercises 1.1 q.3, we know how to construct a T invariant basis,
55
CIRCULAR ENSEMBLES
{ψn } say. Now ψm |Sψn = T ψm |T Sψn
(since T is anti-unitary)
= T ψm |S† T ψn (using (2.5) and T 2 = 1) = ψm |S† ψn (since {ψn } is T invariant) = ψn |Sψm
(from meaning of † and property of inner product).
Hence S must be symmetric. Consider next (2.5) with T = Z2N K and so T 2 = −1. Then Proposition 2.1.2 gives −1 † T −1 D S = T S† T −1 = Z2N KS† K −1 Z−1 2N = Z2N KS KZ2N = Z2N S Z2N =: S .
(2.6)
This constraint is said to specify S as a self-dual quaternion matrix, and in terms of the 2 × 2 blocks in position (jk) and (kj) of S it requires (1) (2) (4) (2) −zkj zkj zjk zjk = (2.7) (3) (4) (3) (1) zjk zjk −zkj zkj (cf. (1.24)). As with Hamiltonians, two matrices S and S modeling scattering matrices and related by a similarity transformation S = U−1 SU of unitary matrices are equally good physical descriptions, provided S and S have the same symmetries. Let us investigate the restrictions on U implied by time reversal symmetry. Suppose first that the system has a time reversal symmetry with T 2 = 1. Then the matrix modeling the scattering matrix must be symmetric. The subgroup of unitary matrices which under a similarity transformation map S to another symmetric matrix is the real orthogonal matrices (or i times the real orthogonal matrices — see Exercises 2.1 q.2). Next suppose the system has a time reversal symmetry with T 2 = −1 and T = Z2N K, so that the matrices modeling the scattering matrix must be self-dual quaternion. The subgroup of unitary matrices, which under a similarity transformation map self-dual quaternion matrices into self-dual quaternion matrices, is the symplectic equivalent matrices of Proposition 1.3.3 (see Exercises 2.1 q.3). Finally, if the system has no time reversal symmetry, S must be unitary but has no further constraints. Under similarity transformations all unitary matrices map S into another unitary matrix. We remark that the element Snm in S as specified by (2.1) has the property that its absolute value squared gives the probability that the state n scatters to the state m. Thus a quantity of interest is the mean value of 2 |Snm | for S drawn from an ensemble of unitary matrices, appropriately constrained in the cases of a time reversal symmetry. This is computed in Section 2.3.2. 2.1.2 Random Floquet operators We consider next evolution operators U (t) := U (t, 0) as specified by (2.3) in the case that the Hamiltonian is periodic of period τ , H(t) = H(t + τ ). Then for n ∈ Z, U (nτ ) = F n , where F := U (τ ). The operator F is called the Floquet operator, which being a special value of the unitary operator U is itself unitary. To model F for periodic chaotic quantum systems such as kicked tops, random unitary matrices have been used (see, e.g., [284]). As in the above discussion of scattering matrices, the structure of a unitary matrix F modeling F is constrained by the requirement of time reversal symmetry. Substituting t = τ , t0 = 0 in (2.4) implies T −1 FT = F−1 = F† . This is identical to the constraint of scattering matrices (2.5). Thus for T 2 = 1, F must be a symmetric unitary matrix, while for T 2 = −1, F must be a self-dual quaternion unitary matrix. E XERCISES 2.1
1. Show from the definitions that a Hermitian quaternion real matrix is self-dual.
2. Let T be a time reversal symmetry such that T 2 = 1. Suppose S and S are two symmetric unitary matrices
56
CHAPTER 2
related by a similarity transformation S = U−1 SU, where U is a unitary matrix, which satisfy the time reversal symmetry constraint S = T S† T −1 and S = T S† T −1 , (2.8) and suppose the only symmetry of S in general is some constant times the identity. Show that U is real orthogonal or i times a real orthogonal matrix. 3. Let T be a time reversal symmetry such that T 2 = −1. Follow the method of Proposition 1.3.3 to show that if S and S are two self-dual quaternion matrices satisfying (2.8) and related by a similarity transformation as in q.2, and the only symmetry of S is in general some multiple of the identity, then U is a symplectic equivalent matrix or i times a symplectic equivalent matrix.
2.2 DEFINITIONS AND BASIC PROPERTIES In keeping with the above findings on the classes of scattering matrices and Floquet operators consistent with time reversal symmetry, we seek a theory of symmetric, unitary, and self-dual quaternion random unitary matrices. This has been given by Dyson [146]. In contrast to the theory of Gaussian random matrices presented in the previous chapter, the theory was developed not from an explicit formula for the matrix elements of the random unitary matrices, but rather from the requirement that there is a uniform measure on each of these spaces. The definitions adopted were (at least implicitly) motivated by the relationship of the spaces of matrices to certain symmetric spaces. Some aspects of this are covered in Section 2.2.3. Here we adopt a simpler approach, by noting that in the case of unitary matrices with no further constraints, the construction of a uniform measure is a classical problem. 2.2.1 The circular unitary ensemble (CUE) Among the spaces of symmetric, unitary and self-dual quaternion unitary matrices, the space of unitary matrices is special because it forms a group. Moreover, this group is compact. As such there are well established ways to specify and determine a volume form which defines a uniform measure on the space (see, e.g., [540]). The volume form, or Haar form in keeping with the terminology of the corresponding measure — the Haar measure — is denoted (dH U) and is required to have the homogeneity property that, for any fixed unitary matrix V, (dH UV) = (dH VU) = (dH U).
(2.9)
In words this says the Haar form is invariant under left and right actions of the group, and in this sense the Haar form implies a uniform measure on the space of unitary matrices. It is a well-known theorem that the Haar form specified by (2.9) exists and is unique up to normalization for all compact groups. P ROPOSITION 2.2.1 The Haar form for the unitary group is (dH U) =
1 (U† dU), C
(2.10)
where C denotes the normalization. Proof. According to the above discussion, we merely have to check (2.9). Now for fixed V, dUV = dU V,
dVU = VdU,
and thus (2.10) implies (dH UV) = Thus we are done if we can show
1 (V† U† dU V), C
(dH VU) =
(V† U† dU V) = (U† dU),
1 (U† dU). C (2.11)
57
CIRCULAR ENSEMBLES
which is in fact immediate from (1.35).
The definition of an ensemble of random unitary matrices can now be made explicit. D EFINITION 2.2.2 The circular unitary ensemble (CUE) is the group of unitary matrices endowed with the volume form (2.10). Since U is unitary, a short calculation shows that the matrix of differentials U† dU has the anti-Hermitian property (U† dU)† = −U† dU. Thus we can write U† dU = i dM2
(2.12)
for some Hermitian matrix M2 . There are no further constraints on the elements of M2 since U has the same number of independent elements as an arbitrary Hermitian matrix. Consequently computing the volume form (U† dU) for U unitary is equivalent to computing (dM2 ) for M2 Hermitian, and thus the methods of the previous chapter are applicable. 2.2.2 The circular orthogonal and symplectic ensembles The spaces of symmetric and self-dual quaternion unitary matrices do not form groups. Thus the invariance (2.9) expressing the homogeneity of the sought volume form must be modified. Consider first the case of symmetric unitary matrices. Two symmetric unitary matrices S1 and S2 can be related by S2 = VT S1 V for some unitary matrix V. This follows since each symmetric unitary matrix S can be written in the form S = UT U
(2.13)
for U unitary; see the proof of Proposition 2.2.4 below. Thus for a uniform distribution we require that (dH S) = (dH VT SV)
(2.14)
for all unitary V. Now, because the explicit formula (2.10) for the Haar volume form of the unitary group has both the left and right invariance properties (2.9), we see that the same formula (2.10) is consistent with (2.14). We also know from (2.11) that the underlying matrix of differentials, δS say, is arbitrary up to a similarity transformation with a unitary matrix. This arbitrariness can be used so that δS like S is symmetric, and thus analogous to (2.12) is of the form i dM1 for M1 real symmetric. The sought form is achieved by setting V = U† in (2.11), where U is such that (2.13) holds, and so obtaining δS = (UT )† dSU† =: i dM1 .
(2.15)
˜ the analogue of (2.13) is the decomposition In the case of self-dual quaternion unitary matrices S ˜ = Z−1 UT Z2N U =: UD U S 2N
(2.16)
˜ 1 and for U a 2N × 2N unitary matrix. It follows from this that two self-dual quaternion unitary matrices S D˜ ˜ ˜ S2 can be related by S2 = V S1 V for some 2N × 2N unitary matrix V. Thus, in this case for a uniform distribution we require ˜ = (dH VD SV). ˜ (dH S) Of course the Haar form (2.10) has this invariance property. As in the case of symmetric unitary matrices, we ˜ is itself self-dual want to choose the arbitrary matrix in (2.11) so that the underlying matrix of differentials δ S (recall (2.6)), and thus analogous to (2.12) is of the form i dM4 for M4 a self-dual quaternion real matrix. Again as in the symmetric case, this is achieved by the choice V = U† , where U is such that (2.16) holds. We then have ˜ = (Z−1 UT Z2N )† dS ˜ U† = (UD )† dS ˜ U† =: i dM4 . δS 2N We are thus led to the following definition.
(2.17)
58
CHAPTER 2
D EFINITION 2.2.3 The circular orthogonal ensemble (COE) is the space of symmetric unitary matrices endowed with the volume form corresponding to (2.15), and similarly the circular symplectic ensemble (CSE) is the space of self-dual quaternion unitary matrices endowed with the volume form corresponding to (2.17). 2.2.3 Relationship to symmetric spaces Notice from (2.15), (2.12) and (2.17) that the infinitesimal examples of real symmetric, Hermitian and selfdual real quaternion matrices occur in the volume forms of the COE, CUE and CSE respectively. As observed in Section 1.3.4, these infinitesimal generators are the Hermitian part of the matrix Lie algebras gl(N, R), gl(N, C) and u∗ (2N ) respectively, and as such are invariant under the Hermitian conjugation mapping. The general theory of symmetric spaces [295] relates these Lie algebras and the Hermitian conjugation mapping to the compact quotient spaces U (N )/O(N ),
U (N ) × U (N )/U (N ),
U (2N )/Sp(2N )
respectively . Here O(N ) := {N × N real unitary matrices} U (N ) := {N × N complex unitary matrices} Sp(2N ) := {N × N real quaternion unitary matrices}, or equivalently O(N ) consists of the set of N × N real orthogonal matrices and Sp(2N ) the set of 2N × 2N unitary symplectic equivalent matrices. In fact, as observed by Dyson [150] these spaces are isomorphic to the space of symmetric unitary matrices, unitary matrices and symplectic self-dual unitary matrices, respectively. The demonstration of this fact exhibits some essential properties of the respective circular ensembles. For definiteness consider the quotient space U (N )/O(N ). P ROPOSITION 2.2.4 We have U (N )/O(N ) ∼ = US (N ), where US (N ) denotes the set of N × N symmetric unitary matrices. Proof. First we recall that in general for a group G and subgroup K, the quotient space H = G/K is the set of equivalence classes with the equivalence relation g1 ∼ g2 meaning that g1 and g2 are related by the transformation K (i.e., g1 = g2 k for some k ∈ K). We proceed by showing g1 ∼ g2 , g1 , g2 ∈ U (N ), if and only if g1 g1T = g2 g2T . The only if direction is immediate. For the if direction we see that g1 g1T = g2 g2T implies (g2−1 g1 )(g2−1 g1 )T = 1N . Thus g1 = g2 k for some k ∈ O(N ) as required. This demonstrates a one-to-one correspondence between elements of U (N )/O(N ) and elements of the form ggT in US (N ). It remains to check that all elements of US (N ) can be written in the form ggT . For h ∈ US (N ) there exists a matrix k ∈ O(N ) such that h = k diag[eiθj ]j=1,...,N kT , so choosing g = k diag[eiθj /2 ]j=1,...,N expresses h in the desired form.
One feature of the proof of Proposition 2.2.4 is the decomposition (2.13). Thus an involutive mapping φ = T (T denoting transpose) is identified which leaves the space K in the quotient G/K unchanged. It is the existence of the map φ which distinguishes the quotient space as a symmetric space. Another crucial aspect of the proof of Proposition 2.2.4 is the diagonalization formula S = R diag[eiθj ]j=1,...,N RT ,
(2.18)
for some real orthogonal matrix R. Comparison with (1.8) shows this to be the circular ensemble counterpart to the diagonalization formula for members of the GOE. As will be seen in the proof of Proposition 2.2.5, (2.18) is the key formula in the determination of the eigenvalue p.d.f. The CSE allows for a description similar to that of the COE, with the role of (2.13) replaced by the decomposition formula (2.16). In particular this formula can be used to generate a member of the CSE from
59
CIRCULAR ENSEMBLES
a member of the CUE. Also, the diagonalization formula for members of the CSE is S = B diag[eiθj 12 ]j=1,...,N BD ,
(2.19)
where B is symplectic equivalent, satisfying (1.25). This is the circular ensemble counterpart of the diagonalization formula for members of the GSE given in Exercises 1.3 q.4. 2.2.4 Eigenvalue p.d.f. for the circular ensembles With each eigenvalue written λj = eiθj , the eigenvalue p.d.f.’s corresponding to the three circular ensembles can be calculated using the methods of Section 1.2. P ROPOSITION 2.2.5 For an appropriate choice of normalization Cβ,N , which is given explicitly in Proposition 2.8.7, 1 |eiθk − eiθj |β , −π < θl ≤ π (2.20) Cβ,N 1≤j
with β = 1, 2 and 4 is the eigenvalue p.d.f. for the COE, CUE and CSE, respectively. Proof. Only the COE will be considered here; the remaining cases of the CUE and CSE are similar (the CUE is considered in Exercises 2.2 q.1). The starting point is the diagonalization formula (2.18). For notational convenience write Θ = diag[eiθj ]j=1,...,N in that formula. Analogous to (1.9), differentiation and minor manipulation gives RT dSR = RT dR Θ − ΘRT dR + iΘdθ, where θ is the diagonal matrix with entries θj (j = 1, . . . , N ). But from (2.15) RT dSR = i(UR)T dM1 (UR), where U is any unitary matrix such that S = UT U. In particular, with U = Θ1/2 RT , where Θ1/2 is the diagonal matrix with entries eiθj /2 (j = 1, . . . , N ), comparison of the two equations for RT dSR gives dM1 = dθ − iΘ−1/2 RT dR Θ1/2 + iΘ1/2 RT dR Θ−1/2 2 dθ1 2 sin((θ2 − θ1 )/2) r1 · dr2 . . . . . . 2 sin((θN − θ1 )/2) r1 · drN 6 ∗ dθ2 . . . 2 sin((θN − θ2 )/2) r2 · drN 6 =6 . .. .. .. 4 .. . . . ∗ ∗ ... dθN
3 7 7 7, 5
where the elements * are chosen so that the matrix is symmetric. Taking the wedge product of the independent elements, and noting that for θk > θj (2.21) 2 sin((θk − θj )/2) = |eiθk − eiθj | gives the result of the proposition for the COE.
Comparison of Proposition 2.2.5 with Proposition 1.4.1 shows immediately that the eigenvalue p.d.f. for the circular ensembles is directly proportional to the Boltzmann factor of the one-component log-potential Coulomb gas on a circle. 2.2.5 Antisymmetric unitary matrices In Exercises 1.3 q.5 the eigenvalue p.d.f. of antisymmetric Gaussian random matrices with pure imaginary elements was calculated. If instead one considers antisymmetric unitary matrices, then the eigenvalue p.d.f. for the CUE, after a rescaling 2θj → θj , is reclaimed. To understand this result, first note that for a unitary
60
CHAPTER 2
antisymmetric matrix S, if λ is an eigenvalue then so is −λ. Because we must also have |λ| = 1 this implies that such matrices must be even dimensional. They can be constructed out of unitary matrices U by forming UT Z2N U. A result of Hua [300] gives the decomposition 0 eiθN 0 eiθ1 T , . . . , RT , U Z2N U = Rdiag −eiθ1 0 −eiθN 0 where R is real orthogonal, and it follows from this that the eigenvalues of UT Z2N U are ±ieiθj (j = 1, . . . , N ). By defining δS = (UT Z2N )† dS U† we can proceed as in Section 2.2.4 to deduce that the eigenvalue p.d.f. for the eigenvalues eiθj (j = 1, . . . , N ) with 0 < θj < π is proportional to |e2iθk − e2iθj |2 1≤j
and thus, after the replacements 2θj → θj , formally equivalent to that of the CUE. E XERCISES 2.2
1. Here the result of Proposition 2.2.5 in the case of CUE will be derived using the metric form approach of Section 1.2.4. (i) Show from the diagonalization formula U = U2 ΘU†2 , where U2 is a unitary matrix with the first component of each of the entries in the first row chosen to be real and positive, that U†2 dU U2 = δU2 Θ − ΘδU2 + iΘdθ, where δU2 := U†2 dU2 = [δu2 jk ]j,k=1,...,N . (ii) Recalling (2.12), and using the fact that Tr((δU2 Θ − ΘδU2 )Θdθ) = 0, show from this that Tr(dM2 dM†2 ) =
N X
|δu2 jk (eiθj − eiθk )|2 +
N X
(dθj )2 .
(2.22)
j=1
j,k=1 j=k
(iii) Read off from (ii) the result Y
(U† dU) =
|eiθj − eiθk |2
V j
dθj (U†2 dU2 ),
(2.23)
j=1
1≤j
where (U†2 dU2 ) = β = 2.
N ^
δur2 jk δui2 jk and note that this is in agreement with Proposition 2.2.5 in the case
2. [458] Suppose for a certain ensemble of N × N unitary matrices the eigenvalue p.d.f. is PN . Let the p.d.f. of the (p) p-th power of the eigenvalues in the same ensemble be denoted PN , so that for any Laurent expandable function iθ1 iθN f (e , . . . , e ), f (eipθ1 , . . . , eipθN )PN = f (eiθ1 , . . . , eiθN )P (p) . N
(i) Show that for PN a Laurent polynomial in {e
iθj
},
“ ” (p) PN (eiθ1 , . . . , eiθN ) = LP PN (eiθ1 /p , . . . , eiθN /p ) , where LP denotes the Laurent polynomial portion of the function, i.e., only terms involving integer powers of the monomials eiθj . (ii) For U (N ), note from Proposition 2.2.5 and the Vandermonde identity (1.173) that PN (eiθ1 , . . . , eiθN ) =
N ” “ X Y 1 iθl (l−Q(l)) ε(Q) e , Asym (2π)N N ! Q∈S l=1 N
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CIRCULAR ENSEMBLES
where Asym denotes the operation of anti-symmetrization (see (4.135) below), SN denotes the set of all permutations of {1, . . . , N } and ε(Q) denotes the corresponding parity. (iii) For a given 0 ≤ μ < p, write l = pl + μ + 1, 0 ≤ l < (N − μ)/p and define Q(μ) (l ) by Q(l) = pQ(μ) (l ) + μ + 1 to show N ” p−1 “ X Y Y“ ε(Q) eiθl (l−Q(l))/p = LP Q∈SN
μ=0
l=1
(N−μ)/p −1
X
ε(Q(μ) )
Y
−Q(μ) (l ))
eiθpl +μ+1 (l
l =0
Q(μ) ∈S(N −μ)/p (p)
(iv) Conclude from the result of (iii) that for U (N ), PN is proportional to the symmetrization of the product Qp−1 (p) μ=0 P (N−μ)/p , and thus in particular that PN is the uniform measure for p ≥ N .
2.3 THE ELEMENTS OF A RANDOM UNITARY MATRIX 2.3.1 Constructing a random unitary matrix With members of the COE and CSE generated from a random unitary matrix according to (2.13) and (2.16), it is of some interest to consider the problem of specifying a random unitary matrix from a numerical generation perspective. In fact it is possible to give an explicit construction of the matrix elements of a member of the CUE [558], a fact which follows from a classical result of Hurwitz [302]. P ROPOSITION 2.3.1 Almost all N × N unitary matrices U have the unique decomposition U = eiα0 UN −1 UN −2 · · · U1 with Uk = U(k,k+1) (φk,k+1 , ψk,k+1 , 0)U(k,k+2) (φk,k+2 , ψk,k+2 , 0) × · · · × U(k,N −1) (φk,N −1 , ψk,N −1 , 0)U(k,N ) (φk,N , ψk,N , αk ), −π ≤ αk < π,
−π ≤ ψjk < π,
0 ≤ φjk ≤ π/2,
where the N × N unimodular matrices U(j,k) (φ, ψ, α) are defined so that all diagonal elements are 1 except for the jth and kth which are equal to cos φ eiα and cos φ e−iα respectively, and all off-diagonal elements are zero except for the element in the jth row and kth column which is equal to sin φ eiψ , and the element in the kth row and jth column which is equal to − sin φ e−iψ (the quantities φ and ψ are referred to as Euler angles). Furthermore (U† dU) =
1 C
d[(cos φjk )2(N −k+1) ]dψjk
N −1
dαj .
j=0
1≤j
Proof. For the decomposition we essentially follow [411]. The strategy is first to choose the parameters φ1,j and ψ1,j (j = 2, . . . , N ) and α1 such that – » 1 01×N−1 , (2.24) UU†1 = 0N−1×1 V where V is an (N − 1)-dimensional unitary matrix, and then to repeat the procedure to reduce V to an analogous form, and so on. To obtain (2.24), φ1,N , ψ1,N and α1 are to be chosen so that UU(1,N)† (φ1,N , ψ1,N , α1 ) has the element 0 in position (1, N ) and element r in position (1, 1) for some 0 < r < ∞. A simple calculation using the definition of U(1,N)† shows this is achieved by requiring tan φ1,N ei(ψ1,N −α1 ) =
u1N , u11
1 r = . cos φ1,N u11 e−iα1
(2.25)
” .
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CHAPTER 2
From the first equation in (2.25) φ1,N is uniquely determined (0 < φ1,N < π/2) as is ψ1,N −α1 (−π < ψ1,N −α1 < π), while the second equation gives that α1 = arg(u11 ) (−π ≤ arg(u11 ) < π) and r = |u11 |/ cos φ1,N . Next φ1,N−1 and ψ1,N−1 are chosen so that UU(1,N)† (φ1,N , ψ1,N , α1 )U(1,N−1)† (φ1,N−1 , ψ1,N−1 , 0)
(2.26)
has an element 0 in position (1, N − 1) and element r in position (1, 1) for some 0 < r < ∞. This requires tan φ1,N−1 eiψ1,N −1 = u1 N−1 /r and 1/ cos φ1,N−1 = r /r which has a unique solution for 0 < φ1,N−1 < π/2, −π ≤ ψ1,N−1 < π and 0 < r < ∞. Note from the definition of U(1,N−1) that since UU(1,N)† has element 0 in position (1, N ), (2.26) must also have element 0 in this position. Continuing this procedure gives that UU†1 has all elements zero in the first row except that in position (1, 1) which is real positive. But UU†1 is unitary so the element in position (1, 1) must equal unity and also the elements below this element in the first column must necessarily vanish. To compute the volume form (2.10) we first note that the decomposition (2.24) gives » – 0 01×N−1 † † † U dU = U1 dU1 + U1 (2.27) U1 . 0N−1×1 V† dV As the contribution to (U† dU) from (U†1 dU1 ) comes entirely from the first column, while the second term in (2.27) contains no such contribution, we obtain the factorization » – “ ” 0 01×N−1 U1 = (U†1 dU1 )(V† dV), (U† dU) = (U†1 dU1 ) U†1 † 0N−1×1 V dV where to obtain the second equality (1.35) has been used. It remains to compute (U†1 dU1 ), as the value of (V† dV) will (1) u1 , say, is independent, we have then follow by induction. Now, since only the first column of dU1 , d (U†1 dU1 ) = (U†1 d u1 ). (1)
Here there are 2N − 1 independent variables α1 , φ1,j , ψ1,j (j = 2, . . . , N ) which is in keeping with the first component u(1) being pure imaginary and the remaining component having independent real and imaginary parts. of U†1 d Now, by definition, up to a sign 3 2 n n n (1) oi (1) oi (1) oi ∂ u1 ∂ u ∂ u U†1 ∂α U†1 ∂φ 1 U†1 ∂ψ1 1,k 1,k 6 1 1 1 7 N 6 n n n ^ (1) or (1) or (1) or 7 ∂ u1 u1 u1 7 6 † ∂ † ∂ (U†1 d u(1) ) = det 6 U†1 ∂α dα dφ1,k dψ1,k , (2.28) U1 ∂φ1,k U1 ∂ψ1,k 7 6 j j j 7 k=2 o n o 5 4 n ∂ u(1) oi n (1) (1) i i ∂ u1 ∂ u1 1 U†1 ∂α U†1 ∂φ1,k U†1 ∂ψ1,k j
j
j
j,k=2,...,N
{ }sj
denotes the real (s = r) or imaginary (s = i) part of the jth component. We observe that the matrix in (2.28) where can be written in the factorized form, giving 1 02 3 2 n (1) or n (1) or n (1) or 3 −{U1 }i1k {U1 }r1k ∂ u1 ∂ u1 ∂ u1 C B6 7 ∂α ∂φ1,k ∂ψ1,k C B 6 j j j 7 (2.29) det B6 {U1 }rjk {U1 }ijk 7 C. 4 n (1) oi n (1) oi n (1) oi 5 4 5 ∂ u1 ∂ u1 ∂ u1 A @ ∂α ∂φ1,k ∂ψ1,k j=1,...,N −{U1 }ijk {U1 }rjk j j j j=2,...,N k=2,...,N k=1,...,N
According to the Binet-Cauchy theorem [8] (see (6.88) below) the determinant of such a product of matrices is equal to the sum of the product of determinants of the matrix obtained by blocking out the lth column of the 2N − 1 × 2N matrix and the lth row of the 2N × 2N − 1 matrix. From the explicit formulas u11 = cos φ1,2 · · · cos φ1,N eiα1 , u31 = − sin φ1,3 e
−iψ1,3
u21 = − sin φ1,2 e−iψ1,2 cos φ1,3 · · · cos φ1,N eiα1 ,
cos φ1,4 · · · cos φ1,N eiα1 ,
...,
uN 1 = − sin φ1,N e−iψ1,N
(2.30)
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CIRCULAR ENSEMBLES
we find 2 n 6 det 4 n
(1)
∂ u1 ∂α
(1) ∂ u1
∂α
or
n
j
n
oi
j
(1)
∂ u1 ∂φ1,k (1) ∂ u1 ∂φ1,k
or
n
j
n
oi
j
(1)
∂ u1 ∂ψ1,k (1) ∂ u1 ∂ψ1,k
or 3(l) j 7 oi 5
= (−1)l−1 {U†1 }s1 [(l+1)/2]
N Y
sin φ1,k (cos φ1,k )2(N−k)+1
k=2 j=1,...,N k=2,...,N
j
where the superscript (l) on the l.h.s. denotes that the lth row of the matrix is to be deleted, and on the superscript s on r.h.s. denotes the real part for l odd and the imaginary part for l even. Thus (2.29) is equal to 2 “
N Y k=2
sin φ1,k (cos φ1,k )2(N−k)+1
2N ”X l=1
−{U1 }i1k
6 r (−1)l−1 {U†1 }s1 [(l+1)/2] det 6 4 {U1 }jk
−{U1 }ijk
{U1 }r1k
3(l)
7 {U1 }ijk 7 5 {U1 }rjk j=2,...,N
k=1,...,N
where now the superscript (l) denotes that the lth column is to be deleted. But the sum over l is precisely the Laplace expansion of det U†1 by the first row where the N × N complex matrix U†1 is written as a 2N × 2N real orthogonal matrix by replacing each complex element by its 2 × 2 real matrix representation (1.36). Since a real orthogonal matrix has determinant equal to ±1, we have that u(1) ) = dα (U†1 d
N ^
sin φ1,k (cos φ1,k )2(N−k)+1 dφ1,k dψ1,k
k=2
and the stated formula for the volume form now follows by induction.
We remark that the precise detail of the decomposition presented in Proposition 2.3.1 differs from the one originally given by Hurwitz [302]. In Proposition 2.3.1 the matrix UU†1 has the structure (2.24), whereas in the decomposition of [302] the analogous matrix product has the unit element in the bottom right corner. Further details are given in Exercises 2.3 q.1. 1/2(N −k+1) From Proposition 2.3.1, we see that with φjk = arccos ξjk , 0 ≤ ξjk ≤ 1, (U† dU) gives the uniform measure (in αj , ψjk and ξjk ) so that all matrices are equally probable, which is what is required for the CUE. Thus with αj , ψjk and ξjk chosen at random with uniform density from their respective intervals, the unitary matrix U formed according to the decomposition in Proposition 2.3.1 will be a member of the CUE. An essential ingredient in the Hurwitz construction is the decomposition (2.24). The analogous decomposition 1 01×N −1 U = U(1) (2.31) 0N −1×1 V has been used in [134] to give a probabilistic generation of U. The idea is that U(1) in (2.31) can be interpreted as an element of the quotient space U (N )/U (N −1). The latter has the geometrical interpretation as the point on the complex (N − 1)-sphere determining the axis of the lower-dimensional complex rotations specified by U (N − 1) (i.e., the image of the unit vector e1 under U (N )). Thus given V ∈ CUEN −1 (CUEn denoting the circular unitary ensemble of n × n random unitary matrices), U ∈ CUEN follows from (2.31) by choosing U(1) as the matrix corresponding to mapping e1 to a random point x(1) on the complex (N − 1)-sphere. The first task is to generate x(1) . Analogous to the result of Exercises 1.2 q.2(iv) this can be achieved by forming the vector z = (z1 , . . . , zN ) in which each (complex) component has distribution zj /|z|, with each zi a complex normal random variable with mean 0 and variance 1. Let z1 = |z1 |eiθ1 , 0 ≤ θ1 < 2π and define (1) v (1) = x1 + eiθ1 e1 , u(1) = v (1) /|v (1) |. Then it is straightforward to check that the matrix U(1) = −eiθ1 (1N − 2u(1) (u(1)∗ )T ),
64
CHAPTER 2
where here ∗ denotes the complex conjugate, is unitary and has the sought property U(1)e1 = x(1) (cf. Exercises 1.9 q.3). We can now proceed inductively. Define x(j) = z(j) /|z(j) | where z (j) = (zj , . . . , zN ) with each zi a standard complex Gaussian chosen independently for a given j. Write zj = |zj |eiθj , 0 ≤ θj < 2π, and define v (j) = x(j) + eiθj ej , and u(j) = v (j) /|v (j) |. From these quantities construct the unitary matrix 0(j−1)×(N −j+1) 1j−1 (j) . U = 0(N −j+1)×(j−1) −e−iθj (1N −j+1 − 2u(j) (u(j)∗ )T ) Then the matrix U = U(1) U(2) · · · U(N ) gives a Haar distributed element of U (N ). The diagonalization of a GUE matrix X also leads to the construction of a member of the CUE. To see this, we know from Exercises 1.3 q.3 that (dX) =
(λk − λj )2
1≤j
N
dλj (U† dU),
j=1
where the columns of the unitary matrix U consist of the normalized eigenvectors of X, chosen, for example, so that their first element is real and positive. This subgroup of unitary matrices can be extended to the full group of unitary matrices by multiplying each eigenvector by an arbitrary phase eiαj , 0 ≤ αj < 2π. Defining U out of these modified eigenvectors, the Haar form (U† dU) in the formula for (dX) is unchanged since it is invariant under all unitary transformations, and thus a random unitary matrix results. A still more efficient way to calculate a member of the CUE via a decomposition of another random matrix is noted below the proof of Proposition 3.2.5. 2.3.2 Integration over the elements of a unitary matrix The scattering matrix problem of Section 2.1.1 motivates the study of the matrix averages ,...,al αl Qab11βα11,...,b := (Ub1 β1 · · · Ubm βm )(Ua1 α1 · · · Ual αl )U∈U (N ) . m βm
(2.32)
We first make note of a number of general properties of this average [398]. P ROPOSITION 2.3.2 The average (2.32) is (a) nonzero only if {aj } = {bj }, and {αj } = {βj }; (b) is unchanged by interchanging the labels of the rows or columns of U; (c) is unchanged by permuting a1 α1 , . . . , al αl or b1 β1 , . . . , bm βm ; (d) satisfies N
a2 α2 ,...,am αm 1 ,a2 α2 ,...,am αm Qnα nβ1 ,b2 β2 ,...,bm βm = δα1 ,β1 Qb2 β2 ,...,bm βm .
n=1
Proof. Let U
(0)
= diag(e
iθ1
, . . . , eiθN ). Since the Haar form is invariant under U → U(0) U, U → UU(0) we must
have a α ,...,a α
a α ,...,a α
Qb11β11,...,bml βlm = ei(θb1 +···+θbm )−i(θa1 +···+θal ) Qb11β11,...,bml βlm ,...,al αl = Qab11βα11,...,b ei(θβ1 +···+θβm )−i(θα1 +···+θαl ) , m βm
which implies (a). Property (b) follows from the invariance of the Haar form underP the interchange of rows or columns, while (c) follows immediately from the definition (2.32). Finally, (d) follows from N ¯nβ1 = δα1 ,β1 which in n=1 unα1 u turn characterizes U as a unitary matrix.
The results of Proposition 2.3.2 can be used to calculate (2.32) for l = m = 1 and l = m = 2 [53].
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CIRCULAR ENSEMBLES
P ROPOSITION 2.3.3 We have 1 Qab11βα11 = δa1 ,b1 δα1 ,β1 , N 1 a1 α1 ,a2 α2 δa1 ,b1 δα1 ,β1 δa2 ,b2 δα2 ,β2 + δa1 ,b2 δα1 ,β2 δa2 ,b1 δα2 ,β1 Qb1 β1 ,b2 β2 = 2 N −1 1 δ δ δ δ + δ δ δ δ − α ,β a ,b α ,β a ,b α ,β a ,b α ,β a ,b 1 1 1 2 2 2 2 1 1 2 1 1 2 1 2 2 . N (N 2 − 1)
(2.33)
(2.34)
11 Proof. The Kronecker deltas in (2.33) follow from (a), while (b) tells us that Qaα aα = Q11 , which when combined with
(d) implies the nonzero value in (2.33) of 1/N . In relation to (2.34), according to (a), (b), (c) it suffices to show that Q11,11 11,11 =
2 , N (N + 1)
Q11,21 11,21 =
1 , N (N + 1)
Q13,24 13,24 =
1 , N2 − 1
Q23,14 13,24 = −
1 . N (N 2 − 1)
(2.35)
The invariance of (2.32) under the mapping U → U(0) U tells us that 11,21 Q11,11 11,11 = 2Q11,21
X
(0)
(0)
|U1m |2 |U1n |2 + Q11,11 11,11
where use has been made of the facts that follows that
=
(0)
|U1n |4 ,
n=1
m =n m1,n1 Qm1,n1
N X
Q11,21 11,21
(m = n),
Qn1,n1 n1,n1
(0) = Q11,11 it 11,11 . Using the unitarity of U
11,21 Q11,11 11,11 = 2Q11,21 .
On the other hand, (d) and (2.33) tell us that 11,21 Q11,11 11,11 + (N − 1)Q11,21 =
1 , N
thereby implying the first two equations in (2.35). The derivation of the remaining equations is similar.
It follows from (2.33) and (2.34) that |Unm |2 U∈U(N ) =
1 , N 1 + δn,m , N +1 2 − δn,m = 2N − 1
|Unm |2 U∈COE = |(UUT )nm |2 U∈U (N ) = |Unm |2 U∈CSE = |(UUD )nm |2 U∈U (2N )
(2.36)
(in the final formula UUD is to be regarded a quaternion real matrix for purposes of computing the absolute values of the elements). For the diagonal entries n = m, the full distribution of |Unm |2 can in fact be computed in each case (see Exercises 3.8 q.3). We remark that an evaluation formula for (2.32) in general, involving a double sum over permutations of N , with summand containing a further summation involving group characters (see Section 11.6.3) known as the Weingarten function, has been derived [114]. E XERCISES 2.3
1. Here the decomposition of Proposition 2.3.1 is modified to correspond with that of [302].
Let Ek := U(k,k+1) (φk,k+1 , ψk,k+1 , 0)U(k−1,k+1) (φk−1,k+1 , ψk−1,k+1 , 0) × · · · × U(2,k+1) (φ2,k+1 , ψ2,k+1 , 0)U(1,k+1) (φ1,k+1 , ψ1,k+1 , αk ). Modify the working of the proof of Proposition 2.3.1 to show that for appropriate choice of the parameters, – » V 0N−1×1 , UE†N−1 = 01×N−1 1
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Figure 2.1 Schematic diagram of a lead with a tunneling barrier coupled to a chaotic barrier, indicating the notation for the various incoming and outgoing waves.
where V is an (N − 1)-dimensional unitary matrix. Hence conclude that an arbitrary unitary matrix U can be decomposed as U = eiα0 E1 E2 · · · EN−1 .
2.4 POISSON KERNEL The consideration of a more general scattering problem [101] than that of Section 2.1.1 leads to an ensemble of random unitary matrices S specified by the Poisson kernel ¯ β(N −1)/2+1 ¯ † S) det(1 − S 1 P (S) = . (2.37) ¯ † S)|2 C | det(1 − S ¯ =: SP (S) (dH S) and thus denotes the mean of S. For the circular ensembles, S ¯ = 0 and (2.37) Here S reduces to the uniform distribution. Generally (2.37) has the reproducing property [300] ¯ f (S) = P (S)f (S) (dH S) (2.38) valid for any analytic function f of the matrix S. In the case N = 1 (2.38) reads 2π 1 (1 − |z|2 ) f (z) = f (eiθ ) dθ, 2π 0 |1 − z¯eiθ |2
(2.39)
which is the Poisson formula on a circle giving the value of a harmonic function f for |z| < 1 in terms of its value on the unit circle. The setting of the scattering problem is as in Figure 2.1. The lead contains a tunneling barrier, through which a portion of the incoming wave specified by the N -component vector I is transmitted and a portion is reflected. The scattering of the incoming wave through the tunneling barrier is described by the equation I r t I O S1 := = , (2.40) t r I I O where r and t denote the N × N transmission and reflection matrix, respectively, for the incoming wave at the left of the barrier, and similarly r and t at the right of the barrier. As well as being unitary, the matrix S1 is required to be symmetric (β = 1) and self-dual (β = 4). Similarly, the scattering at the cavity entrance and the scattering in the total system, are described by the equations = I , S0 O
SI = O.
(2.41)
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CIRCULAR ENSEMBLES
Straightforward manipulation of (2.40) and (2.41) shows that S is related to S0 by S = r + t (1 − S0 r )−1 S0 t.
(2.42)
The Jacobian for the change of variables from S0 to S gives the Poisson kernel (2.37) for a certain choice ¯ [300], [400]. of S ¯ = r we have P ROPOSITION 2.4.1 With S ¯ β(N −1)/2+1 ¯ † S) det(1 − S (dH S0 ) = (dH S) ¯ † S)|2 | det(1 − S
(2.43)
and thus the Jacobian is proportional to the Poisson kernel (2.37). Proof. First, one can check from the unitarity of S1 in (2.40) that t − r t−1 r = (t† )−1 , These equations used in (2.42) imply Differentiating XX
−1
t† r = −r† t .
S0 = t−1 (S − r)(1 − r† S)−1 t† .
= 1 gives dX
−1
−1
(2.44)
−1
= −X dXX and so we deduce from (2.44) that “ ” dS0 = t−1 1 + (S − r)(1 − r† S)−1 r† dS(1 − r† S)−1 t† .
Use of the matrix identity †
(1 − AB)−1 A = A(1 − BA)−1 ,
(2.45)
†
and the formula 1 − rr = t t which follows from the unitarity of the scattering matrix in (2.40), shows that this can be rewritten (2.46) dS0 = t† (1 − Sr† )−1 dS(1 − r† S)−1 t† . Let us suppose now that β = 2, so the volume form is given by (2.10). Writing dS as S S−1 dS in (2.46), using once more the identity (2.45) and multiplying both sides by S†0 shows S†0 dS0 = t(1 − Sr† )−1 SdS(1 − r† S)−1 t† .
(2.47)
The change of variables formula (1.35) in the case β = 2 applied to (2.47) gives (2.43) in the case β = 2, where use is also made of the fact that 1 − r† r = t† t. In the cases β = 1 and β = 4 we start again from (2.46). Recalling the formulas (2.15) and (2.17), we decompose S0 and S in terms of unitary matrices U0 , U, according to (2.13) and (2.16) as appropriate. Using the additional fact that S1 is symmetric (self-dual quaternion) for β = 1 (β = 4) and thus r = rT (r = rD ) we see that † −1 B, U0 dS0 U−1 0 = B UdS U
B := U(1 − r† S)−1 t† U−1 0 .
(2.48)
Furthermore, we can check that B has real elements for β = 1, complex elements for β = 2 and real quaternion elements for β = 4 (see Exercises 2.4 q.1). Applying the change of variables formula (1.35) in the cases β = 1 and β = 4 to (2.48) gives (2.43) in the cases β = 1 and β = 4.
From (2.43) and the fact that P (S0 ) = 1/C we have P (S0 )(dH S0 ) = P (S)(dH S), ¯ = r represents the distribution of the S where P (S) is given by (2.37). Thus the Poisson kernel with S matrices for the lead with a tunneling barrier coupled to a chaotic cavity. The eigenvalue p.d.f. corresponding to the Poisson kernel (2.37) can be specified in the special case ¯ = z1N , when (2.37) is a function only of the eigenvalues eiθj . The eigenvalue p.d.f. is then obtained S by multiplying the Poisson kernel by the eigenvalue p.d.f. (2.20) for the circular ensembles, and is thus given
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CHAPTER 2
by N 1 1 iθl |β(N −1)+2 ˜ |1 − ze C l=1
|eiθk − eiθj |β .
(2.49)
1≤j
This is proportional to the Boltzmann factor of the one-component log-potential Coulomb gas on a circle, with an external charge of strength −(N − 1) + 2/β at the point 1/z in the complex plane. 1. [400] The objective of this exercise is to show that the matrix B in (2.48) has real elements for β = 1 and real quaternion elements for β = 4. This is equivalent to showing B† = BT and B† = BD , respectively.
E XERCISES 2.4
(i) Use (2.44) to show that the formula for B can be rewritten B = U(S − r)−1 t S0 U−1 0 . (ii) Use the fact that S0 = UT0 U0 (β = 1), S0 = UD 0 U0 (β = 4) to rewrite the formula in (i) as B = U(S − r)−1 t UT0 (β = 1),
B = U(S − r)−1 t UD 0 (β = 4).
(iii) Similar to (ii) use the decompositions S = UT U (β = 1), S = UD U (β = 4) to show from the formula in (2.48) that B† = U0 t(S − r)−1 UT (β = 1),
B† = U0 t(S − r)−1 UD (β = 4).
(iv) Use the facts that rT = r, tT = t , ST = S (β = 1), and rD = r, tD = t , SD = S (β = 4) in the results of (iii) and compare with the results of (ii) to conclude that B† = BT (β = 1) and B† = BD (β = 4) as required.
2.5 CAUCHY ENSEMBLE The mapping eiθ =
1 + iλ 1 − iλ
(2.50)
maps each point λ onto the real line to a point eiθ (−π < θ < π) on the unit circle via a stereographic projection. The angle θ can be constructed by drawing the unit circle in the complex plane with center (0, i) and projecting from the point (0, 2i) on the circle to the point λ on the real axis. The intersection of this line with the unit circle gives θ which is measured anticlockwise from the point (0, 0). Changing variables for each θj in (2.20) according to (2.50) gives |eiθk − eiθj |β dθ1 · · · dθN 1≤j
= 2N +βN (N −1)/2
N
(1 + λ2j )−β(N −1)/2−1
j=1
|λk − λj |β dλ1 · · · dλN .
(2.51)
1≤j
In orthogonal polynomial theory the function w(x) = 1/(1 + x2 )α is referred to as the Cauchy weight. The result (2.51) motivates the definition of the Cauchy ensemble of random matrices via the eigenvalue p.d.f. N 1 (1 + λ2j )−α C j=1
1≤j
|λk − λj |β .
(2.52)
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CIRCULAR ENSEMBLES
In the case α = β(N − 1)/2 + 1, β = 1, 2 and 4, (2.51) gives that this is realized by the stereographic projection of the eigenvalue p.d.f. of the circular ensembles onto the real line. In fact Hermitian matrices H with the eigenvalue p.d.f. (2.52) for α = β(N − 1)/2 + 1, β = 1, 2 and 4, can be constructed out of their circular counterparts U by making the Cayley transformation 1N − U . 1N + U
(2.53)
1N + iH 1N − iH
(2.54)
H=i To see this, note from the inverse of this relation, U=
that the eigenvalues eiθj of U are related to the eigenvalues λj of H by (2.50). It follows from (2.51) that the joint p.d.f. P (H) of the matrices H can be specified by −β(N −1)/2−1 1 det(1N + H2 ) P (H) = (2.55) C for H real symmetric (β = 1), complex Hermitian (β = 2) and self-dual quaternion real (β = 4). The Cayley transformation can be used to compute the volume form (U† dU) in terms of the volume form (dH). Since the volume form (dH) has previously been computed in terms of the eigenvalues and eigenvectors of H, and the eigenvalues of H and U are related by (2.50), this gives an alternative way to derive the result (2.20) [300]. The calculation begins by making use of the formula of Exercises 2.5 q.1 to deduce from (2.54) that U† dU = 2i(1N + iH)−1 dH(1N − iH)−1 .
(2.56)
For U symmetric (self-dual quaternion) it follows from (2.53) that H will be symmetric (self-dual quaternion), and thus so will be (1N ± iH)−1 . Hence we can apply the result of Exercises 1.3 q.2 on the r.h.s. of (2.56) to deduce −β(N −1)/2−1 (δU) = 2N (β(N −1)/2+1) det(1N + H2 ) (dH), (2.57) where β = 1, 2 or 4 for the elements of H real, complex or quaternion real, respectively. Changing variables to the eigenvalues and eigenvectors on the r.h.s. using the result of Proposition 1.3.4 and integrating out the eigenvector dependence gives the r.h.s. of (2.51), and thus the eigenvalue p.d.f. (2.20) after the change of variables (2.50). For general β (2.52) represents the Boltzmann factor for a one-component log-gas on a line (the x-axis) subject to a one-body potential βV (λ) = α log(1 + λ2 ), which corresponds to an external charge of strength −2α/β at the point (0, 1) in the xy-plane. In the case 2α/β = N − 1 + 2/β when (2.51) holds, since the particle density in the circular ensemble is uniform (by symmetry), making the change of variables ρ(1) (θ) dθ = ρ(1) (λ) dλ according to (2.50) shows the density in this particular case of the Cauchy ensemble is given by ρ(1) (λ) =
N , π(1 + λ2 )
(2.58)
independent of β. We conclude this section by noting some special properties of P (H) as specified by (2.55) [300], [101]. P ROPOSITION 2.5.1 Let H have the distribution (2.55). Then (a) H−1 is also distributed according to (2.55); (b) every n × n submatrix of H, obtained by omitting N − n rows and corresponding columns of H, is distributed according to the ensemble (2.55) with N replaced by n. Proof. For the result (a), we simply change variables Y = H−1 . Since YH = 1N we see that dH = −Y−1 dY Y−1 .
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Use of (1.35) then gives (dH) = (det Y −1 )β(N−1)+2 (dY), and so ”−β(N−1)/2−1 “ (det Y−1 )β(N−1)+2 , P (Y) ∝ det 1N + Y −2 which after simple manipulation reduces to the r.h.s. of (2.55). To obtain the result (b), suppose for definiteness that the final row and column of H are deleted, and write – » G y , H= y † hNN where G is the (N − 1) × (N − 1) submatrix, y is a vector of length N − 1 with real (β = 1), complex (β = 2) or quaternion real (β = 4) elements, and hNN is real. The task is to compute the distribution of G given that H is distributed according to (2.55). From the identity det(1N + H2 ) = det(1N−1 + G2 + yy † ) i h 1 h 2 y y† y † )−1 0(N−1)×1 (1N−1 + G2 + y N−1 + G + det × det † † 01×(N−1) 1 y G+ y hNN
i G y+ yhNN , 2 † 1 + hNN + y y
in which the product of the first two terms equals unity, we see that y y † )(ah2NN + 2bhNN + c), det(1N + H2 ) = det(1N−1 + G2 +
(2.59)
where a=1 − y † (1N−1 + G2 + y y † )−1 y, 2b = − y † G(1N−1 + G2 + y y † )−1 y− y † (1N−1 + G2 + y y † )−1 G y, c=1 + y† y− y † G(1N−1 + G2 + y y † )−1 G y. Now diagonalize G by G = Udiag(λ1 , . . . , λN−1 )U† and set q ” “q 1 + λ21 , . . . , 1 + λ2N−1 U† T = Udiag so that T† = T and T2 = 1N−1 + G2 . Also, introduce the vector x according to y† = x† T, and note that x) = (d y) = (det T)β (d These results show P (G) :=
“
det(1N−1 + G2 )
”β/2
(d x).
Z
“ ”−β(N−2)/2−1 P (H)(d y)dhNN = det 1N−1 + G2 Z Z ∞ × (d x) det(1N−1 + xx† )−β(N−1)/2−1 dhNN (ah2NN + 2bhNN + c)−β(N−1)/2−1
(2.60)
−∞
where in terms of the vector x x x† )−1 x= a = 1 − x† (1N−1 +
1 , 1+α
x α := x†
(the second equality follows by multiplying (1N−1 + x x† )−1 by ( x x† )( x x† )−1 and using the fact that x† x =: α is a scalar) and similarly x† G ( x† G x x )2 b=− , c=1+α+ . 1+α 1+α
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CIRCULAR ENSEMBLES
√ Now, changing variables y = (a/ ac − b2 )(hNN + b/a) shows that in general Z ∞ Z ∞ dhNN dy p−1 2 1/2−p = a (ac − b ) , 2 p (ah + 2bh + c) (1 + y 2 )p NN −∞ −∞ NN assuming p > 12 , a > 0, ac − b2 > 0. But here we have that ac − b2 = 1, while a is positive and independent of G. Thus (2.60) gives P (G) ∝ det(1N−1 + G2 )−β(N−2)/2−1 , which is the required result.
E XERCISES 2.5
1. The objective of this exercise is to derive the identity d dK (1 − K)−1 = (1 − K)−1 (1 − K)−1 , da da
where 1 denotes the identity operator and it is assumed the operator K is a smooth function of a. (i) Write (1 − K)−1 as a power series in K and use the differentiation formula n−1 d n X j dK n−j−1 K = K K da da j=0
to show
∞ n−1 X X j dK n−j−1 d K . (1 − K)−1 = K da da n=1 j=0
(ii) Change summation labels (n, j) → (l, p), where l = j and p = n − j − 1, to obtain the stated result.
2.6 ORTHOGONAL AND SYMPLECTIC UNITARY RANDOM MATRICES The compact classical groups [540] are the general unitary matrices U (N ), together with their restriction to real and real quaternion elements. Restricting the elements to be real gives the subgroup O(N ), while restricting the 2 × 2 sub-blocks in 2N × 2N unitary matrices to be real quaternion gives matrices equivalent to the unitary symplectic matrices Sp(2N ). While the set of all general unitary matrices coincides with the CUE, the real orthogonal matrices and unitary symplectic matrices do not make up the COE and CSE. Rather members of the COE are diagonalized by real orthogonal matrices, and members of the CSE are diagonalized by symplectic equivalent matrices. Nonetheless, as will be discussed in Chapters 7 and 10, there are applications requiring the eigenvalue distribution of all three classical groups, so we must address the problem of computing the eigenvalue p.d.f. for O(N ) and Sp(2N ). Orthogonal matrices can be divided into two classes, O+ (N ) and O− (N ), according to their determinants equaling +1 or −1 respectively. For orthogonal matrices R in O+ (N ) it is possible to introduce the Cayley transformation 1N + iA (2.61) R= 1N − iA for some antisymmetric Hermitian matrix A (both R and A have 12 N (N − 1) independent elements). It follows from (2.61), and the fact that the eigenvalues of A come in ± pairs, that the eigenvalues of R come in complex conjugate pairs e±iθj (for N odd +1 is also an eigenvalue). Using (2.61) and proceeding as in Section 2.5 we find (see Exercises 2.6 q.1) the eigenvalue p.d.f. for matrices R is equal to 2 2 2(cos θ − cos θ ) (2.62) k j (N/2)!(2π)N/2 1≤j
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CHAPTER 2
for N even and 1 ((N − 1)/2)!(2π)(N −1)/2
(N −1)/2
2(1 − cos θj )
2 2(cos θk − cos θj )
(2.63)
1≤j
j=1
for N odd, where the normalizations follow from Exercises 5.5 q.4 below. It is of interest to interpret the results (2.62) and (2.63) in the context of the calculation of Section 1.2.4. Consider for definiteness (2.62). For an orthogonal matrix there are 21 N (N − 1) independent elements. Now for each eigenvalue pair eiθj , e−iθj there is an eigenvector pair uj , u ¯j . With Θ the diagonal matrix formed from eiθ1 , e−iθ1 , . . . , eiθN/2 , e−iθN/2 the independent elements in δU2 are the strictly upper triangular entries, of which there are 12 N (N − 1), excluding the N/2 entries (δU2 )2j−1 2j = uTj · duj = 0, where the second ¯ j . In the analogue of (2.22) equality follows from the fact that U2 is unitary with columns of the form uj , u the entries would otherwise contribute the factor eiθj − e−iθj , so the analogue of (2.23) reads
T
(R dR) =
N/2
|e
iθj
−e
| |e
iθk 2
iθj
−e
−iθk 2
|
dθj (U†2 , dU2 ),
(2.64)
j=1
1≤j
thus implying (2.62). Consider now members of O− (N ). In the case of N odd, the eigenvalues are as in the case O+ (N ) except that θ → π − θ (this maps the eigenvalue +1 in O+ (N ) to an eigenvalue −1 in O− (N )). Hence from (2.63) the eigenvalue p.d.f. is equal to 1 ((N − 1)/2)!(2π)(N −1)/2
(N −1)/2
2(1 + cos θj )
(2(cos θk − cos θj ))2 .
(2.65)
1≤j
j=1
In the case of N even, members of O− (N ) contain the pair of eigenvalues ±1. Arguing as in the derivation of (2.64) shows the eigenvalue p.d.f. is equal to N/2−1 1 |1 − eiθj |2 |1 + eiθj |2 (N/2 − 1)!(2π)N/2−1 j=1
|eiθj − eiθk |2 |eiθj − e−iθk |2 .
(2.66)
1≤j
where the normalization again follows from Exercises 5.5 q.4 below. It remains to consider the case of unitary symplectic equivalent matrices, that is, 2N × 2N unitary matrices satisfying the additional relation UZ2N UT = Z2N
(2.67)
constraining the elements to be real quaternion. The constraint (2.67) together with the unitarity requirement means that there are N (2N + 1) independent variables. We see from (2.67) that if eiθ is an eigenvalue with eigenvector v , then e−iθ is an eigenvalue with eigenvector Z2N v , thus implying that in the formalism of Section 1.2.4 there are 2N 2 independent elements in δU2 . With the eigenvalues ordered eiθ1 , . . . , eiθN , e−iθ1 , . . . , e−iθN , these can be taken as the real and imaginary parts of the elements (δU2 )jk and (δU2 )j N +k with 1 ≤ j < k ≤ N and 1 ≤ j ≤ k ≤ N , respectively, thus giving for the analogue of (2.23) the formula (U† dU) =
N j=1
|eiθj − e−iθj |2
1≤j
|eiθj − eiθk |2 |1 − ei(θj +θk ) |2
N
dθj (U†2 dU2 ).
(2.68)
j=1
Hence, with 0 ≤ θj ≤ π (j = 1, . . . , N ) the corresponding eigenvalue p.d.f. is precisely the N/2 − 1 → N
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CIRCULAR ENSEMBLES
case of (2.66), and thus equal to N 1 |eiθk − e−iθk |2 N !(2π)N k=1
|eiθj − eiθk |2 |1 − ei(θj +θk ) |2 .
(2.69)
1≤j
An alternative way to derive (2.69) is to note that unitary symplectic equivalent matrices can be constructed out of quaternion real matrices Q with the property that iQ is Hermitian via the Cayley transformation U=
12N − Q . 12N + Q
(2.70)
Changing variables as in Section 2.5, deriving the analogue of (2.71) below and making use of the result of Exercises 1.3 q.8 gives (2.69). The eigenvalue p.d.f.’s computed above for the orthogonal and symplectic groups of random matrices all have interpretations as the Boltzmann factor for certain log-gas systems. This is made explicit at the end of Section 2.9. E XERCISES 2.6
1. The objective of this exercise is to derive the eigenvalue p.d.f.’s (2.62) and (2.63) for orthogonal matrices with determinant equal to +1. (i) In general, for an antisymmetric Hermitian matrix A, Y (dA) = dAijk , j
where the superscript i denotes the imaginary part (the real part is zero). From this, use the method of the proof of Proposition 1.2.5 to show that for a general real N × N matrix B, (BT dA B) = (ii) Deduce from (2.61) that
“
det(BT B)
”(N−1)/2
(dA).
(2.71)
RT dR = 2i(1 + iA)−1 dAT (1 + iAT )−1
(cf. (2.56)), and then use (2.71) to show from this that ”−(N−1)/2 “ (dA). (RT dR) = 2N(N−1)/2 det(1 + A2 ) (iii) Use the result of (ii) together with the formulas of Exercises 1.3 q.5 to write down the eigenvalue p.d.f. for (RT dR) in terms of the eigenvalues λ1 , . . . , λ[N/2] of A. Change variables according to (2.61) to derive (2.62) and (2.63).
2.7 LOG-GAS SYSTEMS WITH PERIODIC BOUNDARY CONDITIONS At the end of Section 2.2.4 the p.d.f. (2.20) was interpreted as the Boltzmann factor for a log-gas system on a circle. An equivalent interpretation is that the log-gas is defined on a line with periodic boundary conditions. To see this, suppose the line is in the x-direction and is of length L. To specify a two-dimensional Coulomb system in this setting, for the pair potential we seek the solution of the Poisson equation (1.40) subject to the semiperiodic boundary condition Φ((x + L, y), (x , y )) = Φ((x, y), (x , y )). Using the further facts that the solution must depend only on x − x and y − y , and that for r ∼ r , Φ(r, r ) ∼ − log |r − r | (which is the solution in free boundary conditions (1.41) with l = 1), we see from the theorem of elementary complex analysis asserting that the real part of an analytic function satisfies Laplace’s equation that Φ(r, r ) = − log | sin(π(x − x + i(y − y ))/L)|(L/π) . (2.72)
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CHAPTER 2
With the particles confined to the segment [0, L] of the x-axis, this reduces to L L = − log |e2πix/L − e2πix /L | , − log | sin(π(x − x )/L)| π 2π thus revealing the equivalence to (1.41) with r, r confined to a circle. In this section three different log-gas systems with periodic boundary conditions will be introduced. Two of these appear in different contexts in subsequent chapters; all three have solvability properties analogous to log-gas systems with unitary symmetry, as to be discussed in Chapter 5. 2.7.1 Transverse semiperiodic boundary conditions Suppose that for a log-gas system interacting via the pair potential (2.72), instead of the particles being confined to the line segment [0, L] in the x-direction, they are confined to the full line in the y-direction. Up to an additive constant, the pair potential is then − log | sinh(π(y − y )/L)|, and if the particles are restrained from repelling to infinity by an attractive harmonic potential, the Boltzmann factor is then of the form N j=1
2
e−βc yj /2
| sinh(π(yk − yj )/L)|β ,
−∞ < yj < ∞.
(2.73)
1≤j
We will see in Section 10.1.5 that (2.73) with β = 1 occurs in the theory of non-intersecting paths. In the case β = 2 the corresponding partition function occurs in Chern-Simon field theory [385, 514]. The quadratic term in (2.73) results from the particle-background interaction ∞ c y 2 + C = ρb (x) log | sinh(π(y − x)/L)| dx. (2.74) −∞
Analogous to the situation with (1.47), this equation cannot be solved for y → ∞, as each side is then of a different order. For x and y confined to a finite interval, reintroducing the variable λ = e−2πy/L the equation (2.74) can be solved according to the method of Section 1.4. However it is simpler to construct an asymptotic solution, valid for N → ∞. This solution is obtained by noting log | sinh(π(y − x)/L)| ∼ π|y − x|/L d2 for |y − x| → ∞, and that π|y − x|/L satisfies the one-dimensional Poisson equation dy 2 (π|y − x|/L) = 2
d (2π/L)δ(x − y). Thus operating on both sides of (2.74) with dy 2 we obtain the asymptotic solution ρb (x) = Lc /π, |x| < N π/2Lc . As an application of this result, the principle of local charge neutrality for Coulomb systems implies that for fixed y and large N the leading order density of the quantities {yj } in (2.73) will be the constant Lc /π for |x| < N π/2Lc and zero otherwise. We remark that changing variables xj = e−2π(yj −y0 )/L , for suitable y0 , shows that (2.73) can be written in the form N l=1
w(xl )
|xk − xj |β ,
(2.75)
1≤j
with w(x) = e−βc(log x) , 2
x > 0.
(2.76)
Interpreted as an (abstract) eigenvalue p.d.f, this form has been used [107] in studies of conductance (see (3.97) below) in regimes for which there is an insulator to conductor transition. 2.7.2 Metal wall Suppose that in addition to periodic boundary conditions in the x-direction, a perfect conductor occupies the region y < 0. This means that the pair potential satisfies Poisson’s equation (1.40) subject to the boundary
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CIRCULAR ENSEMBLES
condition Φ((x, y), (x , y )) = 0 for y = 0. According to the method of images the sought solution is sin(π(x − x + i(y − y ))/L) , Φ(r, r ) = − log sin(π(x − x + i(y + y ))/L) which has the interpretation of there being an image charge of opposite sign created at (x , −y ). Because of the image effect of the metal wall, from a log-gas perspective, it makes sense to impose a uniform background charge density −η, independent of the particle density. The Boltzmann factor for a system of N particles confined to the line y = d is thus specified as follows. P ROPOSITION 2.7.1 From the log-gas system near a metal wall as specified, mobile charges q = 1, the Boltzmann factor is N β/2 sin(π(x − x )/L) β 2 π k j e−β(πη dL−2πηN ) (2.77) . L sinh 2πd/L sin(π(xk − xj + 2id)/L) 1≤j
Proof. The total potential energy consists of the particle-particle energy ˛ ˛ log ˛
X
U1 = −
1≤j
the self-energy
˛ sin(π(xk − xj )/L) ˛ ˛; sin(π(xk − xj + 2id)/L)
U1 U1 :=
N ” “ “ ” π 1X N lim Φ((xj , d), (x , d)) − log |xj − x|) = − log ; 2 j=1 x →xj 2 L sinh 2πd/L
the particle-background energy U2 = η
N Z X j=1
L
˛ ˛ log ˛
0
˛ sin(πx/L) ˛ ˛ dx = −2πdηN ; sin(π(x + 2id)/L)
and the background-background energy U3 = πη 2 dL.
Note that for d → 0 (2.77) becomes independent of {xj }, while for d → 0 the Boltzmann factor of Proposition 1.4.1 is reclaimed. Thus for general d there in an interpolation between a perfect gas and a onecomponent log-gas. For β = 2 we will see in Exercises 11.6 q.2 that the p.d.f. corresponding to (2.77) can be derived from a particular random matrix model.
2.7.3 Doubly periodic boundary conditions The pair potential Φ with doubly periodic boundary conditions, period L and W , must obey the charge neutral Poisson equation ∂2Φ ∂2Φ 2π + = −2πδ(x − x )δ(y − y ) + 2 2 ∂x ∂y LW
(2.78)
(note the integral of the right-hand side over the fundamental rectangle is 0; without this charge neutrality condition a doubly periodic solution is not possible). To solve (2.78) introduce the θ1 function defined by
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CHAPTER 2
[541] ∞
θ1 (z; q) = −i
2
(−1)n q (n−1/2) e2i(n−1/2)z
n=−∞
= 2q 1/4 sin z
∞ ! ! ! 1 − q 2n e2iz 1 − q 2n e−2iz 1 − q 2n .
(2.79)
n=1
With q = eiπτ , Im(τ ) > 0 the facts that θ1 is an entire function, θ1 (z; q) = 0 if and only if z = πm + πτ n, m, n ∈ Z, and θ1 (z; q) ∼ zθ1 (0; q) as z → 0, tell us that ˜ z ) := − log L|θ1 (π(z − z )/L; q)| , q := e−πW/L (2.80) Φ(z, πθ1 (0; q) satisfies the Poisson equation (1.40) for 0 ≤ x, x < L, 0 ≤ y, y < W , with the further specification that ˜ z ) ∼ − log |z − z | as |z − z | → 0. Since Φ(z, θ1 (z + π; q) = −θ1 (z; q) and θ1 (z + πτ ; q) = −q −1 e−2iz θ1 (z; q)
(2.81)
we see that (2.80) has the periodicity properties ˜ ˜ y), (x , y )), Φ((x + L, y), (x , y )) = Φ((x, π ˜ ˜ y), (x , y )). Φ((x, y + W ), (x , y )) = − (2y + W ) + Φ((x, L It follows that the potential πy 2 ˜ z ) + Φ(z, (2.82) LW is doubly periodic, and satisfies the charge neutral Poisson equation (2.78). For a system confined to the x-axis the quadratic term in (2.82) does not contribute, and the pair potential is given by (2.80). Using this, we find for the Boltzmann factor of a one-component system of N unit charges confined to the interval [0, L], in the presence of a smeared out neutralizing background, the expression ∞ πθ (0; q) N β/2 −βN 2 /2 π(x − x ) β k j 1 ;q . q 1/4 (1 − q 2n ) (2.83) θ1 L L n=1 Φ(z, z ) :=
1≤j
2.8 CIRCULAR β-ENSEMBLE 2.8.1 Hessenberg form For non-Hermitian matrices, Householder similarity transformations analogous to (1.139) cannot give a tridiagonal form. Instead what results is a Hessenberg form, in which all entries are zero below the first subdiagonal. By introducing into the conjugations a suitable diagonal unitary matrix, the entries on the first sub-diagonal can be chosen to be real and positive. Thus one can construct a sequence of unitary matrices V(1) , . . . , V(N −1) and diagonal unitary matrices D(1) , . . . , D(N −1) such that for a general complex matrix W = [wij ]i,j=1,...,N , Y(N −1) = D(N −1)† V(N −1)† · · · D(1)† V(1)† WV(1) D(1) · · · V(N −1) D(N −1) is of the sought Hessenberg form. The matrices V(j) have the reflector structure of (1.140), 0j×N −j 1j×j (j) (j) (j)† , = V = 1 − 2v v 0N −j×j UN −j×N −j
(2.84)
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CIRCULAR ENSEMBLES
where v (j)†v (j) = 1 and UN −j×N −j is unitary. The entries of v (j) can be expressed iteratively, starting with V(1) . For this set β wl1 (1) (1) (1) vl = − (l ≥ 3), (2.85) v2 = , v1 = 0, c c where, with l := |w21 |2 + · · · + |wN 1 |2 , √ w21 |w21 | 1/2 |w21 | 1/2 β = −√ 1− , c = 2l 1 − . l l 2|w21 | ˜ (1)† := V(1)† WV(1) has Noting that then v (1)† w (1) = −c/2 shows Y ˜ (1)† )l1 = 0 (l ≥ 3). ˜ (1)† )21 = w21 l, ˜ (1)† )11 = w11 , (Y (Y (Y |w21 |
(2.86)
This is the first column of a Hessenberg form, but with a complex subdiagonal entry. The phase w21 /|w21 | = ˜ (1) D(1) , where eiφ can be cancelled by forming the conjugation Y(1) := D(1)† Y D(1) = diag[1, eiφ , 1, . . . , 1].
(2.87)
The matrices V(j) (j = 2, . . . , N − 2), D(j) (j = 2, . . . , N − 1) are defined by the formulas (2.85), (2.87), (j) (j) the former modified so that v1 = · · · = vj = 0 and the matrix elements wl1 replaced by those in the first column of the bottom right N − j + 1 × N − j + 1 submatrix of Y(j−1) , and the latter modified so that the phase is in position j + 1 on the diagonal. For j = N − 1 we take V(N −1) = 1. In the case that W is unitary, the Hessenberg form is special because the diagonal entries completely determine the matrix. Moreover, the remaining entries can be determined explicitly. P ROPOSITION 2.8.1 Set α−1 := −1, and suppose |αN −1 | = 1. For j = 0, . . . , N − 2 let αj be complex numbers with |αj | < 1, and put ρj = 1 − |αj |2 . The Hessenberg matrix H = [Hij ]i,j=0,...,N −1 with diagonal entries Hii = −αi−1 α ¯ i and subdiagonal entries Hi+1,i = ρi is unitary if ¯j Hij = −αi−1 α
j−1
ρl ,
i < j.
l=i
Proof. One checks that
N−1 X i=0
|Hij |2 = 1,
N−1 X
¯ ij = 0 (j = j ). Hij H
i=0
A special feature of the unitary Hessenberg matrix H as specified in Proposition 2.8.1 is a coupled recurrence satisfied by the characteristic polynomial [271]. P ROPOSITION 2.8.2 Let Hk denote the top k × k block of H, and write χk (λ) = det(λ1k − Hk ). We have ¯ k−1 χ ˜k−1 (λ), χk (λ) = λχk−1 (λ) − α ˜k−1 (λ) − λαk−1 χk−1 (λ) χ ˜k (λ) = χ
(2.88) k
˜0 (λ) = 1. Furthermore χ ˜k (λ) = λ χ ¯k (1/λ) (here χ ¯k denotes the (k = 1, . . . , N ), where χ0 (λ) = χ polynomial χk with its coefficients replaced by their complex conjugates). ¯ j αN −1 (j = 0, . . . , N − 2), the bottom k × k submatrix, Note that with the replacements αj → −α after reflection in the anti-diagonal, becomes identical to the top k × k submatrix but with αj → αN −2−j
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CHAPTER 2
(j = 0, . . . , N − 2). Hence the corresponding characteristic polynomial, χbk (λ) say, satisfies the recurrences ¯ N −1−k α ¯ N −1 χ ˜bk−1 (λ), χbk (λ) = λχbk−1 (λ) + α χ ˜bk (λ) = χ ˜k−1 (λ) + λαk−1 α ¯ N −1−k αN −1 χ ˜bk−1 (λ)
(2.89)
˜b0 (λ) = 1 and χ ˜bk (λ) = λk χ ¯bk (1/λ). (k = 1, . . . , N ), where χb0 (λ) = χ For random unitary matrices, the distribution of the parameters {αk } occurring in Propositions 2.8.2 and (2.89) can be determined explicitly [356], thus allowing the corresponding eigenvalues to be calculated as the zeros of χN (λ), without the need to diagonalize a matrix. This requires introducing a particular distribution of complex numbers. D EFINITION 2.8.3 For ν > 1 and complex numbers |z| < 1, let Θν denote the distribution of complex numbers with p.d.f. ν−1 (1 − |z|2 )(ν−3)/2 , 2π while for complex numbers |z| = 1 let Θ1 denote the uniform distribution. We remark that in the case ν > 1, in terms of z = reiθ , 0 ≤ r < 1, 0 ≤ θ < 2π, Θν is such that θ (ν−3)/2 has uniform distribution, while s := r2 has distribution ν−1 . Also, analogous to the result of 2 (1 − s) Exercises 1.2 q.2(ii), one notes the marginal distribution of a single component of a vector on the complex (N − 1)-sphere, chosen uniformly at random, or equivalently a single entry of a random element of U (N ), has distribution Θ2N −1 . P ROPOSITION 2.8.4 The unitary Hessenberg matrix, with positive elements on the subdiagonal, obtained by applying the augmented Householder transformations (2.84) to a random element of U (N ), has parameters {αj−1 }j=1,...,N distributed according to αN −j−1 ∈ Θ2j+1 (j = 0, . . . , N − 1). Proof. According to Proposition 2.8.1, H00 = α¯ 0 , while (2.86) gives that H11 = w11 , where w11 = W11 , W ∈ U (N ).
As remarked above, the latter has distribution Θ2N−1 . In the construction of H, after determining the first column, we must apply the algorithm to the bottom right N − 1 × N − 1 submatrix of Y (1) . In particular, we seek (Y(1) )22 . The (N − 2)-sphere, constrained to be orthogonal second column of Y (1) can be regarded as a random point on the complex p to the first column of Y (1) , and thus orthogonal to the vector (w11 , 1 − |w11 |2 , 0, . . . , 0). This gives the structure p ¯11 z, w32 , . . . , wN2 )T , ( 1 − |w11 |2 z, −w where (z, w32 , . . . , wN2 ) is a random point on the complex (N − 2)-sphere, telling us that (Y(1) )22 = −w ¯11 z = −α0 z, ¯ 1 = H11 = (Y (1) )22 so α1 ∈ Θ2N−3 . z ∈ Θ2N−3 . But −α0 α In general, with the first k − 1 columns in Hessenberg form, the k-th column of Y (k−1) will have the structure (ρ0 ρ1 · · · ρk−2 z, −α0 ρ1 · · · ρk−2 z, −α1 ρ2 · · · ρk−2 z, . . . , −αk−3 ρk−2 z, −αk−2 z, wk+1 k , . . . , wNk )T where (z, wk+1 k , . . . , wNk ) is a random point on the complex (N − k)-sphere, this being the most general unit vector ¯ k−1 = orthogonal to the first k − 1 columns. Consequently (Y(k−1) )kk = −αk−2 , z ∈ Θ2N−2k+1 . As −αk−2 α Hk−1 k−1 it follows that αk−1 ∈ Θ2N−2k+1 as required.
The situation is now analogous to that which arose after obtaining the tridiagonal matrix of Proposition 1.9.1. Thus for the unitary Hessenberg H of Proposition 2.8.1 we would like to know how to directly change variables from the parameters α = (α0 , . . . , αN −1 ) to λ = (λ1 , . . . , λN ) and q = (q1 , . . . , qN ). Here the iθj λj = e are the eigenvalues of H and the qi the modulus of the first component of the corresponding normalized eigenvectors. The latter must therefore satisfy (1.144). In preparation for the change of variables, we must establish the analogue of (1.148).
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CIRCULAR ENSEMBLES
P ROPOSITION 2.8.5 For the Hessenberg matrix H of Proposition 2.8.1, N −2 (1 − |αl |2 )N −1−l |λi − λj |2 = l=0 N . 2 j=1 qj 1≤i<j≤N
(2.90)
Proof. See Exercises 2.8 q.1. P ROPOSITION 2.8.6 The Jacobian for the change of variables from α to (λ, q) is equal to N −2 2 i=0 (1 − |αi | ) . N qN i=1 qi
(2.91)
Proof. [223] We follow the strategy of Proposition 1.9.3, starting with (1.157) (T → H therein). Equating successive powers of λ on both sides (recalling the explicit form of the matrix elements from Proposition 2.8.1) gives N X
1=
qj2 ,
α ¯0 =
j=1
N X
qj2 λj ,
N X
∗+α ¯ 1 ρ20 =
j=1
∗+α ¯ 2 ρ20 ρ21 =
N X
qj2 λ2j ,
qj2 λ2j ,
j=1
...,
∗+α ¯ N−1 ρ20 ρ21 · · · ρ2N−2 =
j=1
N X
qj2 λN j ,
(2.92)
j=1
where ∗ denotes terms involving only variables already having appeared on the l.h.s. of the preceding equations, thus implying a triangular structure. Recalling that αj (j = 0, . . . , N − 2) have independent real and imaginary parts, while αN−1 := eiφ , λj := eiθj (j = 1, . . . , N ) have unit modulus, we see the number of equations can be made equal to the 2 in the subsequent equations, then forming the complex number of variables by using the first equation to eliminate qN conjugate of all these equations but the last. Taking differentials, then wedge products, of both sides of these 2N − 1 equations shows ρ20 ρ21 · · · ρ2N−2
N−2 Y
ρ4l α ∧ dφ N−l−2 d
l=1
2 = qN
N−1 Y j=1
˛ 2 » – ˛ λjk − λjN ˛ −j 6 ˛ λ−j qj3 ˛det 4 k − λN j,k=1,...,N−1 ˛ N ˛ [λN k − λN ]k=1,...,N−1
»
jλjk −jλ−j k
– j=1,...,N −1 k=1,...,N
[N λN k ]k=1,...,N
3˛ ˛ ˛ 7˛ q. 5˛ dθ ∧ d ˛ ˛
(2.93)
Now it is straightforward to check that the above determinant is a symmetric function of λ1 , . . . , λN , which is homo−(2N−3) geneous of degree N . The highest negative power of λ1 is λ1 , so it must be of the form 1 p(λ1 , . . . , λN ), 2N−3 l=1 λl
QN
where p is a symmetric polynomial of λ1 , . . . , λN of degree 2N (N − 1). We note too that the determinant Q vanishes when j ∂ ) (j = 1, 2, 3). The polynomial p must thus contain as a factor λ1 = λ2 , as does its derivatives (λ1 ∂λ 1≤j
0, sgn(x) := −1 for x < 0). The coefficient of λ−2N+7 1 cofactor coming from multiplying together the (2N − 3, 1) and (2N − 5, N − 1) elements. Proceeding in this manner
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CHAPTER 2
Q 4(l−1)−2N+3 we see that the coefficient of N is (−1)(N−1)(N−2)/2 in the determinant, so in fact the determinant is l=1 λl equal to (2.94) times this sign. Substituting in (2.93), recalling |λl | = 1 and making use of (2.90) gives (2.91).
Knowledge of the Jacobian (2.93) allows a unitary Hessenberg matrix to be specified for which the eigenvalue p.d.f. is distributed as for the circular β-ensemble [356]. P ROPOSITION 2.8.7 Consider the matrix H of Proposition 2.8.1 with parameters {αj−1 }j=1,...,N distributed according to αN −j−1 ∈ Θβj+1
(j = 0, . . . , N − 1).
The eigenvalues and first component of the eigenvectors (which form q) are independent, with the distribution of the eigenvalues given by 1 Γ(βN/2 + 1) |eiθk − eiθj |β d θ, Cβ,N = (2π)N Cβ,N Γ(β/2 + 1) 1≤j
and the distribution of q by (1.161). Proof. Denote the unitary Hessenberg matrix by Hβ , and its joint distribution by P (Hβ ). We have P (Hβ )(dHβ ) =
N−1 1 Y βj α ∧ dφ (1 − |αN−j−1 |2 )βj/2−1 d 2π j=1 2π
=
N−2 Y 1 β N−1 (N − 1)! (1 − |αl |2 )β(N−l−1)/2 d θ ∧ d q Q N (2π)N qN i=1 qi l=0
=
N qiβ−1 β N−1 (N − 1)! πi=1 N (2π) qN
Y
|eiθj − eiθk |β d θ ∧ d q,
1≤j
where the second equality uses (2.91) and the third (2.90). After recalling the normalization in (1.161), and dividing by N ! to effectively eliminate the ordering of the eigenvalues, we have the stated result.
E XERCISES 2.8
1. The objective of this exercise is to derive (2.94).
(i) Use the analogue of (1.147) and follow the derivation of (1.152) to show ˛ Qn ˛ b |χ (λ )| ˛ n i n−1 i=1 Y 2 αj →−α ¯ j αn−1 Q qi = , 2 |λ − λ i j| 1≤i<j≤n i=1 thus reducing the task to that of showing n Y
|χbn−1 (λi )| =
i=1 (p)
(ii) With λj
n−2 Y
(1 − |αl |2 )n−1−l .
(2.95)
l=0 (k)
denoting the jth zero of χbp (λ), substitute λ = 1/λj ¯ (k) ) = χbk (1/λ j
in (2.89) to show
1 ¯ (k) ), (1 − |αn−k−1 |2 )χbk−1 (1/λ j (k) ¯ λj
and then introduce the factorizations χbk−1 (x) =
k−1 Y i=1
(k−1)
(x − λi
),
χ ˜bk (x) =
k Y i=1
(k)
¯ ) (1 − xλ i
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CIRCULAR ENSEMBLES
to deduce
k Y
(k)
¯ ) = (1 − |αn−k−1 |2 )k χbk (1/λ i
i=1
k Y
(k)
¯ )k (1/λ i
i=1
k−1 Y
(k−1)
χ ˜bk−1 (λj
).
j=1
Similarly derive the equation k Y
(k)
χ ˜bk (λi ) = (1 − |αn−k−1 |2 )k
i=1
k−1 Y
(k−1) k−1
¯ (λ j
)
¯ (k−1) ). χbk−1 (1/λ j
j=1
(iii) Show that the final two equations of (ii) imply k Y
¯ (k) )k χbk (1/λ ¯ (k) ) = (λ i i
i=1
k−1 Y
(1 − |αn−l |2 )l+1 .
l=0
(k)
¯ ) using the first equation in (ii) and set k = n to deduce (2.95). Substitute for χbk (1/λ i
2.9 REAL ORTHOGONAL β-ENSEMBLE The real orthogonal and symplectic unitary random matrices can be transformed to upper Hessenberg form. Consider in particular a member of O+ (2N ). Because the elements are real, the parameters {αj }j=0,...,2N −1 are all real, and because the determinant is equal to +1, α2N −1 = −1 (this can be seen by setting α0 = · · · = α2N −2 = 0). Following the strategy of the proof of Proposition 2.8.4 allows the distribution of the remaining αj ’s to be determined explicitly [356]. ˜ β], is specified by the p.d.f. D EFINITION 2.9.1 The beta distribution on (−1, 1), denoted B[α, 1 Γ(α + β) 1 − x α−1 x + 1 β−1 . 2 Γ(α)Γ(β) 2 2 P ROPOSITION 2.9.2 The real orthogonal Hessenberg matrix, with positive elements on the subdiagonal, obtained by applying the augmented Householder transformations (2.84) to a random element of O+ (2N ), has parameters {αj }j=0,...,2n−2 distributed according to ˜ 2N − j − 1 , 2N − j − 1 . αj ∈ B 2 2 This result has the immediate significance of implying that the eigenvalue p.d.f. for O+ (2N ) can be sampled by computing the zeros of the polynomial χ2n (λ) as determined by (2.88) (because each χk is real, χ ¯k (λ) = λk χk (1/λ) and so only the first equation in (2.88) is required). Furthermore it suggests a βgeneralization analogous to Proposition 2.8.7. First, the analogue of Propositions 2.8.5 and 2.8.6 must be noted. In preparation, we note that for a general real orthogonal upper Hessenberg 2N × 2N matrix, there are 2N − 1 independent real parameters α0 , . . . , α2N −2 . In the corresponding eigendecomposition, there are N independent variables qj (j = 1, . . . , N ) where 12 qj2 is the square of the first component of both the ¯j . eigenvalues λj and λ P ROPOSITION 2.9.3 For a 2N ×2N real orthogonal Hessenberg matrix H of determinant +1, parametrized
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CHAPTER 2
in terms of the real parameters {αi }i=0,...,2N −2 , |αi | < 1, we have 2N −2 N 2 (2N −1−l)/2 1 2 2 N l=0 (1 − αl ) |λi − λj | |λi − 1/λj | = 2 , λi − N 2 λi i=1 qi i=1
(2.96)
1≤i<j≤N
N
|1 − λj |2 = 2
2N −2
N
k=0
j=1
(1 − αk ),
j=1
|1 + λj |2 = 2
2N −2
(1 + (−1)k αk ).
(2.97)
k=0
Proof. Analogous to (1.157) we have ((I2N − λH)−1 )11 =
n ” 1 X 2“ 1 1 qj + . ¯ 2 j=1 1 − λλj 1 − λλj
(2.98)
Since the r.h.s. is equal to χb2n−1 (1/λ)/λχb2n (1/λ) it follows that ˛ χb ˛ 1 2 ˛ 2n−1 (λj ) ˛ ˛ ˛ = qj χ2n (λj ) 2
(j = 1, . . . , 2n),
¯ j , qj+N = qj . Taking the product over j = 1, . . . , 2n, making use of (2.95), then taking the square root where λj+N = λ gives (2.96). For the results (2.97), note n Y
|1 − λj |2 = χ2n (1),
j=1
n Y
|1 + λj |2 = χ2n (−1).
j=1
But from (2.88) χk+1 (λ)|λ=±1 = (λ − αk λk )χk (λ)|λ=±1 .
P ROPOSITION 2.9.4 For a real orthogonal upper Hessenberg 2N × 2N matrix of determinant +1, the Jacobian for the change of variables from α to ( θ, q) is equal to 2N −2 2 2N −1 l=0 (1 − |αl | ) . (2.99) N 2N −2 qN i=1 qi k=0 (1 − αk )1/2 (1 + (−1)k αk )1/2 Proof. Expanding (2.98) in powers of λ, analogous to (2.92) we obtain 1=
N X
qj2 ,
α0 =
j=1
N 1X 2 ¯ j ), q (λj + λ 2 j=1 j
∗ + α2N−2 ρ20 · · · ρ22N−3 =
N 1 X 2 2 ¯2 q (λ + λj ), . . . , 2 j=1 j j
∗ + α1 ρ20 =
N 1 X 2 2N−1 ¯ 2N−1 qj (λj + λj ). 2 j=1
2 , taking differentials then wedge products of both sides, shows Using the first of these equations to substitute for qN 2N−3 Y
2(2N−2−l)
ρl
d α
l=0 2 = 2−N qN
N−1 Y j=1
˛ h ˛ j −j qj3 ˛˛det [λjk + λ−j k − (λk + λk )] j=1,...,2N −1 k=1,...,N −1
i˛˛ ˛ q. [j(λjk − λ−j k )] j=1,...,2N −1 ˛ dθ ∧ d k=1,...,N
The determinant is in fact equal to N Y j=1
(λj − 1/λj )
Y 1≤j
(λk − λj )2 (1/λk − 1/λj )2 (λj − 1/λk )2 (1/λj − λk )2 .
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CIRCULAR ENSEMBLES
To see this, note that it is a symmetric rational function of λ1 , . . . , λN , and is antisymmetric under the mapping λi → 1/λi for any i = 1, . . . , N . It must thus be of the form N Y
(λj − 1/λj ) q(λ1 , . . . , λN )
(2.100)
j=1
where q is symmetric and unchanged by the mapping λi → 1/λi . Noting too that the determinant vanishes when λ1 = λ2 , we see that q must contain as a factor Y (λk − λj )2 (1/λk − 1/λj )2 (λj − 1/λk )2 (1/λj − λk )2 . (2.101) 1≤j
The highest order term in degree of (2.101) multiplied by the first factor in (2.100) is The highest order term in degree in the determinant is h i det [λjk − λjN ] j=1,...,2N −1 [jλjk ] j=1,...,2N −1 . k=1,...,N −1
QN j=1
λj
Q
1≤j
− λ j )4 .
k=1,...,N
According to (1.175), up to a sign this is equal to the same expression, so in fact q must equal (2.101) up to this sign. Using (2.96) to write the evaluation in terms of {αi }, {qi } gives the result.
The above results allow a real orthogonal upper Hessenberg 2N × 2N matrix with unit determinant, possessing an eigenvalue p.d.f. which β-generalizes that for O+ (2N ), to be specified [356]. P ROPOSITION 2.9.5 Consider the upper Hessenberg matrix of Proposition 2.9.2, but with ˜ 2N − k − 2 β + a + 1, 2N − k − 2 β + b + 1 , k even, αk ∈ B 4 4 2N − k − 3 2N − k − 1 ˜ αk ∈ B β + a + b + 1, β , k odd, 4 4 k = 0, . . . , 2N − 2. The eigenvalues and first component of the eigenvectors (which form q) are independent, with the distribution of the former given by 1 |1 − eiθl |2a+1 |1 + eiθl |2b+1 CN (a, b; β)
N
l=1
|eiθj − eiθk |β |1 − ei(θj +θk ) |β ,
(2.102)
1≤j
where the normalization is given by CN (a, b; β) = 2(2a+2b+2)N +βN (N −1) SN (a, b, β/2), SN (λ1 , λ2 , λ) =
N −1 j=0
Γ(λ1 + 1 + jλ)Γ(λ2 + 1 + jλ)Γ(1 + (j + 1)λ) . Γ(λ1 + λ2 + 2 + (N + j − 1)λ)Γ(1 + λ)
(2.103)
Proof. With Hβ denoting the Hessenberg matrix, we see P (Hβ )(dHβ ) = KN (a, b; β) ×
2N−2 Y k=0
(1 − α2k )−1+β(2N−k−1)/4
2N−2 Y k=0
(1 − αk )a+1−β/4 (1 + (−1)k αk )b+1−β/4 d α,
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where 2N−2 Y
KN (a, b; β) = 2−(2N−1)
k=0 k even
×
2N−3 Y k=1 k odd
Γ((2N − k − 2)β/2 + a + b + 2)2−(2N−k−2)β/2−a−b Γ((2N − k − 2)β/4 + a + 1)Γ((2N − k − 2)β/4 + b + 1)
Γ((2N − k − 2)β/2 + a + b + 2)2−(2N−k−2)β/2−a−b Γ((2N − k − 3)β/4 + a + b + 1)Γ((2N − k − 1)β/4) Q2N−2
= 2−σ QN−1 r=0
p=N−1
Γ(pβ/2 + a + b + 2) QN−2
Γ(rβ/2 + a + 1)Γ(rβ/2 + b + 1)
s=0
Γ((s + 1)β/2)
,
with σ := ((N − 1)β/2 + a + b + 1)(2N − 1). Changing variables using the results of Propositions 2.9.3 and 2.9.4 gives (2.102) with 1 1 −Nβ/2−(a+b+1−β/2) ΓN (β/2) = 2 KN (a, b; β), CN (a, b; β) N! Γ(βN/2) and this can readily be written in the form (2.103).
The p.d.f. (2.102) permits a log-gas interpretation. P ROPOSITION 2.9.6 Consider a log-gas system of unit charges confined to a half circle 0 < θ < π. Let the coordinates of the particles be θ1 , . . . , θN and suppose there are image charges at −θ1 , . . . , −θN . Suppose too that there are fixed particles at θ = 0, π of charge (2a + 1)/β − 21 , (2b + 1)/β − 12 , respectively. The Boltzmann factor is then proportional to (2.102). Proof. For the log-gas on a half circle with like image charges, the pair potential is
Φ(θ, θ ) = − log |eiθ − eiθ ||1 − ei(θ+θ ) |. The total potential energy then consists of the particle-particle energy X log |eiθj − eiθk ||1 − ei(θj +θk ) |, U1 = − 1≤j
the self energy U1 :=
N N “ ” 1X 1X lim Φ(θ, θj ) − log |eiθ − eiθj | = − log |1 − eiθj ||1 + eiθj | 2 j=1 θ→θj 2 j=1
and the particle-fixed particle energy Uf = −
“ 2a + 1 β
“ 2b + 1 1”X 1”X log |1 − eiθj | − log |1 + eiθj |. − 2 j=1 β 2 j=1 N
−
Forming e−β(U1 +U1 +Uf ) gives (2.102).
N
Chapter Three Laguerre and Jacobi ensembles A Hermitian random matrix X can be formed out of a rectangular Gaussian matrix in the top right block, and its Hermitian conjugate in the bottom left block, with zeros elsewhere. This structure, which defines the chiral ensembles, can be motivated by the consideration of Dirac operators in the context of quantum chromodynamics, and time reversal symmetry distinguishes the cases of real, complex and real quaternion elements. The positive eigenvalues of matrices from the chiral ensembles are the singular values of X, or equivalently the nonzero eigenvalues of X† X. The ratio of the largest to smallest singular value is precisely the condition number of the linear system associated with X. Eigenvalues of the matrix product X† X, in the case that X relates to a data matrix in multivariable statistics, are of basic importance to the method of principal component analysis. With X Gaussian, this matrix product is said to be a Wishart matrix. The eigenvalue p.d.f.’s for the chiral ensembles and Wishart matrices can be calculated using the method of wedge products, or metric forms. For certain values of the parameter α these same eigenvalue p.d.f.’s appear in the study of random matrix models of Hamiltonians for electron and hole wave functions at normal metal/superconductor junctions. The Gaussian ensembles of Chapter 1, the chiral ensembles, and these further random matrix models of Hamiltonians together form a list of ten, which can be identified with the ten infinite families of matrix Lie algebras. Also studied are Jacobi ensembles, defined by a family of eigenvalue p.d.f.’s with each eigenvalue supported on [−1, 1]. Realizations for β = 1, 2 and 4 in terms of Wishart matrices, unitary matrices corresponding to compact symmetric spaces, and singular values of block decompositions of unitary matrices are given. Motivated by an identity between canonical averages, a circular analogue of the Jacobi ensemble is defined. In the last three sections, random matrix realizations of the Laguerre, Jacobi and circular Jacobi ensembles are given for general β > 0.
3.1 CHIRAL RANDOM MATRICES 3.1.1 Random Dirac operators A random matrix theory of Dirac operators, in the context of quantum chromodynamics (QCD), has been introduced by Verbaarschot [529] (for a review see [530]). More precisely, one is considering a massless Dirac particle coupled to a random gauge field, which has Hamiltonian iγμ (∂μ + iAμ ) (with summation over repeated indices implicit). As is well known, the γ-matrices γ1 , . . . , γ4 anticommute with the matrix γ5 , so the nonzero eigenvalues of the massless Dirac operator occur in pairs ±λ, corresponding to the eigenfunctions ψ and γ5 ψ. Now consider a basis, referred to as the chiral basis, which consists of eigenvectors of iγ5 . Since (iγ5 )2 = 1 these eigenvectors have eigenvalue either +1 or −1, and so in forming a matrix representation of the Hamiltonian out of such a basis, the matrix elements between states with the same eigenvalue of γ5 must vanish as γ5 anticommutes with the Hamiltonian, leaving a block structure with nonzero elements in the upper-right and lower-left blocks only. Furthermore, the application to QCD requires that the Dirac operator has a given number, ν say, of zero eigenvalues (this determines the topological charge). A matrix structure consistent with these facts is given by the following result.
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P ROPOSITION 3.1.1 The matrix
H=
0n×n X†
X 0m×m
,
(3.1)
where X is an n × m (n ≥ m) matrix, has in general n − m zero eigenvalues and the remaining eigenvalues given by ± the positive square roots of the eigenvalues of X† X. Proof. Write the eigenvalues of H in block form so that » – » – ψn ψn =λ . H φm φm This is equivalent to the coupled equations m = λψ n , Xφ
m. n = λφ X† ψ
m by −φ m and λ by −λ leaves these equations unchanged, showing that the eigenvalues come in ± pairs. Replacing φ n = λ2 ψ n . But the n × n matrix XX† has rank m, and so has n − m Furthermore, the coupled equations imply XX† ψ zero eigenvalues with the remaining eigenvalues equal to those of the full rank matrix X† X.
Since this result holds independent of the details of X a random matrix hypothesis can be made: the statistical properties of the eigenvalues of the Dirac operator in the QCD problem will be the same as those of a generic matrix of the form (3.1), subject only to the constraints imposed by time reversal symmetry. Since H is Hermitian, the latter are the same as those of a nonrelativistic Hamiltonian. Thus, from Section 1.1, if the Dirac operator has a time reversal symmetry with T 2 = 1, then X can be chosen to have real elements, while if the time reversal symmetry is such that T 2 = −1 with T = Z2(n+m) K, then X must be quaternion real. A Gaussian distribution on the elements of X can be distinguished by the maximum entropy property of Proposition 1.1.3. Due to their origin in studying the Dirac equation with a chiral basis, random matrices of this type have become known as chiral random matrices. D EFINITION 3.1.2 Let X denote an n × m (n ≥ m) random matrix, and suppose the elements of X are determined by a parameter β = 1, 2 or 4. These elements are real, complex or real quaternion independent random variables with Gaussian densities 2 1 1 −|zjk |2 2 −2|zjk |2 2 −2|wjk |2 √ e−xjk /2 , e e e , and π π π 2π in the three cases β = 1, 2 and 4 respectively (recall from (1.20) that a real quaternion is specified by two complex numbers z and w). Use X to form the matrix H according to (3.1). The resulting ensembles of matrices are referred to as the chiral orthogonal ensemble (β = 1), chiral unitary ensemble (β = 2) and chiral symplectic ensemble (β = 4). According to the singular value decomposition, any n × m (n ≥ m) matrix X can be written as X = UΛV† ,
(3.2)
where Λ is an n × m diagonal matrix containing the m positive square roots of the eigenvalues of the matrix X† X, and U and V are m × m and n × n unitary matrices, respectively. This decomposition can be used to determine the eigenvalue p.d.f. of the chiral ensembles [529]. First, one notes that the constraint of real elements (β = 1) and real quaternion elements (β = 4), implies that X = O1 ΛOT2 ,
X = B1 ΛZ2n BT2 Z−1 2n
(3.3)
for O1 , O2 real orthogonal (β = 1) and B1 , B2 unitary symplectic equivalent (β = 4), respectively. The metric forms approach of Section 1.2.4 can be used to calculate the eigenvalue p.d.f.
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P ROPOSITION 3.1.3 The p.d.f. of the positive eigenvalues of the chiral ensembles is proportional to m
−βλj /2 λβα j e 2
j=1
|λ2k − λ2j |β ,
α =n−m+1−
1≤j
1 . β
(3.4)
Proof. Consider first the real case (β = 1), for which the singular value decomposition is given by the first equation
in (3.3). Since X has nm independent elements, and there are m nonzero elements in Λ, the matrices O1 and O2 must together have a total of m(n − 1) independent elements (or combinations of elements). We begin by computing the differential of the singular value decomposition to obtain OT1 dXO2 = δO1 Λ + dΛ − ΛδO2 , where δO := OT dO. Substituting this result and its transpose in Tr(dXdXT ) and simplifying using the cyclic property of the trace and the identities ΛdΛT − dΛ ΛT = 0n×n , shows
ΛT dΛ − dΛT Λ = 0m×m
“ ” Tr(dXdXT ) = Tr − (δO1 )2 ΛΛT − (δO2 )2 ΛT Λ + 2ΛδO2 ΛT δO1 + dΛdΛT ” “ = Tr (ΛδO2 − δO1 Λ)(ΛT δO1 − δO2 ΛT ) + dΛdΛT .
(3.5)
Writing this result in component form, using the fact that δO1 , δO2 are real antisymmetric, gives ”2 ”2 X 1“ X 1“ [δO2 ]kl − [δO1 ]kl (λk + λl )2 + [δO2 ]kl + [δO1 ]kl (λk − λl )2 Tr(dXdXT ) = 2 2 1≤k
+
m “ X
1≤k≤l≤m
[δO2 ]kk − [δO1 ]kk
”2
λ2k +
k=1
n m X X
(δO1 )2kl λ2k +
k=1 l=m+1
m X
(dλk )2 .
(3.6)
k=1
The fact that δO1 and δO2 are real antisymmetric also implies [δO1 ]kk = [δO2 ]kk = 0,
(3.7)
and thus there are a total of m(n − 1) differentials in (3.6) involving elements of δO1 and δO2 , which as noted above is the number required in the singular value decomposition to be contributed by O1 and O2 . These differentials must therefore all be independent, so using (1.14) we read off the factors in (3.4) in the case β = 1 excluding the exponential (the terms involving the exponential result from the Gaussian measure). Consider next the complex case (β = 2). The appropriate singular value decomposition is now given by (3.2). Writing U = U1 , V = U2 in that formula and setting δUi = U†i dUi we see that (3.6) is unchanged if we write δOi → δUi and replace the squares of all quantities which are now complex by their absolute value squared. The formula for the Jacobian in (3.4) then follows by noting [δUi ]jk , j < k has an independent real and imaginary part, while [δUi ]jj has one independent (pure imaginary) component. In the quaternion case (β = 4) the matrix elements [δUi ]jk , j < k, have four independent components, while [δUi ]jj has three independent components.
3.1.2 Singular values and the condition number A fundamental question in numerical analysis relates to the sensitivity of the computed solution to the input data. If an error of measure in the input produces an error of measure c in the computed solution, with c around unity, the problem is said to be well conditioned. However, if c is orders of magnitudes greater than unity, the input error is significantly amplified and the problem is said to be illconditioned. The proportionality c is termed the condition number. To see how these ideas apply in a matrix setting, consider the task of computing the solution x to the N ×N
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linear system Xx = b.
(3.8)
Let |x| denote the usual (complex) Euclidean norm, and define the corresponding matrix norm (the so-called 2-norm) by |X| = sup|x|=1 |Xx|. Some immediate consequences of the definitions are the inequalities |Xx| ≤ |X||x|,
|A||B| ≤ |AB|.
(3.9)
It is also true that |X† X| = |X|2 . From this latter fact, by writing the vector in the definition of |X† X| in terms of the eigenvectors of the Hermitian matrix X† X, it follows that 1/2
|X| = λ1
= μ1 ,
(3.10)
†
where λ1 is the largest eigenvalue of X X and μ1 is the largest singular value of X. We seek to quantify the effect of perturbing either the matrix X or the vector b in (3.8). Suppose first that the matrix X is perturbed by the addition of δX, and use the matrix norm to measure the size of the perturbation according to = |δX|/|X|. With y the solution of the perturbed system (X + δX)y = b,
(3.11)
subtracting (3.11) from (3.8) and using the inequalities (3.9) show |x − y| ≤ κ(X), |y|
κ(X) := |X||X−1 | =
μ1 . μN
(3.12)
Here, with μN denoting the smallest singular value of X, the final equality follows from (3.10). Similarly, for a perturbation δb in the vector b, defining = |δb|/|b|, κ(X) as in (3.12), and with y such that Xy = b + δb, a straightforward calculation shows |y − x| ≤ κ(X). |x|
(3.13)
For X a complex Gaussian matrix, and for X a real Gaussian matrix, the distribution of the condition number κ(X) is calculated in Exercises 8.3 q.3 and Exercises 13.2 q.5, respectively.
3.1.3 Relationship to Lie algebras The matrices of Definition 3.1.2 are the Hermitian part of the matrix Lie algebras " X1 X2 so(p, q) = , all X real, X p × p, X q × q, both skew symmetric , i 1 3 XT2 X3 " Z1 Z2 , all Z complex, Z p × p, Z q × q, both anti-Hermitian , u(p, q) = i 1 3 Z†2 Z3 " Q1 Q 2 , all Qi real quaternion, Q1 p × p, Q3 q × q, both anti-Hermitian , sp(p, q) = Q†2 Q3 with p = n, q = m. In view of the relationship between the Lie algebras of Section 1.3.4 and the symmetric spaces of Section 2.2.3, we would expect that there is a corresponding theory of random unitary matrices. This is indeed the case, as will be considered in Section 3.7.
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3.1.4 Coulomb gas analogy Written in the form m λ2j /2 − α log |λj | − β exp − β j=1
log |λk − λj | + log |λk + λj |
(3.14)
1≤j
we see that the eigenvalue p.d.f. (3.4) has the interpretation as the Boltzmann factor for a one-component log-potential system of unit charges on the half line x > 0, with image charges of like sign in the region x < 0. The charges are confined by a one-body harmonic potential as in the log-gas picture of the p.d.f. of Proposition 1.3.4, and there is also a fixed charge of strength α at the origin. Proceeding as in the proof of Proposition 1.4.4 (see Exercises 3.1 q.1) we can specify the background charge density in the Boltzmann factor of this log-potential system. P ROPOSITION 3.1.4 The Boltzmann factor of the one-component √ log-potential Coulomb system with particles of unit charge at x , . . . , x , confined to the interval [0, 4N ], with image charges of the same sign 1 N √ in the interval [− 4N, 0], a fixed charge of strength α − 12 at the origin, and a background charge density neutralizing the mobile charges √ 4N y2 −ρb (y) = − 1− π 4N is proportional to N
|xj |βα e−βxj /2 2
j=1
|x2k − x2j |β .
1≤j
N In obtaining this result the fact that the finite portion of the self energy term 12 j=1 log |xj | must be included in the total energy has been used (recall the proof of Proposition 2.7.1). The background density again has a semicircle profile, although the independent particles are restricted to x > 0. E XERCISES 3.1
1. For a one-component log-gas of unit charges, with image forces as in (3.14) and subject to a one-body potential V (x), the background charge density −ρb (x) satisfies the integral equation Z a V (x) + C = ρb (t) log |x2 − t2 | dt, x ∈ (0, a), 0
Z
where a is such that
a
ρb (t) dt = N. 0
Show that this is equivalent to the integral equation Z a ρb (|t|) log |x − t| dt, V (x) + C = −a
x ∈ (−a, a),
and so deduce the background density in Proposition 3.1.4 from that in Proposition 1.4.4. 2. [492] Consider the matrix H in (3.1) in the case n = m with X a symmetric complex matrix. (i) Show that the singular value decomposition of X must be of the form X = UΛUT for U unitary. (ii) Follow the proof of Proposition 3.1.3 to show (dX) =
Y 1≤j
|λ2k − λ2j |
n Y j=1
λj dλj (U† dU).
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3.2 WISHART MATRICES 3.2.1 Setting in multivariate statistics In multivariate statistics there may be m variables yk (k = 1, . . . , m), with each variable measured n times. (It is typical in this setting to denote the number of different variables by p (for population) rather than m, but we will persist with m.) For example, yk may represent the noise level at location k at 6 a.m. on a weekday. (j) Measuring the noise levels at this time on n different weekdays gives vectors of data yk := [yk ]j=1,...,n for each variable yk , which in turn can be used to define a data matrix Y := [yk ]k=1,...,m . We remark that another convention has the vectors of data as row vectors, giving then our YT as the data matrix; see, e.g., [279]. The average of the measurements for variable yk is 1 (j) y . n j=1 k n
y¯k :=
With the average data vector for variable yk defined by y¯k = [¯ yk ]j=1,...,n , and the average data matrix by ¯ := [y¯k ]k=1,...,m , the matrix product Y n 1 (j) (j) ¯ T (Y − Y) ¯ = 1 (Y − Y) (y − y ¯ )(y − y ¯ ) , k k 1 2 k2 n∗ n∗ j=1 k1 k1 ,k2 =1,...,m
with n∗ := n − 1, then represents an empirical approximation to the covariance matrix (yk1 − y¯k1 )(yk2 − y¯k2 ) k1 ,k2 =1,...,m
for the variables yk . The eigenvalue-eigenvector decomposition of YT Y is of basic importance in the analysis of the covariance matrix. This comes under the name of principal component analysis (see, e.g., [410]), in which one considers the eigenvectors corresponding to the largest, second largest and subsequent eigenvalues as giving orthogonal linear combinations of variables which account for the successive maximum variations in the data. This then allows the statistically important linear combination of variables to be identified, and so effectively reduces the dimension of the problem. Our interest is in the theoretical setting that the variables yk are drawn from a multivariate Gaussian distribution with variance Σ and mean μ. Then a well-known result (see, e.g., [279]) gives that the distribution of YT Y is the same as the distribution of A := XT X,
(3.15)
where X is an m × n Gaussian matrix with elements of each row drawn from an m dimensional Gaussian distribution having variance Σ but mean zero. We define ensembles of matrices relating to the matrix structure (3.15) according to the following definition. D EFINITION 3.2.1 With n × m random matrices X specified as in Definition 3.1.2 in the three cases β = 1, 2 and 4, define the real (β = 1), complex (β = 2) and quaternion real (β = 4) Wishart ensembles as consisting of matrices X† X. The matrices X† X are referred to as (uncorrelated) Wishart matrices (for the correlated case see Section 3.5). As already remarked, the square roots of the nonzero eigenvalues of (3.15) are the singular values of X, or equivalently the positive eigenvalues of (3.1). The eigenvalue p.d.f. for matrices from the Wishart ensembles can therefore be written down from knowledge of the eigenvalue p.d.f. for the positive eigenvalues of the matrices H as given in Proposition 3.1.1, after the change of variables λ2j → λj in the latter.
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P ROPOSITION 3.2.2 The eigenvalue p.d.f. for the real (β = 1), complex (β = 2) or quaternion real (β = 4) Wishart matrices is given by m
1 Waβm
βa/2 −βλl /2
λl
e
|λk − λj |β ,
λl ≥ 0,
(3.16)
1≤j
l=1
where a = n − m + 1 − 2/β and the normalization constant is given explicitly in (3.134) below. Due to the one body factors of the form λβa/2 e−βλ/2 (3.16) is said to define the Laguerre ensemble of random matrices, or more accurately of eigenvalue p.d.f.’s. In particular, the case β = 1 is referred to as the Laguerre orthogonal ensemble (LOE), the case β = 2 is referred to as the Laguerre unitary ensemble (LUE), while the case β = 4 is referred to as the Laguerre symplectic ensemble (LSE). Note that the corresponding Wishart ensembles realize these eigenvalue p.d.f.’s. Changing variables y 2 → y, x2j → xj in the result of Proposition 3.1.4 gives the log-gas analogy of the Laguerre ensemble p.d.f. (3.16). P ROPOSITION 3.2.3 The Boltzmann factor of the one-component log-potential system with particles of unit charge at x1 , . . . , xN , confined to the interval [0, 4N ] with a background charge density neutralizing the mobile charges −ρb (y) = −
1 (4N − y)1/2 , 2πy 1/2
and with a fixed particle of charge (a − 1)/2 + 1/β at x = 0 is proportional to N
βa/2 −βxj /2
xj
j=1
e
|xk − xj |β .
(3.17)
1≤j
3.2.2 Wireless communication systems At a theoretical level the problem of maximizing the information transfer between antennas and receivers in wireless communication systems relates to complex Wishart matrices [523]. Thus practical methods to achieve maximum rates [489] make use of multiple antennas (MT , say) to transmit distinct bitstreams (signals), which are decoded by multiple receivers (MR , say, with MR ≥ MT ). For the decoding to be possible in the case MR = MT it is necessary that a distinct linear combination of the original signals be received at each antenna, the physical mechanism for which is scattering of the signal as it travels between antenna and receiver. This setting is described by the equation y = Gx + z, where the jth component of y is the received signal at receiver j, while the j-th component of x is the signal sent by transmitter j. The linear combination of transmitted signals received at each antenna is determined by the propagation matrix G, while the vector z denotes the noise at the receivers due to the presence of extraneous signals. The information capacity I measures the number of bits per second per frequency which can be transfered between input at the antenna and output at the receiver. In the case of a single antenna and receiver, this is given in terms of the signal to noise ratio u := |x|/|z| by Shannon’s formula I = log2 (1 + u). In the case of multiple antennas and receivers Shannon’s formula generalizes to I˜ = Tr log(1 + u ˜G† G) =
MT j=1
log(1 + u˜λj ),
(3.18)
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where I˜ := (log 2)I, u ˜ := MR |x|2 /MT |z |2 and the λj denote the eigenvalues of G† G. Because the propagating signals are subject to random scattering, it is reasonable to model G as a random matrix with complex Gaussian entries. If furthermore there is no correlation between the random scattering of the components, we then have that G† G is a complex Wishart matrix. With respect to the corresponding eigenvalues I˜ is then an example of a linear statistic (see Definition 14.3.1 below). In an appropriate scaled limit the distribution of general linear statistics are Gaussians with means given in terms of the eigenvalue density, and O(1) variances (see Sections 14.3 and 14.4). 3.2.3 The distribution of Wishart matrices For X real in (3.15), the task of expressing the volume form (dA) in terms of (dX) was considered long ago by Wishart [547]. The result of this calculation, which we will carry out below in the complex case, can be used to rederive Proposition 3.2.2 for the eigenvalue p.d.f. of the matrices A. We will follow the presentation in [410], generalizing the working to the complex case (only the real case is considered in [410]). According to Definition 3.2.1, we are given that the joint probability distribution of the elements of the n × m complex matrix X is m n
1 π nm
†
e−|zjk | (dX) ∝ e−Tr(X 2
X)
(dX).
(3.19)
j=1 k=1
With A := X† X the strategy is to first use the Gram-Schmidt orthogonalization procedure to write X = U1 T,
(3.20)
U†1 U1
= 1m and T is a m × m upper triangular matrix with diagonal where U1 is an n × m matrix such that entries real and positive. This allows (dX) to be calculated in terms of (dU1 ) and (dT). Noting from (3.20) and the definition of A that A = T† T we then calculate (dT) in terms of (dA). Substituting the result of this second calculation into the first gives (dX) in terms of (dA) as required. Proceeding as in the above outline, we must relate (dT) and (dA). This requires a preliminary result (cf. (1.17)). P ROPOSITION 3.2.4 The Jacobian of the transformation z = Aw, where w, z and A have complex entries, is | det A|2 . Proof. See Exercises 3.2 q.2. This result will be used to establish the relation between volume forms implied by (3.20). P ROPOSITION 3.2.5 With U1 and T defined by (3.20) we have (dX) =
m
2(n−j)+1
tjj
(dT)(U†1 dU1 ),
j=1
where the tjj are the diagonal entries of the matrix T. Proof. Since X = U1 T we have
dX = dU1 T + U1 dT.
Now extend the number of columns of U1 from m to n by defining an (n − m) × n matrix U2 such that U = [U1 U2 ] and U† U = 1. Then # " # " # " U†1 dU1 T + dT U†1 U†1 (dU1 T + U1 dT) = , dX = U† dX = U†2 U†2 (dU1 T + U1 dT) U†2 U1 T where to obtain the last equality the facts U†1 U1 = 1 and U†2 U1 = 0 have been used.
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Consider the above equation. On the l.h.s., from Proposition 3.2.4 we have (U† dX) = | det U† |2m (dX) = (dX), which is the l.h.s. of the assertion. On the r.h.s. consider the matrix product ( u†j d u1 · · · u†j d um )T
(3.21)
U†2 U1
T. The jth row is
(m + 1 ≤ j ≤ n).
Again using Proposition 3.2.4, the wedge product of the elements in this row is | det T|2
m ^
u†j · d uk .
k=1
Hence, the wedge product of all the elements in the matrix product is n ^
m ^
| det T|2
j=m+1
n ^
†j · d u uk = | det T|2(n−m)
m ^
u†j · d uk .
j=m+1 k=1
k=1
It remains to consider the matrix U†1 dU1 T + dT. From U†1 U1 = 1 we see U†1 dU1 = −(U†1 dU1 )† , so the matrix U†1 dU1 is skew symmetric Hermitian. We therefore have 2 6 6 U†1 dU1 T = 6 4
u†1 · d u1 t11 u†2 · d u1 t11 .. . u1 t11 u†m · d
∗ · d u2 t22 .. . u2 t22 + ∗ u†m · d u†2
... ... .. . ...
∗ ∗ .. . u†m · d um tmm
3 7 7 7, 5
where the terms in which u†k · d uj and ( u†k · d uj )† occur are ignored, as indicated by ∗, if u†k · d uj has already appeared † in a previous column. We note that the diagonal terms uj · d uj tjj are pure imaginary. If we now add dT, we see that the wedge product of the resulting matrix is equal to m Y
2(m−i)+1)
tii
i=1
m ^ m ^
u†j · d ui (dT).
i=1 j=1
Hence the wedge product of all the elements on the r.h.s. is | det T|2(n−m)
m Y i=1
Since det T =
Qm
i=1 tii
n ^
2(m−i)+1)
tii
m ^
u†j · d uk
j=m+1 k=1
m ^ m ^
†j · d u ui (dT).
i=1 j=1
and n ^
m ^
†j · d u uk
j=m+1 k=1
m ^ m ^
u†j · d ui =
i=1 j=1
n m ^ ^
u†j · d ui = (U†1 dU1 )
i=1 j=1
this is precisely the r.h.s. of the assertion.
Note Proposition 3.2.5 gives that the elements of T are all independently distributed, while the dependence on U1 is of the form (U†1 dU1 ). Thus in the case n = m the matrices U1 are Haar distributed, and so can be used to generate members of the CUE [157], [402]. The next result can be used to express (dA) in terms of (dT). P ROPOSITION 3.2.6 Let the m × m matrix T be as in (3.20), so that A = T† T. We have (dA) = 2m
m j=1
t2m+1−2j (dT). jj
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Proof. Since A = T† T we have dA = dT† T + T† dT. Hence 2 6 6 dA = 6 4
2t11 dt11 ∗ .. . ∗
t22 dt12 + ∗ 2t22 dt22 .. . ∗
... ... .. . ∗
tmm dt1m + ∗ tmm dt2m + ∗ .. . 2tmm dtmm
3 7 7 7, 5
where * denotes differentials that have already appeared and thus do not contribute to the wedge product. Taking the wedge product of the above elements gives the stated result.
Combining Propositions 3.2.5 and 3.2.6 we can obtain the p.d.f. of A. This result was first given in [269], and is the complex analogue of the classical result obtained by Wishart [547] in the real case. We will include in the statement the analogous result for the real and quaternion real cases, the derivation of which is given in Exercises 3.2 q.4 and q.5 P ROPOSITION 3.2.7 Let A be a real (β = 1), complex (β = 2) or quaternion real (β = 4) Wishart matrix as defined in Definition 3.2.1. The p.d.f. of A is 1 −(β/2)Tr(A) e (det A)(β/2)(n−m+1−2/β) , ˆ Cβ N
(3.22)
where Cˆβ N is a normalization constant. Proof. We are considering the case β = 2. Substituting the result of Proposition 3.2.6 in Proposition 3.2.5 gives (dX) = 2−m (det A)n−m (dA)(U†1 dU1 ), (3.23) Q 2 where we have used the fact that det A = (det T)2 = m j=1 tjj . Substituting this expression for (dX) in the formula P 2 for the joint probability distribution of the elements of X (3.19), noting that TrA = m j,k=1 |xjk | , and integrating over the independent variables in U†1 dU1 gives the stated result.
In Exercises 3.2 q.6, a derivation of the A-dependent portion of the Jacobian in (3.23), valid for all three β values, is given by exploiting functional equations. The matrix A in Proposition 3.2.7 is Hermitian, with real (β = 1), complex (β = 2) or real quaternion elements (β = 4). Changing variables to the eigenvalues and eigenvectors according to the result of Proposition 1.3.4 gives the eigenvalue p.d.f. (3.16). 3.2.4 A matrix integral derivation An alternative derivation [322] of Proposition 3.2.7 in the case β = 2 makes use of a matrix integral evaluation [247]. P ROPOSITION 3.2.8 Let Im,n (Qm ) :=
−n i e 2 Tr(Hm Qm ) det(Hm − μ1m ) (dHm ),
(3.24)
where Hm , Qm are m × m Hermitian matrices, and suppose n ≥ m, Im μ > 0. For Qm positive definite (i.e. all eigenvalues positive) one has i n−m i 2m π m(m+1)/2 im (−1)m(m−1)/2 n Im,n (Qm ) = det Qm e 2 μTr(Qm ) 2 j=n−m+1 Γ(j) while Im,n (Qm ) vanishes if Qm has an eigenvalue less than or equal to zero.
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LAGUERRE AND JACOBI ENSEMBLES
Proof. We proceed in an analogous way to the proof of statement (b) in Proposition 2.5.1. Because the integral (3.24) is
invariant under the transformation Hm → Um Hm U−1 m for Um unitary, Im,n (Qm ) is a function of the eigenvalues of Qm only and so we can take Qm = diag[q1 , . . . , qm ]. Now introduce the decomposition
" Hm =
Hm−1 h†
#
h
,
hmm
where h is a vector of length m − 1 with complex entries, and proceed as in the derivation of (2.59) to show a := μ + h† (Hm−1 − μ1m−1 )−1h.
det(Hm − μ1m ) = det(Hm−1 − μ1m−1 )(hmm − a), This shows
Z Im,n (Qm ) =
i
(dHm−1 ) e 2 Tr(Hm−1 Qm−1 )
Z “ ”−n Z × det(Hm−1 − μ1m−1 ) (dh)
∞ −∞
i
dhmm e 2 qm hmm (hmm − a)−n .
Since by assumption Im(μ) > 0 we have Im(a) > 0, which allows the integral over hmm to be computed by closing the contour in the upper half-plane (for qm > 0) to give Z
∞ −∞
e 2 qm hmm 2πi “ iqm ”n−1 2i qm a dhmm = e . n (hmm − a) Γ(n) 2 i
On the other hand, closing the contour in the lower half-plane shows that for qm < 0 the integral vanishes. The next task then is to evaluate Z i (dh) e 2 qm a . (3.25) where Um−1 is a unitary matrix such that U−1 Changing variables h = U−1 m−1 w m−1 Hm−1 Um−1 = diag[h1 , . . . , hm−1 ] and applying Proposition 3.2.4 to deduce (dh) = (dw), separates (3.25) into 2(m − 1) one-dimensional integrals, Z
i i (dh) e 2 qm a = e 2 qm μ
m−1 Y Z ∞ l=1
i
= e 2 qm μ
−∞
m−1 Y “ l=1
Thus Im,n (Qm ) =
−
dwlr e
iqm 2
(wlr )2 (hl −μ)−1
Z
∞ −∞
dwli e
iqm 2
(wli )2 (hl −μ)−1
“ “ 2πi ”m−1 ” i 2π ” det Hm−1 − μ1m−1 . (hl − μ) = e 2 qm μ iqm qm
2π m i(−1)m−1 “ iqm ”n−m 2i qm μ e Im−1,n−1 (Qm−1 ). Γ(n) 2
Iterating this and noting I0,n (Q0 ) := 1 gives the stated result.
To make use of (3.24), note that with X an n × m complex Gaussian matrix, P (X) = and A = X† X we have
P (A) =
1 π mn
†
e−Tr(X
X)
,
δ(A − X† X)P (X)(dX).
(3.26)
Here δ(A − X† X) is equal to the product of one-dimensional delta functions over the independent real and
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imaginary parts of A. Writing each of these as a Fourier integral shows † 1 eiTr(H(A−X X)) (dH), δ(A − X† X) = (2π)m2
(3.27)
where H is an m × m Hermitian matrix. Substituting this in (3.26) and noting that −n † † e−Tr(X X) e−iTr(HX X) (dX) = π mn det(1 + iH) , which in turn is deduced from the fact that the integral is a function of the eigenvalues of H only and then separating the integration into a product of one-dimensional integrations, gives π 2mn eiTr(HA) P (A) = (dH). 2 (det(1 + iH))n (2π)m Now the result (3.24) can be applied, thus reclaiming (3.22). = Xφ| ψ and the property of the inner † ψ 1. (i) Use the definition of the adjoint φ|X † product φ|φ > 0 for φ = 0 to show that the eigenvalues of X X are non-negative.
E XERCISES 3.2
(ii) Consider the matrix product X† X with X an n × m matrix with n < m. Show that X† X = Y † Y where Y is an m × m matrix obtained from X by the addition of m − n rows of zeros. Hence show that X† X has m − n zero eigenvalues. (iii) By considering the corresponding characteristic polynomials, and making use of (5.26) below, show that in the setting of (ii) the nonzero eigenvalues of X† X and XX† are equal. 2. Here Proposition 3.2.4 will be established. = [uj + ivj ]j=1,...,N and A = [ajk + ibjk ]j,k=1,...,N , show that the (i) With z = [xj + iyj ]j=1,...,N , w equation d z = Adw can be rewritten as the real matrix equation » – » –» – [dxj ]j=1,...,N [ajk ]j,k=1,...,N −[bjk ]j,k=1,...,N [duj ]j=1,...,N = . [dyj ]j=1,...,N [bjk ]j,k=1,...,N [ajk ]j,k=1,...,N [dvj ]j=1,...,N (ii) To evaluate the determinant of the 2N ×2N matrix on the r.h.s. of the above equation, and thus the Jacobian, add i times the blocks in the bottom half to the blocks in the top half. Then subtract i times the blocks in the left half to the blocks in the right half so that the top right block is now the zero matrix. 3.
(i) Show that the problem of calculating the eigenvalue p.d.f. of the Wishart matrices in the case m = 1 is p P 2 2 equivalent to calculating the p.d.f. of βn β/2πe−βxj /2 . j=1 xj where xj has distribution (ii) Calculate the p.d.f. in (i), p(λ) say, according to the formula „ p(λ) =
β 2π
«βn/2 Z
∞
2
dx1 e−βx1 /2 . . .
−∞
Z
∞ −∞
βn “ ” X 2 dxβn e−βxβn /2 δ λ − x2j , j=1
and thus reclaim Proposition 3.2.2 in the case m = 1. 4. [410] The objective of this exercise is to compute the eigenvalue p.d.f. for real Wishart matrices. (i) Let X be a real n × m (n ≥ m) matrix. Suppose the Gram-Schmidt orthogonalization procedure has been used to write X = RT where R is an n × m real matrix such that RT R = 1m and T is an upper triangular m × m real matrix with positive diagonal entries. Use the method of Proposition 3.2.5 to show that (dX) =
m Y j=1
T tn−j jj (dT)(R dR).
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LAGUERRE AND JACOBI ENSEMBLES
(ii) Let A = TT T where T is as in (i). Use the method of the proof of Proposition 3.2.6 to show that (dA) = 2m
m Y
tm+1−j (dT). jj
j=1
(iii) Use the results of (i) and (ii) and Definition 3.2.1 to show that the p.d.f. of a real Wishart matrix is 1 − 12 Tr(A) (det A)(n−m−1)/2 , e C where C is a normalization constant, and then use (1.11) to express this p.d.f. in terms of the eigenvalues of A. 5. The objective of this exercise is to compute the eigenvalue p.d.f. for quaternion real Wishart matrices. (i) Let X be an n × m (n ≥ m) matrix with real quaternion elements. Suppose the Gram-Schmidt orthogonalization procedure has been used to write X = UT where U is an n × m matrix of real quaternions such that U† U = 1 and T is an upper triangular m × m matrix with diagonal entries positive real multiples of 12 and off-diagonal entries real quaternions. Use the method of the proof of Proposition 3.2.5 to show that (dX) =
m Y
4(n−j)+2
tjj
(dT)(U† dU).
j=1
(ii) Let A = T† T where T as in (i) above. Use the method of the proof of Proposition 3.2.6 to show that (dA) = 2m
m Y
4(m−j)+1
tjj
(dT).
j=1
(iii) Use the results of (i) and (ii) and Definition 3.2.1 to show that the p.d.f. of a quaternion real Wishart matrix is 1 −2Tr(A) (det A)2(n−m+1/2) , e C where C is a normalization constant and the operations Tr and det are not to include repeated eigenvalues, and then use (1.27) for β = 4 to express this p.d.f. in terms of the eigenvalues of A. 6. [433] Let the p.d.f. of the n × m matrix X (with real (β = 1), complex (β = 2), real quaternion (β = 4) elements) be of the form F (X† X). The p.d.f. of the elements of A = X† X is then h(A)F (A) for some h. The objective of this exercise is to determine h. (i) Let A = B† VB, where V is positive definite. Making use of (1.35), show that the p.d.f. of V is then F (B† VB)h(B† VB) det(B† B)(β/2)(m−1+2/β) .
(3.28)
(ii) Let X = YB, where Y is such that V = Y† Y. Use an appropriate generalization of Proposition 3.2.4 to show that the p.d.f. of Y is F (B† Y † YB)(det B† B)βn/2 and hence that of V is F (B† VB)h(B† B) det(B† B)βn/2 h(V).
(3.29)
(iii) Equate (3.28) and (3.29) with V = 1 to deduce h(B† B) = h(1)(det B† B)(β/2)(n−m+1−2/β) , and note that h(1) = c for some constant c to conclude h(A) ∝ (det A)(β/2)(n−m+1−2/β) .
(3.30)
7. Let Y, Z be m × m positive definite Hermitian matrices (denoted Y > 0, Z > 0) with real (β = 1), complex (β = 2) or real quaternion (β = 4) elements. By changing variables X = Z1/2 YZ1/2 , making use of (1.35), show Z 1 e−(β/2)Tr(YZ) (det Y)β(a−(m−1)−2/β)/2 (dY) = (det Z)−βa/2 , (3.31) C Y>0
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where C is such that both sides equal unity when Z = 1.
3.3 FURTHER EXAMPLES OF THE LAGUERRE ENSEMBLE IN QUANTUM MECHANICS The chiral ensemble, which from the viewpoint of its eigenvalue distribution is equivalent to the Laguerre ensemble, is motivated from a problem in quantum mechanics. Here some further problems in quantum mechanics are studied which lead to the Laguerre ensemble. As part of this four random Hamiltonians are isolated, which together with those forming the Gaussian and chiral ensembles, make up the ten Hermitian random matrix ensembles in correspondence with the ten matrix Lie algebras associated with infinite families of symmetric spaces. 3.3.1 Eigenvalue statistics of Wigner-Smith delay time matrix In this subsection it will be shown, following [102], that for the quantum cavity problem in Section 2.1.1 the distribution of the scaled reciprocal eigenvalues of ∂S −1/2 S (3.32) ∂E is given by the Laguerre ensemble (3.16) with a = N . Here E is the energy of the waves as they enter the cavity (for a long lead and fixed number of channels N the energy will to leading order be constant). The eigenvalues of QE , denoted τ1 , . . . , τN say, are referred to as the proper delay times, and are also the eigenvalues for the Wigner-Smith matrix Q = −iS−1 ∂S/∂E. A precise formulation of the problem requires a Hamiltonian approach to the coupled lead-cavity system [531]. This can be done by introducing a basis of states {|ai }i=1,...,N for the lead, and a basis of states {μj }j=1,...,M for the cavity (typically M N ). The Hamiltonian is then defined as ¯ ij μi | , |μj Wji ai | + |ai W |ai Eai | + |μj Hjj νj | + H= QE := −iS−1/2
i
j,j
j,i
where the matrix elements Hjj form a random Hermitian M × M matrix H with real (β = 1), complex (β = 2) or quaternion real (β = 4) elements, while the coupling constants form a fixed M × N matrix W. A key point is that for this H the corresponding N × N scattering matrix S can be calculated exactly as [380] S=
1N + iπW† (H − E)−1 W . 1N − iπW† (H − E)−1 W
In the simplest case W can be taken as proportional to the identity, √ ΔM W= 1M×N , π where Δ is the mean level spacing in the cavity. Furthermore it is assumed that the Hermitian matrix H is a Gaussian random matrix with (3.33) P (H) ∝ exp − βπ 2 TrH2 /4Δ2 M . Writing H − E = UM×M diag[E1 − E, . . . , EM − E]U†M×M , where the columns of U consist of the eigenvectors of H, gives ΔM 1N + iK , K= UN ×M diag (E1 − E)−1 , . . . , (EM − E)−1 (UN ×M )† . S= 1N − iK π
(3.34)
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LAGUERRE AND JACOBI ENSEMBLES
Here UN ×M denotes the matrix UM×M with the last M − N rows deleted. Since H has distribution (3.33) and N M , it follows that the distribution of the submatrix UN ×N is given by 1 P √ U ∝ exp − βTr UU† /2 (3.35) M (cf. Exercises 1.3 q.2). The formula (3.34), which expresses S as a function of E, can be used to compute the eigenvalue distribution of (3.32) [102]. But before undertaking this task we will present a matrix integration formula which is required in the course of the derivation. P ROPOSITION 3.3.1 Let X be a N × N random matrix with real elements (β = 1), complex elements (β = 2) or real quaternion elements (β = 4), and let B be an N × N Hermitian random matrix which is real (β = 1), complex (β = 2) or quaternion real (β = 4). Furthermore suppose the joint distribution of X and B is proportional to †
e−βTr(XX
)/2
δ(X†−1 BX−1 ),
where δ denotes the Dirac delta function. Then the distribution of the eigenvalues a1 , . . . , aN of A = XX† , defined up to a multiplicative constant as the eigenvalue dependent factor in −βTr(XX† )/2 δ(X†−1 BX−1 )(dB), e (3.36) is proportional to N
βN/2 −βaj /2
aj
e
j=1
|ak − aj |β .
(3.37)
1≤j
Proof. To perform the integration over B in (3.36) we make the change of variables X†−1 BX−1 = C which in turn requires the wedge product formula (1.35), (dB) = (X† dCX) = This shows that (3.36) reduces to †
e−βTr(XX
)/2
“
“
det(XX† )
det(XX† )
”β(N−1)/2+1
”β(N−1)/2+1
.
(dC).
(3.38)
But we know from our study of Wishart matrices that with A = XX† , (dX) ∝ (det A)β/2−1 (dA). Using this in (3.38) to deduce the distribution of A, and then changing variables to the eigenvalues and eigenvectors of A according to (1.27) gives the stated result.
P ROPOSITION 3.3.2 Let the eigenvalues of the matrix QE (3.32) be denoted τ1 , . . . , τN , and write γj = 1/τj . Then the p.d.f. of the γj is proportional to N l=1
βN/2 −βτH γl /2
γl
e
|γj − γk |β ,
j
where τH := 2π/Δ and β = 1, 2 or 4 according to the scattering matrix S being a symmetric, unitary or self-dual quaternion random unitary matrix. Derivation. Since the Haar form (dH S) has the invariance (2.11) it suffices to consider the neighborhood of S = −1N . According to (3.34) this means that N eigenvalues (at least) of H are almost degenerate with E, so that |E − Eμ | Δ, μ = 1, . . . , N . Hence in fact only this special subclass of Gaussian random matrices make up H. Ignoring the other
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M − N eigenvalues in (3.34), as well as the final M − N components of each eigenvector, gives ˜ 1N + iK , ˜ 1N − iK
˜ = ΔM UN×N diag[(E1 − E)−1 , . . . , (EN − E)−1 ]U† K N×N π π ˜ −1 , K ˜ −1 = ≈ −1N + 2iK UN×N diag[E1 − E, . . . , EN − E]U†N×N . ΔM
S≈
(3.39)
Introducing the scaled matrix Ψ := M 1/2 UN×N , we see from (3.35) that the joint distribution of the elements of Ψ is proportional to exp(−βTrΨΨ† /2). This distribution is invariant under the transformation Ψ → ΨO, where O is an orthogonal (β = 1), unitary (β = 2) or symplectic (β = 4) N × N random unitary matrix with uniform measure. ˜ := O diag(E − E1 , . . . , E − EN )O† , by the assumption that the {Eμ } are near degenerate eigenvalues Defining H ˜ has a uniform distribution near H ˜ = 0. Furthermore, (3.39) then from the Gaussian ensemble, we see from (3.33) that H reads ˜ −1 . S = −1N + (iτH /)Ψ†−1 HΨ Hence at S = −1N the distribution of ΨΨ† is proportional to Z † ˜ −1 ) (dH). ˜ e−βTrΨΨ /2 δ(Ψ†−1 HΨ According to Proposition 3.3.1 the corresponding eigenvalue distribution of ΨΨ† is given by (3.37). The stated result now follows since from (3.32) and (3.39), to leading order in M , QE = τH Ψ†−1 Ψ−1 and so the eigenvalues of ΨΨ† are equal to τH times the reciprocal of the eigenvalues of QE .
3.3.2 Normal metal–superconductor junctions In mesoscopic physics, a situation of interest is conductance through a normal metal–superconductor (NS) junction (see, e.g., [53]). At this junction the phenomenon of retroreflection (also known as Andreev reflection) can take place: an electron from the normal metal may be reflected at the junction as a hole with the same momentum but opposite velocity, and with the missing charge 2e absorbed as a Cooper pair by the superconducting condensate. The theoretical description of the NS junction is via the (matrix) Bogoliubov-deGennes equation h Δ u u , (3.40) = ¯ −hT −Δ v v where Δ = −ΔT (this is a requirement of Fermi statistics) while h = h† , and both Δ and h are 2N × 2N matrices. The vectors u and v represent the electron and hole wavefunctions (their dimension is 2N because for each electron (hole) there are an up and down state so u = (u+ , u− ) and v = (v+ , v− )). Here, following [14], the constraints on the matrix elements of h and Δ in (3.40) due to time reversal symmetry and spin rotation invariance will be considered. It turns out that for this problem the appropriate time reversal operator T is such that T 2 = −1 and has the special structure 1N 0N T = 12 ⊗ K, (3.41) −1N 0N where K denotes the complex conjugation operator. Thus ⎡ ⎤ ⎡ u+ u∗− ⎢ u− ⎥ ⎢ −u∗+ ⎥ ⎢ T⎢ ⎣ u+ ⎦ = ⎣ u∗− u− −u∗+
⎤ ⎥ ⎥, ⎦
LAGUERRE AND JACOBI ENSEMBLES
101
where ∗ denotes complex conjugate. Also, spin-rotation invariance is the requirement that the Hamiltonian matrix in (3.40), H say, commutes with the matrix σk ⊗ 1N 02N (3.42) 02N −σkT ⊗ 1N for each k = x, y, z, where the σk are the Pauli matrices of (1.22). There are four cases to consider, depending on the presence or absence of time reversal symmetry and spin-rotation invariance. (1) No time reversal symmetry or spin-rotation invariance. We have the general structure exhibited in (3.40), A B (3.43) H= ¯ −A ¯ −B ˜ := U0 HU−1 with with A = A† , B = −BT . Introducing the transformation H → H 0 1 12N 12N U0 = √ i1 −i1 2N 2N 2 ˜ =H ˜ T , and hence H ˜ is a Hermitian matrix with pure imaginary elements, or equiv˜ ∗ = −H shows that H alently equal to i times a real antisymmetric matrix. The number of independent elements is equal to the number of independent elements in A and B together, and thus equal to 8N 2 − 2N . This is the number of ˜ belongs to independent elements in a general 4N × 4N antisymmetric Hermitian matrix, so we conclude H this class. (2) Time reversal symmetry, no spin-rotation invariance. For H to commute with T we require that h and Δ have the block structure Δ+− Δ++ h++ h+− h= , Δ = . ¯ +− hT −h −ΔT+− −Δ†++ ++ Rearranging the blocks via an appropriate similarity transformation gives a matrix of the form (3.43) but with the elements of A and B, now real quaternions, and thus consist of N × N lots of 2 × 2 blocks of the form (1.20). The matrices A and B must also be Hermitian and antisymmetric respectively as in (3.43). As in case (1) a further similarity transformation can be made to obtain an antisymmetric matrix, but now with a 2 × 2 sub-block structure of real quaternions in which the entries are pure imaginary. Using a similarity transformation to rearrange appropriate rows and columns shows the latter can be written in the form A B , A = −AT , B = −BT , (3.44) B −A with the elements of A and B pure imaginary. The eigenvalue problem for this matrix is equivalent to the eigenvalue problem for 02N W , (3.45) W† 02N where W is a 2N × 2N antisymmetric matrix with complex elements (see Exercises 3.3 q.1). Also shown in Exercises 3.3 q.1 is the fact that the eigenvalue equation for (3.45) is equivalent to the eigenvalue problem for 02N D , (3.46) D† 02N where D is a 2N × 2N self-dual matrix with complex elements. (3) Spin-rotation invariance, no time reversal invariance.
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The requirement that H commutes with the matrix (3.42) restricts h and Δ to have the block structure 0N Δ+− h++ 0N , Δ= , h= 0N h++ −Δ+− 0N where h++ = h†++ and Δ+− = ΔT+− . Thus H consists of two commuting sub-blocks. Concentrating on one of these gives the structure A B Hr = ¯ (3.47) ¯ B −A with A = A† , B = BT . Multiplying (3.47) by i and using a similarity transformation to rearrange appropriate rows and columns we see iHr = Q,
Q = −Q† ,
(3.48)
where Q is quaternion real. (4) Spin-rotation invariance and time reversal invariance. The constraints of (2) and (3) together imply that the structure of (3) holds with h++ and Δ+− real symmetric. Thus the reduced Hamiltonian (3.47) results with A and B real symmetric. We show in Exercises 3.3 q.2 that the eigenvalue equation for such matrices is equivalent to that for matrices of the form 0N D , (3.49) D† 0N where D is symmetric with complex elements. The matrices in each of the classes (1)–(4) have the property that if λ is an eigenvalue, then so is −λ. This ∗ can be seen from (3.40). Thus given the eigenvector as in (3.40) with eigenvalue , the eigenvector uv ∗ , also satisfies (3.40) but with eigenvalue −, a fact which can be verified by taking the complex conjugate of the latter equation and noting h = h† . Since the matrices are Hermitian, they can all be diagonalized by a unitary matrix, where the elements of the unitary matrix are further constrained due to the special symmetries for each class. In fact these matrices have all appeared in earlier sections, and the Jacobian for the change of variables to the eigenvalues and eigenvectors has been computed. In case (1), the class of matrices is equivalent to 4N × 4N antisymmetric Hermitian matrices. From Exercises 1.3 q.5 the eigenvalue dependent portion of the Jacobian is (λ2k − λ2j )2 . (3.50) 1≤j
Case (2) led to 4N ×4N antisymmetric Hermitian matrices with quaternion real elements (the entries of which are pure imaginary numbers). Since the matrix is Hermitian with quaternion real elements, each eigenvalue is doubly degenerate. From Exercises 1.3 q.7 we read off that the eigenvalue dependent portion of the Jacobian is N
|λj |
j=1
(λ2k − λ2j )4 .
(3.51)
1≤j
The matrices of case (3) can be written in the form (3.48). We read off from Exercises 1.3 q.8 that the eigenvalue dependent portion of the Jacobian is N j=1
λ2j
(λ2k − λ2j )2 .
(3.52)
1≤j
For the matrices of case (4) we read off from Exercises 3.1 q.8 that the eigenvalue-dependent portion of the
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LAGUERRE AND JACOBI ENSEMBLES
Jacobian is N
|λj |
j=1
|λ2k − λ2j |.
(3.53)
1≤j
Choosing the independent real and imaginary parts of the random matrices X to be Gaussian random variables such that their joint distribution is proportional to exp(−βX2 /2), where β is given by the exponent in the product of differences in each of (3.50)–(3.53), we see that the eigenvalue p.d.f.’s for the classes (1)–(4) are examples of the Laguerre eigenvalue p.d.f. written in the form of (3.4). 3.3.3 Relationship to Lie algebras It is emphasized in [14] and [556] that the above classes of matrices are the Hermitian part of the matrix Lie algebras i × so(n, C) := {i times n × n skew symmetric complex matrices}, i × so∗ (2n) := {i times n × n skew symmetric real quaternion matrices}, " Z1 Z2 sp(n, C) := , all Z complex, Z , Z symmetric , i 2 3 Z3 −ZT1 " X2 X1 , all X real, X , X symmetric , sp(n, R) := i 2 3 X3 −XT1 and furthermore these matrix Lie algebras, together with those of Sections 1.3.4 and 3.1.3, complete Cartan’s list [295] of the ten Lie algebras associated with the ten infinite families of symmetric spaces. We summarize the matrix structures corresponding to matrix Lie algebras giving rise to p.d.f.’s of the form (3.4) in Table 3.1. 3.3.4 Entanglement of a random pure quantum state
m The p.d.f. proportional to (3.16), with the further constraint that j=1 λj = 1, occurs in studies of quantum entanglement (see, e.g., [56]). The setting is a finite-dimensional quantum system decomposed into distinct subsystems A and B of dimensions n and m, respectively (for convenience it is assumed that n ≥ m). A state |ψ of the composite system will have the decomposition |ψ =
m n
xi,j |ai ⊗ |bj ,
(3.54)
i=1 j=1
where {|ai }i=1,...,n , {|bi }i=1,...,m are basis of the corresponding subsystems, and the xi,j are scalars. The density matrix corresponding to |ψ is the projection operator ρ = |ψψ|. The reduced density matrix of the subsystem B is defined as the trace of ρ over the states of A, ρB := TrA [ρ] =
n
ai |ρ|ai .
i=1
Substituting in this the definition of ρ, and then substituting (3.54), one obtains the form ρB =
m i,j=1
where X = [xi,j ] i=1,...,n . j=1,...,m
(X† X)i,j |bi bj |
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CHAPTER 3
Matrix structure Xn×m 0n×n (Xn×m )† 0n×n real elements (β = 1) complex elements (β = 2) real quaternion elements (β = 4) 0N ×N XN ×N X†N ×N 0N ×N X antisymmetric complex or X self-dual (N even) X symmetric complex
Parameter values in (3.4)
β = 1, 2 or 4, α = n − m + 1 − 1/β
These matrices are equivalent to antisymmetric matrices with pure imaginary real quaternion elements β = 1, α = 1, m = N
antisymmetric N × N matrices, XT = −X pure imaginary complex elements N odd : m = (N − 1)/2, β = 2, α = 1 pure imaginary real quaternion elements anti-Hermitian N × N matrices, X† = −X real quaternion elements
N even : m = N/2, β = 2, α = 0 N even: m = N/2, β = 4, α = 1/4 N odd: m = (N − 1)/2, β = 4, α = 5/4 iλj → λj , β = 2, α = 1, m = N
Table 3.1 Matrix structures which give rise to the eigenvalue p.d.f. (3.4).
m B On the other hand, the diagonalization of X† X gives the diagonal form ρB = i |vi , where i=1 λ m B † {|vi }i=1,...,m are the normalized eigenvectors of X X. Analogous reasoning gives ρA = i=1 λi |viA , where {|viA }i=1,...,m are the normalized eigenvectors of XX† corresponding to the nonzero eigenvalues. Now writing |ψ as a linear combination of the product states {|viA ⊗ |vjB }i,j=1,...,m it follows from these formulas for the reduced density matrices that |ψ =
m λi |viA ⊗ |viB ,
(3.55)
i=1
which is referred to as √the Schmidt decomposition. The coefficients { λi } can be used to quantify the degree of entanglement of the subsystems A and B m via the Shannon entropy S = − i=1 λi log λi . One extreme is when λ1 = 1, λi = 0 (i = 2, . . . , m), in which case |ψ is a direct product of states from subsystems A and B, and thus disentangled. In this case S = 0. Another extreme is when λi = 1/m (i = 1, . . . , m). This weights all states in (3.55) equally and gives S = log m, which is the maximum allowed value. Suppose the coefficients {xi,j } in (3.54) are independent Gaussians. If the subsystems A and B are invariant under time reversal symmetry with T 2 = 1 then these coefficients are real, while a time reversal symmetry with T 2 = −1 implies the coefficients are real quaternions. If there is no time reversal symmetry, the coefficients will be complex. In such a circumstance, X† X is a Wishart matrix and the eigenvalues, if not further constrained, would have distribution (3.16). However, here there is a further constraint, as the m normalization of |ψ requires that i=1 λi = 1, and thus the eigenvalue distribution is equal to 1
m
ˆ aβm W
l=1
βa/2 λl
1≤j
|λk − λj | δ β
m
λj − 1 ,
λl ≥ 0
(3.56)
j=1
with a = n − m + 1 − 2/β. The explicit form of the normalization can be read off from (4.155) below. The
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LAGUERRE AND JACOBI ENSEMBLES
eigenvalue p.d.f. (3.56) will be said to specify the fixed trace Laguerre β-ensemble. Another setting of this ensemble is given in Exercises 3.3 q.3, while in Exercises 3.3 q.4 the mean value of the Shannon entropy is computed. E XERCISES 3.3
1. The objective of this exercise is to show that the eigenvalue problem for the matrix (3.44) is equivalent to that for the matrices (3.45) and (3.46). (i) With the eigenvector for (3.44) corresponding to the eigenvalue λ denoted » – u + iv , x + i y where u, v , x, y are 2N -component real vectors, equate real and imaginary parts in each block to obtain the coupled equations iAv + iB y = λ u,
−iA u − iB x = λv ,
iBv − iA y = λ x,
−iB u + iA x = λ y.
(ii) Show that the coupled equations in (i) are equivalent to the eigenvalue equation » –» – » – 02N −iA + B u + i x u + i x =λ iA + B 02N −v + i y −v + i y and thus conclude that the eigenvalue problem for (3.44) is equivalent to that for (3.45). (iii) Note that » −1 –» –» – » – Z2N 02N −iA + B Z−1 02N Z2N 02N 02N 2N (−iA + B) = 02N 12N 02N 12N 02N iA + B 02N (iA + B)Z2N while
“
Z−1 2N (−iA + B)
”D
T −1 −1 = Z2N (−iA + B)T (Z−1 2N ) Z2N = Z2N (−iA + B)
and conclude from this that the eigenvalue problem for (3.44) is equivalent to that for (3.46). 2.
(i) Suppose A and B are real symmetric N × N matrices. Show that the real eigenvalue equation » –» – » – A B u u =λ B −A v v is equivalent to the complex equation (A + iB)( u − iv ) = λ( u + iv ), and thus to the complex eigenvalue equation –» – » – » A + iB u + iv u + iv 0N =λ . A − iB 0N u − iv u − iv
(ii) Conclude from (i) that the eigenvalue problem for the matrices of case (4) is equivalent to that for the matrices (3.49). 3. [279]; cf. Exercises 1.3 q.10. Let A = [ajk ]j,k=1,...,m be a Wishart matrix, and thus have p.d.f. (3.22). The matrix C := [ajk /(ajj akk )1/2 ]j,k=1,...,m is then called a correlation matrix, and it has the property Tr C = m. By noting that 1/2 1/2 1/2 A = diag(a11 , . . . , amm )Cdiag(a11 , . . . , a1/2 mm ) Q (m−1)/2 show that the Jacobian for changing variables from A to (a11 , . . . , amm ) and C is m . Use (3.22) to i=1 aii then show that the joint density of a11 , . . . , amm and C is proportional to m Y
(m−1)/2−(β/2)(n−m+1−2/β)
aii
(det C)(β/2)(n−m+1−2/β) δ(c1 + · · · + cm − m)
i=1
and read off from this that the p.d.f. of C is proportional to (det C)(β/2)(n−m+1−2/β) δ(c1 + · · · + cm − m).
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Hence conclude that the eigenvalues of C have distribution (3.56). 4. [560] This exercise develops some of the properties of the fixed trace Laguerre β-ensemble (3.56). (i) Denote the fixed trace Laguerre β-ensemble with one body weight xa and m = N by fLβEa , and the corresponding Laguerre β-ensemble with one body weight xa e−x by LβEa . By writing the delta function constraint as a Fourier integral show that N DX j=1
λpj
E
(βN (N − 1)/2 + N (a + 1) − 1)! D X p E λ . (βN (N − 1)/2 + N (a + 1) + p − 1)! j=1 j LβEa N
fLβEa
=
(ii) Note that for large N the ratio of factorials in (i) has the large N form (βN 2 /2)−p , and conclude from this that for large N ( N (4N − xN 2 )1/2 , 0 < x < 4/N fLβEa 2 LβEa 2 ρ(1) (x) ∼ (βN /2)ρ(1) (βN x/2) ∼ 2πx1/2 0, x > 4/N. (iii) Use the result of (ii) to compute that for large N Z ∞ “ 4 ”ν+1 N 2 Γ(ν + 1/2)Γ(3/2) a xν ρfLβE (x) dx ∼ . (1) N 2π Γ(ν + 2) 0 Read off that the first moment equals unity, as required by the fixed trace condition. Also, by differentiating with respect to ν and making use of the dilogarithm function, show −
“ log N ” E 1 . λj log λj ∼ log N − + O 2 N j=1
N DX
3.4 THE EIGENVALUE DENSITY 3.4.1 Mar˘cenko-Pastur law Proposition 3.2.3 predicts that the eigenvalue density in the Laguerre ensemble is such that 2 (1 − y)1/2 , 0 < y < 1, πy 1/2 lim 4ρ(1) (4N y) = N →∞ 0, y ≥ 1.
(3.57)
This result in fact remains valid for a large class of Wishart type matrices, in which one relaxes the requirement that the elements of the matrix X be Gaussian, and is an example of the Mar˘cenko-Pastur law [388]. Furthermore, it gives a good approximation for finite N , as illustrated in Figure 3.1 where the r.h.s. of (3.57) is compared to the empirical value of ρ(1) (Y ), Y = 4N y for 500 10 × 10 complex Wishart matrices. We know from Proposition 3.1.4 that in the case α √ fixed the support of the density corresponding to (3.14)√ with m = N is to leading order in the interval [0, 4N ]. Consequently, under the change of variable y → N y, the support is in the interval [0, 2]. We see from (3.14) that if we also make the replacement α → αN , both the quadratic term and the log term will then be of the same order, so it is to be expected that the support of the density will be in some interval [c, d], c > 0. According to Exercises 3.1 q.1, the integral equation satisfied by the background charge density for a general potential V (x) in this setting is equivalent to V (x) + C = ρb (t) log |x − t| dt, J
where J = [−d, −c]∪[c, d]. For even potentials, the physically relevant solution of such “double cut” integral
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LAGUERRE AND JACOBI ENSEMBLES
density
5 4 3 2 1 0.2
0.4
0.6
Y
1
0.8
Figure 3.1 Empirical scaled eigenvalue density for 10 × 10 complex Wishart matrices compared with the theoretical scaled density (3.57).
equations can be deduced from the complex electric field approach of Section 1.4.3. One finds [345], [7] # d xV (t) − tV (x) (d2 − x2 )(x2 − c2 ) 2 ρb (x) = 2 dt (3.58) π c t2 − x2 (d2 − t2 )(t2 − c2 ) (cf. (1.65)), where c and d are determined by the conditions d V (t) dt = 0, 2 (d − t2 )(t2 − c2 ) c
d
ρb (x) dx = N.
(3.59)
c
Specializing V (x) to correspond to the chiral ensemble gives the following prediction for the eigenvalue density, known rigorously from [388] (see also next subsection). P ROPOSITION 3.4.1 In the case V (x) = N
x2 2
− α log |x| ,
(3.60)
which corresponds to the chiral ensemble with m = N , the change of scale x → explicit form of (3.58) with the constraints (3.59) is N 1 2 (d − x2 )(x2 − c2 ), c < x < d, ρb (x) = π x where d − c = 2, cd = α.
√
N x and α → N α, the (3.61)
Proof. With the substitution (3.60), we read off from (3.58) and (3.59) that ρb (x) =
2N α p R(x)I1 , π2 x
where R(u) := (d2 − u2 )(u2 − c2 ) and Z d dt p , I1 := R(t) c t
Z I2 := c
I2 − αI1 = 0,
d
t dt p , R(t)
π2 = I1 I3 , 2α Z
I3 := c
d
(3.62)
1p R(t) dt. t
It turns out that after the change of variables s = t2 , the antiderivative of all these integrals can be computed exactly,
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leading to the results I1 =
π , 2cd
I2 =
π , 2
I3 =
π (d − c)2 . 4
Substituting in (3.62) gives (3.61).
3.4.2 Combinatorial derivation of the density We have seen in Section 1.6 and the ensuing exercises how the density (1.52) can be deduced from a combinatorial interpretation of Tr(X2k )GUE . Here we will show how (3.61) can be deduced from consideration of Tr(X† X)k cG , where cG denotes that the average is over n × m complex Gaussian matrices X [535], [240]. Let I, I be finite ordered sets of pairs of indices (j, k), j ∈ [1, n], k ∈ [1, m]. Let P denote a matching between elements of I and elements of I . Then analogous to (1.99) we have z¯i,j zi ,j = ¯ zi,j zi ,j cG . (3.63) cG
(i ,j )∈I
(i,j)∈I
pairings P between I,I
(i,j),(i ,j )
Noting n
Tr(X† X)k =
m
z¯i1 ,jk z¯i2 ,j1 · · · z¯ik ,jk−1 zi1 ,j1 zi2 ,j2 · · · zik ,jk
i1 ,...,ik =1 j1 ,...,jk =1
it follows from (3.63) that Tr(X† X)k
cG
n
=
m
¯ zl,m zl ,m cG ,
(3.64)
i1 ,...,ik =1 j1 ,...,jk =1 pairings (l,l ),(m,m )
where the pairings are between the sets and {(i1 , j1 ), (i2 , j2 ), . . . , (ik , jk )}.
{(i1 , jk ), (i2 , j1 ), . . . , (ik , jk−1 )}
The pairing in (3.64) can be carried out graphically using a construction similar to that in Figure 1.3, where the vertices in the 2k-gon correspond to i1 , j1 , . . . , ik , jk in that order, and the odd (even) labeled edges must connect to odd (even) labeled edges. Each independent vertex corresponding to the is is to be weighted by n, while each independent vertex corresponding to the js is to be weighted by m. If the total number of independent vertices after pairing is ν, then analogous to (1.106) we have
Tr(X† X)k
cG
=
k+1
ν=1
i,j≥0 i+j=ν
aν (k, i, j)ni mj ,
where aν (k, i, j) denotes the number of pairings which have i independent vertices corresponding to the is and j independent vertices corresponding to the js. The equation (1.108) again allows the pairings to be catalogued according to the genus of the underlying map, and we have in particular lim m−k−1 Tr(X† X)k
m→∞
cG
=
n i ak+1 (k, i, k + 1 − i) , m i=1
k
where ak+1 (k, i, k + 1 − i) counts pairings corresponding to planar maps. Another interpretation of the meaning of ak+1 (k, i, k + 1 − i) comes from the graphical picture of the pairings given in Figure 1.2, the planar version of which has no lines intersecting. The independent vertices corresponding to the is and js can be determined by the bi-coloring (white and gray, say) the interior of the 2k-gon according to the rule that the
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LAGUERRE AND JACOBI ENSEMBLES
colors alternate, and the region containing the marked vertex always has the same color (white, say). Then the number of regions colored white gives the value of the i index. Writing Ak (p, q) :=
k
ak+1 (k, i, k + 1 − i)pi q k+1−i ,
(3.65)
i=1
it follows from a recursive argument analogous to the one leading to (1.124) that Ak+1 (p, q) =
k
Aj (q, p)Ak−j (p, q),
A0 (p, q) = q.
j=0
Analogous to (1.125), introducing the generating function ψ(p, q, t) = symmetry ψ(p, q, t) = ψ(q, p, t), we thus have
∞ k=1
Ak (p, q)tk and noting the
ψ(p, q, t) = t(ψ(p, q, t) + p)(ψ(p, q, t) + q) and so, with u = pt, v = qt,
1 1 − u − v − (1 − 2(u + v) + (u − v)2 )1/2 . (3.66) 2 Applying the Lagrange inversion formula of Exercises 1.6 q.3 to the previous equation to expand ψ as a function of t allows Ak (p, q) to be deduced, and expanding this as required by (3.65) shows 1 k k . (3.67) ak+1 (k, i, k + 1 − i) = k i i−1 tψ(p, q, t) =
This is the Narayana number N (k, i − 1) [501]. It follows that the scaled density ρ˜(1) (x) := limm→∞ ρ(1) (mx)|n/m=α+1 is such that
∞
x2k ρ˜(1) (x) dx =
0
k 1 k k (α + 1)i . k i i − 1 i=1
(3.68)
Using the general formula [270] √ √ √ am xm R3/2 b(m + 3/2) − xm−1 R dx − xm R dx, xm+1 R dx = c(m + 3) c(m + 4) c(m + 4) where R = a + bx + cx2 we can verify that ρ˜(1) (x) = ρb (x)/N as specified by (3.61) is consistent with (3.68). Moreover, the moments (3.68) have the property (1.128) and thus uniquely determine ρ˜(1) (x). Comparing (3.68) to (3.67) we see from (3.65) that d Ak (α + 1, 1) = x2k ρ˜(1) (x) dx c
and hence, for |t|d < 1, 2
c
d
ρ˜(1) (x) dx = 1 + ψ(α + 1, 1, t). 1 − tx2
Making use now of (3.66) gives for the complex electric in the region Re(z) > d, d ρ˜(1) (x) α 2(α + 2) α2 1/2 . dx = z 1 − 2 − 1 − + 4 2z 2 2 z z2 z c z −x
(3.69)
We can check using the equations below (3.61) that the term in the square root is negative real for z real and between c and d only, so by analytic continuation this evaluation extends to all z outside this interval.
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CHAPTER 3
E XERCISES 3.4
1. With a > 0 and xi positive, let F (a; x1 , . . . , xN ) :=
N 1X (xj − a log xj ) − 2 j=1
X
log |xk − xj |.
1≤j
By following the method of Exercises 1.4 q.8 show that the location of the minimum of F occurs at the zeros (a−1) of the Laguerre polynomial LN (x). Conclude that for N → ∞ the density of zeros must obey the limit law (3.57).
3.5 CORRELATED WISHART MATRICES As formulated in Section 3.2.1 the n× m data matrix X in (3.15) has each row drawn from an m-dimensional Gaussian with mean zero and variance Σ. Equivalently, the distribution of X is proportional to 1 exp − Tr(XT XΣ−1 ) . (3.70) 2 By the analogue of (3.21), (dX) is invariant under the transformation X → XO for O an m × m real orthogonal matrix. Hence Σ−1 in (3.70) can be replaced by its diagonal form, which in particular prompts us to refer to any situation in which Σ−1 is not the identity as correlated. Application of Wishart matrices to wireless communications (recall Section 3.2.2), in which the analogue of the data matrix X is complex, suggests that (3.70) be generalized to exp − Tr(X† Ω−1 XΣ−1 ) (3.71) (see, e.g., [488]). Setting Y = Ω−1/2 XΣ−1/2 in this show that Y† Y is an uncorrelated complex Wishart matrix. Here we will consider a special feature of the case that Σ−1 = 1m while Ω−1 = diag((1)n−1 , b), where (1)n−1 denotes 1 repeated n − 1 times. Then X† X = A + bxx† ,
(3.72)
where A is an uncorrelated complex Wishart matrix as specified in Definition 3.2.1 with n → n − 1, while x is an m component standard complex Gaussian vector. By an analysis similar to that undertaken in Section 1.8, we can use knowledge of the leading asymptotic form of the eigenvalue density to analyze the behavior of the largest eigenvalue of (3.72) as a function of b. We first note that working analogous to Proposition 1.8.1, and the paragraph preceding that result, shows that with the eigenvalues of A denoted {ai } and x = [xi ] the eigenvalues of (3.72) satisfy 1=b
m |xi |2 , λ − ai i=1
(3.73)
and are constrained by the inequalities (1.135). For convenience, let us scale A by A/m. According to (3.57) √ the eigenvalue density ρ(1) (y) then has support on (0, 4), and is given explicitly by ρ(1) (y) = (m/π y)(1 − y/4)1/2 , 0 < y < 4. Thus with b → b/m in (3.73), and each |xi |2 replaced by its mean value unity, (3.73) assumes the large m form 4 ρ(1) (y) 2b 2 (1 − x2 /4)1/2 1=b dy = dx, λ−y π 0 λ − x2 0 where the second equality follows by changing variables y = x2 . Making use now of (1.131) and (1.126) allows this equation to be solved by λ, and an eigenvalue separation phenomenon analogous to that seen in Proposition 1.8.1 exhibited.
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LAGUERRE AND JACOBI ENSEMBLES
P ROPOSITION 3.5.1 For the rank 1 perturbation (3.72) with b → b/m and A a diagonal matrix formed from the scaled eigenvalues λj /m of the Laguerre ensemble, the bulk of the spectrum is supported on (0, 4) and for b > 2 a single eigenvalue separates from the bulk and occurs at λ=
b2 . b−1
We remark that the same result holds if in (3.71) we set Ω−1 = 1n and Σ−1 = diag((1)m−1 , b) [29]. To see this, we use the fact that the nonzero eigenvalues of Y† ΩY are the same as those for ΩYY† (recall Exercises 3.2 q.1) and then interchange the role of Y and Y† .
3.6 JACOBI ENSEMBLE AND WISHART MATRICES 3.6.1 Correlation coefficients The Laguerre ensemble has been specified as the family of eigenvalue p.d.f.’s (3.16). Likewise, the Jacobi ensemble is defined as the family of eigenvalue p.d.f.’s −1 CabβN
N
(1 − yj )aβ/2 (1 + yj )bβ/2
j=1
|yk − yj |β ,
yj ∈ [−1, 1].
(3.74)
1≤j
Random matrices, constructed out of Wishart matrices, which give rise to (3.74) in the cases β = 1, 2 and 4 and for certain discrete a, b are given by the following result (see, e.g., [118]). P ROPOSITION 3.6.1 Let A = a† a, B = b† b, where a and b are n1 × m and n2 × m random matrices, specified for β = 1, 2 and 4 as in Definition 3.1.2. The eigenvalues x1 , . . . , xm of the matrix A(A + B)−1 (or equivalently the Hermitian matrix (A + B)−1/2 A(A + B)−1/2 ) have p.d.f. (3.74) with N = m,
yj = 1 − 2xj ,
a = n1 − m + 1 − 2/β,
b = n2 − m + 1 − 2/β.
Proof. See Exercises 3.6 q.2. We remark that the matrix A(A + B)−1 can equivalently be written as X† JX(X† X)−1 , where X is the (n1 + n2 ) × m matrix [a b] and J := diag((1)n1 , (0)n2 ). One use of Proposition 3.6.1 relates to correlation coefficients in multivariate statistics. Correlation coefficients arise in canonical correlation analysis (see, e.g., [130]), for which the task is to form linear combinations within two sets of variables, x1 , . . . , xp and y1 , . . . , yq , so that the correlation between the new variables in the different sets is maximized, while the new variables within each of the sets are uncorrelated. Now, as(j) sociating with the variables x1 , . . . , xp the n × p matrix of observed values X = [xk ] j=1,...,n , and with the k=1,...,p
(j)
variables y1 , . . . , yq the matrix of observed values Y = [yk ] j=1,...,n , the empirical covariance matrix has k=1,...,q the block form T X X XT Y . YT X YT Y The correlation coefficients are defined as the square of the eigenvalues of (XT X)−1 XT Y(YT Y)−1 YT X.
(3.75)
The eigenvalue distribution of (3.75) can be computed by reducing the problem down to that of Proposition 3.6.1 [118]. P ROPOSITION 3.6.2 Let X be a n × p matrix, and Y be a n × q matrix, q ≥ p, n ≥ p + q, with Gaussian entries as specified in Definition 3.1.2 for β = 1, 2 and 4. Then the eigenvalues x1 , . . . , xp of the matrix
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(3.75) (with the transpose operation T replaced by the Hermitian conjugate operation † for β = 2 and 4) have the p.d.f. (3.74) with N = p,
yj = 1 − 2xj ,
a = q − p + 1 − 2/β,
b = n − q − p + 1 − 2/β.
Proof. The first step is to form an n × n orthogonal (β = 1), unitary (β = 2) or unitary symplectic (β = 4) matrix U such that its last n − q rows are in the orthogonal complement of the space spanned by the columns of Y, and thus » – Y1 q×q UY = , 0n−q×q
where Y1 is nonsingular. Setting
» UX =: Wn×p =:
we see that we can write X† Y(Y † Y)−1 Y † X − xX† X = W† = W†
» »
Y1 0 1q 0
aq×p
–
b(n−q)×p –
(Y1† Y1 )−1 [Y1† 0]W − xW† W – 0 W − xW† W 0
= a† a − x(a† a + b† b). Now the eigenvalue condition for (3.75) can be written det(X† Y(Y † Y)−1 Y † X − xX† X) = 0 so the above working shows that this is equivalent to det(a† a − x(a† a + b† b)) = 0, which is exactly the eigenvalue condition for the matrix a† a(a† a + b† b)−1 . Furthermore, recalling the distribution of the elements of X is proportional to exp(−(β/2)Tr X† X), and noting that Tr(X† X) = Tr(W† W) = Tr(a† a) + Tr(b† b), and that the matrix X has the same number of elements as the matrices a and b together, we see the elements of a and b are independently distributed as specified in Definition 3.1.2 for β = 1, 2 and 4. The result now follows from Proposition 3.6.1.
3.6.2 Coulomb gas analogy The Jacobi ensemble p.d.f. (3.74) has a Coulomb gas analogy which can be determined by the change of variables y = cos θ. P ROPOSITION 3.6.3 The Boltzmann factor of a one-component log-potential system with particles of unit charge at y1 , . . . , yN , confined to the interval [−1, 1] with a neutralizing background density N (1 − y 2 )−1/2 π and with a fixed particle of charge (a−1)/2+1/β at y = 1 and another fixed particle of charge (b−1)/2+1/β at y = −1, is proportional to the eigenvalue p.d.f. (3.74). −ρb (y) = −
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LAGUERRE AND JACOBI ENSEMBLES
Proof. Changing variables yj = cos θj , 0 < θj < π shows N Y
Y
(1 − yj )aβ/2 (1 + yj )bβ/2
j=1
|yk − yj |β
1≤j
Y
×
N Y
dyj =
j=1
|2 sin((θj − θk )/2) sin((θj + θk )/2)|β
N Y
(2 sin2 θj /2)aβ/2 (2 cos2 θj /2)bβ/2
j=1 N Y
sin θj dθj .
(3.76)
j=1
1<j
The r.h.s. is proportional to (2.102) with the identifications aβ + 1 → 2a + 1,
bβ + 1 → 2b + 1.
This explains the values of the fixed charges. As on the r.h.s. of (3.76) there is no one-body potential from any other source, to leading order the background charge density −ρb (θ) must satisfy Z π ρb (θ) log | cos φ − cos θ| dθ. C= 0
The method of eigenfunction expansions R π detailed below Proposition 1.4.2 shows that the solution of this integral equation is ρb (θ) a constant. Since we require 0 ρb (θ) dθ = N , we must have ρb (θ) = N/π. Changing variables N N 1 p dθ = dy π π 1 − y2 gives the stated form of the background density.
1. Consider the Jacobi ensemble eigenvalue p.d.f. (3.74) with a → N a, b → N b. The aim of this exercise is to use the formulas (1.67), (1.68) to deduce the background density in the corresponding log-gas, or equivalently the global form of the eigenvalue density.
E XERCISES 3.6
(i) Substitute V (x) = −N ((a/2) log(1 − x) + (b/2) log(1 + x)) in (1.67), (1.68) (after changing notation in these equations a → c, b → d) to deduce that p Z 1 aN (y − c)(d − y) d 1 p ρb (y) = 2 dt, π 1 − y2 (t − c)(d − t) c 1−t where c and d are determined by the equations Z d Z d dt dt p p =b , a (t − c)(d − t) (t − c)(d − t) c (1 − t) c (1 + t) Z d Z d a tdt tdt b p p − =1 2π c (1 − t) (t − c)(d − t) 2π c (1 + t) (t − c)(d − t) (ii) [536], [113] Make use of tabulated integrals to simplify the formulas of (i) to read p 2 + a + b (y − c)(d − y) ρb (y) = , 2π 1 − y2 where c and d are determined by p
a b a+b+2 = p = . 2 (1 − c)(1 − d) (1 + c)(1 + d)
(3.77)
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(iii) Show that in the variable X, with y = 1 − 2X 2 , the density (3.77) reads p 2 + a + b (B 2 − X 2 )(X 2 − A2 ) , X ∈ [A, B], ρ˜b (X) = π X(1 − X 2 ) with “ ”2 1−d 1 1/2 1/2 1/2 (a + b + 1) − (b + 1) , (a + 1) = 2 (a + b + 2)2 “ ” 2 1−c 1 (a + 1)1/2 (a + b + 1)1/2 + (b + 1)1/2 . B2 = = 2 (a + b + 2)2 A2 =
From [536] we know that for Re(w) > B 2 , Z
B
2 A
ρ˜b (X) a b ((w − A2 )(w − B 2 ))1/2 dX = − + − (a + b + 2) . 2 w−X w 1−w w(1 − w)
Use this to compute the first moment in X 2 , Z B X 2 ρ˜b (X) dX = A
(3.78)
a+1 . a+b+2
2. In this exercise Proposition 3.6.1 will be proved, following [410, Thm. 3.3.1] where the real case was considered. Let A + B = T† T where T is an upper triangular m × m matrix with positive diagonal elements, and let U be the m × m matrix defined by A = T† UT. (i) Write down from (3.20), (3.23) and their analogue in the real and real quaternion cases the joint distribution of A and B, and conclude from this that the joint distribution of C := A + B and A is proportional to e−(β/2)Tr(C) (det A)(β/2)(n1 −m+χβ ) (det(C − A))(β/2)(n2 −m+χβ ) (dA) ∧ (dC), where χβ := 1 − 2/β. (ii) Note from (1.35) that (T† dUT) = (det T† T)β(m+1)/2 (dU) and use this to show (dA) ∧ (dC) = (det T† T)β(m+1)/2 (dU) ∧ (d(T† T)). (iii) Write det A and det(C−A) in terms of det U and det(1m −U), and use (ii) to show that the joint p.d.f. of T† T and U is proportional to e−(β/2)Tr(T
†
T)
det(T† T)(β/2)(n1 +n2 +2χβ −m+1) (det U)(β/2)(n1 −m+χβ ) (det(1m −U))(β/2)(n2 −m+χβ )
and deduce from this that the p.d.f. of U is proportional to (det U)(β/2)(n1 −m+χβ ) (det(1m − U))(β/2)(n2 −m+χβ )
(3.79)
while the p.d.f. of C = A + B is proportional to e−(β/2)Tr(C) (det C)(β/2)(n1 +n2 +2χβ −m+1) . (iv) Note that (T−1 T† )1/2 U(T−1 T† )−1/2 = (A + B)−1/2 A(A + B)−1/2 , and use (1.35) to deduce that U has the same distribution as (A + B)−1/2 A(A + B)−1/2 , which is thus given by (3.79). Use this result and Proposition 1.3.4 to deduce Proposition 3.6.1. 3. [279] Define A in terms of a according to Proposition 3.6.1. Let X be an m × N Gaussian matrix specified for β = 1, 2 and 4 as in Definition 3.1.2, and set Y = A−1/2 X.
(3.80)
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LAGUERRE AND JACOBI ENSEMBLES
The objective of this exercise is to show that the p.d.f. of Y is proportional to det(1 + Y † Y)−(β/2)(n1 +N) .
(3.81)
(i) With a specified in Proposition 3.6.1, show that the joint density of A and X is proportional to † X)
(det A)βa/2 e−(β/2)(TrA+TrX
.
(ii) Use a generalization of Proposition 3.2.4 and the result of (i) to show that the change of variables (3.80) gives that the joint density of A and Y is proportional to (det A)β(N+a)/2 e−(β/2)Tr(A(1+Y
†
Y))
.
(iii) Use (3.31) to integrate over A > 0, and thus deduce the result. 4. Let A, B be N × N Gaussian matrices with entries for β = 1, 2 and 4 as specified in Definition 3.1.2. Proceed as in q.3 to show that the distribution of Y = A−1 B is proportional to det(1 + Y † Y)−βN .
(3.82)
5. [508], [305] Use the method of Exercises 1.4 q.8 to show that for a, b ≥ 0 the minimum of −
N ” 1 X“ a log(1 − yj ) + b log(1 + yj ) − 2 j=1
X
(a−1,b−1)
occurs at the zeros of the Jacobi polynomial PN
log |yj − yk |,
|yj | < 1 (j = 1, . . . , N )
1≤j
(x).
6. [2] Let a, b be n1 × m, n2 × m (n1 , n2 ≥ m) Gaussian matrices with real (β = 1), complex (β = 2) or quaternion real (β = 4) entries. Use the results of Exercises 1.3 q.1(iv) and the remark below (2.31) to argue that the rectangular matrix » – a (a† a + b† b)−1/2 b is the (n1 + n2 ) × m block of an (n1 + n2 ) × (n1 + n2 ) Haar distributed unitary matrix with real (β = 1), complex (β = 2) or quaternion real (β = 4) entries. Now make use of Proposition 3.6.1 to deduce that the top n1 × m block of such random matrices has the square of its singular values as specified therein.
3.7 JACOBI ENSEMBLE AND SYMMETRIC SPACES Sections 1.3.4, 3.1.3 and 3.3.3 contain the ten Lie algebras associated with the ten infinite families of symmetric spaces. The symmetric spaces corresponding to the matrix Lie algebras of Section 1.3.4 have been made explicit in Section 2.2.3, and have been shown to be isomorphic to the three circular ensembles. In this section the remaining seven symmetric spaces are given explicitly, as are the corresponding unitary matrices and their eigenvalue p.d.f.’s. After the change of variables yj = cos θj , all the eigenvalue p.d.f.’s obtained are found to be examples of the Jacobi ensemble p.d.f. (3.74). 3.7.1 Symmetric spaces isomorphic to O+ (N ) and Sp(2N ) The compact quotient spaces O+ (N ) × O+ (N )/O+ (N ),
Sp(2N ) × Sp(2N )/Sp(2N )
are symmetric spaces, isomorphic to O+ (N ) and Sp(2N ) respectively. The Cayley transformations (2.61), (2.70) show that the corresponding spaces of Hermitian matrices are antisymmetric Hermitian matrices with
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CHAPTER 3
pure imaginary elements, and quaternion real matrices with the property that when multiplied by i they are Hermitian. These in turn are the Hermitian part of the Lie algebras i × (so(n, C)) and sp(n, C). To relate the eigenvalues p.d.f.’s for O(N ) and Sp(2N ) to the Jacobi ensemble p.d.f, we note that in (3.76) the latter has been related to the p.d.f. for the β generalization of the orthogonal and symplectic ensembles (2.102). Recalling then the results of Section 2.6 gives the precise form of this relationship. P ROPOSITION 3.7.1 In terms of the variable yj = cos θj , the eigenvalue p.d.f. for random real orthogonal matrices is given by the Jacobi ensemble (3.74) with β = 2, N replaced by N ∗ and ⎧ (N/2, −1/2, −1/2) for matrices in O+ (N ), N even, ⎪ ⎪ ⎨ ((N − 1)/2, 1/2, −1/2) for matrices in O+ (N ), N odd , (N ∗ , a, b) = ((N − 1)/2, −1/2, 1/2) for matrices in O− (N ), N odd , ⎪ ⎪ ⎩ (N/2 − 1, 1/2, 1/2) for matrices in O− (N ), N even. Similarly, in terms of the variable yj = cos θj the eigenvalue p.d.f. for 2N × 2N random unitary matrices with each 2 × 2 sub-block a real quaternion element (unitary symplectic equivalent matrices) is given by the p.d.f. (3.74) with β = 2, N replaced by N ∗ and (N ∗ , a, b) = (N, 1/2, 1/2). 3.7.2 Symmetric spaces corresponding to the chiral ensembles Consider the block matrix (3.1). Use of the singular value decomposition (3.2) gives V X U 0n×n 0n×m 0m×m ΛT 0m×n = . U† X† V† 0n×m 0m×m 0m×n Λ 0n×n
(3.83)
The block matrix involving V and U† can be characterized as an (n + m) × (n + m) unitary matrix g such that gφ = g−1 , with In,m :=
where
gφ = In,m g† In,m
1n 0m×n
0n×m −1m
(3.84)
.
In the case that the matrix X in Proposition 3.1.1 has real elements or quaternion real elements, we know from (3.3) that the unitary matrix is constrained to be real symmetric and unitary symplectic equivalent respectively. Thus in these cases the second equation in (3.84) should be replaced by gφ = In,m gT In,m , g ∈ O(n + m) and gφ = I2n,2m gD I2n,2m , g ∈ Sp(2(n + m)).
(3.85)
The formulas (2.13) and (2.16) for matrices in the COE and CSE suggest considering the spaces of matrices of the form S = gφ g
(3.86)
for g ∈ O(n + m) with gφ as in the first formula of (3.85) (β = 1), for g ∈ U (n + m) with gφ as in (3.84) (β = 2), and for g ∈ Sp(2(n + m)) with gφ as in the second formula of (3.85) (β = 4). With these specifications of gφ the considerations of the proof of Proposition 2.2.4 shows that the matrices (3.86) are isomorphic to O(n + m)/O(n) × O(m),
U (n + m)/U (n) × U (m),
Sp(2(n + m))/Sp(2n) × Sp(2m),
respectively. A uniform measure on these symmetric spaces is achieved by the Haar form (2.10), but with the matrix of
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LAGUERRE AND JACOBI ENSEMBLES
differentials modified using the invariance (2.11) to be unchanged by the mapping φ. From (2.15) and (2.17) we know we should choose δS = (gφ )† dS g† =: idH
(3.87)
where H is as in (3.1) with real elements (β = 1), complex elements (β = 2) and quaternion real elements (β = 4). Whereas the matrix (3.1) has n − m zero eigenvalues, the matrix S has n − m unit eigenvalues. This can be seen from the Cayley transformation S=
1n+m + iH 1n+m − iH
(3.88)
for a certain H of the form (3.1). Recalling that the nonzero eigenvalues of H occur in ± pairs, we see too that the non-unit eigenvalues of S come in complex conjugate pairs. Now, we can use the facts that H has the eigenvalue decomposition given in the proof of Proposition 3.1.1 and that the elements of the matrix X in (3.1) are independent to deduce from (3.88) that −βm/2 (δS) ∝ det(1n+m + H2 ) (dH) (compare the exponent with that in (2.57)). Knowledge of the eigenvalue p.d.f (3.4) for {H} then implies that the eigenvalue p.d.f. for {S} is given by m 1 sin θj (cos2 θj /2)β(1−2/β)/2 (sin2 θj /2)β(n−m+1−2/β)/2 C j=1
| cos θj − cos θk |β .
(3.89)
1≤j
The change of variables yj = cos θj shows (3.89) to be the special case a = n − m + 1 − 2/β, b = 1 − 2/β of the Jacobi ensemble (3.74). 3.7.3 The symmetric spaces Sp(2N )/U (N ), O+ (2N )/U (N ) Out of the ten sets of Hermitian matrices corresponding to infinite families of matrix Lie algebras, we have so far made a correspondence with eight sets of unitary matrices. It remains to identify the unitary matrices corresponding to Hermitian matrices of the form (3.46), and antisymmetric Hermitian matrices with quaternion real elements. From the theory of Exercises 3.3 q.1 we know that matrices (3.46) satisfy the diagonalization type equation (3.83) with m = n = N , V = (U−1 )T . Now unitary matrices g of the form U 0N 0N (U−1 )T have the property (3.84) and furthermore satisfy (1.26) and so are examples of unitary symplectic matrices. From this observation it can be seen [139] that unitary matrices constructed according to the prescription (3.86) with g ∈ Sp(2N ) are isomorphic to the symmetric space Sp(2N )/U (N ). In particular the equation (3.87) holds with H of the form (3.46). Use of the Cayley transformation (3.88), (2.57) with β = 1, together with the result (3.52) shows that the corresponding eigenvalue p.d.f. is proportional to N j=1
sin θj
| cos θk − cos θj |.
(3.90)
1≤j
Regarding the unitary matrices corresponding to antisymmetric Hermitian matrices with real quaternion elements, we first note that such matrices can be diagonalized by unitary symplectic equivalent matrices with real entries, and these in turn can be identified with matrices U (N ) (recall Exercises 1.3 q.6(ii)). This suggests
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considering the unitary matrices g ∈ O+ (2N ),
S = gD g,
(3.91)
which are isomorphic to the symmetric space O+ (2N )/U (N ) [139]. Note that then (Z2N S)T = −Z2N S, so Z2N S is a 2N × 2N antisymmetric unitary matrix. For the set of matrices (3.91), (3.87) holds with H an antisymmetric Hermitian matrix with real quaternion elements. Use of the Cayley transformation (3.88), (2.57) with β = 4, together with the results of Exercises 1.3 q.7(iii), shows that the corresponding eigenvalue p.d.f. is proportional to
N/2
j=1
sin θj
j=1
N even,
(3.92)
1≤j
(N −1)/2
| cos θj − cos θk |4 ,
sin θj sin4
θj 2
| cos θj − cos θk |4 ,
N odd.
(3.93)
1≤j
In terms of the variable yj = cos θj the eigenvalue p.d.f.’s (3.90), (3.92) and (3.93) are of the form (3.74). Explicitly, with N replaced by N ∗ the parameters are ⎧ ⎨ (N, 0, 0; 1) for matrices in Sp(2N )/U (N ), (N/2, 0, 0; 4) for matrices in O+ (2N )/U (N ), N even, (N ∗ , a, b; β) = ⎩ ((N − 1)/2, 1/2, 0; 4) for matrices in O+ (2N )/U (N ), N odd. 1. [247] Let Q be a 2n × 2n positive definite Hermitian matrix (i.e. all eigenvalues positive) and let I n,n = diag[1n , −1n ]. The objective of this exercise is to note special properties of the diagonalization of the non-Hermitian matrix QI n,n .
E XERCISES 3.7
(i) With the diagonalization formula for Q reading Q = Udiag[λ1 , . . . , λ2n ]U† , √ √ set S = Udiag[ λ1 , . . . , λ2n ]U† so that S2 = Q. Show that QI n,n is similar to SI n,n S, and thus conclude that the eigenvalues of QI n,n are all real. (ii) Let H be a Hermitian n × n matrix, and let T be an arbitrary nonsingular n × n matrix. It can be shown [255] that the number of positive eigenvalues of H is the same as the number of positive eigenvalues of T† HT. Use this result and the result of (i) to show that exactly half the eigenvalues of QI n,n are positive. (iii) With {vj }j=1,...,2n denoting the eigenvectors of QI n,n corresponding to the eigenvalues {μj }j=1,...,2n , assumed distinct and with the first n being positive, set V = [v1 · · · v2n ]. Show that V† I n,n V = diag[sgn(μ1 ), . . . , sgn(μ2n )] = I n,n .
(3.94)
(Matrices with this property form a group denoted U (n, n).)
3.8 JACOBI ENSEMBLE AND QUANTUM CONDUCTANCE The situation to be considered is a quasi one-dimensional conductor containing scattering impurities and having n available scattering channels (which are the different wavenumbers of the plane wave states) at the left-hand edge and m available scattering channels at the right-hand edge. The length of the sample is assumed to be much smaller than the coherence length, but much larger than the mean free path of the electrons, which is the criterion for the metallic phase. At each end of the conductor is an electron reservoir of different chemical potentials, which causes a current to flow. A quantity of interest is then the conductance of the sample.
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3.8.1 Definitions Before presenting the formula for the conductance, it is necessary to revise the theoretical description of the above setting. Similar to the case for the scattering problem involving a lead with a tunneling barrier considered in Section 2.4, fundamental quantities are the electron fluxes at the left- and right-hand edges of I the conductor, which are specified by the 2n component vector O and the 2m component vector OI , and I (O ) denote the n and m component amplitudes of the plane wave states traveling into where I (O) (out of) the left and right sides of the conductor, respectively. For definiteness we will suppose n ≥ m. Flux conservation requires 2 + |I |2 = |O| 2 + |O |2 . |I| By definition, the (n + m) × (n + m) scattering matrix S relates the flux traveling into the conductor to that traveling out, O I (3.95) S := . O I As done in Section 2.1.1 the flux conservation condition can be used to show that S must be unitary, and as in (2.40) the scattering matrix is further decomposed in terms of reflection and transmission matrices by rn×n tn×m S= . (3.96) tm×n rm×m We know from Section 2.1.2 that S must be symmetric if the system has a time reversal symmetry with T 2 = 1, and a self-dual quaternion matrix when there is a time reversal symmetry with T 2 = −1, T = Z2(n+m) K. The conductance G is given in terms of the transmission matrix t (or t ) by the so-called two-probe Landauer formula, G/G0 = Tr(t† t) = Tr(t† t ),
(3.97)
where G0 = 2e2 /h is twice the fundamental quantum unit of conductance. Thus the quantities of interest are the eigenvalues of t† t (or t† t ). In fact the matrix S can be decomposed in a form which isolates the eigenvalues of t† t. 3.8.2 Distribution of singular values Suppose for definiteness that there is no time reversal symmetry, so S is a general unitary matrix. We can then decompose each block of S as in the singular value decomposition (3.2), rn×n = Ur Λr Vr† ,
tm×n = Ut Λt Vt† , etc.
where Ur , Vr† , . . . are unitary matrices and Λr (Λt ) is a rectangular diagonal matrix consisting of the positive square roots of the eigenvalues of r† r (t† t) (these eigenvalues are between 0 and 1 since r† r + t† t = 1) and t† t must have n − m zero eigenvalues since the rank of t is equal to m. The unitarity constraint inter-relates the matrices Ur , Ur , . . . (see Exercises 3.8 q.1) and implies the decomposition † 0 Vr 0 Ur L , (3.98) S= 0 Ur 0 Vr† where
1 − Λt ΛTt L := iΛTt
iΛt 1 − ΛTt Λt
.
(3.99)
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In the case of a time reversal symmetry with T 2 = 1, S must be symmetric, and thus so must be the blocks of rn×n and rm×m in (3.96). These square matrices then permit the singular value decompositions rn×n = Ur Λr UTr ,
rm×m = Ur Λr UTr
and thus (3.99) holds with Vr† = UTr ,
Vr† = UTr .
(3.100)
Similarly in the case of a time reversal symmetry with T = −1, when S is self-dual quaternion, (3.99) holds with 2
Vr† = UD r ,
Vr† = UD r .
(3.101)
It turns out that the distribution of the elements of Λt in the decomposition (3.99) coincides exactly with the distribution of the elements of Λ in the singular value decompositions (3.2) and (3.3), which is given by (3.4) excluding the Gaussian factors [399], [53].
P ROPOSITION 3.8.1 Let S be a (n + m) × (n + m) matrix which is a member of the COE (β = 1), CUE (β = 2) or CSE (β = 4) (in the latter case each element is itself a 2 × 2 matrix). Decomposing S as in (3.99), the distribution of the nonzero elements of Λt , which are equal to the square root of the nonzero eigenvalues of t† t, and thus in the range [0, 1], is proportional to m j=1
λβα j
|λ2k − λ2j |β ,
α := n − m + 1 − 1/β.
(3.102)
1≤j
Proof. Consider for definiteness the case β = 2, so that S is a random unitary (n + m) × (n + m) matrix, and thus
has (n + m) × (n + m) independent elements. We proceed in an analogous way to the proof of Proposition 3.1.3, first computing the differential of (3.98) to obtain – » – » – » – » † 0 Ur 0 0 0 Vr δUr δVr dS = L + dL − L . 0 Vr 0 δUr 0 δVr 0 U†r
Using this to form Tr(dSdS† ), simplifying using the fact that – “ » δV ” 0 r Tr L† dL = 0, 0 δVr and factoring the terms as in (3.5) but retaining the block matrix structure shows Tr(dSdS† ) q q ”“q ”” ““q 1 − Λt ΛTt δVr − δUr 1 − Λt ΛTt 1 − Λt ΛTt δUr − δVr 1 − Λt ΛTt = Tr “ ” “ ” +Tr (Λt δVr − δUr Λt )(ΛTt δUr − δVr ΛTt ) + Tr (ΛTt δVr − δUr ΛTt )(Λt δUr − δVr Λt ) q q ”“q ”” ““q + Tr(dLdL† ). 1 − ΛTt Λt δVr − δUr 1 − ΛTt Λt 1 − ΛTt Λt δUr − δVr 1 − ΛTt Λt +Tr With the diagonal elements of ΛΛT denoted λj , (j = 1, . . . , n), and thus λm+1 = · · · = λn = 0, and the five traces
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above denoted T1 , . . . , T5 , we see that in component form T1 =
n X
k=1
1≤k
“
X
+
1≤k
T2 =
X
(1 − λ2k )|(δVr )kk − (δUr )kk |2 +
m X
1“ 2
q
1 − λ2l +
q ”2 1 − λ2k |(δVr )lk − (δUr )lk |2
n m q ” 1 “q ”2 X X + 1 − λ2l − 1 − λ2k |(δVr )lk + (δUr )lk |2 , 2 k=1 l=m+1
k=1
n1
X
λ2k |(δVr )kk − (δUr )kk |2 +
2
1≤k
(λl + λk )2 |(δVr )lk − (δUr )lk |2
n m o X X 1 + (λl − λk )2 |(δVr )lk + (δUr )lk |2 + λ2k |(δUr )lk |2 , 2 k=1 l=m+1
T3 =
m X k=1
n1
X
λ2k |(δVr )kk − (δUr )kk |2 +
2
1≤k
(λl + λk )2 |(δVr )lk − (δUr )lk |2
n m o X X 1 + (λl − λk )2 |(δVr )lk + (δUr )lk |2 + λ2k |(δVr )lk |2 , 2 k=1 l=m+1
T4 =
m X
k=1
T5 = 2
X
(1 − λ2k )|(δVr )kk − (δUr )kk |2 +
m X (dλj )2 . 1 − λ2j j=1
1“ + 2
q
1 − λ2l −
q
1≤k
1 − λ2k
”2
q q ”2 1“ 1 − λ2l + 1 − λ2k |(δVr )lk − (δUr )lk |2 2
|(δVr )lk + (δUr )lk |2 ,
The task now is to identify the set of independent differentials, as unlike in (3.6) the differentials occurring in T1 –T5 are not all independent. In fact the independent differentials are those in T1 , T4 , T5 and the first term in T3 . Explicitly, these are the imaginary part of (δVr )kk − (δUr )kk ,
1 ≤ k ≤ n,
(δVr )kk − (δUr )kk ,
1 ≤ k ≤ m,
(δVr )kk − (δUr )kk ,
1 ≤ k ≤ m,
(3.103)
and the real and imaginary parts of (δVr )lk − (δUr )lk ,
1 ≤ k < l ≤ n,
(δVr )lk + (δUr )lk ,
1 ≤ k < l ≤ m,
(δVr )lk − (δUr )lk ,
1 ≤ k < l ≤ m,
(δVr )lk + (δUr )lk ,
1 ≤ k < l ≤ m,
1≤ k ≤ m & m+1 ≤ l ≤ n (3.104)
as well as dλj ,
1 ≤ j ≤ m.
This is a total of (n + m) terms as required by the number of independent terms in S. To compute the Jacobian we must make use of the general formula (1.15) rather than the simpler formula (1.14) because the metric form is not diagonal, as will become apparent in the subsequent calculations. Now, for 1 ≤ k < l ≤ m we see from (3.104) that the independent differentials can be taken as the real and imaginary parts of 2
(δUr )lk ,
(δUr )lk ,
(δVr )lk ,
(δVr )lk .
(3.105)
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CHAPTER 3
With z1 := x1 + iy1 , z2 := x2 + iy2 , using the simple formula |z1 − z2 |2 = x21 + x22 + y12 + y22 − 2x1 x2 − 2y1 y2 ,
(3.106)
we see from the expressions for T1 through to T4 that the contribution to the metric form from the differentials (3.105) is X X “ (s) (s) (s) (s) 2((δUr )lk )2 + 2((δUr )lk )2 + 2((δVr )lk )2 + 2((δVr )lk )2 s= real part imag. part
1≤k
n o n o” (s) (s) (s) (s) (s) (s) (s) (s) +2alk (δUr )lk (δVr )lk + (δUr )lk )(δVr )lk + 2blk (δUr )lk (δVr )lk + (δVr )lk (δUr )lk ,
with
q q q ”2 1 “q ”2 1“ 1 − λ2l − 1 − λ2k − 1 − λ2l + 1 − λ2k 2 2 1 1 2 2 blk = (λl − λk ) − (λl + λk ) . 2 2
alk =
This contributes to (det[gjk ])1/2 in (1.15) the factor ˛ ˛ 2 alk 0 ˛ m Y ˛ alk 2 blk ˛ ˛ 0 2 blk k
blk 0 alk 2
˛ ˛ ˛ m ˛ Y ˛∝ (λ2l − λ2k )2 . ˛ ˛ k
(3.107)
For m + 1 ≤ k < l ≤ n, the coefficient of |(δVr )lk − (δUr )lk |2 in T1 is independent of the λs, so for the purposes of computing the p.d.f. of the latter these differentials can be ignored. For 1 ≤ k ≤ m, m + 1 ≤ l ≤ n, we see from (3.104) that the independent differentials can be taken as the real and imaginary parts of (δVr )lk . (δUr )lk , Furthermore, use of (3.106) in the formula for T1 shows that the contribution to the metric form from these differentials is n m “ ” X X X (s) (s) (s) (s) 2((δUr )lk )2 + 2((δVr )lk )2 + 2ck (δUr )lk (δVr )lk , s= real part imag. part
k=1 l=m+1
q ck = −2 1 − λ2k .
where
This contributes to (det[gjk ])1/2 in (1.15) the factor ˛ n m Y Y ˛ ˛ p2 ˛ −2 1 − λ2 k k=1 l=m+1
p −2 1 − λ2k 2
˛ Y m ˛ 2(n−m) ˛∝ λk . ˛
(3.108)
k=1
It remains to calculate the contribution from the differentials (3.103). Because the coefficient of |(δVr )kk −(δUr )kk |2 for m + 1 ≤ k ≤ n is independent of the λ’s, we can restrict attention to 1 ≤ k ≤ m in all the differentials of (3.103). In this range of k values, the first term in T2 couples together the three differentials in (3.103). Thus making use of a formula analogous to (3.106), and recalling that only the imaginary parts of (3.103) are nonzero, we have that the contribution to the metric form is m X
(δXk )2 + (δYk )2 + 2λ2k (δZk )2 − 2λ2k δXk δZk − 2λ2k δYk δZk + 2λ2k δXk δYk ,
k=1
where
” “ δXk := Im (δVr )kk − (δUr )kk ,
” “ δYk := Im (δVr )kk − (δUr )kk ,
” “ δZk := Im (δVr )kk − (δUr )kk .
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LAGUERRE AND JACOBI ENSEMBLES
This contributes to (det[gjk ])1/2 in (1.15) the factor ˛ m ˛ Y ˛ 12 ˛ λk ˛ 2 k=1 ˛ −λk
−λ2k −λ2k 2λ2k
λ2k 1 −λ2k
˛1/2 ˛ m Y ˛ ˛ ∝ λk (1 − λ2k )1/2 . ˛ ˛ k=1
(3.109)
Finally, the contribution to (det[gjk ])1/2 in (1.15) from the coefficients of the terms (dλj )2 in T5 is m Y
1 . 2 1/2 (1 − λ k) k=1
(3.110)
Multiplying together (3.107)–(3.110) gives (3.102) in the case β = 2.
Changing variables λ2j → xj in (3.102), where the xj are the eigenvalues t† t, gives a p.d.f. proportional to m j=1
βa/2
xj
|xk − xj |β ,
xj ∈ [0, 1],
(3.111)
1≤j
where a = n − m + 1 − 2/β as in Proposition 3.2.2. With yj := 1 − 2xj , the p.d.f. (3.111) is an example of the Jacobi ensemble (3.74). 3.8.3 Wishart matrix of a rectangular sub-block of U (N ) In the case β = 2 the distribution of the Wishart matrix of a general n1 × n2 rectangular block of elements from S can be determined [536], [487], [197]. Due to the invariance of the underlying Haar measure for U (N ) with respect to interchanges of rows and columns, this block can be taken to be in the top left corner. P ROPOSITION 3.8.2 Let S ∈ U (N ), and let A denote the n1 × n2 block of elements in the top left. For n1 ≥ n2 , N ≥ n1 + n2 , the distribution of Y := A† A is proportional to (N −n1 −n2 ) . (det Y)(n1 −n2 ) det(1n2 − Y) Proof. Let C denote the block of elements in the first n1 rows and last N − n2 rows not in A. Then the unitarity of S implies
AA† + CC† = 1n1 .
We follow [559] and regard this as a constraint in the space of general complex rectangular matrices A, C. Up to normalization the distribution of A is then given by Z Z Z † † (dC) e−iTr(H(AA +CC −1n2 )) (dH), (3.112) δ(AA† + CC† − 1n2 )(dC) ∝ where the second expression follows upon using (3.27). Following [252], we would like now to interchange the order of integration, and integrate first over C. To get an answer which is integrable around H = 0, we replace H in the exponent by H−iμ1m , μ > 0. Computing the Gaussian integrals by taking H to be diagonal then shows that up to a multiplicative constant (3.112) reduces to Z “ ”−(N−n2 ) † eiTr(H(1n1 −AA )) (dH). det(H − iμ1m ) lim μ→0+
Evaluating this via the matrix integral (3.24) shows that the distribution of A is proportional to (det(1n1 − AA† ))(N−n1 −n2 ) = (det(1n2 − A† A))(N−n1 −n2 ) . The result now follows from (3.23) with X → A, A → Y.
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CHAPTER 3
The corresponding eigenvalue p.d.f. is an example of (3.74) with yj := 1 − 2xj , a = n1 − n2 , b = N − n1 − n2 and β = 2. The derivation of Proposition 3.8.2 can be extended to the cases in which S ∈ O(N ), or S ∈ Sp(2N ) [197] (see also Exercises 3.8 q.2). In particular, with A denoting the n1 × n2 block of elements in the top left corner, the distribution of A is found to be proportional to (det(1n2 − A† A))(β/2)(N −n1 −n2 +1−2/β)
(3.113)
and thus the distribution of Y := A† A is proportional to (β/2)(N −n1 −n2 +1−2/β) . (det Y)(β/2)(n1 −n2 +1−2/β) det(1n2 − Y) The eigenvalue p.d.f. of the latter is an example of (3.74) with yj := 1 − 2xj , a = (β/2)(n1 − n2 + 1 − 2/β),
b = (β/2)(N − n1 − n2 + 1 − 2/β).
1. (i) With S given by (3.96) write down the six equations implied by the unitarity condition SS† = S† S = 12N .
E XERCISES 3.8
(ii) With the singular value decompositions r = Ur Λr Vr etc. show that the equations in (i) allow a solution implied by (3.99). (iii) From (3.96) and the unitarity of S derive the equations rt† = −t r† and t† r = −r† t and thus deduce that Tr(tt† ) = Tr(t t† ). 2. [504] This exercise considers properties of the transfer matrix associated with a scattering problem. (i) Consider the quantum conductance problem described in terms of the scattering matrix S by (3.95), and ˜ relates the states at the left-hand edge to those at suppose n = m = N . The 2N × 2N transfer matrix M the right hand edge, " # " # I I ˜ M = . O O ˜ ∈ U (N, N ), and use this to show that if λ Show from the unitarity of S that in the notation below (3.94) M ˜ is an eigenvalue of M, then so is 1/λ. (ii) Use the decomposition (3.96) in the case n = m = N to show – » (t† )−1 rt−1 ˜ . M= −t−1 r t−1 From this deduce that with the eigenvalues of t† t denoted T1 , . . . , TN , Λ = diag(λ1 , . . . , λN ), λi := ˜ permits the parametrization (1 − Ti )/Ti , and L as in (3.99) M » † – » – 0 Ur 0 ˜ = Vr M L , (3.114) † 0 Vr 0 Ur and hence show
“
˜M ˜ † )−1 ˜M ˜ † + (M 2+M
”−1
=
1 4
»
tt† 0
0 t t
†
– .
Conclude that, with the eigenvalues of M denoted e±2xj (j = 1, . . . , N ), 1 = Tj . cosh2 xj
(3.115)
3. Read off from Proposition 3.8.1 that, for N × N matrices U from the COE (β = 1), CUE (β = 2), CSE (β = 4), the modulus squared |UNN |2 of the bottom right element has distribution proportional to (1 − t)(β/2)(N−1−2/β) ,
0 ≤ t ≤ 1,
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LAGUERRE AND JACOBI ENSEMBLES
and thus, making use of (4.2) below, compute that |UNN |2p =
Γ((β/2)(N − 1) + 1)Γ(p + 1) . Γ((β/2)(N − 1) + p + 1)
3.9 A CIRCULAR JACOBI ENSEMBLE There is a mathematical transformation which relates the normalization of (3.74) for the Jacobi ensemble to the normalization of the p.d.f. in Proposition 2.2.5 for the circular ensemble, and more generally leads to the consideration of a circular Jacobi ensemble. The transformation is accomplished by the following general result [192]. P ROPOSITION 3.9.1 Let f (t1 , . . . , tN ) be a Laurent polynomial in t1 , . . . , tN . For Re() large enough so that the r.h.s. exists, 1/2 1/2 N dθ1 · · · dθN e2πiθl f (−e2πiθ1 , . . . , −e2πiθN ) −1/2
−1/2
=
sin π π
l=1
N
1
dt1 · · ·
0
1
dtN 0
N
t−1+ f (t1 , . . . , tN ). l
l=1
This result follows immediately from term-by-term integration of the Laurent polynomial on both sides. Note that for an integer the Fourier integral is equal to CT{t1 ,...,tN }
N
t l f (−t1 , . . . , −tN )
l=1
where CT{t1 ,...,tN } denotes the constant term, i.e. the term independent of t1 , . . . , tN in the Laurent expansion. Writing tj = 12 (1 − yj ) in (3.74) shows that CabβN = 2N +(a+b)βN/2+βN (N −1)/2SN (aβ/2, bβ/2, β/2), where
SN (λ1 , λ2 , λ) :=
1
dt1 · · ·
0
1
dtN 0
N
tλl 1 (1 − tl )λ2
|tk − tj |2λ .
(3.116)
(3.117)
1≤j
l=1
The integral SN is known as the Selberg integral. The change of variables formula (3.76) gives the trigonometric form π π N (2λ1 +2λ2 +2)N +2λN (N −1) SN (λ1 , λ2 , λ) = dθ1 · · · dθN |1 − eiθl |2λ1 +1 2 ×|1 + e
|
iθl 2λ2 +1
0
0
|e
iθj
−e
l=1
| |1 − ei(θj +θk ) |2λ .
iθk 2λ
(3.118)
1≤j
Comparison with (2.102) and (2.103) shows that SN is thus evaluated by the product of gamma functions as specified in (2.103). This topic will be further developed in the next chapter. Our interest here is that SN is of the class of integrals for which Proposition 3.9.1 applies. Thus with λ, λ2 non-negative integers we have π N MN (a, b, λ), (3.119) SN (λ1 , λ2 , λ) = (−1)N +N (N −1)λ/2 sin πb
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CHAPTER 3
where MN (a, b, λ) 1/2 := dθ1 · · · −1/2
1/2
−1/2
dθN
N
eπiθl (a−b) |1 + e2πiθl |a+b
|e2πiθk − e2πiθj |2λ , (3.120)
1≤j
l=1
with −1−b−λ(N −1) = λ1 and a+b = λ2 . This trigonometric integral is often associated with Morris [407], although it was known to Selberg [227]. For a, b, λ non-negative integers it can be written in the equivalent form N tj λ tk λ 1 MN (a, b, λ) = CT{t1 ,...,tN } (1 − tl )a (1 − )b 1− 1− . (3.121) tl tk tj 1≤j
l=1
In the special case a = b the integrand of (3.120) is real, and after making the replacement 2πθl → θl reads N
|1 + eiθl |2b
|eiθk − eiθj |2λ ,
−π < θj ≤ π.
(3.122)
1≤j
l=1
This is a generalization of the p.d.f. of Proposition 2.2.5 for the circular ensemble. When properly normalized, (3.122) will be referred to as defining the circular Jacobi ensemble [414]. In a random matrix context, (3.122) with b = λ, 2λ = β is the eigenvalue p.d.f. in the circular ensemble given that there is an eigenvalue at θ = π. For general b it is said to describe a spectrum singularity. From the log-gas viewpoint, we see from Proposition 1.4.1 that (3.122) is proportional to the Boltzmann factor for a one-component log-potential system on a circle at coupling β = 2λ, with an impurity charge of strength q = b/λ fixed at the angle π on the boundary of the circle. Applying the mapping (2.50) we see that N
|1 + eiθl |2b
|eiθk − eiθj |2λ dθ1 · · · dθN
1≤j
l=1
= 2λN (N −1)+N (1+2b)
N
(1 + λ2j )−α
j=1
|λk − λj |2λ dλ1 · · · dλN , α = λ(N − 1) + 1 + b,
1≤j
(3.123) so the circular Jacobi ensemble is in fact equivalent to the Cauchy ensemble. The most general choice of parameters for which the integrand in (3.120) is real is a = ¯b =: b1 − ib2 . We then have the (unnormalized) probability N
eb2 θl |1 + eiθl |2b1
|eiθk − eiθj |2λ dθ1 · · · dθN = 2λN (N −1)+N (1+2Re(b))
1≤j
l=1
×
N j=1
1 (1 + iλj
)b+1+λ(N −1) (1
− iλj
)¯b+1+λ(N −1)
|λk − λj |2λ dλ1 · · · dλN , (3.124)
1≤j
where the r.h.s. results from the mapping (2.50). This generalizes the circular Jacobi ensemble and Cauchy ensemble, respectively.
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LAGUERRE AND JACOBI ENSEMBLES
E XERCISES 3.9
1. [225] Show that for q1 , q2 > 0 the minimum of −q1
2N X
log |1 − eiθk | − q2
k=1
2N X
log |1 + eiθk | −
k=1
X
log |eiθk − eiθj |
1≤j
subject to the requirement that 0 < θj < π (j = 1, . . . , N ),
π < θj < 2π (j = N + 1, . . . , 2N )
(q −1/2,q2 −1/2)
occurs at the zeros of the Jacobi polynomial PN 1
(cos θ), 0 < θ < 2π.
3.10 LAGUERRE β-ENSEMBLE Analogous to the Gaussian β-ensemble of Section 1.9, it is possible to construct random tridiagonal matrices which have as their eigenvalue p.d.f. (3.16) for general β > 0 [140]. To gain insight into their form, consider the effect of the Householder reduction (1.139) on matrices XT X, with X an n× m (n ≥ m) rectangular matrix of standard Gaussians. For this purpose, one constructs two sequences of Householder reflector matrices ˜ (j) }j=1,...,m−2 , {U(j) }j=1,...,m−1 , where each U ˜ (j) is n × n and U(j) is m × m, such that (1.140) {U ˜ (m−1)T ˜ (1)T · · · U U(m−2)T · · · U(1)T XT U
(3.125)
˜ (1)T has zeros in the first row for ˜ (1) so that XT U is a bidiagonal matrix. This is done by first choosing U (1)T T ˜ (1)T ˜ k ≥ 2. Because U is a projection, the norm of the first row of X U must be conserved and so the dis˜ (1)T )11 is χ2n , where χ2n is the particular gamma distribution Γ[n/2, 2] (cf. χ tribution of (XT U ˜k as specified in Proposition 1.9.1; by definition the gamma distribution Γ[s, σ] has p.d.f. proportional to (x/σ)s/2−1 e−x/σ ). The elements of U(0)T are independent of the rows 2, . . . , m − 1 in XT , so the distribution of the elements of ˜ (1)T has zeros in the first column these columns is unchanged. Now one constructs U(1)T so that U(1)T XT U (1)T for j ≥ 3. Because of the structure of U from (1.140) with V therein a projector, it leaves invariant ˜ (1)T (which as remarked above has the same the norm of the entries 2, . . . , m of the first column of XT U T ˜ (1)T )12 distribution as the corresponding elements of X ), and consequently the distribution of (U(1)T XT U 2 is χm−1 . Proceeding inductively shows that the real Gaussian m × n matrix XT is mapped via (3.125) to the bidiagonal matrix [486] ⎡ ⎤ χn ⎢ χm−1 χn−1 ⎥ ⎢ ⎥ (3.126) BT1 := ⎢ . ⎥. . .. .. ⎣ ⎦ χ1 χn−m+1 It follows immediately that the m × m random tridiagonal matrix BT1 B1 has the same eigenvalue p.d.f. as real Wishart matrices, and is thus given by (3.16) with β = 1. As in going from Proposition 1.9.1 to Proposition 1.9.4 in the Gaussian case, there is a simple generalization of (3.126) which leads to the eigenvalue p.d.f. (3.16) for general β > 0. P ROPOSITION 3.10.1 Define the real m × m bidiagonal random matrix ⎡ χβn ⎢ χβ(m−1) χβ(n−1) ⎢ BTβ := ⎢ .. .. ⎣ . . χβ
χβ(n−m+1)
⎤ ⎥ ⎥ ⎥, ⎦
(3.127)
128
CHAPTER 3
where n > m − 1 and n may be real, and set Tβ := BTβ Bβ . The eigenvalues and the first component of the eigenvectors (which form the vector q ) are independent, with the distribution of the former given by 1
m
(a,β,m W
j=1
βa/2 −λj /2
λj
e
|λk − λj |β ,
λj ≥ 0,
(3.128)
1≤j
where a = n − m + 1 − 2/β and (a,β,m = 2m(aβ/2+1+(m−1)β/2) W
m Γ(1 + βj/2)Γ(1 + βa/2 + β(j − 1)/2) Γ(1 + β/2) j=1
(3.129)
while the distribution of the latter is given by (1.161) with N → m.
Proof. Let us write 2 BTβ
3
xn
6 ym−1 6 := 6 . .. 4
xn−1 .. . y1
7 7 7, 5 xn−m+1
2
am
6 bm−1 6 6 Tβ := 6 . . . 6 4
bm−1 am−1 .. . b2
3 bm−2 .. . a2 b1
b1 a1
7 7 7 7. 7 5
(3.130)
Now, by definition, the probability measure of the matrix Bβ is P (Bβ )(dBβ ) = cm,n
m−1 Y
β(n−i)−1 −x2 e n−i /2
xn−i
i=0
cm,n := 2−(2m−1)
m−1 Y
Vn i=1
dxn−m+i , d y :=
am = x2n ,
Vm−1 i=1
2
ykβk−1 e−yk /2 d x ∧ d y,
(3.131)
k=1
Γ(jβ/2)
j=1
where d x := (3.130) we see
m−1 Y
m Y
Γ((n − k + 1)β/2),
k=1
dyi . Our first task is to introduce the variables of the matrix Tβ . From
ai = yi2 + x2n−m+i ,
bi = yi xn−m+i+1
(i = m − 1, m − 2, . . . , 1).
(3.132)
To compute the Jacobian for the change of variables from {xi , yj } → {ai , bj } we note from these equations that dam = 2xn dxn , dai = 2(xn−m+i dxn−m+i + yi dyi ), dbi = xn−m+i+1 dyi + yi dxn−m+i+1 (i = m − 1, . . . , 1). Taking the wedge product of both sides of these equations shows m−2 “ Y 2 ”−1 xn−i da ∧ db, d x ∧ d y = 2n xn−m+1
(3.133)
i=0
Vm−1 V where d x, d y are as in (3.131) and da := m i=1 dai , db := i=1 dbi . We now substitute (3.133) in (3.131), and then change variables {ai , bj } → {λi , qj } using Proposition 1.9.3 to deduce −n
(dBβ ) = 2
cm,n e
−
Pm−1 2 Pm−1 2 i=0 xn−i /2 − i=1 yi /2
e
Qm−1 i=0
β(n−i)−2 Qm−1 βi xn−i i=1 yi Q . qm m q i=1 i
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LAGUERRE AND JACOBI ENSEMBLES
Use of (1.148) and the fact that bi = yi xn−m+i+1 shows Qm−1 i m−1−i Qm−1 i Y bi i=0 yi xn−i Qm |λi − λj | = Qi=1 = m q i i=1 i=1 qi 1≤i<j≤m
and thus we have (dBβ ) = 2−n cm,n e− Y
×
Pm−1 2 Pm−1 2 i=0 xn−i /2 − i=1 yi /2
e
|λi − λj |β
1≤i<j≤m
m−1 m ”β(n−m+1)−2 1 Y β−1 “ Y qi xn−i d q ∧ dλ. qm i=1 i=0
Finally, we note that m−1 X
x2n−i +
m−1 X
i=0
yi2 = Tr Tβ =
m X
i=1
λi ,
i=1
m−1 Y
x2n−i = det Tβ =
i=0
m Y
λi
i=1
and make use of the normalization in (1.161) to deduce the stated result.
Note that the evaluation of the normalization (3.129) implies that the normalization Waβm in (3.16) has the evaluation −m(aβ/2+1+(m−1)β/2
Waβm = (β/2)
m−1 j=0
Γ(1 + β(j + 1)/2)Γ(1 + βa/2 + βj/2) . Γ(1 + β/2)
(3.134)
Furthermore, the three-term recurrence (1.154) with k = 1, 2, . . . , m and {ak }, {bk } specified by (3.132) can be used to give a rapid computation of the characteristic polynomial pm (λ). By calculating the zeros we can then sample from (3.128), and the eigenvalues histogrammed according to the method of Section 1.9.3.
3.11 JACOBI β-ENSEMBLE We have seen that the p.d.f.’s (1.160) and (3.128), specifying the Gaussian and Laguerre β-ensembles, respectively, can be realized by the eigenvalues of certain tridiagonal matrices. Equivalently, the characteristic polynomials for the p.d.f.’s can be generated from the three-term recurrence (1.154) with the ak , bk therein chosen according to certain distributions. For the Jacobi β-ensemble (3.74) the same is true [356]. This is a corollary of Proposition 2.9.4 for the real orthogonal β-ensemble. P ROPOSITION 3.11.1 For k ⎧ ⎨ αk ∈ ⎩
= 0, . . . , 2n − 2, in terms of the notation of Definition 2.9.1, let ˜ 2n−k−2 β + a + 1, 2n−k−2 β + b + 1 , k even, B 4 4 2n−k−2 2n−k−2 ˜ β + a + 1, β , k odd, B 4 4
and set α2n−1 = α−1 = −1. The eigenvalues of the tridiagonal matrix (1.143) with an−k = (1 − α2k−1 )α2k − (1 + α2k−1 )α2k−1 , bn−k = {(1 − α2k−1 )(1 − α22k )(1 + α2k+1 )}1/2
(3.135)
are distributed according to the Jacobi β-ensemble (3.74) with aβ/2 → a, bβ/2 → b, N → n. Proof. With the change of variables xl = cos θl ,
(3.136)
(2.102) becomes the Jacobi β-ensemble (3.74) with aβ/2 → a, bβ/2 → b. This transformation maps the eigenvalues from the unit half circle to the interval (−1, 1) (recall (3.76)). A result of Szeg¨o in orthogonal polynomial theory implies
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CHAPTER 3
that for each real upper Hessenberg 2N × 2N matrix with unit determinant, there is a tridiagonal matrix of the form (1.143) with eigenvalues related by (3.136). Moreover, the parameters of the tridiagonal matrix are related to those of the Hessenberg matrix by (3.135), which are known as the Geronimus relations [261]. Thus we see that Proposition 3.11.1 follows as a corollary of Proposition 2.9.5.
E XERCISES 3.11
1. [159] Given Θ = (θn , . . . , θ1 ), Φ = (φn−1 , . . . , φ1 ), define four n×n bidiagonal matrices B11 , B12 , B21 , B22 by – » B11 B12 B= B21 B22 3 2 cn −sn cn−1 sn sn−1 7 6 .. 7 6 . cn−1 sn−1 cn−1 cn−1 sn−1 sn−2 7 6 7 6 . . . 7 6 . . −s c .. .. 2 1 7 6 7 6 c1 c1 s1 7 c1 s1 6 =6 7 −cn cn−1 cn sn−1 7 6 −sn 7 6 7 6 .. 7 6 . −s s −s c c s n−1 n−1 n−1 n−1 n−1 n−2 7 6 7 6 . . . . . −c c .. .. 5 4 2 1 −s1 s1 −s1 c1 c1
where ci = cos θi , si = sin θi , ci = cos φi , si = sin θi . (i) Show that B is orthogonal. (ii) Decompose B into the form (3.98) with n = m, write Γr = diag(σ1 , . . . , σn ) for the singular values of BT11 B11 , and denote the first entries of the corresponding normalized eigenvectors (constrained to be positive) by (v1 , . . . , vn ). Use the working of the proof of Proposition 3.10.1 to show that n “ Y
β(j−1)+1 β(j−1) sj dcj
cj
” n−1 Y“
j=1
” (cj )βj−1 (sj )β(j−1)+2 dcj ,
j=1
=
Y
|σj2 − σk2 |β
j
n “Y
σj dσj
”“ n−1 Y
j=1
” vjβ−1 dvj .
j=1
(iii) With cos2 θj ∈ B[β(a + j)/2, β(b + j)/2] cos2 φj ∈ B[βj/2, β(a + b + 1 + j)/2]
(j = n, . . . , 1) (j = n − 1, . . . , 1),
use the above result to show that {σj2 } are distributed according to the Jacobi β-ensemble n 1 Y β(a+1)/2−1 λ (1 − λj )β(b+1)/2−1 C j=1 j
Y
|λk − λj |β .
1≤j
3.12 CIRCULAR JACOBI β-ENSEMBLE In Section 2.8 a random unitary Hessenberg matrix having eigenvalue p.d.f. realizing the circular β-ensemble was specified. Here, following [93], it will be shown how to modify that construction so the eigenvalue p.d.f. of the circular Jacobi β-ensemble (3.122) is realized. With {αj } relating to a unitary Hessenberg matrix as in Proposition 2.8.1, we begin by introducing some related variables.
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LAGUERRE AND JACOBI ENSEMBLES
D EFINITION 3.12.1 Set ¯k γk = α
k−1 j=0
1 − γ¯j , 1 − γj
k = 0, . . . , N − 1.
(3.137)
We now examine the significance of the quantities (3.137). In (2.88), for k = 1, . . . , N − 1 introduce (k) {zl }l=1,...,k so that (k) k (λ − zj ) χk (λ) = . bk (λ) := χ ˜k (λ) j=1 (1 − λ¯ zj )
(3.138)
Using the definition in (3.138), it follows from (2.88) that γk (λ) := λ −
α ¯k χk+1 (λ) = . χk (λ) bk (λ)
(3.139)
It follows from this definition that χk (λ) =
k−1
(λ − γj (λ))
(k = 1, . . . , N ),
(3.140)
j=0
and this in turn used in (3.138) gives γk (λ) = α ¯k
k−1 j=0
1 − λ˜ γj (λ) , λ − γj (λ)
where γ˜j (λ) := γ¯j (1/λ). Comparison with (3.137) shows γk = γk (1). The γk variables will be chosen to be independently distributed in terms of a generalization of Θν (recall Definition 2.8.3). D EFINITION 3.12.2 Let Θbν+1 denote the distribution of complex numbers |z| < 1 with p.d.f. (Γ(ν/2 + 1 + b))2 (1 − |z|2 )(ν/2−1) |1 − z|2b πΓ(ν/2)Γ(ν/2 + 1 + 2b) while for complex numbers |z| = 1 let Θb1 denote the distribution 1 (Γ(1 + b))2 |1 − z|2b 2π Γ(1 + 2b) (note that Θbν |b=0 = Θν ). P ROPOSITION 3.12.3 Let {αk }k=0,...,N −1 specify a unitary Hessenberg matrix as in Proposition 2.8.1, and let these variables be determined by {γk } according to (3.137), where γN −j−1 is required to have distribution Θbβj/2 . The unitary Hessenberg matrix then has its eigenvalue p.d.f. given by the p.d.f. (3.122) specifying the circular Jacobi β-ensemble. Furthermore, the first component of its eigenvectors are independently distributed as specified by (1.161). Proof. The measure associated with the joint distribution of {γN−j−1 } is proportional to N−2 Y j=0
(1 − |γN−j−1 |2 )βj/2−1
N−1 Y
|1 − γN−j−1 |2b dγ .
(3.141)
j=0
Let us write γk = rk eiθk , (k = 0, . . . , N − 2), γN−1 = eiθN −1 , αk = rk eiψk , (k = 0, . . . , N − 2) and αN−1 =
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CHAPTER 3
eiψN −1 . From the triangular structure of (3.137), the Jacobian for the transformation {r0 , . . . , rN−2 , θ0 , . . . , θN−1 } → {r0 , . . . , rN−1 , ψ0 , . . . , ψN−1 } is seen to equal one. Hence (3.141) is equal to N−2 Y
(1 − |αN−j−1 |2 )βj/2−1
j=0
N−1 Y
|1 − γN−j−1 |2b d α,
(3.142)
j=0
where we have used the fact that |γk | = |αk |. Now, making use of (3.140) with k = N , λ = 1, and the fact that N Y
χN (1) = det(1N − Hβ ) =
(1 − eiθj ),
j=1
shows
N−1 Y j=0
|1 − γN−j−1 |2b =
N Y
|1 − eiθj |2b .
j=1
Furthermore, we can rewrite the remaining factors in (3.142) in terms of {λj } and {qi } using the working of the proof of Proposition 2.8.7.
Chapter Four The Selberg integral The normalization of the Jacobi β-ensemble, known as the Selberg integral, is studied in its own right. Four different derivations of the evaluation of the Selberg integral, including the original one of Selberg, are presented. Two of these derivations give extensions of the Selberg integral which are of use in the calculation of correlation functions considered in subsequent chapters. Furthermore, the derivation due to Anderson has an interpretation in terms of the eigenvalues of a random corank 1 projection of a fixed matrix and leads to a further random three-term recurrence for the characteristic polynomial of the Jacobi β-ensemble. Alternative constructions of the Gaussian and Laguerre β-ensembles also result. Underlying Anderson’s derivation is a further integral evaluation—the Dixon-Anderson integral—which implies that the classical ensembles at β = 1 and β = 4 are related by integrating over every second eigenvalue in the former. This integral evaluation can be generalized, with the consequence that classical β-ensembles at β = 2/(r + 1), integrated over all eigenvalues not labeled by a multiple of r + 1, give back classical β-ensembles with β = 2(r + 1). The Selberg integral can be used to prove Macdonald’s constant term conjectures relating to root systems. Another application, involving explicitly the normalizations for the circular and Gaussian β-ensembles, is the computation of the free energy of the corresponding log-gas systems.
4.1 SELBERG’S DERIVATION The Selberg integral refers to the N -dimensional integral (3.117) 1 1 N SN (λ1 , λ2 , λ) := dt1 · · · dtN tλl 1 (1 − tl )λ2 0
0
l=1
|tk − tj |2λ .
(4.1)
1≤j
In the case N = 1, this integral is the Euler beta integral, and has the evaluation [541] 1 Γ(λ1 + 1)Γ(λ2 + 1) . xλ1 (1 − x)λ2 dx = Γ(λ1 + λ2 + 2) 0
(4.2)
For general N Selberg [483] evaluated (4.1) as the product of gamma functions (2.103), SN (λ1 , λ2 , λ) =
N −1 j=0
Γ(λ1 + 1 + jλ)Γ(λ2 + 1 + jλ)Γ(1 + (j + 1)λ) , Γ(λ1 + λ2 + 2 + (N + j − 1)λ)Γ(1 + λ)
(4.3)
long before the development of the theory of Section 2.9. Note that (4.3) has various poles. Those with the largest real part depend on the sign of Re(λ), being at λ1 = −1 or −1−(N −1)λ, λ2 = −1 or −1−(N −1)λ and λ = −1/N , which is in keeping with (4.1) defining a convergent integral for parameter values to the right of these in the appropriate complex planes. According to (3.119), the evaluation of the Selberg integral for general N gives the evaluation of the Morris
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integral. Thus we find MN (a, b, λ) :=
1/2
−1/2
=
N −1 j=0
dθ1 · · ·
1/2 −1/2
dθN
N
eπiθl (a−b) |1 + e2πiθl |a+b
|e2πiθk − e2πiθj |2λ
1≤j
l=1
Γ(λj + a + b + 1)Γ(λ(j + 1) + 1) , Γ(λj + a + 1)Γ(λj + b + 1)Γ(1 + λ)
(4.4)
where use has been made of the functional equation π . (4.5) sin πz This includes as special cases the normalization of the circular Jacobi ensemble (3.122) and the eigenvalue p.d.f. (2.20) of the circular ensemble. In fact, as will be demonstrated below, the normalization of the Hermite and Laguerre ensembles similarly follow as limiting cases of the Selberg integral. Before showing how (4.3) can be used to calculate the normalizations of the various ensembles, we will consider a number of different derivations of this evaluation in addition to the one already given in Section 2.9, each being of independent interest. The first derivation to be presented is the original one of Selberg [483], but with the last step (analytic continuing off the integers) done differently [23]. To begin, note that for λ a positive integer the product of differences in (4.1) is a multidimensional polynomial of order 2λ(N − 1) and so can be expanded |tk − tj |2λ = cn1 ,...,nN tn1 1 · · · tnNN . (4.6) Γ(z)Γ(1 − z) =
0≤n1 ,...,nN ≤2λ(N −1)
1≤j
Substituting this expansion in the definition of SN , and use of the Euler beta integral evaluation (4.2) shows
SN (λ1 , λ2 , λ) =
cn1 ,...,nN
0≤n1 ,...,nN ≤2λ(N −1)
N Γ(λ1 + 1 + nj )Γ(λ2 + 1) . Γ(λ1 + λ2 + 2 + nj ) j=1
(4.7)
The next stage of Selberg’s derivation of (4.3) consists of evaluating (4.7) as a function of λ1 and λ2 . For this purpose some properties of the integers cn1 ,...,nN occurring in (4.6) are required. P ROPOSITION 4.1.1 The value of the coefficients cn1 ,...,nN in (4.6) is independent of the order of the integers n1 , . . . , nN . For 0 ≤ n1 ≤ n2 · · · ≤ nN the nonzero values of cn1 ,...,nN occur when the integers N n1 , . . . , nN satisfy the summation formula j=1 nj = N (N − 1)λ and the inequalities (j − 1)λ ≤ nj and nj ≤ (N + j − 2)λ, (j = 1, . . . , N ). of the integers n1 , . . . , nN Proof. The fact that the value of the coefficients cn1 ,...,nN is independent of the order Q (N) 2
follows because the function being expanded is symmetric. With Δ (t1 , . . . , tN ) := 1≤j
nj
Now the allowed powers on the r.h.s, say with 0 ≤ n1 ≤ n2 · · · ≤ nN , are given in terms of the allowed powers on the l.h.s. by nj = 2(N − 1)λ − nN+1−j , and must satisfy the inequality derived above for nj , nj ≥ (j − 1)λ. The second stated inequality now follows.
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THE SELBERG INTEGRAL
P ROPOSITION 4.1.2 The expression (4.7) can be evaluated as a function of λ1 and λ2 to give SN (λ1 , λ2 , λ) = cN (λ)
N −1 j=0
Γ(λ1 + 1 + jλ)Γ(λ2 + 1 + jλ) , Γ(λ1 + λ2 + 2 + (N + j − 1)λ)
where cN (λ) remains to be determined. Proof. From the inequalities for nj in Proposition 4.1.1 we see Γ(1 + λ1 + nj ) Γ(1 + λ1 + (j − 1)λ) = qn (λ1 , λ2 ), Γ(2 + λ1 + λ2 + nj ) Γ(2 + λ1 + λ2 + (N + j − 2)λ) j where qnj (λ1 , λ2 ) is a polynomial in λ1 and λ2 of degree (N + j − 2)λ − nj in λ2 . Thus N N Y Y Γ(1 + λ1 + nj )Γ(1 + λ2 ) Γ(1 + λ1 + (j − 1)λ)Γ(1 + λ2 ) = Q(λ1 , λ2 ) , Γ(2 + λ + λ + n ) Γ(2 + λ1 + λ2 + (N + j − 2)λ) 1 2 j j=1 j=1
where the polynomial Q(λ1 , λ2 ) has degree XN j=1
` ´ (N + j − 2)λ − nj = N (N − 1)λ/2
in λ2 . The expression (4.7) is a linear combination of terms of this form and can therefore be written ˜ 1 , λ2 ) SN (λ1 , λ2 , λ) = Q(λ
=
N Y Γ(1 + λ1 + (j − 1)λ)Γ(1 + λ2 ) Γ(2 + λ1 + λ2 + (N + j − 2)λ) j=1
N ˜ 1 , λ2 ) Y Γ(1 + λ1 + (j − 1)λ)Γ(1 + λ2 + (j − 1)λ) Q(λ , R(λ2 ) j=1 Γ(2 + λ1 + λ2 + (N + j − 2)λ)
˜ 1 , λ2 ) is a polynomial of degree at most N (N − 1)λ/2 and where Q(λ R(λ2 ) :=
N Y Γ(1 + λ2 + (j − 1)λ) Γ(1 + λ2 ) j=1
is a polynomial in λ2 of degree N (N − 1)λ/2. Since SN is symmetric in λ1 and λ2 we must have ˜ 1 , λ2 ) ˜ 2 , λ1 ) Q(λ Q(λ = . R(λ2 ) R(λ1 ) ˜ 1 , λ2 ) must be divisible by R(λ2 ). But the Now the r.h.s. of this identity is a polynomial in λ2 , which implies Q(λ ˜ 1 , λ2 ) is the same as the degree of R(λ2 ), so the maximum degree must be maximum allowed degree in λ2 of Q(λ ˜ 1 , λ2 )/R(λ2 ) must be independent of λ2 . By symmetry the quotient must also be independent of λ1 , so attained and Q(λ is in fact only dependent on N and λ.
To specify cN (λ) a limiting case of the Selberg integral can be used. P ROPOSITION 4.1.3 We have lim (1 + λ1 )SN (λ1 , λ2 , λ) = N SN −1 (2λ − 1, λ2 , λ).
λ1 →−1+
Proof. Since the integrand in (4.1) is symmetrical in t1 , . . . , tN we can write Z SN (λ1 , λ2 , λ) = N !
Z
1
1
dtN 0
tN
Z dtN−1 · · ·
1
dt1 t2
N Y l=1
tλl 1 (1 − tl )λ2
Y 1≤j
|tk − tj |2λ .
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CHAPTER 4
The stated result then follows after application of the general formula Z 1 Z 1 Z “ tλ1 f (t) dt = lim (1 + λ1 ) f (0) tλ1 dt + lim (1 + λ1 ) λ1
→−1+
λ1
0
→−1+
0
1
” tλ1 (f (t) − f (0)) dt = f (0),
0
applicable whenever (f (t) − f (0))/t is integrable near t = 0.
Propositions 4.1.2 and 4.1.3 together give the recurrence relation cN (λ) =
Γ(1 + N λ) cN −1 (λ), Γ(1 + λ)
which, since c1 (λ) = 1, has solution cN (λ) =
N −1
Γ(λ(j + 1) + 1) . Γ(λ + 1)
j=0
Substituting this equation in Proposition 4.1.2 gives the evaluation (4.3) of the Selberg integral, proved for all non-negative integers λ. In fact the evaluation is valid for all λ for which (4.1) is defined. To show this Carlson’s theorem (see, e.g., Titchmarsh [515]) can be used, although in [483] Selberg carries out this step from first principles. P ROPOSITION 4.1.4 For Re(z) ≥ 0 suppose f (z) is analytic and has the bound |f (z)| = O(eμ|z| ), μ < π. Suppose also that f (z) = 0 for z ∈ Z+ . Then f (z) = 0 identically. With f (λ − 1) equal to the integral in (4.1) minus the gamma functions in (4.3), all the criteria of the theorem are satisfied (note in particular that both (4.1) and (4.3) are bounded by constants for Re(λ) ≥ 1), so indeed the evaluation (4.3), proved above for non-negative integer λ, is valid for all λ for which (4.1) is defined. E XERCISES 4.1
(i) Change variables x = sin2 θ in the Euler beta integral (4.2) to obtain the integration
1.
formula
Z
π/2
sin2λ1 +1 θ cos2λ2 +1 θ dθ =
0
Γ(λ1 + 1)Γ(λ2 + 1) . 2Γ(λ1 + λ2 + 2)
(ii) Use the change of variable in (i) to note that (3.118) can be written Z
π
SN (λ1 , λ2 , λ) =
Z
dθN
0
0
N Y
sin2λ1 +1 (θj /2) cos2λ2 +1 (θj /2)
j=1
˛2λ ˛ ˛ ˛ ˛ sin((θj − θk )/2) sin((θj + θk )/2)˛ ,
Y
×
π
dθ1 · · ·
1≤j
which is known as the BCN Selberg integral (see Section 4.7.2). Interpret this integral in terms of the partition function for a log-gas on a half-circle 0 < θj < π with image charges of the same sign at 2π − θj (recall the proof of Proposition 3.6.3). 2. Use the change of variables tl → 1/tl in (4.1) to show Z
∞ 1
3.
Z
∞
dt1 · · ·
dtN 1
N Y
Y
tλl 1 (tl − 1)λ2
l=1
|tk − tj |2λ = SN (−(2 + λ1 + λ2 + 2λ(N − 1)), λ2 , λ). (4.8)
1≤j
(i) Make the change of variables tl = 1 − 1/(1 + e−sl ) in (4.1) to obtain Z N
∞
SN (λ1 , λ2 , λ) = 2
−∞
Z ds1 · · ·
∞
dsN −∞
N Y l=1
e−(λ1 −λ2 )sl (2 cosh sl )λ1 +λ2 +2+2λ(N−1)
Y 1≤j
“
”2λ . 2 sinh |sk −sj |
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THE SELBERG INTEGRAL
(ii) Make the change of variables tl = 2/(1 + cosh sl ) to obtain Z
∞
SN (λ1 , λ2 , λ) = −∞
×
Z
Y
∞
N Y
(sinh2 sl )1/2+λ2 (cosh2 sl )λ1 +λ2 +3/2+2λ(N−1) −∞ l=1 ”2λ “ . sinh |sk − sj | sinh |sk + sj |
ds1 · · ·
dsN
1≤j
4.
(i) Use Liouville’s theorem to show that with “ f (u) =
”2 θ1 (u; q) θ1 (u + π/2; q)
one has f (u) − f (v) =
(θ1 (π/2; q))2 θ1 (u + v; q)θ1 (u − v; q) , (θ1 (u + π/2; q)θ1 (v + π/2; q))2
d (θ1 (π/2; q))2 θ1 (2u; q) f (u) = θ1 (0; q) . du (θ1 (u + π/2; q))4 (ii) [459] Change variables tl = f (ul ) in (4.1) using the formulas of (i) to show SN (λ1 , λ2 , λ) = ×
(θ1 (π/2; q))2λN(N−1)+2N(λ2 +1) (θ1 (3π/4; q))2λ2
Z
π/4
Z
0
(θ1 (ul ; q))2λ1 (θ1 (π/4 + ul ; q)θ1 (π/4 − ul ; q))λ2 (θ1 (ul + π/2; q))4+4λ(N−1)+2(λ1 +λ2 )
π/4
du1 · · ·
duN 0
Y
N Y
θ1 (2ul ; q)
l=1
|θ1 (uj + uk ; q)θ1 (uj − uk ; q)|2λ .
1≤j
Interpret the product over pairs in terms of charges interacting via the pair potential (2.80) in the interval ul ∈ [0, π/4], together with image charges of the same sign in ul ∈ [−π/4, 0]. 5. [193] Suppose λ and λ2 are positive integers. (i) Show from (4.3) that SN is the reciprocal of a polynomial in λ1 . (ii) Show from (4.1) that polynomial in (i) occur at the negative of the powers of t1 , . . . , tN in Q the zeros of λthe Q 2λ 2 t (1 − t ) and show that the order of the zeros is equal the expansion of N l l l=1 1≤j
(λ1 + λ2 + 1)! . λ1 !λ2 !
Show that the particular configuration with the yellow point in [t, t + dt] and all the red points are to its left, and all the blue points to its right occurs with probability tλ1 (1 − t)λ2 dt. Use this to deduce (4.2) in the case λ1 , λ2 ∈ Z≥0 .
4.2 ANDERSON’S DERIVATION 4.2.1 A recurrence in N for SN In the study of some conjectured finite field analogues of the Selberg integral (see [17]), Anderson [16] developed a method of relating SN +1 (λ1 , λ2 , λ) to SN (λ1 + λ, λ2 + λ, λ), and thereby evaluating the Selberg
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CHAPTER 4
integral by iteration in N . The key integration formula developed for this purpose can in fact be found in the classical work of Dixon [137]. The integration formula can be derived as a corollary to a result relating to the density of the roots of a certain random rational function. P ROPOSITION 4.2.1 Consider the random rational function R(λ) :=
n i=1
wi , ai − λ
(4.9)
where the wi are distributed according to the Dirichlet distribution n Γ(s1 + · · · + sn ) sj −1 w , Γ(s1 ) · · · Γ(sn ) j=1 j
(4.10)
with w1 , . . . , wn > 0 and nj=1 wj = 1, and to be denoted Dn [s1 , . . . , sn ] (note that only n − 1 of the wi s are independent). The roots of R(λ), denoted {λj }, have the p.d.f. n (λj − λk ) n−1 Γ(s1 + · · · + sn ) 1≤j λ1 > a2 > λ2 > · · · > λn−1 > an .
(4.12)
Proof. Considering the degree of the numerator when written with a common denominator shows that R(λ) has exactly n − 1 roots. That these are real and interlace as specified in (4.12) follows from a graphical argument, P making essential use of each wi being positive (recall the proof of Proposition 1.8.1) Hence, using too the fact that n i=1 wi = 1, we see that (4.9) can be written Qn−1 n X (λ − λl ) wi = Ql=1 , n λ − a (λ − al ) i l=1 i=1 where the {λj } are real. Computing the residue at λ = aj gives Qn−1 (aj − λl ) wj = Qn l=1 . l=1,l =j (aj − al )
(4.13)
The distribution of {wj }j=1,...,n−1 is given by (4.10). We want to change variables to {λj }j=1,...,n−1 . First, from (4.13), up to a sign n−1 n−1 n−1 h 1 i ^ Y ^ dwj = wj det dλj , aj − λl j,l=1,...,n−1 j=1 j=1 j=1 while the determinant can be evaluated according to the Cauchy double alternant identity (4.33) below. Thus we have that the Jacobian is equal to Q n−1 Y 1≤j
Because (4.11) is a p.d.f. in {λj }, integrating it over the region (4.12) must give unity. Rearranging and replacing n by N + 1 gives the integral evaluation, to be referred to as the Dixon-Anderson integral, N +1 +1 N N Γ(si ) dλ1 · · · dλN (λj − λk ) |λj − ap |sp −1 = i=1 (aj − ak )sj +sk −1 , N +1 Γ( s ) X i 1≤j
139
THE SELBERG INTEGRAL
where X is the domain of integration (4.12) with n → N + 1. We remark that this has the interpretation as the partition function of a log-gas at β = 1 in which there are fixed particles of charge sp − 1 at the points ap (p = 1, 2, . . . , N ) interlaced as specified by (4.12). Dixon in fact gave an integration formula more general than (4.15). This is presented in Exercises 4.2 q.2. We can use (4.15) to establish a recurrence formula in N for the Selberg integral (4.1). P ROPOSITION 4.2.2 Let XN +1 denote the region
1 > x1 > y1 > x2 > y2 > · · · > yN > xN +1 > 0. Consider
K(λ1 , λ2 , λ) :=
XN +1
×
dx1 · · · dxN +1 dy1 · · · dyN
+1 N N
N +1
(4.16)
xλl 1 (1 − xl )λ2
l=1
|yi − xj |λ−1
i=1 j=1
N
|yi − yj |
i<j
N +1
|xi − xj |.
(4.17)
i<j
By first integrating over {yj } we obtain K(λ1 , λ2 , λ) = SN +1 (λ1 , λ2 , λ)
(Γ(λ))N +1 (N + 1)!Γ((N + 1)λ)
while first integrating over {xl } gives K(λ1 , λ2 , λ) = SN (λ1 + λ, λ2 + λ, λ)
Γ(1 + λ1 )Γ(1 + λ2 )(Γ(λ))N . N !Γ(2 + λ1 + λ2 + N λ)
Proof. To integrate over {yj } we make use of (4.15) with sp = λ (p = 1, . . . , N + 1). For the integral over {xl }, replace N → N + 1 in (4.15) and set a1 = 1, aN+2 = 0, s1 = λ2 + 1, sN+2 = λ1 + 1, sp = λ (p = 2, . . . , N + 1). Proposition 4.2.2 gives the recurrence formula SN+1 (λ1 , λ2 , λ) = SN (λ1 + λ, λ2 + λ, λ)
(N + 1)Γ((N + 1)λ)Γ(1 + λ1 )Γ(1 + λ2 ) . Γ(λ)Γ(2 + λ1 + λ2 + N λ)
After iteration with the initial condition S0 := 1 the Selberg integral evaluation (4.3) is obtained.
4.2.2 Relationship to an eigenvalue probability density function The result of Proposition 4.2.1 can be used to provide the eigenvalue p.d.f. for matrices of the form M := ΠAΠ,
Π := 1 − xx † ,
(4.18)
where A is a real symmetric, or complex Hermitian, matrix with eigenvalues a1 > a2 > · · · > an having corresponding multiplicities m1 , m2 , . . . , mn and x is a real or complex normalized Gaussian vector of the same number of rows as A. The matrix Π is thus a projection onto the subspace orthogonal to the vector x (for this reason Π is said to have codimension 1), and the matrix M corresponds to A projected onto this subspace. We begin by noting that the eigenvalues ai of A must occur in M with multiplicity mi − 1. This is a corollary of the following formula for the characteristic polynomial. P ROPOSITION 4.2.3 With M defined by (4.18) we have det(M − λ1) = −λ det(A − λ1)Tr((A − λ1)−1 xx † ).
(4.19)
Proof. Because Π is a projector, we can check that [M, AΠ] = 0. The matrices M and AΠ therefore have the same
140
CHAPTER 4
eigenvalues, so we have det(M − λ1) = det(AΠ − λ1) = det(A − λ1 − A x x †) = det(A − λ1) det(1 − (A − λ1)−1 Ax x † ). But the matrix in the second determinant of the final expression is of the form 1 + Y, where Y has rank 1. In this circumstance, det(1 + Y) = 1 + Tr Y as used in (1.136). The result (4.19) now follows after straightforward manipulation, and making use of the fact that Tr( x x † ) = 1.
According to (4.19) there are an eigenvalue λ = 0, and eigenvalues λ satisfying (j) n n mi j=1 ui ml = 0, (al − λ) ai − λ i=1
(4.20)
l=1
(j)
where the ui denote the diagonal elements of xx† . The fact that M has eigenvalues ai with multiplicity no less than mi − 1 follows immediately from (4.20). Of interest is the distribution of the n − 1 eigenvalues differing (in general) from the ai and 0. According to (4.20) these are given by the zeros of the random mi (j) rational function (4.9) with wi = j=1 ui . Now the sum of the squares of s independent real Gaussians with mean zero and standard deviation σ 2 has distribution proportional to xs/2−1 e−x/2σ (recall Exercises 3.2 q.3). This is referred to as the gamma distribution and denoted Γ[s/2, 2σ 2 ], as used in Proposition 1.9.1 and Section 3.10. Also relevant is the fact that a vector of appropriately normalized gamma distributed variables specifies the Dirichlet distribution. P ROPOSITION 4.2.4 Let X1 , . . . , Xn be independent random variables such that each Xj has distribution Γ[sj , γ], and let ρj :=
Xj X1 + · · · + Xn
for j = 1, . . . , n.
Then the p.d.f. for ρ := (ρ1 , . . . , ρn ) is given by the Dirichlet distribution (4.10). It follows from the above theory that w := (w1 , . . . , wn ) has a Dirichlet distribution with si = βmi /2
(4.21)
where β = 1 for x real and β = 2 for x complex. As a corollary of Proposition 4.2.1 we thus can compute the p.d.f. for the perturbed eigenvalues of M [28], [222]. P ROPOSITION 4.2.5 The eigenvalues of M in (4.18) differing from the eigenvalues of A and from 0, are given by the zeros of the random rational function (4.9) and have the p.d.f. (4.11) with the sj specified by (4.21). 4.2.3 Inter-relations between matrix ensembles with β = 1 and β = 4 The Dixon-Anderson integral (4.15) implies that integrating over the eigenvalue p.d.f. of certain matrix ensembles with β = 1, the eigenvalue p.d.f. of β = 4 matrix ensembles results [220], [222]. To state these inter-relations, let OEN (e−V (x) ) denote the matrix ensemble with orthogonal symmetry having eigenvalue p.d.f. proportional to N l=1
e−V (xl )
1≤j
|xk − xj |.
(4.22)
141
THE SELBERG INTEGRAL
Similarly, let us denote by SEN (e−4V (x) ) the matrix ensemble with symplectic symmetry having eigenvalue p.d.f. proportional to N
e−4V (xl )
(xk − xj )4 .
(4.23)
1≤j
l=1
With the eigenvalues in (4.22) assumed ordered according to x1 > x2 > · · · > xN ,
(4.24)
let even(OEN (f )) denote the distribution of the even labeled coordinates in the ensemble OEN (f ). P ROPOSITION 4.2.6 We have the inter-relations between matrix ensembles even(OE2N +1 (x(a−1)/2 (1 − x)(b−1)/2 )) = SEN (xa+1 (1 − x)b+1 ), even(OE2N ((1 − x)(b−1)/2 )) = SEN ((1 − x)b+1 ).
(4.25) (4.26)
Proof. Consider first (4.25). In (4.15) set N → N + 1, a1 = 1, aN+2 = 0, s1 = (b + 1)/2, sN+2 = (a + 1)/2, sj = 2 (j = 2, . . . , N + 1) and multiply both sides by N+1 Y
(a−3)/2
aj
QN+1 j=1
(a−3)/2
aj
Y
(1 − aj )(b−3)/2
j=1
N+1 Y
(a−1)/2
(a−3)/2
aj
(1 − x)
(b−1)/2
j=1
(λj − λk )
N+1 Y N+2 Y
|λj − ap |sp −1
j=1 p=1
1≤j
)
Y
(1 − aj )(b−3)/2
Y
(aj − ak )
1≤j
∝ OE2N+1 (x while
(1 − aj )(b−3)/2 . Noting that then
(aj − ak )sj +sk ∝ SEN (xa+1 (1 − x)b+1 ),
1≤j
we see that (4.25) follows. To derive (4.26), in (4.15) set a1 = 1, s1 = (b + 1)/2, sj = 2 (j = 2, . . . , N ) and multiply both sides by
QN
j=1 (1
− aj )(b−3)/2 .
The change of variables xj → 12 (1 − xj /L) in (4.25) with a = b = L2 and L → ∞ is the Gaussian limit, and one obtains even(OE2N +1 (e−x
2
/2
)) = SEN (e−x ). 2
(4.27)
Note that (4.26) does not permit a Gaussian limit. On the other hand both (4.25) and (4.26) permit the Laguerre limit xj → xj /L, b = L, L → ∞, giving even(OE2N +1 (x(a−1)/2 e−x/2 )) = SEN (xa+1 e−x ), even(OE2N (e
−x/2
)) = SEN (e
−x
),
(4.28) (4.29)
respectively. Furthermore, the method of proof of Proposition 4.2.6 applied to the integration formula following from (4.88) below being a p.d.f. in {xj } gives the inter-relation between Cauchy ensembles ¯ ¯ even(OE2N +1 ((1 + ix)−(α+1)/2 (1 − ix)−(α+1)/2 )) = SEN ((1 + ix)−(α−1) (1 − ix)−(α−1) ).
With the notation Cbβ,N referring to the p.d.f. corresponding to (3.122) with 2λ → β, applying the change of variables (2.50) to this gives (b−1)/2
(b+1)
even CE1,2N +1 = CE4,N .
(4.30)
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CHAPTER 4
Finally, setting b = 3 in (4.30), noting that in general CEββ,N = CE0β,N +1 |θN +1 =2π
(4.31)
and replacing N → N − 1, we obtain [397] alt COE2N = CSEN ,
(4.32)
where alt denotes the distribution of every second eigenvalue ordered around the circle.
E XERCISES 4.2
1. The objective of this exercise is to verify the Cauchy double alternant identity Q h i 1 1≤j
(i) For each row j, extract a factor 1/
QN
k=1 (xj
(4.33)
− yk ), so that the r.h.s. of (4.33) reads
i hY 1 (xj − yl ) . det j,k=1,...,N l=1 j,k=1 (xj − yk ) N
(−1)N(N−1)/2 QN
l=k
Argue that this determinant is a homogeneous polynomial of degree N (N − 1)/2 in both sets of variables.
(ii) Note that the determinant in (i) vanishes when xj = xj or yk = yk and so contains a factor of the two difference products Y (xk − xj )(yk − yj ). 1≤j
Since this is of the required degree in each set of variables, conclude that the determinant is proportional to the two difference products. Determine the proportionality constant by setting xj = yj , and thus verify (4.33).
(iii) Replace yj by 1/yj in (4.33) to write the Cauchy double alternant identity in the form Q i h 1 1≤j
(4.34)
(iv) The permanent of a square matrix A = [ajk ]j,k=1,...,N is specified as the symmetric function of the elements perm A =
N X Y
al,σ(l)
(4.35)
σ∈SN l=1
(cf. the formula (5.22) below for a determinant). By differentiating (4.34) with respect to x1 , . . . , xN , deduce the Borchardt identity Q » – h i 1 1 1≤j
2. [137] Let α0 , . . . , αn , β0 , . . . , βm have positive real parts, and suppose
Pn i=0
αi =
Pm j=0
βj . Using a method
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THE SELBERG INTEGRAL
analogous to that presented in the proof of Proposition 4.2.1, Dixon derived an integration identity equivalent to Z dx1 · · · dxn Rn
n Y
|xi − xj |
i<j
Y
=
Qn
j=0 Γ(αj ) × Qm i=0 Γ(βi )
where Rn and
i=1
(ai − aj )(αi +αj −1)
0≤i<j≤n
Rm
n “Y n Y
|aj − xi |αj −1
m ”“ Y
j=0
Rm
dx1 · · · dxm
”
j=0
Y
(bi − bj )1−βi −βj
m Y
|xi − xj |
i<j
m n Y Y
|bj − ai |αi −βj
i=0 j=0
0≤i<j≤m
Z
|bj − xi |−βj
m “Y n Y i=1
|aj − xi |−αj
m ”“ Y
j=0
” |bj − xi |βj −1 ,(4.37)
j=0
denote the regions
an ≤ xn ≤ an−1 ≤ · · · ≤ a1 ≤ x1 ≤ a0 ,
bm ≤ xm ≤ bm−1 ≤ · · · ≤ b1 ≤ x1 ≤ b0 ,
respectively. Here we will give a derivation of (4.37) due to Rains [457]. (i) By taking b0 → ∞, show that in the case m = 0, (4.37) reduces to (4.15). (ii) Verify that for b0 > b1 > a0 > a1 Z a0 Z b0 dx dx = (a0 − a1 ) (b0 − b1 ) b1 (x − a0 )(x − a1 ) a1 (b0 − x)(b1 − x) and from this conclude that for b0 > · · · > bN > a0 > · · · > aN , N Y j=1
»Z (b0 −bj ) det
a0
ai
dx (b0 − x)(bj − x)
– = i,j=1,...,N
N Y
»Z (a0 −aj ) det
b0 bi
j=1
dx (x − a0 )(x − aj )
– . i,j=1,...,N
(iii) Subtract row (i − 1) from row i (i = N, . . . , 2) in order) then make use of the Cauchy double alternant determinant formula (4.33) to rewrite the final formula of (ii) as the integral identity Q Z Y 1≤i<j≤N (xi − xj ) (bi − bj ) dx1 · · · dxN QN QN RN i=0 j=1 (bi − xj ) 0≤i<j≤N Q Z Y 1≤i<j≤N (xi − xj ) . (ai − aj ) dx1 · · · dxN QN QN = RN i=0 j=1 (ai − xj ) 0≤i<j≤N Pm Pp Pn ∗ := Now write N = i=0 αi = j=0 βj where αi , βj ∈ Z≥0 . With p i=0 αi (p = 0, . . . , n), by taking the limit ap∗ +1 , . . . , ap∗ +αp+1 → ap∗ +1 and then relabeling ap∗ +1 → ap , and performing an analogous merging procedure with the bs, deduce (4.37) in the case αi , βj ∈ Z≥0 . 3. [223] The objective of this exercise is to present circular analogues of aspects of Dixon’s and Anderson’s working. P q = 1, λj = eiθj (θ1 < θ2 < · · · < θn ), t = eiφ , λ = eiψ and consider the rational function Let qi > 0, n j j=1 Cn (λ) := t − (t − 1)λ
n X j=1
qj . λ − λj
(4.38)
(i) Let e1 denote the n × 1 unit vector (1, 0, . . . , 0)T as in Exercises 1.9 q.3, let U be an n × n unitary matrix ˜ denote the matrix which results by multiplying the first with distinct eigenvalues eiθ1 , . . . , eiθn , and let U ˜ are given by (4.38) with qj = |v1j |2 , where v1j denotes the row of U by t. Show that the eigenvalues of U first entry of the eigenvector corresponding to eiθj . (ii) Deduce from the result of (i) that Cn (z) has exactly n zeros, each of unit modulus. With these zeros denoted
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CHAPTER 4
eiψj (j = 1, . . . , n), and after setting λ = eiψ in (4.38) to rewrite it as (t − 1) “ (ψ − θi ) ” φ X ci cot cot − , 2i 2 2 i=1 n
Cn (λ) =
show that the zeros interlace in the sense θi−1 < ψi < θi
(i = 1, . . . , n,
θ0 := θn mod 2π).
(4.39)
(iii) Note from the result of (i) that we can write Qn j=1 Cn (λ) = Qn
˜j ) (λ − λ
j=1 (λ
− λj )
˜ j := eiψj and so deduce the formulas where λ Qn ˜l) (λj − λ Q −(t − 1)λj qj = n l=1 l=1,l =j (λj − λl ) as well as the constraint
n Y
˜l = t λ
l=1
(iv) Let
Qn
j=1 (z
˜ j ) =: −λ
Pn
j=0 (−1)
n−j
n Y
,
(4.40)
(j = 1, . . . , n)
(4.41)
λl .
l=1
F˜j z j . Use the formulas of (iii) to show
˛ ˛ h ∂q i 1 ˛ ˛ j −(n−1) Q , ˛ = |1 − t| ˛ det ˜ ∂ Fl j,l=1,...,n−1 1≤j
J = |1 − t|−(n−1)
Y 1≤j
˛˜ ˜ j ˛˛ ˛ λk − λ ˛ ˛. λk − λj
(4.42)
(v) Use the formulas of (ii) and the fact that each qj > 0 to show “ n−1 Y j=1
” qjα−1 qnα0 −1 =
1 |1 − t|α0 +(n−1)α−n
Qn−1 Qn Q
j=1
l=1
1≤j
˜ l |α−1 |λj − λ
|λl − λj |2(α−1)
Qn ˜ α0 −1 l=1 |λn − λl | . Qn−1 α0 +α−2 l=1 |λn − λl |
Note from this and the results of (iii) that if (q1 , . . . , qn ) is distributed according to the Dirichlet distribution Dn [(α)n−1 , α0 ] (here the notation (α)n−1 denotes α repeated n − 1 times) while t has p.d.f. specified by the measure Γ2 ( 12 (α0 + (n − 1)α + 1)) |1 − t|α0 +(n−1)α−1 dφ, 2πΓ((n − 1)α + α0 ) ˜ 1 := eiψ1 , . . . , λ ˜ n := eiψn given λ1 := eiθ1 , . . . , λn−1 := eiθn−1 , λn := 1 then the conditional p.d.f. of λ is equal to Qn−1 Qn Qn iψl α0 −1 Y |eiθj − eiψl |α−1 | l=1 |1 − e Q j=1 l=1 iθ A Qn−1 |eiψk − eiψj |, (4.43) iθj 2α−1 iθ α +α−1 k 0 l |e − e | | 1≤j
145
THE SELBERG INTEGRAL
A :=
Γ2 ( 21 (α0 + (n − 1)α + 1)) . 2π(Γ(α))n−1 Γ(α0 )
(vi) With A as in (v) but with α = λ, α0 − 1 = a, let Y A(n − 1)! |1 − eiψl |a n−1 (2π) Mn−1 ((a + a1 + λ)/2, (a + a1 + λ)/2, λ) l=1 n
Co(n,n−1) (ψ, θ) = ×
Y
|eiψk − eiψj |
n−1 Y
1≤j
|1 − eiθl |a1
l=1
Y
|eiθk − eiθj |
n n−1 YY
|eiθj − eiψl |λ−1 .
j=1 l=1
1≤j
Show that (4.43) being a conditional p.d.f. implies the integration formula Z (n − 1)! dψ1 · · · dψn Co(n,n−1) (ψ, θ) = (2π)n−1 Mn−1 ((a + a1 + λ)/2, (a + a1 + λ)/2, λ) R ×
n−1 Y
Y
|1 − eiθl |a+a1 +λ
l=1
|eiθk − eiθj |2λ ,
1≤j
where R denotes the region implied by (4.39) with θ0 = 0 and thus conclude Co p.d.f.
(ψ, θ) defines a joint
(vii) Show that (4.30) can be derived directly from the result of (vi).
4.3 CONSEQUENCES FOR THE β-ENSEMBLES The theory of the previous section relating to Anderson’s derivation of the Selberg integral plays a fundamental role in the theory of the various β-ensembles. In particular, it allows a random three-term recurrence for the characteristic polynomial distinct from that which follows from the tridiagonal matrix characterization Proposition 3.11.1; it provides for inductive constructions of both the Gaussian β-ensemble and Laguerre βensemble; it allows too for a three-term recurrence of the characteristic polynomial of the Cauchy β-ensemble, and thus the circular Jacobi β-ensemble. 4.3.1 Random three-term recurrence for the Jacobi β-ensemble Consideration of Anderson’s derivation of the Selberg integral as implied by Proposition 4.2.2 leads one naturally to the joint p.d.f. Jo(n,n−1) (x, y) := ×
n 1 Γ(2 + α + β + (n − 1)λ) xα (1 − xi )β Γ(1 + α)Γ(1 + β)(Γ(λ))n−1 Sn−1 (α + α1 + λ, β + β1 + λ, λ) i=1 i
|xj − xi |
1≤i<j≤n
n−1 i=1
yiα1 (1 − yi )β1
|yj − yi |
n n−1
|xj − yi |λ−1 . (4.44)
i=1 j=1
1≤i<j≤n−1
Corresponding to the integrations noted in the proof of Proposition 4.2.2, this has the properties Xn
1 y α+α1 +λ (1 − yi )β+β1 +λ Sn−1 (α + α1 + λ, β + β1 + λ, λ) i=1 i × |yi − yj |2λ =: Jo( ,n−1) (y), (4.45) n−1
dx1 · · · dxn Jo(n,n−1) (x, y) =
1≤i<j≤n−1
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CHAPTER 4
Xn
dy1 · · · dyn−1 Jo(n,n−1) (x, y) =
1 Sn (α, β, λ)
n
α1 =β1 =0
β xα i (1 − xi )
i=1
|xi − xj |2λ =: Jo(n, ) (x),
(4.46)
1≤i<j≤n
where Xn is specified by (4.16) with n = N − 1. Note that (4.46) is the p.d.f. for the Jacobi 2λ-ensemble on the interval [0, 1]. Here we will show that the relationship of the conditional p.d.f.’s (n,n−1)
Jo
(x, y)|α1 =β1 =0
(n, ) Jo (x)
(n,n−1)
Jo
,
(x, y)
(4.47)
( ,n−1) Jo (y)
to random rational functions allows the characteristic polynomial of the Jacobi 2λ-ensemble to be generated by a random three-term recurrence [222]. The first of the ratios (4.47) is formally identical to (4.11) with s1 = · · · = sn = λ and thus gives the density of zeros of the random rational function (4.9) with (w1 , . . . , wn ) distributed according to the Dirichlet distribution Dn [(λ)n ], with the notation (λ)n denoting λ repeated n times. The second of the ratios (4.47) is formally identical to (4.11) with n → n + 1 (this is to be done by starting the labelling at zero), λ0 = 1, λn = 0, s0 = β + 1, sn = α + 1 and si = λ (i = 1, . . . , n − 1), and thus gives the density of zeros of the random rational function wi ˜ n+1 (x) := w0 + wn+1 + R , x−1 x x − yi i=1 n
(4.48)
with (w0 , . . . , wn ) distributed according to Dn+1 [β + 1, (λ)n−1 , α + 1]. Let us now seek the recursive construction of a polynomial, the zeros of which have joint density given by (4.46), the latter to be referred to as the Selberg density. # # P ROPOSITION 4.3.1 Let A# −1 (x) := 0, A0 (x) := 1 and define the polynomials {Aj (x)}j=1,...,n by the (j)
(j, )
requirement that their zeros {λl }l=1,...,j say have the p.d.f. J0 α(j) := (n − j)λ + α0 , (j, )
(in particular the zeros of A# n (x) have the p.d.f. J0 term recurrence (j)
(x)| α=α(j) with β=β (j)
β (j) := (n − j)λ + β0 (x)| α=α 0 ). These polynomials satisfy the random threeβ=β 0
(j)
(j)
# # # A# j (x) = w2 (x − 1)Aj−1 (x) + w0 xAj−1 (x) + w1 x(x − 1)Aj−2 (x) (j)
(j)
(4.49)
(j)
where (w0 , w1 , w2 ) is distributed according to D3 [β (j) + 1, (j − 1)λ, α(j) + 1]. Proof. Consider the rational function (4.9) with n = j − 1. Let {ai } have the p.d.f. J0(j−1, ) (x)| α=α(j−1) so that β=β (j−1)
(j)
al = λl , and let {wi } be chosen from the Dirichlet distribution Dj−1 [(λ)j−2 , λ]. Then it follows from Proposition 4.2.2 and the integration formula (4.46) that the roots are the zeros of A# j−2 (x) and thus j−1 X
wl
l=1
x − λl
(j−1)
=
A# j−2 (x) A# j−1 (x)
.
(4.50) (j−1, )
Consider next the rational function (4.48) with n = j − 1. Again let {ai } have the p.d.f. J0 (j)
˜i let the wi = w
(x)| α=α(j−1) , and β=β (j−1)
be chosen from the Dirichlet distribution Dj+1 [β (j) + 1, (k)j−1 , α(j) + 1]. Similarly, it then follows
147
THE SELBERG INTEGRAL
from Proposition 4.2.2 and the integration formula (4.45) that the roots occur at the zeros of A# j (x) and thus j−1 (j) (j) X w ˜j w ˜l w ˜ . + 0 + x x − 1 l=1 x − λ(j−1) l (j)
A# j (x) x(x − 1)A# j−1 (x)
=
(4.51) (j)
According to Exercises 4.3 q.2(i) the marginal distribution of each wi in (4.50) is D2 [λ, (j − 2)λ], while each w ˜l (l = 1, . . . , j − 1) in (4.51) has distribution D2 [λ, (j − 1)λ + α(j) + β (j) + 2]. We similarly calculate that the marginal (j) distribution of w1 as specified in the statement of the proposition is D2 [(j − 1)λ; α(j) + β (j) + 2]. Using the result of (j) (j) (j) ˜i . This implies that we can substitute w1 times Exercises 4.3 q.1 it follows that w1 wi has the same distribution as w the l.h.s. of (4.50) for the sum over l in (4.51) to obtain (j)
A# j (x) x(x − 1)A# j−1 (x) (j)
(j)
=
(j)
(j) w ˜j w1 A # w ˜ j−2 (x) , + 0 + # x x−1 Aj−1 (x)
(4.52)
(j)
where it is required that w ˜j + w ˜0 + w ˜1 = 1. (j)
(j)
˜j , Again using the result of Exercises 4.3 q.2(i), we see that the marginal distribution of w2 agrees with that of w and that the marginal distributions of
(j) w0
and
(j) w ˜0
agree. Thus (4.52) is a rewrite of (4.49).
The recurrence (4.49) does not have the structure (1.154) and so does not relate to a single tridiagonal matrix. Thus it is distinct from the recurrence implied by Proposition 3.11.1 which also generates the characteristic polynomial for the Jacobi β-ensemble. In fact (4.49) relates to the generalized eigenvalue problem for two tridiagonal matrices [553]. 4.3.2 Gaussian β-ensemble A limiting form of (4.9) relating to Gaussian ensembles is obtained by first specializing some of the parameters to obtain n w0 wn+1 wi + + , (4.53) x−1 x x − yi i=1 and requiring (w0 , . . . , wn+1 ) to be distributed according to Dn+2 [α/2, s1 , . . . , sn , α/2]. If we then write 1 α = L2 and take L → ∞, the marginal distribution of w0 and wn+1 have the asymptotic form 12 + 2L N[0, 1] 1 while the wi (i = 1, . . . , n) have to leading order the marginal distribution L2 Γ[si , 1]. Hence with x → ai 1 X 1 2 (1 − L ), yi → 2 (1 − L ) we obtain from (4.53) G Rn+1 (X) := X − μ0 −
n j=1
μj , X − aj
(4.54)
where μ0 has distribution N[0, 1] and μj distribution Γ[si , 1] (i = 1, . . . , n). The same change of variables and scaling in the specialization of (4.11) giving the distribution of the zeros of (4.48) specifies the distribution of the zeros of (4.54). P ROPOSITION 4.3.2 The zeros of the random rational function (4.54) have the p.d.f. n n 1 n+1 (λj − λk ) n+1 1 sp −1 2 2 1≤j λ1 > a1 > λ2 > · · · > an > λn+1 > −∞.
(4.56)
148
CHAPTER 4
We can likewise take the Gaussian limit of the joint p.d.f. (4.44), the marginal distribution (4.46) and the random three-term recurrence (4.49), the latter reproducing (1.154) for the computation of the characteristic yi xi polynomial of the Gaussian β-ensemble. Thus in (4.44) let us change variables xi → ( 12 − 2L ), yi → ( 12 − 2L ), 2 2 set α = β = aL , α1 = β1 = a1 L and take L → ∞. We then obtain the joint p.d.f. a 1/2 (2a)(n−1)λ 1 G(n,n−1) (x, y) := o n−1 π (Γ(λ)) mn−1 (λ; 2(a + a1 )) n n−1 n n−1 2 2 × e−axi |xj − xi | e−a1 yi |yj − yi | |xj − yi |λ−1 , (4.57) i=1
i=1
1≤i<j≤n
where
mn (λ; c) :=
∞ −∞
dx1 · · ·
∞
−∞
i=1 j=1
1≤i<j≤n−1
dxn e−(c/2)
Pn l=1
x2l
|xj − xi |2λ
(4.58)
1≤i<j≤n
and the x’s and y’s are interlaced according to ∞ > x1 > y1 > x2 > y2 > · · · > yn−1 > xn > −∞. Also, in this limit (4.45) and (4.46) read dx1 · · · dxn G(n,n−1) (x, y) o
(4.59)
(4.60)
R
= R
n−1 2 1 e−(a+a1 )yi mn−1 (λ; 2(a + a1 )) i=1
dy1 · · · dyn−1 G(n,n−1) (x, y) o
a1 =0
=
|yi − yj |2λ =: G(o ,n−1) (y),
(4.61)
1≤i<j≤n−1
n 2 1 e−axi mn (λ; 2a) i=1
) |xi − xj |2λ =: G(n, (x), (4.62) o
1≤i<j≤n (n, )
as the p.d.f. of the Gaussian where here R refers to (4.59). With a = 12 , λ = β/2 we recognize G0 β-ensemble (1.160). To generate a random polynomial with zero p.d.f. (4.62) we can take the Gaussian limit of the three(j) (j) (j) term recurrence (4.49). First we note that with (w0 , w1 , w2 ) distributed as specified in Proposition 4.3.1, (j) (j) setting α0 = β0 = aL2 and taking L → ∞, the marginal distributions of w0 and w2 have the asymptotic (j) 1 1 form 12 + 2L N[0, √12a ] while w1 has the leading order marginal distribution L12 Γ[(j − 1)λ, 2a ] (cf. the statements above (4.54)). Thus by also writing x → (4.49) reduces to
1 2 (1
−
x L ),
−j # A# j (x) → (−2L) Cj (x), we see that
# # Cj# (x) = (x − r)Cj−1 (x) − s(j−1) Cj−2 (x)
(4.63)
# 1 ], and C−1 (x) = 0, C0# (x) = where r has distribution N[0, √12a ] while s(j−1) has distribution Γ[(j − 1)λ, 2a 1. Setting 2a = 1, λ = β/2 we see that (4.63) coincides with (1.154), the ak , b2k therein specified by (1.164). A different derivation of (4.63) is also possible. This has its origin in considering a sequence of random matrices defined recursively by [222], [200] diag Mn−1 b Mn = (4.64) bT c
where c, b are chosen from suitable distributions. First we will show that if, for fixed n, the eigenvalue dis( ,n−1) tribution of Mn−1 is prescribed to equal Go (y) as specified in (4.60), then the joint distribution of the
149
THE SELBERG INTEGRAL
eigenvalues of Mn and Mn−1 is given by (4.57). To see this, note that with the eigenvalues of Mn−1 denoted {y1 , . . . , yn−1 }, n−1 det(1n x − Mn ) = det(1n−1 x − Mn−1 ) x − c − j=1
b2j . x − yj
(4.65)
Hence the eigenvalues of Mn are specified by the zeros of (4.54) with μ0 = c, μj = b2j . Suppose now that √ c has distribution N[0, 1/ 2a], while b2j has distribution Γ[λ, 1/2a]. It then follows from Proposition 4.3.2 that the√eigenvalue p.d.f. √ of Mn , given the eigenvalues of Mn−1 , is equal to (4.55) with n → n − 1, si = λ, λj → 2axj , aj → 2aaj , and multiplied by (2a)n/2 . In the case that the eigenvalue p.d.f. of Mn−1 is ( ,n−1) given by Go (y), multiplication by this latter quantity shows that the joint eigenvalue p.d.f. is indeed equal to (4.57). Similar working suffices to compute the eigenvalue p.d.f. of Mn defined recursively by (4.64) for n = 1, 2, . . . . P ROPOSITION 4.3.3 Consider the sequence of random matrices (4.64) with c, b2j having distribution N[0,1], Γ[β/2, 1]. The eigenvalue p.d.f. of Mn is equal to the p.d.f. (4.62) (with a = 12 , λ = β/2) specifying the Gaussian β-ensemble, and furthermore the characteristic polynomial satisfies the three-term recurrence (4.63) with a = 1/2, λ = β/2. Proof. With the eigenvalues of Mn , Mn−1 denoted {xi }, {yj }, let Gn−1 ({xi }, {yj }) denote the conditional eigen-
value p.d.f. of Mn . We know then that Gn−1 is given by (4.55) with n → n − 1, si = β/2. The eigenvalue p.d.f. of Mn , pn (x1 , . . . , xn ) say, must satisfy the recurrence Z pn (x1 , . . . , xn ) = dy1 · · · dyn−1 Gn−1 ({xi }, {yj })pn−1 (y1 , . . . , yn−1 ), (4.66) R
where R denotes the region (4.59) and p0 := 1. The integration formula (4.62) shows that the solution of this recurrence is given by setting pn equal to (4.62), with a = 1/2, λ = β/2, as required. Let qn (x) denote the characteristic polynomial of Mn . Then (4.65) gives n X b2j qn+1 (x) . = x−c− qn (x) x − yj j=1
(4.67)
On the other hand, since (4.64) is a symmetric matrix, according to (1.149) we must have qn−1 (x) X μ2j = qn (x) x − yj j=1 n
(4.68)
for μ2j the square of the first component of the eigenvector corresponding to the eigenvalue yi . Choosing μ2j = b2i /
n X
b2j
(4.69)
j=1
we can check from Proposition 4.2.5 and (4.11) that the implied joint distribution of {yi } and the zeros {xi }Pof qn−1 (x) 2 is the same as that implied by (4.66), thus justifying (4.69). Substituting (4.69) in (4.68), and noting that n j=1 bj has distribution Γ[nβ/2, 1] reclaims the three-term recurrence (4.63).
Note that in the case β = 1 the construction (4.64) produces a matrix Mn ∈ GOEn by the appendage of an extra row and column to a matrix Mn−1 ∈ GOEn−1 .
150
CHAPTER 4
4.3.3 Laguerre β-ensemble In the Dirichlet distribution (4.10), set sn = L/σ and scale ρ1 , ρ2 , . . . , ρn−1 so that ρi = xi /L (i = 1, . . . , n − 1). Taking the limit L → ∞ we see that (4.10) reduces to the product of independent gamma distributions n−1 σ −(n−1) (xi /σ)si −1 e−xi /σ , Γ(s1 ) · · · Γ(sn−1 ) i=1
xi > 0.
(4.70)
With the lower terminal in (4.9) set equal to i = 0 (or equivalently n → n + 1), this result suggests scaling wi → wi /L, ai → ai /L (i = 1, . . . , n), λ → λ/L therein, setting a0 = 1 and taking the limit L → ∞. Noting that then w0 → 1, we see the limiting form of (4.9) reads RL (λ) := 1 +
n i=1
wi , ai − λ
(4.71)
where each wi is now distributed according to Γ[si , σ]. The distribution of the roots of (4.71) is given by the corresponding Laguerre type limiting form of the Dixon-Anderson density (4.11) with n → n + 1.
P ROPOSITION 4.3.4 Consider the random rational function (4.71) with the coefficients wi distributed according to Γ[si , 1]. This has exactly n roots, all of which are real, and for given a1 , . . . , an these roots have the p.d.f. Pn 1 e− j=1 (λj −aj ) Γ(s1 ) · · · Γ(sn )
1≤i<j≤n
n (λi − λj ) |λi − aj |sj −1 , (ai − aj )si +sj −1 i,j=1
(4.72)
where λ1 > a1 > λ2 > a2 > · · · > λn > an .
(4.73)
The joint p.d.f. (4.44) has a well-defined Laguerre limit, specified by changing variables xi → xi /L, yi → yi /L, setting β = L/b, β1 = L/b1 and taking the limit L → ∞. This gives (x, y) L(n,n−1) o :=
n 1 1 −xi /b xα |xj − xi | i e n−1 (n−1 (α + α1 + λ, λ; bb1 /(b + b1 )) Γ(1 + α)(Γ(k)) W i=1 1≤i<j≤n
×
n−1 i=1
yiα1 e−yi /b1
|yj − yi |
(n (a, λ; b) = W
∞
dx1 · · ·
0
|xj − yi |λ−1 ,
(4.74)
i=1 j=1
1≤i<j≤n−1
where
n n−1
∞
dxn 0
n l=1
xal e−xl /b
|xk − xj |2λ
1≤j
and the xs and ys are interlaced according to x1 > y1 > x2 > y2 > · · · > yn−1 > xn > 0.
(4.75)
151
THE SELBERG INTEGRAL
We note too the Laguerre limit of the marginal distributions (4.45), (4.46), n−1 1 (n,n−1) dx1 · · · dxn Lo (x, y) = yiα+α1 +λ e−bb1 yi /(b+b1 ) ( ˜ Wn (α + α1 + λ, λ; bb1 /(b + b1 )) i=1 R × |yk − yj |2λ =: L(o ,n−1) (y), (4.76) 1≤j
˜ R
dy1 · · · dyn−1 L(n,n−1) (x, y) o
α1 =0 1/b1 =0
=
n
1
(n (α, λ; b) W i=1
−xi /b xα i e
) |xk − xj |2λ =: L(n, (x), o
1≤j
(4.77) ˜ refers to (4.75). With b = 2, α = βa/2 and λ = β/2 the latter is the p.d.f. specifying the Laguerre where R β-ensemble (3.128). Let us also note the Laguerre limit of the random three-term recurrence (4.49). The Laguerre limit is (j) (j) (j) (j) (j) (j) (j) obtained by scaling x → x/L, w1 → v1 /L, w2 → v2 /L, w0 = 1, where the v1 , v2 are distributed (j) (j) according to the gamma distributions Γ[(j − 1)λ, b], Γ[(n − j)λ + α0 + 1, b], respectively. With v1 , v2 so −j # specified, and introducing the further scaling A# j (x) = L Bj (x) we see that the Laguerre limit of (4.49) reads (j)
(j)
# # Bj# (x) = (x − v2 )Bj−1 (x) − xv1 Bj−2 (x).
(4.78)
# (x) = 0 and B0# (x) = 1. The zeros of This recurrence is to be solved subject to the initial conditions B−1 (n, ) Bn# (x) are then distributed according to Lo (x)|α=α0 . Note that (4.78) differs from the random three-term recurrence (1.154) with the substitutions (3.132). Such recurrences are in fact related to Laurent orthogonal polynomials [499]. In Proposition 4.3.3 the recursive construction of a sequence of n × n matrices, each with eigenvalue p.d.f. specifying a Gaussian β-ensemble, was noted. It is similarly possible to give a recursive construction of a sequence of matrices whose eigenvalues realize the Laguerre β-ensemble p.d.f. [199]. As noted below the proof of Proposition 4.3.3, the recursive construction in the Gaussian case can be motivated by the natural recursive structure of GOE matrices. In the case of the LOE, the eigenvalue p.d.f. (3.16) with a = N − n − 1, and n nonzero eigenvalues, is realized by XT(n) X(n) , where X(n) is an n × N matrix of standard Gaussian entries. These matrices satisfy the recurrence
XT(n+1) X(n+1) = XT(n) X(n) + xxT ,
n = 0, 1, . . . , N,
(4.79)
for x an N × 1 Gaussian column vector. This suggests inductively defining a sequence of N × N positive definite matrices, indexed by the number of nonzero eigenvalues n, according to A(n+1) = diag A(n) + xxT ,
n = 0, 1, . . . , N,
(4.80)
where diag A(n) refers to the diagonal form of A(n) , and A(0) := [0]N ×N . By an appropriate choice of the distribution of the elements of the random vector x, the eigenvalue p.d.f. of each A(n) is given by a Laguerre β-ensemble.
P ROPOSITION 4.3.5 Let the jth entry xj (j = 1, . . . , N ) of the random vector x in (4.79) be such that x2j has distribution Γ[β/2, 1]. The eigenvalue p.d.f. for the nonzero eigenvalues of A(n) is proportional to n l=1
(N −n+1)β/2−1 −λl
λl
e
1≤j
|λk − λj |β .
(4.81)
152
CHAPTER 4
Proof. We see from (4.79) that det(λ1N − A(n+1) ) = det(λ1N − A(n) ) det(1N − (λ1N − A(n) )−1 x xT ) and thus
n X det(λ1N − A(n+1) ) x2j − =1− det(λ1N − A(n) ) λ − aj j=1
PN j=n+1
x2j
λ
,
(4.82)
where {aj }j=1,...,n denotes the eigenvalues of A(n) . The eigenvalues of A(n+1) are thus given by the zeros of (4.82). Proposition 4.3.4 then tells us that the conditional p.d.f. for the eigenvalues of A(n+1) is given by (4.72) with n → n + 1, s1 = · · · = sn = β/2, sn+1 = (N − n)β/2, an+1 = 0. With Ln ({λj }j=1,...,n+1 ; {aj }j=1,...,n ) denoting this conditional p.d.f, the eigenvalue p.d.f. of A(n) can then be specified as the solution of the recurrence (4.66), modified so that n → n + 1, {xj } → {λj }, {yj } → {aj }, Gn → Ln and R equal to the region (4.73) with n → n + 1. The integration formula (4.77) shows that (4.81) satisfies this recurrence.
4.3.4 The circular Jacobi β-ensemble As we have seen, the Dixon-Anderson density is intimately related to the Jacobi β-ensemble, and the limiting forms (4.55) and (4.72) relate to the Gaussian and Laguerre β-ensembles. In Exercises 4.2 q.3 a circular analogue of the Dixon-Anderson density was given. We will show in this section that a stereographic projection of the latter gives rise to a conditional p.d.f. which relates to the generalized Cauchy β-ensemble (3.124) [223]. This in turn can be used to derive a random three-term recurrence for the characteristic polynomial of the circular Jacobi β-ensemble. We begin by equating (4.38) and (4.40) from Exercises 4.2 q.3, extending the lower terminals to zero and ˜ j with modulus unity onto the real line via the stereographic projections projecting the variables λ, t, λj , λ λ=
z−i , z+i
t=
c−i , c+i
Noting from (4.41) that
we obtain
λj =
yj − i (j = 0), yj + i
λ0 = 1,
˜j = xj − i . λ xj + i
n (yl + i) q0 = nl=1 , c+i (x l=0 l + i) n (z 2 +
j=0 (z − xj ) 1) nj=1 (z −
(4.83)
(4.84)
(qj /q0 ) z−c − , 2 q0 (z + 1) j=1 z − yj n
yj )
=
(4.85)
where x0 > y1 > x1 > · · · > yn > xn+1 .
(4.86)
Analogous to the result of Exercises 4.3 q.3(v), with {qj } specified to have a Dirichlet distribution, the distribution of the zeros of (4.85) can readily be determined by making use of (4.84), the formula n |y − xl | q0 n l=0 j qj = 2 |1 + iyj | l=1,l=j |yj − yl | and the Jacobian relation d q ∧ dc =
n J 2 |1 + ic|2 dx, 2 (1 + ixj )(1 − ixj ) j=0
where J is specified by (4.42), rewritten according to (4.83). P ROPOSITION 4.3.6 Consider the rational function (4.85). Let {qj }j=0,...,n−1 have the Dirichlet distribu-
153
THE SELBERG INTEGRAL
tion
n Γ( nj=0 dj ) d −1 n qj j , Γ(d ) j j=0 j=0
n
qj = 1,
(4.87)
j=0
where n
dj + 1 = 2Re γ.
j=0
Let c have the generalized Cauchy distribution Γ(γ)Γ(¯ γ) (1 + ic)−γ (1 − ic)−¯γ . π22(1−Re γ) Γ(2Re γ − 1) The zeros of the random rational function (4.85) have the p.d.f. A˜
n
−γ
(1 + ixj )
−¯ γ
(1 − ixj )
j=0
×
n
(1 + iyj )γ−dj (1 − iyj )γ¯ −dj
j=1 n n
|yj − xl |dj −1
j=1 l=0
|yj − yk |1−dj −dk
1≤j
|xj − xk |,
(4.88)
0≤j
where Γ(γ)Γ(¯ γ) 1 n n A˜ = π22(1−Re γ) Γ(2Re γ − 1 − i=1 di ) j=1 Γ(dj ) and the inequalities (4.86) are assumed. It is now possible to derive the sought three-term recurrence. P ROPOSITION 4.3.7 Let pn (z; γ; d) denote the monic random polynomial of degree n, specified by its zeros {xj }j=1,...,n having p.d.f. proportional to n
(1 + ixj )−γ (1 − ixj )−¯γ
j=1
|xj − xk |2d
(4.89)
1≤j
(the value of the proportionality can be read off from (4.156) below). With B[α, β] denoting the classical beta distribution on [0, 1] as specified in Exercises 4.3 q.1, let bn ∈ B[2Re γ + nd − 1, nd] (n = 0), b0 = 1, and let cn have the Cauchy distribution Γ(γ + nd)Γ(¯ γ + nd) (1 + ic)−(γ+nd) (1 − ic)−(¯γ +nd) . π22(1−nd−Re γ) Γ(2(Re γ + nd) − 1) We have pn+1 (z; γ + nd; d) 1 (z − cn ) (1 + z 2 )pn−1 (z; γ + (n − 2)d; d), pn (z; γ + (n − 1)d; d) + 1 − = bn bn where p0 = 1.
(4.90)
154
CHAPTER 4
Proof. According to (4.85) X (qj /q0 ) pn+1 (z; γ; d) z−c . = − 2 + 1)pn (z; γ − d; d) q0 (z + 1) j=1 z − yj n
(z 2
(4.91)
To obtain a companion identity to (4.91), introduce the random rational function Qn−1 n X (z − uk ) μj Qk=1 , = n (z − y ) z − yj j j=1 j=1 where {μj } have the Dirichlet distribution Dn [(d)n ]. The distribution of {uk } is then given by Proposition 4.2.1, appropriately specialized. With {yj }j=1,...,n again chosen to have distribution given by the zeros of pn (z; γ − d; d), the marginal distribution of {uk } can be calculated by noting from (4.88) that Z A˜d
R
=
dx0 · · · dxn
n Y
(1 + ixj )−γ (1 − ixj )−¯γ
j=0 n Y
|yk − xl |d−1
k=1 l=0
(1 + iyj )−γ+d (1 − iyj )−¯γ +d
j=1
n n Y Y
Y
Y
|xj − xk |
0≤j
|yj − yk |2d−1 ,
(4.92)
1≤j
˜ d =···=dn =d , and is thus equal to the zero distribution of pn−1 (z; γ − where R denotes the region (4.86) while A˜d := A| 1 2d; d). Consequently n X μj pn−1 (z; γ − 2d; d) . (4.93) = pn (z; γ − d; d) z − yj j=1 In (4.91) and (4.93) let us replace γ → γ + nd. From Exercises 4.3 q.2(i) we calculate that qj ∈ B[d, (n − 1)d] (j = 0), q0 ∈ B[2Re γ + nd − 1, nd], μj ∈ B[d, (n − 1)d] P Pn subject to the constraints n j=0 qj = 1, j=1 μj = 1. Taking this into consideration, and comparing the two identities, (4.90) results.
To relate this to the circular Jacobi 2λ-ensemble (3.122), we know from (3.123) that with the change of variables xj = i(1 − eiθj )/(1 + eiθj ) the p.d.f. (4.89) with γ → γ + 2d (γ real) maps to (3.122) with b = 2γ − 2, λ = d. Thus the zeros of the polynomial pn (z; γ + (n − 1)d; d), with γ real, x1 , . . . , xn say, under the inverse of this change of variables give for {θj } the distribution (3.122) with b = 2γ − 2, λ = d. In particular this gives a recurrence for the characteristic polynomial of the circular β-ensemble distinct from (2.89). E XERCISES 4.3
1. The objective of this exercise is to show that with the beta distribution defined as the twodimensional Dirichlet distribution, B[α, β] = D2 [α; β], and x ∈ B[a + b, c], y ∈ B[a, b], then xy ∈ B[a, b + c]. (i) For x ∈ B[c, d], y ∈ B[a, b] note that the joint distribution of xy is proportional to xc−1 (1 − x)d−1 y a−1 (1 − y)b−1 , and from this deduce that the joint distribution of u := x and z := xy is proportional to uc−a−b (1 − u)d−1 z a−1 (u − z)b−1 . (ii) In the case c = a + b integrate over u to deduce that the p.d.f. of z is proportional to z a−1 (1 − z)b+d−1 .
155
THE SELBERG INTEGRAL
2.
(i) If (w0 , . . . , wn ) is distributed according P to Dn+1 [α0 , . . . , αn ], show that the marginal distribution of wj (j = 0, . . . , n − 1) is given by B[αj , n i=0,i =j αi ]. (ii) In the P setting of (i), show that the marginal distribution of wj + wk , (j = k, j, k ≤ n) is B[αj + αk , n i=0,i =j,k αi ].
3.
(i) [45], [222] For X ∈ GUE∗n+1 (the ∗ is used as in Section 1.6.1), and with Y denoting the top n×n principal ˜ where {yj } are the eigenvalues of Y. minor of X, let U be such that U−1 YU = diag(y1 , . . . , yn ) =: Y Show that –» – » –» – » −1 ˜ 0 Y xn+1 U 0 U Y y = , T † 0T 0T 1 xn+1 xn+1,n+1 y a 1 √ √ where y is an n-component complex vector with Gaussian entries N[0, 1/ 2] + iN[0, 1/ 2] and a has distribution N[0,1]. Use the result of Proposition 4.3.2 with sp = 1 (p = 1, . . . , n) to deduce that the joint distribution of the eigenvalues {xj } of X, and the eigenvalues {yj } of its top n × n principal minor is proportional to n+1 Y −x2 /2 Y Y e l (xj − xk ) (yj − yk ), (4.94) l=1
1≤j
1≤j
where x1 > y1 > · · · > xn > yn > xn+1 .
(4.95)
Deduce this same result from Proposition 4.2.5 by noting – » Y 0 0T 0 = diag(1, . . . , 1, 0)Xdiag(1, . . . , 1, 0) ∼ Π diag X Π, where Π is a complex corank 1 random projection, and the symbol ∼ indicates the matrices in question are similar (i.e. have the same eigenvalues). (ii) [45] In the setting of (i), let the eigenvalues of the top j × j principal minor (j = 1, . . . , n + 1) be denoted (j) (j) xl , (l = 1, . . . , j). Extend the working therein to show that the joint p.d.f. of {xl }l=1,...,j for j = 1, . . . , n + 1 is proportional to n+1 Y
(n+1) 2
e−(xl
l=1
) /2
Y
(n+1)
(xj
(n+1)
− xk
)
n Y
χ(x(j+1) > x(j) ),
(4.96)
j=1
1≤j
where χ(x(j+1) > x(j) ) := χx(j+1) >x(j) >···>x(j+1) >x(j) >x(j+1) . 1
1
j
j
j+1
(iii) Use the characteristic equation for the eigenvalues of a general (n + 1) × (n + 1) Hermitian matrix X, implied by the first displayed equation in (i) to deduce the interlacing property (4.95) with the eigenvalues of a principal minor (assuming the latter are distinct). Deduce the same result from the final displayed relation in (i). 4.
(i) [222] Let A = X† X, where X is an n × N (n > N ) complex Gaussian matrix such that A is a member of the LUE with a = n − N . Suppose b times a row of complex Gaussians is appended to X to form Y, and x†x as considered consider B = Y† Y. By noting that this is equivalent to the rank 1 perturbation B = A+b in (3.72), use (3.73) and Proposition 4.3.4 to conclude that the joint distribution of the eigenvalues of A and B is N PN PN Y Y aal e− j=1 aj e−(1/b) j=1 (bj −aj ) (bj − bk )(aj − ak ), (4.97) l=1
1≤j
where b1 > a 1 > b 2 > a 2 > · · · > b N > a N . (ii) [215] Let X(N) denote an N × N complex Gaussian matrix such that
(4.98) X†(N) X(N)
is a member of the (n)
LUE with a = 0. Let X(n) denote the n × N matrix formed from the first n rows of X(N) . Let xj
156
CHAPTER 4
(j = 1, . . . , n) denote the nonzero eigenvalues of X(n) . Proceed as in (i) to show that the joint p.d.f. of (n) {xj }j=1,...,n for n = 1, . . . , N is equal to N Y l=1
Y
(N )
e−xl
(N)
(xj
(N)
− xk )
N−1 Y
χ(x(p+1) > x(p) ),
(4.99)
p=1
1≤j
where χ(x(p+1) > x(p) ) := χx(p+1) >x(p) >···>x(p+1) >x(p) >x(p+1) >0 . 1
p
1
p
(4.100)
p+1
5. [222] Let A be an n × n member of the GSE with variance such that the eigenvalue p.d.f. of the independent eigenvalues is proportional to n Y Y 2 e−yl (yj − yk )4 . (4.101) l=1
1≤j
Regard A as a 4n × 4n real matrix by replacing each complex entry in the real quaternion according to (1.36) so √ that A is now fourfold degenerate. Next border A by one extra row and column specified as b times a vector of standard Gaussians, obtaining eigenvalues {xi }. Use the theory below (4.65) to conclude that the joint distribution of {xi } and {yi } is proportional to n+1 Y
Y
2
e−xl /2b
l=1
1≤j
(xj − xk )
n Y
2
e−yl (1−1/2b)
l=1
Y
(xj − xk )
n n+1 YY
|xj − yk |,
(4.102)
j=1 k=1
1≤j
where x1 > y1 > x2 > · · · > yn > xn+1 . Note that (4.102) in the case b = 1 is identical to the eigenvalue p.d.f. for GOE2n+1 , and so conclude from (4.101) that even(GOE2n+1 ) = GSEn , thus realizing (4.27).
4.4 GENERALIZATION OF THE DIXON-ANDERSON INTEGRAL We know from Proposition 4.2.6 that the Dixon-Anderson integral (4.15) implies an inter-relation between the distribution of every second eigenvalue in a Jacobi orthogonal ensemble, and the eigenvalue PDF of a Jacobi symplectic ensemble. This integral can be generalized to give an integration formula which relates the distribution of every (r + 1)th eigenvalue in a certain Jacobi ensemble with β = 2/(r + 1) to the eigenvalue PDF of another Jacobi ensemble with β = 2(r + 1) [199]. First the generalization of (4.15) will be presented.
P ROPOSITION 4.4.1 Let Ar denote the interlaced region aj > λr(j−1)+1 > λr(j−1)+2 > · · · > λrj > aj+1
(j = 1, . . . , n − 1),
(4.103)
and let Cˆ be specified in terms of the Selberg integral by Cˆ =
n−1 l=1
l 1 Sr sp + 2(l − 1)r/(r + 1) − l, sl − 1, 1/(r + 1) . r! p=1
(4.104)
157
THE SELBERG INTEGRAL
One has 1 Cˆ
dλ1 · · · dλr(n−1) Ar
(λj − λk )2/(r+1)
1≤j
=
r(n−1) n j=1
|λj − ap |sp −1
p=1
(aj − ak )r(sj +sk −2/(r+1)) .
(4.105)
1≤j
Proof. Essential use will be made of the fact, to be established in Exercises 4.4 q.1, that the integral itself, Lr,n ({ap }), say, interpreted as an analytic function of {ap }p=1,...,n , is symmetric in {(ap , sp )} up to a phase in {sp }. With this assumed, we begin by considering Lr,n ({ap }) as an analytic function of a1 in the appropriately cut complex a1 -plane. This function has singularities at a2 , . . . , an . In relation to the singularity at a2 , taking a1 → a2 reveals that the singular behavior results from the integration of (4.103) for j = 1 only, which can effectively be factorized from the rest of the integral. Simple rescaling of the integrand makes this singular behavior explicit, leading to the result Lr,n ({ap }) =
” 1 (a2 − ap )r(sp −1) (a1 − a2 )r(r−1)/(r+1)+r(s1+s2 −1) Sr (s1 − 1, s2 , 1/(r + 1)) r! p=3
n “Y
×Lr,n−1 ({ap }p=2,...,n )|s1 →s1 +s2 +2/(r+1)−1 F (a1 − a2 ; {ap }p=2,...,n ),
(4.106)
where F (z; {ap }p=2,...,n ) is analytic in z and equal to unity at z = 0. Using now the fact that the analytic continuation of Lr,n ({ap }) is symmetric in {(ap , sp )} gives the further factorization formula Lr,n ({ap }) = Rr,n ({ap })G({ap }),
(4.107)
where Rr,n denotes the r.h.s. of (4.105) and G is analytic in {ap } and symmetric in {(ap , sp )}. To determine G, one can check that upon the replacements ap → cap both Lr,n and Rr,n are homogeneous in c of ˆ can be calculated according to the formula the same degree, and so G must be a constant. The constant C ˆ= C
lim
a1 ,...,an →a
Lr,n ({aj })/Rr,n ({aj })
and this in turn can be computed in terms of the Selberg integral by repeated use of (4.106).
To present the generalization of Proposition 4.2.6, notation extending that relating to (4.22) and (4.23) is required. Thus we denote by MEβ,N (g) the eigenvalue p.d.f. proportional to N
g(xl )
l=1
|xk − xj |β ,
(4.108)
1≤j
and suppose the eigenvalues are ordered as in (4.24). Furthermore, we denote by Dr the operation of integrating over all eigenvalues not labeled by a multiple of r (or equivalently observing only those eigenvalues labeled by a multiple of r). P ROPOSITION 4.4.2 We have Dr+1 (ME2/(r+1),(r+1)N +r (xa (1 − x)b )) = ME2(r+1),N (x(r+1)a+2r (1 − x)(r+1)b+2r ), Dr+1 (ME2/(r+1),(r+1)N ((1 − x)b )) = ME2(r+1),N ((1 − x)(r+1)b+2r ).
(4.109) (4.110)
Proof. One specializes the parameters in (4.105), and multiplies by a suitable function of {aj } as in the proof of Proposition 4.2.6.
Taking the Gaussian limit of (4.109) gives Dr+1 (ME2/(r+1),(r+1)N +r (e−x )) = ME2(r+1),N (e−(r+1)x ), 2
2
(4.111)
158
CHAPTER 4
which generalizes (4.27). Taking the Laguerre limit of (4.109) and (4.110) gives Dr+1 (ME2/(r+1),(r+1)N +r (xa e−x )) = ME2(r+1),N (x(r+1)a+2r e−(r+1)x ), Dr+1 (ME2/(r+1),(r+1)N (e
−x
)) = ME2(r+1),N (e
−(r+1)x
),
(4.112) (4.113)
which generalizes (4.28) and (4.29). Let us specify CEbβ,N as above (4.30). The circular analogue of (4.105) given in Exercises 4.4 q.3, with the parameters appropriately specialized, gives (r+1)b+r
(4.114)
altr+1 (CE02/(r+1),(r+1)N ) = CE02(r+1),N ,
(4.115)
Dr+1 (CEb2/(r+1),(r+1)N +r ) = CE2(r+1),N . Setting now b = 1/(r + 1) and recalling (4.31) gives
where altr+1 denotes the distribution of θr+1 , θ2(r+1) , . . . with θr+1 regarded as the origin, generalizing (4.32). 1. [199] Let A = (nj )j=1,...,r+q , nj = 0 or 1 be a sequence of r 0s or q 1s in a line with 1 ≤ q ≤ r. Furthermore let
E XERCISES 4.4
„j K(nj ) =
0, #0 s to the right of nj ,
and in terms of this define K(A) =
r+q X
nj = 0 nj = 1
« = nj
r+q X
(1 − nk ),
k=j+1
K(nj ).
(4.116)
j=1
The objective of this exercise is to show X
e−2πiK(A)/(r+1) = 0.
(4.117)
A
(i) Show that (4.117) is equivalent to the summation X
e−2πi
Pr+q j=1
nj (r+q−j)/(r+1)
= 0.
(4.118)
A
(ii) Note that the l.h.s. of (4.118) is equal to the coefficient of z q in F (z), where F (z) :=
X
e−2πi
Pr+q j=1
nj (r+q−j)/(r+1)
z
Pr+q j=1
nj
.
n1 ,...,nr+q =0,1
Evaluate F (z) as a product, simplifying the product by using the factorization formula N Y (1 − ze2πi(l−1)/N ) = 1 − z N ,
(4.119)
l=1
and then expand the simplified product to show that F (z) contains no term proportional to z q , as required. 2. [199] Consider the integral in (4.105), Lr,n ({ap }), with the factors |λj − ap |sp −1 therein replaced by power functions according to the formula implied by (4.124) below. Note that then, according to Cauchy’s theorem, the positions of {ap } can be moved into the complex plane, with the value of the integral being independent of the contours, provided no contour crosses a branch of the power functions, and so corresponds to the analytic continuation. Here this analytic continuation is to be studied in the case that al and al+1 swap places on the real axis for general l = 1, . . . , n − 1. This is achieved by the deforming of contours as indicated in Figure 4.1.
159
THE SELBERG INTEGRAL
real axis a l+1
a l+2
al
a l−1
~
a l+1
~
real axis
~
a l+2
~
al
a l−1
Figure 4.1 The contours from al+2 to al+1 , al+1 to al , and al to al−1 are deformed to the contours joining the corre˜l = al+1 , a ˜l+1 = al and sponding tilded variables. Our interest is in the limit that a ˜j = aj (j = l, l + 1), a all contours in the second diagram run along the real axis. In the case l = n − 1 the contour from a ˜l+2 to ˜l to a ˜l−1 is to be deleted. a ˜l+1 is to be deleted, while in the case l = 1 the contour from a
(i) To study the integrand of (4.105) in the case of the second configuration of Figure 4.1, for notational con(μ) venience set λ(j−1)r+μ = λj (μ = 1, . . . , r), and refer to these as species j. Moreover, in the second (μ) ˜ (μ) . From the Figure 4.1, note that beconfiguration let the integration variables be tilded so that λj → λ j ˜l+1 there are r coordinates of species l, and for some p, q ∈ {0, . . . , r} there are p coordinates tween a ˜l and a of species l − 1. (ii) Next one would like to show that only configurations with p = q = 0 contribute, due to a cancellation effect. Suppose first that p = 0 while q = 0, and that furthermore the r species l variables are to the left of the q ˜l+1 ). Inspection of (4.105) shows that interchanging the position species l − 1 variables in the interval (˜ al , a of coordinates corresponding to different species leaves the magnitude of the integrand unchanged, but changes the phase by a factor of e−2πi/(r+1) for each interchange of a species l − 1 and left neighbouring species l. Conclude from this that for a general ordering of the r species l variables, and q species l − 1 ˜l+1 ), the phase changes by K(A) as specified variables, amongst a given set of r + q positions in (˜ al , a by (4.116) and so according to (4.117) cancel when summed over all configurations. Make the same final conclusion in the cases q = 0 (p = 0) and p, q = 0. (iii) By comparing the contributing configurations for p = q = 0 with the allowed configurations before al and al+1 were interchanged, together with the corresponding integrands, conclude ˛ ˛ Lr,n ({ap })˛ a ↔a = e−πir(sl +sl+1 −2/(r+1)) Lr,n ({ap }), l l+1 sl ↔sl+1
which is the sought analytic continuation property. 3. Let Rr denote the interlaced region θj−1 < ψ(r−1)j+1 < ψ(r−1)j+2 < · · · < ψrj < θj
(j = 1, . . . , n)
˜ in terms of the constant C ˆ (4.104) and the Morris integral (4.4) according with θ0 := 0, θn := 2π, and define C to ˛ ˜ = Cˆ ˛˛ Mr (a, a, 1/(r + 1)) C with a :=
1 ( 2
Pn p=1
1 ˜ C
{sp }→{αp }
αp + 2(n − 1)r/(r + 1) − n). The objective of this exercise is to show that Z dψ1 · · · dψrn Rr
=
Y 1≤j
Y
|eiψk − eiψj |2/(r+1)
|e
−e
iθj r(αj +αk −2/(r+1))
|
|eiψj − eiθp |αj −1
j=1 p=1
1≤j
n rn Y Y
.
(4.120)
160
CHAPTER 4
(i) Denote the l.h.s. of (4.120) by Qr.n ({eiθp }). Use the method of q.2 to obtain the factorization ˜ Qr,n ({wp }) = Sr,n ({wp })G({w p }), where
““
Y
Sr,n ({wp }) =
1−
1≤j
wj ”“ wk ””r((αj +αk )/2−1/(r+1)) 1− wk wj
˜ with G({w p }) a symmetric function of {(wp , αp )}, analytic in {wp }. Note furthermore that both sides are ˜ is a constant. homogeneous of degree zero and so G ˜ is unity by computing (ii) Show that G lim
θ1 ,...,θn−1 →0
Qr,n ({wp }) . Sr,n ({wp })
4.5 DOTSENKO AND FATEEV’S DERIVATION Dotsenko and Fateev [138] devised a contour integration method to calculate an integral more general than the Selberg integral, which occurred in their study of conformal field theory. Below we will detail a modification of their method as it applies to the original Selberg integral (4.1). The evaluation of the more general integral, which in fact will appear in the calculation of correlation functions in the one-component log-gas system given in Chapter 13, is the subject of Exercises 4.5 q.1. 4.5.1 Strategy Let us suppose for definiteness that 0 < Re(λ) 1 and −1 < Re(λ2 ) 0. As a function of λ1 the Selberg integral (4.1) is then analytic for Re(λ1 ) > −1. The contour integration technique of Dotsenko and Fateev gives the functional relationship SN (λ1 , λ2 , λ) = SN (−λ1 − λ2 − 2 − 2λ(N − 1), λ2 , λ)
N −1 j=0
sin π(λ1 + λ2 + 2 + (N − 1 + j)λ) . (4.121) sin π(λ1 + 1 + jλ)
The r.h.s. is meromorphic for Re(λ1 ) < −, 0 < 1, so this equation provides the analytic continuation of SN as a function of λ1 into the region Re(λ1 ) ≤ −1. To establish the functional relationship (4.121), the integrals ∞ 1 1 ∞ S(p,N −p) (λ1 , λ2 , λ) := dt1 · · · dtp dtp+1 · · · dtN f (t1 , . . . , tN ), (4.122) 0
0
1
1
where f (t1 , . . . , tN ) :=
N l=1
|tl |λ1 |1 − tl |λ2
|tk − tj |2λ
(4.123)
1≤j
are required. The integral S(p,N −p) is related to S(p−1,N −p+1) , and so by iteration S(N,0) := SN is related to S(0,N ) . A change of variables then gives (4.121). From (4.121) it is possible to prove that SN , when divided by (4.3) is a bounded analytic function of λ1 in the finite complex λ1 plane. The next step in the proof is to show that both SN and (4.3) have the same asymptotic behavior as |λ1 | → ∞. This then gives, by Liouville’s theorem, that both sides are the same function of λ1 . It remains to specify the undetermined function of λ2 (which is immediate since SN is symmetric in λ1 and λ2 ), λ and N . This problem has already been solved in Selberg’s method of Section 4.1.
161
THE SELBERG INTEGRAL
4.5.2 Details The following relationship between S(p,N −p) and S(p−1,N −p+1) holds. P ROPOSITION 4.5.1 One has S(p,N −p) (λ1 , λ2 , λ) =
sin π(N − p + 1)λ sin π(λ1 + λ2 + 2 + (N + p − 2)λ) p N −p+1 sin πpλ sin π(λ1 + 1 + (p − 1)λ) ×S(p−1,N −p+1) (λ1 , λ2 , λ).
Proof. The integral (4.122) is studied via a related contour integral in the variable u, with {tj }j=1,...,N / {tp } parameters such that 0 < t1 < · · · < tp−1 < 1 < tp+1 < · · · < tN . The path of integration is the contour C, specified as the path along the real axis on the upper half-plane side, with indentations around the points 0, 1, t1 , . . . , tp−1 , tp+1 , . . . , tN into the upper half-plane consisting of semicircles with vanishing small radii. The integrand is chosen as uλ1 (1 − u)λ2
N Y
(tl − u)2λ |tl |λ1 |1 − tl |λ2 ,
l=1 l=p
Y
|tk − tj |2λ .
1≤j
The multivalued power functions are defined in terms of the principal argument of the variable. Thus if v = reiθ with r > 0, −π < arg θ ≤ π then v μ := r μ eiθμ . Consequently j u < tl , |tl − u|2λ , (4.124) (tl − u)2λ = e−2πiλ |tl − u|2λ , u > tl . With this definition the integrand as a function of u is analytic in the half-plane Im(u) > 0. With the parameters λ2 , λ obeying the restrictions given in the strategy subsection, and with λ1 < 0, the integrand as a function of u decays sufficiently fast that the contour C can be closed in the upper half-plane without changing the value of the integral. But the integrand is analytic in the upper half-plane so we conclude by Cauchy’s theorem that the integral over C vanishes. On the other hand, let us evaluate the integral over C by parametrization. For the portion of C along (−∞, 0), taking into consideration the definition of the power functions, gives Z 0 dtp f (t1 , . . . , tN ) eπiλ1 −∞
with f given by (4.123). Next consider the portion of C along (0, 1) in the limit that the radius of the half-circles around t1 , . . . tp−1 shrink to zero. We obtain p−1 X
e−2πilλ
l=0
Z
tl+1
dtp f (t1 , . . . , tN ), tl
where in the terminals t0 := 0 and tp := 1. Finally, for the integral over (1, ∞) we obtain e−πiλ2
N X
e−2πi(l−1)λ
Z
tl+1
dtp f (t1 , . . . , tN ), tl
l=p
where in the terminals tp := 1 and tN+1 := ∞. The sum of the above three terms equals zero. Multiplying through by e−πiλ1 and taking imaginary parts eliminates the first of these terms and gives X
Z
p−1
−
l=0
tl+1
sin π(λ1 + 2lλ)
dtp f (t1 , . . . , tN ) = tl
N X l=p
Z sin π(λ1 + λ2 + 2(l − 1)λ)
tl+1
dtp f (t1 , . . . , tN ). tl
162
CHAPTER 4
Now integrate over 0 < t1 < · · · < tp−1 < 1 and 1 < tp+1 < · · · < tN < ∞. Since the integrands above are symmetrical in the variables t1 , . . . , tN we see that within the individual sums the integrals are equal. Use of the definition (4.122) then gives −
p−1 “1 X
p
” sin π(λ1 + 2lλ) S(p,N−p) (λ1 , λ2 , λ)
l=0
“ =
N ” X 1 sin π(λ1 + λ2 + 2(l − 1)λ) S(p−1,N−p+1) (λ1 , λ2 , λ). N − p + 1 l=p
Simple manipulation to absorb the minus sign and use of the summation formula sin θ + sin(θ + a) + · · · + sin(θ + na) =
sin((n + 1)a/2) sin(θ + na/2) sin(a/2)
gives the stated result.
The functional relation (4.121) follows from Proposition 4.5.1 by iteration, starting with p = N , then decreasing p down to p = 1, and the change of variables tj → 1/tj (j = 1, . . . , N ) in S(0,N ) , making use of (4.8). Let us now use the functional equation to prove that the Selberg integral (4.1), when divided by the product of gamma functions (4.3), is an analytic function of λ1 in the finite complex λ1 -plane. Since SN is analytic for Re(λ1 ) > −1, the functional equation gives that in general the only possible singularities of SN as a function of λ1 are simple poles at λ1 = −1 − jλ − k,
j = 0, . . . , N − 1, k = 0, 1, . . .
.
On the other hand the product of gamma functions (4.3) has poles of the same order at precisely these points. Furthermore the product of gamma functions (4.3) has in general simple zeros at λ1 = −2 − λ2 − λ(N − 1 + j) − k,
j = 0, . . . , N − 1,
k = 0, 1, . . .
.
which from the functional equation (4.121) are also simple zeros of SN . Thus the possible singularities of (4.1) divided by (4.3) all cancel out, leaving a function analytic in the finite complex λ1 -plane, as claimed. It remains to establish that (4.1) and (4.3) are the same function of λ1 for |λ1 | → ∞. The asymptotic behavior of the Selberg integral can be calculated as follows. P ROPOSITION 4.5.2 For Re(λ1 ) > −1 and |λ1 | → ∞ ∞ −N (1+λ2 )−N (N −1)λ SN (λ1 , λ2 , λ) ∼ λ1 ds1 · · · 0
0
∞
dsN
N
sλl 2 e−sl
|sk − sj |2λ .
1≤j
l=1
Proof. Change variables tl = e−sl /λ1 in (4.1) to obtain SN (λ1 , λ2 , λ) = λ−N 1
Z 0
∞
Z ds1 · · ·
∞
dsN 0
N Y
e−sl (1+1/λ1 ) (1 − e−sl /λ1 )λ2
l=1
Y
|e−sk /λ1 − e−sj /λ1 |2λ .
1≤j
Expanding the exponentials e−s/λ1 to first order in 1/λ1 gives the stated result.
(4.125)
The functional equation (4.121) gives that this asymptotic behavior remains valid for all −π < arg(λ1 ) < π. On the other hand, using the asymptotic formula Γ(x + a) ∼ xa Γ(x)
as |x| → ∞, arg(x) = −π,
(4.126)
163
THE SELBERG INTEGRAL
we see that the product of gamma functions in (4.3) has precisely the same asymptotic power law decay in λ1 . We thus conclude that (4.1) divided by (4.3) is a bounded entire function of λ1 , and thus by Liouville’s theorem is independent of λ1 . By symmetry the ratio must also be independent of λ2 . The remaining unspecified function of λ and N is calculated as in Selberg’s derivation. E XERCISES 4.5
1. The objective of this exercise is to prove the integration formula
J(0,n)(m,0) (α, β; ρ m Z n Z ∞ Y Y dti := i=1
1
j=1
= m!n!ρ2nm ×
m−1 Y j=0
1
dτj fnm ({ti }, {τj }, α, β; ρ)
0
n−1 m n Y Γ(lρ ) Y Γ(jρ − n) Y Γ(1 + β + lρ )Γ(−1 + 2m − α − β − (n − 1 + l)ρ ) Γ(ρ ) j=1 Γ(ρ) Γ(−α − lρ ) l=1 l=0
Γ(1 − n + α + jρ)Γ(1 − n + β + jρ) , Γ(2 − n + α + β + (m − 1 + j)ρ)
where Q 2ρ Q 2ρ ˛ n m ˛Y Y (τj − τj ) ˛ ˛ i
β = −ρ β
ρ = 1/ρ.
This integral was first given explicitly in [192], but is implicit in the work of Dotsenko and Fateev [138]. Let J(p,n−p)(q,m−q) (α, β; ρ) Z p Z 1 n Y Y := P dti i=1
0
∞
dti
1
i =p+1
q Z Y
1
dτj 0
j=1
Z m Y j =q+1
∞
dτj fnm ({ti }, {τj }, α, β; ρ),
1
where P denotes the principal part, and suppose ρ = −ν + iμ
with 0 < |ν| 1, 0 μ < 1 and |ρ| = 1.
A. Analytic continuation of J(n,0)(m,0) in the complex α-plane (i) By considering the behavior of Z τ1α
Z
0
1 2 (τ i=1 1 − ti )
dtp Qp
dt1 . . . 0
with p = n for τ1 → 0, conclude that, up to terms O(ν), J(n,0)(m,0) is analytic for Re(α) > n−1. By considering −α/ρ the same behavior for general p < n as well as the behavior of tn for tn → ∞ conclude that J(p,n−p)(m,0) is an analytic function of α for p − 1 < Re(α) < ∞, Im(α) ≥ 0. (ii) Revise the method of the proof of Proposition 4.5.1 to conclude that the workings are unaffected by the introduction of the τ variables and thus J(p,n−p)(m,0) (α, β; ρ) = J(p−1,n−p+1)(m,0) (α, β; ρ)
p sin π(n − p + 1)ρ sin π(α + β + 2 + (n + p − 2)ρ ) . n−p+1 sin πpρ sin π(α + 1 + (p − 1)ρ )
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CHAPTER 4
Iterate this equation to show J(n,0)(m,0) (α, β; ρ) = J(0,n)(m,0) (α, β; ρ)
n−1 Y j=0
sin π(α + β + 2 + (n − 1 + j)ρ ) . sin π(α + 1 + jρ )
From (i) note that this equation provides the analytic continuation of J(n,0)(m,0) into the region −1 < Re(α) ≤ n − 1, Im(α) ≥ 0. (iii) Change variables τj → 1/τj and ti → 1/ti in the definition of J(0,n)(0,m) to show J(0,n)(0,m) (α, β; ρ) = J(n,0)(m,0) (−α − β + 2(n − 1) − 2ρ(m − 1), β; ρ). (iv) Proceed similarly as in the first part of (ii) to show J(0,n)(q,m−q) (α, β; ρ) = J(0,n)(q−1,m−q+1) (α, β; ρ)
sin π(m − q + 1)ρ sin π(α + β + 2 + (m + q − 2)ρ) q . m−q+1 sin πqρ sin π(α + 1 + (q − 1)ρ)
By iterating this equation, substituting in the final equation of (ii), and using (iii), obtain the functional equation J(n,0)(m,0) (α, β; ρ) =
n−1 Y j=0
m−1 sin π(α + β + 2 + (n − 1 + j)ρ ) Y sin π(α + β + 2 + (m − 1 + j)ρ) sin π(α + 1 + jρ ) sin π(α + 1 + jρ) j=0
×J(n,0)(m,0) (−α − β + 2(n − 1) − 2ρ(m − 1), β; ρ). With Re(β) ≈ n − 1, from (i) note that this equation provides the analytic continuation of J(n,0)(m,0) into the region Re(α) < 0. With Re(β) ≈ n − 1 and Im(β) > 0, from the final statement in (ii) note that the functional equation also provides the analytic continuation into the region 0 ≤ Re(α) ≤ n − 1, Im(α) < 0.
B. Verification of a product of gamma functions ansatz for the evaluation of J(n,0)(m,0) as a function of α and β (i) Use the analytic property in A(i) and the analytic continuations in A(ii) and A(iv) to show that J(n,0)(m,0) (α, β; ρ) = an,m (α, β, ρ)
n−1 Y l=0
×
m−1 Y j=0
Γ(1 + α + lρ )Γ(1 + β + lρ ) Γ(2 − m + α + β + (n − 1 + l)ρ )
Γ(1 − n + α + jρ)Γ(1 − n + β + jρ) , Γ(2 − n + α + β + (m − 1 + j)ρ)
where an,m (α, β, ρ) is analytic in the finite complex α-plane. (ii) Use the method of Proposition 4.5.2 to show that for |α| → ∞ with Re(α) > n − 1, J(n,0)(m,0) (α, β; ρ) ∼ cα2nm−n−m−nβ
−mβ−n(n−1)ρ −m(m−1)ρ
,
where c is nonzero and independent of α. Use the analytic continuation formulas to argue that this asymptotic behavior is valid for all −π < arg(α) < π, and show that the product of gamma functions has the same asymptotic behavior. (iii) Apply Liouville’s theorem, using the results of B(i), (ii), to show that an,m (α, β, ρ) is independent of α. Use the symmetry of J(n,0)(m,0) in α and β to then conclude that an,m (α, β, ρ) is independent of β also, and so an,m (α, β, ρ) = cn,m (ρ). C. Calculation of the function cn,m (ρ) and the final evaluations (i) Use the method of Proposition 4.1.3 to obtain the formula lim (α + 1)J(n,0)(m,0) (α, β; ρ) = mJ(n,0)(m−1,0) (2ρ − 1, β; ρ).
α→−1+
165
THE SELBERG INTEGRAL
(ii) From C(i) and the equations in B(iii) and B(i) obtain the recurrence relation cn,m (ρ) = mρ2n
Γ(mρ − n) cn,m−1 (ρ) Γ(ρ)
and thus show that cn,m (ρ) = m!ρ2nm
m Y Γ(iρ − n) cn,0 (ρ). Γ(ρ) i=1
(iii) Note that J(n,0)(0,0) (α, β; ρ) = Sn (α , β , ρ ) and thus conclude cn,0 (ρ) = n!
n Y Γ(iρ ) . Γ(ρ ) i=1
(iv) Substitute the results of C(i) and C(ii) in the equation in B(iii) and then substitute that result in B(i) to evaluate J(n,0)(m,0) . Substitute the evaluation of J(n,0)(m,0) in the second equation in A(ii) to obtain the desired evaluation of J(0,n)(m,0) . 2. An integration formula of M. Riesz (see, e.g., [444]) states that for Re(α), Re(β), Re(d − α − β) > 0 Z Rd
dξd kα,β = , d−α d−β | z1 − z2 |d−(α+β) | z1 − ξ| | z2 − ξ|
where kα,β =
π d/2 Γ(α/2)Γ(β/2)Γ((d − α − β)/2) . Γ((d − α)/2)Γ((d − β)/2)Γ((α + β)/2)
(i) For d = 1 use the Euler integral evaluation and (4.8) in the case n = 1 to reclaim this result. (ii) The so-called complex Selberg integral Z
Z
AN (α, β, γ) := R2
dr1 · · ·
drN R2
N Y l=1
|rl |2α |rl − e|2β
Y
|rk − rj |2γ ,
1≤j
e an arbitrary unit vector in R2 , has been considered by Aomoto [18] and Dotsenko-Fateev [138], who have obtained the factorization type formula QN “ ”2 j=1 sin π(α + (j − 1)γ/2) sin π(β + (j − 1)γ/2) sin πjγ/2 SN (α, β, γ/2) AN (α, β, γ) = QN N ! j=1 sin π(α + β + (N + j − 2)γ/2) sin πγ/2 where SN denotes the Selberg integral (4.1). Show that in the case N = 1 this result implies the Riesz formula for d = 2.
4.6 AOMOTO’S DERIVATION Our final derivation of the Selberg integral evaluation is due to Aomoto [19]. The objective of Aomoto’s approach is to relate SN (λ1 , λ2 , λ) to SN (λ1 + 1, λ2 , λ). This is done via the auxiliary functions 1 1 (α) Ip (x) := dt1 · · · dtN F (t1 , . . . , tN ; α, p, x), (4.127) 0
0
166
CHAPTER 4
where F (t1 , . . . , tN ; α, p, x) := (t1 − x)α · · · (tp − x)α (tp+1 − x)α−1 · · · (tN − x)α−1 ×
N
Ip(α) (x)[g] := (α)
(α)
1
1
dt1 · · ·
0
|tk − tj |2λ
1≤j
l=1
and
tλl 1 (1 − tl )λ2
dtN g(t1 , . . . , tN )F (t1 , . . . , tN ; α, p, x). 0
(1)
(1)
Note that Ip (x)[1] = Ip (x), I0 (x) = SN (λ1 , λ2 , λ) and IN (0) = SN (λ1 + 1, λ2 , λ). 4.6.1 Method Aomoto’s derivation makes use of the fundamental theorem of calculus. In particular the two equations 1 1 ∂ ! tp+1 F (t1 , . . . , tN ; α, p, x) = 0 dt1 · · · dtN (4.128) ∂tp+1 0 0 and
1
dt1 · · ·
0
1
dtN 0
∂ !2 t F (t1 , . . . , tN ; α, p, x) = 0, ∂tp+1 p+1
(4.129)
which follow from the fundamental theorem of calculus provided tp+1 F vanishes at tp+1 = 0 and 1 (i.e., provided λ1 > −1 and λ2 > 0), are used. After the partial differentiation is performed, the resulting terms are simplified by using symmetry properties of the integrands. With x = 0 an appropriate linear combination of the two equations then gives the recurrence λ1 + 1 + (N − p − 1)λ (1) Ip+1 (0) = I (1) (0). (4.130) λ1 + λ2 + 2 + (2N − p − 2)λ p (1)
(1)
Iteration, and the relationship between between I0 (0), IN (0) and the Selberg integral, give the recurrence SN (λ1 + 1, λ2 , λ) =
N −1 p=0
λ1 + 1 + pλ SN (λ1 , λ2 , λ). λ1 + λ2 + 2 + (N − 1 + p)λ
4.6.2 Derivation of the recurrence We will consider the simplification of (4.128) and (4.129) separately. P ROPOSITION 4.6.1 We have (1) λ2 Ip (1)
where Ip
(1)
(1)
! 1 = λ1 + λ2 + 1 + (N − p − 1)λ Ip(1) , 1 − tp+1 (1)
=: Ip (0) and Ip [g] := Ip (0)[g].
Proof. Setting α = 1 and performing the partial derivatives in (4.128) gives » (λ1 + λ2 + 1)Ip(1) − λ2 Ip(1)
– » – N X 1 tp+1 + 2λ = 0. Ip(1) 1 − tp+1 tp+1 − tk k=1 k=p+1
(4.131)
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THE SELBERG INTEGRAL
(1)
The summation can be simplified. For k > p + 1 the integrand of Ip is symmetrical in tk and tp+1 . Thus » – » – tp+1 tk = Ip(1) . Ip(1) tp+1 − tk tk − tp+1 Taking the arithmetic mean of both sides gives » – » – tp+1 1 tp+1 1 tk = Ip(1) = Ip(1) . Ip(1) + tp+1 − tk 2 tp+1 − tk tk − tp+1 2 (1)
For k ≤ p the integrand of Ip is symmetrical in tk and tp . Thus » – » – » – tp+1 tp+1 tp tp+1 (1) = Ip(1) = Ip−1 . Ip(1) tp+1 − tk tp+1 − tp tp+1 − tp h i tp tp+1 (1) (1) is antisymmetrical under But the integrand of Ip−1 is symmetrical in tp and tp+1 so the integrand of Ip−1 tp+1 −tp the interchange tp ↔ tp+1 and this quantity therefore vanishes. The summand has thus been simplified for all k, and the stated equation follows.
P ROPOSITION 4.6.2 We have ! (1) 1 (1) = λ2 Ip(1) + λ1 + λ2 + 2 + (2N − p − 2)λ Ip+1 . λ2 Ip 1 − tp+1 Proof. Setting α = 1 and performing the partial derivative in (4.129) gives » (λ1 + 2)Ip(1) [tp+1 ] − λ2 Ip(1)
– » – N X t2p+1 t2p+1 + 2λ = 0. Ip(1) 1 − tp+1 tp+1 − tk k=1 k=p+1
(1)
(1)
Now Ip [tp+1 ] = Ip+1 and » Ip(1)
t2p+1 1 − tp+1
–
» = Ip(1)
– » – (tp+1 − 1)(tp+1 + 1) + 1 1 (1) = −Ip(1) − Ip+1 + Ip(1) . 1 − tp+1 1 − tp+1
Furthermore, for k > p + 1 » – » – t2p+1 t2p+1 1 1 t2k (1) Ip(1) = Ip(1) = Ip(1) [tp+1 + tk ] = Ip+1 , + tp+1 − tk 2 tp+1 − tk tk − tp+1 2 while for k < p + 1
» Ip(1)
t2p+1 tp+1 − tk
–
» (1)
= Ip+1
tp+1 tp+1 − tk
– =
1 (1) I . 2 p+1
Substitution of these simplifications and rearrangement gives the stated equation.
The recurrence (4.130) results from subtracting the equation in Proposition 4.6.1 from the equation in Proposition 4.6.2. 4.6.3 Solution of the recurrence Iteration of (4.131) M − 1 times to increase λ1 at each step gives SN (λ1 + M, λ2 , λ) = SN (λ1 , λ2 , λ)
N −1 p=0
Γ(λ1 + 1 + M + pλ)Γ(λ1 + λ2 + 2 + (N − 1 + p)λ) . Γ(λ1 + 1 + pλ)Γ(λ1 + λ2 + 2 + M + (N − 1 + p)λ)
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Taking the limit M → ∞ on both sides using the asymptotic formula of Proposition 4.5.2 on the l.h.s. and the asymptotic formula (4.126) on the r.h.s. determines SN as a function of λ1 , SN (λ1 , λ2 , λ) = AN (λ2 , λ)
N −1 p=0
Γ(λ1 + 1 + pλ) . Γ(λ1 + λ2 + 2 + (N − 1 + p)λ)
But SN is symmetrical in λ1 and λ2 , so SN (λ1 , λ2 , λ) = cN (λ)
N −1 p=0
Γ(λ1 + 1 + pλ)Γ(λ2 + 1 + pλ) . Γ(λ1 + λ2 + 2 + (N − 1 + p)λ)
This is precisely the equation obtained using Selberg’s derivation, so the function cN (λ) is specified as in Section 4.1, and the derivation of (4.3) completed. 4.6.4 A Fuchsian differential equation With multivariable notation t = {t1 , . . . , tN }, the elementary symmetric functions are defined by tp1 · · · tpr . er (t) = er (t1 , . . . , tN ) :=
(4.132)
1≤p1 <···<pr ≤N
In terms of this notation, one has that the multidimensional integral (4.127) is proportional to the q = N case of the family of integrals 1 x x 1 (α) Ip,q (x) := dt1 · · · dtN −q dtN −q+1 · · · dtN G(t1 , . . . , tN ; α, p, x), 0
0
0
0
where G(t1 , . . . , tN ; α, p, x) :=
N
tλl 1 (1 − tl )λ2 |x − tl |α−1
|tk − tj |2λ ep (x − t1 , . . . , x − tN ).
1≤j
l=1
It is shown in Exercises 4.6 q.1 that (4.127) satisfies the linear differential-difference equation (α)
(N − p)Ep Ip+1 (x) = −(Ap x + Bp )Ip(α) (x) + x(x − 1)
d (α) (α) I (x) + Dp x(x − 1)Ip−1 (x), dx p
(4.133)
where Ap = (N − p)(λ1 + λ2 + 2λ(N − p − 1) + 2α), ! Dp = p λ(N − p) + α ,
Bp = (p − N )(λ1 + α + λ(N − p − 1)),
Ep = λ1 + λ2 + 1 + λ(2N − p − 2) + α,
(α)
(α)
and thus so does Ip,N (x). In fact the derivation of (4.133) shows that Ip,q (x) for general q = 0, 1, . . . , N satisfies (4.133) provided Re(α) > 0. The differential-difference system (4.133) is in fact the component form of a single first order matrix differential equation d (α) U(x) (α) I (x) = I (x) dx x(x − 1) (α)
(4.134)
where I(α) (x) = [Ip (x)]p=0,...,N and U(x) is the (N + 1) × (N + 1) tridiagonal matrix with diagonal (a0 , . . . , aN ), leading upper diagonal (b0 , . . . , bN −1 ), and leading lower diagonal (c1 , . . . , cN ) where, with Ap , . . . , Ep as in (4.133) ap = Ap x + Bp ,
bp = (N − p)Ep ,
cp = x(x − 1)Dp .
169
THE SELBERG INTEGRAL
(α) The fact that {Ip,q (x)}p=0,...,N satisfies (4.133) for each q means that the vector solution I(α) (x) in (4.134) (α) can be replaced by the matrix solution [Ip,q (x)]p,q=0,...,N . Because the determinant of this matrix is nonzero (α) (a fact which can be seen by studying the small x behavior of Ip,q (x) [122]), this is a fundamental matrix of solutions for the differential equation. We remark too that all singular points of (4.134) (x = 0, 1, x, ∞) are regular, making it an example of a Fuchsian differential equation. (α)
E XERCISES 4.6
1. [188] With Ip (x) specified by (4.127), the objective of this exercise is to derive the differentialdifference equation (4.133). (i) Using some of the manipulations in the proof of Propositions 4.3.4 and 4.6.2 derive the formulas – » 1 d (α) (α) (α) , Ip = −pαIp−1 − (N − p)(α − 1)Ip(α) Ip(α) [tp+1 ] = Ip+1 + xIp(α) , dx tp+1 − x » – ( x (α) » – ( 1 (α) tp+1 tp+1 I , k < p + 1, − 2 Ip−1 , k ≤ p, (α) (α) 2 p+1 = = Ip Ip+1 (α) x (α) 1 (α) tp+1 − tk t I I , k > p + 1, + I , k > p + 1, p+1 − tk p p p+1 2 2 (α)
where here Ip
(α)
(α)
(α)
:= Ip (x) and Ip [g] := Ip (x)[g].
(ii) Use (i) and the method of the proof of Proposition 4.3.4 to show » – “ α ” ´ ` x d (α) 1 (α) Ip − xp + λ Ip−1 . λ2 Ip(α) = λ1 + λ2 + α + λ(N − p − 1) Ip(α) − 1 − tp+1 N − p dx N −p (iii) Use (i) and the method of the proof of Proposition 4.6.2 to show ` ´ (α) −Ep Ip+1 = (λ1 + 2)x + λ2 (x + 1) + 2xλ(N − p − 1) + 2x(α − 1) Ip(α) » – 1 x2 d (α) “ pαx2 ” (α) Ip−1 − λ2 Ip(α) . − Ip − λpx2 + N − p dx N −p 1 − tp+1 (iv) Combine the results of (ii) and (iii) to derive the stated recurrence. 2. [20] In this exercise a q-integral which reduces to the Selberg integral in the limit q → 1 will be evaluated. The q-integral to be considered is Z J(α) := Φ(t) 1 D(t)w, ˜ ξF
where Φ(t) :=
n Y
α+(j−1)(1−2γ)
tj
j=1
and w ˜ :=
(qtj ; q)∞ (q β tj ; q)∞
Y 1≤i<j≤n
dq t n dq t 1 ··· , t1 tn
(q 1−γ tj /ti ; q)∞ , (q γ tj /ti ; q)∞
Q D(t)
The q-integral over ξF is defined as Z ξF
f (t)w ˜ := (1 − q)n
:=
Y
(z; q)∞ :=
∞ Y
(1 − zq j ),
j=0
(ti − Qtj ).
1≤i<j≤n
X
f (t1 , . . . , tn ),
where the sum is over all points such that t1 = q ν1 , t2 /t1 = q ν2 q γ , t3 /t2 = q ν3 q γ , . . . , tn /tn−1 = q νn q γ with each νj ∈ Z≥0 . In the limit q → 1− J(α) tends to the Selberg integral (4.1) with λ1 = α + n − 2 − (n − 1)γ, λ2 = β − 1, λ = γ, and integration domain 1 ≥ t1 ≥ t2 · · · ≥ tn ≥ 0.
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CHAPTER 4
The strategy is again to seek a recurrence relating J(α + 1) to J(α). Using the fact that the α → ∞ behavior can be computed explicitly (this contrasts with the Selberg integral, for which a multiple integral is obtained), the recurrence uniquely specifies J(α). (i) Show that for α → ∞, the behavior of J(α) is given by the term t1 = 1, t2 = q γ , . . . , tn = q (n−1)γ in its definition as a sum, and thus J(α) ∼ q An where An =
Pn
j=1 (α + (j
n Y Γq (β + (j − 1)γ)Γq (jγ)(1 − q)2(j−1)γ+β , Γq (γ) j=1
− 1)(1 − 2γ) + n − j)((j − 1)γ) and Γq (u) := (1 − q)1−u (q; q)∞ /(q u ; q)∞ .
(ii) Define the q-shift operator of the i-th coordinate of φ(t) by Ti φ(t) = φ(t1 , . . . , qti , . . . , tn ) and define the covariant derivative by ∇i φ(t) := φ(t) −
Ti Φ(t) Ti φ(t). Φ(t)
By noting that Φ(t) vanishes at t1 = q −1 , tj+1 /tj = q −1+γ (j = 1, . . . , n−1), which defines the boundary of ξF , show from the definitions that Z Φ(t)∇i φ(t)w ˜ = 0. ξF
(iii) Let σ ∈ Sn denote a permutation of indices, σf (t) = f (tσ(1) , . . . , tσ(n) ). Show that σΦ(t) = Uσ (t)Φ(t),
where
Y
Uσ (t) =
1≤i<j≤n σ−1 (i)>σ−1 (j)
“ t ”2γ−1 θ(q γ t /t ; q) j j i , ti θ(q 1−γ tj /ti ; q)
with θ(x; q) := (x; q)∞ (q/x; q)∞ (q; q)∞ , and use the property θ(qx) = −(1/x)θ(x) to show that Ti Uσ (t) = Uσ (t) for every i so that Uσ (t) is a constant on ξF . Use these properties and the result of (ii) to show Z Φ(t)σ∇i φ(t)w ˜=0 ξF
and hence conclude that Z Φ(t)Asym(∇i φ(t))w ˜ = 0,
where
Asymf :=
ξF
(iv) Show that
X σ∈Sn
n 1 − q β t1 Y t1 − q −γ tj T1 Φ(t) = qα , Φ(t) 1 − qt1 j=2 t1 − q γ−1 tj
and use this result to show that with φ(t) := (1 − t1 )(t2 · · · tr )
Y 1≤i<j≤n
(ti − q γ tj )
sgn(σ)σf.
(4.135)
171
THE SELBERG INTEGRAL
we have ∇1 φ(t) = a − b − c + d, where Y
a := t2 · · · tr
(ti − q γ tj ),
c := q α+n−1 t2 · · · tr
1≤i<j≤n
b := t1 · · · tr
Y
n Y
k=2
(ti − q γ tj ),
1≤i<j≤n
Y
(t1 − q −γ tk )
k=2
Use (iii) to conclude
(ti − q γ tj ),
2≤i<j≤n
n Y
d := q α+β+n−1 t1 · · · tr
Y
(t1 − q −γ tk )
(ti − q γ tj ).
2≤i<j≤n
Z ξF
Φ(t)Asym(a − b − c + d)w ˜ = 0.
(v) Kadell’s lemma [336, lemma 4] states that for M ⊂ {1, . . . , n} Asym
“ Y
” (Q; Q)|M | (Q; Q)n−|M | tj Q D(t) = Qa(M ) e|M | (t) 1 D(t), (1 − Q)n j∈M
where a(M ) = |{(i, j) : 1 ≤ i < j ≤ n, i ∈ / M, j ∈ M }|,
(Q; Q)n :=
n Y
(1 − Qj )
j=1
and er (t) denotes the elementary symmetric function (4.132). Noting that by the interchanges t2 ↔ t1 , t3 ↔ t2 , . . . , tn ↔ tn−1 , Asym(c) = q α+n−1 q −(n−1)γ Asym(t1 · · · tr−1 qγ D(t)), Asym(d) = q α+β+n−1 q −(n−1)γ Asym(t1 · · · tr−1 tn qγ D(t)) use Kadell’s lemma to show (q γ ; q γ )r−1 (q γ ; q γ )n−r+1 (1 − q α+n−1−(n+r−2)γ )er−1 (t) 1 D(t), (1 − q γ )n (q γ ; q γ )r (q γ ; q γ )n−r Asym(−b + d) = − (1 − q α+β+n−1+(1−r)γ )er (t) 1 D(t). (1 − q γ )n Asym(a − c) = q γ(r−1)
Substitute in the result of (iv) to deduce the recurrence Z Z 1 − q n−r+1 1 − q α+n−1−(n+r−2)γ Φ(t)er (t) 1 D(t)w ˜ = q 1+(r−1)γ Φ(t)er−1 (t) 1 D(t)w, ˜ 1 − qr 1 − q α+β+n+(1−r)γ ξF ξF and iterate this recurrence to obtain the desired recurrence for J(α), J(α + 1) =
n Y r=1
q (r−1)γ
1 − q α+n−1−(n+r−2)γ J(α). 1 − q α+β+n−1+(1−r)γ
(vi) Iterate the recurrence above and make use of the functional equation Γq (x + 1) = [x]q Γq (x),
[x]q :=
1 − qx , 1−q
(4.136)
to get a relationship between J(α + M ) and J(α). Now use the formula Γq (x + p) ∼ ([x]q )p Γq (x) x→∞
(4.137)
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CHAPTER 4
to take the limit M → ∞, and then use the asymptotic result in (i) to obtain the evaluation J(α) = q An
n Y Γq (β + (j − 1)γ)Γq (α + n − 1 − (n + j − 2)γ)Γq (jγ) , Γq (γ)Γq (α + β + n − 1 − (n − j)γ) j=1
where An is specified in (i). 3.
(i) In the case γ ∈ Z>0 show that the integrand in the q-integral defining J(α) in q.2 above is symmetric and thus deduce that in this case Z 1 Z 1 1 J(α) = ··· Φ(t)1 D(t)w ˜ n! 0 0 R1 P d t j where 0 f (t) qt := (1 − q) ∞ j=0 f (q ). (ii) From (i) and Kadell’s lemma (q.2(v) above) with M = ∅, show that in the case γ ∈ Z>0 Z 1 Z 1 1 J(α) = ··· Φ(t) qγ D(t)w. ˜ Γqγ (n + 1) 0 0 (iii) [36] Assuming the identities Z 1
tx+p−1
0
“q
(qt; q)∞ (q x ; q)p Γq (x)Γq (y) , dq t = x+y y (q t; q)∞ (q ; q)p Γq (x + y)
”
(q −b ; q)p (q; q)a+b ; q (q −(b+1) t)p = a+1 , t (q ; q)p (q; q)a (q; q)b b deduce that for any Laurent polynomial f CT (t; q)a
„
Γq (x + y) Γq (x)Γq (y) „ =
«N Z
1
Z
0
(q; q)a (q; q)b (q; q)a+b
1
dq t 1 · · ·
dq t N 0
«N CT
N Y
(tj ; q)a
N Y
tx−1 j
j=1
“q
j=1
tj
” ;q
b
(x; q)a :=
(x; q)∞ , (xq a ; q)∞
(qtj ; q)∞ f (t1 , . . . , tN ) (q y tj ; q)∞
f (q −(b+1) t1 , . . . , q −(b+1) tN ),
where x = −b and y = a + b + 1. (iv) Use (ii), (iii) and the evaluation of J(α) given in q.2 (vi) to deduce the q-Morris identity [552] CT
N Y
(xj ; q)a
j=1
“q ” ;q xj b
Y 1≤j
“x
j
xk
;q
N−1 ” “ x ” Y Γq (a + b + 1 + λl)Γq (1 + λ(l + 1)) k q ;q = . xj Γq (a + 1 + λl)Γq (b + 1 + λl)Γq (1 + λ) λ λ l=0
(v) Take a = b = 0 in the q-Morris identity, simplify the r.h.s. to Γq (λN + 1)/(Γq (λ + 1))N and use the definition of Γq (u) to take the limit λ → ∞ and thus obtain the identity [374] Y “ xj ” “ xk ” 1 CT ;q = . (4.138) q ;q N−1 xk xj ∞ ∞ (q; q) ∞ 1≤j
4.7 NORMALIZATION OF THE EIGENVALUE P.D.F.’S The multidimensional integrals specifying the normalization of the eigenvalue p.d.f.’s for the Gaussian, circular, Laguerre and Jacobi random matrix ensembles can all be calculated as limiting cases of the Selberg integral, as we will now show. Furthermore, in the Gaussian and Laguerre cases the same limiting procedure applied to (4.49) gives a random three-term recurrence for the corresponding characteristic polynomials.
173
THE SELBERG INTEGRAL
4.7.1 The various ensembles Gaussian ensemble From Proposition 1.3.4 the normalization Gβ,N for the eigenvalue p.d.f. of the Gaussian ensemble is given by ∞ ∞ N β 2 Gβ,N = dλ1 · · · dλN e− 2 λl |λk − λj |β −∞
−∞
= β −N/2−N β(N −1)/4
l=1 ∞
−∞
dt1 · · ·
1≤j
∞
−∞
dtN
e− 2 tl
1 2
|tk − tj |β .
(4.139)
1≤j
l=1
To express Gβ,N in terms of the Selberg integral (4.1), change variables in (4.1) tl → 1/2 − tl /2L and put L2 /2 ! 2 2λ = β and λ1 = λ2 = L2 /2. Use of the elementary limit 1 − (tl /L)2 → e−tl /2 as L → ∞ shows lim 2L (2L)N +βN (N −1)/2 SN (L2 /2, L2 /2, β/2) = β N/2+N β(N −1)/4 Gβ,N . 2
L→∞
The limit is computed using the evaluation (4.3) of SN and Stirling’s formula (1.92), and the following result obtained. P ROPOSITION 4.7.1 For all β such that the integral is defined, N !mN (β/2) ∞ := dt1 · · · −∞
∞
−∞
dtN
N
e
l=1
− 12 t2l
|tk − tj | = (2π) β
N/2
N −1 j=0
1≤j
Γ(1 + (j + 1)β/2) Γ(1 + β/2) (4.140)
(this integral is referred to as Mehta’s integral [395]) and thus Gβ,N = β −N/2−N β(N −1)/4 (2π)N/2
N −1 j=0
Γ(1 + (j + 1)β/2) . Γ(1 + β/2)
Note that these results have already been obtained in (1.160) and (1.163). A further direct evaluation is given in Exercises 4.7 q.2 using the p.d.f. (4.55). Circular ensemble Proposition 2.2.5 gives the normalization Cβ,N as π π Cβ,N := dθ1 · · · dθN −π
−π
|eiθk − eiθj |β ,
(4.141)
1≤j
which is referred to as Dyson’s integral [146]. This is a special case of the trigonometric integral MN (a, b, λ) (3.120). Specifically Cβ,N = (2π)N MN (0, 0, β/2). From the evaluation (4.4), the following result is then obtained. P ROPOSITION 4.7.2 For all β such that the integral Cβ,N is defined, we have Cβ,N = (2π)N
Γ(1 + βN/2) . (Γ(1 + β/2))N
This result was obtained in Proposition 2.8.7 as a consequence of the realization of the circular β-ensemble in terms of unitary Hessenberg matrices.
174
CHAPTER 4
Laguerre ensemble Here, from Proposition 3.2.2, the normalization is given by ∞ ∞ N βa/2 −βxl /2 WaβN := dx1 · · · dxN xl e 0
0
|xk − xj |β .
(4.142)
1≤j
l=1
To compute this integral from the Selberg integral, in (4.1) change variables tl → xl /L and put 2λ = β, βL/2 ! → e−βtl /2 as L → ∞ then gives λ2 = βL/2, λ1 = βa/2. Using the elementary limit 1 − tLl lim LN +N aβ/2+βN (N −1)/2SN (βa/2, βL/2, β/2) = WaβN .
L→∞
Use of (4.3) and Stirling’s formula (1.92) allows this limit, and thus WaβN , to be evaluated. P ROPOSITION 4.7.3 For all β such that the integral is defined we have WaβN = (β/2)−N (aβ/2+1+(N −1)β/2)
N −1 j=0
Γ(1 + (j + 1)β/2)Γ(aβ/2 + 1 + jβ/2) . Γ(1 + β/2)
This result has previously been obtained in (3.134).
Jacobi ensemble According to (3.74), the normalization is given by 1 1 N CabβN := dy1 · · · dyN (1 + yl )aβ/2 (1 − yl )bβ/2 −1
−1
|yk − yj |β .
(4.143)
1≤j
l=1
This integral is obtained from the Selberg integral by changing variables tl → (yl + 1)/2 and putting 2λ = β, λ1 = aβ/2 and λ2 = bβ/2. Thus CabβN = 2N +(a+b)βN/2+βN (N −1)/2SN (aβ/2, bβ/2, β/2),
(4.144)
as already noted in (3.116).
Cauchy ensemble The equation (3.123) shows that the Cauchy ensemble is equivalent to the circular Jacobi ensemble. The normalization of the latter is the special case a = b of (4.4). Thus ∞ ∞ N 1 (Cy) NN := |λk − λj |β dλ1 · · · dλN 2 α (1 + λl ) −∞ −∞ l=1
=2
βN (N −1)/2−2(α−1)N
1≤j
π MN (α − β(N − 1)/2 − 1, α − β(N − 1)/2 − 1, β/2). (4.145) N
The normalization of the generalized Cauchy ensemble in (3.124) is given in Exercises 4.7 q.4.
4.7.2 Root systems and the Selberg integral The integrand of the Selberg integral exhibits structures indicative of a wider mathematical setting. To investigate further, a convenient starting point [24] is the evaluation of the Dyson integral (4.141) given by
175
THE SELBERG INTEGRAL
Proposition 4.7.2, written as the constant term identity CT
N
(1 − e
N jλ . ) = λ j=1
tj −tk λ
j=k
(4.146)
Macdonald [375] observed that with each tj regarded as the unit vector in the direction of xj in RN , the vectors tj − tk all lie in the (N − 1)-dimensional real hyperplane x1 + · · · + xN = 0. Moreover, the set of vectors {tj − tk , j = k} has the property of being invariant with respect to reflections in any hyperplane orthogonal to a member of the set. In general, for a set of vectors E, and a nonzero vector α ∈ E, the ∈ E : β, α orthogonal hyperplane is Pα = {β = 0}, while the reflection σα in the orthogonal hyperplane of α is defined by σα (γ ) = γ −
2γ , α α . α, α
(4.147)
Note that this mapping sends α to − α and leaves invariant all vectors in Pα . The set of vectors also has two other properties which allow it to be identified as a crystallographic root system. D EFINITION 4.7.4 Let E be a subspace of RN and let ·, · denote the usual inner product (dot product). Then R is said to form a root system if a. R is finite, spans E and 0 is not in R; b. If α ∈ R, the reflection σα leaves R invariant. The set R is said to form a crystallographic root system if it has the further property ∈ R, then 2β, α c. If α , β / α, α is an integer. In fact there are only a small number of crystallographic root systems. The root system under consideration is referred to as AN −1 , or as the root system of type A. There are four other infinite families of crystallographic root systems. These are denoted BN , CN , DN and BCN . BCN is the set of vectors ±tj , ±2tj , ±(tj ± tk ). The root systems BN , CN and DN are subsets of BCN : BN is obtained by omitting the vectors ±2tj , CN by omitting the vectors ±tj , and DN by omitting both these sets. The root system BCN is said to be nonreduced because it is the union of BN and CN . This suggests considering the constant term of a product of the type in (4.146) where the variables, interpreted as vectors, correspond to one of these other root systems. In the BCN case one is thus led to consider CT
N
(1 − e−tj )λ1 (1 − etj )λ1 (1 − e−2tj )λ2 (1 − e2tj )λ2
j=1
×
(1 − etj −tk )λ (1 − etk −tj )λ (1 − etk +tj )λ (1 − e−(tj +tk ) )λ .
(4.148)
1≤j
Here different exponents have been chosen for the different classes of roots. This allows the constant terms corresponding to the type B, C and D roots to be obtained by setting λ2 , λ1 and both λ1 , λ2 equal to zero, respectively. As a step toward evaluating the specialization of (4.148) for all exponents equal, Macdonald [375] noted that the integers j which occur in the binomial coefficients on the r.h.s. of (4.146) are related to the root system AN −1 : they are the degrees of the homogeneous polynomials in x1 , . . . , xN that are invariant under the Weyl group associated with the root system. For AN −1 the Weyl group is just the permutation group N SN , and consequently the invariant polynomials can be chosen as the (symmetric) power sums l=1 xjl N (j = 2, . . . , N ) (j = 1 is not included because l=1 xl = 0). For all root systems except BCN , the invariant polynomials can be defined similarly and their corresponding degrees calculated. In the case of BCN a more general definition is needed [375]. Either way, one obtains integers dj corresponding to the various root systems listed in Table 4.1.
176
CHAPTER 4
integers dj j (j = 2, . . . , N ) 2j (j = 1, . . . , N ) 2j (j = 1, . . . , N ) N and 2j (j = 1, . . . , N − 1) (2j + 2) (j = 1, . . . , N ) 2,5,6,8,9,12 2,6,8,10,12,14,18 2,8,12,14,18,20,24,30 2,6,8,12 2,6
root system AN −1 BN CN DN BCN E6 E7 E8 F4 G2
positive roots ej − ek (1 ≤ j < k ≤ N ) ej (j = 1, . . . , N ), ej ± ek (1 ≤ j < k ≤ N ) 2ej (j = 1, . . . , N ), ej ± ek (1 ≤ j < k ≤ N ) ej ± ek (1 ≤ j < k ≤ N ) ej , 2ej (j = 1, . . . , N ) ej ± ek , (1 ≤ j < k ≤ N )
Table 4.1 Integers associated with root systems and the positive roots in the cases of the infinite families. The notation ej denotes the jth elementary unit vector with N components.
For the root systems BN , CN , DN and BCN Macdonald conjectured that the corresponding product in (4.148) with all exponents equal has constant terms given by the r.h.s. of (4.146) but with j replaced by dj as specified in Table 4.1. In fact, writing the constant term in (4.148) as a Fourier integral it is easily seen to be proportional to the Selberg integral written in the form noted in Exercises 4.1 q.1 (ii). The evaluation of the Selberg integral (4.3) shows that indeed the Macdonald conjecture is correct. More generally Macdonald conjectured that for the reduced root systems λ N dj λ a Γq (a + 1) i−1 − α i α CT , (4.149) (1 − q e )(1 − q e ) = , := λ q Γq (a − b + 1)Γq (b + 1) b q i=1 j=1 α ∈R+
where R+ denotes the positive roots (defined as all α ∈ R such that for a given v ∈ E, α, v > 0, where it is assumed that v is chosen so that α, v = 0 for all α ∈ R), eej := etj , and the dj are given by the degrees of the independent homogeneous polynomials in the variables x1 , . . . , xN of the associated Weyl group. For the root system AN −1 this is a special case of the q-Morris identity of Exercises 4.6 q.3(iv). In the BCN case, both sides of (4.149) require modifying, giving a conjectured identity which was further generalized by Morris [407] to read CT
N i=1
×
−2 2 2 2 (xi ; q)a (qx−1 i ; q)a (qxi ; q )b (qxi ; q )b
−1 −1 −1 (xj xk ; q)λ (qx−1 j xk ; q)λ (xj xk ; q)λ (qxj xk ; q)λ
1≤j
=
N −1 j=0
(q; q)2a+2b+2λj (q; q)2b+2λj (q; q)λ(j+1) (q 2 ; q 2 )2b+λj (q 2 ; q 2 )a+λj (q; q)a+2b+(N +j−1)λ (q; q)2b+λj (q; q)a+λj (q; q)λ (q 2 ; q 2 )a+b+λj .(q 2 ; q 2 )b+λj
(4.150)
(the case given in [375] corresponds to a = b = λ). This was proved by Gustafson [280] and Kadell [337],
177
THE SELBERG INTEGRAL
the former of whom showed it could be deduced from the more general identity CT
N
4
j=1
(x2j ; q)∞ (x−2 j ; q)∞
−1 k=1 (ak xj ; q)∞ (ak xj ; q)∞
−1 −1 −1 (xj x−1 k ; q)∞ (xj xk ; q)∞ (xj xk ; q)∞ (xj xk ; q)∞
1≤j
−1 −1 −1 (bxj x−1 k ; q)∞ (bxj xk ; q)∞ (bxj xk ; q)∞ (bxj xk ; q)∞
× = 2N N !
N
(b; q)∞ (bN +j−2 a1 a2 a3 a4 ; q)∞ j (b ; q)∞ (q; q)∞ 1≤μ<ν≤4 (aμ aν bj−1 ; q)∞ j=1
(4.151)
(see Exercises 4.7 q.5). The conjecture (4.149), appropriately generalized in the case of BCN , has been proved for all crystallographic root systems (AN −1 , BN , CN , DN , BCN and the exceptional root systems E8 , E7 , E6 , F4 and G2 ) by Cherednik [108]. If the extra condition is imposed that the only multiples of α ∈ R allowed are ± α, then as already noted the root system BCN is excluded, while the other root systems remain. For these remaining root systems, Macdonald conjectured that ∞ ∞ N 2 2 α, t 2γ Γ(1 + γdj ) , (4.152) dt1 e−t1 /2 · · · dtN e−tN /2 = 2−νγ (2π)N/2 α Γ(1 + γ) −∞ −∞ j=1 α ∈R+
where the dj are as above and ν is equal to the degree of α ∈R+ α, t. In the type A case, this conjecture coincides with the Mehta integral of Proposition 4.7.1, while for the types B, C and D root systems it is seen to be a special case of WaβN as defined by (4.142) and evaluated by Proposition 4.7.3 (this identification requires the change of variable xj = yj2 ). E XERCISES 4.7
1. The objective of this exercise is to give Good’s proof [268] of the identity CT
N „ Y j,k=1 j=k
xj 1− xk
«aj =
(a1 + · · · + aN )! , a1 ! · · · aN !
aj ∈ Z≥0 , which was conjectured by Dyson [146]. Note that the case a1 = · · · = aN = β/2 is equivalent to Proposition 4.7.2. (i) In the Lagrange interpolation formula f (x) =
N+1 X j=1
f (xj )
N+1 Y l=1 l=j
x − xl , xj − xl
(4.153)
for f (x) a polynomial of degree ≤ N , put N + 1 → N , f (x) = 1, and choose a suitable value of x to obtain the identity «−1 N „ N Y X xj 1− = 1. xl j=1 l=1 l=j
(ii) Denoting the l.h.s. of the stated identity by GN (a1 , . . . , aN ), use the identity in (i) to show GN (a1 , . . . , aN ) =
N X j=1
GN (a1 , . . . , aj−1 , aj − 1, aj+1 , . . . , aN ).
178
CHAPTER 4
(iii) By considering the occurrence of xj in the product, argue that GN (a1 , . . . , aj−1 , 0, aj+1 , . . . , aN ) = GN−1 (a1 , . . . , aj−1 , aj+1 , . . . , aN ). Verify that the stated multinomial coefficient satisfies the recurrences in (ii) and (iii), together with the initial condition GN (0, . . . , 0) = 1, and conclude that since the constant term is uniquely specified by these conditions, the stated identity holds. 2. [166] In this exercise the Mehta integral will be evaluated in an analogous way to Anderson’s derivation of the Selberg integral. (i) Use the fact that (4.55) is a p.d.f. in {λj } to deduce that Z
dλ1 · · · dλn+1 e−(1/2)
Pn+1 j=1
R
=
√
Y
λ2 j
(λj − λk )
Pn
j=1
Y
a2 j
|λj − ap |c
j=1 p=1
1≤j
2π(Γ(c))n e−(1/2)
n+1 n YY
(aj − ak )2c−1 ,
1≤j
where R is the region (4.56). Conclude from this that Z
da1 · · · dan dλ1 · · · dλn+1 e−(1/2) R
=
√
Pn+1 j=1
Y
λ2 j
(λj − λk )
1≤j
Y
(aj − ak )
n n+1 YY
1≤j
|λj − ap |c
j=1 p=1
n
2π(Γ(c)) mn (c).
(ii) Use (4.15) to compute the integral over {ai } first in the integral over {λi }, {ai } above, and thus conclude that it is also equal to (Γ(c))n+1 mn+1 (c). Γ(c(n + 1)) (iii) Equate the expressions in (i) and (ii) and iterate the resulting difference equation, using the initial condition m0 (c) = 1 to rederive the evaluation (4.140). 3. [395] The objective of this exercise is to derive the integration formula Z dy1 · · · dyN Δ
N Y
N “ ”b−1 X yja−1 1 − yj
j=1
i=1
Y
|yk − yj |β
1≤j
N Y Γ(a + β(N − j)/2)Γ(1 + βj/2) Γ(b) = , Γ(b + aN + βN (N − 1)/2) j=1 Γ(1 + β/2)
where Δ is the region yj > 0 (j = 1, . . . , N ), distribution (4.10).
PN i=1
(4.154)
yi ≤ 1. In the case β = 0 this relates to the Dirichlet
(i) Put βa/2 → a − 1, βxl /2 → λxl in (4.142), move all factors of λ to the l.h.s, multiply both sides by λb−1 e−λ and integrate over λ ∈ [0, ∞) to obtain the evaluation of Z
∞
Z dx1 xa−1 ··· 1
0
∞
N “ ”−b X dxN xa−1 xi 1+ N
0
i=1
Y
|xk − xj |β .
1≤j
P −1 (ii) Let xj = yj (1 − N . Note that the range of integration xj ≥ 0 (j = 1, . . . , N ) maps to the region i=1 yi ) Δ, and evaluate the Jacobian as det
h ∂x i j
∂yk
j,k=1,...,N
N “ ”−(N+1) X = 1− yi . i=1
179
THE SELBERG INTEGRAL
(iii) Apply the change of variables in (ii) to the integral in (i) with b → b − aN − βN (N − 1)/2 to obtain (4.154). P (iv) [560] Consider (4.154) in the case b = 1. Implement the constraint 0 ≤ N i=1 yi ≤ 1 by multiplying the PN integrand by δ(t − i=1 yi ) and integrating over t from 0 to 1. By changing variables yi → tyi , show that the t dependence can be factorized, and hence obtain the integral evaluation Z RN
dy1 · · · dyN
N Y
Y
yja−1
j=1
N “ ” X |yk − yj |β δ 1 − yi i=1
1≤j
N Y Γ(a + β(N − j)/2)Γ(1 + βj/2) 1 . = Γ(aN + βN (N − 1)/2) j=1 Γ(1 + β/2)
4.
(4.155)
(i) Use (3.124) and (4.4) to show Z
Z
∞
−∞
dλ1 · · ·
∞
dλN −∞
N Y l=1
1 (1 + iλl )b (1 − iλl )¯b
Y
|λk − λj |β
1≤j
= 2βN(N−1)/2−2(Re b−1)N π N MN (¯b − β(N − 1)/2 − 1, b − β(N − 1)/2 − 1, β/2), where MN is given by (4.4). Writing 2Re b = b + ¯b, argue that b and ¯b can be regarded as independent complex parameters. (ii) Change variables xl = (a + b)λl /2 + i(a − b)/2, and use the result of (i) to deduce Z
∞ −∞
=
Z dx1 · · ·
N Y
∞
dxN −∞
l=1
1 (a + ixl )c (b − ixl )c¯
(2π)N (a +
b)(c+¯c)N−βN(N−1)/2−N
Y
|xk − xj |β
1≤j
MN (¯ c − β(N − 1)/2 − 1, c − β(N − 1)/2 − 1, β/2).(4.156)
(iii) Let the normalization in (3.4) be denoted Cαβm . Use a simple change of variables to show Cαβm = 2−m Wα−1/β,β,m where WαβN is the integral in (4.142) and thus evaluated by (3.134). (iv) From the result of (iii) show that the normalizations in Exercises 1.3 q.5(iv) have the evaluations ( Q 2−N/2 N/2−1 Γ(j + 2)Γ(j + 1/2), N even, j=0 Q CN = Γ(j + 2)Γ(j + 3/2), N odd, 2−(N−1)/2 (N−1)/2−1 j=0 5.
(4.157)
(i) Verify that (x2 ; q)∞ = (x; q)∞ (xq 1/2 ; q)∞ (−x; q)∞ (−xq 1/2 ; q)∞ and thus show that with b = q λ , a1 = −1, a2 = q a , a3 = −a4 = q 1/2 q b/2 the l.h.s. of (4.150) reads CT
N Y
2 2 −2 (xi ; q)a (x−1 q; q 2 )b i ; q)a (xi q; q )b (x
i=1
×
Y
−1 −1 −1 (xj x−1 k ; q)λ (xj xk ; q)λ (xj xk ; q)λ (xj xk ; q)λ .
1≤j
(ii) In the special case M = ∅, Kadell’s lemma (Exercises 4.6 q.2(v)) can be written X
Y
w∈P 1≤i<j≤N
1 − tw(xj /xi ) (t; t)N = , 1 − w(xj /xi ) (1 − tN )
(4.158)
180
CHAPTER 4
where P is the set of all permutations of {x1 , . . . , xn }. Macdonald [374] has given an identity associated with root systems which in the type A case implies (4.158). In the type B case the identity implies N X Y 1 − q a w(x−1 l ) −1 1 − w(x l ) w∈W l=1
=
Y
−1 −1 λ (1 − q λ w(x−1 j xk ))(1 − q w(xj xk ))
1≤j
−1 −1 (1 − w(x−1 j xk ))(1 − w(xj xk ))
(q 2a ; q 2λ )N (q λ ; q λ )N , (q a ; q λ )N (1 − q λ )N
(4.159)
where W denotes the set of all permutations of {x1 , . . . , xn } together with the inversions xj → x−1 j . Show that upon multiplication of the l.h.s. of (4.159) with the constant term in (i) above, the latter becomes equal to 2N N ! times the constant term in (4.150), while multiplying the r.h.s. of (4.159) by the r.h.s. of (4.151) with parameters as in (i) gives 2N N ! times the r.h.s. of (4.150).
4.8 FREE ENERGY 4.8.1 Analytic evaluation The evaluation of the integral in Proposition 4.7.2 can be used to compute thermodynamic quantities associated with the one-component log-potential system on a circle of Proposition 1.4.1. This follows because the integral is proportional to the partition function, and this in turn is related to the free energy [390]. Thus in the canonical formalism of statistical mechanics, the total dimensionless free energy βF is given by 1 ˆ ZN N! and the corresponding dimensionless free energy per particle βf is given by βF = − log
βf =
lim
N,|Ω|→∞ N/|Ω|=ρ
1 βF. N
(4.160)
(4.161)
The limit N, |Ω| → ∞, N/|Ω| = ρ (fixed) is referred to as the thermodynamic limit. From βf other thermodynamic quantities of interest can be computed by differentiation. In particular 1 ∂βf = lim U =: u (4.162) N,|Ω|→∞ ∂β N N/|Ω|=ρ gives the mean energy per particle. A further differentiation gives the specific heat CV at constant volume (also referred to as the heat capacity), CV /kB = −
1 ∂u . β 2 ∂β
(4.163)
We see from (1.45) and (4.162) that in terms of averages CV /kB =
1 β2
lim
N,Ω→∞ N/|Ω|=ρ
1 2 U − U 2 , N
(4.164)
so in particular CV is non-negative. Also, in thermodynamics, the pressure is related to the free energy by dF = −P d|Ω| − SdT (S denotes the entropy) so one has ∂F ∂f P =− (4.165) ∼ ρ2 , ∂|Ω| T ∂ρ
181
THE SELBERG INTEGRAL
where the final expression holds in the thermodynamic limit. Let’s now apply the formalism to the log-gas on a circle. P ROPOSITION 4.8.1 Let N/2πR := ρ be fixed, and consider the one-component log-potential system on a circle with a uniform neutralizing background and unit charges. We have − log
1 ˆ ZN ∼ N βf + O(1), N!
where βf = (1 − β/2) log 2πρ − (β/2) log(β/2) + β/2 + log Γ(1 + β/2) − log 2π − 1.
(4.166)
Proof. This follows immediately from the form of the Boltzmann factor in Proposition 1.4.1, the evaluation of the partition function given by Proposition 4.7.2 and Stirling’s formula (1.92).
Since βf is known explicitly, the thermodynamic formula (4.162) gives for the mean energy per particle u :=
lim
N,L→∞ N/L=ρ
1 1 1 1 Γ (1 + β/2) U = − log 2πρ − log(β/2) + N 2 2 2 Γ(1 + β/2)
and use of (4.163) then gives for the specific heat per particle 1 1 Γ (1 + β/2)Γ(1 + β/2) − (Γ (1 + β/2))2 1 1 ∂u CV /kB = − 2 =− 2 − + . β ∂β β 2β 4 (Γ(1 + β/2))2
(4.167)
(4.168)
By substituting (4.166) in (4.165) we have that the thermodynamic pressure is given by βP = (1 − β/2)ρ.
(4.169)
Two remarks are in order here. First, we see from Proposition 1.4.1 that the volume dependence factors out of the partition function, so (4.169) can be deduced without explicit evaluation of the latter. Second, for β > 2, the expression (4.169) becomes negative. The reason for this is that in performing the partial derivative to calculate the pressure, the background density as well as the particle density is being varied. Physically one requires that the background be maintained at constant density by some external constraint. We note also that (4.166) can be expressed in terms of the excess quantity βf ex := βf − βf∞ ,
(4.170)
where βf∞ refers to the dimensionless free energy per particle for the perfect gas, βf∞ := log ρ − 1. We have 1 1 1 β g(β) = − log β + + log Γ(1 + β/2). (4.171) βf ex = − log πρ + βg(β), 2 2 2 β 4.8.2 Low temperature limit In the low temperature β → ∞ limit, use of Stirling’s formula (1.92) shows that the exact dimensionless free energy per particle (4.166) has the expansion 1 β 1 −1+O . (4.172) βf ∼ (1 − β/2) log 2πρ + log 2 4π β This expansion can be reproduced by making a harmonic approximation in which the total energy is expanded to second order about the minimum energy configuration (which is attained at zero temperature), and the partition function approximated by ZN ≈ RN e−βU0 β→∞
N −1 l=1
2π 1/2 , βλl
(4.173)
182
CHAPTER 4
where the λl denote the nonzero eigenvalues of [Hjk ] := [∂ 2 U/∂θj ∂θk ] evaluated at the minimum energy configuration (a zero eigenvalue occurs due to the rotational invariance of the ground state). The reasoning which gives rise to (4.173) is given in Exercises 4.8 q.2. For the one-component log-gas on a circle, by symmetry the minimum energy configuration must be equally spaced points θj = ν0 + 2π(j − 1)/N (j = 1, . . . , N ), where 0 ≤ ν0 < 2π/N . For all ν0 U0 = (2πρ)N β/2 ,
Hjj =
N −1 1 1 , 4 n=1 sin2 πn/N
Hjk = −
1 (j = k), 4 sin π(j − k)/N 2
where to calculate U0 use has been made of Exercises 4.8 q.1. Since [Hjk ] is cyclic: Hjk = Hj−k 0 = k can be taken to be ψ k = [ √1 e2πijk/N ]j=0,...,N −1 , and the corresponding Hj+N −k 0 its eigenvectors ψ N eigenvalues computed as λk =
N −1 1 1 − cos 2πkn/N . 4 n=1 sin2 πn/N
For N → ∞, k/N fixed λk ∼
∞ N 2 1 − cos 2πkn/N N2 |k/N | − (k/N )2 , = 2 2 2π n=1 n 2
|k/N | <
1 , 2
(4.174)
(the equality can be verified by expressing |y| − y 2 as a Fourier cosine series) so (4.161) applied to (4.173) gives 1/2 β 1 + βf ∼ (1 − β/2) log 2πρ + log log t(1 − t) dt. 2 4π 0 The integral equals −1, so comparison with (4.172) gives that the exact dimensionless free energy per particle is indeed reproduced up to O(1/β).
4.8.3 High temperature limit In the high temperature limit β → 0 the dimensionless free energy per particle (4.166) has the expansion β β β β log + + Γ (1) − log 2π − 1 + O(β 2 ). (4.175) 2 2 2 2 To understand this requires introducing the two-particle correlation function ρ(2) (r, r ), defined as the canonN ical average (1.45) with f = j=k=1 δ(r − rj )δ(r − rk ), βf ∼ (1 − β/2) log 2πρ −
ρ(2) (r, r ) :=
N
δ(r − rj )δ(r − rk )
j,k=1 j=k
=
N (N − 1) ZˆN
dr3 · · ·
Ω
drN e−βU(r,r
, r3 ,..., rN )
,
(4.176)
Ω
where the last equality is valid for a system of identical particles. Introducing the truncated correlation ρT(2) (r, r ) := ρ(2) (r, r ) − ρ(1) (r)ρ(1) (r ),
(4.177)
183
THE SELBERG INTEGRAL
which has the property of decaying for large separation between r and r , we can check from the definitions that for the log-gas 1 1 ∞ T U = − u := lim ρ(2) (x, 0) log x dx. (4.178) N,L→∞ N ρ 0 N/L=ρ In (4.178) we make the weak coupling approximation to the truncated two-particle correlation function (4.177), ρβ ∞ sin kρx ρT(2) (x, 0) = − dk (4.179) x 0 (k + πβ)2 (the derivation of this result is given in Exercises 14.1 q.3). This gives 1 ∞ sin x 1 log x dx. u = − log πβρ + 2 π 0 x
∞ The integral is equal to π2 Γ (1), which can be deduced from the formula 0 ts−1 e−t dt = Γ(s), so after taking the antiderivative with respect to β and adding βf for a perfect gas, βf∞ = log 2πρ − log 2π − 1, we see that (4.175) is reclaimed up to O(β 2 ). E XERCISES 4.8
1.
(i) Use the factorization formula (4.119) to show
(ii) Use the result of (i) to show
Q 1≤j
QN−1 j=1
(1 − e2πij/N ) = N.
|e2πik/N − e2πij/N |2 = N N .
2. For a one-component system defined on a line with a unique ground state configuration, the harmonic approximation to the partition function is (h.a.)
ZN
= e−βU0
Z dx1 · · · dxN exp
“
−
Ω
N ” β X (0) (0) (xj − xj )Hjk (xk − xk ) , 2 j,k=1
where Hjk is defined as below (4.173), Ω denotes the ordered integration domain x1 < x2 < · · · < xN and (0) the xj denote the equilibrium points. For β → ∞ the significant contribution to the integral comes from xj ≈ (0)
xj , so the domain of integration can be extended over (−∞, ∞)N without affecting the asymptotic limit. By diagonalizing [Hjk ] obtain the formula (h.a.)
ZN
= e−βU0
N−1 Y “ l=0
2π ”1/2 , βλl
where the λl denote the eigenvalues of [Hjk ]. 3.
(i) Consider the integral equation (1.47). Show that if ρb (y) is a solution then Z ∞ x2 + C + N log l = ρb (y/l) log |x − y| dy. 2l2 −∞ Use this result Proposition 1.4.4 to show that for a one-component system p of unit charges confined to the √ √ and √ interval [− 2N l, 2N l] with background charge density −( 2N /lπ) 1 − (y 2 /2N l2 ) the Boltzmann factor is N “ βX ” Y (xj /l)2 |(xk − xj )/l|β , Al−βN/2 exp − 2 j=1 1≤j
where A is given in Proposition 1.4.4. √ (ii) Choose l = 2N /πρ so that as N → ∞ with y fixed the background charge density equals −ρ. Furthermore suppose that with the Boltzmann factor above, the particles can move anywhere on the real line. Use
184
CHAPTER 4
the definition of Gβ,N in (4.139) to show that the partition function for this system is given by 1 Al−βN/2+N Gβ,N . N! (iii) Let c be a positive integer. Use the duplication formula Γ(z)Γ(z + 1/c) · · · Γ(z + (c − 1)/c) = c1/2−cz (2π)(c−1)/2 Γ(cz)
(4.180)
to show that N−1 Y
Γ(α + 1 + jc) = ccN(N−1)/2+N(α+1/2) (2π)−N(c−1)/2
j=0
c N−1 ” Y Y “α + p Γ +j . c p=1 j=0
(iv) The Barnes G-function [44] is defined by ∞ “ ` ´Y z ”k exp(−z + z 2 /k) 1+ G(z + 1) = (2π)z/2 exp −z/2 − (γ + 1)z 2 /2 k k=1
(4.181)
(γ denotes Euler’s constant), which has the special values G(1) = G(2) = G(3) = 1,
G(1/2) = π −1/4 21/24 e(3/2)ζ
(−1)
= 0.6032442812 · · ·
and satisfies the functional relation G(z + 1) = Γ(z)G(z).
(4.182)
Use (4.182) to rewrite the identity in (iii) as [184] N−1 Y
Γ(α + 1 + jc) = ccN(N−1)/2+N(α+1/2) (2π)−N(c−1)/2
j=0
c Y G(N + (α + p)/c) . G((α + p)/c) p=1
(4.183)
Use this formula, and the asymptotic expansions [44] x2 x 3 1 log x − x2 + log 2π − log x + ζ (−1) + o(1) x→∞ 2 4 2 12
log G(x + 1) ∼ and log
„
G(N + a + 1) G(N + b + 1)
« ∼ (b − a)N +
N→∞
(4.184)
„ « a−b a 2 − b2 log 2π + (a − b)N + log N + o(1) (4.185) 2 2
to show that for β even log
N Y
” N β “ N2 N Nβ 3 log − N2 + log N + log N 2 2 2 4 2 2
Γ(1 + βj/2) ∼
j=1
+
N β β N 1 β (1 + ) log + log 2π − (1 + )N + O(log N ). 2 2 2 2 2 2
(4.186)
(v) To establish the validity of (4.186) for general β > 0, with c = β/2 substitute the expansion log Γ(1 + cj) = cj log j + c(log c − 1)j +
1 log j + log(2πc) + O(1/j), 2
uniform in j ≥ 1, and sum over j. (vi) Use the above results to show that the dimensionless free energy per particle of the system in (i) with β even is given by (4.166) with ρ replaced by 21 e1/2 ρ. 4. [184] Let βF (n, q) denote the dimensionless free energy of the one-component plasma on a circle of radius R
185
THE SELBERG INTEGRAL
with n mobile unit charges, an impurity particle of charge q at θ = 0, and a uniform neutralizing background. The dimensionless chemical potential βμ∗n due to the introduction of the impurity charge is defined by βμ∗n = βF (n, q) − βF (n, 0). (i) Use the result of Proposition 1.4.1 and (4.160) to show ∗
e−βμn = R−q where Zn [g(θ)] :=
1 (2π)n
Z
π −π
Z dθ1 · · ·
2
+ eiθ |qβ ] , Zn [1]
β/2 Zn [|1
π
dθn −π
n Y
Y
g(θl )
l=1
|eiθk − eiθj |β ,
1≤j
and use (4.4) to obtain the gamma function evaluation ∗
e−βμn = R−q
2
c fn (2cq, c) , fn2 (cq, c)
where
fn (α, c) :=
n−1 Y j=0
(α + jc)! , (jc)!
c := β/2.
(4.187)
(ii) Use (4.183) to show that for c ∈ Z+ we can write fn (α, c) = cαn
c−1 Y p=0
G(n + (α − p)/c + 1)G(−p/c + 1) G(n − p/c + 1)G((α − p)/c + 1)
and apply (4.185) to deduce that for large n 2
fn (α, c)∼e(αn log n) cαn e−αn n−(c−1)α/2+α
/2c
c−1 Y p=0
G(−p/c + 1) . G((α − p)/c + 1)
(iii) Suppose c is rational, c = s/r for s, r ∈ Z+ . Show from the definition that frn (α, s/r) =
r−1 Y ν=0
fn (α + sν/r, s) fn (sν/r, s)
and use this together with the asymptotic expansion of (ii) to deduce that lim
n→∞ n/2πR=ρ
∗
2
e−βμn = (2πρ)q c r −(q ×
r−1 Y s−1 Y ν=0 p=0
2
−b2 )β/2
G((q + b)/r + ν/r − p/s + 1)G2 ((q − b)/r + ν/r − p/s + 1) . (4.188) G(2q/r + ν/r − p/s + 1)G(ν/r − p/s + 1)
Chapter Five Correlation functions at β = 2 At the special coupling β = 2, the general n-particle correlation function for the one-component log-gas can be expressed as an n × n determinant involving orthogonal polynomials. A number of different viewpoints on this result are presented. These include integration formulas, functional differentiation and a formulation in terms of an integral equation. Yet another is to first deduce a determinant formula for the canonical average of what in the random matrix interpretation corresponds to ratios of characteristic polynomials. A general element of the n × n determinant, referred to as the correlation kernel, is independent of n. It is written as a sum of orthogonal polynomials, which can be summed using the Christoffel-Darboux formula. The resulting structure takes on particular significance in the study of spacing distributions, undertaken in Chapter 9. In the case of a general one-body potential with a power series expansion it is the integrability properties of the corresponding configuration integral which are of interest, and in particular its relation to the KP hierarchy from soliton theory. A further generalization isolates the product of two determinants formed out of linearly independent functions, and this leads to the study of multiple orthogonal polynomials. The topic of the final section is correlation functions for multicomponent systems, in which the joint p.d.f. is a product of determinants. As well as when all species consist of an equal number of particles, we treat too the case in which each species j consists of j particles. This latter case is applicable to the GUE minor process.
5.1 SUCCESSIVE INTEGRATIONS 5.1.1 n-point correlations and truncated correlations The definition of the one-point and two-point correlations have been given in (1.46) and (4.176). More generally, the n-point (or n-particle) correlation function ρ(n) (r1 , . . . , rn ), is specified as a canonical average (1.45) according to n ρ(n) (r1 , . . . , rn ) := δ(rl − rjl ) j1 =···=jn l=1
N (N − 1) · · · (N − n + 1) = ZˆN
drN e−βU(r1 ,...,rN ) ,
drn+1 · · · Ω
(5.1)
Ω
where the equality is valid for a system of identical particles. We remark that the ratio ρ(n) (r1 , . . . , rn )/ρ(n−1) (r1 , . . . , rn−1 ) can be interpreted as the density at point rn given that there are particles at points r1 , . . . , rn−1 . The correlation function ρ(n) does not decay for large separation between particles. However, by adding and subtracting appropriate combinations of ρ(1) , . . . , ρ(n−1) to ρ(n) we can obtain a quantity, denoted ρT(n) and called the truncated n-particle correlation function, which will decay when two or more particles are at large separation. The truncated two-particle correlation is given by (4.177), while the truncated three-particle
187
CORRELATION FUNCTIONS AT β = 2
correlation is given by ρT(3) (r1 , r2 , r3 ) := ρ(3) (r1 , r2 , r3 ) − ρ(1) (r1 )ρ(2) (r2 , r3 ) − ρ(1) (r2 )ρ(2) (r1 , r3 ) −ρ(1) (r3 )ρ(2) (r1 , r2 ) + 2ρ(1) (r1 )ρ(1) (r2 )ρ(1) (r3 )
(5.2)
and in general, ρT(n) (r1 , . . . , rn ) :=
n m (−1)m−1 (m − 1)! ρ(|Gj |) (rgj (1) , . . . , rgj (|Gj |) ) m=1 G
(5.3)
j=1
where the sum over G is over all subdivisions of {1, 2, . . . , n} into m subsets G1 , . . . , Gm with Gj = {gj (1), . . . , gj (|Gj |)}. For example, when n = 3 the m = 1 term corresponds to G1 = {1, 2, 3}, the m = 2 term to G1 = {1}, G2 = {2, 3} or G1 = {2}, G2 = {1, 3} or G1 = {3}, G2 = {1, 2}, and the m = 3 term to G1 = {1}, G2 = {2}, G3 = {3} (rearrangements of the Gj are not considered distinct). The inverse of this formula is given in Exercises 5.1 q.3. 5.1.2 Orthogonal polynomials The one-component log-gas confined to a line and subject to a one-body potential V (x) with corresponding Boltzmann factor wβ (x) := e−βV (x) has itself a Boltzmann factor proportional to the p.d.f. MEβ,N (wβ (x)) (recall (4.108)). The problem of computing the corresponding n-particle correlation for β = 2 is intimately related to the theory of orthogonal polynomials. We first note that the p.d.f. can be written in terms of orthogonal polynomials associated with w2 (x). P ROPOSITION 5.1.1 With pk (x) a polynomial of degree k which is furthermore monic (i.e. the coefficient of xk is unity), let {pk (x)}k=0,1,2,... be the orthogonal polynomials associated with the weight function w2 (x), ∞ w2 (x)pj (x)pk (x) dx =: (pj , pk )2 = (pj , pj )2 δj,k . (5.4) −∞
We have N
w2 (xl )
(xk − xj )2
1≤j
l=1
=
N
2 w2 (xl ) det[pk−1 (xj )]j,k=1,...,N
l=1
=
N (pl−1 , pl−1 )2 det (w2 (xj ))1/2 l=1
=
pk−1 (xj ) det[(w2 (xk ))1/2 pj−1 (xk )]j,k=1,...,N (pk−1 , pk−1 )2 j,k=1,...,N
N (pl−1 , pl−1 )2 det[KN (xj , xk )]j,k=1,...,N ,
(5.5)
l=1
where −1 1/2 N pν (x)pν (y) . KN (x, y) := w2 (x)w2 (y) (pν , pν )2 ν=0
Proof. The Vandermonde determinant identity (1.173) says ]j,k=1,...,N = det[xk−1 j
Y 1≤j
(xk − xj ).
(5.6)
188
CHAPTER 5
The first column in the determinant consists of all 1s and thus is equal to [p0 (xj )]j=1,...,N . Adding an appropriate multiple of this column to the 2nd column gives [p1 (xj )]j=1,...,N . Next we add appropriate multiples of columns 1 and 2 to column 3 to obtain [p2 (xj )]j=1,...,N . Proceeding in this fashion we obtain the first line of the stated identity. The second line follows by elementary manipulation of the first line and the general fact that det A = det AT , while the final expression is obtained by matrix multiplication of the determinants in the second line.
The utility of the expression (5.5) lies in the properties ∞ KN (x1 , y)KN (y, x2 )dy = KN (x1 , x2 ) and
−∞
∞
−∞
KN (y, y)dy = N,
(5.7)
which in turn follow from the orthogonality of {pk (x)}. Indeed combining (5.5) and (5.7) allows the nparticle correlation function to be computed [150]. P ROPOSITION 5.1.2 We have ∞ ∞ n 2 N! 1≤j
= det[KN (xj , xk )]j,k=1,...,n where KN (x, y) is defined by (5.6). The corresponding truncated n-particle correlation function is given by KN (xi1 , xi2 )KN (xi2 , xi3 ) · · · KN (xin , xi1 ), ρT(n) (x1 , . . . , xn ) = (−1)n−1 cycles length n
where the sum is over all distinct cycles i1 → i2 → · · · → in → i1 of {1, . . . , n} which are of length n. Proof. From Proposition 5.1.1 the integral over xN in both top and bottom line is equivalent to the m = N case of Z
∞
−∞
det[KN (xj , xk )]j,k=1,...,m dxm .
To compute this integral we make a Laplace expansion of the determinant along the bottom row, and multiply each factor KN (xm , xk ) (k = 1, . . . , m − 1) in the expansion into the (m − 1)st column of the corresponding cofactor, det[KN (xj , xk )]j,k=1,...,m m−1 h i X = (−1)m+k det KN (xj , xl ) KN (xj , xl ) KN (xj , xm )KN (xm , xk ) k=1
j=1,...,m−1 l=1,...,k−1 l =k+1,...,m−1
+KN (xm , xm ) det[KN (xj , xk )]j,k=1,...,m−1 . Since all the dependence on xm is in the last column of the determinants in the sum, the integration over xm can be performed in each element of this column. Using (5.7) then gives Z
∞
det[KN (xj , xk )]j,k=1,...,m dxm = −∞
m−1 X
h (−1)m+k det KN (xj , xl )
i KN (xj , xl ) KN (xj , xk )
k=1
j=1,...,m−1 l=1,...,k−1 l =k+1,...,m−1
+N det[KN (xj , xk )]j,k=1,...,m−1 = (−(m − 1) + N ) det[KN (xj , xk )]j,k=1,...,m−1 , where the last line follows by noting that interchanging columns in the sum over k so that the final column is moved to column k shows that all terms in the sum are the same and equal to − det[KN (xj , xl )]j,l=1,...,m−1 . Repeatedly applying this result, first with m = N, N − 1, . . . , n + 1, then with m = N, N − 1, . . . , 1, allows both integrals in the definition of ρ(n) to be computed and the stated result follows.
189
CORRELATION FUNCTIONS AT β = 2
To obtain the formula for the truncated correlation, we recall that in general [8] det[ajk ]j,k=1,...,n =
X
(−1)n−l
l Y
P ∈Sn
aαβ aβγ · · · aδα ,
(5.8)
1
where the sum is over all permutations of {1, . . . , n} consisting of l cycles of the form (α → β → δ → · · · → δ → α). Comparing (5.8) with the final formula of Exercises 5.1 q.3 we see that in general if ρ(n) (x1 , . . . , xn ) = det[ajk ]j,k=1,...,n then X ρT(n) (x1 , . . . , xn ) = (−1)n−1 ai1 i2 ai2 i3 · · · ain i1 , (5.9) cycles length n
and hence the result.
The summation KN (x, y) as defined in (5.6) can be performed explicitly, according to the ChristoffelDarboux formula [508]. P ROPOSITION 5.1.3 We have KN (x, y) =
(w2 (x)w2 (y))1/2 pN (x)pN −1 (y) − pN −1 (x)pN (y) . (pN −1 , pN −1 )2 x−y
(5.10)
Proof. Our strategy is to evaluate Z f (x, y) := (w2 (x)w2 (y))1/2
∞ −∞
Z dx1 · · ·
∞
dxN−1 −∞
N−1 Y
Y
w2 (xl )
l=1
(xk − xj )2
1≤j
N−1 Y
(xj − x)(xj − y)
j =1
in two different ways. From the first equality in (5.5) we have Y
(xk − xj )
1≤j
N−1 Y
» (xj − u) = det
j =1
pk−1 (u) pk−1 (xj )
– j=1,...,N −1 k=1,...,N
(u = x, y).
Multiplication of the determinant with u = x by the determinant with u = y gives for the product of the integrand and (w2 (x)w2 (y))1/2 – » N Y KN (x, xk ) KN (x, y) (pl−1 , pl−1 )2 det . KN (xj , y) KN (xj , xk ) j,k=1,...,N−1 l=1
Performing the integration as in the proof of Proposition 5.1.2 gives f (x, y) = (N − 1)!
N Y (pl−1 , pl−1 )2 KN (x, y). l=1
On the other hand, in the integrand we can write 2 3 pk−1 (y) 1 (xk − xj ) (xj − x)(xj − y) = det 4 pk−1 (x) 5 x−y 1≤j
N−1 Y
k=1,...,N +1
and
Y
(xk − xj ) = det[pk−1 (xj )]j,k=1,...,N−1 .
(5.11)
1≤j
Since both factors are antisymmetric in x1 , . . . , xN−1 , the second determinant can be replaced by its diagonal term in
190
CHAPTER 5
the integral, provided we multiply the integrand by (N − 1)!. Thus Z Z ∞ (w2 (x)w2 (y))1/2 ∞ dx1 · · · dxN−1 x−y −∞ −∞ 2 3 N−1 pk−1 (y) Y 5 × w2 (xl ) det 4 pk−1 (x) , l=1 pj−1 (xj )pk−1 (xj ) j=1,...,N −1
f (x, y) = (N − 1)!
(5.12)
k=1,...,N +1
where we have multiplied the jth diagonal term into row j + 2 of the first determinant. The integration can now be performed row by row. For each row j = 3, . . . , N + 1 only the term in column j − 2 is nonzero (and equal to (pj −1 , pj −1 )2 , j = j − 2 = 1, . . . , N − 1). Expanding by these elements gives f (x, y) = (N − 1)!
N−1 Y
(pj −1 , pj −1 )2
j =1
” (w2 (x)w2 (y))1/2 “ pN (x)pN−1 (y) − pN−1 (x)pN (y) . x−y
Equating the two expressions for f (x, y) gives the required summation formula.
The above proof is nonstandard; the conventional proof is given in Exercises 5.1 q.2. By taking the limit y → x in Proposition 5.1.3, the summation formula KN (x, x) := w2 (x)
N −1 ν=0
(pν (x))2 w2 (x) pN (x)pN −1 (x) − pN −1 (x)pN (x) = (pν , pν )2 (pN −1 , pN −1 )2
(5.13)
results (the dashes denote differentiation). It is used to evaluate the diagonal elements of the determinant in Proposition 5.1.2. The second method of evaluation of f (x, y) in the proof of Proposition 5.1.3 gives a multidimensional integral formula for pN (x) due originally to Heine [508], and this in turn leads to the evaluation of a related multidimensional integral. P ROPOSITION 5.1.4 We have pN (x) =
∞
−∞
1 C
∞
−∞
dx1 · · ·
1 pN (x) w2 (x) dx = y−x (N + 1)C
where
∞
C= −∞
dx1 · · ·
∞
−∞
−∞
dxN
dx0 · · ·
∞ −∞
N
∞
dxN
N
w2 (xl )(x − xl )
∞
dxN
w2 (xl )
(xk − xj )2 ,
1≤j
l=1
−∞
N l=0
w2 (xl ) y − xl
(xk − xj )2 ,
0≤j
(xk − xj )2 .
1≤j
l=1
Proof. It remains to consider the second identity. From the first we see that Z
∞
pN (x) w2 (x) dx y−x Z ∞ Z ∞ Z N N Y Y 1 1 1 ∞ dx0 dx1 · · · dxN w2 (xl ) = C −∞ x − x y − x0 0 l −∞ −∞ l=0
−∞
l =1
Y 0≤j
(xk − xj )2 .
191
CORRELATION FUNCTIONS AT β = 2
Now the integrand is symmetric apart from the factor N Y l =1
1 1 =: gy (x0 ; x1 , . . . , xN ). x0 − xl y − x0
Symmetrizing allows this factor to be replaced by N N ˛ ” X 1 Y 1 1 “ ˛ gy (x0 ; x1 , . . . , xN )˛ , gy (x0 ; x1 , . . . , xN ) + = N +1 N +1 y − xl x0 ↔xj j=1 l=0
where the last equality can be verified by performing a partial fractions expansion of the r.h.s. regarded as a function of y.
1. Let {pn (x)}n=0,1,... be the set of monic orthogonal polynomials with respect to the weight function w2 (x), and write Cn := (pn−1 , pn−1 )2 /(pn−2 , pn−2 )2 . The objective of parts (i)–(iii) of this exercise is to establish the three-term recurrence
E XERCISES 5.1
pn (x) = (x + Bn )pn−1 (x) − Cn pn−2 (x)
(5.14)
for an appropriate Bn . (i) Observe that pn (x) − xpn−1 (x) is a polynomial of degree n − 1 and thus can be written as a linear combination of p0 (x), . . . , pn−1 (x). Use the property of the inner product, (xpn−1 , pk )2 = (pn−1 , xpk )2 ,
(5.15)
to show that the coefficients of p0 (x), . . . , pn−3 (x) in this linear combination vanish. (ii) Take the inner product of both sides of (5.14) with respect to pn−2 and use (5.15) to deduce the value of Cn . (iii) By comparing coefficients of xn−1 on both sides of (5.14), show that in the Gaussian, Laguerre and Jacobi cases, respectively, Bn(G) = 0,
Bn(L) = −(2n + a − 1),
Bn(J) =
a 2 − b2 . (2n + a + b − 2)(2n + a + b)
(5.16)
(iv) Starting with the three-term recurrence (5.14) for {pn (x)}n=0,1,... deduce that n Z ∞ p (x ) o n w (x ) dx 2 n=0,1,... −∞ x − x satisfies the same three-term recurrence. Hence conclude that the general solution of (5.14) is a linear combination of this sequence and {pn (x)}n=0,1,... . (v) With w2 (x) = e−2V (x) , by considering “ d ” pn , pn−1 dx 2 show that
and
2(V (x)pn , pn−1 )2 = n(pn−1 , pn−1 )2 ,
“ d ” pn , pn dx 2 (V (x)pn , pn )2 = 0.
(5.17)
(vi) For V (x) an even polynomial of degree 2k, show from the three-term recurrence (5.14) with Bn = 0 that the first equation in (5.17) implies a nonlinear recursion of order 2k − 2 for {Cn }n=0,1,... . For the specific potential a2 2 a4 4 a6 6 V (x) = x + x + x 2 4 6
192
CHAPTER 5
obtain the explicit recurrence [308] “ n = Cn a2 + a4 (Cn−1 + Cn + Cn+1 ) + a6 (Cn−1 + Cn + Cn+1 )2 2 ” +a6 (Cn−2 Cn−1 − Cn−1 Cn+1 + Cn+1 Cn+2 ) . 1/2
1/2
1/2
(vii) With qn (x) = pn (x)/(pn , pn )2 , bn = (pn+1 , pn+1 )2 /(pn , pn )2 , show from (5.14) that for suitable {aj }, j = 0, 1, . . . , bj−1 qj−1 (x) + aj qj (x) + bj qj+1 (x) = xqj (x), where q−1 (x) := 0. Use this to deduce that for all real infinite symmetric tridiagonal matrix (Jacobi matrix) 2 a 0 b0 6 6 b 6 0 a1 T=6 6 6 0 b1 4 .. .. . .
x (q0 (x), q1 (x), . . . )T is an eigenfunction of the 0 b1 a2 .. .
3
··· .. . .. . .. .
7 7 7 7 7 7 5
with eigenvalue x. If x is a zero of pn (x) show that (q0 (x), . . . , qn−1 (x))T is an eigenfunction of the top n × n sub-block of T. 2. [508] Here the conventional proof of the Christoffel-Darboux formula (5.10) will be given. (i) Use (5.14) to show that ” “ pn+1 (x)pn (y) − pn (x)pn+1 (y) = (x − y)pn (x)pn (y) + Cn+1 pn (x)pn−1 (y) − pn−1 (x)pn (y) . (ii) In the identity of (i) divide through by (x − y)(pn , pn )2 , substitute the value of Cn+1 from q.1, and sum over n to obtain (5.10). 3. Define Z
Z dr1 · · ·
un [a] = Ω
drn Ω
n Y
Z a(rl )ρ(n) (r1 , . . . , rn ),
Z dr1 · · ·
vn [a] = Ω
l=1
drn Ω
n Y
a(rl )ρT(n) (r1 , . . . , rn )
l=1
and introduce the generating functions U [z; a] = 1 + (i) Show that U [z; a] =
1 ZˆN
∞ X un [a] n z , n! n=1
Z
Z dr1 · · · Ω
(ii) Note from (5.3) that vn [a] = n!
drN Ω
n X (−1)m−1 m m=1
V [z; a] =
∞ X vn [a] n z . n! n=1
N Y (1 + za(rl ))e−βU ( r1 ,..., rN ) . l=1
X k1 ,...,km ≥1 k1 +···+km =n
(iii) Use the result of (ii) to show V [z; a] = log U [z; a],
m Y ukj [a] . kj ! j=1
193
CORRELATION FUNCTIONS AT β = 2
and from this deduce that un [a] = n!
n X
X
m=1 k1 ,...,km ≥1
1 vk1 [a] · · · vkm [a] , m! k1 ! · · · k m !
k1 +···+km =n
or equivalently
m n XY X
ρ(n) (x1 , . . . , xn ) =
ρT(|Gj |) (xgj (1) , . . . , xgj (|Gj |) ).
m=1 G j=1
5.2 FUNCTIONAL DIFFERENTIATION AND INTEGRAL EQUATION APPROACHES 5.2.1 Functional differentiation method An alternative method to derive Proposition 5.1.2 is to introduce a generalized partition function and make use of the functional differentiation formula ∞ δ a(y)f (y) dy := f (x). (5.18) δa(x) −∞ With A[a] a linear functional of a, and δx (y) := δ(x − y) the operation δ/δa(x) is the special case ψ = δx of the operation Dψ A[a] := lim
→0
A[a + ψ] − A[a] .
The generalized partition function ZˆN [a] is defined as in (1.39) for the ordinary partition function, except that there is an arbitrary one body potential with Boltzmann factor a(r) included in the integrand, ZˆN [a] := dr1 a(r1 ) · · · drN a(rN )e−βU(r1 ,...,rN ) . (5.19) Ω
Ω
It then follows from (5.1) and (5.18) that ρ(n) (r1 , . . . , rn ) =
δn ZˆN [a] . ˆ δa( r ) · · · δa( r ) a=1 1 n ZN [1] 1
(5.20)
In the case of the log-gas at β = 2, the generalized partition function can be written as a determinant. P ROPOSITION 5.2.1 For Ω equal to the real line and e−βU(x1 ,...,xN ) =
N l=1
(5.19) can be written as ZˆN [a] = N ! det
∞ −∞
w2 (xl )
(xk − xj )2 ,
1≤j
w2 (x)a(x)pj−1 (x)pk−1 (x) dx
j,k=1,...,N
,
(5.21)
where the pj (x) are as in Proposition 5.1.1. Proof. Begin by rewriting the Boltzmann factor according to the first equality in (5.5). Expanding the determinants using the general formula [8] (cf. (5.8)) det[uj,k ]j,k=1,...,N =
X P ∈SN
ε(P )
N Y l=1
ul,P (l) ,
(5.22)
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CHAPTER 5
where ε(P ) denotes the signature of the permutation, gives e−βU (x1 ,...,xN ) =
N Y
X
w2 (xl )
P ∈SN
l=1
X
ε(P )
ε(Q)
Q∈SN
N Y
pP (l)−1 (xl )pQ(l)−1 (xl ),
l=1
and this implies X
ZˆN [a] =
X
ε(P )
P ∈SN
ε(Q)
Q∈SN
N Z Y l=1
∞ −∞
w2 (x)a(x)pP (l)−1 (x)pQ(l)−1 (x) dx.
The stated formula now follows from the general formula X
X
ε(P )
P ∈SN
ε(Q)
Q∈SN
N Y
uP (l),Q(l) = N ! det[uj,k ]j,k=1,...,N ,
(5.23)
l=1
which itself is a corollary of (5.22).
Due to the orthogonality of {pj (x)}j=0,1,... we see immediately from (5.21) that ZˆN [a]
a=1
= N!
N (pl−1 , pl−1 )2 .
(5.24)
l=1
In Exercises 5.4 q.1(i) below this same result is deduced from the proof of Proposition 5.1.2. We also see from (5.21) that the functional differentiations required according to (5.20) to compute ρ(n) can be performed row by row. For a nonzero contribution this operation must act on n distinct rows. Setting a = 1, the remaining N − n rows unaffected by the functional differentiation are nonzero only in their diagonals. Expanding the determinant by these elements then gives ρ(n) (x1 , . . . , xn ) =
N j1 ,...,jn =1 j1 =···=jn
n (pjl −1 , pjl −1 )−1 2 det[w2 (xμ )pjμ −1 (xμ )pjγ −1 (xμ )]μ,γ=1,...,n . l=1
Minor manipulation reclaims the formula of Proposition 5.1.2. 5.2.2 Integral equation approach Here we will show how the formula for ρT(n) in Proposition 5.1.2 can be derived directly. We require the formula δn log ZˆN [a] ρT(n) (x1 , . . . , xn ) = (5.25) δa(x1 ) · · · δa(xn ) a=1 (cf. (5.20)), which in turn follows immediately from the formulas of Exercises 5.1 q.3. Also required is a formula which rewrites the expression (5.21) for ZˆN [a] in terms of an integral operator [259]. P ROPOSITION 5.2.2 We have
N −1 (pl−1 , pl−1 )2 det l=1
∞
−∞
w2 (y)a(y)pj−1 (y)pk−1 (y) dy
= det δj,k + (pj−1 , pj−1 )−1 2 = det(1 + Ka ),
∞ −∞
j,k=1,...,N
w2 (y)(a(y) − 1)pj−1 (y)pk−1 (y) dy
j,k=1,...,N
195
CORRELATION FUNCTIONS AT β = 2
where 1 denotes the identity operator, Ka is the integral operator on (−∞, ∞) with kernel (a(y)−1)KN (x, y) and KN (x, y) is given by (5.6). Proof. The first equality follows from the orthogonality of {pj (y)}. To derive the second equality, consider the eigenvalue equation
Z
∞ −∞
(1 − a(y))KN (x, y)fl (y) dy = λl fl (x).
P Seeking solutions of the form fl (x) = (w2 (x))1/2 ∞ j=0 clj pj (x) and equating coefficients of pk (x) gives that nonzero eigenvalues can only occur if clj = 0 for j ≥ N , in which case λl clk =
Z ∞ N−1 X 1 clj w2 (y)(1 − a(y))pk (y)pj (y) dy, (pk , pk )2 j=0 −∞
(l = 0, . . . , N − 1).
This is equivalent to the eigenvalue problem for an N × N matrix, the corresponding characteristic polynomial being » Z det λδj,k + (pj−1 , pj−1 )−1 2
∞ −∞
– w2 (y)(a(y) − 1)pj−1 (y)pk−1 (y) dy
= j,k=1,...,N
N−1 Y
(λ + λl ).
l=0
Setting λ = 1 gives the stated result. An alternative approach [522] is to make use of the general operator identity det(1 + AB) = det(1 + BA)
(5.26)
(see, e.g., [128], and for the special case that A and B are matrices see Exercises 5.2 q.2), which can be verified from the second equality in (5.27) below. Let A be the column vector-valued integral operator on (−∞, ∞) with kernel [(w2 (y))1/2 (a(y) − 1)pj−1 (y)]j=1,...,N , and let B be the operator which multiplies by the row vector [(w2 (x))1/2 pk−1 (x)/(pk−1 , pk−1 )2 ]k=1,...,N . We can then identify the l.h.s. of the stated identity with det(1 + AB), and the r.h.s. by det(1 + BA).
Substituting Proposition 5.2.2 in (5.21) shows log
ZˆN [a] = log det(1 + Ka ) = Tr log(1 + Ka ). ZˆN [1]
(5.27)
We remark that because of its appearance in the kernel of an integral operator, KN (x, y) is often referred to as a correlation kernel. Substituting in (5.25), and using the general formulas δ δKa 1 log(1 + Ka ) = , δa(x) 1 + Ka δa(x)
δKa δ 1 1 1 =− δa(x) 1 + Ka 1 + Ka δa(x) 1 + Ka
(5.28)
δKa |a=1 = KN (·, x) while (for the latter formula, see Exercises 2.5 q.1), together with the fact that δa(x) T Ka |a=1 = 0, reclaims the formula for ρ(n) in Proposition 5.1.2. This working shows that generally if
log
ZˆN [a] = log det(1 + Ka ), ZˆN [1]
(5.29)
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CHAPTER 5
where Ka is an operator with kernel of the form (a(y) + c)K(x, y), then G(xi1 , xi2 )G(xi2 , xi3 ) · · · G(xi1 , xin ) ρT(n) (x1 , . . . , xn ) = (−1)n+1 cycles length n
G(xi , xi ) := xi K(1 + Ka )−1
a=1
xi ,
(5.30)
with x|A|y denoting the kernel of the integral operator A and K denoting the integral operator with kernel K(x, y). The working around (5.9) in the proof of Proposition 5.1.2 then gives ρ(n) (x1 , . . . , xn ) = det[G(xj , xk )]j,k=1,...,n .
(5.31)
The identity of Proposition 5.2.2 can also be used to derive the formula of Proposition 5.1.2 for ρ(n) directly, without having first to deduce the formula for ρT(n) . For this one requires the Fredholm expansion formula from the theory of integral equations [541]. P ROPOSITION 5.2.3 Let K(x, y) be a continuous function of x, y in the interval [a, b]. Let K be the integral operator on [a, b] with kernel K(x, y). One has b b ∞ 1 1+ dx1 · · · dxN det[K(xj , xl )]j,l=1,...,N = det(1 + K). (5.32) N! a a N =1
From the definition of Ka in Proposition 5.2.2, it follows from (5.32) that ∞ ∞ ∞ dx1 (a(x1 ) − 1) · · · dxk (a(xk ) − 1) det[K(xα , xβ )]α,β=1,...,k . det(1 + Ka ) = 1 + k=1
−∞
−∞
Substituting this for ZN [a] in (5.20) we see that the functional derivatives can be computed immediately to reclaim the sought determinant formula for ρ(n) . P 1. Consider the integral operator on [a1 , a2 ] with kernel K(x, y) = N−1 j=0 φj (x)φj (y), where {φj (x)}j=0,1,... are a complete set of real linearly independent functions. From the proof of Proposition 5.2.2 we know that this integral operator admits N eigenfunctions {ψj (x)}j=0,...,N−1 of the form
E XERCISES 5.2
ψj (x) =
N−1 X
ckj φk (x),
k=0
with corresponding eigenvalues {λk }k=0,...,N−1 specified by N−1 X
Z gij cjk = cik λk ,
a2
gij :=
φi (x)φj (x) dx. a1
j=0
(i) Note from the above equation that the matrix [cjk ] has columns which consist of the eigenvectors of [gij ]. (ii) Use (i) and the fact that since K(x, y) is real and symmetric, its eigenfunctions are orthogonal and can be chosen to be real, to note that [cjk ] can be chosen to be real orthogonal. From this conclude that N−1 X j=0
ψj (x)ψj (y) =
N−1 X j=0
Z
a2
φj (x)φj (y) =: K(x, y),
ψj2 (x) dx = λj .
a1
2. The aim of this exercise is to prove the identity det(1p + Ap×q Bq×p ) = det(1q + Bq×p Ap×q ), which is the special case of (5.26) in which the operators A and B are matrices.
(5.33)
197
CORRELATION FUNCTIONS AT β = 2
(i) Note the factorizations » – » –» – Ap×q 0p×q 1p + Ap×q Bq×p Ap×q 1p 1p = , 0q×p 1q −Bq×p 1q Bq×p 1q » – » –» – 1p Ap×q 0p×q Ap×q 1p 1p = . 0q×p 1q + Bq×p Ap×q Bq×p 1q −Bq×p 1q (ii) Take the determinant of the identities in (i) to deduce (5.26).
5.3 RATIOS OF CHARACTERISTIC POLYNOMIALS Let us denote by ·UEN (w2 ) the canonical average (recall (1.45)) for a general one-component log-gas at β = 2, or equivalently a matrix ensemble with unitary symmetry, so that ∞ ∞ N 1 f UEN (w2 ) := dx1 · · · dxN f (x1 , . . . , xN ) w2 (xl ) (xk − xj )2 . ˆ ZN −∞ −∞ l=1 1≤j
(5.34)
in the special cases (K, Q) = (1, 0), (0, 1). Because in the case of w2 (x) classical {xj } can be interpreted as N the eigenvalues of a random matrix X, one has l=1 (u − xl ) = det(u1 − X), and (5.34) is then the average of ratios of characteristic polynomials. Here (5.34) will be evaluated as a (K + Q) × (K + Q) determinant [252], [32]. The special case K = Q can be used to reclaim Proposition 5.1.2. P ROPOSITION 5.3.1 Let {pj (x)}j=0,1,... denote the set of monic orthogonal polynomials with respect to the weight w2 (x), and let ∞ pk (t) w2 (t) dt, Δ({yj }j=1,...,p ) := Δ(y) = (yk − yj ). (5.35) hk (x) := −∞ x − t 1≤j
For Q ≤ N
⎡
hN −Q (v1 ) ⎢ .. ⎢ . ⎢ ⎢ hN −Q (vQ ) (−1)Q(Q−1)/2 AK,Q (u, v) = N −1 det ⎢ ⎢ pN −Q (u1 ) (p , p ) Δ(u)Δ(v) ⎢ j j 2 j=N −Q ⎢ .. ⎣ .
··· ··· ···
pN −Q (uK ) · · ·
⎤ hN +K−1 (v1 ) ⎥ .. ⎥ . ⎥ hN +K−1 (vQ ) ⎥ ⎥. pN +K−1 (u1 ) ⎥ ⎥ ⎥ .. ⎦ . pN +K−1 (uK )
(5.36)
Proof. We base our proof on workings in [116]. In the notation D({aj }j=1,...,m ; {bk }k=1,...,n := D(a; b) =
n m Y Y
(bk − aj )
(5.37)
j=1 k=1
we have AK,Q (u, v) =
1 C
Z
∞ −∞
Z dx1 · · ·
∞
dxN −∞
N Y l=1
w2 (xl )
D(x; u) (Δ(x))2 . D(x; v)
(5.38)
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CHAPTER 5
With a b denoting the concatenation of the sequences (ai ) and (bj ) we have Δ(a b) = Δ(a)D(a; b)Δ(b) and thus Δ(x)Δ(v) D(x; u) 1 (Δ(x))2 = Δ(x u). D(x; v) Δ(u)Δ(v) D(x; v) The first line of (5.5) tells us that " Δ(x u) = det
[pk−1 (xj )] [pk−1 (uj )]
j=1,...,N k=1,...,N +K j=1,...,K k=1,...,N +K
# ,
(5.39)
while according to the identity (5.45) of Exercises 5.3 q.1 below, for N ≥ Q h i Δ(x)Δ(v) 1 1 ··· . = (−1)Q(Q−1)/2 det p0 (xj ) p1 (xj ) · · · pN−Q−1 (xj ) D(x; v) v1 − xj vQ − xj j=1,...,N
(5.40)
Let us now substitute (5.40) and (5.39) in (5.38). Because of the orthogonality of {pk−1 (x)}k=1,2,... with respect to w2 (x) we see that we can replace (5.39) by # " [pk−1 (xj )] j=1,...,N k=1,...,N +K (5.41) det [0]K×(N−Q) [pk−1 (uj )] j=1,...,K k=N −Q+1,...,N +K
without changing the value of the integral. And in this, the elements chosen from columns 1, . . . , N − Q must be from rows 1, . . . , N for a nonzero contribution. Let us denote the rows so selected by 1 ≤ σ(1) < σ(2) < · · · < σ(N − Q) ≤ N . For the remainder of the first N rows we use the labels 1 ≤ σ(N − Q + 1) < · · · < σ(N + K) ≤ N where all the σ(k) must be distinct, and we denote the set of such σ’s by ΣN−Q,Q . By first expanding the determinant (5.41) by elements in row σ(j) of column j (j = 1, . . . , N − Q), then antisymmetrizing in the ordering of these σ(j) we obtain " # [pk−1 (xσ(j) )] j=N −Q+1,...,N X k=N −Q+1,...,N +K ε(σ) det[pj−1 (xσ(k) )]j,k=1,...,N−Q det , [pk−1 (uj )] j=1,...,K ΣN −Q,Q
k=N −Q+1,...,N +K
which is an example of a Laplace expansion Q by multiple columns [8]. In the integrand the first determinant in this expression can be replaced by (N − Q)! N−Q j=1 pj−1 (xσ(j) ). Multiplying this with (5.40) and integrating over xσ(j) (j = 1, . . . , N − Q) row by row in the determinant, due to the orthogonality, the determinant can be expanded down columns σ(j) to show that (5.38) is equal to Z
N−Q X (−1)Q(Q−1)/2 (N − Q)! Y (pj−1 , pj−1 )2 CΔ(u)Δ(v) j=1 Σ
N −Q,Q
Z
∞ −∞
dxσ(N−Q+1) w2 (xσ(N−Q+1) ) · · ·
h
i 1 × dxσ(N) w2 (xσ(N) ) det det vi − xσ(j) j=Ni=1,...,Q −∞ −Q+1,...,N ∞
"
[pk−1 (xσ(j) )] [pk−1 (uj )]
j=N −Q+1,...,N k=N −Q,...,N +K
j=1,...,K k=N −Q,...,N +K
# .
QQ
− xσ(N−Q+j) ), and the integrations carried out “ ” , row by row. As the resulting expression is independent of the set ΣN−Q,Q the sum over this quantity contributes N Q while C is given by (5.24), we see that (5.36) results.
In this expression the first determinant can be replaced by Q!/
j=1 (vj
We have the following general formula relating AK,Q (u, v) in the case K = Q = k to the k-point correlation function [91]. P ROPOSITION 5.3.2 Let Res v=y
∞
−∞
f (t) 1 dt := lim+ Im v−t →0 π
∞
−∞
f (t) dt = f (y). y − i − t
(5.42)
199
CORRELATION FUNCTIONS AT β = 2
We have ρ(k) (y1 , . . . , yk ) = Res · · · Res v1 =y1
vk =yk
∂k Ak,k (u, v) . ∂u1 · · · ∂uk u=v
(5.43)
Proof. This follows upon noting N Y k ˛ Y uj − xl ˛ ∂k = ˛ ∂u1 · · · ∂uk vj − xl u=v j=1 l=1
N X j1 ,...,jk =1 j1 =···=jk
1 . (u p − xjp ) p=1
Qk
It should be noted that (5.43) holds with UEN (w2 ) in the definition of AK,Q replaced by a general p.d.f. It is shown in Exercises 5.3 q.2 that with K = Q (5.36) can be rewritten to read ∞ (−1)Q(Q−1)/2 ∞ AQ,Q (u, v) = ds · · · dsQ Δ(s)D(s; u) 1 Δ(u)Δ2 (v) −∞ −∞ Q 1 w2 (sl ) 1/2 × det KN (uj , sk ) . (5.44) vl − sl w2 (ul ) j,k=1,...,Q l=1
With Q = k, to get a nonzero contribution from the operation of differentiating with respect to u1 , . . . , uk k and setting u = v we see that only the factor j=1 (uj − sj ) in D(s; u) need be differentiated. Furthermore the operations Res are immediate, and the result of Proposition 5.1.2 follows. E XERCISES 5.3
1. [48] In terms of the notation (5.35), (5.37), here the identity # " h 1 i Δ(a)Δ(b) k−1 = (−1)N(N−1)/2+N(M −1) det [aj ] j=1,...,M , k=1,...,M −N aj − bk j=1,...,M D(b; a) k=1,...,N
(5.45)
where it is assumed N ≤ M , will be derived. (i) In the Cauchy double alternant identity (4.33) replace N → M . Take the limit y1 → ∞ to deduce (5.45) in the case N = M − 1. (ii) Subtract the first column of the determinant obtained in (i) from the second column, then take the limit y2 → ∞ to obtain (5.45) in the case N = M −2. Subtract the first and second columns of the corresponding determinant from the third, take the limit as y3 → ∞ to obtain (5.45) in the case N = M − 3, and proceed analogously in the general case. 2. [32] The objective of this exercise is to derive (5.44). (i) Generalize the workings of the proofs of Propositions 5.1.2 and 5.1.3 to show m N Y DY
(uμ −xj )(vμ −xj )
j=1 μ=1
QN+m−1
E UEN (w2 )
=
l=N
h N+m−1 X pν (uj )pν (vk ) i (pl−1 , pl−1 )2 det . Δ(u)Δ(v) (pν , pν )2 j,k=1,...,m ν=0
(ii) Rewrite the determinant in (5.36) in the form 2 Z
∞ −∞
Z ds1 · · ·
∞
dsQ −∞
6 6 6 Q Y 6 w2 (sl ) det 6 6 v l − sl 6 l=1 6 4
pN−Q (s1 ) .. . pN−Q (sQ ) pN−Q (u1 ) .. . pN−Q (uK )
··· ··· ··· ···
pN+K−1 (s1 ) .. . pN+K−1 (sQ ) pN+K−1 (u1 ) .. . pN+K−1 (uK )
3 7 7 7 7 7. 7 7 7 5
200
CHAPTER 5
Use (5.36) in the case Q = 0 to show that the determinant in this expression is equal to Δ(s u)AQ+K,0 (s u, ·)|N→N−Q . (iii) Substitute for AQ+K,0 (s u, ·)|N→N−Q according to the result of (i), substitute this for the determinant above, and finally substitute in (5.36) with K = Q to deduce (5.44).
5.4 THE CLASSICAL WEIGHTS 5.4.1 The Gaussian, Laguerre and Jacobi weights From Proposition 1.3.4, (3.16) and (3.74), the one-body Boltzmann factor w2 (x) for the Gaussian, Laguerre and Jacobi ensembles is given by w2 (x) = e−x , 2
(G)
w2 (x) = xa e−x , (L)
−∞ < x < ∞,
x > 0, a > −1
and (J)
w2 (x) = (1 − x)a (1 + x)b ,
−1 < x < 1, a, b > −1
respectively (in the latter two cases w2 (x) = 0 outside the specified intervals). The monic orthogonal polynomials associated with these weight functions are proportional to the Hermite, Laguerre and Jacobi classical polynomials respectively (see, e.g., [508]). Explicitly −n p(G) Hn (x), n (x) = 2 n a p(L) n (x) = (−1) n!Ln (x), Γ(a + b + n + 1), (a,b) n P (x) p(J) n (x) = 2 n! Γ(a + b + 2n + 1) n
(5.46)
where n (2m)! n−2m x (−1) 2 , Hn (x) = 2m 2m m! m=0 n n + a xm , Lan (x) = (−1)m n − m m! m=0 m n x−1 n + a (n + a + b + 1)m n (a,b) Pn (x) = (a + 1)m 2 m n m=0
[n/2]
m n−m
(5.47)
(the notation (u)p is defined in (5.83) below). The corresponding normalizations are (G)
(pn , pn )2
= π 1/2 2−n n!,
(L)
(pn , pn )2 = Γ(n + 1)Γ(a + n + 1), Γ(n + 1)Γ(a + b + 1 + n)Γ(a + 1 + n)Γ(b + 1 + n) (J) . (pn , pn )2 = 2a+b+1+2n Γ(a + b + 2n + 1)Γ(a + b + 2n + 2)
(5.48)
Using the above formulas in Proposition 5.1.2, with KN (x, y) evaluated according to Proposition 5.1.3 and (5.13), gives an explicit expression for the n-particle correlation in each of the ensembles.
201
CORRELATION FUNCTIONS AT β = 2
5.4.2 Circular ensembles and the Cauchy weight By making the transformation (2.50) we know from (2.51) that N j=1
wβ (eiθj )
|eiθk − eiθj |β dθ1 · · · dθN
1≤j
= 2βN (N −1)/2+N
N
wβ
j=1
1 + iλ j (1 + λ2j )−β(N −1)/2−1 1 − iλj
|λk − λj |β dλ1 · · · dλN .
1≤j
(5.49) In the particular case wβ (z) = |1 + z|βa ,
z := eiθ
(5.50)
this coincides with (3.123), so the effective real weight on the r.h.s. of (5.49) is then wβ (λ) = (1 + λ2 )−α ,
α = β(N + a − 1)/2 + 1,
(5.51)
which specifies the Cauchy ensemble. Our present interest is in the case β = 2, when the Cauchy weight is (1 + x2 )−(N +a) . It is known [470], [464] (see also Exercises 5.4 q.2) that the orthogonal polynomials associated with this weight are the Jacobi polynomials (−a−N,−a−N )
{Pj
(ix)}0≤j
(5.52)
(the bound on j is required because (5.51) has only a finite number of bounded moments). From the final formula in (5.46) it follows that the monic polynomials corresponding to the set (5.52) are (x) := i−n p(J) . (5.53) p(Cy) n n (ix) a=b a →−N −a
For the normalization one has (see Exercises 5.4 q.2) (Cy) (J) (pn , pn )2 = (−1)n−1 tan πa (pn , pn )2 = π2−2(N −n)−2(a−1)
a=b a →−N −a
Γ(n + 1)Γ(2(N + a) − 2n)Γ(2(N + a) − 1 − 2n) . Γ(2(N + a) − n)(Γ(N + a − n))2
(5.54)
The n-point correlation for the weight (5.51) on the real line thus follows immediately from Propositions 5.1.2 and 5.1.3. To deduce from this ρ(n) for the weight (5.50) we use the fact that for θj (= 2πxj /L) and λj related by (2.50), n 4π ρ (λ , . . . , λ ) ρ(n) (x1 , . . . , xn ) = 1 n (n) 2L |1 + z | λj =i(1−zj )/(1+zj ) j j=1 =
n π n 1
L
2(N −1+a) |1 + zj | (1 + zj )2
2 ⎤ ⎡ (Cy) 1−z (Cy) (Cy) (Cy) 1−zj 1−zk 1−zk i 1+zjj pN −1 i 1+z − p i p i pN N N −1 1+zk 1+zj k ⎦ × det ⎣ (Cy) 2i(pN −1 , pN −1 )2 (zj − zk ) j=1
(5.55)
j,k=1,...,n
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CHAPTER 5
5.4.3 Structural properties associated with the classical weights The weight functions
w2 (x) =: e
−2V (x)
⎧ 2 ⎪ e−x , ⎪ ⎨ a −x x e (x > 0), = a ⎪ (1 − x) (1 + x)b (−1 < x < 1), ⎪ ⎩ 2 −α (1 + x ) ,
Hermite, Laguerre, Jacobi, Cauchy,
(5.56)
all have the property that their logarithmic derivative is a rational function. Thus, if we write g(x) w2 (x) = −2V (x) =: − , w2 (x) f (x) where f and g have no common factors, and f > 0, then ⎧ (1, 2x), ⎪ ⎪ ⎨ (x, x − a), (f, g) = (1 − x2 , (a − b) + (a + b)x), ⎪ ⎪ ⎩ (1 + x2 , 2αx),
(5.57)
Hermite, Laguerre, Jacobi, Cauchy.
(5.58)
Note that in (5.58) degree f ≤ 2,
degree g ≤ 1.
(5.59)
We will call any weight function with this property classical. An equivalent characterization of a classical polynomial, which has an immediate consequence regarding the Christoffel-Darboux sum, and future consequence in the construction of skew orthogonal polynomials to be undertaken in the next chapter, follows by consideration of the operator [4] f − g d d −2V A := f + = e2V f . (5.60) fe dx 2 dx First let us consider a general property of A. P ROPOSITION 5.4.1 Assuming e−2V (x) vanishes at the endpoints of its support, (φ, Aψ)2 = −(Aφ, ψ)2 .
(5.61)
Proof. This follows immediately upon using the second form of A in (5.60) and integration by parts. Consider now the matrix [(pj , Apk )2 ]j,k=0,...,N −1 , where {pj }j=0,1,... are the monic orthogonal polynomials associated with the weight function e−2V (x) . By (5.61) this matrix is antisymmetric. Furthermore, the bound on the order of the polynomial pairs (f, g) given in (5.59) and the definition (5.60) give that ck Apk (x) = − pk+1 (x) + polynomials of lower degree. (5.62) (pk+1 , pk+1 )2 These two facts together imply that in the classical cases ⎡ 0 c0 0 ⎢ −c0 0 c1 ⎢ ⎢ 0 −c1 0 ⎢ ⎢ 0 0 −c 2 A := [(pj , Apk )2 ]j,k=0,...,N −1 = ⎢ ⎢ .. .. .. ⎢ . . . ⎢ ⎣
0 0 c2 0 .. .
⎤
··· ··· ··· ··· .. . 0 −cN −1
cN −1 0
⎥ ⎥ ⎥ ⎥ ⎥ ⎥, ⎥ ⎥ ⎥ ⎦
(5.63)
203
CORRELATION FUNCTIONS AT β = 2
or equivalently Apk (x) = −
ck ck−1 pk+1 (x) + pk−1 (x). (pk+1 , pk+1 )2 (pk−1 , pk−1 )2
From the explicit forms (5.58), the definition (5.60) and the fact that {pk } are monic we find ⎧ 1, Hermite, ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ 1, Laguerre, ck 2 = 1 ⎪ (pk+1 , pk+1 )2 ⎪ 2 (2k + 2 + a + b), Jacobi, ⎪ ⎪ ⎪ ⎩ α − 1 − k, Cauchy,
(5.64)
(5.65)
with the explicit values of (pk+1 , pk+1 )2 given by (5.48) and (5.54). Writing ψk := e−V pk , it follows from (5.64) and the three-term recurrence (5.14) that {ψk , ψk−1 } satisfies the first order system ψk (x) A11 (x) A12 (x) ψk (x) = (5.66) f (x) ψk−1 A21 (x) A22 (x) ψk−1 (x) (x) with
ck 1 (x + Bk+1 ) + f , (pk+1 , pk+1 )2 2 ck−1 ck (pk , pk )2 A12 (x) = + , (pk−1 , pk−1 )2 (pk+1 , pk+1 )2 (pk−1 , pk−1 )2 ck−1 ck−2 A21 (x) = + , (pk , pk )2 (pk−1 , pk−1 )2 1 ck−2 (x + Bk ). A22 (x) = − f + 2 (pk−1 , pk−1 )2
A11 (x) = −
Using (5.65), (5.48), (5.58) and (5.16) one obtains from this that in the Hermite case (G)
(G)
A11 (x) = −A22 (x) = −x,
(G)
A12 (x) = k,
(G)
A21 (x) = 2;
(5.67)
in the Laguerre case 1 (L) (L) A11 (x) = −A22 (x) = − (x − 2k − a), 2 while in the Jacobi case
(L)
A12 (x) = k(a + k),
b 2 − a2 1 (J) (J) , A11 (x) = −A22 (x) = − (2k + a + b)x + 2 2(2k + a + b) k(a + b + k)(a + k)(b + k) (J) A12 (x) = , 2(a + b + 2k + 2)(a + b + 2k + 1)(a + b + 2k)
(L)
A21 (x) = 1;
(5.68)
(J)
A21 (x) = 2k + a + b − 1. (5.69)
A consequence of the matrix differential equation (5.66) is integral formulas for the Christoffel-Darboux sum (5.10) in the classical cases [519], [334], [221].
P ROPOSITION 5.4.2 Let ψk := e−V pk . Then in the Gaussian case ∞ 1 (G) (G) (G) (G) (G) ψ (x + t)ψ (y + t) + ψ (x + t)ψ (y + t) dt; (5.70) KN (x, y) = N N −1 N −1 N (G) 0 (pN −1 , pN −1 )2
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CHAPTER 5
in the Laguerre case
1
(L)
KN (x, y) = −
(L)
2(pN −1 , pN −1 )2
and in the Jacobi case (J)
KN (x, y) =
(J)
2(pN −1 , pN −1 )2
0
(L) (L) (L) (L) ψN (xt)ψN −1 (yt) + ψN −1 (xt)ψN (yt) dt;
y+s (J) x + s (J) ψN −1 ψN 1 + xs 1 + ys 0 x+s y + s ds (J) (J) ψ . + ψN −1 1 + xs N 1 + ys (1 + xs)(1 + ys)
(2N + a + b)
1
(5.71)
1
(5.72)
Proof. In general “ ∂ ∂ ” ψN (x)ψN−1 (y) − ψN (y)ψN−1 (x) + ∂x ∂y x−y ” ∂ ”“ 1 “ ∂ + ψN (x)ψN−1 (y) − ψN (y)ψN−1 (x) = x − y ∂x ∂y – » –» 1 “ ∂ ∂ ” [ ψN (x) ψN−1 (x) ] 0 1 ψN (y) . = + −1 0 ψN−1 (y) x − y ∂x ∂y Now specializing to the Hermite case, we see that (5.66) can be used to compute the derivatives, thus implying that it is equal to 3 2 (G) (G) (G) (G) A21 (x) − A21 (y) A11 (x) + A22 (y) " # − (G) (G) 7 ψ (G) (y) [ ψN (x) ψN−1 (x) ] 6 x−y x−y N 7 6 4 A(G) (x) + A(G) (y) A(G) (x) − A(G) (y) 5 ψ (G) (y) 11 12 12 N−1 − 22 x−y x−y # – " (G) » (G) (G) ψN (y) 0 −1 [ ψN (x) ψN−1 (x) ] = (G) −1 0 ψN−1 (y) ” “ (G) (G) (G) (G) = − ψN (x)ψN−1 (y) + ψN−1 (x)ψN (y) , where the second line follows by using the explicit form of the matrix elements given in (5.67). Replace x → x + t, y → y + t and note that (∂/∂x + ∂/∂y) can be replaced by ∂/∂t. Integrating both sides from t = 0 to ∞ gives (5.70). In the Laguerre case, one begins by making use of the general formula “ x
∂ ∂ ” (xy)1/2 (xy)1/2 “ ∂ ∂ ” +y f= x +y f. ∂x ∂y x − y x−y ∂x ∂y
Proceeding as in the Gaussian case then gives “ x
” “ (xy)1/2 ∂ ” ∂ (L) (L) (L) (L) (L) +y (xy)1/2 KN (x, y) = − ψN (x)ψN−1 (y) + ψN−1 (x)ψN (y) . (L) ∂x ∂y 2(pN−1 , pN−1 )2
Changing variables x = e−t , y = e−s , and arguing as in the Gaussian case, we see from this that 1
KN (e−t , e−s ) = − (L)
(L)
2(pN−1 , pN−1 )2 Z ∞ ” “ (L) (L) (L) (L) × e−u ψN (e−t−u )ψN−1 (e−s−u ) + ψN−1 (e−t−u )ψN (e−s−u ) du, 0
which is equivalent to (5.71).
205
CORRELATION FUNCTIONS AT β = 2
Finally, in the Jacobi case, we make use of the general formula “ ∂ ∂ ” (1 − x2 )1/2 (1 − y 2 )1/2 ∂ ∂ ” (1 − x2 )1/2 (1 − y 2 )1/2 “ (1 − x2 ) f, (1 − x2 ) + (1 − y 2 ) f= + (1 − y 2 ) ∂x ∂y x−y x−y ∂x ∂y and proceed as in the Hermite and Jacobi cases to deduce “ ∂ ∂ ” (J) (1 − x2 ) + (1 − y 2 ) (1 − x2 )1/2 (1 − y 2 )1/2 KN (x, y) ∂x ∂y ” (2N + a + b) (1 − x2 )1/2 (1 − y 2 )1/2 “ (J) (J) (J) (J) ψN (x)ψN−1 (y) + ψN−1 (x)ψN (y) . =− (J) 2 (pN−1 , pN−1 )2 Changing variables x = tanh u, y = tanh v we deduce from this that 1 (J) K (tanh u, tanh v) cosh u cosh v N Z ∞ 1 (2N + a + b) = (J) cosh(u + t) cosh(v + t) 2(pN−1 , pN−1 )2 0 “ ” (J) (J) (J) (J) × ψN (tanh(u + t))ψN−1 (tanh(v + t)) + ψN−1 (tanh(u + t))ψN (tanh(v + t)) dt. Another change of variables gives (5.72).
With f (x) as in (5.58) and x0 = 0, 1, 0 for the Hermite, xLaguerre and Jacobi cases, the change of variables in the proof of Proposition 5.4.2 can be written u(x) = x0 dz/f (z), so that ⎧ Hermite, ⎨ (u, 1), (x(u), f (x(u))) = (eu , eu/2 ), Laguerre, ⎩ (tanh u, 1/ cosh u), Jacobi. (·) (·) Writing ψ˜k (u) := f (x(u))ψk (x(u)), and writing c˜k for (5.65), one sees that the results in the proof of Proposition 5.4.2 can be written in the unified form [332] (·) f (x(u)) f (x(v))KN (x, y) ∞ c˜N −1 (·) (·) (·) (·) ψ˜N (u + t)ψ˜N −1 (v + t) + ψ˜N −1 (u + t)ψ˜N (v + t) dt. (5.73) = (·) (pN −1 , pN −1 )2 0 E XERCISES 5.4
1. Z
∞ −∞
(i) From the proof of Proposition 5.1.2 show that Z
dx1 w2 (x1 ) · · ·
Y
∞
dxN w2 (xN ) −∞
(xk − xj )2 = N !
1≤j
N Y
(pk−1 , pk−1 )2 .
(5.74)
k=1
(ii) Use the integral evaluations of Propositions 4.7.1 and 4.7.3, (4.144) and (4.145) in (i) to provide alternative (G) (L) evaluations of (pn , pn )2 , (pn , pn )2 , (pn , pn )(J) and (pn , pn )(Cy) . (iii) Make use of the Vandermonde determinant (1.173), and (5.22), (5.23) to show Z ∞ Z ∞ hZ ∞ i Y dx1 w2 (x1 ) · · · dxN w2 (xN ) (xk − xj )2 = N ! det w2 (x)xj+k−2 dx −∞
Z
L/2 −L/2
−∞
Z dx1 w2 (z1 ) · · ·
L/2
dxN w2 (zN ) −L/2
Y 1≤j
j,k=1,...,N
−∞
1≤j
|zk − zj |2 = N ! det
hZ
L/2 −L/2
w2 (z)z j−k dx
i
,
(5.75)
j,k=1,...,N
,
(5.76)
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CHAPTER 5
where in (5.76) z = e2πix/L . (iv) Use the result of Proposition 2.2.5 in the case β = 2, together with the fact from Proposition 4.7.2 that C2,N therein equals (2π)N N ! (or alternatively derive this using (5.76) with L = 2π, w2 (z) = 1), to show that the final identity in (iii) can be written N DY
w2 (eiθl )
l=1
E U (N)
= det
h 1 Z π i w2 (eiθ )eiθ(j−k) dθ . 2π −π j,k=1,...,N
(5.77)
(v) Use the Vandermonde determinant formula (1.173) in the multidimensional integral formula of Proposition 5.1.4 for pN (x), together with elementary row operations in the resulting determinant to show Z ∞ N! cj := w2 (x)xj dx. (5.78) pN (x) = det[cj+k x − cj+k+1 ]j,k=0,...,N−1 , C −∞ 2.
(i) Show from the Euler beta integral (4.2) that for q ∈ Z≥0 Z 1 x2q Γ(1/2 + q)Γ(α − q − 1/2) . dx = (−1)q−1 cot πα (1 − x 2 )α Γ(α) −1 (ii) Make the change of variables x = y/(1 + y) in the Euler beta integral, then put y → x2 to deduce that for q ∈ Z≥0 Z ∞ x2q Γ(1/2 + q)Γ(α − q − 1/2) dx = . 2 )α (1 + x Γ(α) −∞ (iii) By comparing (i) and (ii) deduce that for f an analytic function such that the integrals are defined Z ∞ Z 1 f (x) f (ix) dx = − cot πα dx, 2 )α (1 − x (1 + x 2 )α −1 −∞ where the integrals are to be interpreted as their analytic continuation in α, and from this deduce (5.54).
3. [291] The objective of this exercise is to compute the integral in (1.97) and so extract information on the coefficients ak+1−2g (k) in (1.106). (i) Note from Proposition 5.1.2, (5.46) and (5.48) that in the GUE (Hermite case) ρ(1) (λ) = e−λ
2
N−1 X ν=0
(Hν (λ))2 . π 1/2 2ν ν!
(ii) Substitute the result of (i) in (1.97) and use the expansions (Hν (λ))2 =
ν X
ep,ν H2p (λ),
ep,ν =
p=0
λ2k =
k X
cl,2k H2k−2l (λ),
cl,2k =
l=0
2ν−p ν! “ ν ” , p! p
(2k)! 22k l!(2k − 2l)!
together with the orthogonality of {Hj (λ)}j=0,1,... to show that Tr(X2k )GUE∗ = (2k − 1)!!
k X p=0
2p
“ k ”“ N ” p p+1
and use this to rederive (1.112). Read off too the value of ak−1 (k).
207
CORRELATION FUNCTIONS AT β = 2
(iii) Substitute the result of (ii) in (1.106) to deduce the recurrence ak+1−2g (k) =
4k − 2 (k − 1)(2k − 1)(2k − 3) ak−2g (k − 1) + ak+1−2g (k − 2), k+1 k+1
where al (k) = 0 for l > k + 1 and al (0) = δl,1 .
5.5 CIRCULAR ENSEMBLES AND THE CLASSICAL GROUPS 5.5.1 Orthogonal polynomials on the unit circle The formalism of Section 5.4.2 relating the circular ensemble to an ensemble on the real line has the virtue of unifying the treatment of the two classes of problems. This is particularly useful in the next chapter, when the correlations at β = 1 and 4 are calculated. However, restricting attention to β = 2, the circular ensembles can readily be studied without recourse to the mapping (2.50) [421]. Write zj := e2πixj /L , −L/2 ≤ xj ≤ L/2, and let {pk (z)}k=0,1,... be the family of monic orthogonal polynomials associated with the real, non-negative weight function w2 (z) so that analogous to (5.4) we have
pk (z) = z k + αk−1 z k−1 + · · · + α0 , L/2 −L/2
(C)
w2 (z)pj (z)pk (z) dx := (pj , pk )2
(C)
= (pj , pj )2 δj,k .
(5.79)
A Gram-Schmidt construction of these polynomials from {z k }k=0,1,... shows that each pj (z) has real coefficients, provided w2 (z) = w2 (¯ z ), and thus provided w2 is even in x. This condition will be assumed henceforth. Proceeding as in the proof of (5.5) we see that N l=1
w2 (zl )
|zk − zj |2 =
N
(C)
(pj−1 , pj−1 )2
det[KN (zj , z¯k )]j,k=1,...,N ,
j=1
1≤j
−1 1/2 N pν (z)pν (¯ u) ¯) := w2 (z)w2 (u) . KN (z, u (C) ν=0 (pν , pν )2
(5.80)
From this and (5.79), the proof of Proposition 5.1.2 shows that the corresponding correlation function is given by ρ(n) (x1 , . . . , xn ) = det[KN (zj , z¯k )]j,k=1,...,n .
(5.81)
¯) can be summed explicitly via a result analogous to the Christoffel-Darboux Also, the function KN (z, u formula (5.10) [508]. P ROPOSITION 5.5.1 For |u| = |z| = 1 we have KN (z, u ¯) =
¯N pN (u)z N pN (¯ z ) − pN (¯ u)pN (z) (w2 (z)w2 (u))1/2 u . (C) 1−u ¯z (pN , pN )
(5.82)
2
A derivation of this result, modeled on the derivation of (5.10), is given in Exercises 5.5 q.1. Consider now the weight function (5.50). In this case the polynomials pn (z) can be obtained explicitly, being polynomial examples of the hypergeometric function 2 F1 . The latter is defined in general by the infinite series ∞ (α)p (β)p p x , F (α, β; γ; x) := (u)p := u(u + 1) · · · (u + p − 1), (5.83) 2 1 (γ)p p! p=0
208
CHAPTER 5
where unless the series terminates, it is required that |x| < 1 for absolute convergence of the series. P ROPOSITION 5.5.2 With w2 (z) given by (5.50) and z = e2πix/L , the monic polynomials a (z) := p(CJ) 2 F1 (−n, a + 1; −n + 1 − a; z) n a+n have the orthogonality property L/2 (CJ) w2 (z)pj (z)pk (¯ z ) dx = (pj , pj )2 δj,k ,
(5.84)
−L/2
(CJ)
(pn , pn )2
=L
Γ(n + 2a + 1)Γ(n + 1) . (Γ(n + a + 1))2
(5.85)
Proof. This is established in Exercises 5.5 q.3. 5.5.2 The classical groups We know from Section 2.6 that the eigenvalue p.d.f.’s for the orthogonal and symplectic groups, after the change of variables cos θj = yj , coincide with the Jacobi ensemble in the case β = 2 and a, b appropriately chosen. Because the Jacobi polynomial in the variable cos θ is simple for these cases, one finds simple expressions for KN after substituting back the original variables. P ROPOSITION 5.5.3 In terms of the original variables {θi }, the k-point correlation for the eigenvalues of the classical groups on the upper half-circle (0 < θi < π) is given by ρ(k) (θ1 , . . . , θk ) = det[KN (θj , θl ]j,l=1,...,k , where
(5.86)
1 sin((2N − 1)(θ − θ )/2) sin((2N − 1)(θ + θ )/2) + , 2π sin((θ − θ )/2) sin((θ + θ )/2) sin N (θ + θ ) 1 sin N (θ − θ ) O± (2N +1) ∓ , (θ, θ ) = KN 2π sin((θ − θ )/2) sin((θ + θ )/2) 1 sin((2N + 1)(θ − θ )/2) sin((2N + 1)(θ + θ )/2) Sp(2N ) KN − , (θ, θ ) = 2π sin((θ − θ )/2) sin((θ + θ )/2) O+ (2N )
KN
(θ, θ ) =
O− (2N +2)
(θ, θ ) = KN
KN
Sp(2N )
(θ, θ ).
(J) Proof. Now KN (θ, θ ) in (5.86) is related to KN (y, y ) for the corresponding Jacobi ensemble by
KN (θ, θ ) = (sin θ sin θ )1/2 KN (cos θ, cos θ ). (J)
For definiteness consider O+ (2N ). According to Proposition 3.7.1 and the Christoffel-Darboux sum (5.10) with w2 (x) = (J) (1 − x2 )−1/2 , pN (x) = pN (x)|a=b=−1/2 , we have O + (2N)
KN
(θ, θ ) =
pN (cos θ)pN−1 (cos θ ) − pN (cos θ )pN−1 (cos θ) ˛˛ . ˛ cos θ − cos θ a=b=−1/2 (J)
1 (J)
(pN−1 , pN−1 )2
(J)
(J)
(J)
But from (5.46) and [508] n p(J) n (x)|a=b=−1/2 = 2 n!
Γ(n) (−1/2,−1/2) Pn (x) = 2−(n−1) Tn (x), Γ(2n)
n ≥ 1,
where Tn (x) denotes the Chebyshev polynomial of the first kind. Hence after recalling (5.48) and using the identity
209
CORRELATION FUNCTIONS AT β = 2
Tn (cos θ) = cos nθ we have O + (2N)
KN
(θ, θ ) =
1 cos N θ cos(N − 1)θ − cos N θ cos(N − 1)θ . π cos θ − cos θ
Use of trigonometric identities reduces this to the stated form. The other cases can be derived similarly. An alternative approach is given in Exercises 5.5 q.4.
The function KN in (5.86) in the case of the CUE, or equivalently the classical group U (N ), also takes on a simple form. The CUE corresponds to the case w2 (z) = 1 in (5.79). The circular orthogonal polynomials (C) for this weight are simply pj (z) = z j with normalization (pj , pj )2 = L. With L = 2π and so using angular variables θ, −π < θ ≤ π, we see that (5.82) simplifies to give
CUE iθ −iθ KN (e , e )=
E XERCISES 5.5
1 i(θ−θ )(N −1)/2 sin(N (θ − θ )/2) e . 2π sin((θ − θ )/2)
(5.87)
1. In this exercise (5.82) will be derived according to the strategy of the proof of Proposition
5.1.3. Let Z f (z, u ¯) := (w2 (z)w2 (u))1/2
L/2 −L/2
Z dx1 · · ·
L/2
dxN−1 −L/2
N−1 Y
Y
w2 (zl )
l=1
|zk −zj |2
N−1 Y
(zj −z)(¯ zj −¯ u).
j=1
1≤j
(5.88) (i) Proceed as in the proof of the first part of the proof of Proposition 5.1.3 to show that f (z, u ¯) := (N − 1)!
N Y
(C)
(pj−1 , pj−1 )2
KN (z, u ¯).
(5.89)
j=1
Q Q (ii) Write N−1 zj − u ¯)(zj − z) = (−¯ u)N−1 N−1 ¯j (zj − u)(zj − z), and proceed as in the second part j=1 (¯ j=1 z of the proof of Proposition 5.1.3, appropriately modified by use of (5.22), to deduce that f (z, u ¯) := (N − 1)!
N Y
(C)
(pj−1 , pj−1 )2
j=1
×
N−1 X
pν (z)pν (0)pN (u) − pν (u)pν (0)pN (z) (C)
(pν , pν )2
ν=0
(iii) Consider now f (z, 0). By writing show
(w2 (z)w2 (u))1/2 N−1 u ¯ u−z
QN−1 j=1
f (z, 0) = (N − 1)!
N−1 Y
z¯j (zj − z) = (−z)N−1 (C)
(pj−1 , pj−1 )2
QN−1 j=1
.
(5.90)
(¯ zj − z¯) and proceeding as in (ii)
(w2 (z)w2 (0))1/2 z N−1 pN−1 (¯ z ),
j=1
and thus, comparing with (5.90), conclude N−1 X
pν (z)pν (0)
ν=0
(pν , pν )2
(C)
or equivalently
N−1 X
pν (z)pν (0)
(C) ν=0 (pν , pν )2
=
= z N−1
pN−1 (¯ z) (C)
(pN−1 , pN−1 )2
z) z N pN (¯ (C) (pN , pN )2
−
,
pN (z)pN (0) (C)
(pN , pN )2
.
Now substitute (5.91) in (5.90) and compare with (5.89) to deduce the summation (5.82).
(5.91)
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CHAPTER 5
2. [508] Suppose w2 (z) = |1 − αz|−2 , |α| < 1, z := e2πix/L . Verify that pn (z) = z n (1 − α/z), (n ≥ 1), p0 (z) = 1 are monic orthogonal polynomials with respect to the inner product (5.79) with (C)
(p0 , p0 )2
=
L , 1 − α2
(C)
(pn , pn )2
= L, n ≥ 1.
Use (5.79) to deduce from this that Z
Z
L/2
−L/2
3.
dx1 · · ·
L/2
dxN −L/2
N Y
Y
|1 − αzl |−2
l=1
|zk − zj |2 =
1≤j
LN . 1 − α2
(i) Use Proposition 3.9.1 to obtain the identity 1 C
Z
1
Z dt1 · · ·
0
=
1 C
Z
N Y
1
dtN
tλj 1 (1 − tj )λ2 (x − tj )
0
j=1
1/2
Z
−1/2
dθ1 · · ·
Y
(tk − tj )2
j
dθN −1/2
N Y
e2πiθl |1 + e2πiθl |λ2 (x + e2πiθl )
l=1
Y
|e2πiθk − e2πiθj |2 ,
j
where = λ1 + λ2 /2 + N and C and C are constants so that the coefficient of xN on both sides is unity. (ii) The monic orthogonal polynomials corresponding to the weight function xλ1 (1 − x)λ2 , (0 < x < 1) are proportional to Pn(λ1 ,λ2 ) (1 − 2x) =
“n + λ ” 1
λ1
2 F1 (−n, n
+ λ1 + λ2 + 1; λ1 + 1; x).
(5.92)
Use this fact together with the result of (i) and the first identity of Proposition 5.1.4 to deduce the orthogonality property stated in Proposition 5.5.2. (iii) Use the analogue of (5.74) for periodic weight functions, together with the Morris integral evaluation (4.4), to deduce the normalization in Proposition 5.5.2. 4.
(i) Verify the so-called C and D type Vandermonde formulas [452] det[zjk − zj−k ]j,k=1,...,n =
n Y j=1
det[zjk−1
+
−(k−1) zj ]j,k=1,...,n
=2
Y
(zj − zj−1 )
1≤j
“ (zk − zj ) 1 −
“ (zk − zj ) 1 −
Y
1≤j
1 ” , zj zk
1 ” . zj zk
(ii) By appropriate choice of zj , rewrite the formulas of (i) to read det[sin kθj ]j,k=1,...,n =
n Y
Y
sin θj
j=1
(2 cos θk − 2 cos θj ),
1≤j
Y
det[cos(k − 1)θj ]j,k=1,...,n = 21−n
(2 cos θk − 2 cos θj ).
1≤j
Use the orthogonality of {sin kθ}k=1,2,... and {cos(k − 1)θ}k=1,2,... on [0, π] to deduce that Z
π 0
Z
π
dθ1 · · ·
dθn
Z
0
0
π
n Y
sin2 θj
j=1
Y
2
(cos θk − cos θj )2 = n!2−n π n ,
1≤j
Z
π
dθ1 · · ·
dθn 0
Y
1≤j
2
(cos θk − cos θj )2 = n!2−(n−1) π n .
211
CORRELATION FUNCTIONS AT β = 2
Recalling (2.62) and (2.66), similarly show that for N even O − (N)
ρ(n)
O + (N)
ρ(n)
(θ1 , . . . , θn ) = det
(θ1 , . . . , θn ) = det
h 2 N/2 X π h1 π
i sin lθj sin lθk
l=1
+
j,k=1,...,n
,
N/2 i 2X cos lθj cos lθk π j,k=1,...,n l=1
and thus rederive the corresponding results in Proposition 5.5.3. (iii) Starting with the B type Vandermonde formulas k−1/2
det[zj
−k+1/2
− zj
]j,k=1,...,n =
n Y
1/2
(zj
−1/2
− zj
j=1 k−1/2
det[zj
−k+1/2
+ zj
]j,k=1,...,n =
n Y
Y
)
“ (zk − zj ) 1 −
1≤j
(zj
−1/2
+ zj
Y
)
j=1
“
1 ” , zj zk
“ zk − zj ) 1 −
1≤j
1 ” zj zk
show det[sin(k − det[cos(k −
n Y 1 θj sin )θj ]j,k=1,...,n = 2 2 j=1 n Y 1 θj cos )θj ]j,k=1,...,n = 2 2 j=1
Y
(2 cos θk − 2 cos θj ),
1≤j
Y
(2 cos θk − 2 cos θj ).
1≤j
Now use the orthogonality of {sin(k − 1/2)θ}k=1,2,... and {cos(k − 1/2)θ}k=1,2,... on [0, π] to deduce that Z π Z π n Y Y θj dθ1 · · · dθn sin2 (cos θk − cos θj )2 2 0 0 j=1 1≤j
Recalling (2.63), (2.65) use these results to show that for N odd O + (N)
ρ(n)
O − (N)
ρ(n)
(θ1 , . . . , θn ) = det
(θ1 , . . . , θn ) = det
h2 π h2 π
X
sin(l +
i 1 1 , )θj sin(l + )θk 2 2 j,k=1,...,n
cos(l +
i 1 1 )θj cos(l + )θk , 2 2 j,k=1,...,n
(N−1)/2−1
l=0
X
(N−1)/2−1
l=0
and thus rederive the corresponding results in Proposition 5.5.3. 5. The objective of this exercise is to derive the factorizations [545], [34] det[ai−j ]i,j=1,...,2N = det[ai−j + ai+j−1 ]i,j=1,...,N det[ai−j − ai+j−1 ]i,j=1,...,N ,
(5.93)
1 det[ai−j + ai+j−2 ]i,j=1,...,N+1 det[ai−j − ai+j ]i,j=1,...,N , 2
(5.94)
det[ai−j ]i,j=1,...,2N+1 = where it is required that a−i = ai .
(i) Consider the l.h.s. of (5.93). For each i = 1, . . . , N replace each row i by row i minus row 2N + 1 − i, then for each j = 1, . . . , N replace column 2N + 1 − j by column 2N + 1 − j minus column j. Note that the
212
CHAPTER 5
»
determinant then has the structure
A B
0N C
–
for A, B, C N × N matrices and thus deduce the factorization as given by the r.h.s. of (5.93). (ii) Show that an analogous procedure yields (5.94). 6. [34] Here the determinants in (5.93), (5.94) will be related to averages over the classical groups. (i) Suppose a(eiθ ) = a(e−iθ ) and set aj =
1 2π
Z
π
a(eiθ )e−iθj dθ.
−π
Note that det[aj−k + aj+k−1 ]j,k=1,...,N Z π Z π N “ ”2 Y 1 1 iθj = dθ · · · dθ a(e ) det[2 cos θ (k − 1/2)] 1 N j j,k=1,...,N N ! (2π)N 0 0 j=1 and obtain similar formulas for all the determinants on the r.h.s’s of (5.93) and (5.94). Now make use of these formulas, substituting for the determinants according to formulas presented in q.4 and recognising the integrals as averages over orthogonal groups according to the results of Section 2.6, to deduce that det[aj−k + aj+k−1 ]j,k=1,...,N =
N DY
a(eiθj )
E O − (2N+1)
j=1
det[aj−k − aj+k−1 ]j,k=1,...,N =
N DY
a(eiθj )
,
E O + (2N+1)
j=1
,
DY E 1 a(eiθj ) , det[aj−k + aj+k−2 ]j,k=1,...,N = 2 O + (2N) j=1 N
det[aj−k − aj+k ]j,k=1,...,N =
N DY
a(eiθj )
E
j=1
O − (2N+2)
,
where the symbol edenotes that only eigenvalues 0 < θj < π are considered in the average. (ii) Use the above results, together with (5.77), to show that (5.93) and (5.94) can be rewritten n DY j=1
a(eiθj )
E U (n)
=
D [(n−1)/2] Y j=1
a(eiθj )
E
D [n/2] Y O− (n+1)
a(eiθj )
j=1
E O+ (n+1)
(5.95)
for n = 2N, 2N + 1 respectively, where again it is required that a(eiθ ) = a(e−iθ ).
5.6 LOG-GAS SYSTEMS WITH PERIODIC BOUNDARY CONDITIONS 5.6.1 Semiperiodic boundary conditions In addition to the classical weight functions, there are a large number of weight functions for which the corresponding orthogonal polynomials are known explicitly. Here we will make use of one such case to specify the correlation functions corresponding to (2.73) with β = 2 [192]. First the change of variables
213
CORRELATION FUNCTIONS AT β = 2
e2π(yj +πN/Lc )/L = uj shows that N
2
e−c yj
j=1
| sinh π(yl − yj )/L|2 dy1 · · · dyN
1≤j
∝
N
e−k
2
log2 uj
j=1
(ul − uj )2 du1 · · · duN ,
(5.96)
1≤j
where k 2 = c L2 /(2π)2 . But for the weight function w(u; q) := π −1/2 ke−k
2
log2 u
,
q = e−1/(2k ) , 2
it is known (see, e.g., [508]) that the Stieltjes-Wigert polynomials l 2 l (−1)l q l/2+1/4 Sl (u; q) := q ν (−q 1/2 u)ν , {(1 − q)(1 − q 2 ) · · · (1 − q l )}1/2 ν=0 ν q
where
(5.97)
l (1 − q l )(1 − q l−1 ) · · · (1 − q l−ν+1 ) := (1 − q ν )(1 − q ν−1 ) · · · (1 − q) ν q
are the corresponding set of orthonormal polynomials. Thus, by making use of the Christoffel-Darboux formula, the correlation functions are specified in terms of this class of polynomial.
5.6.2 Metal wall In statistical mechanics, in addition to the canonical formulation, in which there are a fixed number N of particles, there is also a grand canonical formulation, in which the number of particles is a variable controlled by the so called fugacity ζ. Thus the probability density of there being N (distinguishable) particles at r1 , . . . , rN is postulated to be ζ N −βU(r1 ,...,rN ) e , Ξβ (ζ) where
∞ ζN dr1 · · · drN e−βU(r1 ,...,rN ) . Ξβ (ζ) := N! Ω Ω
(5.98)
N =0
The latter is referred to as the grand partition function. Note that N = ζ
∂ log Ξβ (ζ). ∂ζ
(5.99)
It turns out that the log-gas near a metal wall is best suited for study in this formulation, a fact which is in keeping with there being no need for a neutralizing background, due to the presence of image charges of opposite sign. Set β = 2 in (2.77), and introduce the scaled fugacity πe4πdη ζ. (5.100) L The corresponding generalized grand partition function, defined as in (5.98) but with an additional factor of ξ=
214
CHAPTER 5
N
a(rl ) in the integrand (cf. (5.19)) is then given by L N ∞ ξN L a(xl ) −2πη 2 dL dx1 · · · dxN Ξ2 [a](ξ) = e N! 0 sinh 2πd/L 0 l=1
N =0
l=1
1≤j
sin π(xk − xj )/L 2 . sin π(xk − xj + 2id)/L
(5.101) The product over pairs in (5.101) can be written in terms of the Cauchy double alternant (4.33). Thus setting xj = e2πi(xj +id)/L , yj = e2πi(xj −id)/L in (4.33) shows N l=1
1 sinh 2πd/L
1≤j
sin π(xk − xj )/L 2 1 . = iN det sin π(xk − xj + 2id)/L sin π(xj − xk + 2id)/L j,k=1,...,N
(5.102) To proceed further, observe that (5.102) substituted into (5.101) gives an expression which has be summed according to the identity (5.32). Thus Ξ2 [a](ξ) = e−2πη
2
dL
det(1 + iξK),
(5.103)
where K is the integral operator on [0, L] with kernel K(x, y) =
a(y) . sin π(x − y + 2id)/L
This shows that (5.29) is valid, so using (5.31) gives ρ(n) (x1 , . . . , xn ) = det[G(xj , xk )]j,k=1,...,n , where G(x, y) = x|iξK(1 + iξK)−1 |a=1 |y. The integral operator G with kernel G(x, y) must satisfy G(1 + iξK) = iξK, and consequently
G(x, y) + iξ 0
L
G(x, u) iξ du = . sin π(u − y + 2id)/L sin π(x − y + 2id)/L
(5.104)
This integral equation can be used to determine G(x, y) explicitly. P ROPOSITION 5.6.1 We have G(x, y) = 2ξ
∞ e−4πd(n+1/2)/L e2πi(n+1/2)(x−y)/L . 1 + 2ξLe−4πd(n+1/2)/L n=0
(5.105)
Proof. Taking the complex conjugate of (5.104) shows G(y, x) = G(x, y). This together with the periodicity of the system suggests we seek a solution of the form G(x, y) =
∞ X
gn e2πi(n+1/2)(x−y)/L .
n=−∞
Noting that for d > 0
∞ X 1 e−4πdn/L e2πiz(n+1/2)/L , = −2ie−2πd/L sin π(z + 2id)/L n=0
we can equate Fourier coefficients and so deduce (5.105).
215
CORRELATION FUNCTIONS AT β = 2
5.6.3 Doubly periodic boundary conditions In the case β = 2 it is possible to calculate the n-point correlation corresponding to (2.83) by writing the product over theta functions in terms of a determinant [507], [182], [181]. In addition to θ1 (u; q) as specified by (2.79), also required is the further Jacobi theta function θ3 (u; q) :=
∞
2
q n e2iun .
(5.106)
n=−∞
P ROPOSITION 5.6.2 Define the Jacobi theta functions θ1 and θ3 by (2.79) and (5.106) and let −1)(N −2)/2 . fN (q) := N N/2 q −(N −1)(N −2)/24 (q 2 ; q 2 )−(N ∞
Then with s = 3 for N odd and s = 1 for N even we have 1 det θs (π(xj + α − k/N ); q 1/N ) dα = fN (q) j,k=1,...,N
0
θ1 (π(xk − xj ); q).
(5.107)
(5.108)
1≤j
Proof. Only the case N odd will be considered; the N even case follows similarly. We proceed as in [182]. Both the l.h.s. and r.h.s. of the stated equation are antisymmetric functions of x1 , . . . , xN that vanish whenever two of the variables are equal. It thus suffices to check that both sides are the same function of x1 . This is done by studying the periodicity properties of both sides, and using Liouville’s theorem. It is straightforward to check that both sides are periodic under the mapping x1 → x1 + 1. Now write q = eπiτ , where Im(τ ) > 0, and consider the effect of the mapping x1 → x1 + τ . From (2.79) we see that the r.h.s. remains the same apart from a factor N Y q −(N−1) e−2πix1 (N−1) e2πixk . k=2
On the l.h.s., since
θ3 (π(x + τ ); q) = q −1 e−2πix θ3 (πx; q)
(5.109)
the kth term of the first row of the determinant can be written q −(N
2
−1)/N −2πi(x1 −α−k/N)(N−1)
e
θ3 (π(x1 − α + τ /N − k/N ); q 1/N ),
while the kth term of the j th row (j = 2, . . . , N ) can be written q 1/N e2πi(xj −α−k/N) θ3 (π(xj − α + τ /N − k/N ); q 1/N ). In the first row we note that e2πik(N−1)/N = e−2πik/N and thus a common factor of e−2πik/N can be removed from the kth column (the product over k of such factors equals 1). Furthermore, removing obvious common factors from each row of the determinant, the l.h.s. becomes q −(N−1) e−2πix1 (N−1)
N Y k=2
Z
1
e2πixk
det[θ3 (π(xj − α + τ /N − k/N ); q 1/N )]j,k=1,...,N dα.
0
This integral is in fact the same as in the l.h.s. of the proposition, as the line integral along the path α + iIm(τ )/N , 0 < α < 1 is the same as that along the same path 0 < α < 1 by Cauchy’s theorem, and the periodicity of the integrand under α → α + 1. Thus under the mapping x1 → x1 + τ the l.h.s. and the r.h.s. have the same quasi-periodicity property. Finally, consider the ratio r.h.s./l.h.s. From the above results, this is a doubly periodic function with periods 1 and τ . Furthermore, since the zeros of the l.h.s. are cancelled by the zeros of the r.h.s. (due to two rows of the determinant being equal), we have that r.h.s./l.h.s. is a doubly periodic entire function and is thus by Liouville’s theorem independent of x1 . The result of the Proposition now follows, except for the evaluation of fN (q). This latter task is taken up in Exercises 5.6 q.1.
The formula (5.108) can be generalized so that the variable α is not integrated over. In fact knowledge of
216
CHAPTER 5
(5.108) is the key to deriving the generalization. P ROPOSITION 5.6.3 Let fN (q) be given by (5.107). For N odd det θ3 π(xj + α − k/N ); q 1/N
j,k=1,...,N
= θ3 π
N
(xj + α); q fN (q)
j=1
while for N even
θ1 π(xk − xj ); q ,
(5.110)
θ1 π(xk − xj ); q .
(5.111)
1≤j
det θ1 π(xj + α − k/N ); q 1/N
j,k=1,...,N
N = θ4 π (xj + α); q fN (q) j=1
1≤j
In the latter formula the Jacobi theta function θ4 is defined by ∞
θ4 (u; q) =
2
(−1)n q n e2iun .
(5.112)
n=−∞
integrand in (5.108) as a function of α. Since it is Proof. We will give the details in the N even case [181]. PConsider the 2πipNα I e , where periodic of period 1/N , it can be expanded in the form ∞ p p=−∞ Z
1
Ip =
e−2πipNα det[θ1 (π(xj + α − l/N ); q 1/N ]j,l=1,...,N dα.
0
Shifting the contours of integration from the unit interval to γ − τ p/N (0 < γ < 1, q = eπiτ ) (this leaves the value of the integral unchanged due to the periodicity of the integrand and Cauchy’s theorem) and using the second property in (2.81) gives PN 2 (5.113) Ip = q p e2πip j=1 xj I0 and thus
i h det θ1 (π(xj + α − l/N ); q 1/N
j,l=1,...,N
N ” “ X = θ3 π (xj + α); q I0 . j=1
The final step is to substitute for I0 using (5.108).
Although the functions {θs (π(x − k/N ); q 1/N )}k=1,...,N forming the columns in the determinant (5.110) and (5.111) are not an orthogonal set of functions, they can readily be transformed into one. For this, multiply both sides of (5.110) with α = 0 by det[e2πilk/N ]
l=1,...,N k=0,...,N −1
= N N/2 i(N −1)(3N/2+1)
and multiply both sides of (5.111) with α = −πτ /2 by det[e2πil(k+1/2)/N ] to obtain det[hk (xj )]j,k=1,...,N = q −
l=1,...,N k=0,...,N −1
PN −1
= N N/2 iN +1 i(N −1)(3N/2+1)
N −N/2 i(N −1)(3N/2+1) fN (q) N ×θs − π xj ; q θ1 (−π(xk − xj ); q) . m=0
m2 /N
j=1
1≤j
(5.114)
217
CORRELATION FUNCTIONS AT β = 2
where as in Proposition 5.6.2 s = 3 for N odd and s = 1 for N even. Here hk (x) := e−2πi(k−1)x θ3 (π(N x − τ (k − 1)); q N ),
(5.115)
1
which for k = 1, . . . , N form an orthogonal set with respect to the inner product f |g := 0 f¯(x)g(x) dx. It follows from (5.114) that for both N even and N odd, 2 1 ¯ k (xj + α/N ) θ1 (π(xk − xj ); q) ∝ det hk (xj + α/N ) det h dα. j,k=1,...,N
0
1≤j
j,k=1,...,N
Proceeding now as in the proofs of Propositions 5.1.1 and 5.1.2, we have that for the ensemble (2.83) with β = 2, N 1 −1 ρ(n) (x1 , . . . , xn ) = det e−2πip(xj −xk )/L θ3 (π(N xj /L − τ (p − 1) + α); q N ) 0
p=0
×
θ3 (π(N xk /L + τ (p − 1) + α); q N ) Lθ3 (2πτ (p − 1); q 2N )
dα,
(5.116)
j,k=1,...,n
where q = eπiτ , τ = iW/L. E XERCISES 5.6 Y
1. (i) Use the infinite product expansion (2.79) to deduce that “x ” “ x ” j k ; q2 q2 ; q2 xk xj ∞ ∞
1≤j
= iN(N−1)/2 q −N(N−1)/8 (q 2 ; q 2 )−N(N−1)/2 ∞
N Y
Y
eπi(N−2j+1)uj
j=1
θ1 (π(uk − uj ); q),
1≤j
where xj = e2πiuj . (ii) Suppose N is odd. Make use of (5.108) to show Z
1
fN (q)
Z
0
=q =q
N Y
1
du1 · · ·
duN 0
PN
j=1
j=1 ((N+1)/2−j)
2
/N
Y
eπi(N−2j+1)uj
θ1 (π(uk − uj ); q)
1≤j
h i det e2πik((N+1)/2−j)/N
j,k=1,...,N
PN
2 j=1 ((N+1)/2−j) /N −N(N−1)/2
i
N
N/2
,
and obtain an analogous formula for N even. (iii) Read off from the identity (4.138) that Z 1 Z 1 Y du1 · · · duN 0
0
1≤j
“x
j
xk
; q2
” “ ∞
q2
1 xk 2 ” ;q = 2 2 N−1 , xj ∞ (q ; q )∞
and compare this to the evaluation of (ii), recalling the equation of (i), to deduce (5.107).
5.7 PARTITION FUNCTION IN THE CASE OF A GENERAL POTENTIAL The β = 2 log-gas partition function ∞ n P j ∞ 1 ∞ dx1 · · · dxn e j=1 tj xl Zn [{ti }] := n! −∞ −∞ l=1
1≤j
(xk − xj )2 ,
(5.117)
218
CHAPTER 5
considered as a function of {ti }, is intimately related to the theory of integrable systems [5], [362]. First we will demonstrate that Zn satisfies the Toda lattice equation Zn+1 Zn−1 ∂2 log Zn = . ∂t21 Zn2
(5.118) P∞
i
Let {pj (x; {ti })}j=0,1,... be the monic orthogonal polynomials with respect to the weight function e i=1 ti x so that ∞ P ∞ j e j=1 tj x pj (x; {ti })pk (x; {ti }) dx =: (pj , pk ) = h2j δj,k . (5.119) −∞
According to (5.74) we have Zn [{ti }] =
n−1
h2j .
(5.120)
j=0
Let p˜(x; {ti }) =
1 hj pj (x; {ti })
be the corresponding orthonormal polynomials. Introduce the matrix 1 (xpj , pk ) , L= hj hk j,k=0,1,...
so that p0 , p˜1 , . . . )T . L(˜ p0 , p˜1 , . . . )T = x(˜
(5.121)
It then follows from the three term recurrence (5.14) that this matrix has the symmetric tridiagonal structure ⎤ ⎡ 1 h1 (xp0 , p0 ) h0 h20 ⎥ ⎢ h1 h2 1 (xp1 , p1 ) ⎥ ⎢ h0 h1 h21 ⎢ L=⎢ h2 h3 ⎥ 1 ⎥. (xp , p ) 2 2 2 h1 h2 ⎦ h2 ⎣ .. .. .. . . . Furthermore (xpn , pn ) =
∂ ∂ 2 ∂ (pn , pn ) − 2 pn , pn = h , ∂t1 ∂t1 ∂t1 n
where the second equality follows upon noting ∂t∂1 pn must be of degree n − 1 since pn is monic and thus the second inner product in the first equality vanishes. The matrix L has the special property that its evolution with respect to {ti } can be determined explicitly. P ROPOSITION 5.7.1 We have ∂L 1 = [(Li )+ − (Li )− , L], ∂ti 2
(5.122)
where the notation (A)+ ((A)− ) denotes the strictly upper (lower) triangular portion of the matrix A. Proof. We must have
X (i) ∂ p˜k (x) = Akl p˜l (x) ∂ti k
l=0
(i)
for some Akl and thus
∂ (˜ p0 (x), p˜1 (x), . . . )T = A(i) (˜ p0 (x), p˜1 (x), . . . )T , ∂ti
(5.123)
219
CORRELATION FUNCTIONS AT β = 2
where A(i) is lower triangular. Now for l < k, by orthogonality Z ∞ P∞ i ∂ p˜k (x)˜ pl (x)e i=1 ti x dx 0= ∂ti −∞ Z ∞ Z ∞ P∞ P∞ i i ∂ p˜k (x) p˜l (x)e i=1 ti x dx + xi p˜k (x)˜ pl (x)e i=1 ti x dx, = ∂ti −∞ −∞ where to obtain the second equality use has been made of the fact implied by the differentiation vanishes. It follows from this that
∂p ˜l (x) ∂ti
(5.124)
has degree l so the integral involving this term
(A(i) )− = −(Li )− . Similarly, for l = k 0=
∂ ∂ti
Z
∞
(˜ pk (x))2 e
P∞
i=1 ti x
i
−∞
(i)
dx = 2Akk + (Li )kk
and so
1 i (5.125) (L )0 , 2 where the notation (B)0 denotes the diagonal part of the matrix B. It follows immediately upon multiplying (5.123) by x that ” ∂ “ p0 (x), p˜1 (x), . . . )T L(˜ p0 (x), p˜1 (x), . . . )T = A(i) L(˜ ∂ti and thus ∂ L = [A(i) , L] = [−(A(i) )T , L], ∂ti A(i) = −(Li )− −
where the second equality follows from the first and the fact that final two expressions gives the stated result.
∂L/ ∂ti
is symmetric. Taking the arithmetic mean of the
When i = 1, equating diagonal entries in (5.122) gives h 2 ∂ 1 ∂ 2 hn+1 2 hn−1 2 n h = + − 2 , ∂t1 h2n ∂t1 n hn hn−2 hn−1
n ≥ 2.
Using (5.120) we see that this implies the Toda lattice equation (5.118). The next integrability property of (5.117) to be discussed relates to the construction of annihilating operators {Lp }p=0,1,... which satisfy the Virasoro algebra [Lp , Lq ] = (p − q)Lp+q .
(5.126)
P ROPOSITION 5.7.2 The operators p ∞ ∂ ∂ ∂ + ktk , Lp := ∂tk ∂tp−k ∂tp+k k=0
k=1
∂ := N ∂t0
(5.127)
have the property that L p ZN = 0
(5.128)
and furthermore satisfy (5.126). Proof. As in Aomoto’s approach to the Selberg integral discussed in Section 4.6, we start by making use of the funda-
220
CHAPTER 5
mental theorem of calculus to deduce N DX ∂ 1 E ∂xi z − xi ZN i=1 Z ∞ Z ∞ N P∞ Y j 1 := dx1 · · · dxN e j=1 tj xi N ! −∞ −∞ i=1
Y
(xj − xi )2
1≤i<j≤N
N X ∂ 1 = 0. ∂x i z − xi i=1
Integrating by parts, and using the simple identity N “ X i=1
we have that
N N ” X X 1 1 1 + 2 = , (z − xi )2 (x − x )(z − x ) (z − x )(z − xj ) i i i k k=1 i,j=1 k=i
N D X i,j=1
Equating coefficients of z
−p
X X ntn xn−1 E 1 i = 0. + (z − xi )(z − xj ) i=1 n=1 z − xi ZN N
∞
for p > 1 shows p−2 N X D X
xki xp−2−k + j
i,j=1 k=0
∞ N X X
ntn xn+p−2 i
i=1 n=1
E ZN
= 0.
The l.h.s. of this expression is easily identified with Lp−2 ZN . Regarding (5.126), we note from the explicit form (5.127) that p “ X k [Lp , Lq ]f ({ti }) = 2 k=0
q ∞ ” X X ∂ ∂2 ∂2 −2 k + (p − q) ktk f ({ti }). ∂tp−k ∂tq+k ∂tq−k ∂tp+k ∂tp+q+k k=0 k=1
We remark that the above proof shows (5.128), referred to as Virasoro constraints, remains valid for p = −1, in which case the first term in (5.127) is not present. Our final point on integrability aspects of (5.117) relates to the KP hierarchy of equations. To specify these equations, introduce polynomials sj ({ti }) according to the expansion e
P∞
j=1
tj xj
=
∞
sk ({ti })xk ,
(5.129)
k=0
so that sk ({ti }) =
ν1 ,ν2 ,···≥0 ν1 +2ν2 +3ν3 +···=k
tν11 tν22 tν33 · · · . ν1 !ν2 !ν3 ! · · ·
For functions f ({ti }) =: f (t), g({ti }) =: g(t) define the Hirota symbol by + ∂ , + ∂ , f ◦ g := A f (t + y)g(t − y) . A ∂ti ∂yi y=0 In terms of this notation we will show that for k = 1, 2, . . . + 1 ∂ , 1 ∂2 − ZN ◦ ZN = 0. sk+3 i ∂ti 2 ∂t1 ∂tk+2
(5.130)
(5.131)
221
CORRELATION FUNCTIONS AT β = 2
P ROPOSITION 5.7.3 We have ∞ ∞ ∂ 1 ∂ Zn [{ti }]Zn [{ti }] = 0, (ti − ti )z i exp − − [z −1 ] exp i ∂t iz ∂t i i i=1 i=1
(5.132)
where [z −1 ] denotes the coefficient of z −1 in the formal Laurent expansion. Proof. Suppose m < n. Observe that [z −1 ] exp Z
∞ “X
∞ −∞ ∞
= −∞
∞
−∞
i=1
= Z
Z ” (ti − ti )z i pn (z; {ti })
pm (u, {t })e
P∞
i i=1 ti u
[z −1 ] exp
P∞
pm (u; {ti })e z−u ∞ “X
(ti − ti )z i
i=1
pm (u; {ti })pn (u; {ti })e
P∞
i=1 ti u
i
i i=1 ti u
du
” p (z; {t }) n i du z−u
du = 0,
(5.133)
where to obtain the second equality use has been made of the general result that [z −1 ]f (z)/(z − u) = f (u) for f (z) analytic, while the final equality uses the fact that m < n and the orthogonality of {pn (u; {ti })}n=0,1,... with respect to P∞ i e i=1 ti u . On the other hand, we know from Proposition 5.1.4 that pn (z; {ti }) = Z
∞
−∞
P∞
pm (z; {ti })e z−u
i i=1 ti u
du =
n E h DY 1 zn 1 i Zn {ti − i } , (z − xj ) = Zn [{ti }] j=1 Zn [{ti }] iz Zn [{ti }]
D m+1 E hn Y z −(m+1) 1 oi 1 1 = + Z t . m+1 i Zm [{ti }] j=1 (z − xj ) Zm+1 [{ti }] Zm [{ti }] iz i
Substituting these formulas in (5.133) with n = N , m = N − 1 gives hn P∞ i 1 oi hn 1 oi = 0. [z −1 ]e i=1 (ti −ti )z ZN ti + i ZN ti − i iz iz
(5.134)
But from the definitions ZN
∞ hn “X 1 ∂ ” 1 oi ti + i = exp ZN [{ti }], i ∂t iz iz i i=1
ZN
∞ hn “ X 1 ∂ ” 1 oi ti − i = exp − ZN [{ti }], iz iz i ∂ti i=1
which when substituted in (5.134) gives (5.132).
Let us now show how (5.132) can be used to derive (5.131). First, we note from (5.129) that (5.132) can be written ∞ + 1 ∂ 1 ∂ , ZN [{ti }]ZN [{ti }] = 0. sk ({ti − ti })sk+1 − i ∂ti i ∂ti k=0
Introducing the new variables uj := 12 (tj − tj ), wj := 12 (tj + tj ) this reads ∞ k=0
sk ({−2ui })sk+1
+ 1 ∂ , ZN [{wj + uj }]ZN [{wj − uj }] = 0. i ∂ui
222
CHAPTER 5
But
+ 1 ∂ , ZN [{wj + uj }]ZN [{wj − uj }] i ∂ui + 1 ∂ , ZN [{wj + uj + yj }]ZN [{wj − uj − yj }] = sk+1 i ∂yi y=0 ∞ + 1 ∂ , ∂ exp ZN [{wj + yj }]ZN [{wj − yj }] = sk+1 uj i ∂yi ∂y y=0 j j=1
sk+1
= sk+1
∞ + 1 ∂ , ∂ exp ZN [{wj }] ◦ ZN [{wj }] uj i ∂wi ∂wj j=1
where to obtain the final line use has been made of the definition (5.130). Noting too that + 1 ∂ , ∂ = s1 , sk ({−2ui }) = −2uk + O(u2 ), k ≥ 2, i ∂wi ∂w1 and expanding in {ui } then shows ∞ ∞ + 1 ∂ , ∂ ∂ 1+ + (−2uk + O(u2 )) sk+1 uj + O(u2 ) ZN [{wj }] ◦ ZN [{wj }] = 0. ∂w1 i ∂wi ∂wj j=1 k=1
Equating terms linear in {ui } we read off from this the equation (5.131) with k replaced by k − 2, valid for k = 1, 2, . . . . The fact that ∂ 2p−1 ZN [{wj }] ◦ ZN [{wj }] = 0 ∂wj2p−1
(5.135)
for all j, p ≥ 1 shows that this form of (5.131) is trivially satisfied for k = 1 and 2, leaving the cases as stated. Note the case k = 1 of (5.131), after making use of (5.129), (5.130) and (5.135), can be written as ∂ 4 2 ∂ 2 ∂2 ∂2 log ZN + 6 +3 −4 log Z = 0. (5.136) N ∂t1 ∂t2 ∂t1 ∂t3 ∂t21 E XERCISES 5.7 R∞
1. With ψj (x) =
1 hj
e−V (x)/2 pj (x), V (x) = −
P∞
j=1 tj x
j
and the inner product (f, g) :=
f (x)g(x) dx, introduce the matrix −∞ P=
”i h“ d . ψj (x), ψk (x) dx j,k=0,1,...
(i) Use integration by parts to conclude that P is antisymmetric. Note too that [L, P] = 1, where L is as in (5.121). (ii) Differentiate using the product rule, and use (5.121) to show ”i h1 i 1 h“ (V (L)ψ(x)) = (pj (x)e−V (x)/2 , ψk (x)) . P+ j , ψk (x) 2 hj j,k=0,1,... j,k=0,1,... P By noting pj (x)e−V (x)/2 = hj−1 jψj−1 (x) + j−2 l=0 γjl ψl (x) for some γjl , show that the entries in the matrix on the r.h.s. vanish for k ≥ j. Now make use of the first result of (i) to conclude ” 1“ P = − V (L)+ − V (L)− , 2 where V (L) := [(V (L)ψ(x)) j , ψk (x))]j,k=0,1,... and furthermore (V (L))jj = 0.
(5.137)
223
CORRELATION FUNCTIONS AT β = 2
(iii) By noting that the entries of the matrix on the r.h.s. of the first equation in (ii) is equal to jhj−1 /hj for (j, k) = (j, j − 1), deduce that (V (L))j,j−1 = jhj−1 /hj . (5.138) Relate (5.137) and (5.138) to (5.17).
5.8 BIORTHOGONAL STRUCTURES The first equality in (5.5) expresses the Boltzmann factor for the one-component log-gas at β = 2 in terms of a product of two determinants. With {ξj (x)}j=1,2,... and {ηj (x)}j=1,2,... separately a sequence of linear independent functions, the Boltzmann factor is a special case of the functional form N 1 w2 (xj ) det[ξj (xk )]j,k=1,...,N det[ηj (xk )]j,k=1,...,N C j=1
(5.139)
(take ξj (x) = ηj (x) = xj−1 ). Like (5.5), the n-particle correlation function for (5.139) can itself be written as a determinant [80]. In general a many particle system for which the n-point correlations are n×n determinants is said to be a determinantal point process [495]. P ROPOSITION 5.8.1 With gjk := (ηj , ξk )2 , let [gjk ]j,k=1,...,n be invertible for each n = 1, 2, . . . , define cjk by −1 , (5.140) [cjk ]j,k=1,...,N = [gjk ]j,k=1,...,N and set KN (x, y) = (w2 (x)w2 (y))1/2
N
cjk ξj (x)ηk (y).
(5.141)
j,k=1
We have ρ(n) (x1 , . . . , xn ) = det[KN (xj , xk )]j,k=1,...,n .
(5.142)
Proof. Assume, deforming the ξj if necessary, that [gjk ] can be decomposed as the product of a lower triangular and upper triangular matrix. It is then possible to construct functions ζi ∈ Span(ξ1 , . . . , ξi ),
ψi ∈ Span(η1 , . . . , ηi )
(5.143)
with the biorthogonality property (ζj , ψk )2 = δj,k .
(5.144)
According to the method of the proof of Proposition 5.1.2, we can deduce that (5.142) holds with KN (x, y) = (w2 (x)w2 (y))1/2
N X
ζl (x)ψl (y).
(5.145)
l=1
This has the reproducing property Z ∞ (w2 (y))1/2 KN (x, y)ξl (y) dy = (w2 (x))1/2 ξl (x)
(l = 1, . . . , N ).
(5.146)
−∞
On the other hand, because of (5.143) the decomposition (5.141) must hold for some cjk , and so Z
∞ −∞
(w2 (y))1/2 KN (x, y)ξl (y) dy = (w2 (x))1/2
N X j,k=1
cjk gkl ξj (x).
(5.147)
224
CHAPTER 5
Equating (5.146) and (5.147) gives (5.140).
An example of the functional form (5.139), generalizing the β = 2 Boltzmann factor as given on the l.h.s. of (5.5), is N 1 det[e−aj xk ]j,k=1,...,N w2 (xj ) C j=1 1≤j
(xk − xj ).
(5.148)
1≤j
Thus in the limit aj → 0 (j = 1, . . . , N ) (5.148) reduces to the l.h.s. of (5.5). We will see in Section 11.6.4 (see also Exercises 5.8 q.2(iv)) that with w2 (x) given by either the Gaussian or Laguerre weight, (5.148) has an interpretation as an eigenvalue p.d.f. Our present interest is to show that for these particular classical weights the correlation kernel (5.141) can be written as a double contour integral. For this we first establish the following preliminary result. P ROPOSITION 5.8.2 With w2 (x) = e−x , 2
1 ηj (x) = √ Hj−1 (x) π
ξk (x) = e−ak /4−ak x , 2
(5.149)
we have N
cjk z k−1 =
k=1
N (z + al ) , (a l − aj ) l=1
(5.150)
l=j
while with w2 (x) = xa e−x ,
ξk (x) = (1 + ak )a+1 e−ak x ,
ηj (x) =
xj−1 Γ(a + j)
(5.151)
we have N
cjk z −(k−1) =
k=1
N (1 + aj )N −1 z − (1 + al ) . z N −1 a j − al l=1
(5.152)
l=j
Proof. In the case of (5.149) 2 1 gjk = √ e−ak /4 π
Z
∞
e−x
−∞
2
−ak x
Hj−1 (x) dx = (−ak )j−1 ,
(5.153)
where the second equality follows from a well-known classical identity [508]. Since by definition N X
cjk gkl = δj,l
(5.154)
k=1
P k−1 = δj,l . Hence the l.h.s. of (5.150), which is a polynomial of degree N − 1, vanishes it follows that N k=1 cjk (−al ) for z = −al (l = j), and equals unity for z = −aj , and thus must be given by the r.h.s. In the case of (5.151) Z (1 + ak )a+1 ∞ a+j−1 −(1+ak )x x e dx = (1 + ak )−(j−1) . gjk = Γ(a + j) 0 P −(k−1) = δj,l and (5.152) follows. According to (5.154) N k=1 cjk (1 + al )
225
CORRELATION FUNCTIONS AT β = 2
In addition to these formulas, also required are the integral identities ∞ 2 2k Hk (x) = √ e−y (x + iy)k dy, π −∞ z a+j−1 ewz 1 = dw, Re(z) > 0, Γ(a + j) 2πi CH wa+j
(5.155) (5.156)
where CH is a simple contour starting at −∞ + i ( > 0), staying in the upper half-plane until it is fully in the right half-plane, turning around and staying in the lower half-plane until it reaches −∞ − i (Hankel loop). P ROPOSITION 5.8.3 [554], [29] For the p.d.f. (5.148) with w2 (x) = e−x the n-point correlation ρ(n) is given by (5.142) with i∞ 2 2 N dv dt e−(v −t )/4−vx−ty (t + al ) KN (x, y) = − , (5.157) t+v (al − v) C{a} 2πi −i∞ 2πi 2
l=1
where C{a} is a simple contour which encircles {a1 , . . . , aN } anti-clockwise and stays entirely in the right half-plane. For the p.d.f. (5.148) with w2 (x) = xa e−x , (5.142) holds with N dv dw e−vx+wy v a+N (w − (1 + al )) KN (x, y) = , (5.158) w (v − (1 + al )) C{a} 2πi CH 2πi w − v l=1
where C{a} is as in (5.157) with the additional constraint that it does not intersect CH . Proof. Substituting (5.149) in (5.141) and using the integral representation (5.155) shows, after a simple change of variables in the latter, KN (x, y) =
e−x
2
/2+y 2 /2
2πi
N X
2
cjk e−aj /4−aj x
Z
i∞
et
2
/4−ty k−1
t
dt.
−i∞
j,k=1
The sum over k can be performed according to (5.150) to give KN (x, y) = e−x
2
/2+y 2 /2
N X j=1
2
e−aj /4−aj x
Z
i∞ −i∞
et
2
/4−ty
N Y (t + al ) dt , (a l − aj ) 2πi l=1 l=j
while the sum over j can be done by way of a contour integral to yield (5.157), but with an additional factor of 2 2 e−x /2+y /2 . However this factor cancels out of the determinant and so can be ignored. The derivation of (5.158) is similar, using (5.156) and (5.152).
Although (5.157) and (5.158) were calculated without requiring explicit knowledge of the corresponding biorthogonal polynomials (functions), the double integral representations are in fact intimately related to integral representations of the latter. In general so-called multiple orthogonal polynomials [22] allow one to construct functions from span{e−bi x xk } k=0,...,j , ψj (x), say, and polynomials ζi (x) of degree i, which have i=1,...,D the biorthogonality property (5.144). To see this, let us suppose that of the parameters a = (a1 , . . . , aN ) only D are distinct, so that a = bm := D mp m1 mD ∗ ∗ (b1 , . . . , bD ), where bp means bp repeated mp times. Note that i=1 mi = N . Let m = (m1 , . . . , m∗D ) D be such that 0 ≤ m∗j ≤ mj and set |m ∗ | = p=1 m∗p . Construct a family of monic polynomials Pm ∗ (x) of
226
CHAPTER 5
degree |m ∗ |, referred to as multiple orthogonal polynomials of type II, with the properties ∞ j −bk x w2 (x)Pm dx = 0 (j = 0, . . . , mk − 1) ∗ (x)x e
(5.159)
−∞
(j)
∗ for each k = 1, . . . , D. Furthermore, let Qm ∗ (x) be a polynomial of degree mj − 1, referred to as multiple orthogonal polynomials of type I, such that the function
Qm ∗ (x) := w2 (x)
D
−bj x Qm ∗ (x)e (j)
j=1
satisfies
∞
−∞
j
w2 (x)Qm ∗ (x)x dx =
j = 0, . . . , |m ∗ | − 2, ∗ j = |m | − 1.
0, 1,
(5.160)
∗(N ) such that |m ∗(j) | = j and m ∗(j) ≤ m ∗(j+1) (this Choose a sequence of multi-indices m ∗(0) , . . . , m inequality must hold for each component), and define ζj (x) = Pm ∗(j) (x),
ψj (x) = Qm ∗(j+1) (x).
(5.161)
A direct calculation gives that the polynomials {ζi (x)} and the functions {ψi (x)} have the biorthogonality property (5.144). For the Hermite and Laguerre weights, the biorthogonal polynomials and functions can be determined explicitly for general indices m as certain contour integrals. Let us consider first the Laguerre case. To avoid complication with the contours, it will be assumed that the parameter a in the Laguerre weight is an integer. Following [68], [132], let the multiple Laguerre function and polynomial of type I and II be specified by D e−xz (1 + z)|m|+a−1 dz a −bi x ˜a ˜ Lm , a ∈ Z, (5.162) e Lm,i (x) = (x) = D {−1} m i 2πi C i=1 (z − bi ) i=1 b D exw dw (|m| + a)! −a a Lm x , a ∈ Z. (5.163) (w − 1 − bi )mi (x) = |m|+a+1 |m|! 2πi w C{0} i=1 {−1}
In (5.162) C denotes a simple closed contour which encircles anticlockwise b1 , . . . , bD but not, for |m| + b a − 1 ≤ −1, the point −1 (otherwise this latter restriction is not necessary), while in (5.163) C{0} is a simple closed contour which encircles the origin anticlockwise. These functions and polynomials have the property of reducing to the classical Laguerre polynomials in the limit b → 0. P ROPOSITION 5.8.4 We have a (x), lim L˜am (x) = L|m|−1
b→0
a lim Lam (x) = L|m| (x).
b→0
Proof. The first formula follows from the definitions (5.162), upon comparison with the integral representation of the Laguerre polynomials
Z Lan (x) =
C{0}
e−xw dw (1 + w)n+a , wn+1 2πi
a ∈ Z.
(5.164)
For the second formula we modify (5.164) in accordance with the identity a n!(−x)−a L−a n (x) = (n − a)!Ln−a (x),
a ∈ Z.
Furthermore, the Laguerre functions and polynomials satisfy orthogonality relations of the type required
227
CORRELATION FUNCTIONS AT β = 2
by (5.160) and (5.159). P ROPOSITION 5.8.5 Let a ∈ Z≥0 and b1 , . . . , bD > −1. We have ∞ 0, j = 0, . . . , |m| − 2, −x a+j ˜a e x Lm (−1)|m|−1 (|m|+α−1)! (x) dx = QD , j = |m| − 1, 0 (1+b )mi i
i=1
and
0
∞
e−x−bi x xa+j Lam (x) dx = 0
(i = 1, . . . , D) (j = 0, . . . , mi − 1).
Thus, with w2 (x) = xa e−x , the multiple orthogonal polynomials and functions are, respectively, D mi (−1)|m|−1 |m|! (−1)|m| a i=1 (1 + bi ) Lm (x), Qm Pm L˜am (x) = D (x) = (x). mi (| m| + a − 1)! (1 + b ) i i=1
(5.165)
Proof. Substituting (5.162), then performing the x-integration first, shows Z 0
∞
e−x xa+j L˜am (x) dx = (a + j)!
Z {−1} b
C
(1 + z)|m|−j−2 dz . QD mi 2πi (z − b ) i i=1
For j = |m| − 1, use of the residue theorem applied to the region outside the contour (this is valid since the integrand decays sufficiently fast), in which there is a simple pole at z = −1, gives the stated result. For j < |m| − 1 the point z = −1 is analytic and so the residue theorem gives that the integral vanishes. The same strategy suffices to establish the second orthogonality relation.
With mi = 1, bi = ai (i = 1, . . . , N ), D = N and m ∗(j) having its first j components equal to 1, and the remaining N − j components equal to 0, the multiple orthogonal functions (5.165) substituted in (5.161) give the correlation kernel in terms of biorthogonal functions according to (5.145). Suppose all but r of the aj are zero. By choosing these to be the final r of the aj s, but labeling them a1 , . . . , ar (note that (5.148) is symmetric in the aj ) we see that the first N − r terms in the correlation kernel (5.145) are independent of the aj . This structure can also be exhibited from the double contour integral form (5.158). First note that up to a factor which leaves (5.142) unchanged, (5.158) with N − r aj zero reads r dv dw e−xv+yw 1 + v N +a w N −r w − ai ¯ a (x, y), =: K (5.166) N {−1} 2πi 1+w v v − a i C a C{−1} 2πi w − v i=1 where the Hankel loop has been deformed to a closed contour upon the assumption that a is an integer. Because (5.148) reduces to the l.h.s. of (5.5) in the limit a → 0, (5.166) must, up to a factor which leaves (5.142) unchanged, be equal to the Laguerre case of (5.7). Tracing through the calculation of (5.166) for this factor shows # w2 (y) (L) a KN K (x, y), (x, y) = w2 (u) = ua e−u . (5.167) w2 (x) N a=0 We can write (5.166) in terms of (5.167), and so called incomplete multiple Laguerre functions of type I and II (special cases of (5.162) and (5.163)), j−1 exw wN −r k=1 (w − ak ) dw e−xz (1 + z)N +a dz (j) ˜ (j) (x) = , Λ . (x) = Λ j 2πi (1 + w)N +a 2πi C{0,a1 ,...,aj } z N −r C{−1} k=1 (z − ak ) (5.168)
228
CHAPTER 5
P ROPOSITION 5.8.6 We have a+r a (x, y) = KN KN −r (x, y)
a=0
+
r
˜ (i) (x)Λ(i) (y), Λ
(5.169)
i=1
which will be referred to as the perturbed Laguerre kernel. Proof. First one can use induction to check that r r Qi−1 X (w − ak ) 1 1 Y w − ai = . + Qk=1 i w − z i=1 z − ai w−z k=1 (z − ak ) i=1
Substituting this in (5.166), the result follows from the appropriate definitions.
The analogue of Proposition 5.8.6 in the Hermite case is given in the exercises below. E XERCISES 5.8
1. Follow the method of proof of Proposition 5.1.2 to show that Z
Z
∞
−∞
dx1 · · ·
N Y
∞
dxN −∞
hZ
w2 (xl ) det[ξj (xk )]j,k=1,...,N det[ηj (xk )]j,k=1,...,N
l=1
i
∞
= N ! det −∞
w2 (x)ξj (x)ηk (x) dx
j,k=1,...,N
.
(5.170)
2. [132] Let KN (x, y) refer to (5.157). (i) Use the fact that for a → 0, (5.148) reduces to the l.h.s. of (5.5), and follow the proof of Proposition 5.8.3 to conclude ˛ “ w (y) ”1/2 2 ˛ 2 (H) KN (x, y)˛ = KN (x, y), w2 (u) = e−u , w2 (x) a= 0 (H)
where KN refers to (5.6) in the Hermite case. (ii) Let Z
2
˜ (j) (x) := Γ C{0,a ,...,a } 1 j
dz (j) e−xz−z /4 , Γ (x) := Qj N−r 2πi z (z − a ) k k=1
Z
i∞
exw+w
2
Y
j−1 /4
wN−r
−i∞
(w−ak )
k=1
dw . 2πi
(5.171) Use the method of the proof of Proposition 5.8.6 to show that ˛ ˛ KN (x, y)˛
ai =0 (i=r+1,...,N)
˛ ˛ = KN−r (x, y)˛
a= 0
+
r X
˜ (j) (x)Γ(j) (y). Γ
(5.172)
j=1
(iii) Use the integral representations of the Hermite polynomial Z Hn (x) = 2n n! C{0}
2
√ 2 exz−z /4 dz = πex z n+1 2πi
Z
i∞ −i∞
z n e−xz+z
2
/4
dz 2πi
to show that 2 1 Γ(1) (x) = √ e−x HN−1 (−x), π
N−2 −xa1 −a2 1 /4 HN−2−p (−x) 1 X 1 ˜ (1) (x) = e Γ − . p N−2−p N−1 a a 2 (N − 2 − p)! a1 1 p=0 1 2
(iv) The discussions around (11.1) and in Section 11.6.4 below tell us that (5.148) with w2 (x) = e−x is the eigenvalue p.d.f. for the matrix H = X + 12 H(0) . Here X is a member of the GUE while H(0) is a fixed matrix with eigenvalues {−a1 , . . . , −aN }, so that H is a shifted mean GUE matrix. Note that in the case all
229
CORRELATION FUNCTIONS AT β = 2
entries of H(0) are equal to 2a/N , and thus H equal to a GUE matrix where the Gaussian entries all have mean a/N , one then has that a1 = −2a, a2 = · · · = aN = 0. Use the results of (ii) and (iii) to show that for large a 2 ˛ e−(x−a) ˜ (1) (x)Γ(1) (x)˛˛ √ Γ ∼ a1 =−2a π and interpret this in the context of the result of Proposition 1.8.2.
5.9 DETERMINANTAL k-COMPONENT SYSTEMS 5.9.1 All components having an equal number of particles Consider the p.d.f. specified by (4.97) and (4.98). This may be viewed as a two-species particle system as determined by {ai } and {bj }. Furthermore, the constraint (4.98) can be replaced by a determinant according to the following identity. P ROPOSITION 5.9.1 Let χT = 1 if T is true and χT = 0 otherwise. Then for x1 > · · · > xN ,
y1 > · · · > yN ,
(5.173)
we have det[χxj −yk >0 ]j,k=1,...,N = χx1 >y1 >···>xN >yN .
(5.174)
Proof. For the ordering x1 > y1 > · · · > xN > yN the determinant is triangular with 1s down the diagonal, so (5.174) is correct in this case. All other orderings must have at least two xs (or two ys) in succession. The corresponding rows (or columns) in the determinant will then be equal so the determinant vanishes. Use of this together with the Vandermonde identity (1.173) shows the p.d.f. in question to be a particular 2-component case of the p.d.f. for a k-component system k−1 1 (1) (l) (l+1) (k) det[φi (xj )]i,j=1,...,N det[Wl (xi , xj )]i,j=1,...,N det[ψi (xj )]i,j=1,...,N , C
(5.175)
l=1
(l)
where the coordinates of species l are {xj }j=1,...,N . Following [88], to compute the corresponding correlation functions, a discretization M of the domain [a, b], say to which the particles are confined will be introduced. Also, use will be made of theory relating to an abstract point process referred to as an L-ensemble. D EFINITION 5.9.2 With M a discrete set let L = [L(mi , mj ]mi ,mj ∈M . For X ⊂ M, denote by LX the symmetrically labeled submatrix LX = [L(xi , xj )]xi ,xj ∈X , and suppose det LX is non-negative. The L-ensemble is specified by setting Prob (X) =
det LX . det(1 + L)
Note that when confined to a lattice, according to (5.102) the log-gas near a metal wall for β = 2 is an example of an L-ensemble. Now, in a lattice setting correlations are in general specified by ρ(Y ) = Prob(X|Y ⊂ X). X⊂M
For the L-ensemble, according to the derivation of (5.32) ρ(Y ) = det(L(1 + L)−1 |Y ). Also required is the notion of a conditional L-ensemble.
230
CHAPTER 5
D EFINITION 5.9.3 Let Y ⊂ M. For U ⊂ Y define the conditional L-ensemble by setting Prob(U ) =
det LU∪Y¯ , det(1Y + L)
(5.176)
where 1Y is the |M| × |M| matrix with all but the diagonal entries corresponding to the sublattice Y equal to zero, and Y¯ denotes the complement of the set Y. (This has the interpretation that the sites Y¯ are required to be fully occupied.) P ROPOSITION 5.9.4 For the conditional L-ensemble ρ(Y ) = det KY with K = 1Y − (1Y + L)−1 |Y .
(5.177)
Proof. Use will be made of a result of Jacobi [8] which states that for B = A−1 , det BX = det AX¯ / det A.
(5.178)
By expanding (5.177) about the diagonal we see that for Y ⊂ Y X
det KY =
(−1)|X| det((1Y + L)−1 )X =
=
¯ Y ¯ X⊃
(−1)|X|
¯ X⊃Y
X⊂Y
X
X
(−1)
|X|
¯ = (Prob all points are in X)
X
det(1Y + L)X¯ det(1Y + L)
(−1)|X| (Prob no points are in X) = ρ(Y ),
X⊂Y
where the last equality follows from the inclusion–exclusion principle.
(j)
Consider next k discretizations M1 , . . . , Mk of the interval [a, b], and let xi ∈ Mj for each j = (j) 1, . . . , k and i = 1, . . . , N . A particular choice of {xi } is said to define a configuration. Suppose each (1) configuration occurs with probability (5.175). Associate with [φi (xj )]i,j=1,...,N the N × |M1 | matrix Φ = (1)
[φi (xj ))] i=1,...,N , and similarly define |Ml | × |Ml+1 | matrices Wl (l = 1, . . . , k − 1) and a |Mk | × N xj ∈M1
matrix Ψ. Then one sees from the Cauchy-Binet formula (6.88) below that C = det M, where M := ΦW1 · · · Wk−1 Ψ. We seek the multispecies correlation functions associated with (5.175). P ROPOSITION 5.9.5 Suppose det M = 0. Let Yi ⊂ Mi , and set Wi · · · Wj−1 , i < j, W[i,j) = 0, i ≥ j.
(5.179)
We have ρ(Y1 , . . . , Yk ) = det[Kij |Yi ×Yj ] where ˜ [i,k) ΨM−1 ΦW ˜ [1,j) − W ˜ [i,j) Kij = W
(5.180)
˜ [1,1) = W ˜ [k,k) = 1, and W ˜ [i,j) = W[i,j) otherwise. with W Proof. Set M = {1, . . . , N }∪M1 ∪· · ·∪Mk and consider the conditional L-ensemble on M with Y = M1 ∪· · ·∪Mk
231
CORRELATION FUNCTIONS AT β = 2
and
2 6 6 6 6 L=6 6 6 4
0 0 0 .. . 0 Ψ
Φ 0 0 .. . 0 0
0 −W1 0 .. . 0 0
0 0 −W2 .. . 0 0
··· ··· ··· .. . ··· ···
0 0 0 .. . −Wk−1 0
3 7 7 7 7 7. 7 7 5
(5.181)
From this, symmetrically labeled submatrices must be formed which include the first N rows. But the only such submatrices with nonzero determinant are blocks of size N × N , which reduces (5.176) to (5.175). The correlation functions are therefore specified by (5.177), which requires us to invert 1Y + L. For this we use the general identity »
A C
–−1
B D
»
−E−1 D CE−1
=
−1
D
−1
E−1 BD−1 − D−1 CE−1 BD−1
– ,
(5.182)
where A and D are square matrices not necessarily of the same dimension, and E := BD−1 C − A. With A = [0]N×N , we use (5.182) with D replaced by 1 + D and D the square matrix resulting from blocking out the first block row and column of L, and B and C correspondingly defined to relate to L. Then one has 3 2 1 W[1,2) W[1,3) · · · W[1,k) 6 0 1 W[2,3) · · · W[2,k) 7 7 6 6 0 −1 0 1 · · · W[3,k) 7 (5.183) (1 + D) = 6 7 7 6 . .. .. .. .. 5 4 .. . . . . 0 0 0 ··· 1 and E = ΦW[1,k) Ψ = M. Consequently (5.177) reduces to (5.180). (1)
(2)
We return now to the case k = 2 of (5.175) (with xi → xi , xi → yi for notational convenience) and take a continuum limit in which the discretizations M1 and M2 are dense in [a, b]. For special choices of φi , Wl , ψi this was studied directly in [168], [416], while a direct approach to the general case can be found in [293]. The following result is then obtained [456]. P ROPOSITION 5.9.6 In the above specified setting ρ(k1 ,k2 ) (x1 , . . . , xk1 ; y1 , . . . , yk2 ) [Koo (xj , xl )]j,l=1,...,k1 = det [Keo (yj , xl )] j=1,...,k2 l=1,...,k1
where Koo (x, x ) =
N
−t φj (x) Mjk
j,k=1
Koe (x, y) =
N
Keo (y, x) = −W (x, y) + Kee (y, y ) =
j,k=1
l=1,...,k2
[Kee (yj , yl )]j,l=1,...,k2
,
(5.184)
W (x , u)ψk (u) du,
a
−t φj (x) Mjk ψk (y),
j,k=1
N
b
[Koe (xj , yl )] j=1,...,k1
a
N j,k=1
b
a
b
−t W (u, y)ψj (u) du Mjk
−t W (v, y)φj (v) dv Mjk ψk (y ).
b
W (x, v)φk (v) dv ,
a
(5.185)
232
CHAPTER 5
Here X−t denotes the operation of taking the transpose of the inverse of X and [Mjk ] is the matrix with entries b b Mjk = dx φj (x) dy ψk (y)W (x, y). (5.186) a
a
At this stage the formulas of Proposition 5.9.6 could be applied to the p.d.f. (4.97), with the main technical task being to choose {φi } and {ψi } in the Vandermonde determinants so that they are biorthogonal with respect to the inner product (5.186). This is done in [221], but will not be reported on here. Rather these formulas will be applied in Section 11.6 to the computation of dynamical correlations. 5.9.2 Components with an unequal number of particles In contrast to the p.d.f. (4.97), the two-component system with p.d.f. specified by (4.94) and (4.95) has one species consisting of n + 1 particles, and the other species consisting of n particles. Extending (4.94) to the (k) joint p.d.f. for the eigenvalues {xj }j=1,...,k of nested k × k minors (k = 1, . . . , n + 1) of X ∈ GUE∗N gives the form (4.99) in which component k consists of k particles. Making use of the identity (5.174) shows that (4.99) is of the determinantal form N −1 1 (l) (l+1) (N ) det[Wl (xi , xj )]i,j=1,...,l+1 det[ΨN ]i,j=1,...,N i−1 (xj C
(5.187)
l=0
(l)
(0)
with xl+1 = −∞ and W0 (x1 , x) = 1. Analogous to the result of Proposition 5.9.5, the general r-point cor∞ relation of (5.187) can be expressed as an r ×r determinant [84]. With (a∗b)(x, y) := −∞ a(x, z)b(z, y) dz, this requires the quantities (Wn1 ∗ · · · ∗ Wn2 −1 )(x, y), n1 < n2 , W (n1 ,n2 ) (x, y) = (5.188) 0, n1 ≥ n2 , and, for n = 1, . . . , N − 1, j = 1, . . . , N , Ψnn−j (x) = (W (n,N ) ∗ ΨN N −j )(x). Furthermore, define functions
{Φnj (x)}j=0,...,n−1 ,
(5.189)
n = 1, . . . , N , constructed from
˜ (i,n) (x(i−1) , x)}i=1,...,n , span{Wi−1 ∗ W i ˜ (i,m) = W (i,m) for i < m and Wi−1 ∗ W ˜ (i,m) = 1 for i = m, by the orthogonality requirement where W ∞ Φnj (x)Ψnk (x) dx = δj,k . (5.190) −∞
P ROPOSITION 5.9.7 Suppose (n)
Φn+1 (x) ∝ Wn (xn+1 , x). 0
(5.191)
For the p.d.f. (5.187) the correlation between eigenvalues of species sj at positions yj (j = 1, . . . , r) has the determinantal form ρ({(sj , yj )}j=1,...,r ) = det[K(sj , yj ; sk , yk )]j,k=1,...,r with the kernel K given in terms of the quantities W (n1 ,n2 ) (x, y), Ψnj (x), Φnj (x) specified above according to sk s K(sj , yj ; sk , yk ) = −W (sj ,sk ) (yj , yk ) + Ψsjj −l (yj )Φsskk −l (yk ). (5.192) l=1
233
CORRELATION FUNCTIONS AT β = 2
Proof. We follow [84], which in turn follows the method of the proof of Proposition 5.9.5. Before beginning this strategy proper, we consider the significance of (5.191). Introduce the m × m matrix Bm according to its elements (i−1) ˜ (i,m) ∗ Ψm ). (Bm )ij = (Wi−1 ∗ W m−j )(xi
(5.193)
Because of the discretization of the domain which is part of the method of the proof of Proposition 5.9.5, ∗ is now interpreted as a summation operation. From the definition (5.189) we see that for 1 ≤ i, j ≤ m, (Bm )ij = (BN )ij . This tells us that BN is upper triangular if and only if (m)
(Bm+1 )m+1,j = (Wm ∗ Ψm+1 m+1−j )(xm+1 ) = cm δj,m+1
(cm = 0).
Expanding Wm in terms of {Φm+1 }j=0,...,m and using (5.190) shows that this condition is equivalent to requiring j (m)
Wm (xm+1 , x) = cm Φm+1 (x), which is the condition (5.191). 0 We now implement the method of the proof of Proposition 5.9.5. The support of each of the species (j) is discretized to domains Mj . Set M = {1, . . . , N }∪M1 ∪· · ·∪MN and consider the conditional L-ensemble with Y = M1 ∪· · ·∪Mk and 2 3 E1 E2 · · · EN−1 0 E0 6 0 7 0 −W1 0 ··· 0 6 7 6 0 7 · · · 0 0 0 −W 2 L=6 7, 6 . 7 . . . . . .. .. .. .. .. 4 .. 5 Ψ 0 0 0 ··· 0 where Ψ is the |MN | × N matrix, and Wl the |Ml | × |Ml+1 | matrix N (xi )] [ψj−1
xi ∈MN j=1,...,N
,
(l)
(l+1)
Wl = [Wl (xi , xj
)]x(l) ∈M i
(l+1) ∈Ml+1 i ,xj
(i)
, (i+1)
(i+1)
respectively, while Ei is the N × |Mi+1 | matrix with entries in row i + 1 given by Wi (xi+1 , xj ), xj ∈ Mi+1 , and entries in all other rows zero. As with (5.181), from this choice of L, symmetrically labeled submatrices must be formed which include the first N columns. The structure of L is such that the only submatrices of this type which have a nonzero determinant have determinant of the form (5.187). The task now is to compute (5.177). Writing L in the form » – B 0N , C 1 + D0 where B = [E0 , . . . , EN−1 ], C = [0, . . . , 0, Ψ]T , it follows from (5.182) that the inverse in (5.177) can be computed to give K = 1 − D−1 + D−1 CM−1 BD−1 , where D := 1 + D0 , M := BD−1 C. Now D−1 is given explicitly by the r.h.s. of (5.183) with k = N . This tells us that 1 − D−1 = −[W[i,j) ] (recall the notation (5.188)), and furthermore D−1 C = [W[1,N] Ψ, . . . , W[N−1,N) Ψ, Ψ]T , BD−1 = [E0 , E0 W[1,2) + E1 , . . . ,
N−1 X
Ek−1 W[k,N) + EN−1 ].
k=1
Hence the (n, m) block of K is equal to −W[n,m) + W[n,N) ΨM−1
“ m−1 X k=1
” Ek−1 W[k,m) + Em−1 ,
(5.194)
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CHAPTER 5
(here W[n,N) is to be replaced by 1 for n = N ) while M=
“ N−1 X
” Ek−1 W[k,N) + EN−1 Ψ.
(5.195)
k=1 (n)
(m)
Furthermore, it follows from (5.179) and (5.188) that (W[n,m) )i,j = W (n,m) (xi , xj n shows (W[n,N) Ψ)i,j = Ψn n−j (xi ). Similarly, from the definitions “ m−1 X
Ek−1 W[k,m) + Em−1
k=1
j
” (m) i,xj
(i−1)
˜ (i,m) )(x (Wi−1 ∗ W i 0,
=
(m)
, xj
),
), and this together with (5.189)
1 ≤ i ≤ m, m + 1 ≤ i ≤ N.
These facts applied to (5.195) give (i−1)
˜ (i,N) ∗ ΨN M = [Wi−1 ∗ W N−j (xi
)]i,j=1,...,N .
Next, define the matrix Bm such that (i−1)
˜ (i,m) (x Wi−1 ∗ W i
, x) =
m X
(Bm )i,l Φm m−l (x).
l=1
It follows from the orthogonality (5.190) that Bm is also given by (5.193) and so in particular M = BN . And with the N × m matrix Φm specified by j m (m) Φm−i (xj ), 1 ≤ i ≤ m, (Φm )i,j = 0, m + 1 ≤ i ≤ N, we have
m−1 X
» Ek−1 W[k,m) + Em−1 =
k=1
Bm 0
0 0(N−m)×(N−m)
– Φm .
The considerations of the first paragraph tell us that this multiplied on the left by M−1 gives Φm , and consequently the elements of (5.194) are precisely those implied by (5.192).
5.9.3 GUE minor process (j)
As recalled at the beginning of the previous subsection, the joint p.d.f. for the eigenvalues {xl }l=1,...,j , (j) j = 1, 2, . . . , of the top j × j sub-block of matrices from GUE∗ is given by (4.96). Changing scale xl → √ (j) 2xl this gives the same joint p.d.f. for matrices from the GUE. Here we seek the correlation function ρ(r) ({(sj , xj )}j=1,...,r ) between eigenvalues of value xj from the top sj × sj sub-block [333], [423]. In fact this can be obtained as an example of a result which holds for each of the Gaussian, Laguerre and Jacobi weights [215]. P ROPOSITION 5.9.8 Let w(x) be one of the classical weight functions, Gaussian, Laguerre or Jacobi (with the latter defined on (0, 1) and so given by xa (1 − x)b ), and consider the joint p.d.f. proportional to N l=1
(N )
w(xl
)
1≤j
(N )
(xj
(N )
− xk )
N −1
χ(x(s+1) > x(s) ).
(5.196)
s=1
We have ρ(r) ((sj , xj )j=1,...,r ) = det[f ((sj , xj ); (sl , xl ))]j,l=1,...,r ,
(5.197)
235
CORRELATION FUNCTIONS AT β = 2
where
⎧ ⎪ ⎪ ⎪ ⎪ ⎨
(s) (t) t 1/2 es−k ps−k (x)pt−k (y) w(s) (x)w(t) (y) , s ≥ t, (t) et−k Nt−k k=1 (5.198) f ((s, x), (t, y)) = (s) (t) 0 1/2 ⎪ es−k ps−k (x)pt−k (y) ⎪ (s) (t) ⎪ , s < t. ⎪ ⎩ − w (x)w (y) (t) et−k Nt−k k=−∞ ∞ Here pj (x) denotes the monic orthogonal polynomial, Nj := −∞ w(x)(pj (x))2 dx is the normalization (n)
(n)
while ej = (−1)j , j!, 2j j! in the three cases respectively. The superscripts (n) in w(n) (x), pj (x) and Nj indicate that a → a + n (Laguerre case), a → a + n, b → b + n (Jacobi case), while in the Gaussian case they have no effect. Proof. After recalling (5.174), and introducing the convention that x(s) s+1 := −∞, we see that (5.196) can be written in the form of (5.187) with W (x, y) = χy>x , ψjN (x) = w(x)cj pj (x) (cj is a proportionality constant chosen for convenience). Substituting these in the definitions (5.188) and (5.189) gives 1 χy>x (y − x)n2 −n1 −1 , (n2 − n1 − 1)! Z ∞ cN−j w(y)pN−j (y)(y − x)N−n−1 dy. Ψn n−j (x) = (N − n − 1)! x
W (n1 ,n2 ) (x, y) =
(5.199) (5.200)
The Rodrigues formula for orthogonal polynomials gives cj pj (y) =
” dj “ 1 w(y)(Q(y))j =: p˜j (y), j ej w(y) dy
(5.201)
where Q(y) equals 1, y, y(1 − y) (thus in the notation of (5.57), Q(y) = g(y)) and the proportionality constant cj has (a,b) (1 − 2y) in the three cases, respectively. been chosen so the r.h.s. is equal to Hj (y), Laj (y), Pj Substituting (5.201) in (5.200) and integrating by parts shows that for j ≥ 0 (n = N ) N−n Ψn j (x) = (−1)
ej (n) w(n) (x)˜ pj (x) eN−n+j
(5.202)
while for j < 0
Z ∞ (−1)N−n+j 1 (y − x)−j−1 w(y)(Q(y))N−n+j dy. eN−n+j (−j − 1)! x In view of (5.202), the requirement (5.190) is satisfied by Ψn j (x) =
N−n Φn j (x) = (−1)
˜ (n) := where N j
R∞ −∞
(5.203)
eN−n+j 1 (n) p˜ (x), ˜ (n) j ej N j
(5.204)
(n)
w(n) (x)(˜ pj (x))2 dx. (n)
(n)
Note from (5.204) that φn+1 (x) is a constant. Also, due to the convention that xn+1 := −∞, we have W (xn+1 , x) = 0 (n) χx>−∞ = 1. Thus φn+1 (x) ∝ W (xn+1 , x), which is the condition (5.191) in Proposition 5.9.7, and so the correlations 0 are specified by the kernel (5.192). For s ≥ t it is immediate that f ((s, x), (t, y)) =
a(s, x) K(s, x; t, y), a(t, y) (t)
(5.205)
with a(s, x) = (−1)s /(w(s) (x))1/2 . By writing W (s,t) (x, y) as a basis in pk (y) this same equation can be checked for s < t. Because the factor a(s, x)/a(t, y) does not effect the determinant, (5.197) follows.
Chapter Six Correlation functions at β = 1 and 4 At β = 1 and 4, generalizations of the orthogonal polynomial technique used to calculate the correlation functions of the log-gas at β = 2 are possible. These generalizations involve introducing skew orthogonal polynomials and quaternion determinants, or equivalently Pfaffians. For classical one-body potentials, the required skew orthogonal polynomials can be written in terms of the corresponding orthogonal polynomials, and the correlation kernel can be expressed as the Christoffel-Darboux summation plus a correction involving these polynomials. To obtain the quaternion determinant form both integration formulas and the method of functional differentiation, known from the studies of the previous chapter, are used. For β = 1 the method of integration over alternate variables plays an important role. In Chapter 4 the Dixon-Anderson integral was used to show that the classical ensembles at β = 1 and β = 4 are related by integrating over every second eigenvalue in the former. Here some different perspectives on this result are given, and it is further shown that the random superposition of two classical β = 1 ensembles, and then integration over every second eigenvalue, gives back a classical β = 2 ensemble. Also studied is a two-component log-gas, which interpolates between the one-component log-gas on a circle with β = 1 and β = 4.
6.1 CORRELATION FUNCTIONS AT β = 4 6.1.1 Skew orthogonal polynomials and quaternion determinants We begin with β = 4 rather than β = 1 because in the latter case there are complications due to the need to treat separately the cases N even and N odd. The first point to note is that whereas the calculation of the correlations at β = 2 requires orthogonal polynomials, the calculation of the n-particle correlation function at β = 4 requires so called (monic) skew orthogonal polynomials. D EFINITION 6.1.1 A skew symmetric inner product ·|·s has the property f |gs = −g|f s . Monic skew orthogonal polynomials, {Un (x)}n=0,1,... , are a family of monic polynomials which satisfy the skew orthogonality U2m |U2n+1 s = −U2n+1 |U2m s = um δm,n ,
U2m |U2n s = U2m+1 |U2n+1 s = 0.
The quantity um is referred to as the normalization. From Definition 6.1.1 we see that the skew orthogonality property still holds if we make the replacement U2m+1 (x) → U2m+1 (x) + γ2m U2m (x)
(6.1)
for arbitrary γ2m . A Gram-Schmidt type procedure shows that the monic skew orthogonal polynomials are unique up to this mapping. Note also that for a set of 2N skew orthogonal polynomials {Un (x)}n=0,...,2N −1 ,
237
CORRELATION FUNCTIONS AT β = 1 AND 4
the skew orthogonality condition can be written in matrix form as ⎡ 0 u0 ⎢ −u0 0 ⎢ ⎢ 0 u1 ⎢ ⎢ −u 0 1 [Uj |Uk s ]j,k=0,...,2N −1 = ⎢ ⎢ ⎢ ⎢ ⎣
⎤
..
. 0 −uN −1
uN −1 0
⎥ ⎥ ⎥ ⎥ ⎥ ⎥, ⎥ ⎥ ⎥ ⎦
(6.2)
where the entries not explicitly shown are all zero. The Boltzmann factor for the particle-particle interaction at β = 4 can be expressed in a useful form in terms of the polynomials {Qn (x)}n=0,1,... (with normalizations {qm }), which are skew orthogonal with respect to the inner product 1 ∞ −4V (x) f |g4 := e (f (x)g (x) − f (x)g(x))dx 2 −∞ 1 ∞ −2V (x) d d = f (x) (e−2V (x) g(x)) − g(x) (e−2V (x) f (x)) dx, e (6.3) 2 −∞ dx dx by use of quaternion determinants [150], or equivalently Pfaffians. The former will be considered first. D EFINITION 6.1.2 The determinant qdet of an N × N self-dual quaternion matrix Q (recall (2.6)) is defined as qdet Q =
P ∈SN
N −l
(−1)
l (qab qbc · · · qda )(0) .
(6.4)
1
Here the superscript (0) denotes the operation 12 Tr, or equivalently the scalar part (i.e., the number α0 in the expansion q = α0 1 + α1 e1 + α2 e2 + α3 e3 ; recall Section 1.3.2), P is any permutation of the indices (1, . . . , N ) consisting of l disjoint cycles of the form (a → b → c · · · → d → a) and (−1)N −l is equal to the parity of P . Recall from Section 1.3.2 that a quaternion, denoted qαβ in the above definition, can be regarded as a 2 × 2 matrix. It follows from (6.4) that qdet Q is unchanged if each element qαβ is replaced by A−1 qαβ A,
(6.5)
where A is an arbitrary non-singular 2 × 2 matrix. In the case that each quaternion is a multiple of the identity, qαβ = aαβ 12 , the above definition gives qdet Q = det[aαβ ]α,β=1,...,N as specified by (5.8). The motivation for choosing (6.4) as the definition of the determinant of a self-dual quaternion matrix is that the product over cycles occurs naturally in the calculation of correlation functions (recall Proposition 5.1.2). From a more abstract viewpoint, it is possible to define the determinant of a general quaternion matrix in a number of non-equivalent ways (the determinant function is first specified by some appropriate axioms generalizing the fundamental properties of detA for a complex matrix A; see [151] and [25]). The definition (6.4) is due to Moore [406]. Suppose in (6.4) we insist that the order of the cycles in the product be the same for permutations which differ by the directions of some or all of the cycles. Now, for a particular cycle a → b → c · · · → d → a, reversing its direction gives the cycle a → d → · · · → c → b → a and because the matrix is assumed self-dual this gives the product qad · · · qcb qba = (qab qbc · · · qda )D , where use has been made of the general property of the dual, (AB)D = BD AD . With this convention of the
238
CHAPTER 6
order assumed, it follows that
(−1)N −l
P ∈SN
l (qab qbc . . . qda ) 1
is a multiple of the identity, and we can write qdet Q =
(−1)N −l
l (0) (qab qbc . . . qda ) .
P ∈SN
(6.6)
1
A crucial result is that qdet Q can be expressed in terms of det Q (in qdet Q, Q is to be regarded as an N × N matrix with quaternion elements, while in det Q, Q is to be regarded as a 2N × 2N matrix with complex elements) [150]. P ROPOSITION 6.1.3 For a self-dual quaternion matrix we have ! 2 qdet Q = det Q.
(6.7)
Proof. Since (6.7) is a polynomial identity it suffices to consider the case that Q has real elements. Thus a typical element qjk of Q has the form
» qjk =
ajk −bjk
bjk ajk
–
for ajk , bjk real. But as noted in (1.36) such 2 × 2 matrices are isomorphic to complex numbers zjk := ajk + ibjk . Substituting in (6.6) gives X
qdetQ =
(−1)N−l
P ∈SN
= det Z,
l Y
zab zbc · · · zda
1
Z := [zjk ]j,k=1,...,N .
(6.8)
But the result of Exercises 3.1 q.2(ii) tells us that (det Z)2 = det Q
(6.9)
(since Q is self-dual, Z is Hermitian, so there is no need to take the absolute value on the l.h.s. of (6.9)).
At this stage we introduce Pfaffians. D EFINITION 6.1.4 Let X = [αij ]i,j=1,...,2N , where αji = −αij so that X is an antisymmetric matrix of even degree. Then the Pfaffian of X, denoted Pf X, is defined by ∗ Pf X = ε(P )αP (1)P (2) αP (3)P (4) · · · αP (2N −1)P (2N ) P (2l)>P (2l−1)
1 = N ε(P )αP (1)P (2) αP (3)P (4) · · · αP (2N −1)P (2N ) , 2 N!
(6.10)
P ∈S2N
where in the first summation the ∗ denotes that the sum is restricted to distinct terms only (i.e., only one term from the N ! ways of permuting the pairs of indices is to be included). We note that for Q a self-dual quaternion matrix, QZ−1 2N (where now Q is regarded as a 2N × 2N matrix with complex elements) is an antisymmetric matrix, so its Pfaffian is well defined. In fact the Pfaffian of this antisymmetric matrix is equal to the quaternion determinant of Q. P ROPOSITION 6.1.5 For a self-dual quaternion matrix Q qdet Q = Pf QZ−1 2N .
(6.11)
239
CORRELATION FUNCTIONS AT β = 1 AND 4
Proof. For X an antisymmetric matrix, a classical result (see Exercises 6.1 q.1) says (Pf X)2 = detX. Noting det Z−1 2N = 1, it follows that
(6.12)
2 (Pf QZ−1 2N ) = det Q,
and comparing this with (6.7) we see (6.11) is correct up to a sign. Taking Q the identity verifies the sign.
A useful corollary of (6.7) is the formula det Q = (det QQD )1/2 = qdet(QQD ),
(6.13)
valid for a general 2N × 2N matrix Q with a positive determinant (the fact that qdet(QQD ) is positive follows from (6.8)). We can use (6.13) to express the Boltzmann factor for the log-gas with β = 4 in terms of the monic skew symmetric polynomials orthogonal with respect to the inner product (6.3) [381].
P ROPOSITION 6.1.6 Let χk (x) :=
Q2k (x) Q2k (x)
Q2k+1 (x) Q2k+1 (x)
,
C(x) :=
1 2V (x)
0 1
,
where {Qn (x)} are monic skew orthogonal polynomials with respect to the inner product (6.3), with corresponding normalizations {qn }, and let χD k (x) denote the dual of χk (x). We have N N xk−1 j e−4V (xl ) (xk − xj )4 = e−4V (xl ) det (k − 1)xk−2 j=1,...,N j l=1 1≤j
=
N −1
2ql qdet[f4 (xj , xk )]j,k=1,...,N ,
(6.14)
l=0
where f4 (x, y) := e−2(V (x)+V (y))
N −1 k=0
with S4 (x, y) =
m=0
S4 (x, y) D4 (x, y)
I4 (x, y) S4 (y, x)
e−2V (x) d −2V (y) d −2V (y) e e Q2m (x) Q2m+1 (y) − Q2m+1 (x) Q2m (y) , 2qm dy dy
N −1
I4 (x, y) = −
1 D C(x)χk (x)χD k (y)C (y) = 2qk
y
S4 (x, y ) dy ,
x
∂ S4 (x, y). ∂x Note that the matrix [f4 (xj , xk )]j,k=1,...,N is self-dual. D4 (x, y) =
(6.15)
Proof. The first stated identity is the case p = 2 of (1.174). By adding suitable multiples of columns 1, 2, . . . , j − 1 to
240
CHAPTER 6
row j (j = 2, . . . , 2N in order) and using the fact that the Qn are monic we see that # " N Y xk−1 j −4V (xl ) e det (k − 1)xk−2 j=1,...,N j l=1 2
k=1,...,2N
3 e−2V (xj ) √ Q2k−1 (xj ) 7 2qk−1 7 . 7 −2V (xj ) 5 e √ Q2k−1 (xj ) 2qk−1 j,k=1,...,N
−2V (xj )
e N 6 √2qk−1 Q2k−2 (xj ) Y 6 = 2ql−1 det 6 4 e−2V (xj ) l=1 √ Q2k−2 (xj ) 2qk−1
(6.16)
Next multiply the matrix in (6.16) by diag[C(xj )]j=1,...,N on the left. Since the determinant of this latter matrix is unity, (6.16) is unchanged. Now applying (6.13) gives the second stated result.
It follows from the skew orthogonality property of {Qn (x)}n=0,1,... and the definition of f4 involving χk (x) and χD k (y) that ∞ ∞ 1 0 and f4 (x, x)dx = N f4 (x, y)f4 (y, z)dy = f4 (x, z). (6.17) 0 1 −∞ −∞ Due to the similarities between (6.17) and (5.7) we might expect that the integrations required to compute the n-particle correlation can be carried out according to the method of the proof of Proposition 5.1.2. This would require making a Laplace expansion of qdet, which is possible [406], [151] but due to the noncommutativity of the quaternions there are some complicating factors. Alternatively, the integrations can be carried out directly, as we will now demonstrate [150], [381]. P ROPOSITION 6.1.7 We have ∞ dx1 · · · −∞
∞
−∞
dxN
N
e−4V (xl )
(xk − xj )4 = N !
1≤j
l=1
N −1
2qk ,
k=0
ρ(n) (x1 , . . . , xn ) = qdet[f4 (xj , xk )]j,k=1,...,n , (0) ρT(n) (x1 , . . . , xn ) = (−1)n−1 f4 (xi , xi ) , cycles length n
(i,i )
where f4 is defined in Proposition 6.1.6. Proof. Consider qdet[f4 (xj , xk )]j,k=1,...,m . According to (6.6), this can be expanded as a sum over all permutations of {1, . . . , m} with summand (−1)m−l S1 S2 · · · Sl , (6.18) where each Sp consists of a particular product of the f4 (xj , xk ) in which the indices form a cycle. Let us compare this with the expansion of qdet[f4 (xj , xk )]j,k=1,...,m−1 . Analogous to (6.18), this can be written as a sum over all permutations of {1, . . . , m − 1} with summand (6.19) (−1)m−1−l A1 · · · Al , where again each Ap consists of a particular product of the f4 (xj , xk ) in which the indices form a cycle. Now, for each term (6.19) with l = l − 1 there are m ways to extend it to form a product of cycles of the form (6.18). One of these is to introduce an extra identity cycle m → m which gives (−1)m−l A1 · · · Al−1 f4 (xm , xm ).
(6.20)
The other m − 1 possibilities are to introduce m within an existing cycle, which gives (m)
(−1)m−l+1 A1 · · · Aj
· · · Al−1 ,
(6.21)
241
CORRELATION FUNCTIONS AT β = 1 AND 4
(m)
where Aj := f4 (xα , xβ ) · · · f4 (xδ , xm )f4 (xm , xγ ) · · · f4 (xτ , xα ). The sum over all permutations of {1, . . . , m − 1} with (6.20) and the m − 1 terms of the form (6.21) as the summand gives qdet[f4 (xj , xk )]j,k=1,...,m . Integrating (6.20) over xm using (6.17) gives N (−1)m−l A1 · · · Al−1 , while integrating the sum of the m − 1 terms of the form (6.21) over xm gives −(m − 1)(−1)m−l A1 · · · Al−1 .
(6.22)
According to (6.19) the sum total of these two terms is just (N − (m − 1)) times a typical term in the expansion of qdet[f4 (xj , xk )]j,k=1,...,m−1 , and so we have Z ∞ qdet[f4 (xj , xk )]j,k=1,...,m dxm = (N − (m − 1))qdet[f4 (xj , xk )]j,k=1,...,m−1 . (6.23) −∞
The first two results follow. The expression for ρT(n) follows from that for ρ(n) in an analogous way to the deduction of ρT(n) from ρ(n) in Proposition 5.1.2.
The n-point correlation for the β = 4 log-gas system is thus given by an n × n quaternion determinant, or equivalently a 2n × 2n Pfaffian. This will be shown also to be the case for β = 1 log-gas systems. In keeping with the definition of a determinantal point process, a many particle system for which the n-point correlation can be written as a 2n × 2n Pfaffian with entries independent of n is said to be a Pfaffian point process.
6.1.2 Functional differentiation method Historically (see [392]), the normalization integral and correlations in Proposition 6.1.7 were first calculated without making explicit use of quaternion determinants. The method used was to make use of the functional derivative (5.18) and Pfaffian (6.10) instead. However the implementation was such that only the one- and two-point correlations were obtained. More recently [522] it has been shown how to obtain the general npoint correlation via this method. Analogous to Proposition 5.2.1, the strategy is to write the generalized partition function in terms of the determinant of an integral operator. First the analogue of the l.h.s. of the identity of Proposition 5.2.1 is required. P ROPOSITION 6.1.8 For Ω equal to the real line and e−βU(x1 ,...,xN ) =
N l=1
in (5.19) we have
where
e−4V (xl )
(xk − xj )4
(6.24)
1≤j
1/2 ZˆN [a] = N !2N Pf[βjk ]j,k=1,...,2N = N !2N det[βjk ]j,k=1,...,2N
(6.25)
1 ∞ −4V (x) e a(x) Qj−1 (x)Qk−1 (x) − Qk−1 (x)Qj−1 (x) dx 2 −∞ d d 1 ∞ −2V (x) e a(x) Qj−1 (x) (e−2V (x) Qk−1 (x)) − Qk−1 (x) (e−2V (x) Qj−1 (x)) dx. = 2 −∞ dx dx (6.26)
βjk :=
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CHAPTER 6
Proof. According to the first equality in (6.14), and (6.16), we have 2
Y
(xk − xj ) = det 4 4
1≤j
X
=
Q2k−2 (xj )
Q2k−1 (xj )
Q2k−2 (xj )
Q2k−1 (xj )
ε(P )
P ∈S2N
N Y
3 5 j,k=1,...,N
QP (2l−1)−1 (xl )QP (2l)−1 (xl ).
l=1
The dependence on each integration variable is now decoupled, and this allows us to write ZˆN [a] =
X
ε(P )
P ∈S2N
N Y
Z αP (2l−1),P (2l) ,
αj,k :=
∞
−∞
l=1
e−4V (x) a(x)Qj−1 (x)Qk−1 (x) dx.
Introducing the restriction P (2l) > P (2l − 1), which can be done if we include all interchanges of this type in the summand by writing αP (2l−1),P (2l) → αP (2l−1),P (2l) − αP (2l),P (2l−1) gives X
ZˆN [a] = 2N
P ∈S2N P (2l)>P (2l−1)
ε(P )
N Y
βP (2l−1),P (2l) .
l=1
Comparing this expression with the definition (6.10) of the Pfaffian, and using (6.12), gives the stated result.
The strategy now is to make use of (5.26) and so introduce the sought determinant of an integral operator. P ROPOSITION 6.1.9 With f4 as in Proposition 6.1.6, the expression (6.25) for the generalized partition function can be rewritten as ZˆN [a] = N !2N
N −1
1/2 ql det[12 + f4T (a − 12 )] ,
(6.27)
l=0
where f4T (a − 12 ) is the 2 × 2 matrix integral operator with kernel f4T (x, y)diag[a(y) − 1, a(y) − 1]. Proof. With βjk = βjk [a] write h i [βjk [a]]j,k=1,...,2N = βjk [1] + βjk [a − 1]
j,k=1,...,2N
.
From the definition (6.26) of the βjk and the skew orthogonality of {Qj (x)} we see that [βjk [1]] is antisymmetric and of the form (6.2). Taking out an appropriate factor from each row and interchanging rows 2j − 1 and 2j for each j = 1, . . . , N shows 0 1 » – N−1 −1 Y 2 −q β [a − 1] 2j,k j−1 A. (6.28) qj det @12N + det[βjk [a]]j,k=1,...,2N = −1 qj−1 β2j−1,k [a − 1] j=1,...,N j=0
k=1,...,2N
In preparation for using (5.26) we note that with F2j−1 (x) = −e−2V (x) Q2j−1 (x),
F2j (x) = e−2V (x) Q2j−2 (x)
the integrand in the element (jk) of the final matrix in (6.28) consists of (a(x) − 1)/q[j/2] times ” d “ −2V (x) d Qk−1 (x) − e−2V (x) Qk−1 (x) (Fj (x)) Fj (x) e dx dx – » – » d −2V (x) d (e Qk−1 (x)) Fj (x) − dx (Fj (x)) dx = e−2V (x) Qk−1 (x)
243
CORRELATION FUNCTIONS AT β = 1 AND 4
This shows that with A the 2N × 2 matrix valued integral operator on (−∞, ∞) with kernel (a(y) − 1)
h 1 Fj (y) q[j/2]
−
1 q[j/2]
i d Fj (y) , dy
and B the 2 × 2N matrix multiplication operator " ” # d “ −2V (x) Qk−1 (x) e , dx −2V (x) Qk−1 (x) e the determinant in (6.28) can be written det(12N + AB). Now using (5.26) gives the stated result. The analogous result in the case β = 2, Proposition 5.2.2, suggests an alternative strategy. This is to consider the 2 × 2 matrix integral/ eigenfunction equation # " – " (1) # Z ∞ » (1) cψl (x) S4 (x, y) I4 (x, y) ψl (y) (1 − a(y)) dy = λ , (6.29) l (2) S4 (y, x) ψl(2) (y) ψl (x) −∞ D4 (x, y) and to seek eigenfunctions of the form # " " ∞ (1) X ψl (y) = cl,ν (2) ψl (y) ν=0
e−2V (y) Qν (y) d (e−2V (y) Qν (y)) dy
# .
(6.30)
Substituting (6.30) in (6.29) and equating coefficients of the vector in the νth term of (6.30) shows cl,ν = 0 for ν ≥ 2N while for ν < 2N 2N−1 X 1 cl,ν βν +1,ν+2 [1 − a(y)]. λl cl,ν = q[(ν+1)/2] ν =0
This is the eigenvalue problem for the 2N × 2N antisymmetric matrix i h 1 βj,k [a(y) − 1] , q[(j−1)/2] j,k=1,...,2N which can therefore be substituted for f4T (a − 12 ) in (6.27), reclaiming (6.25).
Substituting (6.27) in (5.25) and proceeding as in the working which led to (5.30) we find n 1 f4 (xi , xi ), ρT(n) (x1 , . . . , xn ) = (−1)n+1 Tr 2 cycles length n
(i,i )
which is equivalent to the final formula in Proposition 6.1.7. More generally, we note that if K is a 2 × 2 matrix integral operator depending on a(x) and 1/2 ZˆN [a] ∝ det(12 + K) we have n 1 G(xi , xi ), ρT(n) (x1 , . . . , xn ) = (−1)n+1 Tr 2 cycles length n
(i,i )
∂K (1 + K)−1 xi G(xi , xi ) = xi ∂a(x) a=1 (cf. (5.30)).
(6.31)
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CHAPTER 6
The formula (6.27) can also be used to compute ρ(n) directly. First we make use of (6.7) to note 1/2 det[12 + f4T (a − 12 )] = qdet [12 + f4 (a − 12 )]. But analogous to (5.32) we have for K4 a 2 × 2 matrix integral operator with kernel (1) (2) K4 (x, y) K4 (x, y) , K4 (x, y) := (3) (1) K4 (x, y) K4 (y, x) (2)
(3)
and K4 , K4 antisymmetric with respect to interchange of x and y so that [K4 (xj , xl )]j,l=1,...,k is a selfdual quaternion matrix ∞ ∞ 1 ∞ qdet [12 +K4 (a−12 )] = 1+ dx1 (a(x1 )−1) · · · dxk (a(xk )−1) qdet [K4 (xj , xl )]j,l=1,...,k . k! −∞ −∞ k=1 (6.32) From this the functional derivative required by (5.20) can be computed to give the second formula in Proposition 6.1.7. E XERCISES 6.1
1. Let { xj }j=1,...,2N be 2N component vectors and let A be a 2N × 2N antisymmetric matrix
so that Pf[ xj · (A xk )]j,k=1,...,2N
(6.33)
is well defined. (i) Let σ be a permutation of {1, 2, . . . , 2N }. Show from the definition (6.10) that Pf[ xσ(j) · (A xσ(k) )]j,k=1,...,2N = sgn(σ)Pf[ xj · (A xk )]j,k=1,...,2N . Note also from the definition (6.10) that (6.33) is a linear function in the components of {xj }. Since the above equation shows the linear function is antisymmetric, conclude that for some f xk )]j,k=1,...,2N = f (A) det[ x1 x2 · · · x2N ]. Pf[ xj · (A By choosing xj = ej (an elementary unit vector), show that f (A) = Pf A. (ii) Let A denote the 2N × 2N antisymmetric matrix obtained from A by interchanging two rows and the corresponding two columns. Show from the first equation in (i) that Pf A = −Pf A.
(6.34)
(iii) Replace xj by Bej , for B a general 2N × 2N matrix, in the second equation of (i) to conclude Pf(BT AB) = det B Pf A.
(6.35)
T (iv) Set A = Z−1 2N , and X = B AB to deduce from (6.35) the result (6.12).
(v) By considering the coefficient of αr,2N in (6.10), deduce the Laplace type expansion Pf A =
2N−1 X
(−1)r+1 αr,2N Pf r,2N (A),
(6.36)
r=1
where Pf r,2N is the Pfaffian of the (2N − 2) × (2N − 2) antisymmetric matrix obtained by deleting rows and columns r, 2N . 2. [456], [89] Let B, C be 2J × 2J and 2K × 2K invertible antisymmetric matrices, and A be a matrix of size
245
CORRELATION FUNCTIONS AT β = 1 AND 4
2J × 2K. The aim of this exercise is to show that Pf((B−1 )T − ACAT ) Pf((C−1 )T − AT BA) = . Pf((C−1 )T ) Pf((B−1 )T )
(6.37)
(i) By writing Ap×q → Ap×q Rq×q , Bq×p → Bq×p Mp×p , where R and M are assumed invertible, in (5.33) obtain the determinant identity −1 det Mp×p det(M−1 p×p + Ap×q Rq×q Bq×p ) = det Rq×q det(Rq×q + Bq×p Mp×p Ap×q ).
(6.38)
(ii) In (6.38), note that (C−1 )T = −C−1 , (B−1 )T = −B−1 , then square both sides, write in terms of determinants using (6.12), and verify the resulting identity using (6.37). This establishes (6.38) up to a sign. Show that the sign is correct by setting A = 0. 3. [167, 263] The objective of this exercise is to derive the formulas Q2n (x) =
Q2n+1 (x) =
1 ˆ Zn 1 ˆ Zn
Z
∞ −∞
Z
∞ −∞
dx1 e−4V (x1 ) · · · dx1 e−4V (x1 ) · · ·
Z
∞
dxn e−4V (xn )
−∞
Z
∞ −∞
n Y (x − xl )2 l=1
Y
(xk − xj )4 ,
1≤j
n n “ ”Y X dxn e−4V (xn ) x + 2 xl (x − xl )2 l=1
l=1
Y
(xk − xj )4
1≤j
(cf. first formula of Proposition 5.1.4). (i) Show that n Y (x − xl )2 l=1
2
Y
(xk − xj )4
1≤j
3 xk−1 j 5 = det 4 (k − 1)xk−2 j k−1 x
“ x+2
n X l=1
xl 2
n ”Y (x − xl )2 l=1
2
j=1,...,n k=1,...,2n+1
Y
3 Qk−1 (xj ) = det 4 Qk−1 (xj ) 5 Qk−1 (x)
(xk − xj )4
1≤j
xk−1 j
= det 4 (k − 1)xk−2 j xk−1
, j=1,...,n k=1,...,2n+1
2 3 x2n+1 Qk−1 (xj ) j 5 = det 4 Qk−1 (xj ) (2n + 1)x2n j Qk−1 (x) x2n+1 j=1,...,n−1
3 Q2n+1 (xj ) Q2n+1 (xj ) 5 Q2n+1 (x)
k=1,...,2n
, j=1,...,n k=1,...,2n+1
where in obtaining the final equality the arbitrary constant γ2m in (6.1) is chosen so that the coefficient of x2m in U2m+1 (x) vanishes. (ii) By proceeding as in the proof of Proposition 6.1.8, and using the notation therein, show that the first of the multiple integrals is equal to 2n
X P ∈S2n+1 P (2l)>P (2l−1)
ε(P )
n Y
βP (2l−1),P (2l) |a=1 QP (2n+1)−1 (x),
l=1
and show that this formula also holds for the second integral provided Q2n (x) is replaced by Q2n+1 (x) throughout. From the skew orthogonality property of {Ql (x)}l=0,1,... note that βP (2l−1),P (2l) |a=1 equals ql δP (2l),P (2l+1)+1 and thus derive the stated formulas. 4. [217] Let ·|·s denote a general skew symmetric inner product. Let {pj (x)}j=0,1,... denote a general family of
246
CHAPTER 6
monic polynomials in which pj (x) is of degree j. With J mn := pm |pn s let i h Dn = det[J 2n−j 2n−k ]j,k=1,...,2n , En = − det [J 2n−j 2n+1 ]j=0,...,2n [J 2n−j 2n−k ] j=0,...,2n . k=1,...,2n
Verify that the monic polynomials of degree 2n, 2n + 1, h i U2n (x) = Dn−1 det [C2n−j (x)]j=0,...,2n [J 2n−j 2n−k ] j=0,...,2n , k=1,...,2n i h U2n+1 (x) = En−1 det [J 2n+1−j 2n+1 C2n+1−j (x)]j=0,...,2n+1 [J 2n+1−j 2n+1−k ] j=0,...,2n+1 + γ2n U2n+1 (x) k=2,...,2n+1
satisfy the skew orthogonality properties of Definition 6.1.1 by showing U2n |pj s = 0, U2n+1 |pj s = 0 (j = 0, . . . , 2n − 1).
6.2 CONSTRUCTION OF THE SKEW ORTHOGONAL POLYNOMIALS AT β = 4 6.2.1 Special properties of the classical cases Polynomials skew orthogonal with respect to (6.3) can, for the classical weight functions of Section 5.4.1, be calculated explicitly [419]. Here we will follow a subsequent treatment [3] which emphasizes the relationship with particular orthogonal polynomials and also inter-relates the construction of the skew orthogonal polynomials at β = 4 with the construction at β = 1 to be undertaken below. The same general formulas apply to the Cauchy weight (5.51). In the theory of [3], it is convenient to work with the inner product 1 ∞ −2V (x) f |g4 := e (f (x)g (x) − f (x)g(x))dx 2 −∞ d d 1 ∞ −V (x) f (x) (e−V (x) g(x)) − g(x) (e−V (x) f (x)) dx, e (6.39) = 2 −∞ dx dx which differs from (6.3) by the replacement e−4V (x) → e−2V (x) . For the classical weight functions (5.56), this simply corresponds to rescaling x and/or the parameters a, b and α. It turns out that the operator A (5.60) relates the inner product (6.39) to the β = 2 inner product in (5.4).
P ROPOSITION 6.2.1 Assuming e−2V (x) vanishes at the endpoints of its support, we have . (φ, Aψ)2 = φ|ψ4 2V (x) →2V (x)−log f (x)
Proof. It follows from the definition of A (5.60) and the general formula Z
d F (x) = dx that we have
−∞
Z
q Aψ[x] =
∞
∞
e2V (x) f (x) −∞
δ (x − y)F (y) dy
δ (x − y)
q f (y)e−2V (y) ψ(y) dy.
(6.40)
247
CORRELATION FUNCTIONS AT β = 1 AND 4
This gives Z (φ, Aψ)2 =
Z
∞
q
∞
dx −∞ Z ∞
dy −∞
f (x)e−2V (x) φ(x)δ (x − y)
q f (y)e−2V (y) ψ(y)
q “ ∂ q ” f (x)e−2V (x) ψ(x) f (x)e−2V (x) φ(x) dx =− ∂x −∞ Z q “ ∂ q ” 1 ∞ f (x)e−2V (x) ψ(x) f (x)e−2V (x) φ(x) dx =− 2 −∞ ∂x Z q “ ∂ q ” 1 ∞ f (x)e−2V (x) φ(x) f (x)e−2V (x) ψ(x) dx + 2 −∞ ∂x ˛ ˛ . = φ|ψ4 ˛ 2V →2V −log f
The result of Proposition 6.2.1 can be used in conjunction with the simple structure exhibited by (5.63) to specify the classical skew orthogonal polynomials at β = 4 in terms of their orthogonal polynomial ˜ j } denote the skew orthogonal polynomials corresponding to the inner product (6.39), counterparts. Let {Q modified by the replacement 2V (x) → 2V˜4 (x) := 2V (x) − log f (x)
(6.41)
and write {˜ ql } for the corresponding normalizations. Next introduce the lower triangular transition matrix X(4) such that ˜ j (x)]j=0,...,N −1 = X(4) [pj (x)]j=0,...,N −1 , [Q Use of (6.40) then gives
˜ ˜ ˜ := Qj , Qk 4 q
(4)
Xjj = 1.
(6.42)
= X(4) AX(4) T , 2V (x) →2V (x)−log f (x) j,k=0,...,N −1
where A is specified by (5.63), or equivalently ˜ (X(4) T )−1 = A. X(4)−1 q
(6.43)
(4) The equation (6.43) can be solved for the matrix X(4)−1 := [β˜jk ]j,k=0,...,N −1 , which is lower triangular with 1s along the diagonal. This can be done by multiplying both sides of (6.43) by X(4) T on the right, and then equating the strictly lower triangular parts of both sides. On the l.h.s. we have ⎡ ⎤ ∗ ⎢ −˜ ⎥ q0 ∗ ⎢ ⎥ (4) (4) ⎢ ⎥ −˜ q0 β˜21 q˜0 β˜20 ∗ ⎢ ⎥ (4)−1 ⎢ ⎥ ˜ X q = ⎢ −˜ , (6.44) ˜(4) q˜1 β˜(4) ⎥ q −˜ q ∗ β 1 1 31 30 − ⎢ ⎥ (4) (4) (4) (4) ˜ ˜ ⎢ −˜ ⎥ q1 β˜43 q˜1 β˜42 ∗ ⎣ q1 β41 q˜1 β40 −˜ ⎦ .. .. .. .. .. . . . . . 2N ×2N
where the subscript in ( )− denotes the strictly lower triangular entries, while on the r.h.s. = (A)− . AX(4)T −
(6.45)
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CHAPTER 6
Equating (6.44) and (6.45) gives p = 0, . . . , N − 1,
c2p = q˜p , (4) β˜2p+1,j
= 0,
j = 0, . . . , 2p − 1,
(4) β˜2p,j
= 0,
j = 0, . . . , 2p − 1, j = 2p − 2,
(4) qp−1 = −c2p−1 /c2p−2 β˜2p,2p−2 = −c2p−1 /˜
(6.46)
(4)
while β˜2p+1,2p is undetermined. Thus we have ˜ 2j+1 (x) + β˜(4) ˜ p2j+1 (x) = Q 2j+1,2j Q2j (x), ˜ 2j (x) − c2j−1 Q ˜ 2j−2 (x), p2j (x) = Q c2j−2 q˜p = c2p .
(6.47)
The nonuniqueness of the skew orthogonal polynomials up to the transformation (6.1) implies we can (4) ˜ 2j (x) in the second choose β˜2j+1,2j = 0 (this gives the simplest results). Doing this, and solving for Q equation of (6.47) gives ˜ 2j+1 (x) = p2j+1 (x), Q j l−1 j−1 c2p+1 c2p ˜ 2j (x) = p2l (x), Q c2p c p=0 p=0 2p+1 l=0
q˜p = c2p .
(6.48)
˜ 2j (x) in terms of an indefinite integral involving p2j+1 (x). Furthermore, it is possible to write Q P ROPOSITION 6.2.2 With the notation (6.41) we have in the classical cases ∞ q˜m ˜ ˜ ˜ 2m (x) = − e−2V (t)+V4 (t) p2m+1 (t) dt. e−V4 (x) Q (p2m+1 , p2m+1 )2 x
(6.49)
Proof. We can check from the second formula in (6.39) that ˛ ” ˜ d “ −V˜4 (y) ˜ ˜ 2m (x)4 ˛˛ Q2m (y) = eV4 (y) δ(x − y)|Q . e ˜4 dy V →V But the completeness of {pj (x)} shows δ(x − y) = e−2V (y)
∞ X pn (x)pn (y) . (pn , pn )2 n=0
(6.50)
Furthermore substituting for pn (x) by using the expansion pn (x) =
n X
(4) ˜ β˜nj Q j (x),
(4) β˜nn = 1,
(6.51)
j=0
˜ j } that we conclude from the skew orthogonality of {Q ∞ ” X d “ −V˜4 (y) ˜ ˜ pν (y) ˜(4) Q2m (y) = −˜ qm eV4 (y)−2V (y) βν,2m+1 . e dy (p ν , p ν )2 ν=2m+1
(6.52)
The stated formula now follows after noting from (6.46) that the only nonzero value of β˜ν,2m+1 is when ν = 2m + 1, in (4) which case β˜ = 1. (4)
ν,2m+1
249
CORRELATION FUNCTIONS AT β = 1 AND 4
Strictly speaking, the derivation of (6.50) given in the above proof does not apply to the Cauchy weight: according to (5.52) the set of orthogonal polynomials is then finite and so (6.50) cannot be justified. However an alternative proof of (6.50) which uses only (6.48) and (5.64), and so is applicable to the Cauchy case, can be given. This is done by taking (6.52) as given and verifying it as an identity. In the classical cases the ˜ 2m (x) using (6.48) and computes the r.h.s. consists of a single term only. On the l.h.s. one substitutes for Q derivative using (5.64); simplification yields the same single term as resulted on the r.h.s. Combining Proposition 6.2.2 and (6.47) we thus have that the monic skew orthogonal polynomials and corresponding normalization with respect to the inner product (6.39), and with the classical weight functions
e
˜4 (x) −2V
⎧ 2 e−x , ⎪ ⎪ ⎨ a+1 −x x e , = (1 − x)a+1 (1 + x)b+1 , ⎪ ⎪ ⎩ (1 + x2 )−(α−1) ,
Hermite, Laguerre, Jacobi, Cauchy
(6.53)
(recall (5.56), (5.58) and (6.41)) are given in terms of the corresponding monic orthogonal polynomials of Section 5.4.1 according to ˜ 2j+1 (x) = p2j+1 (x), Q ˜
c2j eV4 (x) (p2j+1 , p2j+1 )2 q˜j = c2j ,
˜ 2j (x) = − Q
∞
˜
e−2V (t)+V4 (t) p2j+1 (t) dt,
x
(6.54)
where c2j is specified by (5.65). 6.2.2 Summation formula From Propositions 6.1.6 and 6.1.7 we have ⎡ ⎢ ρ(n) (x1 , . . . , xn ) = qdet ⎢ ⎣
S4 (xj , xk ) ∂ S4 (xj , xk ) ∂xj
−
xk
⎤
S4 (xj , y) dy ⎥ ⎥ ⎦ S4 (xk , xj )
xj
,
(6.55)
j,k=1,...,n
where S4 is specified in (6.15). It is possible to rewrite S4 in a form which separates the β = 2 correlation kernel (5.6). P ROPOSITION 6.2.3 With S˜4 (x, y) defined by S4 (x, y) in (6.15) but modified so that V → 12 V˜4 as specified ˜ k (x), and {pj (x)}j=0,1,... the set (assumed complete) of monic orthogonal polynomials by (6.41), Qk (x) → Q associated with the weight function e−2V (x) we have 2N −1 ∞ 2N −1 pn (x)pn (y) pn (y) ˜(4) ˜ 1 ˜ ˜ S˜4 (x, y) = e−2V (y)−V4 (x)+V4 (y) + (6.56) βnk Qk (x) . 2 (pn , pn )2 (pn , pn )2 n=0 n=2N k=0
Proof. The key to this summation is the formula (6.52) and its counterpart ∞ ” X ˜ pν (x) ˜(4) d “ −V˜4 (x) ˜ Q2m+1 (x) = q˜m eV4 (x)−2V (x) βν,2m , e dx (p ν , p ν )2 ν=2m
(6.57)
which is derived in an analogous manner. Substituting in the first equation of (6.15) (modified as specified) and performing minor manipulation gives (6.56).
The first sum in (6.56) is evaluated by the Christoffel-Darboux formula (5.10), so it remains to simplify
250
CHAPTER 6
the second sum. Although this is complicated in general, in the classical cases simplification occurs. Thus (4) choosing β˜2j+1,2j = 0, as done in obtaining (6.48), we see from (6.46) that in the classical cases the only (4) nonzero value of β˜ for n > k is nk
c2n−1 (4) , β˜2n,2n−2 = − c2n and thus 1 ˜ ˜ S˜4 (x, y) = e−(V (y)−V4 (y)) eV (x)−V4 (x) K2N (x, y) 2 c2N −1 p2N (y) ˜ 1 ˜ ˜ + e−V4 (x)+V4 (y)−2V (y) Q2N −2 (x). 2 c2N (p2N , p2N )2 (6.58) ˜ 2N −2 (x) according to (6.54) eliminates all reference to the skew orthogonal polynomials Substituting for Q in the final summation formula [3]. P ROPOSITION 6.2.4 In the classical cases 1 ˜ ˜ S˜4 (x, y) = e−(V (y)−V4 (y)) eV (x)−V4 (x) K2N (x, y) 2 ∞ ˜ eV4 (y)−2V (y) c2N −1 p2N (y) ˜ − e−2V (t)+V4 (t) p2N −1 (t) dt. (6.59) 2(p2N , p2N )2 (p2N −1 , p2N −1 )2 x E XERCISES 6.2 1. Use the first equation in Proposition 6.1.7 and the definitions (4.139), (4.142) and (4.143) to note that G4,N (G) = 2N qN−1 , G4,N−1
Wa,4,N (L) = 2N qN−1 , Wa,4,N−1
Ca,b,4,N (J) = 2N qN−1 , Ca,b,4,N−1
(Cy)
Nα,4,N (Cy)
Nα,4,N−1
(Cy)
= 2N qN−1
for the Gaussian, Laguerre, Jacobi and Cauchy ensembles, respectively. From the explicit evaluations Proposition 4.7.1 and 4.7.3, and (4.144), (4.145), verify the corresponding formulas for qn are consistent with (6.54). 2. [522] In this exercise the analogue of (5.140) for S4 (x, y) will be derived. (i) In the formula (6.15) for S4 write e−2V (x) Qp (x) = φp (x), and then put φp (x) =
2N−1 X
αpj ψj (x),
ψj (x) = e−2V (x) uj (x),
j=0
where the span of {uj (x)}j=0,1,...,2N−1 is equal to the space of polynomials of degree ≤ 2N − 1. Show that S4 (x, y) =
2N−1 X 2N−1 X
μl1 l2 ψl1 (x)ψl2 (y), (4)
l1 =0 l2 =0 (4)
μl1 l2 :=
N−1 X m=0
1 (α2m l1 α2m+1 l2 − α2m l2 α2m+1 l1 ). 2qm
(ii) Writing α = [αjk ]j,k=0,...,2N−1 and M = [uj |uk 4 ]j,k=0,...,2N−1 , check that „» – » –« 0 q0 0 qN−1 2k , ,..., M α2k+1 = −qk α αT Mα = diag −q0 0 −qN−1 0 and from this show
[μl1 ,l2 ]l1 ,l2 =0,...,2N−1 = M−1 . (4)
251
CORRELATION FUNCTIONS AT β = 1 AND 4
3. [544] In this exercise S4 will be related to KN from β = 2 theory. Throughout we consider the original β = 4 problem modified so that V (x) → V (x)/2, N → N/2, and we denote S¯4 (x, y) = S4 (y, x)| N →N/2 . V →V /2
−2V (x)
(i) Define KN (x, y) as in (5.6) and set w2 (x) = e . With S¯4 , K denoting the integral operators with kernels S¯4 (x, y), KN (x, y), show using the form of S¯4 (x, y) implied by q.2(i) above that S¯4 K = S¯4 . (ii) With M defined as in q.2(ii), and assuming {ψj (x)} vanishes at the endpoints of the support of e−2V (x) , R∞ show mjk = −∞ ψj (x)ψk (x) dx, where here ψj (x) := e−V (x) uj (x) (since V (x) → V (x)/2). Use this and the result of (i) to show S¯4 Kψj = ψj and thus conclude that for S¯4 restricted to functions in H = Span{ψi }i=0,...,2N−1 we have S¯4 |H = D(KDH )−1 , where D denotes differentiation, and DH differentiation restricted to H. Note also that S¯R4 |H⊥ = 0, where H⊥ denotes the orthogonal complement of ∞ H with respect to the inner product (f, g) := −∞ f (x)g(x) dx. (iii) Let (x) = 21 sgn(x), and denote by the integral operator with kernel (x − y). Verify that when restricted to functions in H, (KDH )−1 = (IH − K(I − K)DH )−1 KH . Substitute this in the formula obtained in (ii) for S¯4 |H to deduce S¯4 |H = IH + AH (IH − BA)−1 B|H = (IH+DH − AB)−1 |H where A = (I − K)D, B = K. Conclude that in general S¯4 = (IH+DH − (I − K)DK)−1 K.
(6.60)
6.3 CORRELATION FUNCTIONS AT β = 1 The identity necessary to compute the correlations at β = 1 is more difficult to derive than the corresponding identities at β = 2 and 4, although the general strategy is to adopt an approach analogous to that used for β = 4. The starting point is to introduce appropriate skew orthogonal polynomials. We require the function sgn(x) where sgn(x) = 1 (x > 0), sgn(x) = −1 (x < 0) and sgn(0) = 0. D EFINITION 6.3.1 Define a skew symmetric inner product ·|·1 by ∞ 1 ∞ −V (x) dx e f (x) dy e−V (y) g(y)sgn(y − x), f |g1 := 2 −∞ −∞
(6.61)
and let {Rn (x)}n=0,1,... be a corresponding family of monic skew orthogonal polynomials so that R2m |R2n+1 1 = −R2n+1 |R2m 1 = rn δmn ,
R2m |R2n 1 = R2m+1 |R2n+1 1 = 0.
The Boltzmann factor of the log-gas on a line at β = 1 can be expressed [381] as a quaternion determinant involving these polynomials, which in turn can be used to compute the n-particle correlation. The N even and N odd cases must be treated separately. 6.3.1 N even Use will be made of the identity [123] sgn(xk − xj ) = Pf[sgn(xk − xj )]j,k=1,...,2n .
(6.62)
1≤j
To see the validity of (6.62), we note from (6.10) that Pf[sgn(xk − xj )]j,k=1,...,2n =
∗ P (2l)>P (2l−1)
ε(P )
n l=1
sgn(xP (2l) − xP (2l−1) ).
(6.63)
252
CHAPTER 6
In the case that x1 < x2 < · · · < x2n all terms in the product are positive and so the r.h.s. reads ∗ ε(P ). P (2l)>P (2l−1)
But there are (2n − 1)!! terms in this sum, which can be constructed systematically from the identity permutation by successive elementary transpositions, each of which reverses the sign of the signature. Since (2n − 1)!! is odd the sum then adds to 1, verifying (6.62) in this case. For other orderings we note that the l.h.s. of (6.62) changes sign if xi and xi+1 are interchanged. On the r.h.s. this interchange is equivalent to interchanging rows i and i + 1, then columns i and i + 1, which we know from (6.34) also changes the sign.
P ROPOSITION 6.3.2 Let N be even and suppose Φk (x),
rk (> 0) and H = [h(xj , xk )]j,k=1,...,N (h(xk , xj ) = −h(xj , xk ))
are arbitrary, and let Rk (x) be such that det[Rk (xj )]j,k=1,...,N =
(xk − xj )
(6.64)
1≤j
(here the skew orthogonality property (6.61) is not assumed). Set Φ2k+1 (x) Φ2k (x) φk (x) := . R2k (x)e−V (x) R2k+1 (x)e−V (x) We have N
e−V (xj )
j=1
N/2−1
(xk − xj )Pf H =
rp qdet[f1 (xj , xk )]j,k=1,...,N ,
(6.65)
p=0
1≤j
where [f1 (xj , xk )] is a self-dual quaternion matrix specified by
N/2−1
f1 (x, y) :=
k=0
1 φk (x)φD k (y) + rk
0 h(x, y) 0 0
=
S1 (x, y) D1 (x, y)
I˜1 (x, y) S1 (y, x)
with e−V (y) Φ2k (x)R2k+1 (y) − Φ2k+1 (x)R2k (y) , rk
N/2−1
S1 (x, y) =
k=0
e−(V (x)+V (y)) R2k (x)R2k+1 (y) − R2k+1 (x)R2k (y) , rk
N/2−1
D1 (x, y) =
k=0
N/2−1
I˜1 (x, y) =
k=0
1 Φ2k+1 (x)Φ2k (y) − Φ2k (x)Φ2k+1 (y) + h(x, y) rk
=: I1 (x, y) + h(x, y).
Proof. We use a method first introduced in the more general parameter-dependent problem [417]. Let D1 := [D1 (xj , xk )]j,k=1,...,N ,
I1 := [I1 (xj , xk )]j,k=1,...,N ,
S1 := [S1 (xj , xk )]j,k=1,...,N .
253
CORRELATION FUNCTIONS AT β = 1 AND 4
With this notation, we begin by noting from (6.64) that N Y
e−2V (xl )
l=1
Y
(xk − xj )2 =
1≤j
N Y
e−2V (xl ) det[Rk−1 (xj )]j,k=1,...,N det
»
l=1
Y
R2j−1 (xk ) −R2j−2 (xk )
– j=1,...,N/2 k=1,...,N
N/2
=
2 rj−1 det D1 .
j=1
Since D1 is antisymmetric we can make use of (6.12) to conclude N Y
e−V (xj )
j=1
Y
Y
N/2
(xk − xj ) =
rj−1 Pf D1
j=1
j
(there is a choice of signs in taking the square root; that the correct sign has been taken can be checked by considering 0 the construction of the term xN−1 xN−2 N N−1 · · · x2 x1 on the r.h.s.). It follows from this that N Y
e−V (xj )
j=1
Y
Y
»
N/2
(xk − xj )Pf H = (−1)N/2
rj−1 Pf
j=1
j
D1 0
0 −H
– .
(6.66)
To introduce the matrices S1 and I1 into (6.66) we introduce the nonunique (x-dependent) matrix α = [αij ]i,j=1,...,N such that N X Φk (xj ) = − αji e−V (xi ) Rk (xi ). i=1
It then follows from the definitions that αD1 αT = −I1 .
αD1 = −S1 ,
We make use of these relations by adding to the second block row of the matrix on the r.h.s. of (6.66) the first block row multiplied by α on the left, and adding to the new second block column the first block column multiplied by αT on the right. This leaves the value of the Pfaffian unchanged and gives N Y
e−V (xj )
j=1
Y
Y
»
N/2
(xk − xj )Pf H = (−1)N/2
rj−1 Pf
j=1
j
D1 −S1
ST1 −˜I1
– .
Interchanging the first block row with the second and using (6.11) gives the sought result.
The identity (6.62) implies that with h(x, y) = (the factor of
1 2
1 sgn(y − x) 2
(6.67)
is for later convenience), (6.65) reads N j=1
e
−V (xj )
N/2−1
|xk − xj | =
2rp qdet[f1 (xj , xk )]j,k=1,...,N .
(6.68)
p=0
j
The use of (6.68) is that with the Rk (x) chosen to have the skew orthogonality property of Definition 6.3.1, the rk chosen as the corresponding normalization and ∞ h(y, x)Rk (y)e−V (y) dy, (6.69) Φk (x) := −∞
the quantity f1 in Proposition 6.3.2 has integration properties analogous to those of f4 given in the previous
254
CHAPTER 6
section. One finds [381] ∞
0 0 1 f1 (x, z) + f1 (x, z) f1 (x, y)f1 (y, z)dy = 0 1 0 −∞ ∞ 1 0 f1 (x, x) dx = N 0 1 −∞
0 0
, (6.70)
(cf. (6.17)). These formulas, together with the method of the proof of Proposition 6.1.7, allow the integrations required to compute the n-particle correlations to be performed. P ROPOSITION 6.3.3 Let {Rn (x)}n=0,1,... , {rn }n=0,1,... be as in Definition 6.3.1, h(x, y) as in (6.67) and Φk (x) as in (6.69). In terms of these quantities specify f1 as in Proposition 6.3.2. Then for N even we have
∞
−∞
dx1 · · ·
∞
−∞
dxN
N j=1
e
−V (xj )
N/2−1
|xk − xj | = N !
1≤j
2rk ,
k=0
ρ(n) (x1 , . . . , xn ) = qdet[f1 (xj , xk )]j,k=1,...,n , (0) ρT(n) (x1 , . . . , xn ) = (−1)n−1 f1 (xi , xi ) . cycles length n
Furthermore, D1 (x, y) =
∂ S1 (x, y), ∂x
I1 (x, y) =
1 2
∞
−∞
(i,i )
S1 (x, z)sgn(z − y)dz = −
y
S1 (x, z) dz. x
Proof. We see that all equations in the proof of Proposition 6.1.7 up to (6.22) remain valid. For the replacement of (6.22), note that integrating Aj , as defined below (6.21), over xm gives » – » 0 0 1 f1 (xα , xβ ) · · · f1 (xδ , xγ ) · · · f1 (xτ , xα ) + f1 (xα , xβ ) · · · f1 (xδ , xγ ) 0 1 0
0 0
– · · · f1 (xτ , xα ).
If the cycle Aj is of length p, then summing this expression over the p possible ways of inserting xm into the cycle gives » – » – 1 0 0 0 + (p − 1)Aj . Aj + Aj 0 0 0 1 Now, in the definition (6.4) of qdet, only the scalar part of each cycle is required. Since in general Tr(BC) = Tr(CB), we see that the scalar part of the above expression is the same as the scalar part of pAj . Thus (6.22) in the proof of Proposition 6.1.7 remains valid, provided we take the scalar part. Consequently (6.23) again holds, which implies the stated results. The formulas for D1 and I1 are a consequence of the definitions in Proposition 6.3.2 together with the explicit formulas (6.67) and (6.69).
6.3.2 Integration over alternate variables As with the corresponding results at β = 4, historically the normalization and one- and two-particle correlation functions in Proposition 6.3.3 were first evaluated in a way that did not make explicit use of quaternion determinants. Instead, use was make of functional differentiation, and an expression for the generalized partition function in terms of a Pfaffian obtained via the method of integration over alternate variables [392]. P ROPOSITION 6.3.4 Let N be even. For Ω equal to the real line and e−βU(x1 ,...,xN ) =
N l=1
e−V (xl )
1≤j
|xk − xj |
(6.71)
255
CORRELATION FUNCTIONS AT β = 1 AND 4
in (5.19) we have
1/2 , ZˆN [a] = N !2N/2 Pf[γjk ]j,k=1,...,N = N !2N/2 det[γjk ]j,k=1,...,N ∞ ∞ 1 γjk := dx e−V (x) a(x)Rj−1 (x) dy e−V (y) a(y)Rk−1 (y)sgn(y − x). 2 −∞ −∞
(6.72)
Proof. Ordering the integration variables −∞ < x1 < · · · < xN < ∞ gives Z ZˆN [a] = N ! Z
−∞<x1 <···<xN <∞
= N! −∞<x1 <···<xN <∞
dx1 · · · dxN
N Y
e−V (xl ) a(xl )
l=1
Y
(xk − xj )
1≤j
dx1 · · · dxN det[e−V (xj ) a(xj )Rk−1 (xj )]j,k=1,...,N ,
Q where to obtain the second equality use has been made of the Vandermonde identity (1.173) modified to read 1≤j
The determinant is now symmetrical in x2 , x4 , . . . , xN , so we can replace the domain of integration by the entire real line in each of the variables, provided we divide by (N/2)!. Expanding out the determinant and integrating over each x2l gives N/2 Y X N! ZˆN [a] = ε(P ) μP (2l−1),P (2l) , (N/2)! P ∈S l=1
N
Z
where
∞
μj,k :=
dx e−V (x) a(x)Rk−1 (x)
−∞
Z
x
dy e−V (y) a(y)Rj−1 (y).
−∞
As in the proof of Proposition 6.1.8, if we restrict P (2l) > P (2l − 1), we can generate all other terms in the sum over permutations by replacing each μP (2l−1),P (2l) by γP (2l−1),P (2l) := μP (2l−1),P (2l) − μP (2l),P (2l−1) . Thus ZˆN [a] =
N! (N/2)!
X P (2l)>P (2l−1)
Y
N/2
ε(P )
γP (2l−1),P (2l) ,
l=1
which after comparison with the definition (6.10) of a Pfaffian implies the stated result.
An alternative way to derive (6.72) is to make use of (6.62) and (6.64). In fact as with Proposition 6.3.2 there is a general integration formula [123] for the product Pf[h(xj , xk )]j,k=1,...,2n det[Rj−1 (xk )]j,k=1,...,2n .
256
CHAPTER 6
P ROPOSITION 6.3.5 Let N be even. We have N N dx1 · · · dxN e−V (xl ) a(xl ) det[Rk−1 (xj )]N j,k=1 Pf[h(xj , xk )]j,k=1 −∞<x1 <···<xN <∞
l=1
∞ ∞ 1 −V (x1 ) N = dx1 e a(x1 ) · · · dxN e−V (xN ) a(xN ) det[Rk−1 (xj )]N j,k=1 Pf[h(xj , xk )]j,k=1 N ! −∞ −∞ ∞ ∞ N −V (x) = Pf dx e a(x) dy e−V (y) a(y)Rj−1 (x)h(x, y)Rk−1 (y) . (6.73) −∞
j,k=1
−∞
Proof. The integrand is symmetric in the xs, so the second multiple integral is equal to N ! times the first. To derive the final equality note that since both the determinant and Pfaffian are antisymmetric in the xs, we can write the second line as Z ∞ Z ∞ N Y dx1 e−V (x1 ) a(x1 ) · · · dxN e−V (xN ) a(xN ) Rj−1 (xj )Pf[h(xj , xk )]j,k=1,...,N . (6.74) −∞
−∞
j=1
Recalling (6.10), this can be rewritten Z
X
ε(P )
dx1 e
QN j=1
−V (x1 )
−∞
P (2l)>P (2l−1)
But
∞
Rj−1 (xj ) =
Z a(x1 ) · · ·
∞
dxN e−V (xN ) a(xN )
−∞
N Y
Y
N/2
Rj−1 (xj )
j=1
h(xP (2l−1) , xP (2l) ).
l=1
(6.75)
QN
RP (j)−1 (xP (j) ) so (6.75) is equal to Z ∞ Z X ε(P ) dx1 e−V (x1 ) a(x1 ) · · ·
j=1
P (2l)>P (2l−1)
Y
−∞
∞
dxN e−V (xN ) a(xN )
−∞
N/2
×
RP (2l−1)−1 (xP (2l−1) )h(xP (2l−1) , xP (2l) )RP (2l)−1 (xP (2l) ),
l=1
and upon noting, by changing the order of integration, that Z ∞ Z ∞ dx1 e−V (x1 ) a(x1 ) · · · dxN e−V (xN ) a(xN ) −∞
=
−∞
N/2 “ Z ∞ Y l=1
−∞
dxP (2l−1) e−V (xP (2l−1) ) a(x2l−1 )
Z
∞ −∞
” dxP (2l) e−V (xP (2l) ) a(x2l ) ,
this in turn reduces to X P (2l)>P (2l−1)
ε(P )
N/2 Z ∞ Y l=1
dx e−V (x) a(x)
Z
−∞
∞ −∞
dy e−V (y) a(y)RP (2l−1)−1 (x)h(x, y)RP (2l)−1 (y),
which from (6.10) we recognize as the r.h.s. of (6.73).
As with the β = 4 theory of Section 6.1.2 it is possible to obtain from (6.72) the n-point correlation of Proposition 6.3.3 [522]. P ROPOSITION 6.3.6 The expression (6.72) for the generalized partition function can be rewritten as
N/2−1
ZˆN [a] = N !2N/2
1/2 rk det[12 + f1T (a − 12 )] ,
k=0
where f1T (a − 12 ) is the matrix integral operator with kernel f1T (x, y)diag[a(y) − 1, a(y) − 1].
(6.76)
257
CORRELATION FUNCTIONS AT β = 1 AND 4
Proof. In the definition of γjk write a := f + 1,
ψj (x) := e−V (x) Rj−1 (x),
(6.77)
and denote by the integral operator with kernel 12 sgn (x − y) to obtain Z ∞ “ ˛ ” ˛ f (x)ψj (x)ψk [x] − f (x)ψk (x)ψj [x] − f (x)ψk (x)(f ψj )[x] dx. γjk = γjk ˛ − a=1
−∞
Because of the skew orthogonality of {Rj (x)}j=0,1,... , [γjk ]|a=1 is skew symmetric with the structure of (6.2) (except that N → N/2). With G2j (x) := −ψ2j−1 (x) (6.78) G2j−1 (x) := ψ2j (x), we see that factoring out [γjk ]|a=1 , multiplying the even rows by −1, and interchanging odd and even rows gives Y
N/2
det[γjk ]j,k=0,...,N−1 = Z ×
∞ −∞
“
j=1
h 2 rj−1 det δj,k +
1 r[(j−1)/2]
” i f (x)Gj (x)ψk [x] − f (x)ψk (x)Gj [x] − f (x)ψk (x)(f Gj )[x] dx
j,k=1,...,N
.
(6.79)
Now the determinant in (6.79) can be written as det[1N + AB], where A is the N × 2 matrix valued integral operator on (−∞, ∞) with kernel i h f (y)Gj [y] , − f (y)Gj [y] − f (y)(f Gj )[y] j=1,...,N
while B is the operator which multiplies by the 2 × N matrix – » ψk (y) . ψk [y] k=1,...,N To apply (5.26), we note that with these definitions of A and B, the operator 1 + BA is the 2 × 2 matrix integral operator ” “ 3 2 PN P 1− N j=1 ψj ⊗ f Gj + ψj ⊗ f (f Gj ) j=1 ψj ⊗ f Gj “ 5 4 P P − N 1+ N j=1 ψj ⊗ f Gj + ψj ⊗ f (f Gj ) j=1 ψj ⊗ f Gj #» " P PN – 1− N ψj ⊗ f Gj ψj ⊗ f Gj 1 0 j=1 j=1 PN PN , (6.80) = f 1 − j=1 ψj ⊗ f Gj − f 1 + j=1 ψj ⊗ f Gj where the notation a ⊗ b denotes the integral operator with kernel a(x)b(y). The determinant of the last matrix equals 1, while recalling (6.78) and (6.77) we can identify the first matrix with the r.h.s. of (6.76).
With (6.76) established, the formula in Proposition 6.3.3 for ρT(n) now follows as an example of (6.31), or from the analogue of (6.32).
6.3.3 N odd The N odd analogue of Proposition 6.3.2 can be derived [150], [238], provided H is replaced by the N + 1 × N + 1 antisymmetric matrix [h(xj , xk )]j,k=1,...,N [F (xj )]j=1,...,N . (6.81) X := −[F (xk )]k=1,...,N 0
258
CHAPTER 6
The relevance of this structure is that with h(x, y) given by (6.67) and F (x) = sgn(xk − xj ), Pf X = 2−(N +1)/2
1 2
we have
1≤j
which follows from (6.62) by taking x2n → ∞. P ROPOSITION 6.3.7 Let N be odd, and suppose ˆ k (x), rˆk (= 0), F (x) and H = [h(xj , xk )]j,k=1,...,N , (h(xk , xj ) = −h(xj , xk )) Φ ˆ k (x) be such that are arbitrary, and let R
ˆ k (xj )]j,k=1,...,N = det[R
(xk − xj ).
1≤j
Then, with X defined by (6.81) N j=1
(N −1)/2
e−V (xj )
(xk − xj )Pf X =
rˆp qdet[f1odd (xj , xk )]j,k=1,...,N ,
p=0
1≤j
where
f1odd (x, y) :=
I˜1odd (x, y) S1odd (y, x)
S1odd (x, y) D1odd (x, y)
with (N −1)/2−1
S1odd (x, y) =
k=0 (N −1)/2−1
D1odd (x, y) =
k=0 (N −1)/2−1
I˜1odd (x, y) =
k=0
+
−V (y) e−V (y) ˆ ˆ 2k+1 (y) − Φ ˆ 2k+1 (x)R ˆ 2k (y) + e ˆ N −1 (y), Φ2k (x)R F (x)R rˆk rˆ(N −1)/2 e−V (x) e−V (y) ˆ ˆ 2k+1 (y) − R ˆ 2k+1 (x)R ˆ 2k (y) , R2k (x)R rˆk 1 ˆ ˆ 2k (y) − Φ ˆ 2k (x)Φ ˆ 2k+1 (y) + h(x, y) Φ2k+1 (x)Φ rˆk !
1 rˆ(N −1)/2
ˆ N −1 (x)F (y) − F (x)Φ ˆ N −1 (y) . Φ
Proof. As in the analogous result for N even, here we will use a method first introduced in the more general parameterˆ k (x)} that dependent problem [417]. The first step is to note from the defining property of {R N Y
Y
e−2V (xj )
j=1
=
(xk − xj )2
1≤j
2 »
ˆ k−1 (xj )]j,k=1,...,N det 6 e−2V (xj ) det[R 4
3
– j=1,...,(N −1)/2 k=1,...,N
7 5
ˆ N−1 (xk )]k=1,...,N [R
j=1
Y
ˆ 2j−1 (xk ) R ˆ 2j−2 (xk ) −R
(N+1)/2
=
2 rˆj−1 det[Dodd + rrT ], 1
j=1
ˆ N−1 (xj )/r(N−1)/2 ]j=1,...,N . But in general for a column = [D1odd (xj , xk )]j,k=1,...,N , r := [e−V (xj ) R where Dodd 1
259
CORRELATION FUNCTIONS AT β = 1 AND 4
vector v = [vj ]j=1,...,N , and N × N matrix A, it is easy to check that » A det[A + vv T ] = det −v T
v 1
– (6.82)
(use elementary column operations to make the terms in the final row, except that in the final column, zero). Now with A = Dodd 1 , A is an antisymmetric matrix of odd dimension and so det[A] = 0. This means the element 1 in the lower right-hand corner on the r.h.s. of the above expression can be replaced by 0, giving an antisymmetric matrix. Making use of (6.12) then shows – » odd (N+1)/2 N Y Y Y r D1 e−V (xj ) (xk − xj ) = rˆj−1 Pf −rT 0 j=1
j=1
j
:= [F (xj )]j=1,...,N , and in particular, with F 2
N Y j=1
e−V (xj )
Y
Y
(N+1)/2
(xk − xj ) Pf X = (−1)(N+1)/2
j=1
j
Y
Dodd 1 6 −rT rˆj−1 Pf 6 4 "
(N+1)/2
= (−1)
(N+1)/2
rˆj−1 Pf
j=1
Dodd 1 rT −F
3
r 0
7 7 −F 5 0
−H T F #
T rF −H
,
(6.83)
where the validity of the second equality can be checked from the definition (6.10). Consider now the quantities S1 , D1 and I1 as defined in Proposition 6.3.2, all with N → N − 1 and N odd (note that then D1 = D1odd ). In terms of these quantities define the matrices D1 := [D1 (xj , xk )]j,k=1,...,N ,
I1 := [I1 (xj , xk )]j,k=1,...,N ,
S1 := [S1 (xj , xk )]j,k=1,...,N .
To transform between these matrices we introduce the (nonunique x-dependent) matrix α = [αij ]i,j=1,...,N such that Φk (xj ) = −
N X
αji e−V (xi ) Rk (xi ).
i=1
It then follows from the definitions that αD1 = −S1 ,
αD1 αT = −I1 .
We make use of these relations by multiplying the first block row on the r.h.s. of (6.83) by α on the left, and adding to the second block row, then multiplying the first block column by αT on the right and adding the second block column, to obtain » odd – (N+1)/2 N T Y Y Y D1 (Sodd −V (xj ) (N+1)/2 1 ) e (xk − xj ) Pf X = (−1) rˆj−1 Pf . −Sodd −˜Iodd 1 1 j=1 j=1 j
Interchanging the first block row with the second gives the stated result.
By an appropriate choice of the arbitrary terms in Proposition 6.3.7 the analogue of Proposition 6.3.3 can be established.
260
CHAPTER 6
P ROPOSITION 6.3.8 Let {Rn (x)}n=0,1,... and {rn }n=0,...,(N −1)/2−1 be as in Definition 6.3.1, and let ∞ rˆ(N −1)/2 := e−V (x) F (x)RN −1 (x) dx, rˆn := rn (n = 0, . . . , (N − 1)/2 − 1), −∞ ∞ −1 −V (x ) ˆ e F (x )Rn (x ) dx RN −1 (x) (n = 0, . . . , N − 2), Rn (x) := Rn (x) − rˆ(N −1)/2 −∞
ˆ N −1 (x) := RN −1 (x), R ∞ ˆ n (y) dy ˆ n (x) := e−V (y) h(y, x)R Φ
(n = 0, . . . , N − 1),
−∞
h(xj , xk ) = Then we have ∞ −∞
1 sgn(xk − xj ), 2
dx1 · · ·
∞
−∞
dxN
F (x) =
N
e
1 . 2
−V (xj )
j=1
(N −1)/2
|xk − xj | = N !
1≤j
2ˆ rk ,
k=0
ρ(n) (x1 , . . . , xn ) = qdet[f1odd (xj , xk )]j,k=1,...,n . Proof. As in the proof of Proposition 6.3.3, this is a consequence of the validity of the two formulas (6.70) (with f1odd replacing f1 ). The formulas are checked from the skew orthogonality property of {Rj (x)} and the properties Z ∞ Z ∞ ˆ m (x)R ˆ n (x) dx = rˆ(N−1)/2 δn,N−1 , ˆ n (x) dx = Rm |Rn 1 . e−V (x) F (x)R e−V (x) Φ −∞
−∞
The qdet formula of Proposition 6.3.8 for ρ(n) =: ρN (n) in the case N odd can also be deduced as a corollary of the corresponding result in Proposition 6.3.3 for N even [210]. To see how this comes about, suppose N +1 is odd and consider the correlation ρN (n+1) . According to the definition +1 ρN (n+1) (x1 , . . . , xn+1 )
∞ ∞ n+1 N +1 N +1 (N + 1)! = w(xl ) dxn+2 · · · dxN +1 w1 (xl ) |xk −xj |, ZˆN +1 (N + n)! −∞ −∞ 1
l=1
l=n+2
j
and consequently lim
xn+1 →∞
+1 (w(xn+1 ))−1 (xn+1 )−N ρN (n+1) (x1 , . . . , xn+1 ) = (N + 1)
ZˆN N ρ(n) (x1 , . . . , xn ). ˆ ZN +1
+1 In the qdet expression for ρN (n+1) implied by Proposition 6.3.3, the only dependence on xn+1 is in the final row and column. The leading asymptotic form of the entries can readily be determined and the limit computed. The qdet expression of Proposition 6.3.8 results after appropriate elementary reduction of the limiting final row and column.
E XERCISES 6.3
1. The objective of this exercise is to apply the method of integration over alternate variables in the case N odd and thus show that then » – [γjk ]j,k=1,...,N [νj ]j=1,...,N ZˆN [a] = N !2(N+1)/2 Pf , −[νk ]k=1,...,N 0
where γjk is as in Proposition 6.3.4 while νk :=
1 2
Z
∞ −∞
e−V (x) a(x)Rk−1 (x) dx.
261
CORRELATION FUNCTIONS AT β = 1 AND 4
(i) Proceed as in the proof of Proposition 6.3.4 to show Z
Y
(N−1)/2
ZˆN [a] = N ! −∞<x2 <x4 <···<xN −1 <∞
2 » R x2j
6 × det 4
=
dx2l e−V (x2l ) a(x2l )
l=1
e−V (x) a(x)Rk−1 (x) dx −∞ Rk−1 (x2j )
–
3 j=1,...,(N −1)/2
k=1,...,N R∞ [ −∞ e−V (x) a(x)Rk−1 (x) dx]k=1,...,N
7 5
“ (N−1)/2 ” X Y N! ε(P ) μP (2l−1),P (2l) 2νP (N) ((N − 1)/2)! P ∈S l=1 N
=
N! 2(N+1)/2 ((N − 1)/2)!
X
ε(P )
“ (N−1)/2 Y
” γP (2l−1),P (2l) νP (N) .
l=1
P (2l)>P (2l−1) l=1,...,(N −1)/2
(ii) Write νP (N) =: νP (N),N+1 = −νN+1,P (N) in the above expression, and make use of (6.36) to identify this expression as proportional to the stated Pfaffian. 2. [123] Let X be given as in (6.81). Derive the analogue of (6.73) for N odd, Z ∞ Z ∞ dx1 e−V (x1 ) a(x1 ) · · · dxN e−V (xN ) a(xN ) det[Rk−1 (xj )]j,k=1,...,N Pf X −∞
−∞
»
[ajk ]j,k=1,...,N −[bk ]k=1,...,N
= N !Pf where
Z ajk =
∞
[bj ]j=1,...,N 0
dx e−V (x) a(x)
−∞ Z ∞
bj =
Z
∞
−∞
– ,
(6.84)
dy e−V (y) a(y) Rj−1 (x)h(x, y)Rk−1 (y),
e−V (x) a(x)F (x)Rj−1 (x) dx.
−∞
3. Use an argument similar to that used in the verification of (6.62) to show Pf[sgn(j − i)xi yj ]i,j=1,...,2n =
n Y
x2i−1
i=1
n Y
y2j .
(6.85)
j=1
4. Assume that N ≤ M are positive integers and N is even. Let T = [tik ] i=1,...,N be an N × M matrix and k=1,...,M
A = [akl ]k,l=1,...,M be an M × M skew symmetric matrix. Then the minor summation formula states that [304] X Pf[AJ,J ] det[TJ ] = Pf[TAT T], (6.86) J ⊂{1,...,M}, |J |=N
where AJ,J denotes the N × N square matrix which results from restricting A to the rows and columns indexed by J, while TJ denotes the N ×N square matrix which results by restricting the columns of T to J. The objective of this exercise is to prove (6.86) and derive as a consequence the Binet-Cauchy formula. (i) [34] In (6.73) suppose the integration measure is uniform and discrete, Z
∞ −∞
dxj e−V (xj ) a(xj ) →
M X
,
nj =1
and write Rj−1 (nk ) =: rjk ,
h(nj , nk ) =: ajk .
(6.87)
262
CHAPTER 6
With the determinant in (6.73) replaced by its transpose, identify (6.73) with (6.86). (ii) [303] Show that with M = 2m, N = 2n and » – 0m 1m A= , −1m 0m one has
j
Pf[AJ,J ] =
(−1)n(n−1)/2 , 0,
» T=
Xm×n 0m×n
0m×n Ym×n
– ,
if J = {k1 < · · · < kn < k1 + m < · · · < kn + m}, otherwise,
and for J as in the first case det[TJ ] = det[XK ] det[YK ], where K = {k1 , . . . , km }. (iii) Note that with A and T as in (ii), Pf[TAT T] = (−1)n(n−1)/2 det[XT Y] and thus substitute this together with the results of (ii) in (6.86) to deduce X det[XK ] det[YK ] = det[XT Y],
(6.88)
K⊂{1,...,m}, |K|=n
which is the Binet-Cauchy formula. (iv) Relate (6.88) to the discretization (6.87), with e−V (x) a(x) replaced by w2 (x), of (5.170). QN 5. [91] For eigenvalues {xi }i=1,...,N let P (y) := i=1 (y − xi ) denote the characteristic polynomial and let OEN (e−V (x) ) be defined as above (4.22). Let ZN := ZˆN /N ! denote the corresponding canonical partition function (recall (1.39)). One has the result that with k + m even, k − m = 2s, s > 1 − N , D Qk P (α ) E D(β; α) Z2N+2s i (1) Qi=1 = Pf [WN ({α, β}|{α, β})], m −V (x) ) Z2N Δ(−α)Δ(−β) i=1 P (βi ) OE2N (e (1)
where use has been made of the notation (5.35), (5.37), and WN is a skew symmetric (k + m) × (k + m) matrix with rows and columns parametrized by {α, β} and with matrix elements D E Z2N+2s−2 (αi − αj ) P (αi )P (αj ) , Z2N+2s OE2N +2s−2 (e−V (x) ) D E P (αi ) 1 (1) , WN (αi , βj ) = αi − βj P (βj ) OE2N +2s (e−V (x) ) E D Z2N+2s+2 1 (1) (βi − βj ) . WN (βi , βj ) = Z2N+2s P (βi )P (βj ) OE2N +2s+2 (e−V (x) ) (1)
WN (αi , αj ) =
(i) Use Proposition 5.3.2 to deduce from this that for OE2N (e−V (x) ), (1)
ρ(n) (x1 , . . . , xn ) = Pf [K2N (xi , xj )]i,j=1,...,n (1)
where K2N (x, y) =
k12 i is the 2 × 2 skew symmetric matrix with elements k22
D P (x) E Z2N−2 1 (x − y)P (x)P (y)OE2N −2 (e−V (x) ) , k12 = , Res Z2N x − y z=y P (z) OE2N (e−V (x) ) D E 1 Z2N+2 = −k12 , k22 = (x − y) Res , z=y Z2N P (x)P (z) OE2N +2 (e−V (x) )
k11 = k21
h k 11 k21
263
CORRELATION FUNCTIONS AT β = 1 AND 4
with use having been made of (5.42). (ii) Use the method of integration over alternate variables to show that for N even P (x)P (y)OEN −2 (e−V (x) ) =
rN/2−1 D1 (x, y). x−y
Adopting the strategy of the proof of Proposition 5.1.4, use this to deduce D P (x) E P (y) D
X R2i+1 (x)h2i (y) − R2i (x)h2i+1 (y) + 1, ri i=0
N/2−1 OEN (e−V (x) )
= (x − y)
E 1 P (x)P (y) OEN +2 (e−V (x) ) =
1 rN/2
N/2−1 ˛ 1 “ X h2i+1 (x)h2i (y) − h2i (x)h2i+1 (y) D 1 ˛ 1 E ” + , ˛ x−y ri x−t y−t 1 i=0
where hj (x) := Rj (t)|1/(x − t)1 . (iii) Combine the results of (i) and (ii), and make use of (6.11) to reclaim the evaluation of ρ(n) given in Proposition 6.3.3.
6.4 CONSTRUCTION OF THE SKEW ORTHOGONAL POLYNOMIALS AND SUMMATION FORMULAS 6.4.1 Skew orthogonal polynomials The operator (5.60) can also be used to facilitate the calculation of the monic skew orthogonal polynomials with respect to the inner product (6.61) in the classical cases. This follows because of a relationship between the inner products (6.61) and (5.4) involving the operator A [6]. P ROPOSITION 6.4.1 We have
(φ, A−1 ψ)2 = −φ|ψ1
V (x) →V (x)+log f (x)
.
(6.89)
Proof. From (5.60) we see that A−1 = (f (x)e−2V (x) )−1/2 while it is easy to check that
“ d ”−1 (e2V (x) f (x))−1/2 dx
Z “ d ”−1 1 ∞ ψ(x) = sgn(x − y)ψ(y) dy. dx 2 −∞
Thus A−1 ψ[x] = (f (x)e−2V (x) )−1/2
1 2
Z
∞ −∞
e−V (y) sgn(x − y)(f (y))−1/2 ψ(y) dy
and the result is immediate.
To make use of Proposition 6.4.1 we first note that it is possible to write the matrix [(pj , A−1 pk )2 ]j,k=0,...,N −1 in terms of A as specified by (5.63). First note that with D := diag((pj , pj )2 )j=0,...,N −1 we have
[(pj , A−1 pk )2 ]j,k=0,...,N −1 D−1 A
jk
=−
N −1 l=0
(A−1 pj , pl )2 (pl , Apk ) (pl , pl )2
(6.90)
264
CHAPTER 6
where use has been made of the first equation in Proposition 6.2.1. But according to (5.64) N −1 l=0
(pl , Apk )2 pl (x) = Apk (x) (pl , pl )2
for k = N − 1,
which when substituted back in (6.90) implies [(pj , A−1 pk )2 ]j,k=0,...,N −1 D−1 A = D ,
(6.91)
where D = D + Ck with Ck having all entries zero except for the final column. These latter entries are easy to compute but it turns out their explicit form is not needed. The formula (6.91) together with (6.89) allows the determination of the classical skew orthogonal polyno˜ j (x) and corresponding normalizations r˜j , where the tilde denotes that the underlying inner product mials R is the modification of (6.61) specified by V (x) → V˜1 (x) := V (x) + log f (x). (1)
Thus with X
:= [αjk ]j,k=0,...,N −1 denoting the lower triangular transition matrix defined by ˜j (x)]j=0,...,N −1 = X(1) [pj (x)]j=0,...,N −1 , [R
we have
(6.92)
˜ k 1 ˜j , R ˜r := R V (x) →V (x)+log f (x) j,k=0,...,N −1 (1) pj , pk 1 =X
αjj = 1,
V (x) →V (x)+log f (x) j,k=0,...,N −1
X(1)T
= −X(1) [(pj , A−1 pk )2 ]j,k=0,...,N −1 X(1)T = −X(1) D A−1 DX(1)T ,
(6.93)
where in obtaining the first equality of the final line (6.89) has been used, while in the final equality (6.91) has been used. Taking inverses of both sides of (6.93) gives the equivalent formula X(1)T ˜r−1 X(1) = −D−1 AD
−1
.
(6.94)
(1)
In the classical cases the equation (6.94) can be solved for X . It is sufficient to premultiply both sides by (X(1)T )−1 and then equate the strictly lower triangular entries on both sides. Using (6.2) we see that on the l.h.s. ⎡ ⎤ ∗ ⎢ 1/˜ ⎥ r0 ∗ ⎢ ⎥ ⎢ −α30 /˜ ⎥ r −α /˜ r ∗ 1 31 1 ⎢ ⎥ ⎢ α20 /˜ ⎥ −1 (1) r α /˜ r 1/˜ r ∗ 1 21 1 1 ˜r X =⎢ . (6.95) ⎥ ⎢ −α50 /˜ ⎥ − r −α /˜ r −α /˜ r −α /˜ r ∗ 2 51 2 52 2 53 2 ⎢ ⎥ ⎢ α40 /˜ ⎥ r2 α41 /˜ r2 α42 /˜ r2 α43 /˜ r2 1/˜ r2 ∗ ⎣ ⎦ .. .. .. .. .. .. . . . . . . 2N ×2N On the r.h.s. the fact that (X(1)T )−1 is an upper triangular matrix with 1’s along the diagonal and the explicit
265
CORRELATION FUNCTIONS AT β = 1 AND 4
form (5.63) of A together show −1 −1 = − D−1 AD − (X(1)T )−1 D−1 AD − − ⎡ ∗ ⎢ γ0 ∗ ⎢ ⎢ = ⎢ 0 γ1 ∗ ⎢ 0 0 γ2 ∗ ⎣ .. .. .. .. . . . .
⎤ ⎥ ⎥ ⎥ ⎥, ⎥ ⎦
(6.96)
where γj := cj /(pj+1 , pj+1 )2 (pj , pj )2 .
(6.97)
Equating (6.95) and (6.96) gives rp (p = 0, . . . , N − 1), γ2p = 1/˜ α2p,l = 0 (l = 0, . . . , 2p − 1), α2p+1,l = 0 (l = 0, . . . , 2p − 2), α2p+1,2p−1 = −γ2p−1 r˜p = −γ2p−1 /γ2p , while α ˜ 2p+1,2p is left unspecified. Hence ˜ 2j (x) = p2j (x), R ˜ 2j+1 (x) = p2j+1 (x) + α2p+1,2p p2j (x) − γ2j−1 p2j−1 (x), R γ2j r˜p = 1/γ2p .
(6.98)
˜ 2j+1 (x) is consistent with the skew orthogonal Note that the fact that α2p+1,2p is arbitrary in the formula for R polynomials being nonunique up to the transformation (6.1). It is simplest to choose α2p+1,2p = 0. Then, in light of (6.97) and (5.65) all quantities in (6.98) are known explicitly. Note that with the choice α2p+1,2p = 0 it follows from (5.64) and (6.98) that d −V˜4 (x) 1 ˜ ˜ 2j+1 (x) = − e eV1 (x) p2j (x) . (6.99) R γ2j (p2j , p2j )2 dx In summary we thus have that skew orthogonal polynomials and corresponding normalizations with respect to the skew symmetric inner product (6.61), with weight functions ⎧ −x2 /2 e , Gaussian, ⎪ ⎪ ⎨ (a−1)/2 −x/2 ˜ e (x > 0), Laguerre, x e−V1 (x) = (6.100) ⎪ (1 − x)(a−1)/2 (1 + x)(b−1)/2 (−1 < x < 1), Jacobi, ⎪ ⎩ Cauchy (1 + x2 )−(α+1)/2 , are given by (6.98) with α2p+1,2p = 0. 6.4.2 Summation formulas — N even Analogous to the situation at β = 4 for the classical weight functions, the fundamental quantity S1 (x, y), defined in Proposition 6.3.2, determining the correlation function at β = 1 can be summed explicitly. The details depend on the parity of N . P ROPOSITION 6.4.2 Let N be even. Define S˜1 (x, y) as in Proposition 6.3.2, but with V → V˜1 and similarly ˜k , Φ ˜ k and r˜k . Let {pj (x)}j=0,1,... be the set of monic orthogonal polynomials (assumed complete) define R
266
CHAPTER 6
associated with the weight function e−2V (x) and also write pl (x) =
l
(1) ˜ β˜lj R j (x),
(1) β˜ll = 1.
(6.101)
j=0
We have ˜ ˜ S˜1 (x, y) = e−2V (x)+V1 (x)−V1 (y)
N −1 n=0
∞ N −1 pn (x)pn (y) pn (x) ˜(1) ˜ + β Rk (y) . (pn , pn )2 (pn , pn )2 nk
(6.102)
n=N k=0
Proof. From the definition (6.61) of the skew inner product at β = 1, and the definition in Proposition 6.3.2 of Φk (x), both modified as specified, we see that it is possible to write ˛ ˜ k (x ) = eV˜1 (x ) R ˜ k (x)|δ(x − y)1 ˛˛ Φ
˜1 V →V
.
˜ k (y)} that Making use of (6.50) and substituting for pn (y) therein by (6.101) we see from the skew orthogonality of {R ∞ X
˜ 2k (x) = r˜k eV˜1 (x) e−2V (x) Φ
ν=2k+1 ˜ ˜ 2k+1 (x) = −˜ Φ rk eV1 (x) e−2V (x)
∞ X ν=2k
pν (x) ˜(1) , β (pν , pν )2 ν 2k+1
(6.103)
pν (x) ˜(1) β . (pν , pν )2 ν 2k
(6.104)
Substituting these results in the formula of Proposition 6.3.2 for S1 (x, y) we obtain (6.102) after minor manipulation.
The first sum in (6.102) is evaluated according to the Christoffel-Darboux formula (6.3). To evaluate the (1) second sum requires knowledge of the transition coefficients β˜lj . In the case of the classical polynomials these coefficients can be determined from (6.98). Setting α2p+1,2p = 0 and comparing with the definition (6.101) we see that (1) β˜2l,j = 0 (j = 0, . . . , 2l − 1), (1) β˜2l+1,2j = 0 (j = 0, . . . , l), l ak γ2k+1 (1) β˜2l+1,2j+1 = jk=1 , ak := . γ2k k=1 ak
(6.105)
(1) In fact we don’t require the explicit form of the β˜lj , but rather their factorization property (1) (1) (1) β˜nk = β˜n,N −1 β˜N −1,k ,
n ≥ N,
(6.106)
which is evident from the above formulas. Substituting (6.106) in the double summation in (6.102) and recalling (6.101) shows ∞ N ∞ −1 pn (x) ˜(1) ˜ pn (x) ˜(1) pN −1 (y) βnk Rk (y) = β (pn , pn )2 (pn , pn )2 n,N −1 n=N k=0 n=N pN −1 (x) 1 ˜ ˜ N −2 (x) − pN −1 (y), e−V (x)+2V (x) Φ = r˜N/2−1 (pN −1 , pN −1 )2
where the second equality follows from (6.103) and the fact that N is assumed even. Thus ˜ ˜ ˜ ˜ N −2 (x)pN −1 (y). S˜1 (x, y) = e−(V (x)−V1 (x)) e(V (y)−V1 (y)) KN −1 (x, y) + γN −2 e−V1 (y) Φ
˜ N −2 in the definition of ΦN −2 according to (6.98) allows all reference to the skew orthogSubstituting for R
267
CORRELATION FUNCTIONS AT β = 1 AND 4
onal polynomials to be eliminated in the final summation formula. P ROPOSITION 6.4.3 For the classical cases ˜ ˜ S˜1 (x, y) = e−(V (x)−V1 (x)) e(V (y)−V1 (y)) KN −1 (x, y) 1 ∞ ˜ ˜ sgn(x − t)pN −2 (t)e−V1 (t) dt, +γN −2 e−V1 (y) pN −1 (y) 2 −∞
(6.107)
where γN −2 is specified in terms of (pN −2 , pN −2 )2 by (6.97) and (5.65). Because the completeness of the orthogonal polynomials has been assumed in the derivation of (6.107), and in the Cauchy case the polynomials are a finite set and so not complete, the validity of (6.107) in the Cauchy case remains to be established. In fact (6.107) can be verified directly in all the classical cases by making use of (5.64) and (6.99) in the formula of Proposition 6.3.2 for S˜1 (x, y).
6.4.3 Summation formulas — N odd In the case of N odd, we see from the definitions of Proposition 6.3.7 and Proposition 6.3.8 that ˜ N −1 (y) R ˜ + e−V (y) S˜1odd (x, y) = S˜1 (x, y) 2˜ sN −1 N →N −1 (N −1)/2−1 −V ˜ ˜ N −1 (x) e 1 (y) Φ ˜ 2k+1 (y) + s˜2k+1 R ˜ 2k (y) − s˜2k R + (1) s˜N −1 r˜k k=0 ˜ ˜ N −1 (y) (N −1)/2−1 e−V1 (y) R ˜ 2k+1 (x) + s˜2k+1 Φ ˜ 2k (x) , − s ˜ Φ − 2k (1) s˜N −1 r˜ k=0
where 1 s˜k := 2
k
∞
−∞
˜
˜ k (x) dx. e−V1 (x) R
(6.108)
The quantity S˜1 (x, y)|N →N −1 is evaluated by (6.107). Furthermore, from the definitions we see that (N −1)/2−1
k=0
e−V1 (y) ˜
(1) r˜k
˜ 2k+1 (y) + s˜2k+1 R ˜ 2k (y) − s˜2k R
= − lim S˜1 (x, y) x→∞
˜
N →N −1
= −γN −3 s˜N −3 e−V1 (y) pN −2 (y),
(6.109)
where the second equality follows from (6.107), while (N −1)/2−1
1 (1)
k=0
=
1 2
r˜k ∞ −∞
˜ 2k+1 (x) + s˜2k+1 Φ ˜ 2k (x) − s˜2k Φ
sgn(x − y) − lim S˜1 (x , y) x →∞
N →N −1
dy = −γN −3 s˜N −3 φ˜N −2 (x),
where the second equality follows from (6.109) and 1 ∞ −V˜1 (y) ˜ e sgn(x − y)pj (y) dy. φj (x) := 2 −∞
(6.110)
(6.111)
268
CHAPTER 6
Hence
S˜1odd (x, y) = S˜1 (x, y)
˜
N →N −1
−γN −3 s˜N −3
+ e−V1 (y)
pN −1 (y) 2˜ sN −1
˜ e−V1 (y) ˜ φN −1 (x)pN −2 (y) − pN −1 (y)φ˜N −2 (x) , s˜N −1
(6.112)
˜ N −1 (y) = pN −1 (y), which follows from (6.98), and the fact that N is odd have been where the fact that R used. For future reference we note that s˜n for n even can be expressed in terms of the γk . Thus it follows from ˜ n (x)/Φ ˜ n−2 (x) gives (6.103), (6.105) and (6.98) that taking the limit x → ∞ in the ratio Φ s˜n−2 γn−1 = , s˜n γn−2 and thus s˜2l = s˜0
l−1
γ2j . γ j=0 2j+1
Recalling (6.98) and (6.108) this is equivalent to the evaluation ∞ l−1 γ2j ˜ c˜2l = c˜0 , c˜k := e−V1 (t) pk (t) dt. γ −∞ j=0 2j+1
(6.113)
We note also that taking the limit x → ∞ in (6.104) and using (6.98) implies c˜2l−1 = 0. E XERCISES 6.4
(6.114)
1. [167, 263] Use the method of integration over alternate variables to show that
R2n (x) =
R2n+1 (x) =
1 ˆ Z2n 1 ˆ Z2n ×
Z
∞ −∞
Z
∞ −∞
dx1 e−V (x1 ) · · · dx1 e−V (x1 ) · · ·
2n Y
Y
l=1
1≤j
(x − xl )
Z
∞
dx2n e−V (x2n )
−∞
Z
∞
2n Y (x − xl ) l=1
Y
|xk − xj |,
1≤j
2n “ ” X dx2n e−V (x2n ) x + xj
−∞
j=1
|xk − xj |
(cf. Exercises 6.1 q.2), where in R2n+1 (x) the arbitrary constant γ2m in (6.1) is chosen so that the coefficient of x2n vanishes. For this purpose, as well as the Vandermonde determinant identity, make use of the determinant formula " k−1 # 2n 2n ”Y “ X Y xj x2n+1 j = x+ xj (x − xl ) |xk − xj |. det k−1 2n+1 x x j=1,...,2n j=1 l=1 1≤j
2. [544] In this exercise the β = 1 analogue of (6.60) is obtained. (i) In an analogous notation to Exercises 6.2 q.3, show S1 K = S1 . Use this result to show S1 Kψi = ψi and thus S1 |H = (K|H )−1 . (ii) Verify that the equations of Exercises 6.2 q.3(iii) hold with and D interchanged, and thus S1 = (IH+H − (I − K)KD)−1 K.
269
CORRELATION FUNCTIONS AT β = 1 AND 4
6.5 ALTERNATE CORRELATIONS AT β = 1 We know from the inter-relations of Section 4.2.3, and also from the method of integration over alternate variables (recall Section 6.3.2), that at β = 1 a special role is played by every second eigenvalue in the sequence. With the coordinates ordered according to (4.24) (this is the reverse of the ordering convention used in Section 6.3.2), it is possible to calculate the (n1 , n2 )-point correlation function ρ(n1 ,n2 ) , where n1 coordinates are odd numbered and n2 coordinates are even numbered [456]. We will see that in the classical cases the distribution of the even labeled coordinates can be identified with a β = 4 distribution, as is consistent with the results of Section 4.2.3.
6.5.1 N even As usual in β = 1 theory, the N even and N odd cases must be treated separately. We begin assuming N to be even. Let us write xj = w2j−1 , yj = w2j (j = 1, . . . , N/2) where it is assumed N is even. Let {Rj (w)}j=0,1,... be such that (wk − wj ) = det[Rj−1 (wk )]j,k=1,...,N = det[Rj−1 (xk ) Rj−1 (yk )] j=1,...,N . k=1,...,N/2
1≤j
Then it follows from Proposition 5.9.1 that χx1 >y1 >···>xN/2 >yN/2
N j=1
e−V (wj )
(wj − wk )
1≤j
N/2
=
e−V (xj )−V (yj ) det[Rj−1 (xk ) Rj−1 (yk )]
j=1
j=1,...,N k=1,...,N/2
det[−χxj −yk >0 ]j,k=1,...,N/2 ,
(6.115)
assuming the inequalities (5.173). On the l.h.s. of this identity is the Boltzmann factor for the ordered log-gas at β = 1. The r.h.s. naturally separates the even and odd labeled coordinates, and has the further significant feature of being a symmetric function of {xi } and {yj } separately. The identity (6.115) can be used to compute ρ(n1 ,n2 ) . P ROPOSITION 6.5.1 Let ρ(n1 ,n2 ) (x1 , . . . , xn1 ; y1 , . . . , yn2 ) denote the (n1 , n2 )-point correlation function corresponding to (6.115) for n1 particles with odd numbered w-coordinates and n2 particles with even numbered w-coordinates. With x, x odd numbered w-coordinates and y, y even numbered w-coordinates let 1 hoo (x, x ) = hee (y, y ) = 0, hoe (x, y) = −heo (y, x) = χx−y>0 , ∞ ∞2 o −V (y) e Φk (x) = heo (y, x)Rk (y)e dy, Φk (y) = hoe (x, y)Rk (x)e−V (x) dx, −∞
−∞
and let {Rk (t)} denote the monic skew orthogonal polynomials of Definition 6.3.1, with corresponding normalizations {rj }. Substitute these quantities in Proposition 6.3.2 and denote the corresponding expressions ˜ respectively. Then we have for S1 , D1 and I˜1 by the symbols S, D and I, ρ(k1 ,k2 ) (x1 , . . . , xk1 ; y1 , . . . , yk2 ) [foo (xj , xl )]j,l=1,...,k1 = qdet [feo (yj , xl )] j=1,...,k2 l=1,...,k1
[foe (xj , yl )] j=1,...,k1 l=1,...,k2
[fee (yj , yl )]j,l=1,...,k2
,
(6.116)
270
CHAPTER 6
where with p, p ∈ {e, o}
fpp (w, z) =
Spp (w, z) I˜pp (w, z) Dpp (w, z) Spp (z, w)
.
Proof. This result can be deduced from Proposition 6.3.2. It is necessary first to replace each integration R
(6.117) R
dw over the appropriate w-variables in (6.115) by a Riemann sum approximation dμ(w). In the latter the lattice points which make up the domain of integration are chosen to be the set X if w is odd numbered (an x variable), and the set Y if w is even numbered (a y variable). These lattice points must interlace so that for coordinates xj restricted to X and yj restricted to Y (j = 1, . . . , N/2) it’s always possible to have xj > yj . Now consider N Y
e−V (wj ) det[Rj−1 (wk )]j,k=1,...,N Pf[−h(wj , wk )]j,k=1,...,N .
(6.118)
j=1
It follows from the definition of h(w, w ) = h·,· that (6.118) vanishes unless exactly half the wk variables are on X and the other half are on Y . In the latter case (6.118) equals 2−N/2 times (6.115). The expression (6.118) has a structure identical to the l.h.s. of (6.65), although the variables are confined to lattice points. We can therefore use Propositions 6.3.2 and 6.3.3 to write down the (n1 , n2 )-point correlation function, where n1 coordinates are on X (odd numbered) and n2 coordinates on Y (even numbered). The polynomials Rk (z) must be skew orthogonal with respect to the inner product Z Z dμ(w) dμ(z) e−V (w)−V (z) f (w)g(z)h(w, z) f, g = X∪Y X∪Y Z Z “ ” = dμ(x) dμ(y) e−V (x)−V (y) f (x)g(y)χx−y>0 − f (y)g(x)χy−x>0 , X
Y
which in the continuum limit is identical to (6.61).
Propositions 6.5.1 and 6.3.2 together give x x e−V (x ) R2k+1 (x ) e−V (t) R2k+1 (t) dt , e−V (t) R2k (t) dt − R2k (x ) 2rk −∞ −∞ k=0 x ∂ 1 I˜oo (x, x ) = − Soo (x, t) dt, Doo (x, x ) = 2 Soo (x, x ), ∂x 2 x ∞ ∞ N/2−1 −V (y ) e −V (t) See (y, y ) = − R2k+1 (y ) e R2k (t) dt − R2k (y ) e−V (t) R2k+1 (t) dt , 2rk y y k=0 y ∂ 1 I˜ee (y, y ) = − See (y, t) dt. (6.119) Dee (y, y ) = 2 See (y, y ), ∂y 2 y N/2−1
Soo (x, x ) =
Next we will show that in the classical cases See (y, y ) is closely related to S4 (y, y ). First we note, by comparing the definition of See in (6.119) and the definition of S1 in Proposition 6.3.2, with h, Φ therein given by (6.67), (6.69), that in general 1 See (y, y ) = S1 (y, y ) − S1 (∞, y ) (6.120) 2 ˜
(in the Jacobi case S1 (∞, y ) means S1 (1, y )). With e−V → e−V1 (recall (6.100)), let the corresponding summations in (6.120) be denoted See → S˜ee , S1 → S˜1 . Now S˜1 is summed in all the classical cases by (6.107). Furthermore, taking x → ∞ in (6.107) we see ∞ ˜1 (y ) ˜ −V ˜ S1 (∞, y ) = γN −2 e pN −1 (y ) pN −2 (t)e−V1 (t) dt. −∞
271
CORRELATION FUNCTIONS AT β = 1 AND 4
Thus we have
˜
˜
2S˜ee (y, y ) = e−(V (y)−V1 (y)) eV (y )−V1 (y ) KN −1 (y, y ) ∞ ˜ ˜ pN −2 (t)e−V1 (t) dt. −γN −2 e−V1 (y ) pN −1 (y ) We remark that, after replacing (y, y ) by (x, x ) and
∞
(6.121)
y
x
, the same formula applies for S˜oo (x, x ). The expression (6.121) has the same structure as the summation formula (6.56) for S˜4 (x, y) (recall here that the tilde denotes the modified weight functions (6.53)) in the classical cases. Noting that in general ˜ ˜ e−2V (t)+V4 (t) = e−V1 (t) we thus observe = S˜ee (y, y ) . (6.122) S˜4 (y, y ) N →N/2
y
by −
−∞
N →N +1
We will see in the next subsection that the r.h.s. of (6.122) does indeed correspond to S˜ee (y, y ) for N odd, or equivalently that (6.121) remains valid for N odd. Recalling the analogous structure of (6.14) and the formula (6.116) for ρ(0,n) (which too remains valid for N odd), we therefore have the following result [220]. P ROPOSITION 6.5.2 Let OEN (e−V (x) ), SEN (e−4V (x) ) and the operation even be as in Section 4.2.3. We have ˜
˜
ρ(n) (y1 , . . . , yn ; SEN/2 (e−2V4 (x) )) = ρ˜(n) (y1 , y2 , . . . , yn ; even(OEN +1 (e−V1 (x) ))).
(6.123)
The identity (6.123) with n = N/2 says that integrating the p.d.f. at β = 1, with weight function (6.100) and N + 1 particles, over every odd numbered coordinate gives the p.d.f. at β = 4 with weight function (6.53). Equivalently, in terms of the distributions, ˜
˜
even(OEN +1 (e−V1 (x) )) = SEN/2 (e−2V4 (x) ).
(6.124)
In the case N → 2n, this summarizes in one formula the results of Section 4.2.3 relating to eigenvalue p.d.f.’s on the real line. In fact a simple direct derivation of (6.124) based on (5.64) can be given (see Exercises 6.5 q.1). The result (6.124) raises the question as to the relationship of ρ(0,n) to a β = 4 correlation in the cases when the weight function is not classical. It was proved in [220] that the only continuous weight functions for which (6.124) holds are the classical weight functions (or weight functions related to the classical weight functions by a fractional linear transformation). Another question of interest is the relationship of the distribution of every even labeled coordinate for OEN , N even, to a β = 4 distribution. The following result was proved in [220]. P ROPOSITION 6.5.3 Let N be even. We have even(OEN (f )) = SEN/2 ((g/f )2 ) for
(f, g) =
x > 0, (e−x/2 , e−x ), ((1 − x)(a−1)/2 , (1 − x)a ), 0 < x < 1,
and up to linear fractional transformations these pairs of weights are unique. It remains to consider the N odd case, and to verify that indeed (6.121) still holds true.
(6.125)
(6.126)
272
CHAPTER 6
6.5.2 N odd The computation of ρ(n1 ,n2 ) for N odd yields to a similar strategy as does the N even case. First one notes that in the limit yn → −∞, (5.174) reads det [χxj >yk ] j=1,...,N [1]j=1,...,N = χx1 >y1 >···>yn−1 >xn , k=1,...,N −1
valid for x1 > · · · > xn , Furthermore, with the latter ordering det [χxj >yk ] j=1,...,N [1]j=1,...,N k=1,...,N −1 ⎡ 0 χx1 >y1 0 ⎢ −χx1 >y1 0 −χ x2 >y1 ⎢ ⎢ 0 χx2 >y1 0 = Pf ⎢ ⎢ −χx1 >y2 0 −χ x2 >y2 ⎣ .. .
χx1 >y2 0 χx2 >y2 0
y1 > · · · > yn−1 .
0
−χx3 >y1 0 −χx3 >y2
··· ··· ··· ···
χx1 >yN −1 0 χx2 >yN −1 0
0
−χxN >y1 0 −χxN >y2
1 1 1 1 .. .
⎤ ⎥ ⎥ ⎥ ⎥, ⎥ ⎦
and this extends to the setting (6.118), thereby allowing Proposition 6.3.7 to be used in an analogous way to the use made of Proposition 6.3.2 in the derivation of Proposition 6.5.1. P ROPOSITION 6.5.4 Let x, x be odd numbered w-coordinates and y, y be even numbered w-coordinates as in Proposition 6.5.1. Let 1 1 hoo (x, x ) = hee (y, y ) = 0, hoe (x, y) = −heo (y, x) = χx−y>0 , F (x) = F (y) = , 2 ∞ 2∞ 1 1 o −V (y) e −V (x) ˆ k (x) = ˆ k (y)e ˆ k (y) = ˆ k (x)e Φ h(y, x)R dy, Φ h(x, y)R dx, 2 −∞ 2 −∞ ˆ k (t)} be specified in terms of the monic skew orthogonal polynomials of Definition 6.3.1 as in Propolet {R sition 6.3.8, and similarly specify {ˆ rj }. Substitute these quantities in Proposition 6.3.7 and denote the corresponding expressions for S1odd , D1odd and I˜1odd by the symbols S odd , Dodd and I˜odd , respectively. Then the formulas (6.116) and (6.117) again apply, where in the latter the superscript “odd” is to be attached to the matrix elements. Comparing the above specification of S odd with the formula for S1odd in Proposition 6.3.7 we see that 1 odd odd S1 (y, y ) − S1odd (∞, y ) . (y, y ) = See 2 For the modified weight functions (6.100), indicated by the use of a tilde, we have the summation formula (6.107) of S˜1 . Substituting this in the above formula gives ˜ e−V1 (y ) odd (y, y ) = 2S˜ee (y, y ) + γN −3 s˜N −3 2S˜ee s˜N −1 N →N −1 ∞ ∞ ˜ ˜ × pN −2 (y ) e−V1 (t) pN −1 (t) dt − pN −1 (y ) e−V1 (t) pN −2 (t) dt , (6.127) y
y
where S˜ee (y, y ) is specified by (6.121). The formula (6.127) can be further simplified. The first step is ˆ N −3 (x)/Φ ˆ N −1 (x) = to note from (6.103), (6.105) and (6.98) that taking the limit x → ∞ in the ratio Φ
273
CORRELATION FUNCTIONS AT β = 1 AND 4
φ˜N −3 (x)/φ˜N −1 (x) implies γN −3
s˜N −3 = γN −2 . s˜N −1
(6.128)
Substituting this in (6.127) and recalling (6.121) we see that odd (y, y ) = S˜ee (y, y ), S˜ee
(6.129)
provided ˜ γN −3 φ˜N −3 (x) − γN −2 φ˜N −1 (x) = e−2V (x)+V1 (x)
pN −2 (x) (pN −2 , pN −2 )2
(6.130)
for N odd. This latter identity is verified by checking that both sides agree for x → ∞ and that it reduces to (6.131) below upon differentiation. E XERCISES 6.5
1.
(i) Note from (5.60) and (5.64) that ” ˜ d “ −V˜4 (x) cl cl−1 pl (x) = − pl+1 (x) + pl−1 (x) eV1 (x) e dx (pl+1 , pl+1 )2 (pl−1 , pl−1 )2
(6.131)
for l = 0, 1, . . . , where c−1 := 0. (ii) For N odd let I(x2 , x4 , . . . , xN−1 ) :=
1“ C
(N+1)/2 Z x Y 2l l=1
x2l−2
dx2l−1
N ”Y j=1
˜
e−V1 (xj )
Y
|xk − xj |,
1≤j
where x0 := −∞, xN+1 := ∞. With x1 < x2 < · · · < xN use the Vandermonde formula (1.173) to write N Y
|xk − xj |
j
» – h i (pl , pl )2 ” ck−1 ck−2 det [p0 (xj )]j=1,...,N − pk (xj ) + pk−2 (xj ) j=1,...,N cl−1 (pk , pk )2 (pk−2 , pk−2 )2 k=2,...,N l=1 – » N−1 ”i “ ” h “ Y ˜ ˜ d (pl , pl )2 = − det [p0 (xj )]j=1,...,N eV1 (xj ) e−V4 (xj ) pk−1 (xj ) j=1,...,N , c dx l−1 k=2,...,N l=1
=
N−1 Y “
−
where the second equality follows from (i). Substitute this in the integrand, then use the method of integration over alternate variables to show I(x2 , x4 , . . . , xN−1 ) N−1 1 Y “ (pl , pl )2 ” − = C cl−1 l=1 # " # 2 " Rx ˜ ˜ 2j −V e−V4 (x2j ) pk−1 (x2j ) e 1 (x) p0 (x) dx −∞ ˜ 6 ˜ d e−V4 (x2j ) pk−1 (x2j ) j=1,...,(N −1)/2 e−V1 (x2j ) p0 (x2j ) × det 6 dx2j j=1,...,(N−1)/2 4 k=2,...,N R ∞ −V˜ (x) 1 e p (x) dx [0] 0 k=2,...,N −∞ # " Z ˜ N−1 ” e−V4 (x2j ) pk−1 (x2j ) 1 Y “ (pl , pl )2 ”“ ∞ −V˜1 (x) = e p0 (x) dx det . − ˜ d C cl−1 e−V4 (x2j ) pk−1 (x2j ) j=1,...,(N −1)/2 −∞ dx2j l=1
k=2,...,N
3 7 7 5
274
CHAPTER 6
(iii) Deduce from the final result in (ii) that Y
Y
(N−1)/2
(N−1)/2
I(x2 , x4 , . . . , xN−1 ) ∝
˜
e−2V4 (x2l )
l=1
(x2k − x2j )4 ,
j
which with N → N + 1 (and N now even) is the result (6.124). 2. [397] In this exercise it will be shown that integrating the distribution COE2N over alternate angles θ1 , θ3 , . . . , θ2N−1 in the region (j = 0, . . . , N − 1), (6.132) θ2j < θ2j+1 < θ2j+2 where θ0 := θ2N − 2π gives the distribution CSEN , and thus rederiving (4.32).
(i) Use the identity
|eiθk − eiθj | = i−1 e−i(θk +θj )/2 (eiθk − eiθj ),
θk > θj
to show that the p.d.f. COE2N is proportional to det[eipθj ]
j=1,...,2N p=−(N −1/2),...,N −1/2
.
(6.133)
(ii) Integrate the determinant in (i) over the region (6.132) using the method of integration over alternate variables to obtain – » (ip)−1 (eipθ2j + eipθ2N ) . det eipθ2j j=1,...,N p=−(N −1/2),...,N −1/2
−1
Now subtract (ip) times the final row from each of the odd numbered rows, then take out a factor (ip)−1 from each column to show that the determinant is proportional to » ipθ – Y e 2j det = |eiθ2k − eiθ2j |4 , ipθ2j pe j=1,...,N p=−(N −1)/2,...,(N −1)/2
1≤j
where the final equality can be deduced by taking a confluent limit in (6.133) (recall Exercises 1.9 q.2).
6.6 SUPERIMPOSED β = 1 SYSTEMS The superimposed ensemble is formed out of a system of N1 particles distributed according to (6.71) with N → N1 , and a distinct set of N2 particles distributed accordingto (6.71) with N → N2 . A p.d.f. of 2 N1 +N2 coordinates x1 , . . . , xN1 +N2 is formed by summing over the N1N+N distinct ways of choosing N1 1
coordinates out of {x1 , . . . , xN1 +N2 } to be distributed according to (6.71) with N → N1 , and the remaining N2 coordinates to be distributed according to (6.71) with N → N2 . Symbolically, one writes OEN1 (e−V (x) )∪ OEN2 (e−V (x) ) for the resulting ensemble. In the cases (N1 , N2 ) = (N, N ) or (N, N + 1) the p.d.f. has a special structure in which the dependence on the even and odd numbered coordinates factorizes [278], [220]. P ROPOSITION 6.6.1 Let S = {s1 , s2 , . . . , sl }, s1 > s2 > · · · > sl ≥ 1 denote a set of positive integers and write Δ(xS ) := (xsj − xsk ). 1≤j
275
CORRELATION FUNCTIONS AT β = 1 AND 4
Then we have
Δ(xS )Δ(x{1,...,2N }−S ) = 2N Δ(x{1,3,...,2N −1} )Δ(x{2,4,...,2N } ),
(6.134)
Δ(xS )Δ(x{1,...,2N +1}−S ) = 2N Δ(x{1,3,...,2N +1} )Δ(x{2,4,...,2N } ).
(6.135)
S⊂{1,...,2N } |S|=N
S⊂{1,...,2N +1} |S|=N
Proof. Consider the effect of exchanging xi and xi+2 in a term of either equation. Since Δ is alternating, we see immediately that for i, i + 2 ∈ S or i, i + 2 ∈ /S Δ(xS )Δ(x{1,...,l}−S ) → −Δ(xS )Δ(x{1,...,l}−S ). For other values of i, i + 2 we see that every factor xk − xj , k > j, is taken to another such factor, except for the factor xi+1 − xi or xi+2 − xi+1 , whichever is present (exactly one of these factors must be present due to the assumption on i, i + 2) which changes sign. This means that in fact each term in the sum is taken to the negative of a term in the sum. It follows from this that the sum is alternating under parity-preserving permutations; the degrees of each side imply the sum must therefore be proportional to the given r.h.s. The proportionality constant can be calculated by computing the coefficient of an appropriate monomial in the sum.
According to the result (6.135), the distribution which results from superimposing a system with Boltzmann factor (6.71) with another system with Boltzmann factor (6.71) but with N → N + 1 is proportional to P2N +1 e− j=1 V (xj ) (x2j−1 − x2k−1 ) (x2j − x2k ), 1≤j
1≤j
where x1 > x2 > · · · > x2N +1 . The factorized form of this distribution allows the odd numbered coordinates (for example) to be integrated out, thus giving the distribution of the even numbered coordinates in the superimposed ensemble. Doing this, by making use of the Vandermonde formula and integrating the resulting determinant row by row, we see that the corresponding distribution is proportional to ∞ PN e− j=1 V (x2j ) det e−V (t) tk−1 dt (x2j − x2k ) (6.136) j,k=1,...,N +1
x2j
1≤j
˜
˜
where x2N +2 := −∞. By making the replacements e−V (x) → e−V1 (x) in this expression, and setting e−V1 (x) equal to one of the classical forms (6.100) the first determinant can be considerably simplified. P ROPOSITION 6.6.2 The distribution (6.136), with weight function one of the modified classical forms, is proportional to PN (x2k − x2j )2 , (6.137) e−2 j=1 V (x2j ) 1≤j
where V and V˜1 are related by (6.92). Equivalently, in the classical cases and with the coordinates ordered x1 > x2 > · · · > x2N +1 ˜ ˜ even OEN (e−V1 (x) ) ∪ OEN +1 (e−V1 (x) ) = UEN (e−2V (x) ). (6.138) Proof. With {pj }j=0,1,... denoting the monic orthogonal polynomials corresponding to the weight e−2V (x) , we first add together appropriate linear combinations of columns and thus obtain the identity i hZ ∞ ˜ e−V1 (t) tk−1 dt det x2j
hZ
j,k=1,...,N+1
∞
= det x2j
˜
e−V1 (t) dt
Z
∞ x2j
” i “ ˜ e−V1 (t) pl (t) + μl−2 pl−2 (t) dt j=1,...,N +1 l=1,...,N
(6.139)
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CHAPTER 6
where p−1 := 0. Now, for appropriate μl−2 , the formulas (6.98) and (6.99) together imply Z ∞ ” “ ˜ ˜ e−V1 (t) pl (t) + μl−2 pl−2 (t) dt = αl e−V4 (x2j ) pl−1 (x2j )
(6.140)
x2j
for some αl . Recalling x2N+2 := −∞ we see that the only nonzero term in the final row of the determinant on the r.h.s. of (6.139) is in the first column. Expanding by the first column we thus have that (6.139) is proportional to N Y
h i ˜ e−V4 (x2j ) det pl−1 (x2j )
j,l=1,...,N
j=1
=
N Y j=1
˜
e−V4 (x2j )
Y
(x2k − x2j ),
1≤j
which implies the stated formula.
If we write (6.138) as even OEN (f ) ∪ OEN +1 (f ) = UEN (g) then (6.124) reads
˜
(f, g) = (e−V1 (x) , e−2V (x) ),
even OEN +1 (f ) = SEN/2 ((g/f )2 )
(6.141)
(6.142)
valid for N even. In fact it is proved in [220] that the first equation in (6.141) and the equation (6.142) are equivalent. Similarly, it is proved that equation (6.125) is equivalent to (6.143) even OEN (f ) ∪ OEN (f ) = UEN (g) and thus holds when (f, g) are specified by (6.126). We will see in the next chapter that the inter-relationships (6.123) and (6.138), and their circular ensemble counterparts (4.32), and (6.154) below, in turn imply inter-relationships between spacing probabilities for the couplings β = 1, 2 and 4. 6.6.1 Distribution of the odd numbered coordinates In addition to the distribution on the l.h.s. of (6.138), it is also of interest to consider the distribution ˜ ˜ odd OEN (e−V1 (x) ) ∪ OEN +1 (e−V1 (x) ) (6.144) in the classical cases. In this distribution the particle closest to the edges of the support of the density is not integrated out. With the ordering of Proposition 6.6.2 this feature is also true of ˜ ˜ (6.145) odd OEN (e−V1 (x) ) ∪ OEN (e−V1 (x) ) for the particle at the right edge and
˜ ˜ even OEN (e−V1 (x) ) ∪ OEN (e−V1 (x) )
(6.146)
for the particle at the left edge. Thus for purposes of computing the scaled limit of (6.144) at the right hand soft edge of the Gaussian ensemble it suffices to consider (6.145), while to compute the scaled limit of (6.144) at the hard edge of the Laguerre ensemble it suffices to consider (6.146). It turns out that the n-point correlation of (6.145) and (6.146) is simpler than that of (6.144). ˜ ˜ According to (6.134) the distribution OEN (e−V1 (x) ) ∪ OEN (e−V1 (x) ) is proportional to P2N ˜ (x2j−1 − x2k−1 ) (x2j − x2k ), (6.147) e− j=1 V1 (xj ) 1≤j
1≤j
277
CORRELATION FUNCTIONS AT β = 1 AND 4
where x1 > x2 > · · · > x2N . Consider now the distribution (6.145), obtained by integrating out the even numbered coordinates in (6.147). Analogous to (6.136) we see that the corresponding distribution is proportional to x2j−1 PN ˜ ˜ e− j=1 V1 (x2j−1 ) det e−V1 (t) tk−1 dt (x2j−1 − x2k−1 ). (6.148) j,k=1,...,N
−∞
1≤j
With V˜1 (x) one of the classical forms, the procedure of the proof of Proposition 6.6.2 shows that the determinant in (6.148) is in fact proportional to x2j−1 PN ˜ ˜ ˜ e−V1 (x) dx e− j=1 V4 (x2j−1 ) det [pk−1 (x2j−1 )] j=1,...,N eV4 (x2j−1 ) k=1,...,N −1 j=1,...,N −∞ PN ˜ (6.149) = e− j=1 V4 (x2j−1 ) det [pk−1 (x2j−1 )] j=1,...,N [FN −1 (x2j−1 )]j=1,...,N k=1,...,N −1
where ∞ (pl , f )2 pl (x), FN −1 (x) = (pl , pl )2
f (x) := e
˜4 (x) V
l=N −1
Now write
(x2k−1 − x2j−1 ) = det pk−1 (x2j−1 )
1≤j
x −∞
˜
e−V1 (t) dt.
(6.150)
j,k=1,...,N
and substitute this together with (6.149) in (6.148) to obtain a p.d.f. of the form (5.139). Noting that ∞ e−2V (x) pj (x)FN −1 (x) dx = 0 (j = 0, . . . , N − 2) −∞
we see that furthermore the columns of the two determinants have the biorthogonality property (5.144) (up to normalization), and the correlation kernel is therefore given by (5.145). P ROPOSITION 6.6.3 For the distribution (6.145) the n-point correlation is given by N n −2 pl (xj )pl (xk ) pN −1 (xj )FN −1 (xk ) odd(OEN )2 −2V (xj ) ρ(n) (x1 , . . . , xn ) = e det + . (pl , pl )2 (pN −1 , FN −1 )2 j,k=1,...,n j=1 l=0
(6.151) The formulas (6.103), (6.105) and (6.113) allow the coefficients (pl , f )2 in (6.150) to be computed explicitly. P ROPOSITION 6.6.4 In the classical cases l−1 ν ∞ ∞ 1 j=0 (γ2j /γ2j+1 ) 2 l=1 (γ2l−1 /γ2l ) p2ν+1 (x) + 2˜ c0 p2l (x), (6.152) FN −1 (x) = γ0 (p2ν+1 , p2ν+1 )2 (p2l , p2l )2 ν=[(N −1)/2]
l=[N/2]
where c˜0 is defined by (6.113) and the γj by (6.97). Proof. According to (6.113) and (6.111) Z
x
˜ e−V1 (t) dt = φ˜0 (x) + c˜0 .
−∞
Substituting in (6.150) and making use of (6.103), (6.105) and (6.113), (6.114) gives (6.152).
E XERCISES 6.6
1. [148], [278] In this exercise the circular analogue of (6.138) will be derived.
(6.153)
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CHAPTER 6
(i) With S as in Proposition 6.6.1, let ˜ S ) := Δ(θ
Y
sin((θsk − θsj )/2)
1≤j
(cf. the definition of Δ(xS ) in Proposition 6.6.1). By writing xj = eiθj in (6.134) show X ˜ S )Δ(θ ˜ {1,3,...,2N−1} )Δ(θ ˜ {1,...,2N}−S ) = 2N Δ(θ ˜ {2,4,...,2N} ). Δ(θ S⊂{1,...,2N } |S|=N
(ii) With the operation alt defined in (4.32) q.3 show that analogous to (6.138) “ ” alt COEN ∪ COEN = CUEN .
(6.154)
2. In the case a = 0, b = 2, recognize (4.97) as LOEn |a=0 ∪ LOEn |a=0 . On the other hand, with a = 0 note that the definition of A in the derivation of (4.97) gives that {ai } has distribution LUEn |a=0 and thus obtain a realization of (6.143) with (f, g) = (e−x/2 , e−x ).
6.7 A TWO-COMPONENT LOG-GAS WITH CHARGE RATIO 1:2 In this section a generalization of integration techniques introduced in Sections 6.1.2 and 6.3.2 will be used to calculate the free energy and correlation functions for the two-component log-gas with charge ratio 1:2, on a line with periodic boundary conditions, at the special coupling β = 1. In the limit that the concentration of the +2 charges goes to zero, this system reduces to the one-component log-gas at β = 1, while in the limit of zero concentration of the +1 charges, the one-component log-gas at β = 4 is reclaimed.
6.7.1 The generalized partition function The two-component log-gas to be considered consists of N1 particles of charge q = 1 and N2 particles of charge q = 2, with coordinates x1 , . . . , xN1 and xN1 +1 , . . . , xN1 +N2 , respectively, interacting via the logarithmic potential φ(xi , xj ) = −qi qj log (e2πixi /L − e2πixj /L )(L/2π) , (6.155) on a circle of radius L/2π. The coordinates xi and xj are scaled angles, and the constant L/2π is included in (6.155) so that φ(xi , xj ) ∼ − log |xi − xj | for |xi − xj | → 0. The circle carries a neutralizing background charge density −ρb , where ρb := (N1 + 2N2 )/L.
(6.156)
An equivalent interpretation of (6.155) is as the potential resulting from solving the two-dimensional Poisson equation in periodic boundary conditions (recall Section 2.7). The total energy of the system is calculated in a way analogous to that of the one-component system on a circle, detailed in the proof of Proposition 1.4.1. One finds U = (2N2 + N1 /2) log(L/2π) − log |e2πixj /L − e2πixk /L |qj qk . (6.157) 1≤j
279
CORRELATION FUNCTIONS AT β = 1 AND 4
Thus the generalized partition function for this system at β = 1 is given by L L N1 N1 +N2 C ZN1 ,N2 [u, v] := dx1 · · · dxN1 +N2 u(xl ) v(xl ) N1 !N2 ! 0 0 l=1 l=N1 +1 |e2πixj /L − e2πixk /L |qj qk , ×
(6.158)
1≤j
where C = (2π/L)(2N2 +N1 /2) . Using a special case of the confluent alternant determinant in Exercises 1.9 q.2, together with some integration techniques from Sections 6.1.2 and 6.3.2, an expression for ZN1 ,N2 suitable for computing the partition function and correlation functions can be obtained.
P ROPOSITION 6.7.1 For N1 even we have ZN1 ,N2 [u, v] = C[ζ N1 /2 ] Pf[β−˜jk˜ − iζα−˜jk˜ ]j˜,k=−(N ˜ 1 /2+N2 −1/2),...,N1 /2+N2 −1/2 1/2 = C[ζ N1 /2 ] ± det[βj˜k˜ − iζαj˜k ]j˜,k=−(N , ˜ 1 /2+N2 −1/2),...,N1 /2+N2 −1/2 (6.159) where [ζ N1 /2 ]f denotes the coefficient of ζ N1 /2 in the power series expansion of f , the tildes on j˜, k˜ indicate they are half odd integers, the sign in the second equality is chosen so that when u = v = 1 the determinant is positive and L L ˜ αj˜,k˜ := dx u(x) dy u(y)sgn(x − y)e2πikx/L e−2πi˜jy/L , 0
βj˜,k˜ := (k˜ + j˜)
0
L
˜
v(x)e2πi(k−˜j)x/L dx.
(6.160)
0
Proof. Recalling (2.21), it follows that for 0 ≤ x1 < x2 · · · < xN1 ≤ L, Y
|e2πixj /L − e2πixk /L |qj qk = (−i)N1 /2 Y
×
e−πixl (N1 +2N2 −1)/L
l=1
1≤j
N1 Y
Y
e−2πixl (N1 +2N2 −2)/L
l=N1 +1
(e2πixk /L − e2πixj /L )qj qk ,
1≤j
while according to Exercises 1.9 q.2 we have 2
Y
(e2πixj /L − e2πixk /L )qj qk
1≤j
3 [e2πixj (k−1)/L ] j=1,...,N1 k=1,...,N +2N 1 – 2 6 » 2πix (k−1)/L 7 j 7. = det 6 e 4 5 2πixj (k−2)/L (k − 1)e j=N1 +1,...,N1 +N2 k=1,...,N1 +2N2
Substituting these formulas in (6.158), applying the method of integration over alternate variables (recall Section 6.3.2) to integrate over x1 , x3 , . . . , xN1 −1 , and expanding the resulting determinant to integrate over all other variables gives ZN1 ,N2 [u, v] = C(−i)N1 /2
1 (N1 /2)!N2 !
X P ∈SN1 +2N2
Y
N1 /2
ε(P )
l=1
Y
N1 /2+N2
aP (2l−1),P (2l)
l=N1 /2+1
bP (2l−1),P (2l) ,
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CHAPTER 6
where Z aj,k :=
Z
L
dx u(x) 0
x
dy u(y)e2πiy(j−N1 /2−N2 −1/2)/L e2πix(k−N1 /2−N2 −1/2)/L ,
0
Z
L
bj,k := (k − 1)
v(x)e2πix(j+k−N1 −2N1 −1)/L dx.
0
˜ ˜ ∈ {−(N1 /2 + N2 − 1/2), . . . , (N1 /2 + N2 − 1/2)} (a set of half Q(l) Now put P (l) → N1 /2 + N2 + 1/2 + Q(l), ˜ ˜ odd integers). Making the restriction Q(2l) > Q(2l − 1), as in the proof of Proposition 6.1.8 we then have ZN1 ,N2 [u, v] = C(−i)N1 /2
1 (N1 /2)!N2 !
X
Y
˜ ˜ Q(2l)> Q(2l−1)
Y
N1 /2+N2
N1 /2
˜ ε(Q)
α−Q(2l−1), ˜ ˜ Q(2l)
l=1
β−Q(2l−1), , ˜ ˜ Q(2l)
l=N1 /2+1
where α and β are defined as in the statement of the proposition. The first formula in (6.159) now results as a consequence of the definition (6.10) of a Pfaffian, while the second equality follows from (6.12) and interchanging rows in the matrix.
With u = v = 1 we see from (6.160) that αj˜,k˜ = −
L2 δ ˜, j˜πi j˜,k
βj˜,k˜ = 2L˜ jδj˜,k˜ ,
(6.161)
and thus ZN1 ,N2
L N1 /2 (N1 /2 + N2 )! N1 /2 [ζ =C (8L)N1 /2+N2 ] 2π (N1 + 2N2 )!
N1 /2+N2
l=1
1 2 ζ+ l− . 2
(6.162)
6.7.2 Evaluation of the correlation functions For the one-component log-gas the general n-particle correlation function has been denoted ρ(n) (x1 , . . . , xn ). For the present two-component log-gas we will denote the correlation function formed by specifying the coordinates of n1 particles of charge +1 and n2 particles of charge +2 by ρ+1n1 ,+2n2 . Analogous to (5.20) these correlations can be calculated from the generalized partition function ZN1 ,N2 [u, v] according to the formula ρ+1n1 ,+2n2 (x1 , . . . , xn1 ; y1 , . . . , yn2 ) δ n1 +n2 1 ZN1 ,N2 [u, v] = . ZN1 ,N2 δu(x1 ) · · · δu(xn1 )δv(y1 ) · · · δv(yn2 ) u=v=1
(6.163)
Our ability to compute (6.163) relies on a formula for ZN1 ,N2 [u, v] analogous to (6.27) and (6.76). P ROPOSITION 6.7.2 Let
c˜l := (β˜l,˜l − iζα˜l,˜l )
u=v=1
Lζ = 2L ˜l + , 2π˜l
and put u(x) = 1 + f (x), Also let ej˜ denote the integral operator with kernel e kernel 12 sgn (x − y). Then we have
v(x) = 1 + g(x). 2πi˜ jy/L
(6.164)
, and let be the integral operator on [0, L] with
1/2 ZN1 ,N2 [1 + f, 1 + g] = [ζ N1 /2 ]Z˜N1 ,N2 det X ,
(6.165)
281
CORRELATION FUNCTIONS AT β = 1 AND 4
where Z˜N1 ,N2 is given by (6.162) but with the operation [ζ N1 /2 ] therein removed, and X is the integral operator with 4 × 4 matrix kernel 2π δj,k + bj ˜l a ˆ˜lk − δj,2 δk,1 f , L j,k=1,...,4 ˜ l
ζL f e−˜j πcj˜j˜ ⎡ ⎤ ek˜ ⎢ 1˜ ek˜ ⎥ ⎥ ik [bj k˜ ]4×(N1 +2N2 ) := ⎢ ⎣ e˜ ⎦ k ˜ ˜ ike
−
[ˆ aj˜k ](N1 +2N2 )×4 :=
k
ζL f e−˜j iπcj˜
j˜ ge−˜j cj˜
1 ge−˜j , icj˜ (N1 +2N2 )×4
.
(6.166)
4×(N1 +2N2 )
Proof. As in Propositions 6.3.6 and 6.1.9 the formula relies on the identity (5.26). The first step is to substitute (6.164) in (6.160) and simplify to obtain „ Z ˜ + j˜) Z L ˜ ˜ (k ζ“L L” L g(x)e2πi(k−˜j)x/L dx + f (x)e2πi(k−˜j)x/L dx βj˜,k˜ − iζαj˜,k˜ = cj˜ δj˜,k˜ + + ˜ cj˜ c π˜ j j˜ πk 0 0 « Z ˜ 2iζ L 2πikx/L −2πiy˜ j/L − f (x)e (f e ) dx . (6.167) cj˜ 0 Factoring out cj˜, we see the determinant of (6.167) can be written in the form det(1 + AB), where A is the integral operator with (N1 + 2N2 ) × 4 matrix kernel aj˜k ] − [ [aj˜k ] = [ˆ
2iζ f (f e−˜j ) 0 0 cj˜
0],
with a ˆj˜k specified by the first equation in (6.166), and B is the 4 × (N1 + 2N2 ) matrix multiplication operator specified by the second equation in (6.166). On the other hand, BA is the integral operator with 4 × 4 matrix kernel i hX bj ˜l a˜lk j,k=1,...,4
˜ l
and furthermore, analogous to (6.80), we have the factorization h i X bj ˜l a˜lk δj,k + ˜ l
j,k=1,...,4
2 h i X 6 2π 6 = δj,k + bj ˜l a ˆ˜lk − δj,2 δk,1 f L j,k=1,...,4 4 ˜ l
1 2π f L
0 0
0 1 0 0
0 0 1 0
3 0 0 7 7. 0 5 1
Since the determinant of the final matrix is unity, we deduce (6.165) upon using the identity (5.26).
Using (6.32) the functional derivatives required by (6.163) can be computed. First, for consistency with (6.27) and (6.76) we take the transpose of the matrix X in (6.165), and also introduce the notation N1 /2+N2 −1/2
Sj (x) =
˜ l=−(N1 /2+N2 −1/2)
˜
e2πixl/L . c˜˜lj l
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CHAPTER 6
Then, making use of (6.6), we find δ n1 +n2 ZN1 ,N2 [u, v] δu(x1 ) · · · δu(xn1 )δv(y1 ) · · · δv(yn2 ) u=v=1 N2 +N [a(xj − xk )]j,k=1,...,n1 [b(xj − yk )] j=1,...,n1 1 /2 k=1,...,n2 = [ζ N1 /2 ] cl qdet [c(yj − xk )] j=1,...,n2 [d(yj − yk )]j,k=1,...,n2 l=1
where ζL a(x) = π
S1 (x) iS0 (x)
c(x) =
,
(6.168)
k=1,...,n1
2
− Lπ2 ζ sgn(x) − iS2 (x) , S1 (x) ζL S−1 (x) −iS0 (x) , iS0 (x) S1 (x) π
b(x) = d(x) =
ζL π
−iS0 (x) S−1 (x) −iS−2 (x) . S−1 (x)
S1 (x) iS0 (x)
S−1 (x) iS0 (x)
This, together with (6.162), explicitly determines ρ+1n1 ,+2n2 . E XERCISES 6.7
1. In the case N2 = 1, use the Morris integral evaluation (4.4) to evaluate ZN1 ,N2 [1, 1] as specified by (6.158), and verify that this agrees with the value given by (6.162).
Chapter Seven Scaled limits at β = 1, 2 and 4 In the previous two chapters the general n-point correlations for log-gas systems at β = 1, 2 and 4 have been evaluated as determinants or quaternion determinants with entries given in terms of certain correlation kernels. In the classical cases the explicit form of the latter can readily be analyzed in certain scaling limits. Our main technique is to make use of known asymptotic expansions of the classical polynomials. The bulk, soft edge and hard edge scaling limits all refer to moving the origin to the respective portions of the support of the density, choosing the length scale to be of the order of the mean inter-particle spacing, then taking N → ∞. Knowledge of the scaled form of the bulk two-, three- and four-point correlations is used to obtain the first order expansion in β − β0 , β0 = 1, 2 and 4 for the two-point correlations. The density itself is also analyzed in terms of a global coordinate, which maps the support of the density to a finite interval, and this allows corrections to the Wigner and Mar˘cenko-Pastur laws to be determined. In the case of the GUE minor process, when the minors differ by O(N 2/3 ) in size, a soft edge correlation which occurs in the Dyson Brownian motion model of the GUE is obtained. We also compute the free energy and bulk correlations for the log-gas system with charge ratio 1:2. This requires the use of a particular local limit theorem relating to the coefficients of a polynomial given in factorized form. Asymptotic expansions of the various two-point scaled correlations are determined, revealing some simple expressions which are later explained from a macroscopic viewpoint in Chapter 14. The correlations in the bulk at β = 2 have been conjectured to be identical to the correlations between the large Riemann zeros in the theory of prime numbers; an introduction to this topic is given within this chapter.
7.1 SCALED LIMITS AT β = 2 — GAUSSIAN ENSEMBLES 7.1.1 Bulk correlations Proposition 1.4.4 predicts the semicircle √ law for the global density of the Gaussian ensembles. It follows that with the change of scale xj → πρxj / 2N , the bulk density (i.e., density in the neighborhood of the origin) approaches ρ as√N → ∞. With this scale, the corresponding Christoffel-Darboux sum, multiplied by the scale factor πρ/ 2N , tends to a well defined limit as N → ∞. (G)
(G)
(G)
P ROPOSITION 7.1.1 Denote by KN (x, y) (5.6) with pn → pn , w2 → w2 (x). We have πρ πρy sin πρ(x − y) (G) πρx K bulk(x, y) := lim √ KN √ ,√ = N →∞ π(x − y) 2N 2N 2N and thus for the Gaussian ensemble with β = 2 ρbulk (n) (x1 , . . . , xn ) n πρx πρ sin πρ(xj − xk ) πρxn 1 ρ(n) √ , ,..., √ = ρn det := lim √ N →∞ πρ(xj − xk ) j,k=1,...,n 2N 2N 2N where the diagonal elements in the determinant are equal to unity. The corresponding scaled truncated n-
284
CHAPTER 7
particle correlation function is given by ρT(n)bulk (x1 , . . . , xn ) = (−1)n−1 ρn
sin πρ(xi − xi ) sin πρ(xin − xi1 ) 1 2 ··· , πρ(x − x ) πρ(xin − xi1 ) i1 i2 cycles length n
where the sum is over all distinct cycles i1 → i2 → · · · → in → i1 of {1, . . . , n} which are of length n. Proof. The first result follows immediately from the asymptotic formula [508] √ Γ(n/2 + 1) −x2 /2 Hn (x) = cos( 2n + 1x − nπ/2) + O(n−1/2 ), e Γ(n + 1)
(7.1)
together with Stirling’s formula, applied to the r.h.s. of the Christoffel-Darboux formula (5.10) (a simple trigonometric formula is also required). The second result follows from the first and Proposition 5.1.2. The formula for the truncated correlation follows from (5.9).
Letting ρbulk (2) (x, y) now denote the bulk scaled two-point correlation, Proposition 7.1.1 gives 2 sin πρ(x − y) 2 2 (x, y) = ρ − ρ . ρbulk (2) πρ(x − y)
(7.2)
We know that for the Gaussian ensemble with β = 2, ρ(2) (x, y) can be realized by calculating the twopoint correlation function for the eigenvalues of random Hermitian matrices. Of course the limit N → ∞ is not accessible, so we consider instead a finite value (N = 15) and calculate ρ(2) empirically using the middle (8th) eigenvalue as the origin, and scaling the mean spacing between the middle eigenvalue and its two neighbors to unity (i.e., ρ = 1). To compute ρ(2) empirically the quantities p(n; s) (n = 0, . . . , 3) — the p.d.f.’s for the event that there are exactly n eigenvalues a distance s from the middle eigenvalue — are computed from the eigenvalues of 5,000 computer-generated random Hermitian matrices, and added together according to the second formula of (8.18) below. The results of the empirical calculation are compared with the theoretical prediction in Figure 7.1. A straightforward calculation using (7.2) allows the dimensionless Fourier transform 1 ∞ T . S(k) := 1 + ρ (x, 0)eiρxk dx, (7.3) ρ −∞ (2) referred to as the dimensionless structure function, to be calculated as - |k| 2π , |k| < 2π, . S(k) = 1, |k| ≥ 2π.
(7.4)
7.1.2 Perturbation about β = 2 in the bulk Knowledge of the higher order correlation function at β = 2 can be used to compute the first order correction, in powers of (β − 2), to ρ(2) (0, x) at β = 2 in the bulk [207]. Explicitly, introducing the dimensionless correlation g(n) (x1 , . . . , xn ) := ρ(n) (x1 , . . . , xn )/ρn we can use knowledge of g(n) (x1 , . . . , xn ) for n = 2, 3 and 4 at β = β0 to expand g(2) (x1 , x2 ) = g(2) (x1 , x2 ; β) about β = β0 (here β0 = 2) to first order in β − β0 . Beginning with the definition of g(2) (x1 , x2 ; β) in the finite one-component log-gas on a circle, expanding the integrand in the numerator and denominator about β0 to first order, then making use again of the definition of the correlations gives that in
285
SCALED LIMITS AT β = 1, 2 AND 4
g(s) 1 0.8 0.6 0.4 0.2
0.5
1
1.5
2
s
Figure 7.1 Comparison between the theoretical prediction for the two-point correlation function g(s) := ρ(2) (s, 0)/ρ2 for infinite GUE matrices in the bulk, scaled so that the density is unity, and the empirical two-point correlation function for 5,000 computer-generated 15 × 15 matrices from the GUE.
the bulk limit, up to terms O((β − β0 )2 ) [313] g(2) (x1 , x2 ; β) = g(2) (x1 , x2 ) + (β − β0 ) − g(2) (x1 , x2 )Φ(x1 , x2 ) ∞ −2ρ g(2) (x1 , x2 , x3 ) − g(2) (x1 , x2 ) Φ(x1 , x3 ) dx3 −∞ 1 2 ∞ ∞ g(2) (x1 , x2 , x3 , x4 ) − g(2) (x1 , x2 )g(2) (x3 , x4 ) − g(2) (x1 , x2 , x3 ) − ρ 2 −∞ −∞ " (7.5) −g(2) (x1 , x2 , x4 ) + 2g(2) (x1 , x2 ) Φ(x3 , x4 ) dx3 dx4 , where Φ(x1 , x2 ) := − log |x1 − x2 | and the correlations on the right hand side are evaluated at β = β0 . For β0 = 2 in (7.5), substituting for the correlations using Proposition 7.1.1 and simplifying by expanding out the determinants shows that up to terms O((β − 2)2 ) 2 g(2) (x1 , x2 ; β) = 1 − P2 (x1 , x2 ) + (β − 2) − (1 − (P2 (x1 , x2 ))2 )Φ(x1 , x2 ) ∞ −2 − (P2 (x2 , x3 ))2 − (P2 (x1 , x3 ))2 + 2P2 (x1 , x2 )P2 (x2 , x3 )P2 (x3 , x1 ) Φ(x1 , x3 ) dx3 −∞ 1 ∞ ∞ 4P2 (x1 , x3 )P2 (x3 , x4 )P2 (x4 , x1 ) − 4P2 (x1 , x2 )P2 (x2 , x3 )P2 (x3 , x4 )P2 (x4 , x1 ) − 2 −∞ −∞ " 2 2 P2 (x2 , x4 ) Φ(x3 , x4 ) dx3 dx4 −2P2 (x1 , x3 )P2 (x3 , x2 )P2 (x2 , x4 )P2 (x4 , x1 ) + 2 P2 (x1 , x3 ) (7.6) where P2 (x, y) :=
sin πρ(x − y) . πρ(x − y)
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It turns out that these integrals can be computed exactly in terms of elementary functions, together with the sine and cosine integrals ∞ ∞ sin t cos t dt, ci (x) := − dt, si (x) := − t t x x with the final result being sin πρx 2 g(2) (0, x; β) = 1 − πρx 1 sin πρx 2 sin 2πρx + ci (2πρx) +(β − 2) − 2 πρx 2πρx " 1 + (log 2πρ|x| + C) cos 2πρx − ci (2πρx) + O((β − 2)2 ) (7.7) 2(πρx)2 (here C denotes Euler’s constant). Moreover, the corresponding dimensionless structure function is given in terms of elementary functions only, ⎧ " ⎪ |k| |k| 2π+|k| |k| 1 k2 ⎪ + + (β − 2) log 1 − log − ⎨ 2π 2 (2π)2 4π 2π−|k| 4π , |k| < 2π, ˜ β) = " S(k; (7.8) ⎪ |k|+2π |k| (2π)2 π ⎪ , |k| > 2π, − |k| ⎩ 1 + (β − 2) 12 log |k|−2π + 4π log 1 − k2 up to terms O((β − 2)2 ). 7.1.3 Soft edge scaling limit
√ The edges of the spectrum are predicted by Proposition 1.4.4 to occur at ± 2N . Since the eigenvalues are not confined by a wall at these points but have a nonzero √ density on either side, each edge is referred to as a soft edge. For x in the neighborhood of the right edge ( 2N ) we have the large N asymptotic formula [508] (7.9) exp(−x2 /2)HN (x) = π 1/4 2N/2+1/4 (N !)1/2 N −1/12 πAi(t) + O(N −2/3 ) , where x = (2N )1/2 + 2−1/2 N −1/6 t and with Ai(x) denoting the Airy function. The Airy function in turn can be specified by the integral representation 3 dv Ai(x) = e−xv+v /3 , (7.10) 2πi A where the contour starts at e−πi/3 ∞ and finishes at eπi/3 ∞, following the corresponding rays asymptotically, staying in the sector −π/3 < arg z < π/3. The expansion (7.9) suggests that in order to evaluate the √ Christoffel-Darboux sum in the neighborhood of x = 2N we should make the change of variables x = (2N )1/2 +
X , 21/2 N 1/6
y = (2N )1/2 +
Y 21/2 N 1/6
(7.11)
(the factors of 1/21/2 are chosen for later convenience). Using the above asymptotic formula and Stirling’s formula, the Christoffel-Darboux sum is then readily evaluated [189]. P ROPOSITION 7.1.2 We have
X Y 1 (G) 1/2 1/2 (2N ) K + , (2N ) + N →∞ 21/2 N 1/6 N 21/2 N 1/6 21/2 N 1/6 Ai(X)Ai (Y ) − Ai(Y )Ai (X) , = X −Y
K soft (X, Y ) := lim
(7.12)
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SCALED LIMITS AT β = 1, 2 AND 4
(G)
(G)
where KN (x, y) is given by Proposition 5.1.3 with pN (x) = pN (x). We see from the formula of Proposition 5.1.2 that to specify ρ(n) we must also calculate K soft (X, X). This can be deduced from (7.12) by taking the limit Y → X and simplifying using the fact that Ai(x) satisfies the differential equation y (x) = xy(x). This gives K soft (X, X) = −X(Ai(X))2 + (Ai (X))2 .
(7.13)
7.1.4 Soft edge scaling of the perturbed Hermite kernel The perturbed Hermite kernel is given by (5.172), and thus consists of the unperturbed kernel, which has the soft edge scaling (7.12), plus a sum of r correction terms. The latter depend on parameters a1 , . . . , ar , which to give a well-defined soft edge limit must be replaced by the parameters s1 , . . . , sr according to the scaling [442], [132] √ √ ak = − 2N + 2N 1/6 sk . To specify the limiting functional form, we introduce a class of incomplete multiple Airy functions 3 (j) e−xv+v /3 dv ˜ , Ai (x) := j 2πi A{s1 ,...,sj } k=1 (v − sk ) j−1 3 dv Ai(j) (x) := (−1)j . e−xv+v /3 (v + sk ) 2πi A
(7.14)
k=1
Here A{s1 ,...,si } is a contour which starts at e−πi/3 ∞ and finishes at eπi/3 ∞, following the corresponding rays asymptotically, staying in the sector arg z > π/3, arg z < −π/3, and crossing the real axis to the left of {sk }. The contour A is defined as in (7.10). P ROPOSITION 7.1.3 For large N 2 ˜ (j) (x) √ e−x /2 Γ = (−1)N +r+1 x= 2N +X/21/2 N 1/6 2 ex /2 Γ(j) (x)
ak =
x=
√
2N (−1+sk /N 1/3 )
√ 2N +X/21/2 N 1/6 √ 2N (−1+sk /N 1/3 )
ak =
N (j−1)/3 ˜ (j) (x) 1 + O(N −1/3 ) , Ai (2N )(N +j−r−1)/2 (2N )(N +j−r)/2 (j) −1/3 = (−1)N +r+1 Ai (x) 1 + O(N ) . (7.15) N j/3
Proof. Consider the first formula. In the definition (5.171) make the change of variables z → This shows e−x
2
/2 ˜ (j)
Γ
˛ ˛ (x)˛
x=
√
2N (−1 + w/N 1/3 ).
√ 2N +X/21/2 N 1/6 √ 2N (−1+sk /N 1/3 )
ak =
N (j−1)/3 = (−1)N−r √ ( 2N )N−r+j−1
Z
2/3
C
{N 1/3 ,s1 ,...,sr }
2
1/3
dw e−wX e−N w e−w N /2 . Qj 1/3 N−r 2πi (1 − w/N ) k=1 (w − sk )
But for large N 2/3
2
1/3
3 e−N w e−w N /2 ∼ ew /3 (1 + O(N −1/3 )), (1 − w/N 1/3 )N−r and the method of steepest descent says we must deform the contour to rays such that the exponent is minimized. This occurs along the rays arg w = ±π/3, giving the first formula in (7.15). The second asymptotic formula is derived similarly.
It follows immediately from these asymptotic formulas, and (7.12), that with the scaled variables X, Y ,
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{sk } specified by √ x = 2N + X/21/2 N 1/6 ,
y=
√ 2N + Y /21/2 N 1/6 ,
ak =
√ 2N (−1 + sk /N 1/3 ),
(7.16)
the perturbed Hermite kernel (5.172) has the soft edge scaled form r 2 2 1 ˜ (j) (X)Ai(j) (Y ). e−x /2+y /2 KN (x, y) = K soft (X, Y ) + Ai lim √ N →∞ 2N 1/6 j=1
We see from (7.10) and (7.14) that ˜ (1) (x) = Ai
Ai(1) (x) = Ai(x),
(7.17)
x
es1 (x−t) Ai(t) dt, −∞
so in the case r = 1 this simplifies to read
K soft (X, Y ) + Ai(Y )
X
−∞
es1 (X−t) Ai(t) dt.
(7.18)
7.1.5 Soft edge of GUE minor process The correlations for the GUE minor process are given by (5.198), with all quantities referring to the Gaussian case. They permit a soft edge scaling, in which the species are separated by O(N 2/3 ) [215]. The limiting correlation is the so-called dynamical extension of the Airy kernel, specified by soft ρsoft ((τj , yj ), (τk , yk ))]j,k=1,...,n , (n) ((τ1 , y1 ), . . . , (τn , yn )) = det[K
where K
soft
⎧ ⎪ ⎪ ⎨ ((τx , x), (τy , y)) =
∞
0
⎪ ⎪ ⎩ −
e−(τy −τx )u Ai(x + u)Ai(y + u) du, 0
−∞
(7.19)
τy ≥ τx ,
e−(τy −τx )u Ai(x + u)Ai(y + u) du, τy < τx
(see Section 11.7 below). P ROPOSITION 7.1.4 In the Gaussian case of (5.198), introduce the scalings si = N − 2ci N 2/3 , One has
lim
N →∞
Yi yi = (2si )1/2 + √ 1/6 . 2si
r 1 √ ρ(r) ({(sj , yj )}j=1,...,r ) = det[K soft ((cj , Xj ), (ck , Xk ))]j,k=1,...,r . 2N 1/6
(7.20)
Proof. Substituting in (5.198) the appropriate Gaussian quantities we obtain, for cl ≥ cj , 2 sl 1 e−yj X Hs −k (yj )Hsl −k (yl ), K(sj , yj ; sl , yl ) = √ π k=1 2sl −k (sl − k)! j
(7.21)
while for cl < cj the r.h.s. is to be modified by multiplying by −1 and changing the summation to k ∈ Z≤0 . Only the former case will be considered explicitly, as the latter is essentially the same. For the analysis of the sum (7.21) we
289
SCALED LIMITS AT β = 1, 2 AND 4
substitute (7.9) to obtain K(sj , yj ; sl , yl ) ∼ e−N ×
1/3
(Yj −Yl ) −(cj −cl )N 2/3 21/2 N −1/6
N „ X k=1
2
(N − 2cj N 2/3 − k)! (N − 2cl N 2/3 − k)!
«1/2 Ai(Yj + 2k/(2N )1/3 )Ai(Yl + 2k/(2N )1/3 ).
(7.22)
Noting that the leading order contribution to the summation comes from k of order N 1/3 , then using Stirling’s formula to simplify the ratio of factorials in this regime, we can recognize the sum as the Riemann sum approximation to the first integral in (7.20).
7.1.6 Global limit of density The scaled global density (1.53) was computed in Exercises 1.6 q.1. Here, following [257], we will show how this same result can be derived from the formula ρ(1) (x) = KN (x, x), where KN (x, x) is given by (5.13) √ with the quantities therein specified in the Gaussian case of Section 5.4.1, but with x scaled x → 2Nx, so 2 the weight becomes e−2N x . With this scaling the relevant monic orthogonal polynomials are √ −3n/2 −n/2 N Hn ( 2Nx). p˜(G) n (x) = 2 Making use of the differentiation formula for the Hermite polynomials we have that d (G) (G) p˜ (x) = n˜ pn−1 (x), dx n and thus e−2N x
(G)
KN (x, x) =
2
(G) (G) (G) N (˜ pN −1 (x))2 − (N − 1)˜ pN −2 (x)˜ pN (x) .
(7.23) (G) (G) (˜ pN −1 , p˜N −1 )2 √ The asymptotic form of Hn ( 2N x) for n near N is given by the Plancheral-Rotach formula [508]. It tells us that for x ∈ (−1, 1) 1 2 2−m (G) (G) (G) N x2 , p˜N +m (x) = (˜ p , p ˜ ) e cos f (x) − m arccos x 1 + O 2 N N N π (1 − x2 )1/4 N π 1 fN (x) := N x 1 − x2 − N + arccos x + . (7.24) 2 4 Substituting this in (7.23), with N → N + 1 therein, and making use of the trigonometric identity cos2 a − cos(a − b) cos(a + b) = 1 − cos2 b, we reclaim (1.54). Asymptotic analysis can also be carried out on integral representations of (7.23) [341], [258]. This allows correction terms to the Wigner semicircle law to be obtained. In particular, with |x| < 1 fixed, one finds 1 √ 2 cos(2N πP (x)) 1 +O , (7.25) 2/N ρ(1) ( 2Nx) = ρW (x) − 3 W 2 π (ρ (x)) N N2 where ρW (x) =
2 1 − x2 , π
P (x) = 1 +
x W 1 ρ (x) − arccos x. 2 π
(7.26)
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The analysis of [341] also applies at the soft edge, where it gives 1 X ρ(1) ((2N )1/2 + 1/2 1/6 ) 21/2 N 1/6 2 N 1 1 1 2 2 2 (7.27) = ρsoft,2 + O (1) (X) − 20 3X (Ai(X)) − 2X(Ai (X)) − 3Ai(X)Ai (X) N N 2/3 with ρsoft,2 (1) (X) given by (7.13) (the 2 in the superscript soft,2 indicates β = 2; below we will encounter ρsoft,β (1) (X) for other values of β). E XERCISES 7.1
1. Show from (7.8) that ∞ “ |k| ”2j ˜ β) ˛˛ ∂ S(k; 1 |k| X =− , + ˛ ∂β 4π j=1 2j(2j − 1) 2π β=2
|k| < 2π.
(7.28)
7.2 SCALED LIMITS AT β = 2 — LAGUERRE AND JACOBI ENSEMBLES 7.2.1 Laguerre ensemble — hard edge The global density for the Laguerre ensemble is predicted by Proposition 3.2.3. In particular the density of the large eigenvalues is identical to the density of the large eigenvalues in the Gaussian ensemble. In contrast, the small eigenvalues in the Laguerre ensemble bunch together as the origin is approached, giving rise to the x−1/2 divergence. Due to the hard wall at x = 0 in the Laguerre ensemble this is referred to as the hard edge of the spectrum, whereas the boundary of the support of the large eigenvalues is a soft edge, as there is no wall in the statistical mechanics interpretation. To compute the correlation functions in the neighborhood of the origin, we first change scales so that the average inter-particle (inter-eigenvalue) spacing is O(1) in the N → ∞ limit. From the large N asymptotic formula [508, p.199] Γ(N + a + 1) Ja (2(M x)1/2 ) + O(N a/2−3/4 ), N! ∼ N a/2 Ja (2(N x)1/2 ),
e−x/2 xa/2 LaN (x) ∼ M −a/2
a > −1, (7.29)
where M = N + (a + 1)/2 and Ja (z) denotes the Bessel function, we see that the appropriate choice of scale is provided by the change of variable x=
X 4N
(7.30)
(the factor of 14 is chosen for convenience). With this scale, by using the above asymptotic formula and Stirling’s formula, the Christoffel-Darboux summation is readily evaluated in the N → ∞ limit [189]. P ROPOSITION 7.2.1 We have
1 (L) X Y KN , N →∞ 4N 4N 4N Ja (X 1/2 )Y 1/2 Ja (Y 1/2 ) − X 1/2 Ja (X 1/2 )Ja (Y 1/2 ) = , 2(X − Y )
Kahard (X, Y ) := lim
(L)
(7.31)
(L)
where KN (x, y) is given by Proposition 5.1.3 with pN (x) = pN (x). By taking the limit Y → X in the above formula and using the Bessel function identities α uJα (u) = αJα (u) − uJα+1 (u), Jα (u) = Jα−1 (u) − Jα (u), u
(7.32)
291
SCALED LIMITS AT β = 1, 2 AND 4
we see from (7.31) that
1 (Ja (X 1/2 ))2 − Ja+1 (X 1/2 )Ja−1 (X 1/2 ) . 4 The Bessel function identities also show that (7.31) can be rewritten Kahard (X, X) =
Kahard (X, Y ) =
(7.33)
X 1/2 Ja+1 (X 1/2 )Ja (Y 1/2 ) − Y 1/2 Ja+1 (Y 1/2 )Ja (X 1/2 ) . 2(X − Y )
(7.34)
7.2.2 Laguerre ensemble — soft edge It was remarked above that at the soft edge the global density in the Laguerre ensemble is the same as the global density at the spectrum edge in the Gaussian ensemble. In fact, use of the asymptotic expansion [508, p.201] e−x/2 Lan (x) = (−1)n 2−a−1/3 n−1/3 Ai(t) + O(n−2/3 ) , (7.35) where x = 4n + 2a + 2 + 2(2n)1/3 t, shows that (L)
lim 2(2N )1/3 KN (4N + 2(2N )1/3 X, 4N + 2(2N )1/3 Y ) = K soft (X, Y ).
N →∞
(7.36)
Thus, with this scaling of the coordinates, all correlation functions are those of the Gaussian ensemble at the soft edge. As for the Gaussian kernel, correction terms to the l.h.s. of (7.36) regarded as a function of N can (L) be computed using integral representations of KN . For the corresponding density, one finds [341], [258] 2 a (L) Ai(X) (X) + + O(N −2/3 ). (7.37) 2(2N )1/3 ρ(1) (4N + 2(2N )1/3 X) ∼ ρsoft,2 (1) (2N )1/3
7.2.3 Laguerre ensemble — global density (L)
The result (3.57) for the global density can be derived by computing the large N limit of KN (4N y, 4N y). One approach is to make use of the differentiation formula for the Laguerre polynomials, as given in (7.130) (L) below, to write the formula for KN in terms of {Lα n } for certain n and α, and then to make use of the Laguerre analogue of the Plancheral-Rotach formula (see, e.g., [202]) 1 (L) xa/2 e−x/2 Lan+m (x)|x=4nX = (−1)n+m (2π X(1 − X))−1/2 na/2−1/2 gm,n , (X) + O n √ √ (L) gm,n (X) = sin 2n( X(1 − X) − arccos X) − (2m + a + 1)arccos X + 3π/4 . (7.38) (L)
An alternative approach is to make use of a contour integral form of KN . This has the advantage of allowing corrections to (3.57) to be computed, giving the expansion up to terms O(N −2 ) [258] cos((2N + a)πP (X) − aπ(1 + XρMP (X))) 1 a MP 4ρ(1) (4N X) ∼ ρMP (X) − , − 4πX(1 − X) 2π X(1 − X) N (7.39) where √ 2 1−x 2 MP , PMP (x) := 1 + xρMP (x) − arccos x, ρ (x) := π x π valid for 0 < X < 1.
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7.2.4 Hard edge scaling of the perturbed Laguerre kernel Analogous to the situation with the perturbed Hermite kernel, its Laguerre counterpart (5.169) admits a scaling at both the soft and hard edges [29], [132]. At the soft edge the eigenvalues {xj }j=1,...,k and parameters {aj }j=1,...,r are to be scaled according to xj = 4N + 2(2N )1/3 Xj ,
sj 1 aj = − + , 2 2(2N )1/3
(7.40)
with the result r ˜ (l) (Xα )Ai(l) (Xβ ) Ai lim (2(2N )1/3 )k ρ(k) (x1 , . . . , xk ) = det K soft (Xα , Xβ ) +
N →∞
l=1
α,β=1,...,k
, (7.41)
as found in (7.17). We will concentrate on the hard edge scaling. The limiting functional form involves incomplete multiple Bessel functions exw−1/4w p−1 e−xz+1/4z z a+r dz (p) (p) k=1 (w − sk ) dw ˜ p , J (x) = . J (x) = a+r w 2πi C{0,s1 ,...,sp } C{0} k=1 (z − sk ) 2πi Explicitly, the following result holds [132]. P ROPOSITION 7.2.2 For large N , with ak = 4N sk (k = 1, . . . , r) X 1 ˜ (i) X = J˜(p) (X) + O 1 , (4N )a+r−p Λ(i) = J (p) (X) + O . (7.42) (4N )p−a−r−1 Λ 4N N 4N N Consequently r Y (a+r)/2 1 a X Y hard KN , = K (X, Y ) + (7.43) J˜(p) (X)J (p) (Y ). lim a+r N →∞ 4N 4N 4N ak =4N sk X p=1 Proof. The first term in (7.43) follows immediately from (5.169), (5.167) and (7.31), while the sum in (7.43) follows from (5.169) and (7.42). To derive (7.42), substitute ak = 4N sk in (5.168), change variables z → 4N z, w → 4N w, scale the contours and make use of the elementary limit (1 + u/N )N → eu .
From the integral representation of the Bessel function −1 ex(z−z )/2 dz , Jα (x) = z α+1 2πi C{0} together with the formula J−α (x) = (−1)α Jα (x), both valid for α ∈ Z, we can check from the definitions that √ J (1) (x) = (4x)(a+r−1)/2 Ja+r−1 ( x), x √ t−(a+r+1)/2 es1 (t−x) Ja+r+1 ( t) dt J˜(1) (x) = −2−(a+r+1) −∞ x √ √ −(a+r)/2 Ja+r ( x) − s1 2−(a+r) t−(a+r)/2 es1 (t−x) t−(a+r)/2 Ja+r ( t) dt. = (4x) −∞
Here the final line follows on integrating by parts, making use of the identity 1 d −α (u Jα (u)). u du Furthermore, making use of the form (7.34), together with the Bessel function three-term recurrence u−α−1 Jα+1 (u) = −
tJα+2 (t) = 2(α + 1)Jα+1 (t) − tJα (t),
(7.44)
293
SCALED LIMITS AT β = 1, 2 AND 4
it is straightforward to verify that Y 1/2 1 hard Ka+1 (X, Y ) = Kahard (X, Y ) − X −1/2 Ja+1 (X 1/2 )Ja (Y 1/2 ). X 2 As a consequence of these facts, in the case r = 1 (7.43) can be written in terms of Bessel functions according to X Y a/2 √ √ s1 Kahard (X, Y ) − Ja ( Y )X a/2 es1 (t−X) t−(a+1)/2 Ja+1 ( t) dt . X 2 −∞ 7.2.5 Jacobi ensemble For the Jacobi ensemble, Proposition 3.6.3 predicts the same global eigenvalue density in the neighborhood of the edges x = ±1 as for the Laguerre ensemble in the neighborhood of the edge x = 0. From the Jacobi polynomial large N asymptotic formula [508, p.197] x x1/2 −a (a,b) PN 1− ∼ Ja (x1/2 ) (7.45) 2N 2 2N X we see that by making the shift of origin and change of scale x = 1 − 2N 2 the scaled Christoffel-Darboux summation can be evaluated in the N → ∞ limit to give the same expression as in Proposition 7.2.1. Thus the scaled correlations in the neighborhood of the edges x = 0 and x = 1 of the Laguerre and Jacobi ensembles, respectively, are the same. This is to be expected as in the neighborhood of these points the corresponding Boltzmann factors are proportional.
7.2.6 Circular Jacobi ensemble — spectrum singularity Consider the n-point correlation function (5.55). Our interest is in the neighborhood of the point z = −1, which is analyzed by writing xj → xj + L/2 or equivalently zj → −zj . In preparation for taking the thermodynamic limit, we make use of the formula 2 Γ(β + 1 + 2n + α) x − 1 n , Pn(α,β) (x) = 2 F1 − n, −n − α; −2n − α − β; − n!Γ(β + 1 + n + α) 2 x−1 where 2 F1 denotes the Gauss hypergeometric function (5.83), to write 1 + z 2 N (CJ) = i−N − i pN 2 F1 (−N, a; 2a; 1 − z), 1−z 1−z 1 + z 2 N −1 (CJ) = i−(N −1) − pN −1 i 2 F1 (−(N − 1), a + 1; 2a + 2; 1 − z). 1−z 1−z To take the thermodynamic limit we use the formulas lim 2 F1 (−n, b; c; t/n) = 1 F1 (b; c; −t),
n→∞
1 F1 (b; c; −t) :=
∞ (b)n (−t)n , (c) n! n n=0
1 x −(a−1/2) ix F (a; 2a; 2ix) = Γ a + e Ja−1/2 (x), 1 1 2 2 which give
1 + z (−1)N 1 j (πρx/2)−(a−1/2) Ja−1/2 (πρx), i ∼ Γ a+ N 1 − zj α=N +a (sin πx/L) 2 1 + z (−1)N −1 3 j (CJ) (πρx/2)−(a+1/2) Ja+1/2 (πρx). ∼ Γ a + pN −1 i 1 − zj α=N +a (sin πx/L)N −1 2 (CJ)
pN
(7.46)
(7.47)
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Using (5.48) to evaluate the normalization in (5.55) and using the duplication formula (4.180) we find that in the thermodynamic limit [418] s.s. ρ(n) (x1 , . . . , xn ) = det K s.s. (xj , xk ) , j,k=1,...,n Ja+1/2 (πρx)Ja−1/2 (πρy) − Ja+1/2 (πρy)Ja−1/2 (πρx) , K s.s. (x, y) := (πρx)1/2 (πρy)1/2 2(x − y) (7.48) where ρ := N/L, which is valid for all values of x1 , . . . , xn in the case a a non-negative integer, while it is valid for x1 , . . . , xn all positive otherwise. In the limit y → x we have (πρ)2 x 2a (Ja−1/2 (πρx))2 + (Ja+1/2 (πρx))2 − Ja−1/2 (πρx)Ja+1/2 (πρx) . (7.49) K s.s. (x, x) = 2 πρx Using the Bessel function formulas 2 1/2 2 1/2 sin x, J−1/2 (x) = cos x, J1/2 (x) = πx πx together with the addition formula for the sine function, we see that sin πρ(x − y) =: K bulk (x, y). = K s.s. (x, y) π(x − y) a=0
(7.50)
This is expected because with a = 0 the circular Jacobi ensemble corresponds to Dyson’s circular ensemble, and the latter is locally identical to the bulk of the Gaussian ensemble. Another relationship between (7.48) and the bulk correlation of Proposition 7.1.1 is given in Exercises 7.2 q.3.
7.2.7 The classical groups There are three possible scalings of k-point correlations for the classical groups specified by Proposition 5.5.3 and (5.87). One is the bulk scaling with unit density. For the CUE this is achieved by θ → 2πx/N , while for the cases of Proposition 5.5.3 one requires θ → π/2 + πx/N . A simple calculation gives ρbulk (k) as specified by the result of Proposition 7.1.1 with ρ = 1 as the scaled k-point correlation. In addition to the bulk scaling, there are two possible edge scalings—when there is an eigenvalue at the endpoint, and when there is not. Typical of the former case is O+ (2N + 1) in the neighborhood of θ = 0. A straightforward calculation shows π k + O (2N +1) ρ+ ρ(k) (πx1 /N, . . . , πxk /N ) (k) (x1 , . . . , xk ) := Nlim →∞ N sin π(x − x ) sin π(x + x ) j l j l − . (7.51) = det π(xj − xl ) π(xj + xl ) j,l=1,...,k Typical of the latter is O− (2N + 1) in the neighborhood of θ = 0. It follows from Proposition 5.5.3 that π k − O (2N +1) (x , . . . , x ) := lim ρ(k) (πx1 /N, . . . , πxk /N ) ρ− 1 k (k) N →∞ N sin π(x − x ) sin π(x + x ) j l j l + . (7.52) = det π(xj − xl ) π(xj + xl ) j,l=1,...,k The two edge scalings can be related to the hard edge result (7.31). Thus we see that with Kach (X, Y ) := 2π 2 (XY )1/2 Kahard ((πX)2 , (πY )2 )
(7.53)
295
SCALED LIMITS AT β = 1, 2 AND 4
determining the edge correlations in the ensemble (3.4) with β = 2 and the bulk density unity, ch (X, Y ) = K±1/2
sin π(X − Y ) sin π(X + Y ) ± , π(X − Y ) π(X + Y )
(7.54)
where use has been made of (7.50). E XERCISES 7.2
1. Use the asymptotic formulas (7.9) and (7.29) to deduce from (5.70) and (5.71) that Z ∞ Z √ √ 1 1 K soft (X, Y ) = Ai(X + t)Ai(Y + t) dt, Kahard (X, Y ) = Ja ( Xt)Ja ( Y t) dt. 4 0 0
Note too the integral formula K bulk (x, y) =
ρ 2
Z
1
eπiρ(x−y)t dt.
−1
2. The objective of this exercise is to show the connection between the quantities K soft (X, Y ), Kahard (X, Y ), K s.s. (X, Y ) and K bulk (X, Y ). (i) Use the asymptotic expansions (7.69) and (7.74) below to deduce that Z
√ c+πρx/ c
lim
c→∞
Z ρsoft (1) (−X) dX = ρx,
√ c+2πρx c
lim
c→∞
c
ρhard (1) (X) dX = ρx
c
so the densities are asymptotically constant in the variable x. (ii) Use the asymptotic expansions (7.68) and (7.73) to show √ √ πρ lim √ K soft (−(c + πρx/ c), −(c + πρy/ c)) c √ √ √ = lim 2πρ cKahard (c + 2πρx c, c + 2πρy c) = K bulk (x, y).
c→∞
c→∞
(iii) Use the asymptotic expansion [539] Ja (x) ∼
“ 2 ”1/3 a
“ 21/3 (a − x) ” Ai , x1/3
(7.55)
valid for a and x large such that the argument of the Airy function is order one, to show that lim 2a(a/2)1/3 Kahard (a2 − 2a(a/2)1/3 x, a2 − 2a(a/2)1/3 y) = K soft (x, y).
a→∞
p (iv) With φ(x) := x/2Ja+1/2 (x), show by using the Bessel function identities (7.32) that for ρ = 1/π (7.48) can be rewritten φ(x)yφ(y) − φ(y)xφ (x) K s.s. (x, y) = . x−y (v) Use the asymptotic expansion (7.55) in the result of (iv) to show that ” “ lim (a/2)1/3 K s.s. a − (a/2)1/3 x, a − (a/2)1/3 y = K soft (x, y). a→∞
pj (x)} (vi) Consider the chiral ensemble (3.4) with β = 2 and α = a + 12 (note that this is defined on R+ ). Let {˜ 2
be orthonormal polynomials with respect to the weight function w2 (x) = |x|2a+1 e−x , x ∈ R. Show that KN (x, y) = (w2 (x)w2 (y))1/2
N−1 X
p˜2ν (x)˜ p2ν (y)
ν=0
= (w2 (x)w2 (y))1/2
2N−1 X “ ν=0
” pν (y) + p˜ν (x)˜ pν (−y) , p˜ν (x)˜
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CHAPTER 7
where the second line follows from the parity of the p˜ν (x). Conclude from this that “ ”˛ ˛ 2(XY )1/2 K hard (X 2 , Y 2 ) = K s.s. (X, Y ) + K s.s. (X, −Y ) ˛ a →a+1/2 . ρ=1/π
p2μ−1 (x) = p˜2μ (x)|a→a−1 and use this to show (vii) With {˜ pj (x)} as in (vi), note that x˜ ˛ ˛ ˛ “ ˛ ˛ ˛ K s.s. (X, Y )˛ = (XY )1/2 K hard (X 2 , Y 2 )˛ + K hard (X 2 , Y 2 )˛ a→a−1/2
ρ=1/π
3. Argue that
˛ ˛ s.s. ρbulk (n+1) (x1 , . . . , xn , 0)/ρ = ρ(n) (x1 , . . . , xn )˛
a=1
” a→a+1/2
.
,
(7.56)
s.s. where ρbulk (n+1) refers to the correlation function in Proposition 7.1.1 while ρ(n) refers to the correlation function (7.48) for the spectrum singularity. Use the trigonometric formula for J1/2 (x) in (7.50), an analogous formula for J3/2 (x) deducible from (7.50) and the three term recurrence (7.44) together with elementary row operations in the determinant formula for the l.h.s. to check this directly.
4. [215] In this exercise the soft edge limit of the Laguerre case of (5.198) will be analyzed. Explicitly, it will be shown that with the scalings si = N − s˜i , s˜i := 2ci (2N )2/3 , one has
“ lim
N→∞
2(2N )2/3 )
”r
yi = 4si + 2(a + N − si ) + 2(2N )1/3 Yi ,
ρ(r) ({(sj , yj )}j=1,...,r ) = det[K soft ((cj , Yj ), (ck , Yk ))]j,k=1,...,r ,
(7.57)
where K soft is specified by (7.19).
(i) Substitute in (5.198) the appropriate Laguerre quantities to obtain, for s˜j ≤ s˜l , a+˜ sj −yj
K(N − s˜j , yj ; N − s˜l , yl ) = yj
e
X Γ(N − s˜j − k + 1) (a+˜s ) (a+˜ s ) j (yj )LN−˜sll−k (yl ), (7.58) L Γ(N − k + a + 1) N−˜sj −k
N−˜ sl
k=1
and note that for s˜j > s˜l the r.h.s. is to be modified by multiplying by −1 and changing the summation to k ∈ Z≤0 . (ii) Use the generalization of (7.35) applicable for a = o(n) [334], p xa/2 e−x/2 Lan−k (x) = (−1)n−k (2n)−1/3 (n − k + a)!/(n − k)! ( „ “ 1/3 2k ” O(e−k/n ), −2/3 ) + O(n × Ai X + 1/3 (2n) O(1),
k≥0 k<0
« (7.59)
with n = N − s˜i in (7.58) to show that for large N K(N − s˜j , yj ; N − s˜l , yl ) ∼ e−(2N)
1/3
(Yj −Yl ) (2N )−2/3 „ N−˜ Xsl (N − s˜j − k)! «1/2 “ 2k ” “ 2k ” × Ai Yj + Ai Yl + . 1/3 (N − s˜l − k)! (2N ) (2N )1/3 k=1
Proceed now as in the analysis of (7.22) to deduce (7.57).
297
SCALED LIMITS AT β = 1, 2 AND 4
7.3 LOG-GAS SYSTEMS WITH PERIODIC BOUNDARY CONDITIONS 7.3.1 Semiperiodic boundary conditions For the p.d.f. (5.96) the relationship with the Stieltjes-Wigert polynomials Sn (y; q) shows that ρ(n) (y1 , . . . , yn ) = det[LN (yj , yk )]j,k=1,...,n , 2
(7.60)
where, with q = e−2π /L c , u = ec (y+πN/Lc ) and an denoting the coefficient of y n in Sn (y; q), 1/2 a 2π N −1 SN (y; q)SN −1 (y ; q) − SN (y ; q)SN −1 (y; q) uu w(u; q)w(u ; q) . (7.61) LN (y, y ) = L aN y − y 2
The evaluation of the N → ∞ limit (all other parameters fixed) is given by the following result [191]. P ROPOSITION 7.3.1 We have
c 1/2 q −1/8 (y; q)(−y ; q) − (y ; q)(−y; q) L(y, y ) := lim LN (y, y ) = N →∞ π (q; q)3∞ 2 sinh π(y − y )/L ˆ ˆ ; q)(−y; ˆ ˆ 1 ; q) − (y q) 1 (y; q)(−y = , L θ1 (0; (q )2 ) 2 sinh π(y − y )/L
where ∞
(y; q) := eξπy/2L
2
(−1)ν q (ν+1/4−c Ly/2π) ,
ν=−∞
ˆ q) := e−ξπy/2L θ1 π 1 + c Ly ; q , (y; 4 2π
with θ1 denoting the Jacobi theta function (2.79), q = e−c L odd.
2
/2
and ξ = 1 for N even and ξ = −1 for N
Proof. Reading off the value of aN from (5.97) shows that aN−1 /aN ∼ q −2N for N → ∞. This estimate, together with the fact that
»
N −1 ν
–
» ∼ (1 − q N−ν )
q
N ν
– q
shows N h X N i ν 2 +ν/2 aN−1 q −N SN (u; q)SN−1 (u ; q) ∼ − q (−u)ν N−1 aN (1 − q) · · · (1 − q ) ν=0 ν q – N » X 2 N × (1 − q N−ν )q ν +ν /2 (−u )ν . ν q ν =0
Since we have to subtract the same term with u and u interchanged we see that the factor (1 − q N−ν ) in the sum over ν can be replaced by −q N−ν . Doing this, and completing the square in ν and ν gives exponents (ν − N/2 + 1/4 − 2 c Ly/2π) and (ν − N/2 − 1/4 − c Ly/2π)2 . The remaining step is to change the summation variable ν − [N/2] → ν, and similarly for ν , and note that – » N 1 ∼ . ν + [N/2] q (q; q)∞ This gives the first expression for the limiting value of L(y, y ). The second expression follows by applying the conjugate modulus transformation from the theory of the Jacobi theta functions [541].
Here the density is a periodic function, period 1/η, η = c L/2π, so the system is in a crystalline state. This is to be expected, because as noted in Section 2.7.1 the pair potential is asymptotically that of the one-
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CHAPTER 7
dimensional Coulomb system (|x| potential), which is crystalline for all couplings [367]. The dependence on the parity of N is also understood as a consequence of the ordered state, and symmetry of the system about the origin: for N odd y = 0 is a maximum of the density while for N even y = 0 is a minimum. 7.3.2 Metal wall Reinstating the fugacity (5.100) into (5.105) we see that the summation therein is a Riemann sum approximation to an integral which becomes exact for L → ∞. Thus ∞ −4πdt+2πit(x−y) e ˜ y) := lim G(x, y) = 2πζe4πdη G(x, dt. (7.62) L→∞ 1 + 2πζe4πdη e−4πdt 0 With x = y this gives the density as a transcendental function of the fugacity. For large d (7.62) simplifies to 1 sin πη(x − y) ˜ y) , = e2πit(x−y) dt = ηeπiη(x−y) G(x, πη(x − y) d→∞ 0 reclaiming the bulk scaling result of Proposition 7.1.1 with ρ = η, as expected. 7.3.3 Doubly periodic boundary conditions With N, L → ∞ and N/L = ρ fixed the summation defining the correlation kernel in (5.116) tends to a Riemann integral, giving that 1 ρ(n) (x1 , . . . , xn ) = ρn det H(ρxj + α, ρxk + α) dα, (7.63) j,k=1,...,n
0
where, with q˜ = e−πW ρ ,
H(X, Y ) = 0
1
e−2πi(X−Y )
θ3 (π(X − iW ρu); q˜)θ3 (π(Y + iW ρu); q˜) du. θ3 (2πiW ρu; q˜2 )
(7.64)
As is to be expected from the underlying pair potentials, the state defined by the correlation kernel of Proposition 7.3.1, with c L/2pi = ρ and L = W is closely related to the state defined by (7.63) [191]. In fact, use of residue calculus in (7.64) shows that then H(x, y) = L(x, y), telling us in turn that the correlations for the doubly periodic (DP) system are related to the semiperiodic/free boundary condition (SP) system by averaging over a period, 1/ρ DP ρ(n) (y1 , . . . , yn ) = ρ ρSP (n) (y1 + α, . . . , yn + α) dα. 0
7.4 ASYMPTOTIC BEHAVIOR OF THE ONE- AND TWO-POINT FUNCTIONS AT β = 2 In this section the large separation behavior of the exact truncated two-particle correlation functions, and the behavior of the density at large distances from the edge will be computed. Throughout, the notation ρ(1) and ρ(2) will refer to the appropriate scaled values. 7.4.1 Bulk It follows from (7.2) that ρT(2) (x, 0) = −
1 cos 2πρx + . 2(πx)2 2(πx)2
(7.65)
299
SCALED LIMITS AT β = 1, 2 AND 4
This exhibits that the leading non-oscillatory form is −1/2(πx)2 , which is in keeping with the |k| → 0 form of (7.4). For small x (7.2) gives (πρx)2 2(πρx)4 ρ(2) (x, 0) = ρ2 − − ··· . (7.66) 3 45 The fact that the leading term is proportional to x2 follows from the fact that the Boltzmann factor vanishes as |x − y|2 for particles at positions x and y, in the limit x → y. 7.4.2 Soft edge For the scaled one- and two-point functions near the soft edge, Propositions 5.1.2 and 7.1.2 give ρT(2) (X, Y ) = −(K soft (X, Y ))2 ,
ρ(1) (X) = K soft (X, X),
(7.67)
where K soft (X, X) and K soft (X, Y ) are given by (7.13) and (7.12), respectively. From the asymptotic expansion [136] Ai(−x) ∼
1
x→∞
π 1/2 x1/4
cos(2x3/2 /3 − π/4)
(7.68)
we deduce that [189], [318]
√ 1 X cos(4X 3/2 /3) − +O , ρ(1) (−X) ∼ X→∞ π 4πX X 5/2 X +Y 1 √ ρT(2) (−X, −Y ) ∼ − , X,Y →∞ 4π 2 XY (X − Y )2
(7.69) (7.70)
where in (7.70) only non-oscillatory terms have been written down. The asymptotic expansion e−2x /3 Ai(x) ∼ √ 1/4 x→∞ 2 πx 3/2
(7.71)
tells us 4 (7.72) log ρ(1) (X) ∼ − X 3/2 . X→∞ 3 It is possible to exhibit a matching between the asymptotic expansion (7.69), and the global asymptotic expansion (7.25) expanded in the neighborhood of the edge (see Exercises 7.4 q.1). 7.4.3 Hard edge Here, according to Propositions 5.1.2 and 7.2.1 ρ(1) (X) = Kahard (X, X),
ρT(2) (X, Y ) = −(Kahard (X, Y ))2 ,
where Kahard (X, X) and Kahard (X, Y ) are given by (7.33) and (7.31) respectively. From the asymptotic formula [165] 2 1/2 Jα (x) ∼ cos(x − πα/2 − π/4) (7.73) x→∞ πx we find [189], [51] for the leading non-oscillatory behavior 1 , 2πX 1/2 X +Y 1 √ ρT(2) (X, Y ) ∼ − . 2 X,Y →∞ (X − Y )2 4π XY ρ(1) (X)
∼
X→∞
(7.74) (7.75)
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CHAPTER 7
Note in particular that (7.75) is identical to the behavior (7.70) found for the same quantity at the soft edge. It is also of interest to consider the hard edge correlations in the squared variables of (7.53). Then we obtain the asymptotic behavior ρ(1) (X) ∼ 1, X→∞
ρT(2) (X, Y )
∼
X,Y →∞
−
1 . 2π 2 (X − Y )2
The second of these is the non-oscillatory bulk behavior seen from (7.65). 7.4.4 Spectrum singularity In the neighborhood of a spectrum singularity the correlation functions are given by (7.48). In particular ρ(1) (x) = K s.s. (x, x) ∼ ρ + ρ x→∞
a2 (a2 − 3) , 4(πρx)2
where only non-oscillatory terms have been written down. Also 2 ∼ − ρT(2) (x, y) = − K s.s. (x, y) x,y→∞
1 , 2π 2 (x − y)2
(7.76)
(7.77)
where again oscillatory terms have been ignored. The latter is the non-oscillatory bulk behavior seen from (7.65). 7.4.5 Semiperiodic boundary conditions We have seen that the state of the system in this case is crystalline, and the correlations are given in terms of Jacobi theta functions. Simplified results can be obtained by supposing L 1, and then averaging over the period of the density oscillations. First, since θ1 (u; q) ∼ 2q 1/4 sin u, (7.60) and Proposition 7.3.1 give that for L 1 1 2 π 2 π + πηy sin + πηy ρT(2) (y, y ) ∼ − 2 e−π(y−y )/L 2 sin 2 4 4 2L sinh π(y − y )/L π π + πηy sin2 + πηy eπ(y−y )/L − cos 2πηy cos 2πηy . +2 sin2 4 4 Now introducing the smoothed truncated correlation 1/η T ρT(2) (y + y , y ) dy ρ¯(2) (y, 0) = η 0
and substituting the above asymptotic form we obtain [208] ρ¯T(2) (y, y ) ∼ −
cosh(π(y − y )/L) . 2 sinh2 (π(y − y )/L)
(7.78)
Again with L 1, a different translationally invariant approximation can be obtained directly from the kernel of Proposition 7.3.1, rather than first computing the two-point function. Thus by approximating the factor of e−ξπy/2L in ˆl(y; q) by unity one obtains the kernel ˜ y ) := L(y,
sin πη(x − y) , L sinh π(x − y)/L
(7.79)
which is a functional form first obtained in a random matrix calculation in [413]. Note that (7.79) reduces to the bulk scaling sine kernel of Proposition 7.1.1 in the limit L → ∞. For small L, only the values x ≈ y are important and so only the diagonal terms in the determinant of Proposition 5.1.2 contribute, which is Poisson behavior. Thus the kernel interpolates between these limiting cases.
301
SCALED LIMITS AT β = 1, 2 AND 4
7.4.6 Metal wall Integration by parts of the limiting kernel (5.105) gives that for |x − y| → ∞, ˜ y) ∼ G(x,
1 ζe4πdη . 1 + 2πζe4πdη i(x − y)
Thus the corresponding decay of ρ(2) (x, y) is an inverse square, with a negative coefficient and no oscillatory term. E XERCISES 7.4
1. [258] Here a type of connection formula between the bulk and soft edge densities in the scaled GUE will be exhibited. 2
(i) Note that with respect to the scaled Gaussian weight function w2 (x) = e−2Nx , the asymptotic formula (7.25) reads “ 1 ” 2 cos(2N πP (x)) 1 + O ρ(1) (x) = ρW (x) − 2 π 3 ρW (x) N N2 while (7.27) gives
“ 1 ” “ N 1/3 ξ ” soft,2 (ξ) + O = ρ . ρ(1) 1 + (1) 2 2N 2/3 N 2/3
(ii) Show that substituting x = 1 + ξ/2N 2/3 in the first expression for ρ(1) (x) in (i), multiplying by N 1/3 and expanding in N keeping terms O(1) only gives the first two terms of the ξ → −∞ form of the second expression, as implied by (7.69).
7.5 BULK SCALING AND THE ZEROS OF THE RIEMANN ZETA FUNCTION 7.5.1 Montgomery pair correlation conjecture From Chapter 1 we know that the eigenvalues of a chaotic quantum system without time reversal symmetry will have, after appropriate rescaling, the same statistical properties as the eigenvalues from the (infinite) GUE. In this section, following mainly [353], [74], we will discuss the occurrence of the GUE distributions in the study of the statistical properties of the Riemann zeros of large modulus. The Riemann zeros are the zeros of the analytic continuation of the Riemann zeta function ∞ 1 ζ(s) = , s n n=1
Re(s) > 1,
on the critical line Re(s) = 12 . Indeed, it has been noted [62] that there are formal similarities between a certain representation of the density of zeros of ζ(s) on the critical line (Proposition 7.5.4 below), and the Gutzwiller trace formula for the density of states in a chaotic quantum system without time reversal symmetry, so from this viewpoint the occurrence of the GUE distributions is not unexpected. On the other hand we know that in the large N bulk scaling the GUE distributions coincide with those for the CUE or equivalently U (N ). In fact, as will be seen subsequently, it is U (N ) eigenvalues which more accurately models the Riemann zeros. First we recall that ζ(s) is closely related to the prime numbers, as is seen from a product formula due to Euler. P ROPOSITION 7.5.1 For Re(s) > 1, ζ(s) =
(1 − p−s )−1 , p
where the product is over all primes p (= 2, 3, 5,. . . ).
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CHAPTER 7
Proof. Subtracting 2−s ζ(s) from the definition of ζ(s) gives ∞ X
ζ(s)(1 − 2−s ) =
n=1 n=2m,m∈Z+
1 . ns
Similarly, subtracting from this series 3−s times the series gives ζ(s)(1 − 2−s )(1 − 3−s ) =
∞ X ∗ 1 = 1−s + 5−s + 7−s + · · · , s n n=1
where the * indicates all terms n which are multiples of 2 or 3 are to be omitted. Subtracting all the primes to the power of −s in this fashion gives Y ζ(s) (1 − p−s ) = 1−s = 1 p
as required.
Our interest is in the statistical properties of the zeros of ζ(s) in the complex plane. The famous Riemann hypothesis asserts that all the zeros of ζ(s) with nonzero imaginary part lie on the so-called critical line Re(s) = 21 . As noted above such zeros are called the Riemann zeros. The Riemann hypothesis dates back to 1859. Over a century later, beginning with a conjecture of Montgomery [405] for the pair correlation function ∞ d(E)d(E + ε), d(E) := j=1 δ(E − Ej ) (Ej denotes the (positive) imaginary part of the jth zero), the statistical properties of the Riemann zeros with large imaginary part have been the subject of study. The Montgomery conjecture states that 1 + S(ˆ ε) := lim
E→∞ ε ˆ fixed
d(E)d(E + ε) ε sin2 πˆ = δ(ˆ ε) + 1 − , 2 d(E)d(E + ε) (πˆ ε)
(7.80)
¯ where the averages · are over a region [E, E + ΔE] such that 1 ΔE E, and εˆ = ε/d(E) with ¯ d(E) denoting the mean spacing between zeros at point E on the critical line. Since in general S(x) = δ(x) + ρT(2) (x, 0)/ρ2 , we see from (7.65) that (7.80) is identical to S(x) for the GUE in the bulk limit. Indeed the so called Montgomery-Odlyzko law [352] asserts that after scaling the large Riemann zeros have the same distribution as the bulk eigenvalues for large GUE matrices. This conjecture has supporting evidence from large scale numerical computations [426], involving the accurate evaluation of the zero number of order 1020 along the critical line, and over 107 of its neighbors. Moreover for test functions f (x) with Fourier transform fˆ(k) supported on |k| < 2π, Montgomery rigorously proved that 2π 2π ˆ fˆ(k) dk = 1 |k|fˆ(k) dk, (7.81) S(k) 2π −2π −2π ˆ where Sˆ denotes the Fourier transform of S, which is consistent with (7.80) since S(k) is given by (7.4) (note ˆ ˜ that S(k) = S(k) for ρ = 1). This rigorous result has been extended to higher order correlations in [472]. By abandoning rigor, the general n-point correlation function for the limiting zeros along the critical line can be calculated exactly [73], and the result of Proposition 7.1.1 for the GUE obtained. Here we will give the latter argument in the simplest case (n = 2), and so derive (7.80). 7.5.2 Rigorous theory Since Proposition 7.5.1 gives that there are no zeros of ζ(s) for Re(s) > 1, to study the zeros it is necessary to analytically continue ζ(s). For this purpose a contour integral representation can be used. Furthermore, this representation can be used to establish a functional equation satisfied by ζ(s).
303
SCALED LIMITS AT β = 1, 2 AND 4
P ROPOSITION 7.5.2 We have ζ(s) = −
Γ(1 − s) 2πi
C
(−z)s−1 dz ez − 1
where C is any simple contour starting at z = ∞ + iμ (μ > 0), looping around z = 0, and finishing at z = ∞ − iμ without crossing the real axis, and |arg(−z)| < π. Furthermore ζ(s) = 2s π s−1 sin(πs/2)Γ(1 − s)ζ(1 − s). Proof. For Re(s) > 1, the first result follows by writing on the portion of C with Re(z) > 0 X −kz 1 e , = ez − 1 k=1 ∞
noting that the contribution from other portions of C can be made arbitrarily small, then changing variables kz → z and integrating term by term using the formula [541] Z 1 1 =− (−z)s−1 e−z dz Γ(1 − s) 2πi C (cf. (5.156)). For Re(s) ≤ 1, the integral representation is the analytic continuation. The second result follows by supposing Re(s) < 0 and expanding C to a circle of infinite radius about the origin (the resulting contour integral vanishes), and calculating the original contour integral as 2πi times the sum of the residues of the poles at z = ±2πi, ±4πi, . . . .
It follows from Proposition 7.5.2 that the only singularity of ζ(s) is a simple pole at s = 1, that there are zeros (trivial zeros) at s = −2, −4, . . . and that the only other zeros of ζ(s) must occur in the critical strip 0 ≤ s ≤ 1. To investigate other zeros, Riemann introduced the entire function 1 1 s(s − 1)ζ(s)Γ(s/2)π −s/2 , (7.82) s := − it. 2 2 The Riemann hypothesis is equivalent to the statement that all zeros of ξ(t) are on the real t-axis. Use of the functional equation for ζ(s), the functional equation (4.5) for the gamma function and the duplication formula (1.111) shows that ξ(t) :=
ξ(t) = ξ(−t),
¯ = ξ(t). and thus ξ(t)
This last equation implies ξ(t) is real for real t. Using ξ(t), the number of zeros N (E) of ζ(s) in the critical strip with 0 ≤ Im(s) < E can be computed (see, e.g., [516]). P ROPOSITION 7.5.3 We have ¯ (E) + Nosc (E), N (E) = N where ¯ (E) := 1 − 1 E log π − 1 Im log Γ N 2π π
1 iE − 4 2
,
1 Nosc (E) := − Im log ζ π
1 − iE . 2
Proof. In terms of ξ(t), N (E) is equal to the number of zeros of ξ(t) in the region − 21 ≤ Im(t) ≤ 12 , 0 ≤ Re(t) < E. According to the residue theorem, this is given by N (E) =
1 1 2 2πi
Z C
1 ξ (t) dt = Im ξ(t) 4π
Z C
ξ (t) dt, ξ(t)
where C is the rectangular contour with sides parallel to the real and imaginary t-axes, which passes through the points ±E and ±i( 12 + α), with α > 0. It is assumed that E is not a zero of ξ(t). The factor of 1/2 comes from the fact that ¯ ξ(t) is even. Since ξ(t) is even and ξ(t) = ξ(t), the integral over C can be replaced by 4 times the integral over C1 ,
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CHAPTER 7
where C1 is the contour over the portion of C in the upper right quadrant of the complex t plane. This gives N (E) =
1 Δ arg ξ(t), π
where Δ argξ(t) refers to the change in argument of ξ(t) in going from t = E to t = i( /2 + α) along C1 . But at t = i( 21 + α), ξ(t) > 0 and thus N (E) = −
1 1 arg ξ(E) = − Im log ξ(E). π π
The result now follows by substituting (7.82).
7.5.3 Heuristic argument The above theory is entirely rigorous. However, to achieve our goal of establishing (7.80), it is necessary to proceed heuristically. We begin with a formula for the density of zeros, and assume the validity of the Riemann hypothesis, which gives that all zeros of ξ(t) are real. P ROPOSITION 7.5.4 Let E1 , E2 , . . . denote the positive zeros of ξ(E). We have d(E) :=
∞
δ(E − Ej ) =
j=1
d ¯ N (E) = d(E) + dosc (E), dE
where E 1 ¯ log , d(E) ∼ 2π 2π
dosc (E) = −
∞ 1 log p cos(Ek log p). π p pk/2 k=1
¯ ¯ (E) Derivation. To derive the formula for d(E), which is rigorous, we simply use Stirling’s formula in the formula for N in Proposition 7.5.3 and differentiate. To derive the formula for dosc (E), which is heuristic, we substitute the Euler product for ζ( 12 − iE) in the formula for Nosc (E) in Proposition 7.5.3 (note this is not justified since the product only converges for Re(s) > 1). Expanding log(1 − p−s ) and differentiating gives the stated result.
In an obvious notation, this result gives Nosc (E) = −
∞ 1 exp(− k2 log p) sin(Ek log p). π p k
(7.83)
k=1
It should be remarked [63], [64] that in the study of classically chaotic quantum systems, the so-called trace formula gives a formula of the same structure for the oscillating part of the quantum spectrum, as calculated from the classical data. Thus, after a simplifying approximation, one has for → 0 ∞ k 1 exp(− k2 λp˜Tp˜) πk Nosc (E) ∼ sin Sp˜(E) − μp˜ . (7.84) π k 2 p˜ k=1
Here p˜ labels primitive periodic orbits (i.e., orbits traversed once), k labels their repetitions, Sp˜(E) is the action of the primitive orbit with the property that the period of the latter is given by Tp˜ = ∂Sp˜/∂E, λp˜ is the instability exponent and μp˜ is the Maslov phase. The formulas (7.83) and (7.84) coincide (apart from an overall minus sign; see [69] for a discussion of the possible significance of this) with the primes as primitive periodic orbits of period log p, while λp˜ = 1 and μp˜ = 0. Also, because time reversal symmetry would require all non-self-retracing orbits to have a partner, the fact that the prefactor in (7.83) is 1/π rather than 2/π implies the analogous quantum system does not possess time reversal symmetry. Note that if we average d(E) over some region [E, E + ΔE] with 1 ΔE E, Proposition 7.5.4 gives ¯ d(E) = d(E). Use of Proposition 7.5.4 also allows d(E)d(E + ε), where · denotes this same average,
305
SCALED LIMITS AT β = 1, 2 AND 4
to be reduced into a form suitable for further analysis. P ROPOSITION 7.5.5 For large E we have ¯ d(E ¯ + ε) + dosc (E)dosc (E + ε) d(E)d(E + ε) ∼ d(E) ¯ d(E ¯ + ε) + dosc (E)dosc (E + ε)diag + dosc (E)dosc (E + ε)off , = d(E) where log2 p 1 e−iε log p , Re 2π 2 p p / 0 log2 p −iε log p 1 e dosc (E)dosc (E + ε)off ∼ 2 Re cos Eh/p , π p h p
dosc (E)dosc (E + ε)diag ∼
hp
p denoting a prime, and where h is such that p + h is prime. Derivation. The first asymptotic equality follows by substituting the formula of Proposition 7.5.4 for d(E) and d(E + ε), ¯ ¯ ¯ and noting that due to the averaging d(E)d osc (E + ε) ∼ 0. The averaging does not affect d(E)d(E + ε) so the first term of the equality follows. This leaves the final asymptotic expressions. Now + * ∞ 1 X X log p log p dosc (E)dosc (E + ε) = 2 cos(Ek log p) cos((E + ε)k log p ) . π pk/2 pk /2 p,p k,k =1
Due to the averaging, we can replace the cosine terms by 1 cos(Ek log p − (E + ε)k log p ) 2 (the term involving (1/2) cos(Ek log p + (E + ε)k log p ) will vanish). Furthermore, again due to the averaging, only terms with k log p ∼ k log p are relevant in the limit E → ∞. Now for either k > 1 or k > 1 the sum over these contributions converges, so we expect the sum to be dominated by the terms with k = k = 1, which formally diverges. Thus + * X log p log p iE(log p−log p )−iε log p 1 . Re e dosc (E)dosc (E + ε) ∼ 2π 2 p1/2 p1/2 p,p P P P Now split the double sum into two parts p,p = p=p + p =p . The first sum gives the first of the last two equations in the proposition. To obtain the last equation, make the ordering p < p (and thus multiply the sum by 2) and notice that the leading order contribution comes from primes p ∼ p. Writing p = p + h, expanding the logarithm to first order in h/p, and noting that for h p, e−iEh/p ∼ cos Eh/p, gives the final stated equation.
A straightforward (heuristic) analysis gives the limiting value of the term ·diag in Proposition 7.5.5 (a rigorous analysis can also be given; in fact the result (7.81) was obtained in such a way). This requires use of the prime number theorem, which asserts that to leading order the number of prime numbers less than X is given by X/ log X. P ROPOSITION 7.5.6 We have lim
E,ε→∞ ¯ ε/d(E)= ε ˜
1 1 dosc (E)dosc (E + ε)diag ∼ − . 2 ¯ 2(πˆ ε)2 (d(E))
Derivation. The basic idea is to replace the sum in the formula ·diag given in Proposition 7.5.5 by an integral. According to the prime number theorem, to leading order the density of primes is 1/ log X, and so Z ∞ X log2 p −iε log p Z ∞ log X −iε log X ∼ dX = (2π)2 τ e−2πiτ ε dτ, e e p X 1 0 p
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CHAPTER 7
where the last equation follows from the change of variables log X = 2πτ . Finally note that for Re(z) > 0 Z ∞ Z ∞ 1 1 τ e−τ z dτ = 2 τ e−τ dτ = 2 , z z 0 0
(7.85)
and using this as an analytic continuation to evaluate the integral.
To mimic the strategy used to derive Proposition 7.5.6 requires an asymptotic formula not for the number of primes less than X, but for the more complicated quantity πm (X), the number of primes p not exceeding X and such that (p − m) is a prime. For this one uses the Hardy-Littlewood conjecture [290], which asserts πm (X) ∼ α(m)
X , log2 X
where α(m) (which was given explicitly by Hardy and Littlewood) has after averaging the large m behavior [353] α(m) ∼ 1 −
1 . 2m
(7.86)
P ROPOSITION 7.5.7 We have lim
E,ε→∞ ¯ ε/d(E)= ε ˆ
1 cos 2πˆ ε dosc (E)dosc (E + ε)off ∼ . 2 2 ¯ 2(πˆ ε) (d(E))
Derivation. The first step is to use the Hardy-Littlewood conjecture to replace the sums over p and h in the expression of Proposition 7.5.5 for ·off by the corresponding density α(k)/ log2 p to give Z ∞ Z dp −iε log p ∞ 1 e dh α(h) cos Eh/p. dosc (E)dosc (E + ε)off ∼ 2 Re π p 1 l(p) Here l = l(p) = O(log p), and the upper terminal of integration in the second integral has been set equal to infinity as only the region h p contributes to the leading order. ¯ ¯ Now make two changes of variables: w = log p/(2π d(E)) and y = (2π)w E 1−w h. Since 2π d(E) ∼ log(E/2π) we w have p ∼ (E/2π) and y ∼ Eh/p. Thus the change of variables gives Z ∞ ¯ 2d(E) E w−1 ¯ dw exp(−2πid(E)εw) dosc (E)dosc (E + ε)off ∼ Re π (2π)w 0 fi „ w−1 «fl Z ∞ yE cos y. × dy α (2π)w w 1−w l(2π) E Consider the case w < 1. When E → ∞, E 1−w → ∞ so the second integral vanishes. When w > 1, E 1−w → 0 and E w−1 → ∞ so we can use (7.86) to conclude fi „ w−1 «fl (2π)w yE ∼1− α . w (2π) 2yE w−1 Multiplying by cos y and integrating, we take the first term to integrate to zero, while the second term is to leading order given by the small y behavior Z ∞ cos y ¯ dy ∼ (w − 1) log E ∼ (w − 1)2π d(E). y l(2π)w E 1−w Thus
Z ∞ ¯ d(E) ¯ ¯ e−2πid(E)εw (w − 1)(2π d(E)) dw. Re π 1 Evaluating the integral by changing variables w − 1 → w and using (7.85) establishes the result. dosc (E)dosc (E + ε) ∼ −
307
SCALED LIMITS AT β = 1, 2 AND 4
Combining the results of Propositions 7.5.5–7.5.7 gives the statement (7.80) of the Montgomery conjecture for εˆ = 0 (the delta function at εˆ = 0 follows from the definition). The above heuristic analysis can also be carried out without taking the limit E → ∞, although E is still assumed to be large. Then it is found that [64], [81] ∞ 1 − m im log pε 2 1 ∂2 dosc (E)dosc (E + ε)diag = − 2 2 log |ζ(1 + iε)|2 exp e , 4π ∂ε m2 p m p m=1 1 (1 − piε )2 ¯ dosc (E)dosc (E + ε)off = 2 |ζ(1 + iε)|2 Re e2πid(E)ε ) . (1 − 2π p−1 p Expanding as a function of ε gives [72] 1 (γ 2 + 2γ1 + c0 ) − 0 + O(ε2 ), 2 2 2π ε 2π 2 1 1 ¯ 2 2 2πid(E)ε , dosc (E)dosc (E + ε)off = 2 Re + (γ + 2γ + c ) + iQ + O(ε ) e 1 0 0 2π ε2 where γ0 , γ1 are specified by dosc (E)dosc (E + ε)diag = −
∞
ζ(1 + x) =
1 (−1)n + γn xn x n=0 n!
while c0 :=
p
(log p)2
∞ (n − 1) , pn n=1
Q :=
log3 p . (p − 1)2 p
¯ With εˆ := ε/d(E) this implies that for E → ∞ ε 1 sin2 πˆ Λ 1 2 , d (E)d (E + ε) = 1 − − sin (πˆ ε ) + O osc osc 2 2 3 ¯ ¯ ¯ (πˆ ε)2 (d(E)) (π d(E)) (d(E))
(7.87)
where Λ := γ02 + 2γ1 + c0 = 1.57314... On the other hand (5.87) and (5.86) give that for matrices from U (N ), with lengths scaled so that ρ = 1, 1 sin2 πx 1 2 ρ(2) (x, 0) = 1 − . (7.88) − sin πx + O (πx)2 3N 2 N4 Comparison of (7.87) and (7.88) gives agreement of the first three terms provided one makes the identification E ¯ 1 π d(E) . (7.89) =√ log N= √ 2π 3Λ 12Λ It should be noted that the U (N ) type correction term in (7.87) can be seen in Odlyzko’s numerical data [72]. 7.5.4 L functions The Riemann zeta function is an explicit example of a general class of functions referred to as L-functions. Such functions are Dirichlet series ∞ an L(s) = (7.90) ns n=1 with the an such that the series converges for Re(s) > 1, and with three additional properties. Briefly, these relate to the analytic continuation, functional equation and Euler product, which are all required to be similar
308
CHAPTER 7
to that for ζ(s). The generalized Riemann hypothesis asserts that all complex zeros of an L-function lie on the critical line Re(s) = 12 . Furthermore there is analytic [472] and numerical [471] evidence to suggest that the statistics of these zeros for large imaginary part agrees with the statistics of eigenvalues from large U (N ) matrices. In addition to this, Katz and Sarnak [352], [351] have related the statistical properties of the zeros on the critical line nearest the real axis for families of L-functions to the eigenvalues of large random matrices from the classical groups near θ = 0. One example of a family of L-functions are those associated with the quadratic Dirichlet characters, an = χd (n) in (7.90), for all allowed d. Here d is the fundamental discriminant, which can be either positive or negative. It has the further property of being square free and congruent to 1 mod 4 if it is odd, and of being 4 times a square free integer congruent to 2 or 3 mod 4 if it is even. The so called character χd (n) only takes on values ±1, 0 and has the periodicity property χd (n + |d|) = χd (n), with |d| being referred to as the conductor. (−2) (−1) Consider now the zeros on the critical line closest to the real axis, 1/2 + iγ (n) with · · · γd < γd < (1) (2) (j) 0 < γd < γd < · · · . Define the scaled variable αd log(|d|/2π), and define an ensemble by allowing d to vary and averaging. Analytic [352], [351] and numerical [471] evidence suggests that the distribution of the resulting zeros is that of the eigenvalues of matrices from the classical group Sp(2N ) for N large, about θ = 0. Moreover, an analogous result is expected to hold true for general families of L-functions, but with the correspondence to L-functions now relating to one of the classical groups U (N ), O+ (2N ), O− (2N + 1) or Sp(2N ). We refer to [403], [115] as starting points for studies into this topic.
7.6 SCALED LIMITS AT β = 4 — GAUSSIAN ENSEMBLE 7.6.1 The bulk ˜
In the Gaussian ensemble e−4V (x) = e−2V4 (
√
2x)
, which implies √ √ √ S4 (x, y) = 2S˜4 ( 2x, 2y).
(7.91)
In the bulk we must scale x and y as in Proposition 7.1.1. It turns out that then only the Christoffel-Darboux term in (6.59) contributes to the leading order asymptotics. P ROPOSITION 7.6.1 For the Gaussian ensemble with β = 4 πρ n πρx πρxn 1 ρ(n) √ ,..., √ lim √ N →∞ 2N 2N 2N ⎡ 2πρ(xj −xk ) sin 2πρ(xj − xk ) sin x 1 dx ⎢ 2πρ(x − x ) 2πρ x j k ⎢ 0 = ρn qdet ⎢ ⎣ ∂ sin 2πρ(xj − xk ) sin 2πρ(xj − xk ) ∂xj
2πρ(xj − xk )
⎤ ⎥ ⎥ ⎥. ⎦
2πρ(xj − xk )
Proof. In view of the above remarks, the only point which requires checking is that the second term in the expression (6.59) for S˜4 (x, y) does not contribute to
√ √ √ lim (πρ/ 2N )S4 (πρx/ 2N , πρy/ 2N ).
N→∞
This is a consequence of the N → ∞ asymptotic estimate [420] with (7.1).
R∞ x
e−t
2
/2
H2N−1 (t)dt = O(22N (N − 1)!) together
309
SCALED LIMITS AT β = 1, 2 AND 4
7.6.2 Properties of the two-point function With ρbulk (2) (x, y) now denoting the scaled two-point correlation function in the bulk, Proposition 7.6.1 gives 2 1 ∂ sin 2πρ(x − y) 2πρ(x−y) sin t T bulk 2 sin 2πρ(x − y) 2 dt. (7.92) +ρ ρ(2) (x, y) = −ρ 2πρ(x − y) 2πρ ∂x 2πρ(x − y) t 0 It follows from this that for small x 2 ρbulk (2) (x, 0) = ρ
while for large x,
(2πρx)4 135
−
2(2πρx)6 2(2πρx)8 + + O((ρx)10 ) , 4725 165375
(7.93)
1 1 3 π cos 2πρx − 1 + O + 2 2πρx (2πρx)2 2(2πρx)4 (ρx)2 1 1 cos 4πρx π sin 2πρx 1+O +O , + − 2 (2πρx)2 2(2πρx)4 (ρx)2 (ρx)5
ρT(2)bulk (x, 0) = ρ2
(7.94)
where the terms O(1/x2 ) do not contain any oscillatory factors. Also, making use of the 2 × 2 determinant form of ρT(2) implied by Proposition 7.6.1, the dimensionless Fourier transform (7.3) can be computed as (see Exercises 7.6 q.1) - |k| |k| |k| 4π − 8π log |1 − 2π |, |k| ≤ 4π, . S(k) = (7.95) 1, |k| ≥ 4π. Note in particular the small |k| behavior |k| k2 . S(k) ∼ + + O(|k|3 ). 4π 16π 2
(7.96)
7.6.3 Perturbation about β = 4 in the bulk The expansion (7.5) tells us the first order correction in (β − β0 ) to the two-point correlation function at coupling β0 in terms of higher order correlations at β0 . As was the case at β0 = 2, these higher order correlations are known for β0 = 4. Also, according to (6.4) the expansion of a quaternion determinant is formally the same as that of an ordinary determinant allowing (7.5) in the case of β0 = 4 to be written in a form analogous to (7.6) and computed exactly. Moreover the Fourier transform of this first order correction can be computed exactly in terms of elementary functions together with the dilogarithm x log t dilog(x) = dt. 1 1−t It has a different analytic form for each of the regions |k| < 2π, 2π < |k| < 4π, |k| > 4π. For |k| < 2π, the exact expression (which is rather lengthy) implies the expansion [207] . β) ∂ S(k; |k|3 |k| 5k 4 3|k|5 27k 6 + = − + + + ∂β β=4 16π 256π 3 3072π 4 4096π 5 81920π 6 37|k|7 1273k 8 887|k|9 4423k 10 + + + 245760π 7 18350080π 8 27525120π 9 293601280π 10 1949|k|11 + + ··· . 275251200π 11 +
(7.97)
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CHAPTER 7
7.6.4 Global limit of the density The global density ρ˜(1) (x) for general Gaussian β-ensembles with eigenvalue p.d.f. (1.28) is defined by (1.53). We read off from (6.55) and (7.91) that for β = 4 √ √ √ √ 2/Nρ(1) ( 2N x) = (2/ N )S˜4 (2 N x, 2 N x). To compute the large N form of this expression, we see from (6.59) that in addition the global limit of ∞to −t 2 K2N , which is already known from (7.25), we must also compute the global limit of x e /2 H2N −1 (t) dt. For this, use of the integral evaluation implied by (7.146) below gives ∞ x 2 π (2N − 1)! −t2 /2 − e H2N −1 (t) dt = e−t /2 H2N −1 (t) dt. 2 (N − 1/2)! x 0 The global limit of the integral on the r.h.s. can now be analyzed using the Plancheral-Rotach formula (7.24), and the following result obtained [341], [202]. √ P ROPOSITION 7.6.2 Let −1 < x < 1, set AN (x) := cos(2N x 1 − x2 + (2N + 12 )arcsin x − N π) and let ρW (x) denote the Wigner semicircle density. One has 1 √ 1 1 (−1)N cos AN (x) √ 2/Nρ(1) ( 2N x) = ρW (x) − √ + + O(N −3/2 ). (7.98) + 2 1/4 2πN (1 − x ) 4πN 1 − x2 2πN 7.6.5 Soft edge Here we want to study (7.91) in the scaled limit with x and y given by (7.11), and thus to compute 1 X Y S4soft (X, Y ) := lim 1/2 1/6 S4 (2N )1/2 + 1/2 1/6 , (2N )1/2 + 1/2 1/6 . N →∞ 2 N 2 N 2 N Now, use of the asymptotic formula (7.68) shows that 1 X Y (G) lim 1/2 1/6 K2N 21/2 ((2N )1/2 + 1/2 1/6 ), 21/2 ((2N )1/2 + 1/2 1/6 ) N →∞ 2 N 2 N 2 N −1/3 soft 2/3 2/3 K (2 X, 2 Y ), =2 where K soft is given by (7.12). Writing out the explicit form of the second term in (6.59), substituting in (7.91) and introducing the scaled variables gives ∞ √ √ 2 2−2N −1/2 −y2 −√ e−t H2N −1 ( 2t) dt x →(2N )1/2 +X/21/2 N 1/6 =: A4 (X, Y ). e H2N ( 2y) N Γ(2N ) x y →(2N )1/2 +Y /21/2 N 1/6 Use of the asymptotic expansion (7.9) shows 1 1 A4 (X, Y ) = − 2/3 Ai(22/3 Y ) N →∞ 21/2 N 1/6 2
∞
lim
Ai(22/3 v) dv.
X
Hence S4soft (X, Y ) =
1
2
K soft (22/3 X, 22/3 Y ) − 1/3
1
2
Ai(22/3 Y ) 2/3
∞
Ai(22/3 v) dv.
(7.99)
X
The density is obtained by setting X = Y in (7.99). According to (7.68) the second term decays as X → −∞. This implies √ 1 X soft 2/3 2/3 ρsoft , (7.100) (−X) ∼ K (−2 X, −2 X) ∼ (1) X→∞ 21/3 X→∞ π which is the same leading order behavior as that exhibited at β = 2 (recall (7.69)). Another feature of interest
311
SCALED LIMITS AT β = 1, 2 AND 4
relating to the density is the correction terms to the soft edge scaled form. According to (7.27) the leading correction to the first term in (6.59) is O(N −2/3 ). The correction to the second term is found by making use of (7.9). It is O(N −1/3 ), and gives the expansion [202] 2 1 X 1 soft,4 1/2 2/3 (2N ) ∼ ρ Ai(2 ρ + (X) + X) + O(N −2/3 ). (7.101) (1) (1) 21/2 N 1/6 21/2 N 1/6 24/3 (2N )1/3 From Proposition 6.1.7 we have that the truncated two-particle correlation is given by ∂ Y T soft soft soft soft S (X, Y ) ρ(2) (X, Y ) = −S4 (X, Y )S4 (Y, X) − S4soft (X, Y ) dY . ∂X 4 X
(7.102)
We can check from (7.68) that only K soft in the formula (7.99) for S4soft contributes to the leading nonoscillatory behavior of ρT(2) (X, Y ) for X, Y → −∞. One finds ρT(2)soft (−X, −Y )
∼
X,Y →∞
−
8π 2
X +Y 1 √ , XY (X − Y )2
(7.103)
which is exactly one half of the asymptotic expression (7.70) found for the same quantity at β = 2. E XERCISES 7.6
1.
(i) Let
Z FT f = fˆ(k) :=
∞
f (x)eikx dx.
−∞
Verify that
Z
∞ −∞
f (y − x)g(x)eikx dx =
(ii) Let
1 2π
2
sin 2πx 6 2πx f4 (x) := 4 ∂ sin 2πx ∂x 2πx ( 1 , sin 2πx 2 FT = 2πx 0,
Use the formula
to check that
Z
2
1 6 2 FT f4 (x) = 4 ik − 2
3 i 2k 7 (|k| < 2π), 1 5 2
∞ −∞
Z
e−ily fˆ(l − k)ˆ g (l) dl.
(7.104)
3 sin 2πt dt 7 2πt 5. sin 2πx 2πx
x 0
|k| < 2π, |k| > 2π » FT f4 (x) =
0 0
0 0
–
(iii) Use the result (7.104) with y = 0, together with the result of (ii) to show “ ” FT f4 (x)f4 (−x) = 8 2 3 ˛ ˛ |k| |k| ˛ |k| ˛ > > 1− + log ˛1 − 0 ˛ > 7 > 6 4π 8π 4π > < 4 ˛ ˛ 5, |k| ˛ |k| |k| ˛ + log ˛1 − 0 1− ˛ > 4π 8π 2π » – > > > 0 0 > : , |k| > 4π 0 0
(|k| > 2π).
|k| < 4π, (7.105)
and thus deduce (7.95). 2. [221] Use the integral form of the kernels given in Exercises 7.1 q.1 to show that for scaled = bulk or soft Z ∞ 1 1 ∂ S˜4scaled (X, Y ) = K scaled (X, Y ) + K scaled (t, Y ) dt, (7.106) 2 2 ∂Y X
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CHAPTER 7
where S˜4bulk (x, y) = S4bulk (x/2, y/2), S˜4soft (x, y) = 21/3 S soft (x/22/3 , y/22/3 ) and thus 1“ ∂ ∂ ˜scaled ∂ ” scaled (X, Y ) = (X, Y ), S4 − K ∂X 2 ∂X ∂Y Z Y Z ∞ Z ∞ ” “ 1 S˜4 (X, u) du = − K scaled (X, t) dt − K scaled (Y, t) dt . 2 Y X X
(7.107)
7.7 SCALED LIMITS AT β = 4 — LAGUERRE AND JACOBI ENSEMBLES 7.7.1 Hard edge In the Laguerre case
˜ e−4V (x) ∝ e−2V4 (2x)
a →2a−1
and so
S4 (x, y) = 2S˜4 (2x, 2y)
a →2a−1
.
(7.108)
Furthermore, the explicit form of S˜4 is, from (6.59), ∞ 1 x 1/2 (L) (2N )! y (a−1)/2 e−y/2 La2N (y) K2N (x, y) + t(a−1)/2 e−t/2 La2N −1 (t) dt. S˜4 (x, y) = 2 y 4Γ(a + 2N ) x (7.109) The scaling at the hard edge is as specified in Proposition 7.2.1, so we seek to compute X Y 1 S4hard (X, Y ) := lim S4 , . N →∞ 4N 4N 4N Now, according to Proposition 7.2.1 1 (L) X Y K2N , = 2K hard(4X, 4Y ), lim N →∞ 4N 2N 2N while the asymptotic form of the second term in (7.109) is deduced from (7.29), giving in total X 1/2 J2a−1 (2Y 1/2 ) ∞ S4hard (X, Y ) = 2 K hard (4X, 4Y ) + J2a−1 (2t) dt. (7.110) Y a →2a−1 2Y 1/2 X 1/2 Bessel function identities applied to this give the alternative form S4hard (X, Y
) = 2K
hard
(4X, 4Y )
J2a−1 (2Y 1/2 ) − a →2a 2Y 1/2
X 1/2
J2a+1 (2t) dt
(7.111)
0
(see also Exercises 7.7 q.3). Note that in the case a = 0 the integral in (7.111) can be evaluated, and thus we have the simplified result 1 1/2 1/2 S4hard (X, Y ) = 2K hard (4X, 4Y ) − J (2Y ) J (2X ) − 1 . (7.112) 1 0 a=0 a=0 4Y 1/2 In the case of the Jacobi ensemble a similar calculation shows 1 ˜(J) 1 ˜(L) X Y X Y , , = lim ,1 − S4 1 − lim S4 2 2 2 N →∞ 2N N →∞ 4N 2N 2N 4N 4N which is to be expected in accordance with the remarks following (7.45).
313
SCALED LIMITS AT β = 1, 2 AND 4
7.7.2 Asymptotics of the one- and two-point functions From Proposition 6.1.7 the density in the scaled limit near the hard edge is given by hard (X, X) ρhard (1) (X) = S4 1 (J2a+1 (2X 1/2 ))2 − J2a+2 (2X 1/2 )J2a−2 (2X 1/2 ) = 2 ∞ 1 1/2 J2a−1 (2X ) J2a−1 (2s) ds, + 2X 1/2 X 1/2
(7.113)
where the second equality follows from (7.111) and (7.33). For large X, use of (7.73) shows that the leading order decay of S4hard (X, X) is given by the leading order behavior of 2K hard (4X, 4X). Use of (7.74) then gives ρhard (1) (X)
∼
X→∞
1 , 2πX 1/2
(7.114)
which is identical to the behavior exhibited at β = 2. The truncated two-particle correlation is given by (7.102) with S4soft replaced by S4hard . For the leading asymptotics, we again see from (7.73) that the leading behavior of S4hard (X, Y ) is given by the leading behavior of 2K hard (4X, 4Y ). Substituting this in the modified form of (7.102) shows that ρT(2)hard (X, Y )
∼
X,Y →∞
−
8π 2
X +Y 1 √ XY (X − Y )2
(7.115)
independent of a. This is the same behavior as found in (7.103) for the soft edge at β = 4.
7.7.3 Laguerre ensemble — global and soft edge densities For finite N the density is given by (7.108) with x = y. The global density is then four times this expression with x → 4N X (recall (3.57)). The first term in (7.109) is the Christoffel-Darboux kernel, which has been analyzed in this limit to give (7.39). For the second term, use of (7.131) below allows the integral over (x, ∞) to be replaced by minus the same integral over (0, x). This latter integral can be analyzed using the Plancheral-Rotach type asymptotic expansion (7.38), with the final result being [202] (L)
4ρ(1) (4N X) ∼ ρ
MP
g0,2N (X)|a →2a−1 1 a (X) − + + o(N −1 ), 2(πN )1/2 X 3/4 (1 − X)1/4 2πN X(1 − X)
(7.116)
(L)
where g0,2N (X) is specified by (7.38), and ρMP (X) is as in (7.39). At the soft edge, the expansion of the first term in (7.109) is given by (7.37), while that of the second term is determined by making use of (7.35). With ρsoft (1) (X) as in (7.101), one finds [202] 2(2N )1/3 ρ(1) (4N + 2(2N )1/3 X) ∞ a 2/3 2/3 2 (Ai(2 ∼ ρsoft,4 (X) + X)) + Ai (2 X) Ai(t) dt + O(N −2/3 ). (7.117) (1) 2N 1/3 2/3 X 2
7.7.4 Circular Jacobi ensemble — spectrum singularity We recall from (3.123) that to study the log-gas on a circle with weight (5.50), it is sufficient to study the log-gas on a line with Cauchy weight as specified by (5.56). According to (3.123) the parameter α in (5.56) is dependent on N , β as well as the parameter a in (5.50). For purposes of using the general formula (6.59),
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CHAPTER 7
it is convenient to define α independent of β, by setting e−2V (x) =
1 , (1 + x2 )α
α = N + a,
(7.118)
which according to (3.123) is indeed the correct choice at β = 2. Then (6.53) gives ˜
e−2V4 (x) =
1 , (1 + x2 )α−1
(7.119)
and comparison with (3.123) shows that the correct weight function for the Cauchy ensemble equivalent to ˜ the circular Jacobi ensemble at β = 4 is e−2V4 (x) | α →2α . This means that the quantity S˜4 (x, y)| α →2α gives α=N +a α=N +a the sought n-point correlation at β = 4. Now, with the weight function (7.118), use of (7.119) and (5.65) in (6.59) shows 1 1 + x2 1/2 (C) S˜4 (x, y) = K2N (x, y) 2 1 + y2 ∞ (1 + y 2 )−(α+1)/2 p2N −1 (t) (α − 2N )p2N (y) dt, (7.120) − 2(p2N −1 , p2N −1 )2 (1 + t2 )(α+1)/2 x where (C)
(CJ) K2N (x, y)
(C)
(C)
(C)
p2N (x)p2N −1 (y) − p2N (y)p2N −1 (x) 1 . := x−y (1 + x2 )α/2 (1 + y 2 )α/2
As in the analysis leading to (7.48), our task is to compute the thermodynamic limit of S˜4 with 1+z 1+w , z := e2πix/L , w = e2πiy/L . y → i (7.121) 1−z 1−w Analogous to (7.46) we have 1 + z 2 2N (CJ) p2N i α →2α = i−2N − 2 F1 (−2N, 2a; 4a; 1 − z), 1 − z α=N +a 1−z 1 + z 2 2N −1 (C) p2N −1 i α →2α = i−(2N −1) − 2 F1 (−(2N − 1), 2a + 1; 4a + 2; 1 − z), 1 − z α=N +a 1−z (7.122) x → i
(C)
while (p2N −1 , p2N −1 )2
is specified by (5.54). Making use of the asymptotic formula (7.47) shows 1 + z 1 + w 2πρ|xy| s.s. (CJ) ,i K (2x, 2y) , (7.123) K2N i α →2α ∼ 1 − z 1 − w α=N +a N a →2a
where K s.s. is specified by (7.48). Regarding the second line in (7.120), making the substitutions (7.121) and (7.122), a similar analysis which leads to (7.123) shows that the leading asymptotic form is πρx a − (πρy)3/2 J2a−1/2 (2πρy) s−1/2 J2a+1/2 (2s) ds. N 0 Hence
1 + z 1 + w πρy 2 s.s. ,i S (x, y), S˜4 i α →2α ∼ 1 − z 1 − w α=N +a N 4 S4s.s. (x, y) := K s.s. (2x, 2y)
a →2a
− aπρ
J2a−1/2 (2πρy) (πρy)1/2
0
πρx
s−1/2 J2a+1/2 (2s) ds.
(7.124)
(7.125)
315
SCALED LIMITS AT β = 1, 2 AND 4
˜ 4 as specified by (6.15). One The result (7.124) can be used to deduce the asymptotic behavior of I˜4 and D finds y 2 ∂ 1 + z 1 + w ˜ 4 i 1 + z , i 1 + w ∼ πρxy I˜4 i ,i ∼− S s.s. (x, y). S4s.s. (x, u) du, D 1−z 1−w 1−z 1−w N ∂x 4 x We substitute these asymptotic forms in the expression of (6.14) for f4 , form qdet according to Proposition 6.1.7, and extract common factors from the odd numbered columns and even numbered rows which give an overall contribution n πρ 2 x . (7.126) N j j=1 This procedure gives the leading asymptotic form in the thermodynamic limit of ρ(n) as specified by the first equality in (5.55). The factor (7.126) cancels with the prefactors on the r.h.s. of the first equality in (5.55), leaving the final expression [213] x − xjk S4s.s. (xj , u) du S4s.s. (xj , xk ) s.s. ρ(n) (x1 , . . . , xn ) = qdet . (7.127) ∂ s.s. S4s.s. (xk , xj ) ∂xj S4 (xj , xk ) j,k=1,...,n
When a = 0 the circular Jacobi ensemble reduces to the circular ensemble of Dyson. In this case (7.127) should agree with the scaled n-point correlation given in Proposition 7.6.1 for the Gaussian ensemble in the bulk. Indeed, we see from (7.125), (7.48) and (7.50) that sin 2πρ(x − y) S4s.s. (x, y) , =ρ 2πρ(x − y) a=0 which when substituted in (7.127) gives the result of Proposition 7.6.1. Another case of interest is a = 1. Then the formula (7.56) must apply, with the l.h.s. given by Proposition 7.6.1, and the r.h.s. by (7.127). In particular, taking n = 1 we must have sin 2πρx 2 1 ∂ sin 2πρx 2πρx sin t S4s.s. (x, 0) dt. (7.128) =ρ−ρ +ρ 2πρx 2πρ ∂x 2πρx t a=1 0 This can be checked using (7.125) and (7.49). E XERCISES 7.7
1. Analogous to Exercises 7.1 q.2, verify that √ √ πρ lim √ S4soft (−(α + πρx/ α), −(α + πρy/ α)) α √ √ √ = lim 2πρ aS4hard (a + 2πρx a, a + 2πρy a) = S4bulk (x, y),
α→∞
a→∞
lim 2a(a/2)1/3 S4hard (a2 − 2a(a/2)1/3 x, a2 − 2a(a/2)1/3 y) = S4soft (x, y),
a→∞
and with ρ = 1/π lim (a/2)1/3 S4s.s. (a − (a/2)1/3 x, a − (a/2)1/3 y) = S4soft (x, y).
a→∞
2. [3] The objective of this exercise is to verify that (7.109) can be rewritten as [544], [216] ˛ ˛ (L) (L) 2S˜4 (x, y) = K2N (x, y)˛ a→a+1 (a+1)/2 −y/2 La+1 (y) − La+1 (y) Z x e d (2N )!y 2N 2N−1 + t(a+1)/2 e−t/2 La+1 (t) dt. (7.129) 2Γ(2N + a + 1) y dt 2N 0 (y), k = 0, . . . , 2N in (7.109) and (7.129). The general strategy is to compare coefficients of y (a−1)/2 e−y/2 La+1 k
316
CHAPTER 7
This requires using the Laguerre polynomial identities La−1 (x) = Lan (x) − Lan−1 (x), n d a Ln (x) = −La+1 n−1 (x), dx a+1 a+1 xLa+1 (x) + (2n + a)La+1 n−1 (x) = −nLn n−1 (x) − (n + a)Ln−2 (x),
(7.130)
and the definite integral (see (6.113) and (6.114)) Z
(
∞
x
a/2 −x/2
e
La+1 (x) dx n
0
=
Γ((n+3)/2))Γ(a+n+2) , 2a/2−1 Γ(n+2)Γ((n+a+3)/2)
n even,
0,
n odd.
(7.131)
(i) Use the first and third identities in (7.130) to show that the coefficient of y (a−1)/2 e−y/2 La2N (y) in (7.129) is Z x 2 a+1 ” “d (2N )! 1 (a+1)/2 −x/2 ((2N − 1)!) L2N−1 (x) e t(a+1)/2 e−t/2 x (−2N )+ La+1 2N (t) dt, 2 Γ(2N )Γ(a + 1 + 2N ) 4Γ(2N + a + 1) 0 dt and read off that the same coefficient in (7.109) is Z ∞ (2N )! t(a+1)/2−1 e−t/2 La2N−1 (t) dt. 4Γ(a + 2N ) x Use the definite integral (7.131) to show that both these expressions agree when x = 0. Then use the identities (7.130) to show that the two expressions have the same derivative. (ii) Proceed as in (i) to show that for k < 2N , the coefficient of y (a−1)/2 e−y/2 Lak (y) in (7.129) is ” “ k! a+1 (x) x(a+1)/2 e−x/2 La+1 k+1 (x) − Lk 2Γ(a + 1 + k) while in (7.109) the coefficient is k! x(a+1)/2 e−x/2 Lak (x), 2Γ(a + 1 + k) and note from the first identity in (7.130) that these expressions are equal. (iii) Use (7.129) to show that an alternative expression for (7.110) is (7.111).
7.8 SCALED LIMITS AT β = 1 — GAUSSIAN ENSEMBLE 7.8.1 The bulk In the Gaussian ensemble V˜1 = V and so S˜1 (x, y) = S1 (x, y). Thus from (6.107) we read off that for N even 2 2 e−y /2 HN −1 (y) ∞ (G) (G) ˜ S1 (x, y) = KN −1 (x, y) + N 1/2 sgn(x − t)e−t /2 HN −2 (t) dt. (7.132) 2 π (N − 2)! −∞ Proceeding as in the proof of Proposition 7.6.1 we can obtain from this formula the scaling limit of the n-point correlation function in the bulk.
317
SCALED LIMITS AT β = 1, 2 AND 4
P ROPOSITION 7.8.1 For the Gaussian ensemble at β = 1 we have πρ n πρx πρxn 1 ρ(n) √ , . . . , √ lim √ N →∞ N N N ⎡ πρ(xj −xk ) 1 sin πρ(xj − xk ) sin t 1 dt − sgn(xj − xk ) ⎢ πρ(x − x ) πρ t 2ρ n j k 0 = ρ qdet ⎢ ⎣ ∂ sin πρ(xj − xk ) sin πρ(xj − xk ) ∂xj πρ(xj − xk ) πρ(xj − xk )
⎤ ⎥ ⎥ ⎦
.
j,k=1,...,n
7.8.2 Properties of the two-point function The truncated two-particle correlation, according to Proposition 7.8.1, is given by πρx sin t sin2 πρx 1 d sin πρx π T 2 ρ(2) (x, 0) = ρ − − sgn x + dt . + (πρx)2 πρ dx πρx 2 t 0 This implies the small x expansion ρ(2) (x, 0) = ρ2
π 6
|πρx| −
π 1 |πρx|3 + (πρx)4 − · · · 60 135
and the large x asymptotic expansion 1 cos 2πρx 1 1 3 1 + O + 1 + O , + ρT(2) (x, 0) = ρ2 − (πρx)2 2(πρx)4 x2 2(πρx)4 x2
(7.133)
(7.134)
(7.135)
where the terms O(1/x2 ) do not contain any oscillatory factors. Analogous to (7.95), the corresponding dimensionless Fourier transform can be computed in terms of elementary functions [395], ⎧ ⎨ |k| − |k| log 1 + |k| , |k| ≤ 2π, π 2π π . S(k) = (7.136) ⎩ 2 − |k| log |k|/π+1 , |k| ≥ 2π, 2π |k|/π−1 which exhibits the small |k| expansion |k| k2 . S(k) ∼ − 2 + O(k 3 ). π 2π
(7.137)
7.8.3 Perturbation about β = 1 in the bulk As in the cases of β0 = 2 and β0 = 4, the explicit form of the n-point correlation (n = 2, 3, 4) can be used to compute the first order perturbation of the dimensionless two-point correlation about β0 = 1, or equivalently the first order perturbation of the dimensionless structure function. However, as far as the |k| < π form of the ˜ β) in Section 13.7.4 below shows that latter goes this is superfluous, as an integral formula obtained for S(k; if we set πβ f (k; β) := S(k; β), 0 < k < min (2π, πβ) (7.138) |k| and define f for k < 0 by analytic continuation, then
2k 4 ; . f (k; β) = f − β β
(7.139)
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CHAPTER 7
Thus if we write ∞ |k| j πβ ˜ Aj (β/2) , S(k; β) = 1 + |k| πβ j=1
|k| < min(2π, πβ)
(7.140)
the coefficients Aj (β/2) must satisfy the functional relation Aj (1/x) = (−x)−j Aj (x).
(7.141)
˜ β)/∂β|β=4 we can use this to deduce the corresponding In particular, knowing the expansion (7.97) of ∂ S(k; ˜ β)/∂β|β=1 . expansion of ∂ S(k; 7.8.4 Global limit of the density Suppose for definiteness that N is even, and write ΦN (x) = 2
−N
x
e−t
2
/2
HN (t) dt.
(7.142)
0
Using this in (7.132) gives, after simple manipulation, 2−(N −1) e−x e−x /2 (G) (HN −1 (x))2 + √ HN −1 (x)ΦN −2 (x). ρ(1) (x) = KN (x, x) − √ π(N − 1)! 2 π 2
2
(7.143)
(G)
The global asymptotic expansion of KN (x, x) is given by (7.25), while that of the other terms can be deduced by making use of (7.24). These together show [341] √ 1 √ 2/N ρ(1) ( 2N X) = ρW (X) − + O(N −2 ). (7.144) 2πN 1 − X 2 Inspection of (7.25), (7.98) and (7.144) shows that with oscillatory terms ignored they can be combined into the single formula √ 1 1 1 1 √ − 2/Nρ(1) ( 2N X) ∼ ρW (X) − , −1 < X < 1. (7.145) πN β 2 1 − X2 Macroscopic arguments will be used in Section 14.2 below to derive this formula for general β, and to extend it to all real X = 1. 7.8.5 Soft edge The task here is to evaluate the scaled limit of (7.132) with coordinates (7.11). For the first term the limit is equal to K soft (X, Y ) as specified by (7.12). To compute the scaled limit of the second term we make use of (6.113) to rewrite (7.142) as ∞ 2 π N! −N ΦN (x) = − 2 e−t /2 HN (t) dt, (7.146) N 2 2 (N/2)! x and then use the asymptotic expansion (7.9). This gives ∞ 1 1 √ −y 2 /2 1 √ HN −1 (y)ΦN −2 (x) x → 2N +X/21/2 N 1/6 = Ai(Y ) 1 − Ai(t) dt , lim 1/2 1/6 e √ N →∞ 2 2 π 2 N X y → 2N +Y /21/2 N 1/6 and thus, in an obvious notation S1soft (X, Y
)=K
soft
∞ 1 (X, Y ) + Ai(Y ) 1 − Ai(t) dt . 2 X
(7.147)
319
SCALED LIMITS AT β = 1, 2 AND 4
The density is given by (7.147) with X = Y . Since then the second term decreases as X → −∞, we see that √ X soft , (7.148) ρ(1) (−X) ∼ K (−X, −X) ∼ X→∞ X→∞ π which is identical to the leading asymptotic behavior of the soft edge density at β = 2 and β = 4. The asymptotic expansion of the density at the soft edge can be computed starting from (7.143). We know from (7.27) that the corrections to the first term therein are O(N −2/3 ). Using (7.146) and (7.9) shows the remaining terms have corrections O(N −1/3 ), giving the result [202] ∞ X 1 1 soft,1 1/2 (2N ) ∼ ρ ρ + (X) + Ai (X) 1 − Ai(t) dt + O(N −2/3 ). (1) (1) 21/2 N 1/6 21/2 N 1/6 2N 1/3 X (7.149) From Propositions 6.3.2 and 6.3.3 the truncated two-particle correlation is given by 1 ∂ soft S (X, Y ) ρT(2) (X, Y ) = −S1soft (X, Y )S1soft (Y, X) + 2 ∂X 1 ∞ × S1soft (X, z)sgn(z − Y ) dz − sgn(X − Y ) .
(7.150)
−∞
Use of (7.68) shows that only K soft (X, Y ) in the formula (7.147) contributes to the leading non-oscillatory behavior of (7.150), which one calculates as X +Y . (7.151) (X − Y )2 XY Note that this is the same asymptotic behavior found in (7.70) and (7.103) provided we replace the factor of two in the denominator by 2β. ρT(2) (−X, −Y )
E XERCISES 7.8
∼
X,Y →∞
−
2π 2
1 √
1. From the definition (7.138) read off from (7.136) and (7.95) that “ “ 1 1 k” k ” , f (k; 4) = 1 − log 1 − f (k; 1) = 1 − log 1 + 2 π 2 2π
(7.152)
and thus illustrate (7.139) for β = 1. 2. [221] Use the integral form of the kernel given in Exercises 7.1 q.1 to show that for scaled = bulk or soft Z X 1 1 ∂ K scaled (t, Y ) dt (7.153) S1scaled (X, Y ) = K scaled (X, Y ) − 2 2 ∂Y −∞ (cf. (7.106)) and thus the equations (7.107) hold with S˜4scaled replaced by S1scaled .
7.9 SCALED LIMITS AT β = 1 — LAGUERRE AND JACOBI ENSEMBLES 7.9.1 Hard edge
In the Laguerre case at β = 1 we have V (x) = V˜1 (x)
a →a+1
S1 (x, y) = S˜1 (x, y)
and so
a →a+1
.
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CHAPTER 7
Making use of (6.107) we thus have that for N even x 1/2 (L) KN −1 (x, y) S1 (x, y) = y a →a+1 ∞ (N − 1)! a/2 −y/2 a+1 a/2 −u/2 y e − LN −1 (y) sgn(x − u)La+1 e du N −2 (u)u 4Γ(a + N ) 0 x 1/2 (N − 1)! a/2 −y/2 a+1 (L) y e = KN −1 (x, y) LN −1 (y) − y 4Γ(a + N ) a →a+1 x Γ((N + 1)/2)Γ(a + N ) a/2 −u/2 , × 2 La+1 (u)u e du − N −2 2a/2−1 Γ(N )Γ((N + 1 + a)/2) 0
(7.154)
where the second equality follows from (7.131). Making use of (7.29) then shows X Y 1 S1 , S1hard (X, Y ) := lim N →∞ 4N 4N 4N X 1/2 X 1/2 Ja+1 (Y 1/2 ) hard = 1 − K (X, Y ) + J (v) dv , (7.155) a+1 Y a →a+1 4Y 1/2 0 where K hard is given by (7.31). Bessel function identities applied to this expression give the alternative form [216] X 1/2 Ja+1 (Y 1/2 ) hard hard S1 (X, Y ) = K 1 − (X, Y ) + J (v) dv (7.156) a−1 4Y 1/2 0 (see also Exercises 7.9 q.1). Both (7.155) and (7.156) have the feature that in the case a = 0 the integrals therein can be evaluated explicitly. Doing this in (7.156) gives [418] √ √ 1 S1hard (X, Y ) = K hard (X, Y ) − √ J0 ( X)J1 ( Y ). (7.157) a=0 a=0 4 Y As in the β = 4 theory, for the Jacobi ensemble it can be shown that [415] 1 (J) 1 (L) X Y X Y S1 , . = lim S1 1 − ,1 − lim 2 2 2 N →∞ 2N N →∞ 4N 2N 2N 4N 4N 7.9.2 Asymptotics of the one- and two-point functions From Propositions 6.3.3, 6.3.2 and (7.33), (7.155), we have that for general a > −1 the scaled density at a point X from the hard edge is given by 1 1/2 2 1/2 1/2 ρhard (J (X) = (X )) − J (X )J (X ) a+1 a+2 a (1) 4 √ √X Ja+1 ( X) √ Ja+1 (u) du − 1 . (7.158) − 4 X 0 The asymptotic expansion (7.73) allows the leading large X non-oscillatory behavior of (7.158) to be computed, analogous to the determination of the asymptotics of (7.113). Again the result (7.74) is obtained. This suggests a universality property: in general the large X behavior of ρ(1) (X) near the hard edge will be given by (7.74), independent of the value of β and a. The truncated two-particle correlation function is given by the formula (7.150) with S1soft replaced by hard S1 and the lower terminal of integration −∞ replaced by 0. From the fact that the large X, Y asymptotics of S1hard come from K hard (X, Y ), the (modified) formula (7.150) shows that the asymptotics (7.151) persists independent of a.
321
SCALED LIMITS AT β = 1, 2 AND 4
7.9.3 Laguerre ensemble — global and soft edge densities For finite N (assumed even) the density is given by setting x = y in (7.154). Following the strategy detailed in Section 7.7.3 for the β = 4 Laguerre ensemble density, this expression can be analyzed globally by setting x = 4N X, and at the soft edge by setting x = 4N + 2(2N )1/3 X. The final results are [341], [202] 4ρ(1) (4N X) ∼ ρMP (X) +
a 1 + o(N −1 ), 2πN X(1 − X)
(7.159)
2(2N )1/3 ρ(1) (4N + 2(2N )1/3 X) ∞ a soft,1 2 Ai + O(N −2/3 ). (7.160) ∼ ρ(1) (X) − (X)(1 − Ai(s) ds) − (Ai(X)) 2(2N )1/3 X Comparison of these formulas with their Laguerre counterparts at β = 2 and 4 reveals some common features. Consider first the global density. Ignoring oscillatory terms, inspection of (7.39), (7.116) and (7.159) shows that they all satisfy a 4ρ(1) (4N X) ∼ ρMP (X) + 0 < X < 1. (7.161) + o(N −1 ), 2πN X(1 − X) As discussed in Section 14.2 below, this is closely related to the expansion (7.145). In relation to the expansion of the density at the soft edge, we see from (7.37), (7.117) and (7.160), together with the explicit functional forms of ρsoft,β that they all satisfy (1) 2(2N )1/3 ρ(1) (4N + 2(2N )1/3 X) ∼ ρsoft,β (1) (X) −
d soft,β a ρ (X). 1/3 (2N ) dX (1)
(7.162)
This is in distinction to the analogous expansions for the Gaussian ensembles at the soft edge, which reveal no such structure.
7.9.4 Circular Jacobi ensemble — spectrum singularity Considerations analogous to those of Section 7.7.4 show that the correct weight function for the Cauchy ensemble equivalent to the circular Jacobi ensemble is ˜
e−V1 (x) =
1 , (1 + x2 )(α+1)/2
α = N + a.
(7.163)
Thus the n-point correlation can be computed from the quantity S˜1 (x, y)|N +α=a with e−2V (x) given by (7.118). Use of (7.118), (7.163) and (5.65) in (6.107) shows this is given by 1 + x2 1/2 (C) S˜1 (x, y) = KN −1 (x, y) 1 + y2 N +a=α ∞ (C) (C) sgn(x − t)pN −2 (t) (a + 1)pN −1 (y) dt. (7.164) + (C) 2 (α+1)/2 (pN −2 , pN −2 )2 (1 + y 2 )(α+1)/2 −∞ (1 + t ) Regarding the thermodynamic limit, proceeding as in the derivation of (7.123) we readily find 1 + z 1 + w πρxy (C) ,i Ks.s. (x, y) KN −1 i ∼ . 1 − z 1 − w α=N +a N a →a+1 The asymptotic analysis of the second line in (7.164) is complicated by the region of integration being the whole real line. Analogous to the manipulations in (7.146) and (7.154), with x → −i(1 + z)/(1 − z) =
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CHAPTER 7
− cot πx/L we write ∞ ∞ (C) (C) sgn(x − t)pN −2 (t) sgn(x − t)pN −2 (t) dt =− dt 2 (N +a+1)/2 2 (N +a+1)/2 x →− cot πx/L x →cot πx/L −∞ (1 + t ) −∞ (1 + t ) =−
∞
−∞
(C)
pN −2 (t) dt + 2 (1 + t2 )(N +a+1)/2
(C)
∞
cot πx/L
pN −2 (t) dt. (1 + t2 )(N +a+1)/2
(7.165)
The definite integral is evaluated according to the general formula (6.113), which after recalling (6.97) and (5.65) gives N/2−2 ∞ ∞ (C) N + a − 1 − 2j (p2j+1 , p2j+1 )(C) pN −2 (t) dt 2 dt = 2 (N +a+1)/2 2 (N +a+1)/2 (C) N + a − 2j −∞ (1 + t ) −∞ (1 + t ) (p2j , p2j )2 j=0 = 2a+2 N −a−2
Γ(a/2 + 1)Γ(a + 5/2) , Γ(a/2 + 3/2)
(7.166)
where to obtain the first equality use has been made of (5.54), while the duplication formula (1.111) has been used in obtaining the final line. The second integral in (7.165), along with the terms outside the integral in (7.164), are analyzed as in Section 7.2.6. Combining results, and substituting in (7.164) we find the asymptotic behavior 1 + z 1 + w πρy 2 s.s. S˜1 i ,i S (x, y), ∼ (7.167) 1 − z 1 − w α=N +a N 1 S1s.s. (x, y) = Ks.s. (x, y)
πρ(a + 1)Γ(a/2 + 1) (πρy)−1/2 Ja+1/2 (πρy) 23/2 Γ(a/2 + 3/2) πρx 1/2 Γ(a/2 + 3/2) × 1−2 s−1/2 Ja+3/2 (s) ds . Γ(a/2 + 1) 0
a →a+1
+
(7.168)
Substituting (7.167) in the formulas of Proposition 6.3.2 for D1 and I˜1 and substituting this n-point function in (5.55), after cancellation of the prefactor in the first equality of (5.55), we see that in the thermodynamic limit the n-point correlation about the spectrum singularity at β = 1 is given by [213] x − xjk S1s.s. (xj , u) du − 12 sgn(xj − xk ) S1s.s. (xj , xk ) ρ(n) (x1 , . . . , xn ) = qdet . ∂ s.s. S1s.s. (xk , xj ) ∂xj S1 (xj , xk ) j,k=1,...,n
(7.169) Analogous to the corresponding correlation at β = 4 (7.127), we can check from (7.168) that with a = 0 the result of Proposition 7.8.1 is obtained, while with a = 1 the analogue of (7.128) holds.
7.9.5 Superimposed β = 1 systems The formula (6.152) can be used to compute the scaled limit of (6.151) at the soft edge of the Gaussian ensemble. Proceeding as in Sections 7.6.5 and 7.8.5 one finds [195] odd(OEsoft)2
ρ(n)
(x1 , . . . , xn ) n √ 1 x1 xn odd(GOEN )2 √ := lim ρ(n) 2N + 1/2 1/6 , . . . , 2N + 1/2 1/6 1/2 1/6 N →∞ 2 N 2 N 2 N ∞ Ai(xk − v) dv . = det K soft (xj , xk ) + Ai(xj )
0
j,k=1,...,n
(7.170)
323
SCALED LIMITS AT β = 1, 2 AND 4
Note that this corresponds to the kernel (7.18) with s1 = 0. Consideration of the distribution (6.146) only requires minor modification to the above working. In particular we see the formula (6.149) applies but with ∞ ˜ ˜4 (x) V f (x) = e e−V1 (t) dt x
in (6.150). This means that formula (6.151) formally applies for even(OEN )2
ρ(n)
(x1 , . . . , xn ),
(7.171)
the only difference being that the first summation in (6.152) must be multiplied by a minus sign, which in turn follows from the fact that ∞ ˜ e−V1 (t) dt = −φ˜0 (x) + c˜0 x
(cf. (6.153)). This modified form of (6.152) can then be used to compute the scaled limit of (7.171) at the hard edge of the superimposed Laguerre orthogonal ensembles. Proceeding as in Sections 7.7.1 and 7.9 one finds [195] 1 n xn even(LOEN |a →a−1 )2 x1 odd(OEhard)2 ρ(n) (x1 , . . . , xn ) := lim ρ(n) ,..., N →∞ 4N 4N 4N ∞ √ J ( x ) a j = det K hard (xj , xk ) + √ J (t) dt (7.172) a √ 2 xk j,k=1,...,n xk (the notation odd(OE)2 on the l.h.s. refers to the labeling x1 < x2 < · · · which is natural at the hard edge in the scaled limit). E XERCISES 7.9
1. Using Exercises 7.7 q.2 as a guide, verify that (7.154) can be rewritten as [544], [216] N! (L) (L) y a/2 e−y/2 S1 (x, y) = KN (x, y) + 4Γ(N + a) Z ∞ ×La+1 (y) sgn(x − u)ua/2−1 e−u/2 La−1 N−1 N (u) du. 0
Deduce from this (7.156).
7.10 TWO-COMPONENT LOG-GAS WITH CHARGE RATIO 1:2 7.10.1 A local limit theorem and the evaluation of the free energy To compute the free energy per particle from the closed form expression (6.162) for the partition function requires the asymptotic behavior of the coefficient of ζ N1 /2 in the polynomial
N1 /2+N2
gN1 /2+N2 (ζ) :=
l=1
1 ζ + (l − )2 . 2
(7.173)
This can be deduced as a special case of a local limit theorem of Bender [55], which relates to the asymptotic behavior of the coefficients of general polynomials given in a factorized form. P ROPOSITION 7.10.1 [55, Th. 2] Let Pn (x) =
n k=0
an (k)xk = an (n)
n j=1
(x + rn (j))
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CHAPTER 7
be a polynomial in x whose roots are all real and nonpositive. Associate with Pn (x) the normalized double sequence pn (k) :=
an (k) . Pn (1)
Then with the mean and variance given by μn =
n j=1
1 , 1 + rn (j)
σn2 =
n j=1
rn (j) , (1 + rn (j))2
the an (k) satisfy a local limit theorem
2 1 σn pn ([σn x + μn ]) − √ e−x /2 = 0 n→∞ x∈(−∞,∞) 2π sup
lim
(7.174)
provided σn → ∞ as n → ∞. (The square brackets in [σn x + μn ] denote the integer part.) When x = 0, (7.174) gives an ([μn ]) ∼
Pn (1) √ , σn 2π
(7.175)
provided σn → ∞ as n → ∞. This formula can be used to deduce the asymptotics of [ζ N1 /2 ]gN1 /2+N2 (ζ). P ROPOSITION 7.10.2 We have [ζ N1 /2 ]gN1 /2+N2 (ζ) ∼
|Γ(iα + N1 /2 + N2 + 12 )|2 cosh πα √ παN1 2πσ 2
(7.176)
where α=
N2 + N1 /2 , ν
(7.177)
with ν specified as the solution of the equation N1 arctan ν , = N1 + 2N2 ν and σ2 ∼
(7.178)
1 N2 + N1 /2 arctan ν − . 2 ν 1 + ν2
Proof. To use (7.175) to compute the coefficient of ζ N1 /2 in gN1 /2+N2 (ζ) requires that we scale ζ by a suitable function
of N1 /2 + N2 so that μN1 /2+N2 = N1 /2. Thus we choose Pn (x) = α−N1 gn (α2 x) in Proposition 7.10.1, where α is to be so determined. From the formula for μn we can check that for large n = N1 /2 + N2 , with α given by (7.177) μN1 /2+N2 ∼ α arctan ν which implies μN1 /2+N2 = N1 /2 provided (7.178) is satisfied. The specification of σ 2 follows similarly to the formula for μn , and we note that indeed σ 2 → ∞ as N1 /2 + N2 → ∞, which is required for the validity of (7.175). The stated asymptotic formula now follows from (7.175) after noting from (7.173) that √ √ 1 cosh πα x gN1 /2+N2 (α2 x) = |Γ(iα x + N2 + N1 /2 + )|2 , 2 π where use has been make of (4.5).
Use of Stirling’s formula in (7.176), together with the definitions (7.177) and (7.178) allows (7.176) to be
325
SCALED LIMITS AT β = 1, 2 AND 4
simplified to read
[ζ N1 /2 ]gN1 /2+N2 (ζ) ∼ exp N2 log(1 + 1/ν 2 ) + 2N2 log(N2 + N1 /2) +(N1 /2) log(1 + ν 2 ) − 2N2 + O(1) .
(7.179)
Substituting this result in (6.162), and making a further use of Stirling’s formula specifies the asymptotics of ZN1 ,N2 . Substituting the resulting expression in the general formula (4.160) gives the free energy per particle [179]. P ROPOSITION 7.10.3 Let x1 :=
lim
N1 ,N2 →∞
N1 , N1 + N2
x2 :=
lim
N1 ,N2 →∞
N2 , N1 + N2
ρb =
lim
N1 ,N2 ,L→∞
N1 + 2N2 L
and specify ν by (7.178). Then − log ZN1 ,N2 ∼ (N1 + N2 )βf + O(1), where βf =
1 x1 ρb 1 2 x1 log − x + x2 . 1 + − log ρ (2π) 2 b 2 4(1 + ν 2 ) ν2 2
(7.180)
The quantities x1 and x2 are referred to as concentrations. Note that in the limit x1 → 1, so that ν → 0, ρb → ρ and x2 → 0, ρ 1 1 log − , 2 4 2 which agrees with the β = 1 case of (4.166) for the free energy per particle for the one-component log-gas at β = 1. Similarly, in the limit x2 → 1, so that ν → ∞, ρb → 2ρ and x1 → 0, βf ∼ − log 2ρ(2π)2 + 1, βf ∼
which agrees with the β = 4 case of (4.166) for the free energy per particle of the one-component log-gas. 7.10.2 The correlation functions The local limit theorem Proposition 7.10.1 can also be used to compute the thermodynamic limit of the correlation function (6.163). According to (6.163) and (6.168) it suffices to compute the asymptotics of
N1 /2+N2
[ζ
N1 /2−q
]
l=1 l=l1 ,...,lp
gN1 /2+N2 (ζ) 1 (ζ + (l − )2 ) =: [ζ N1 /2−q ] p 2 2 i=1 (ζ + (li − 1/2) )
for p and q fixed. Application of Proposition 7.10.1 gives
N1 /2+N2
[ζ N1 /2−q ]
l=1 l=l1 ,...,lp
1 α2q N1 /2 (ζ + (l − )2 ) ∼ p ] gN1 /2+N2 (ζ), 1 2 [ζ 2 2 (α + (l − ) ) i i=1 2
where α is given by (7.177). As a result, with 1 j t sin πρb xt sj (x) := dt, t2 + 1/ν 2 0
cj (x) := 0
(7.181)
1 j
t cos πρb xt dt, t2 + 1/ν 2
(7.182)
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CHAPTER 7
g(x)
1
0.8
0.6
0.4
0.2
0.5
1
1.5
2
2.5
3
3.5
x
Figure 7.2 Plot of g(x) = ρ+1,+1 (x, 0)/(ρ+1 )2 (first dashed curve), ρ+1,+2 (x, 0)/ρ+1 ρ+2 (second dashed curve) and ρ+2,+2 (x, 0)/(ρ+2 )2 in the case ρ+1 = ρ+2 = 1/3, ρb = 1. Note the different position of the first maximum in each case.
we obtain from (6.163) and (6.168) the limiting formula ρ+1n1 ,+2n2 (x1 , . . . , xn1 ; y1 , . . . , yn2 ) [A(xj − xk )]j,k=1,...,n1 [B(xj − yk )] j=1,...,n1 k=1,...,n2 = qdet [C(yj − xk )] j=1,...,n2 [D(yj − yk )]j,k=1,...,n2
,
(7.183)
k=1,...,n1
where ρb A(x) = 2 ν
ρb − π2 ν 2 sgn(x) + s−1 (x) c0 (x) s1 (x) , B(x) = √ , c0 (x) ν 2 −s1 (x) c2 (x) ρb ρb c2 (x) s3 (x) c2 (x) s1 (x) , D(x) = . C(x) = √ −s −s (x) c (x) 2 1 0 1 (x) c2 (x) ν 2
c0 (x) −s1 (x)
Note that in the limit ν → 0 this expression reduces to the result of Proposition 7.8.1 for the correlations at β = 1, while it in the limit ν → ∞ reduces to the result of Proposition 7.6.1 for the correlations at β = 4. The determinant form of (7.183) allows the general truncated correlation to be written down. Thus we have (0) ρ+1n1 ,+2n2 (x1 , . . . , xn1 ; y1 , . . . , yn2 ) = (−1)n1 +n2 −1 f (ri , ri ) , (7.184) cycles length n1 +n2
(i,i )
where f (ri , ri ) := A(xi −xi ), i, i ∈ [1, n1 ]; f (ri , ri ) := B(xi −yi −n1 ), i ∈ [1, n1 ], i ∈ [n1 +1, n1 +n2 ]; f (ri , ri ) := C(yi−n1 − xi ), i ∈ [n1 + 1, n1 + n2 ], i ∈ [1, n1 ]; f (ri , ri ) := D(yi−n1 − yi −n1 ), i, i ∈ [n1 + 1, n1 + n2 ]. This of course agrees with the results for the truncated correlations for the one-component log-gas at β = 1 and β = 4, as given by Propositions 7.8.1 and Propositions 7.6.1, respectively, in the appropriate limits.
327
SCALED LIMITS AT β = 1, 2 AND 4
For the two-particle correlation functions (7.183) and (7.184) give [179] πρ2 ρ2 ρT+1,+1 (x) = − b4 (c0 (x))2 + s1 (x)s−1 (x) + 2b s1 (x), ν 2ν 2 ρ ρT+1,+2 (x) = − b2 (s1 (x))2 + c0 (x)c2 (x) , 2ν 2 ρ (7.185) ρT+2,+2 (x) = − b (c2 (x))2 + s1 (x)s3 (x) . 4 Plots of these quantities for a particular value of the parameter ν are given in Figure 7.2. Note that the value of the first maximum is characteristic of local charge neutrality. E XERCISES 7.10
1. Show that to leading order the large x expansions of (7.185) are ρT+1,+1 (x) ∼ − x→∞
1 ρ2b , (1 + ν 2 )2 (πxρb )2
ρ2b ν 4 ρT+2,+2 (x) ∼ − x→∞ 4(1 + ν 2 )2
1 . (πxρb )2
ρT+1,+2 (x) ∼ − x→∞
1 ρ2b ν 2 , 2(1 + ν 2 )2 (πxρb )2 (7.186)
Chapter Eight Eigenvalue probabilities — Painleve´ systems approach The generating function for the probability that there are exactly n eigenvalues in an interval J of a classical matrix ensemble with unitary symmetry can, for certain J, be identified with the τ -function of a Painlev´e system. In particular, this is possible whenever J is a single interval containing an endpoint of the support of the density. This allows the distribution of certain eigenvalue probabilities relating to the largest and smallest eigenvalue, and the bulk spacing, to be characterized in terms of the solution of nonlinear equations of Painlev´e type. A practical consequence is the rapid computation of the power series expansions of the spacing distribution, and the high precision numerical tabulations which follow from these. To obtain the Painlev´e nonlinear equations we give a self-contained development of the Hamiltonian formulation of the Painlev´e theory, which makes essential use of B¨acklund transformations and the Toda lattice equation. In the case of J = [−a, a] and the weight being classical and even, the gap probability for β = 1 can be related to a β = 2 gap probability. This allows for a Painlev´e characterization of the gap probability of the COE and its bulk scaling limit. Inter-relationships between the COE, CUE and CSE known from earlier chapters then imply the analogous result for the CSE. In the last section, the theory of orthogonal polynomial systems on the unit circle is used to characterize some random matrix averages over U (N ) in terms of discrete Painlev´e equations.
8.1 DEFINITIONS From experimental data of energy spectra, it is a simple matter to construct the histogram corresponding to the p.d.f. for the spacing between consecutive energy levels. For the spectra of heavy nuclei, this was first done in the late 1950s [447]. In accordance with our discussion in Chapter 1, the theoretical p.d.f. should be identical to that of the bulk eigenvalues in the GOE, appropriately scaled. Here we will discuss the mathematical theory of Painlev´e systems as it relates to the calculation of such eigenvalue spacing distributions. In general, for a continuous, one-dimensional statistical mechanical system the p.d.f. for the spacing distribution is simply related to the probability of a particle-free interval (sometimes called the gap, or hole, probability).
D EFINITION 8.1.1 We denote by EN,β (n; J) the probability that there are exactly n particles in the interval J of a continuous, one-dimensional statistical mechanical system at inverse temperature β with N particles. With J = (a− , a+ ) we denote by pN,β (n; J) the p.d.f. for the event that given there is a particle at a− , there is a particle at a+ , with exactly n particles in between. Similarly, we use the notation pN,β (n; a± ) to denote the p.d.f. for the event that there is a particle at a± and exactly n particles in total to the right (left).
329
EIGENVALUE PROBABILITIES — PAINLEVE´ SYSTEMS APPROACH
P ROPOSITION 8.1.2 Define the generating functions ∞ ∞ N 1 (l) EN,β (J; ξ) := dx1 · · · dxN (1 − ξχJ ) e−βU(x1 ,...,xN ) , ˆ ZN −∞ −∞ l=1 ∞ ∞ N N (N − 1) (l) pN,β (J; ξ) := dx3 · · · dxN (1 − ξχJ )e−βU(a− ,a+ ,x3 ,...,xN ) , ρ(1) (a1 )ZˆN −∞ −∞ pN,β (a± ; ξ) := (l) χJ
N ZˆN
∞
−∞
dx2 · · ·
l=3
∞
−∞
dxN
N
(1 − ξχJ )e−βU(a± ,x2 ,...,xN ) , (l)
l=2
(l) = 1 if xl ∈ J and χJ = 0 otherwise, in = (a+ , ∞) or (−∞, a− ). For a continuous
where formula J particles, we have
the second formula J = (a− , a+ ), and in the third one-dimensional statistical mechanical system of N
(−1)n ∂ n E (J; ξ) , N,β n! ∂ξ n ξ=1 n n (−1) ∂ pN,β (n; J) = pN,β (J; ξ) n! ∂ξ n ξ=1 2 ∂ 1 =− EN,β (n; J) + 2pN,β (n − 1; J) − pN,β (n − 2; J), ρ(1) (a− ) ∂a− ∂a+ (−1)n ∂ n d p (a ; ξ) = ± EN,β (n; J) + pN,β (n − 1; J), pN,β (n; a± ) = N,β ± n! ∂ξ n da ξ=1 EN,β (n; J) =
(8.1)
(8.2) (8.3)
where pN,β (k; J) = 0 for k < 0 and in (8.2) J = (a− , a+ ), while in (8.3) J = (a+ , ∞) or (−∞, a− ).
Proof. The first equalities in (8.1)–(8.3) follow from the definitions of the generating functions and the defining equations Z Z Z Z 1 “N ” dx1 · · · dxn dxn+1 · · · dxN e−βU (x1 ,...,xN ) , ¯ ZˆN n J J J¯ Z ZJ Z Z 1 1 N! pN,β (n; J)= dx3 · · · dxn+2 dxn+3 · · · dxN ρ(1) (a− ) ZˆN n!(N − n − 2)! J J J¯ J¯
EN,β (n; J)=
×e−βU (a−,a+ ,x3 ,...,xN ) , Z Z Z Z N! 1 pN,β (n; a± )= dx2 · · · dxn+1 dxn+2 · · · dxN e−βU (a±,x2 ,...,xN ) , ZˆN n!(N − n − 1)! J J J¯ J¯ where J¯ := (−∞, ∞) − J. For the second equalities in (8.2) and (8.3), we note that it follows from the first equalities that EN,β (J; ξ) =
∞ X
(1 − ξ)n EN,β (n; J),
n=0 ∞ X
pN,β (a± ; ξ) =
pN,β (J; ξ) =
∞ X
(1 − ξ)n pN,β (n; J),
n=0 n
(1 − ξ) pN,β (n; a± ),
(8.4)
n=0
while inspection of the definitions of the generating functions shows pN,β (J; ξ) = −
∂2 EN,β (J; ξ), (1) (a− ) ∂a− ∂a+ 1
ξ2ρ
pN,β (a± ; ξ) = ±
Substituting (8.4) in (8.5) and equating powers of (1 − ξ) gives the second equalities.
1 d EN,β (J; ξ). ξ da±
(8.5)
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In the case of a matrix ensemble with unitary symmetry specified by the eigenvalue p.d.f. N 1 g(xj ) C j=1
(xk − xj )2
(8.6)
1≤j
(as in Section 5.3 this is to be denoted UEN (g)), the generating functions EN,2 (J; ξ), in the case J = (−∞, a− ) or (a+ , ∞), and pN,2(a± ; ξ), are the particular cases μ = 0, μ = 2 (the latter after multiplication by a suitable constant times g(a± )) of (μ)
EN,2 (J; ξ; g) :=
N (l) (1 − ξχJ )|a± − xl |μ l=1
UEN (g)
.
(8.7)
Specifically (0)
EN,2 (J; ξ) = EN,2 (J; ξ; g),
pN +1,2 (a± ; ξ) =
ZˆN (2) g(a± )EN,2 (J; ξ; g). ˆ ZN +1
(8.8)
Following [228], [230], [233], the theme of this chapter is to relate the general average (8.7) for classical weights g to the theory of Painlev´e systems. Consequently it will be possible to characterize such averages as the solution of certain nonlinear equations. We will see that the Painlev´e systems are Hamiltonian systems associated with the Painlev´e equations. There are six Painlev´e equations. These are special second order nonlinear equations, labelled PI–PVI, given explicitly by PI y = 6y 2 + t, PII y = 2y 3 + ty + α, 1 1 1 δ PIII y = (y )2 − y + γy 3 + (αy 2 + β) + , y t t y 1 2 3 3 β 2 2 (y ) + y + 4ty + 2(t − α)y + , PIV y = 2y 2 y 1 2 1 β γy δy(y + 1) 1 (y − 1) PV y = + (y )2 − y + + + , αy + 2 2y y − 1 t t y t y−1 1 1 2 1 1 1 11 + + (y ) − + + y PVI y = 2 y y−1 y−t t t−1 y−t βt γ(t − 1) δt(t − 1) y(y − 1)(y − t) α + . + + + t2 (t − 1)2 y2 (y − 1)2 (y − t)2
(8.9)
Following [310], let us make a few remarks as to their historical context. Leading up to the work of Painlev´e and the isolation of the equations (8.9) were the studies of Fuchs and Poincar´e in the late nineteenth century on first order differential equations of the form P (y , y, t) = 0,
(8.10)
where P is a polynomial in y , y with coefficients meromorphic in t. In general, the solution of such equations have moveable singularities, by which one means singularities which depend on the initial condition. Of interest is to distinguish equations with moveable poles from those with moveable essential singularities. Thus for example dy = y2 + 1 (8.11) dt has the general solution y = tan(t + c), where c determines the initial condition. Hence in this case all the
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EIGENVALUE PROBABILITIES — PAINLEVE´ SYSTEMS APPROACH
singularities are moveable (first order) poles. In contrast the equation dy 1 = , dt αy α−1
α = 2, 3, . . .
has the general solution y = (t − c)1/α , for which the singularity is a movable branch point (essential singularity). Fuchs and Poincar´e were able to classify all equations of the form (8.10) which are free from movable essential singularities — they must be reducible (i.e., equivalent after an analytic change of variables, or linear fractional transformation) to either the differential equation of the Weierstrass P-function, dy 2 = 4y 3 − g2 y − g3 , (8.12) dt or of the Riccati equation dy = a(t)y 2 + b(t)y + c(t), dt where a, b, c are analytic in t (note that (8.11) is of the latter form). Painlev´e then took up the problem of classifying second order differential equations of the form y = R(y , y, t),
(8.13)
(8.14)
where R is a rational function in all arguments, which are free from movable essential singularities. In general nonlinear equations free from movable essential singularities are now said to possess the Painlev´e property. It was shown that the only equations of the form (8.14) with the Painlev´e property which could not be reduced to the first order equations (8.12), (8.13) or to a linear differential equation are the equations (8.9). In studying (8.7) we will encounter transcendents associated with PII–PVI. In fact we will typically not encounter these nonlinear equations directly, but rather the so-called Jimbo-Miwa-Okamoto σ-form of the Painlev´e equations, 2 2 σPII (σII (σII ) + 4σII ) − tσII + σII − a2 = 0, 1 (v1 − v2 )2 = 0, 43 2 σPIV (σIV ) − 4(tσIV − σIV )2 + 4σIV (σIV + 2α1 )(σIV − 2α2 ) = 0, 2 σPV (tσV )2 − σV − tσV + 2(σV )2 + (ν0 + ν1 + ν2 + ν3 )σV
2 2 σPIII (tσIII ) − v1 v2 (σIII ) + σIII (4σIII − 1)(σIII − tσIII ) −
+4(ν0 + σV )(ν1 + σV )(ν2 + σV )(ν3 + σV ) = 0, 4 2 2 σPVI σV I t(1 − t)σV I + σV I (2σV I − (2t − 1)σV I ) + v1 v2 v3 v4 = (σV I + vk2 ). k=1
(8.15) We will see in the next section that the σ-form is the differential equation satisfied by certain auxiliary Hamiltonians in the Hamiltonian formulation of PII–PVI. (Analogous to the origin of the Painlev´e equations (8.9), they also occur in classifying second order second degree equations with the Painlev´e property [120].)
E XERCISES 8.1
1.
(i) Use the second equality for pN,β (n; J) in (8.2) to deduce that pN,β (n; J) = −
n X 1 ∂2 (n − j + 1)EN,β (j; J) ρ(1) (a− ) ∂a− ∂a+ j=0
(8.16)
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and show that in a translationally invariant state, ρ(1) (a1 ) = ρ, with |a1 − a2 | = s it reduces to pN,β (n; (0, s)) =
n 1 d2 X (n − j + 1)EN,β (j; (0, s)). 2 ρ ds j=0
(8.17)
N (ii) Use the second and third formulas in (8.4), together with the definitions of pN,β and ρN (1) , ρ(2) (in the latter the superscript N denotes a finite system of N particles) to deduce the sum rules N−1 X
N−2 X
pN,β (n; a) = ρN (1) (a),
n=0
pN,β (n; (a1 , a2 )) =
n=0
ρN (2) (a1 , a2 ) . ρN (1) (a1 )
(8.18)
2. For a perfect gas on a line, show that in the thermodynamic limit Eβ ((a1 , a2 ); ξ) = e−ξρ|a1 −a2 | and hence evaluate Eβ (n; (a1 , a2 )) and pβ (n; (a1 , a2 )). 3. [447] (Wigner surmise) The p.d.f. for the Gaussian β-ensemble in the case N = 2 is p(x1 , x2 ) =
1 −βc(x21 +x22 )/2 e (x2 − x1 )β , G2
G2 = 2β π 1/2 (βc)−β/2−1 Γ((β + 1)/2),
(8.19)
where the extra factor c is a scale factor, and the eigenvalues are ordered so that x2 > x1 . Define the averaged spacing p.d.f. by Z ∞ (s) := p(x, x + s) dx pW β with the value of c in (8.19) chosen so that
R∞ 0
−∞
spW β (s) ds
= 1 (mean spacing equals unity).
(i) For general c show that pW β (s) =
1 β −cβs2 /4 s e , ˜ G2
˜2 = G
“ βc ”1/2 π
G2 .
(ii) Use the constraint that the mean spacing be unity to show “ Γ( β + 1) ”2 cβ 2 = 4 Γ( β2 + 12 ) and thus in particular pW 1 (s) =
πs −πs2 /4 , e 2
pW 2 (s) =
32s2 −4s2 /π e , π2
pW 4 (s) =
218 s4 −64s2 /9π e . 36 π 3
(8.20)
W (iii) [65] Show from (8.16) in the case n = 0 that the gap probability associated with pW 1 is E1 (0; s) = R ∞ −πt2 /4 e dt. Use this and your answer to q.2 to show that for a system of unit density, consisting of a s fraction c1 of particles which are a perfect gas (PG), and fraction 1 − c1 of particles obeying the β = 1 Wigner surmise, the probability that the interval (0, s) is free of particles is given by Z ∞ 2 e−πt /4 dt. E1PG (0; c1 s)E1W (0; (1 − c1 )s) = e−c1 s (1−c1 )s
4. [330] Suppose that in a general continuous one-dimensional statistical mechanical system of N particles, the right most particle is labeled x1 , the second rightmost x2 and so on. Set a0 = ∞, suppose a1 > a2 > · · · and put Ij = (aj , aj−1 ). Denote by EN,β ({(nr , Ir )}r=1,...,l ) the probability that exactly nr of the particles are in Ir
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EIGENVALUE PROBABILITIES — PAINLEVE´ SYSTEMS APPROACH
(r = 1, . . . , l). Show that X
Pr(x1 < a1 , . . . , xl < al ) =
EN,β ({(nr , Ir )}r=1,...,l ),
(n1 ,...,nl )∈Ll
where Ll = {(n1 , . . . , nl ) ∈ Zl≥0 :
r X
nj ≤ r − 1 (r = 1, . . . , l)}.
j=1
5. Show that with the change of variables t → (is + 1)/2, σV I (t) → i˜ σV I (s)/2, the σPVI equation in (8.15) reads 4 “ “ ”2 ”2 Y σV I + 4 σ ˜V I (˜ σV I − s˜ σV I ) − iv1 v2 v3 v4 + 4 (˜ σV I + vk2 ) = 0. σ ˜V I (1 + s2 )˜
(8.21)
k=1
(This will be referred to as the σ ˜ PVI equation.)
6. [120] In the general theory of Painlev´e equations it is known that any differential equation of the form (y )2 = −
4 n c1 (ty − y)3 + c2 y (ty − y)2 + c3 (y )2 (ty − y) g 2 (t) + c4 (y )3 + c5 (ty − y)2 + c6 y (ty − y) + c7 (y )2 + c8 (ty − y) + c9 y + c10
o (8.22)
where g(t) := c1 t3 + c2 t2 + c3 t + c4 , referred to as the master Painlev´ e equation, is integrable in terms of Painlev´e transcendents. Show that only four of the ten parameters in (8.22) are essential because the equation retains its form under the gauge transformations at + b t¯ = , ct + d
y¯ =
hy + kt + m . ct + d
8.2 HAMILTONIAN FORMULATION OF THE PAINLEVE´ THEORY 8.2.1 The auxiliary Hamiltonian In the Hamiltonian approach to the Painlev´e equations PII–PVI, one presents a Hamiltonian H = H(p, q, t; v), where the components of v are parameters, such that after eliminating p in the Hamilton equations q =
∂H , ∂p
p = −
∂H , ∂q
(8.23)
q and p denoting derivatives with respect to t, the equation in q is the appropriate Painlev´e equation. This was first achieved by Malmquist in 1922 [383]. However the consequences of the Hamiltonian formulation to be presented below, in particular the B¨acklund transformations (transformations of p and q which conserve the Hamiltonian structure) and the associated sequences of special solutions, were not explored until the work of Okamoto in the 1980s [428].
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CHAPTER 8
The forms of the Hamiltonians given in the work of Okamoto, after some renaming of the parameters, are 1 v1 − v2 q, HII = − (2q 2 − p + t)p − 2 2 1 tHIII = q 2 p2 − (q 2 + v1 q − t)p + (v1 + v2 )q, 2 HIV = (2p − q − 2t)pq − 2(v1 − v2 )p + (v3 − v2 )q, tHV = q(q − 1)2 p2 − {(v1 − v2 )(q − 1)2 − 2(v1 + v2 )q(q − 1) + tq}p t(t − 1)HV I
+ (v3 − v2 )(v4 − v2 )(q − 1), = q(q − 1)(q − t)p2 − (v3 + v4 )(q − 1)(q − t) + (v3 − v4 )q(q − t) − (v1 + v2 )q(q − 1) p + (v3 − v1 )(v3 − v2 )(q − t),
(8.24)
where the parameters herein relate to those in (8.9) according to PII
v1 + v2 = 0,
PIII
α = −4v2 ,
1 α = v1 − , 2 β = 4(v1 + 1),
γ = 4,
δ = −4,
α = 1 + 2v3 − v1 − v2 , β = −2α21 , 1 1 1 PV v1 + v2 + v3 + v4 = 0, α = (v3 − v4 )2 , β = − (v1 − v2 )2 , γ = 2v1 + 2v2 − 1, δ = − , 2 2 2 1 1 1 1 2 2 2 2 PVI α = (v1 − v2 ) , β = − (v3 + v4 ) , γ = (v3 − v4 ) , δ = (1 − (1 − v1 − v2 ) ). 2 2 2 2 These Hamiltonians can be systematically derived from the isomonodromy deformation theory associated with the Painlev´e equations [323], [427] (see also Section 9.9.2), although the details are somewhat complicated and will not be presented here. In the Hamiltonian HIII , q satisfies the differential equation PIV
v1 + v2 + v3 = 0,
y =
1 2 1 αy 3 1 δ (y ) − y + 2 + 2 (βy 2 + γt) + y t 4t 4t 4y
(8.25)
for suitable parameters α, . . . , δ. This equation, referred to as PIII , reduces to PIII in (8.9) upon making the replacements t → t2 , y → ty. A feature of the Hamiltonians in (8.24) is that they involve parameters v = (v1 , v2 , . . . ) instead of the parameters α, β, . . . in the original Painlev´e equations (8.9). An advantage of introducing the parameters v is that the Hamiltonians display certain symmetries in these variables. These symmetries are revealed by the second order second degree differential equation satisfied by simple modifications of the Hamiltonians.
P ROPOSITION 8.2.1 Define the auxiliary Hamiltonians hII (t) = HII , 1 1 hIII (t) = tHIII + v12 − t, 4 2 hIV (t) = HIV − 2v2 t, hV (t) = tHV + (v3 − v2 )(v4 − v2 ) − v2 t − 2v22 , 1 hV I (t) = t(t − 1)HV I + e2 [−v1 , −v2 , v3 ]t, − e2 [−v1 , −v2 , v3 , v4 ], 2 where ep [a1 , . . . , as ] := 1≤j1 <···<jp ≤s aj1 aj2 · · · ajp . These auxiliary Hamiltonians satisfy the nonlinear
335
EIGENVALUE PROBABILITIES — PAINLEVE´ SYSTEMS APPROACH
equations
1 v1 − v2 2 = 0, 4 2 1 2 v1 + v22 = 0, (thIII )2 + v1 v2 hIII − (4(hIII )2 − 1)(hIII − thIII ) − 4 3 (hIV + 2vk ) = 0, (hIV )2 − 4(thIV − hIV )2 + 4 (hII )2 + 4(hII )3 + 2hII (thII − hII ) −
k=1
(thV )2 − (hV − thV + 2(hV )2 )2 + 4
4
(hV + vk ) = 0,
k=1
hV I (t(1 − t)hV I )2 + (hV I {2hV I − (2t − 1)hV I } + v1 v2 v3 v4 )2 =
4
(hV I + vk2 ).
(8.26)
k=1
Proof. We will give the derivation of the differential equation for hV I only; the derivations in the other cases are similar. Using the general fact that for Hamiltonians H = H(p, q, t), dH ∂H = dt ∂t (a consequence of the Hamilton equations (8.23)), we see that hV I = −q(q − 1)p2 + (v3 (2q − 1) − v4 )p − v32 .
(8.27)
q(q − 1)(hV I + v32 ) = −(q(q − 1)p)2 + (v3 (2q − 1) − v4 )q(q − 1)p,
(8.28)
This can be rewritten
which shows a differential equation for hV I will follow if we can express q and q(q − 1)p in terms of hV I and its derivatives. To do this we note from (8.27) and the definition of hV I that hV I − thV I = q(−hV I + e2 [−v1 , −v2 , v3 ]) + (v1 + v2 )q(q − 1)p −
1 e2 [−v1 , −v2 , v3 , v4 ], 2
and this differentiated, making use of the Hamilton equations, implies t(t − 1)hV I = 2q(e1 [−v1 , −v2 , v3 ]hV I − e3 [−v1 , −v2 , v3 ]) − 2q(q − 1)p(hV I − v1 v2 ) −e1 [−v1 , −v2 , v3 , v4 ]hV I + e3 [−v1 , −v2 , v3 , v4 ]. These last two equations are linear in q and q(q − 1)p. Solving for them and substituting in (8.28) gives the equation for hV I in (8.26).
The equations (8.26) are simply related to the σ-form of the Painlev´e equations (8.15). Thus substituting , σII (t) = −21/3 hII (−21/3 t) (v1 ,v2 )=(a,−a)
v1 v2 t , σIII (t) = −hIII (t/4) + + 8 4 σIV (t) = hIV (t) + 2v2 t σV (t) = hV (t) + v2 t + σV I (t) = hV I (t) in (8.15) gives (8.26).
,
(1+v3 −v1 ,v1 −v2 ,v2 −v3 )=(α0 ,α1 ,α2 ) 2 2v2 , νj−1 = vj − v2 (j = 1, . . . , 4),
(8.29)
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CHAPTER 8
8.2.2 Affine Weyl group symmetries Inspection of (8.26) shows the differential equations are invariant with respect to certain simple mappings of the parameters, and thus so are the auxiliary Hamiltonians. The generators of the mappings can be taken to be hII : s1 (v1 , v2 ) = (v2 , v1 ), hIII : s1 (v1 , v2 ) = (v2 , v1 ), s2 (v1 , v2 , t) = (v1 , −v2 , −t), hIV : s1 (v1 , v2 , v3 ) = (v2 , v1 , v3 ), s2 (v1 , v2 , v3 ) = (v1 , v3 , v2 ), hV : s1 (v1 , v2 , v3 , v4 ) = (v2 , v1 , v3 , v4 ), s3 (v1 , v2 , v3 , v4 ) = (v1 , v2 , v4 , v3 ),
s2 (v1 , v2 , v3 , v4 ) = (v1 , v3 , v2 , v4 ),
hV I : s1 (v1 , v2 , v3 , v4 ) = (v2 , v1 , v3 , v4 ), s2 (v1 , v2 , v3 , v4 ) = (v1 , v3 , v2 , v4 ), s3 (v1 , v2 , v3 , v4 ) = (v1 , v2 , v4 , v3 ), s4 (v1 , v2 , v3 , v4 ) = (−v2 , −v1 , v3 , v4 ).
(8.30)
Geometrically the above mappings are reflections in the hyperplane perpendicular to the following vectors, 1 := e1 − e2 , hII : α 1 := e1 − e2 , hIII : α 1 := e1 − e2 , hIV : α
α 2 := e2 , α 2 := e2 − e3 ,
1 := e1 − e2 , hV : α hV I : α 1 := e1 − e2 ,
α 2 := e2 − e3 , α 2 := e2 − e3 ,
α 3 = e3 − e4 , α 3 = e3 − e4 ,
α 4 = e3 + e4 .
(8.31)
These vectors form a base for the root systems A1 , B2 , A2 , A3 , D4 , respectively. α1 , . . . , α l }, α i ∈ R D EFINITION 8.2.2 Let R be a root system in a vector space E. We say that Δ = { (i = 1, . . . , l) is a base for R if Δ is a basis for E with the further requirement that for each α ∈ R, in l the expansion α = i=1 ζi α i the coefficients ζi are integers which are either all non-negative or all nonpositive. The former are called the positive roots, while thelatter are the negative roots (this latter definition l is consistent with that below (4.149), with the choice v = i=1 α i , say). Next we note a further reflection operator closely related to the root system structure revealed in (8.31), A1 : B2 :
s0 (v1 , v2 ) = (2 + v2 , −2 + v1 ), s0 (v1 , v2 ) = (−1 − v2 , −1 − v1 ),
A2 : A3 :
s0 (v1 , v2 , v3 ) = (1 + v3 , v2 , −1 + v1 ), s0 (v1 , v2 , v3 , v4 ) = (1 + v4 , v2 , v3 , v1 − 1),
D4 :
s0 (v1 , v2 , v3 , v4 ) = (1 − v2 , 1 − v1 , v3 , v4 ).
(8.32)
To see the geometric significance of these operators requires the notion of the highest root. D EFINITION 8.2.3 With a positive root defined in Definition 8.2.2, one defines a partial order < on R by −α the statement that α < β if β is a sum of positive roots. The maximal root with respect to this ordering is called the highest root. From this definition the highest roots are α 0 =
e1 − el for Al−1 e1 + e2 for B2 , D4 .
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EIGENVALUE PROBABILITIES — PAINLEVE´ SYSTEMS APPROACH
0 0 A 0
0
1 1
A
1
2 B
2
2
0
1
1 2
3
4 D4
1
2
2 A
3
3
Figure 8.1 Dynkin diagrams for affine root systems relevant to Painlev´e systems.
· x = c} defined by With reflections in the hyperplane {x : φ σc,φ (γ ) = γ −
2γ · φ + 2cφ φ ·φ ·φ φ φ
· x = 2} for A1 , (cf. (4.147)), the mappings (8.32) then correspond to reflections in the hyperplanes {x : φ x = 1} for A2 , A3 and D4 , where φ =α · x = −1} for B2 , {x : φ, 0 . The mappings (8.32) together {x : φ with (8.30) generate the so-called affine Weyl group corresponding to the respective root system. The algebra specifying the affine Weyl groups for the root systems Al , Bl , Dl are given by the Coxeter relations (si sj )mij = 1,
0 ≤ i, j ≤ l.
(8.33)
The exponents mij are positive integers which can be read off from the Dynkin diagrams associated with the affine root systems as presented in Figure 8.1. The Dynkin diagrams are constructed by the rule that if α i and α j are perpendicular (i.e., have dot product equal to zero) then the vertices i and j are not connected; if the angle between α i and α j is π/3 then the vertices i and j are connected by a single line; if the angle between α i and α j is π/4 then the vertices i and j are connected by a double line. The root system A1 is special because α 0 and α 1 are the same. In this case the two vertices are to be connected by a double line. To construct the mij in (8.33) from the Dynkin diagram, the rule is that mii = 1, mij = 2 if the corresponding vertices in the Dynkin diagram are not joined, mij = 3 if they are joined by one line and mij = 4 if they are joined by two lines. Related to the matrix [mij ] is the Cartan matrix [aij ]. It is constructed from the rule that aii = −2, aij = 0, −1, −2 according to the vertices i and j being connected by zero, one or two lines, respectively. The explicit forms of [mij ] and [aij ] are listed in Table 8.1. The introduction of the Cartan matrix allows the action in (8.30) and (8.32) to be written in a succinct way. For this purpose we replace the parameters v1 , v2 , . . . by α0 , α1 , . . . according to A1 : α0 = 1 + (v2 − v1 )/2, α1 = (v1 − v2 )/2, 1 B2 : α0 = − (1 + v1 + v2 ), α1 = v1 − v2 , α2 = v2 , 2 A2 : α0 = 1 + v3 − v1 , α1 = v1 − v2 , α2 = v2 − v3 , A3 : α0 = 1 + v4 − v1 , D4 : α0 = 1 − v1 − v2 ,
α1 = v1 − v2 , α1 = v1 − v2 ,
α2 = v2 − v3 , α2 = v2 − v3 ,
α3 = v3 − v4 , α3 = v3 − v4 ,
α4 = v3 + v4 . (8.34)
j · v = 0, while α0 = 0 defines the hyperplane Note that for j = 1, 2, . . . , αj = 0 defines the hyperplane α α 0 · v = c (c = 2 for A1 , c = −1 for B2 , c = 1 for A2 , A3 , D4 ). These are the hyperplanes with respect to which sj and s0 act as reflection operators. From this geometrical interpretation, it follows that si αj must be
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CHAPTER 8
Root system A1
B2
A2
A3
D4
[mij ] 1 4 ⎡ 1 ⎣ 2 ⎡ 4 1 ⎣ 3 ⎡ 3 1 ⎢ 3 ⎢ ⎣ 2 ⎡ 3 1 ⎢ 2 ⎢ ⎢ 3 ⎢ ⎣ 2 2
4 1 2 1 4 3 1 3 3 1 3 2 2 1 3 2 2
4 4 1 3 3 1 2 3 1 3 3 3 1 3 3
⎤ ⎦ ⎤ ⎦ ⎤ 3 2 ⎥ ⎥ 3 ⎦ 1 ⎤ 2 2 2 2 ⎥ ⎥ 3 3 ⎥ ⎥ 1 2 ⎦ 2 1
Cartan matrix 2 −2 −2 2 ⎡ 2 0 −2 ⎣ 0 2 −2 −2 −2 2 ⎡ 2 −1 −1 ⎣ −1 2 −1 ⎡ −1 −1 2 2 −1 0 ⎢ −1 2 −1 ⎢ ⎣ 0 −1 2 ⎡ −1 0 −1 2 0 −1 ⎢ 0 2 −1 ⎢ ⎢ −1 −1 2 ⎢ ⎣ 0 0 −1 0 0 −1
⎤ ⎦ ⎤ ⎦ −1 0 −1 2 0 0 −1 2 0
⎤ ⎥ ⎥ ⎦ 0 0 −1 0 2
⎤ ⎥ ⎥ ⎥ ⎥ ⎦
Table 8.1 Specification of [mij ] and [aij ].
a linear combination of αj and αi , which we can check is given by the explicit formula si αj = αj − αi aij ,
(8.35)
where [aij ] is the corresponding Cartan matrix. The introduction of the variables (8.34) leads to another elementary mapping which turns out to correspond to a B¨acklund transformation of the Painlev´e system [424], [339]. This mapping relabels the αj in a way which conserves the relative geometry of the root system. Equivalently, these mappings correspond to automorphisms of the Dynkin diagram which conserve the relative structure. Thus one introduces the operators π, r1 , r2 such that Al : παi = αi+1 π i = 0, . . . , l (αl+1 := α0 ), D4 : r1 (α0 , α1 , α2 , α3 , α4 ) = (α1 , α0 , α2 , α4 , α3 ), r2 (α0 , α1 , α2 , α3 , α4 ) = (α3 , α4 , α2 , α0 , α1 ).
(8.36)
Here the B2 affine root system is excluded, the reason being that the natural diagram automorphism r(α0 , α1 , α2 ) = (α1 , α0 , α2 ) is incompatible with the constraint −1 = 2α0 + α1 + 2α2 . The operators in (8.36) obey the algebraic relations Al : π l = 1, πsj = sj+1 π (sl := s0 ), 2 2 D4 : r1 = r2 = 1, r1 s2 = s2 r1 , r2 s2 = s2 r2 , s1 = r1 s0 r1 ,
s4 = r1 s3 r1 ,
s3 = r2 s0 r2 ,
s4 = r2 s1 r2 .
8.2.3 B¨acklund transformations For each of the operators, O, say, introduced above the mappings α ¯ i := Oαi can be combined with particular birational mappings p¯ = Op, q¯ = Oq such that the Hamilton equations (8.23) remain valid in the variables
339
EIGENVALUE PROBABILITIES — PAINLEVE´ SYSTEMS APPROACH
PII
α0
α1
p
s0
−α0
α1 + 2α0
s1
α0 + 2α1
−α1
p
π
α1
α0
−f
p+
q
2α2 4α0 q + 20 f f
α0 f α1 q+ p −q q+
Table 8.2 B¨acklund transformations for PII. Here α0 + α1 = 1, f = p − 2q 2 − t.
PIII
α0
α1
α2
p 1 q t q(p − 1) −
s0
−α0
α1
α2 + 2α0
s1
α0
−α1
α2 + 2α1
p
s2
α0 + 2α2
α1 + 2α2
−α2
1−p
α1 2
2
+1 q−
q
t
− qt
t
α1 2(p − 1)
t
−q
−t
Table 8.3 B¨acklund transformations for PIII . Here 2α0 + α1 + 2α2 = −1.
¯ := OH. Birational mappings which conserve the Hamilton equations are called B¨acklund transforp¯, q¯, H mations. A systematic way to construct p¯, q¯, valid in all cases except PIII , has been given by Noumi and Yamada [424]. However, to proceed along those lines would require introducing a number of new concepts, which we refrain from doing here. Instead we will be content with a statement of the final results. Before doing so, some further theory relating to the Hamiltonian of the PV system must be introduced. Thus it turns out [339] that the natural variables in relation to B¨acklund transformations for PV are not those in the Hamiltonian tHV of (8.24), but rather a transformation of these variables. For ease of notation, let us first put p → P , q → Q, tHV → tKV in the formula for tHV in (8.24) so that it reads tKV = Q(Q−1)2 P 2 −{(v1 −v2 )(Q−1)2 −2(v1 +v2 )Q(Q−1)+tQ}P +(v3 −v2 )(v4 −v2 )(Q−1). (8.37) Introducing the new variables p and q (not to be confused with p and q in the definition of tHV ) in (8.37) via [538] (q − 1)(Q − 1) = 1,
(q − 1)p + (Q − 1)P = v3 − v2
(8.38)
shows tKV = tHV ∗ + (v3 − v2 )(v1 − v4 ),
(8.39)
tHV ∗ := q(q − 1)p(p + t) − (v1 − v2 + v3 − v4 )pq + (v1 − v2 )p + (v2 − v3 )tq.
(8.40)
where
Here 1 + 1/(q + 1) satisfies the PV equation. The Hamiltonian (8.40) and more particularly the associated variables p and q have simple images under B¨acklund transformations. With the PV system so modified, to form what will be referred to as the PV∗ system, the B¨acklund transformations of PII–PVI are given by Tables 8.2–8.6. Using the B¨acklund transformation tables, the action of the elementary operators on the Hamiltonians are
340
CHAPTER 8
α0
α1
α2
s0
−α0
α0 + α1
α0 + α2
s1
α0 + α1
−α1
α2 + α1
s2
α0 + α2
α1 + α2
−α2
p
π
α1
α2
α0
− 21 f
PIV
p
q 2α0 q− f
α0 p+ f α1 p− q
q α2 p −2p
q+
Table 8.4 B¨acklund transformations for PIV. Here α0 + α1 + α2 = 1, f = p − q − 2t.
PV∗
α0
α1
α2
α3
p
q
s0
−α0
α1 + α0
α2
α3 + α0
p
q + pα+0 t
s1 s2
α0 + α1 α0
−α1 α1 + α2
α2 + α1 −α2
α3 α3 + α2
p − αq1 p
q q + αp2
s3
α0 + α3
α1
α2 + α3
−α3
p − q α−3 1
q
π
α1
α2
α3
α0
t(q − 1)
p −t
Table 8.5 B¨acklund transformations for PV∗ . Here α0 + α1 + α2 + α3 = 1.
PVI
α0
α1
α2
α3
α4
p
s0
−α0
α1
α2 + α0
α3
α4
s1
α0
−α1
α2 + α1
α3
α4
α0 p− q−t p
s2
α0 + α2
α1 + α2
−α2
α3 + α2
α4 + α2
p
s3
α0
α1
α2 + α3
−α3
α4
s4
α0
α1
α2 + α4
α3
−α4
r1
α1
α0
α2
α4
α3
r2
α3
α4
α2
α0
α1
α3 q−1 α4 p− q p(q − t)2 + α2 (q − t) − t(t − 1) q − (qp + α2 ) t p−
Table 8.6 B¨acklund transformations for PVI. Here α0 + α1 + 2α2 + α3 + α4 = 1.
q q q q+
α2 p
q q q − 1 t q−t t q
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EIGENVALUE PROBABILITIES — PAINLEVE´ SYSTEMS APPROACH
Operator
Definition
T (α0 , α1 , . . . )
T v
TII TIII
πs1 s0 s2 s1 s2
(α0 + 1, α1 − 1) (α0 − 1, α1 , α2 + 1)
(v1 − 1, v2 + 1) (v1 + 1, v2 + 1)
TIV
π −1 s1 s2
(α0 + 1, α1 , α2 − 1)
(v1 − 1/3, v2 − 1/3, v3 + 2/3)
TV
π −1 s1 s2 s3
(α0 + 1, α1 , α2 , α3 − 1)
(v1 − 14 , v2 − 14 , v3 − 14 , v4 + 34 )
TV I
r1 s0 s1 s2 s3 s4 s2
(α0 + 1, α1 + 1, α2 − 1, α3 , α4 )
(v1 , v2 − 1, v3 , v4 )
Table 8.7 Shift operators with the property T H = H|T (α0 ,α1 ,... )
computed as α0 , s1 HII = HII , πHII = HII + q, f 1 1 − α0 , s0 (tHIII ) = tHIII − q(p − 1) + α1 2 2 1 1 α0 + α1 , s2 (tHIII ) = tHIII − t − (α1 + α2 )q, s1 (tHIII ) = tHIII − 2 2 2α0 , s1 HIV = HIV + 2α1 t, s2 HIV = HIV − 2α2 t, s0 HIV = HIV − f πHIV = HIV + 2p − 2α2 t, t + α0 (α2 − 1), s1 (tHV ∗ ) = tHV ∗ + α1 t + α1 α3 , s0 (tHV ∗ ) = tHV ∗ + α0 p+t s2 (tHV ∗ ) = tHV ∗ − α2 t + α2 (α0 − 1), s3 (tHV ∗ ) = tHV ∗ + α1 α3 , π(tHV ∗ ) = tHV ∗ + (q − 1)p − α2 t, t(t − 1) + α0 (α3 − 1)t + α0 (α4 − 1)(t − 1), s0 t(t − 1)HV I = t(t − 1)HV I − α0 q−t s1 t(t − 1)HV I = t(t − 1)HV I ,
PII : s0 HII = HII + PIII :
PIV :
PV∗ :
PVI :
s2 t(t − 1)HV I = t(t − 1)HV I + α2 (1 + α1 − α0 )t − α2 (α1 + α2 + α3 ), s3 t(t − 1)HV I = t(t − 1)HV I − α3 (1 − α0 )t, s4 t(t − 1)HV I = t(t − 1)HV I − α4 (1 − α0 )(t − 1), r1 t(t − 1)HV I = t(t − 1)HV I − q(q − 1)p − α2 q + α2 (α1 − α0 )t + α2 (α0 + α2 + α4 ), (8.41) r2 t(t − 1)HV I = t(t − 1)HV I + (1 − t)qp + α2 (α0 + α2 + α4 )(1 − t). By composing suitable combinations of the elementary operators in the B¨acklund transformation tables, infinite order shift operators T can be constructed, with action on the parameters α0 , α1 , . . . which changes their value by ±1 or 0. Certain of the shift operators T have the further property that T H := H T (α ,α ,... ) = H , (8.42) 0 1 T p,T q
T (α0 ,α1 ,... )
so that the action of T effectively leaves p and q unchanged. Specifically, one can check from (8.41), making use too of (8.39) in the PV case and the B¨acklund transformation tables, that the operators in Table 8.7, amongst other examples, have this property.
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CHAPTER 8
Painlev´e system PII PIII PIV PV PVI
Restriction on the αj α1 = 0 α1 + 2α2 = 0 α2 = 0 α2 + α3 = 0 α2 = 0
Restriction on the vj v1 − v2 = 0 v1 + v2 = 0 v2 − v3 = 0 v2 − v4 = 0 v2 − v3 = 0
Table 8.8 Given these restrictions on the parameters, the Painlev´e system permits a solution with p = 0, H = 0.
Let us write T n H =: H[n],
T n h =: h[n].
(8.43)
Then it follows from (8.42) that h[n] satisfies the appropriate equation of (8.26) with parameters T nv . On the other hand, if we introduce the τ -function by d d log τ, H[n] = log τ [n], (8.44) dt dt it turns out that (8.42) together with the explicit form of T p and T q deduced from the B¨acklund transformation tables allows τ [n], multiplied by an appropriate elementary function, to be identified as an integral of the form (8.7). In this way the integrals (8.7) can be characterized as solutions of nonlinear equations of the type (8.15). H=
8.2.4 Classical solutions When N = 0 it is natural to assign the averages (8.7) the value unity. To relate the Painlev´e systems to the averages in this case we must show that for a specialization of the parameters it is permissable to choose H[0] = 0, which from the definition (8.44) permits τ [0] = 1 as required. In fact inspection of (8.24) shows that each of the Painlev´e systems PII–PVI permits a solution with H = 0,
p = 0,
(8.45)
provided the parameters are specified as in Table 8.8. Next we want to show that τ [1], in the cases PIV, PV∗ and PVI, is equal to a particular average (8.7) in the case N = 1. Now, from (8.44) we seek H[1] := T H[0]. Using (8.42), (8.24), (8.45) and Tables 8.7, 8.8, with v [0] =: v (0) we note that HII [1] = qII [0], tHV [1] =
(0) (v3
−
tHIII [1] = qIII [0], (0) v2 )(qV
[0] − 1),
HIV [1] = qIV [0], (0)
(0)
t(t − 1)HV I [1] = (v3 − v1 )(qV I [0] − t).
(8.46)
On the other hand q[0] satisfies the Hamilton equation q [0] =
∂H ∂p p=0,v=v[0]
with parameters v [0] specified by the constraints in Table 8.8. This is a first order nonlinear differential equation in q[0], which is seen to be of the Riccati type (8.13). First substituting for q[0] according to (8.46), then substituting for H[1] according to (8.44) gives a second order differential equation in τ [1], which furthermore in the cases of PIV, PV and PVI permits integral solutions of the type (8.7) with N = 1. P ROPOSITION 8.2.4 Let v [0] be specified by the constraint in Table 8.8 in each case, and let T v [0] then be
343
EIGENVALUE PROBABILITIES — PAINLEVE´ SYSTEMS APPROACH
Painlev´e system PII
Differential equation Airy differential equation u + 12 tu = 0
PIII
tu +
(0) (v1
PIV
Bessel differential equation + 1)u − 14 u = 0, u(t) = τIII [1](t/4)
Hermite-Weber equation (0) (0) u + 2tu + 2(v1 − v2 )u = 0
PV
Confluent hypergeometric differential equation (0) (0) (0) (0) tu + (v3 − v1 + 1 + t)u + (v3 − v2 )u = 0
PVI
Gauss hypergeometric differential equation t(1 − t)u + (c − (a + b + 1)tu − abu = 0 (0) (0) (0) (0) (0) (0) a = v3 − v1 , b = 1 − v1 − v3 , c = 1 − v1 + v4
Table 8.9 Classical differential equations satisfied by τ [1] := T τ [0].
specified by Table 8.7. With τ [1] specified by
H
T v [0]
=
d log τ [1], dt
τ [1] satisfies the linear differential equations of Table 8.9. For parameter values such that the integral converges, the general solution of the Hermite-Weber equation is proportional to ∞ ∞ (0) (0) 2 |t − u|−(v1 −v2 ) e−u du; −ξ (8.47) −∞
t
for the confluent hypergeometric equation it is proportional to 1 ∞ e−tu (1 − u)v2 −v1 uv3 −v2 −1 du; −ξ 0
(8.48)
0
and for the Gauss hypergeometric equation it is proportional to t 1 ua−c (1 − u)c−b−1 |t − u|−a du −ξ 0
or
∞
−∞
−ξ
∞
(8.49)
0
(1 + iu)a−c (1 − iu)c−b−1 |s − u|−a du,
s
t=
1 + is . 2
(8.50)
Proof. It remains to consider the derivation of the integral solutions. For definiteness, consider the case of the Gauss hypergeometric differential equation. We use the fact [541] that in general Z s ua−c (1 − u)c−b−1 (t − u)−a du, r
where r and s are any of the values 0, 1, t, ν∞ (|ν| = 1) with the parameters such that the integrand vanishes at these points, satisfies this differential equation. Forming from this a linear combination with (r, s) = (0, 1) and (r, s) = (0, t) gives (8.49). Similarly, choosing (r, s) = (−∞, ∞) and (t, ∞) and suitably deforming these contours in the complex plane gives (8.50).
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CHAPTER 8
Painlev´e system PII PIII
a(t) 1 t
PIV
1
PV∗
g[n] τII [n] 2 (tn /2 τIII [n])|t →t/4 e−t
2
2
tn
t
e
τIV [n]
(0) (0) (−α2 −α3 +n)t (0)
τV ∗ [n]
(0)
(n−α2 )(n+α0 −1)
t(t − 1)
PVI
/2
(0)
(α2 −n)
(t(t − 1))
τV I [n]
Table 8.10 Specifications for the Toda lattice equation (8.51).
It follows from Proposition 8.2.4 that there are solutions of the PIV, PV and PVI systems such that 2 (μ) , E1,2 ((t, ∞); ξ; e−x ) = τIV (t) α0 =μ+2,α1 =−μ,α2 =−1 (μ) E1,2 ((0, t); ξ; xa e−x ) = ta+μ+1 τV (t) ν =−μ,ν =0, , νj−1 := vj − v2 , 0 1 ν2 =a+1,ν3 =1 (μ) E1,2 ((0, t); ξ; xa (1 − x)b ) = τV I (t) v1 =(a+b)/2+1+μ,v2 =(a+b)/2, , v3 =(a+b)/2+1,v4 =(b−a)/2
(μ) E1,2 ((t, ∞); ξ; (1
2 −α
+x )
1 + it ) = τV I v1 =μ+1−α,v2 =−α, . 2 v3 =1−α,v4 =0
8.2.5 Toda lattice equation With the cases n = 0 and n = 1 now settled, the key ingredient in relating τ [n] to the n-dimensional integrals implied by (8.7) for general n ≥ 2 is the Toda lattice equation (cf. (5.118)) δ 2 log g[n] =
g[n + 1]g[n − 1] , g 2 [n]
δ := a(t)
d dt
(8.51)
satisfied by the product of τ [n] and an elementary function. P ROPOSITION 8.2.5 The τ -functions satisfy the Toda lattice equation (8.51) with g[n] and a(t) therein as specified by Table 8.10. Proof. We will explicitly consider the PV case only, although the same strategy suffices in all cases [339]. In this case it is convenient to work with the PV∗ Hamiltonian (8.40). According to (8.39) and (8.44) the τ -function of the latter, τV ∗ say, is related to the τ -function for (8.37) by (0)
τV ∗ [n] = tα2
(0)
(1−α0 −n)
τV [n].
(8.52)
From the definitions t
” “ ” “ d τV ∗ [n − 1]τV ∗ [n + 1] log = TV (tHV ∗ [n]) − tHV ∗ [n] − tHV ∗ [n] − TV−1 (tHV ∗ [n]) . 2 dt τV ∗ [n]
On the other hand, it follows from the property (8.42), Table 8.7 and (8.40) that TV (tHV ∗ [n]) − tHV ∗ [n] = q[n]p[n] and thus the r.h.s. of (8.53) is equal to
q[n]p[n] − (TV−1 q[n])(TV−1 p[n]).
(8.53)
EIGENVALUE PROBABILITIES — PAINLEVE´ SYSTEMS APPROACH
345
With TV−1 given in terms of the elementary operators by Table 8.7, use of the B¨acklund transformation tables for PV∗ , together with the Hamilton equations for HV ∗ , allows this term to be identified with t
” “ d (0) (0) (0) log q[n](q[n] − 1)p[n] + α2 q[n] + (−α2 − α3 + n)t dt ” ” “d “ (0) (0) d d d d (0) (0) = t log tHV ∗ [n] + (−α2 − α3 + n)t = t log t log e(−α2 −α3 +n)t τV ∗ [n] . dt dt dt dt dt (8.54)
Equating (8.54) to the l.h.s. of (8.53) implies (8.51) with the a(t) and g(t) as in Table 8.10.
Let us suppose the parameters are initially constrained as in Table 8.8. We know that then τ [0] = 1 in all cases. Inspection of Table 8.10 shows that we also have g[0] = 1 in all cases. This is significant because an identity of Sylvester (see Exercises 8.2 q.1) gives that if g[0] = 1, then the general solution of the Toda lattice equation (8.51) is given by g[n] = det δ j+k g[1] . (8.55) j,k=0,...,n−1
It therefore remains to show that with g[1] given by the elementary function of t specified in Table 8.10, multiplied by the appropriate integral formula (8.47)–(8.50), substitution in (8.55) leads to n-dimensional integrals of the form (8.7). P ROPOSITION 8.2.6 There are solutions of the PIV, PV and PVI systems such that 2 (μ) , EN,2 ((t, ∞); ξ; e−x ) = τIV (t) α0 =1+μ+N,α1 =−μ,α2 =−N 2 (μ) EN,2 ((0, t); ξ; xa e−x ) = t(a+μ)N +N τV (t) ν =−μ,ν =0, , νj−1 := vj − v2 , 0 1 ν =a+N,ν =N
2 3 (μ) EN,2 ((0, t); ξ; xa (1 − x)b ) = τV I (t) v1 =(a+b)/2+N +μ,v2 =(a+b)/2, , v3 =(a+b)/2+N,v4 =(b−a)/2
(μ) EN,2 ((t, ∞); ξ; (1
2 −α
+x )
1 + it ) = τV I v1 =μ+N −α,v2 =−α, . 2 v3 =N −α,v4 =0
Proof. For definiteness, let us consider the PV case. Then, according to Table 8.10, (8.52), (8.55) and (8.48) we have 2
τV [n] = t−n
g[1] = t1/2 et =t
1/2
h“ d ”j+k i t g[1] , dt j,k=0,...,n−1 Z 1” ∞ −ξ e−tu (1 − u)v2 −v1 uv3 −v2 −1 du
/2 −nt
e “Z
“Z
0 ∞
−1
det
Z −ξ
0 0 ”
e−tu (−u)v2 −v1 (u + 1)v3 −v2 −1 du.
−1
Now a general property of the Toda lattice equation (8.51) with a(t) = t is that if {g[n]}n=0,1,... is a solution with g[0] = 1, then {tnκ g[n]} is also a solution which is given by the determinant formula (8.55) with g[1] → tκ g[1]. With Z 0 ” “Z ∞ −ξ (8.56) F (a, c; t) := e−tu (−u)a−1 (u + 1)c−a−1 du, −1
−1
where a = v2 − v1 + 1, c = v3 − v1 + 1, choosing κ = −1/2 shows 2
t−n/2 τV [n] = t−n
/2 −nt
e
det
h“ d ”j+k i t F (a, c; t) . dt j,k=0,...,n−1
(8.57)
Starting with this formula, our strategy is to use elementary row and column operations to eliminate the operator δ j+k , d δ := t dt .
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CHAPTER 8
Now “Z
Z
”“ d ” t e−tu (−u)a−1 (u + 1)c−a−1 du dt −1 −1 Z 0 ”“ “Z ∞ d −tu ” e =− −ξ (−u)a (u + 1)c−a−1 du du −1 −1 Z 0 ” “Z ∞ = −aF (a, c; t) + (c − a − 1) −ξ e−tu (−u)a (u + 1)c−a−2 du ∞
δF (a, c; t) =
−ξ
0
−1
−1
= −aF (a, c; t) + (c − a − 1)F (a + 1, c; t),
(8.58)
using integration by parts. This identity can be used to eliminate the operation δ k from the r.h.s. of (8.57). Thus substituting (8.58) in column k, and adding a times column k − 1, (k = n, n − 1, . . . , 2) in that order shows h i det[δ j+k F (a, c; t)]j,k=0,...,n−1 ∝ det δ j F (a, c; t) δ j+k−1 F (a + 1, c; t) j=0,...,n−1 . k=1,...,n−1
Repeating this procedure on column k (k = n − 1, n − 2, . . . , k ) for each of k = 2, 3, . . . , n − 2 in that order shows det[δ j+k F (a, c; t)]j,k=0,...,n−1 ∝ det[δ j F (a + k, c; t)]j,k=0,...,n−1 .
(8.59)
Analogous to (8.58) we have Z 0 ” “Z ∞ d ue−tu (−u)a−1 (u + 1)c−a−1 du = −F (a, c + 1; t) + F (a, c; t). −ξ F (a, c; t) = − dt −1 −1 Substituting this in row j of (8.59) (j = n − 1, n − 2, . . . , 1 in this order) and subtracting row j − 1 shows – » F (a + k, c; t) j+k F (a, c; t)]j,k=0,...,n−1 ∝ det . det[δ δ j−1 tF (a + k, c + 1; t) j=1,...,n−1 k=0,...,n−1
Repeating this procedure on row j, j = n − 1, n − 2, . . . , j , for each of j = 2, 3, . . . , n − 2 in that order then gives det[δ j+k F (a, c; t)]j,k=0,...,n−1 ∝ det[tj F (a + k, c + j; t)]j,k=0,...,n−1 . Substituting this in (8.57) we thus have τV [n] ∝ e−nt det[F (a + k, c + j; t)]j,k=0,...,n−1 .
(8.60)
We next substitute the integral form of F from (8.56) in (8.60). This shows Z 0 ” “Z “Z ∞ τV [n] ∝ e−nt −ξ du1 e−tu1 (−u1 )a−1 (u1 + 1)c−a−n · · · −1
−1
∞
−1
Z −ξ
0
” dun
−1
×e−tun (−un )a−1 (un + 1)c−a−n det[(−uj+1 )k (uj+1 + 1)n+j−k−1 ]j,k=0,...,n−1 .
(8.61)
But using elementary column operations it follows that the determinant in this expression is equal to n−1 Y
(1 − uj+1 )j det[ukj+1 ]j,k=0,...,n−1 .
(8.62)
j=0
All factors in the integrand of (8.61), apart from (8.62), are symmetric in {uj } so we can symmetrize (8.62) and then substitute it in (8.61). Noting that Sym
“ n−1 Y j=0
” Y (1 − uj+1 )j det[ukj+1 ]j,k=0,...,n−1 = ± (uj − uk )2 j
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EIGENVALUE PROBABILITIES — PAINLEVE´ SYSTEMS APPROACH
for some sign ±, substituting this for the determinant in (8.61), and changing variables uj → uj − 1, then uj → uj /t, (μ) gives the sought identification between τV (t) and EN,2 ((0, t); ξ, xa e−x ).
E XERCISES 8.2
1. [527] The objective of this exercise is to show that (8.55) satisfies the Toda lattice equation
(8.51). (i) Let D = det[aij ]i,j=1,...,n , and define ∂D , ∂aij
Dij :=
Dij,pq :=
∂2D . ∂aip ∂ajq
Note that the Laplace expansion by row i gives n X
n X
aij Dij = D,
j=1
aij Dkj = 0 (k = i).
(8.63)
j=1
(Hint: the second equation corresponds to the Laplace expansion when two rows of [aij ] are equal.) (ii) Let denote any derivative operation. By making use of the identity n X
ars
s=1
n “ D ” X Dis is =− ars , D D s=1
which follows from (8.63), and (8.63) itself, verify that −
n X n X
ars
r=1 s=1
Choose =
∂ ∂apq
“ D ” Dis Drj ij = . D2 D
to deduce from this that 1 ∂ “ Dij ” = − 2 Diq Dpj . ∂apq D D
On the other hand, show directly from the definitions that ∂ “ Dij ” 1 = 2 (DDip,jq − Dij Dpq ) ∂apq D D »
and thus conclude det
Dij Dpj
Diq Dpq
– = DDip,jq .
(8.64) (n)
(n)
d (iii) Let D(n) := det[δsi δtj g(s, t)]i,j=0,...,n−1 , where δu := a(u) du , and define Dij and Dij,pq as in (i) with (n) D therein replaced by D . Note that (n+1)
δs D(n) = −Dnm (n+1) D(n) = Dm,m ,
,
(n+1)
δt D(n) = −Dmn , (n+1) D(n−1) = Dnm,nm ,
(n+1)
δs δt D(n) = Dnn
,
where m = n + 1. Substitute these formulas in (8.64) with D replaced by D(n+1) , i = j = n, p = q = m to deduce that » – δs δt D(n) δs D(n) det = D(n+1) D(n−1) , δt D(n) D(n) and note that this can be rewritten to read δs δt log D(n) =
D(n+1) D(n−1) . (D(n) )2
(8.65)
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CHAPTER 8
(iv) Let g(s, t) = g(s − t, 0) =: g(u) so that D(n) = (−1)n(n−1)/2 det[δui+j g(u)]i,j=0,...,n−1 . Use this in (8.65) to deduce (8.55) satisfies (8.51). 2. [230] In this exercise a determinant and integral solution of the PIII system will be presented. √ (0) (i) Note that u(t) = t−μ/2 Iμ ( t), μ = v1 , is a solution of the Bessel differential equation in Table 8.9. (ii) Conclude from (i), Table 8.10 and (8.55) that ˛ h“ d ”j+k √ i 2 ˛ = det t t(1−μ)/2 Iμ ( t) , tn /2 τIII [n]˛ dt t→t/4 j,k=0,...,n−1 and upon using the theory of the third sentence of the proof of Proposition 8.2.6 show that this can be rewritten as ˛ h“ d ”j+k √ i 2 ˛ = det t Iμ ( t) . tn(μ−1)/2 (tn /2 τIII [n])˛ dt t→t/4 j,k=0,...,n−1 (iii) In the determinant change variables t = s2 and make use of the Bessel function identities s s
d Iμ (s) = sIμ+1 (s) + μIμ (s), ds
d j (s Iμ+j (s)) = sj+1 Iμ+j−1 (s) − μsj Iμ+j (s) ds (0)
(0)
together with elementary row and column operations to conclude that with v1 = −v2 √ τIII [n](t/4) = t−nμ/2 det[Iμ+j−k ( t]j,k=1,...,n .
=μ (8.66)
(iv) Use the integral representation Iν (s) :=
1 2π
Z
π
e−iνθ es cos θ dθ,
−π
ν∈Z
together with (5.76) to deduce from (8.66) that for μ ∈ Z Z π Z π √ √ t−nμ/2 t cos θ1 −iμθ1 dθ e · · · dθn e t cos θn −iμθn τIII [n](t/4) = 1 n n!(2π) −π −π = t−nμ/2
n DY
e
E √ t cos θj −iμθj CUEn
j=1
(8.67)
Y
|eiθj − eiθk |2
1≤j
.
(8.68)
3. [230] The objective of this exercise is to derive a τ -function sequence for the P V ∗ system which involves an average over CUEn analogous to that in (8.68), and has the feature of being singular in terms of the coordinates of the PV system. Write ∗
∗
(0)
TVn H0V = HnV ,
(0)
(0)
(0)
TVnv (0) = (v1 − n/4, v2 − n/4, v3 − n/4, v4 + 3n/4)
∗
and define τnV (up to normalization) by the requirement that ∗
HnV = (0)
∗ d log τnV . dt ∗
(0)
∗
(i) Show that with q0 = 1, v3 − v4 = 0, the Hamiltonian tH0V reduces to tH0V = (v1 − v3 )t. Note that the transformation formulas (8.38) between {p, q} and {P, Q} are singular at q = 1. Show too that (0)
τ0 = e(v1
(0)
−v3
)t
,
p0 = t
∗ d (0) (0) log τ1V + (v3 − v1 )t, dt
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EIGENVALUE PROBABILITIES — PAINLEVE´ SYSTEMS APPROACH
Function
σ Painlev´e equation
G UN (t; μ; ξ)
σPIV
L UN (t; a, μ; ξ)
σPV
ν0 = 0, ν1 = −μ, ν2 = N + a, ν3 = N
J UN (t; a, b, μ; ξ)
σPVI
v1 = 12 (a + b) + N + μ, v2 = 12 (a + b) v3 = 12 (a + b) + N, v4 = 12 (b − a)
Cy UN (t; α, μ; ξ)
σ ˜ PVI
v1 = μ + N − α, v2 = −α v3 = N − α, v4 = 0
Parameters α1 = −μ,
α2 = −N
Table 8.11 The differential equations σPIV–σPVI are given in (8.15), while σ ˜ PVI is given in (8.21).
(0)
where e−(v1
(0)
−v3
)t V ∗ τ1
satisfies the confluent hypergeometric differential equation ty + (v1 − v2 + 1 + t)y + (v1 − v3 )y = 0. (0)
(0)
(0)
(0)
(ii) Consider the solution of the above equation analytic at the origin, (0)
e−(v1
(0)
−v3
)t V ∗ τ1
(0)
(0)
(0)
(0)
= 1 F1 (v1 − v3 , v1 − v2 + 1; −t).
Write this in an integral form by verifying, using (4.4) in the case N = 1, that with z = eiθ Z π Γ(μ + 1)Γ(ν + 1) 1 z (μ−ν)/2 |1 + z|μ+ν etz dθ. 1 F1 (−ν, μ + 1; −t) = Γ(μ + ν + 1) 2π −π (0)
(0)
(iii) Proceed as in the proof of Proposition 8.2.6 to deduce from (ii) that with v1 − v3 ∗
τnV = e−νt
“
(0)
n n−1 DY E Γ(μ + 1) ”n Y (μ−ν)/2 Γ(ν + l + 1) zl |1 + zl |μ+ν etzl . Γ(μ + ν + 1) CUEn l=0
(0)
= −ν, v1 − v2
=μ
(8.69)
l=1
8.3 σ-FORM PAINLEVE´ EQUATION CHARACTERIZATIONS 8.3.1 Finite N ensembles In Proposition 8.2.6 the averages (8.7) in the four classical cases have been identified as τ -functions of certain Painlev´e systems. By the definition (8.44), the logarithmic derivative of the τ -function is the Hamiltonian, and we know from Proposition 8.2.1 that for each Hamiltonian there is an auxiliary Hamiltonian, which from (8.29) is simply related to a σ-function satisfying the corresponding second order second degree differential equation in (8.15). Thus after multiplying (8.7) in the classical cases by an appropriate function of N , a(t) (as specified in Table 8.10) times its logarithmic derivative satisfies a Painlev´e equation in σ-form. Provided a boundary condition can be specified, we can therefore characterize the averages (8.7) in the classical cases in terms of particular solutions of the equations (8.15). Cy G L J P ROPOSITION 8.3.1 Let UN (t; μ; ξ), UN (t; a, μ; ξ), UN (t; a, b, μ; ξ), UN (t; α, μ; ξ) satisfy the differential equations as specified in Table 8.11, and write
1 J FNJ (t; a, b, μ; ξ) := UN (t; a, b, μ; ξ) − e2 [−v1 , −v2 , v3 ]t + e2 [−v1 , −v2 , v3 , v4 ], (8.70) 2 where e2 is specified as in Proposition 8.2.1. Also let LN (a) := Wa,2,N as used in (4.142), Jn (a, b) =
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CHAPTER 8
Sn (a, b, 1) where Sn denotes the Selberg integral (4.1) and Mn (a) := Mn (a, a, 1) as specified by (4.4). In the notation (8.7) we have ∞ 2 Nμ (μ) G dt , UN EN,2 ((s, ∞); ξ; e−x ) = sμN exp − (t; μ; ξ) − t s s dt LN (a + μ) (μ) L EN,2 ((0, s); ξ; xa e−x ) = exp , UN (t; a, μ; ξ) + μN LN (a) t 0 s JN (a + μ, b) dt (μ) EN,2 ((0, s); ξ; xa (1 − x)b ) = exp , FNJ (t; a, b, μ; ξ) JN (a, b) t(t − 1) 0 EN,2 ((s, ∞); ξ; (1 + x2 )−α ) = (1 + s2 )N μ/2 ∞ Cy UN × exp − (t; α, μ; ξ) − te2 [−v1 , −v2 , v3 ] − tN μ (μ)
s
dt . 1 + t2
(8.71)
Also, with (μ)
EN,2 (π − x, π); ξ; |1 + eiθ |2ω π N 1 π (l) := dθ1 · · · dθN (1 − ξχπ−x,π )|e−ix − eiθl |2μ |1 + eiθl |2ω C −π −π l=1
N (l) =: (1 − ξχ(π−x,π) )|e−ix − eiθl |2μ l=1
(μ)
|eiθk − eiθj |2
1≤j
cJUE
we have EN,2 ((π − x, π); ξ; |1 + eiθ |2ω ) =
MN (ω + μ) exp MN (ω)
− 0
x/2
−(e2 [−v1 , −v2 , v3 ]
Cy UN (cot φ; N + ω + μ/2, μ; ξ) + N μ) cot φ dφ . (8.72)
α=N +ω+μ/2
Proof. In addition to the reasoning noted above the statement of the proposition, one must also consider the limiting (μ) behavior of the particular EN,2 (J; ξ; g) as |J| → 0. The evaluation (8.72) follows from the final equation in (8.71) by the mapping (3.123) between the Cauchy ensemble and the circular Jacobi ensemble. We are still faced with the task of specifying the boundary conditions associated with the solutions of the nonlinear equations in Proposition 8.3.1. For general parameters this can be done by analyzing the |J| → 0 (μ) limit of the individual elements in the determinant form of the EN,2 [235]. For μ = 0, a different approach is possible, for then we have the general expansion ξ2 (0) EN,2 (J; ξ; g(x)) = 1 − ξ ρN (t) dt + dt dt2 ρN (8.73) 1 (1) (2) (t1 , t2 ) + · · · , 2! J J J which is a consequence of (9.1) below. It implies
(0)
log EN,2 (J; ξ; g(x)) ∼ −ξ |J|→0
J
ρN (1) (t) dt.
(8.74)
Small s expansion — Jacobi ensemble The formulas of Proposition 8.3.1 can readily be used to generate power series expansions in the Laguerre, Jacobi and circular Jacobi cases. By use of the formulas (8.8), (8.1) and (8.2) the power series expansions of the corresponding probabilities EN,2 (n; J) and p.d.f.’s pN,2 (n; a) then follow. The results of Proposition 8.3.1
351
EIGENVALUE PROBABILITIES — PAINLEVE´ SYSTEMS APPROACH
also allow the latter to be numerically tabulated or statistical quantities such as the variance to be computed. An example of the latter type of calculation is given in Section 8.3.4. Here the Painlev´e characterization will J be used to compute terms in the power series expansion of EN,2 ((0, s); ξ) for general a, b (the superscript J denotes Jacobi). J We know from Proposition 8.3.1 that UN (t; a, b, 0; ξ) satisfies the σPVI equation in (8.8) with v1 = v3 = N + (a + b)/2, v2 = (a + b)/2, v4 = (b − a)/2. With FNJ specified by (8.70), let us put J (t; a, b; ξ) := FNJ (t; a, b, 0; ξ). fN
(8.75)
J J in terms of fN in the σPVI equation, using the fact that in the case of interest v1 = v3 , Substituting for UN J shows fN satisfies the equation
(t(1 − t)f )2 − 4t(1 − t)(f )3 + 4(1 − 2t)(f )2 f + 4f f 2 − 4f 2 v12 +(f )2 4tv12 (1 − t) − (v2 − v4 )2 − 4tv2 v4 + 4f f (−v12 + 2tv12 + v2 v4 ) = 0.
(8.76)
Furthermore, from (8.73) we know that this equation is to be solved subject to the boundary condition f (t) ∼ + −ξtρJ(1),N (t) ∼ + −ξcJN (a, b)ta+1 , t→0
cJN (a, b) =
t→0
Γ(N + a + b + 1)Γ(N + a + 1) , Γ(a + 1)Γ(a + 2)Γ(N )Γ(N + b)
where the second asymptotic equality follows by noting from the definitions that ρJ(1),N (t) ∼ + ta N t→0
SN −1 (a + 2, b, 1) , SN (a, b, 1)
Sn denoting the Selberg integral. Substituting this boundary condition in (8.76) generates the small t expansion a(1 − b) − 2N (a + b + N ) −f (t) ∼ ξcJN (a, b)ta+1 1 + t + O(t2 ) 2+a t→0+ (ξcJN (a, b))2 2a+2 2 + 6a + 3a2 − (3 + 2a)(2N 2 + 2N (a + b) + ab) 2 t + 1+ t + O(t ) 1+a (2 + a)2 4 + 10a + 5a2 − (4 + 3a)(2N 2 + 2N (a + b) + ab) (ξcJ (a, b))3 3a+3 2 1 + t t + O(t ) + N (1 + a)2 (2 + a)2 +··· . Substituting the above expansion (with the explicit form of the terms O(t2 ) included) in the formula for which we read off from the μ = 0 case of the appropriate formula in Proposition 8.3.1, one finds after integrating and exponentiating that 1+a ξcJ (a, b) a+1 J EN,2 s 1− ((0, s); ξ) = 1 − N (−2 − 2a + ab + 2aN + 2bN + 2N 2 )s + O(s2 ) 2 1+a (2 + a) J 2 (ξc (a, b)) (N − 1)(N + a + 1)(N + b − 1)(N + a + b + 1) 2a+4 s (1 + O(s)) + · · · . + N (a + 2)4 (a2 + 4a + 3)2 (8.77) J EN,2 ((0, s); ξ),
We remark that one can anticipate from the expansion (8.73), together with the small t1 , . . . , tn expansion ρN (n) (t1 , . . . , tn ) ∼ C
n j=1
taj
(tk − tj )2
1≤j
for some C independent of the tj , that the leading term in s accompanying the power ξ k will be proportional 2 to ska+k in agreement with (8.77).
352
CHAPTER 8
Small s expansions — the classical groups The CUE, or equivalently the classical group U (N ), is the special case ω = 0 of the cJUE. Furthermore, the case ω = 1 of the cJUE corresponds to the CUE with an eigenvalue fixed at θ = π. For definiteness we will consider the generating function (μ) cJ EN,2 ((π − x, π); ξ) := EN,2 ((π − x, π); ξ; |1 + eiθ |2ω ) . ω=0
μ=ω=0
Cy This is characterized by UN (t; N, 0; ξ), which according to Proposition 8.3.1 satisfies the σ ˜ PVI equation with v1 = v3 = v4 = 0, v2 = −N . The general expansion (8.73) shows the corresponding boundary Cy condition is UN (t; N, 0; ξ) ∼ c as t → ∞, where c = ξN/π, and this substituted into the σ ˜ PVI equation generates the large t expansion 1 c3 c2 c2 (−2 − N 2 + 9c2 ) Cy UN + 2 + . (8.78) (t; N, 0; ξ) ∼ c + + O t→∞ t t 9t3 t4 cJ Keeping terms through to O(1/t8 ), substituting in the formula for EN,2 ((π − x, π); ξ)|ω=0 which we read 9 off from (8.72), expanding the integral to O(x ) then exponentiating, again keeping terms to O(x9 ) one finds
(N 2 − 1)(2N 2 − 3) 2 6 (N 2 − 1)c2 x4 − c x 36 1350 (N 2 − 1)(N 2 − 2)(3N 2 − 5) 2 8 (N 2 − 4)(N 2 − 1)2 3 9 c x − c x + O(x10 ).(8.79) + 52920 291600 The results of Section 2.6 show that the eigenvalue p.d.f. for the classical groups O± (N ) and Sp(2N ), after the change of variables xj = cos2 θj /2, corresponds to the eigenvalue p.d.f. for the JUE with weight xa (1 − x)b , 0 < x < 1, and a, b = ±1/2. For example, O− (2N +1) J ((0, φ); ξ) = EN,2 ((0, sin2 φ/2); ξ) a=−1/2 , E2 b=1/2 O+ (2N +1) 2 J ((0, φ); ξ) = EN,2 ((0, sin φ/2); ξ) a=1/2 . (8.80) E2 CUE EN,2 ((0, 2x); ξ) = 1 − cx +
b=−1/2
Proceeding as in the derivation of (8.77) one can show (4N 2 − 1) 3 (48N 4 − 40N 2 + 7) 5 c˜x − c˜x 36 3600 (4N 4 − 5N 2 + 1) 2 6 (192N 6 − 336N 4 + 196N 2 − 31) 7 c˜ x + c˜x + 2025 211680 (48N 6 − 112N 4 + 77N 2 − 13) 2 8 c˜ x + O(x9 ), − 198450 (4N 2 − 1) 3 (4N 2 − 1)(12N 2 − 7) 5 O+ (2N +1) c˜x + c˜x ((0, x); ξ) = 1 − E2 36 3600 (4N 2 − 1)(48N 4 − 72N 2 + 31) 7 c˜x + O(x9 ), − 211680 O− (2N +1)
E2
((0, x); ξ) = 1 − c˜x +
(8.81) where c˜ := 2N ξ/π =: 2c.
353
EIGENVALUE PROBABILITIES — PAINLEVE´ SYSTEMS APPROACH
8.3.2 Soft edge Scaled limits — methodology Painlev´e himself noted that upon a suitable limiting procedure, the Painlev´e transcendents degenerate downwards (see, e.g., [310]). Thus there is a limit transition from PVI to PIII for example, but not the other way around. This can be seen at the level of the corresponding differential equations, where changing variables as appropriate for the limit and equating the highest order terms exhibits the differential equation of the limiting Painlev´e transcendent. It is this general approach we will take in characterizing the scaled limits of the finite N quantities EN,2 , known in terms of σ Painlev´e transcendents from Proposition 8.3.1. We know from Section 7.1 that with the replacement (7.11) the Gaussian ensemble scales to the soft edge. From this it can be rigorously established that 2 (0) lim EN,2 ((2N )1/2 + t/21/2 N 1/6 , ∞); ξ; e−x , N →∞
which is the limiting generating function for the probability that the interval (t, ∞) at the soft edge contains n eigenvalues, is well defined (see Section 9.1 below). Furthermore, with suitable C independent of t 2 2 (2) lim Ce−t EN,2 ((t, ∞); ξ; e−x ) 1/2 1/2 1/6 N →∞
t →(2N )
+t/2
N
is the generating function for the p.d.f. of the event that the interval (t, ∞) at the soft edge contains n eigenvalues and there is an eigenvalue at t. The fact that these scaled averages are well defined suggests that for general μ, again with suitable C independent of t, 2 2 (μ) lim Ce−μt /2 EN,2 ((t, ∞); ξ; e−x ) 1/2 1/2 1/6 N →∞
t →(2N )
+t/2
N
will be well defined. Recalling Proposition 8.3.1 and taking the logarithmic derivative, this in turn suggests that 1 G − μt + UN us (t; μ; ξ) := lim √ (t; μ; ξ) (8.82) N →∞ t →(2N )1/2 +t/21/2 N 1/6 2N 1/6 will also be well defined. P ROPOSITION 8.3.2 [519], [228] The function (8.82) satisfies the σPII equation in (8.15) with a = μ. For μ = 0 the boundary condition is us (t; 0; ξ) ∼ ξρsoft (1) (t). t→∞
We have
us (s; 0; ξ) ds , t ∞ soft soft soft p2 (t; ξ) = ρ(1) (t) exp − us (s; 2; ξ) − ρsoft (1) (s)/ρ(1) (s) ds ,
E2soft ((t, ∞); ξ) = exp
−
(8.83)
∞
(8.84)
t
where us (s; 0; ξ) and us (s; 2; ξ) are related by us (t; 2; ξ) =
d log us (t; 0; ξ) + us (t; 0; ξ). dt
(8.85)
Proof. The fact that (8.82) satisfies the σPII equation with a = μ is deduced by recalling from Proposition 8.3.1 that G G G ˜N (t; μ; ξ) satisfies the σPIV equation in (8.15) with α1 = α2 = −N , replacing UN by U + μt, changing UN √ −μ,1/6 1/2 1/2 1/6 G s ˜ variables t = (2N ) + s/2 N , then replacing UN by 2N u (s; μ; ξ) and taking the limit N → ∞. One sees that the σPIV equation degenerates to σPII.
354
CHAPTER 8
For the boundary conditions in the case μ = 0 one uses (8.73), while the inter-relation (8.85) follows from the fact that psoft 2 (t; ξ) =
1 d soft E2 ((t, ∞); ξ), ξ dt
(8.86)
which in turn is an example of (8.5). As noted in the text above the statement of the proposition we have “ ” 2 (0) E2soft ((t, ∞); ξ) = lim EN,2 (2N )1/2 + t/21/2 N 1/6 , ∞); ξ; e−x N→∞
˛ 2 ˛ −t2 (2) psoft EN,2 ((t, ∞); ξ; e−x )˛ 2 ((t, ∞); ξ) = lim Ce
and
t→(2N)1/2 +t/21/2 N 1/6
N→∞
.
2
Recalling the formula of Proposition 8.3.1 for EN,2 (μ; (t, ∞); ξ; e−x ) immediately gives the first formula in (8.84). For the second formula we must take into consideration the asymptotic form (8.83) to ensure that the integrand is integrable soft at infinity, and fix the undetermined proportionality constant by requiring that psoft 2 ((t, ∞); ξ) ∼ ρ(1) (t). t→∞
The formulas (8.84) can be used to compute a tabulation of psoft 2 (n; t), provided n is small. However we defer this task until Section 9.4 when we have available an even simpler Painlev´e transcendent evaluation of E2soft ((t, ∞); ξ).
8.3.3 Hard edge With the variables scaled as specified by (7.30), the Laguerre ensemble scales to the hard edge, and we expect LN (a) t hard (μ) (μ) a −x E ; ξ; x e 0, (8.87) ((0, t); ξ; a) := lim E2 N →∞ LN (a + μ) N,2 4N (here LN (a) = Wa,2,N is as used in (8.71); note that the constants are chosen so that at t = 0 it equals 1) to be well defined. Recalling the second formula in (8.71) it follows that t ds hard (μ) E2 (8.88) ((0, t); ξ; a) = exp uh (s; a, μ; ξ) , s 0 where uh (t; a, μ; ξ) := lim
N →∞
L UN
t ; a, μ; ξ + μN . 4N
(8.89)
L In this limit the σPV equation characterizing UN degenerates to a variant of the σPIII equation.
P ROPOSITION 8.3.3 [520], [230] The function uh specified by (8.89) satisfies the differential equation (tu )2 − (μ + a)2 (u )2 − u (4u + 1)(u − tu ) − and consequently
μ(μ + a) μ2 u − 2 =0 2 4
uh (t; a, μ; ξ) = − σIII (t) + μ(μ + a)/2 ,
(8.90)
(8.91)
where σIII satisfies the σPIII’ equation in (8.15) with parameters v1 = a + μ, v2 = a − μ. For μ = 0 the boundary condition is uh (t; a, 0; ξ) = −σIII (t) ∼ −ξtρhard (1) (t) ∼ −ξ t→0+
t→0+
t1+a . 22+2a Γ(1 + a)Γ(2 + a)
(8.92)
355
EIGENVALUE PROBABILITIES — PAINLEVE´ SYSTEMS APPROACH
We have
t σIII (s) E2hard ((0, t); ξ; a) = exp −
ds , μ=0 s 0 t ds ta hard exp − , σIII (s) p2 ((0, t); ξ; a) = 2a+2 +a+2 2 Γ(a + 1)Γ(a + 2) s μ=2 0 (8.93)
where σIII (s)|μ=0 and σIII (s)|μ=2 are related by σIII (t)|μ=2 = σIII (t)|μ=0 − 1 − t
σIII (t)|μ=0 . σIII (t)|μ=0
(8.94)
Proof. Making the replacement σV → σV − N μ in the σPV equation of (8.15), changing variables t → t/4N , σV (t/4N ) → u(t) and equating terms of order N 2 (which is the leading order) on both sides gives (8.90). The μ = 0 boundary condition follows from the general formula (8.73), while (8.94) follows by noting ((0, t); ξ; a) = − phard 2
1 d hard E2 ((0, t); ξ; a). ξ dt
(8.95)
The first formula in (8.93) follows immediately from the fact that hard(0)
E2hard ((0, t); ξ; a) = E2 hard(0)
together with the evaluation of E2 that
((0, t); ξ; a),
implied by (8.88) and (8.91). For the second formula note from (8.8) and (8.87)
phard ((0, t); ξ; a) = ta lim 2
N→∞
““ ” LN (a + 2) 1 t ” (2) a −x E e 0, , ; ξ; x LN+1 (a) (4N )a N,2 4N
then make use of (8.88) and (8.91). hard(μ)
The case a ∈ Z≥0 of E2 (s; ξ = 1; a) is special, because then an a-dimensional determinant expression can be found. This is a consequence of the identity (8.119) in Exercises 8.3 q.1(v). First we note that a simple change of variables shows that for J = (0, t) or J = (t, ∞) (μ) (a) EN,2 (J, ξ = 0; xa e−x ) = eN t EN,2 ((0, t), ξ = 1; xμ e−x ). t →−t
We then apply (8.119) below to conclude that for a ∈ Z≥0 EN,2 ((0, t), ξ = 1; xμ e−x ) ∝ e−N t (a)
a (1 + zl )μ (1 + 1/zl )N etzl l=1
CUEa
.
The hard edge limit (8.87) can be taken on the r.h.s. by regarding each integration over zl as a contour integral over |zl | = 1, then shifting the contour of integration to |zl | = 2N/t1/2 . This shows that for a, μ ∈ Z≥0 hard (μ)
E2
((0, s); ξ = 1; a) = a!
a a (j + μ − 1)! −s/4 2 aμ μ √s(zl +z−1 )/2 l e zl e j! s1/2 CUEa j=1
(8.96)
l=1
(cf. (8.68)), where the proportionality constant has been specified by the requirement that both sides equal unity as s → 0; see (13.27) below. Making use of the integral representation (8.67) and the identity (5.77) shows that this can equivalently be written as the determinant formula a √ (j + μ − 1)! −s/4 2 aμ hard (μ) E2 e ((0, s); ξ = 1; a) = a! det I ( s) , (8.97) μ+j−k j! s1/2 j,k=1,...,a j=1 where the restriction μ ∈ Z≥0 present in (8.96) can be removed. It follows from this that for a ∈ Z≥0 we
356
CHAPTER 8
have [204]
√ E2hard (n = 0; t; a) = e−t/4 det[Ij−k ( t)]j,k=1,...,a , √ 1 (n = 0; t; a) = e−t/4 det[I2+j−k ( t)]j,k=1,...,a , phard 2 4
(8.98) hard(2)
where the latter formula follows from the fact that phard (0; t; a) is proportional to E2 (s; 1; a), and fixing 2 the proportionality constant by requiring that the small t behavior be as displayed in (8.93). We remark that integration techniques similar to those of the second part of Proposition 5.1.3 allow us to give alternative derivations of the identities (8.98), and also to derive that for a ∈ Z≥0 [204] t (1,2) hard p2 (n = 1; t; a) = p2 (u, t) du, (8.99) 0
(1,2) p2 (t1 , t2 )
where minant form
— the joint distribution of the smallest and second smallest eigenvalue — has the deter-
hard (1,2) p2 (t1 , t2 )
=2
−4 −t2 /4
e
a
(t2 /t1 ) det
1/2
[Ij−k+2 (t2 )]
j=1,...,a
k=1,...,a+2√ 1 (k−j)/2 [( t2t−t ) Ij−k+2 ( t2 − t1 )] 2
.
(8.100)
j=1,2 k=1,...,a+2
8.3.4 High precision numerical computation One use of the Painlev´e transcendent evaluations of the spacing distributions is that it allows for their high precision numerical computation, provided the number of eigenvalues conditioned to be in the spacing is small. As an example we will consider the scaled distribution of the first and second eigenvalues as measured anticlockwise from θ = 0, for the orthogonal groups O+ (2N + 1) and O− (2N + 1). These are in fact related to hard edge distributions. Thus we see from (8.80), and the discussion around (7.45) relating to the hard edge scaling of the Jacobi ensemble, that − 1 O− (2N +1) lim E2 ((0, φ); ξ) := E2O ((0, s); ξ) = E2hard ((0, (πs)2 ); − ; ξ), N →∞ 2 φ=πs/N + + 1 O (2N +1) (8.101) lim E2 ((0, φ); ξ) := E2O ((0, s); ξ) = E2hard ((0, (πs)2 ); ; ξ). N →∞ 2 φ=πs/N According to (8.88) and Proposition 8.3.3 ∓ E2O ((0, s); ξ) = exp −
0
(πs)2
1 dt , v t; ± ; ξ 2 t
(8.102)
where v(t; ± 21 ; ξ) satisfies the equation 1 (tv )2 − (v )2 + v (4v − 1)(v − tv ) = 0, 4 with the transcendents v(t; a; ξ) =: v(t) distinguished by the boundary conditions, v(t) ∼ + t→0
ξt1/2 1 (a = − ), π 2
v(t) ∼ + t→0
ξt3/2 1 (a = ). 3π 2
(8.103)
(8.104)
Noting from (8.95) that ±
pO 2 ((0, s); ξ) = −
1 d O± E ((0, s); ξ) ξ ds
(8.105)
EIGENVALUE PROBABILITIES — PAINLEVE´ SYSTEMS APPROACH
we want to use (8.102) to compute ±
pO 2 (n; s) =
(−1)n ∂ n O± p ((0, s); ξ) . 2 n! ∂ξ n ξ=1
357
(8.106)
This will be done by computing power series solutions of (8.103) about the origin. For this purpose, we begin by noting that a crucial feature of the boundary conditions (8.104) is that (8.103) generates unique power series expansions of v(t) in terms of t1/2 , v(t) = cta+1 +
∞
cp ta+(p+3)/2 .
(8.107)
p=0
In particular direct substitution of (8.107) (with a specified) into (8.103) allows a general coefficient cp to be expressed in terms of a polynomial function of {c, c1 , . . . , cp−1 }, which in turn can be written as an explicit system of equations. These equations are well suited to solving via computer algebra, and in this way we can obtain several hundred coefficients, all of which are polynomials in ξ. It turns out that for a = − 12 , cp is of degree p + 2 in ξ, while for a = 12 , cp is of degree p/3 (approximately) in ξ. For this reason it is more difficult to compute cp in the former case than the latter, and 436 and 700 coefficients were calculated respectively. Let this number be denoted by M . We next substitute the truncated series in (8.105) and compute the first M + 2 terms of the power series of the exponential. Working in integer arithmetic, this gives the exact form of each of these M + 2 terms. Theoretically the fact of E2hard ((0, (πs)2 ); ξ) being a τ -function implies it is an entire function of s. Because a large number of terms in its power series expansion have been computed, high precision computation ∞ q O± ± O± of pO 2 (n; s) is possible. We want to use this to compute the moments 0 s p2 (n; s) ds. Since p2 (n; s) α q O± is given as a truncated power series, we compute term by term 0 s p2 (n; s) ds, where α is a cutoff tuned ∞ ± to get a value for the normalization 0 pO 2 (n; s) ds as close as possible to, but not exceeding, 1. For general n the number of accurate digits can be estimated by reducing the value of the degree M in the truncation of + O− the power series. For pO 2 (n; s) with n > 5 and p2 (n; s) with n > 4 the power series are not large enough to get useful information. For smaller values of n, high precision evaluations of the moments are obtained. From these we compute for the mean, variance and higher order statistical quantities γ1 :=
(X − X)3 , (Var X)3/2
γ2 :=
(X − X)4 − 3, (Var X)2
(8.108)
referred to as the skewness and (excess) kurtosis, respectively, the values as listed in Table 8.12. Table 8.12 has been extended to values of n beyond those accessible using the power series. This has been possible due to a computational procedure based on the Fredholm determinant formula (9.81) below (see Section 9.2 for the required methodology). 8.3.5 Spectrum singularity and bulk The circular Jacobi average of Proposition 8.3.1 is expected to have a well-defined scaled limit after the replacement φ → X/N and multiplication by a suitable constant (i.e., term independent of X). To avoid this latter problem, one considers the scaled logarithmic derivative X d (μ) u(X; ω, μ; ξ) := lim X log EN,2 π − , π ; ξ; |1 + eiθ |2ω . (8.109) N →∞ dX N It follows from (8.72) that d (μ) −2 log EN,2 (π − φ, π); ξ; |1 + e2πiθ |2ω dφ ˜ − (e2 [−v1 , −v2 , v3 ] + N μ) cot φ, = U Cy (cot φ; N + ω + μ/2; ξ) N
358
CHAPTER 8
p.d.f.’s 2
1.5
1
0.5
1
0.5
1.5
2
2.5
3
s
−
Figure 8.2 Plot of pO 2 (n; t) for n = 0 (leftmost curve) and n = 1.
p.d.f. − pO 2 (0; s) − pO 2 (1; s) − pO 2 (2; s) − pO 2 (3; s) − pO 2 (4; s) − pO 2 (5; s) − pO 2 (6; s) − pO 2 (7; s) − pO 2 (8; s) − pO 2 (9; s) +
pO 2 (0; s) + pO 2 (1; s) + pO 2 (2; s) + pO 2 (3; s) + pO 2 (4; s) + pO 2 (5; s) + pO 2 (6; s) + pO 2 (7; s) + pO 2 (8; s) + pO 2 (9; s)
mean
variance
skewness
kurtosis
0.32138 26639
0.06016 64291
1.03522 47759
1.01497 61256
1.27026 53928
0.13347 95698
0.27104 49106
−0.03359 26930
2.26131 89835
0.16459 27733
0.14246 20157
−0.03907 67354
3.25782 76925
0.18375 20687
0.09485 95413
−0.03218 00266
4.25597 96097
0.19759 69896
0.07058 23711
−0.02733 17251
5.25483 70464
0.20844 38843
0.05597 43856
−0.02404 98007
6.25406 10064
0.21736 42245
0.04625 77696
−0.02171 86929
7.25349 95584
0.22494 12584
0.03934 55728
−0.01998 19739
8.25307 45177
0.23152 77403
0.03418 57830
−0.01863 62447
9.25274 15598
0.23735 33656
0.03019 20747
−0.01756 00843
0.78271 57582
0.10658 96782
0.45629 16499
0.05534 84703
1.76454 67894
0.15135 50698
0.18800 06679
−0.04163 48906
2.75925 65680
0.17507 73887
0.11409 64039
−0.03542 19954
3.75678 01658
0.19114 91594
0.08100 77394
−0.02951 00917
4.75534 80131
0.20331 19066
0.06246 53950
−02554 06905
5.75441 51870
0.21310 11204
0.05067 01710
−0.02279 24207
6.75375 94329
0.22129 48617
0.04253 19349
−0.02079 12811
7.75327 32947
0.22834 18447
0.03659 05822
−0.01926 91981
8.75289 85058
0.23452 44998
0.03206 90553
−0.01807 00081
9.75260 07410
0.24003 21874
0.02851 66965
−0.01709 81199
∓
−
+
O O Table 8.12 Statistical properties of pO 2 (n; t) for various n. In the cases of p2 (n; t), n ≥ 2, and p2 (n; t), n ≥ 3, these are from [79].
359
EIGENVALUE PROBABILITIES — PAINLEVE´ SYSTEMS APPROACH
Cy where UN (t; α, μ; ξ) satisfies the σ ˜ PVI equation (8.21) with parameters as noted in Table 8.11. Changing variables and equating terms of leading order in N shows that (8.109) satisfies the σPV equation [230].
P ROPOSITION 8.3.4 We have that h(t) := u(it; ω, μ; ξ) + ωμ satisfies the σPV equation in (8.15) with s.s. (t). ν0 = 0, ν1 = μ, ν2 = ω + μ/2, ν3 = −ω + μ/2, h(−it)|μ=0 ∼ + −ξtρ(1) t→0
Furthermore
E2s.s. ((0, t); ξ) = exp
2πt
h(−is)|μ=0 0
p2s.s. ((0, t); ξ) = (2πt)2ω
ds , s
Γ2 (ω + 1) exp Γ(2ω + 1)Γ(2ω + 2)
2πt
h(−is)|μ=2 − 2ω
ds
0
s
, (8.110)
where h(u)|μ=0 and h(u)|μ=2 are related by h(u)|μ=2 = −1 + h(u)|μ=0 + u
h (u)|μ=0 . h(u)|μ=0
Proof. The formulas (8.110) follow from (8.109) and the respective finite N formulas in Proposition 8.3.1. The boundary condition for μ = 0 follows from the general formula (8.74) and the inter-relation from the fact that ps.s. 2 ((0, t); ξ) = −
From the definitions
E2s.s. (n; (0, t))
ω=0
= E2bulk (n; t),
1 d s.s. E2 ((0, t); ξ). ξ dt
p2s.s. (n; (0, t))
ω=1
= pbulk (n; t), 2
where the bulk state has ρ = 1. Writing σ(s; ξ) := h(−2is)|μ=0 , u(s; ξ) := h(−is)|μ=2 − 2, it follows from Proposition 8.3.4 that [324], [233] πt σ(s; ξ) E2bulk ((0, t); ξ) = exp ds, s 0 2πt π2 2 ds pbulk ((0, t); ξ) = exp u(s; ξ) t (8.111) 2 3 s 0 where σ and u satisfy the equations
(sσ )2 + 4(sσ − σ) sσ − σ + (σ )2 = 0, (su )2 + (u − su )(u − su + 4 − 4(u )2 ) − 16(u )2 = 0
(8.112)
with boundary conditions ξ2 ξ σ(s; ξ) ∼ − s − 2 s2 , s→0 π π 1 2 ξ 5 s + O(s7 ) . u(s; ξ) ∼ − s + O(s4 ) − (8.113) s→0 15 8640π From (8.111)–(8.113), proceeding as in the derivation of (8.79) we obtain the power series expansion for the
360
CHAPTER 8
p.d.f.
mean
variance
skewness
kurtosis
pbulk (0; s) 2 pbulk (1; s) 2 pbulk (2; s) 2 pbulk (3; s) 2 pbulk (4; s) 2 pbulk (5; s) 2 pbulk (5; s) 2 pbulk (7; s) 2 pbulk (8; s) 2 bulk p2 (9; s)
1 2 3 4 5 6 7 8 9 10
0.17999 38776 0.24897 77536 0.29016 98290 0.31944 35563 0.34214 08054 0.36067 45961 0.37633 63928 0.38989 74631 0.40185 51105 0.41254 86854
0.49706 36204 0.24167 43158 0.15542 00591 0.11334 61773 0.08871 43069 0.07263 43907 0.06135 08835 0.05301 56552 0.04661 73337 0.04155 73856
0.12669 98480 −0.01494 23984 −0.02317 40428 −0.02150 23114 −0.01914 18388 −0.01714 28515 −0.01555 25979 −0.01428 79010 −0.01326 81121 −0.01243 20513
Table 8.13 Statistical properties of pbulk (n; s) for various n. In the cases n ≥ 3 these are from [79]. 2
generating function of (8.111) ξ 2 π 4 t6 ξ 2 π 6 t8 ξ 3 π 6 t9 ξ 2 π 8 t10 ξ 3 π 8 t11 ξ 2 π 2 t4 − + − − + + · · · (8.114) 36 675 17640 291600 637875 4961250 which we observe is consistent with (8.79). Consequently, making use also of (8.17), and reinstating a general density ρ, we have E2bulk ((0, t); ξ) = 1 − ξt +
1 bulk 1 d2 E2bulk (t; ξ)|ξ=1 p2 (0; t) = 2 ρ ρ dt2 2 2 2π 4 (ρt)4 π 6 (ρt)6 π 6 (ρt)7 2π 8 (ρt)8 11π 8 (ρt)9 π (ρt) − + − − + + ··· , = 3 45 315 4050 14175 496125 1 bulk 1 d2 E2bulk (t; ξ)|ξ=1 d2 ∂ bulk p2 (1; t) = 2 2 E2 (t; ξ) − 2 2 ρ ρ dt dt ∂ξ ξ=1 6 7 8 9 π (ρt) 11π (ρt) = − + ··· (8.115) 4050 496125 The numerical evaluation of pbulk (n; t) for small n and the corresponding moments can be carried out 2 using the expression for pbulk ((0, t); ξ) in (8.111) according to the method detailed in Section 8.3.4. Results 2 of such a calculation, obtained from computing 700 terms in the power series expansion of E2bulk ((0, t); ξ), are given in Table 8.13. As to be discussed in Section 9.2, for general n ≥ 0 a numerical procedure based on the Fredholm determinant formula (11.26) can be used to provide accurate determination of the pbulk (n; t) 2 [79]. This has allowed our table to extended beyond the cases accessible from the power series. E XERCISES 8.3
1. [230] In this exercise averages over the classical group will be related to averages over the JUE, and so given evaluations in terms of Painlev´e transcedents. (i) Show from the definitions that “ (μ) (μ) EN,2 ((t, ∞); ξ; xa e−x ) = (1 − ξ)N EN,2 (0, t);
” ξ ; xa e−x . ξ−1
Conclude from Proposition 8.3.1 that VNL (t; a, μ; ξ) := t
d (μ) log EN,2 ((t, ∞); ξ; xa e−x ) − μN, dt
L like UN , satisfies the σPV equation in (8.15) with ν0 = 0, ν1 = −μ, ν2 = N + a, ν3 = N .
(8.116)
361
EIGENVALUE PROBABILITIES — PAINLEVE´ SYSTEMS APPROACH
(ii) Show from the definitions that EN,2 ((t, ∞); 1; xa e−x ) ∝ t(a+μ)N+N (μ)
2
D
e−t
PN
j=1
xj
E JUEN |b →μ
,
where here JUEN refers to the ensemble (8.6) with weight xa (1 − x)b , 0 < x < 1. Use the result of (i) to then obtain the evaluation Z t L D E PN VN (s, a, μ; 1) − aN − N 2 ds e−t j=1 xj = exp s JUEN |b →μ 0 and deduce from Proposition 8.2.6 that there is a solution of the PV system such that ˛ D E PN ˛ e−t j=1 xj = τV (t)˛ ν =−μ,ν =0 (νj−1 = vj − v2 ). JUEN |b →μ
0 1 ν2 =a+N,ν3 =N
(iii) With (N ∗ , a, b) specified as in Proposition 3.7.1 note from the results therein that for G = O+ (N ), O− (N ) or Sp(2N ), D E ∗
etTr(U) U∈G = e−2tN eχN ∗ t e4t
PN ∗
j=1
λj
JUEN ∗
,
(8.117)
where χN ∗ = 0 for G = Sp(2N ) and O+ (N ), O− (N ) (N even), while χN ∗ = ±1 for G = O+ (N ), O− (N ) (N odd), respectively. Use the result of (ii) to deduce from this that Z 4t L ∗ VN ∗ (s; a, b; 1) − aN ∗ − N ∗2 eχN ∗ t e−tTr(U) U∈G = e2tN exp ds, s 0 where VNL∗ is specified as in (i) together with the boundary condition VNL (t; a, b; 1) ∼ aN + N 2 + O(t). t→0+
(iv) Use Proposition 3.9.1 and Carlson’s theorem, Proposition 4.1.4, to deduce that D
et
PN
j=1
xj
E
˛ ˛ ˛
JUEN b→μ
∝
N DY
2πixl
eπixl (a −b ) |1 + e2πixl |a +b e−te
l=1
E CUEN
,
(8.118)
where a = N + a + μ, b = −(N + a) and thus from the result of (ii) conclude N DY
eπixl (a −b ) |1 + e2πixl |a
l=1
=
MN (a , b , 1) exp MN (0, 0, 1)
Z
t 0
+b te2πixl
e
E CUEN
VNL (s, a, μ; 1) − aN − N 2 ˛˛ ds˛ a=−(N +b ) , s μ=a +b
where MN denotes the Morris integral. (v) For a , b ∈ Z≥0 , use the identity (a −b )/2
zl
|1 + zl |a +b = (1 + zl )a (1 + 1/zl )b ,
zl := e2πixl
(which has implicitly been used previously in going from (3.120) to (3.121)) to deduce that the CUEN average in (iv) is a polynomial of degree b N in t, and combine (i) and (iv) to deduce that DY E −2πixl d eπixl (a −b ) |1 + e2πixl |a +b ete log t−b N dt CUEN l=1 N
t
satisfies the σPV equation in (8.15) with {ν0 , ν1 , ν2 , ν3 } = {0, a + b , b , −N } (this also requires using
362
CHAPTER 8
the fact that if σ(t) satisfies the σPV equation with parameters {ν0 , ν1 , ν2 , ν3 }, then σ(−t) satisfies the same equation with parameters {−ν0 , −ν1 , −ν2 , −ν3 }). Show that this result could also be deduced from the result of Exercises 8.2 q.3(iii). Note from the definitions that with μ ∈ Z≥0 , J = (t, ∞) or (0, t), (μ) (μ) EN,2 (J, 0; xa e−x ) is a polynomial of degree μN . Recalling too the characterization of EN,2 from (i), combine these facts to conclude that for μ ∈ Z≥0 EN,2 (J, 0; xa e−x ) ∝ (μ)
μ E DY (1 + zl )a (1 + 1/zl )N e−tzl l=1
(8.119)
CUEμ
(note the dual role played by N and μ on the two sides of this identity). 2. [233] The aim of this exercise is to derive a duality formula between certain averages over the CUE. (i) Show that analogous to (8.116), “ (μ) (μ) EN,2 ((t, 1); ξ; xa (1 − x)b ) = (1 − ξ)N (−1)μN EN,2 (0, t),
” ξ ; xa (1 − x)b . ξ−1
J (t; a, b, μ; ξ), Conclude from Proposition 8.3.1 that, like UN (1)
VNJ (t; a, b, μ; ξ) := e2 t − (1)
1 (2) d (μ) e + t(t − 1) log EN,2 ((t, 1); ξ; xa (1 − x)b ), 2 2 dt
(2)
where e2 := e2 [−v1 , −v2 , v3 ] and e2 := e2 [−v1 , −v2 , v3 , v4 ], satisfies the σPVI equation with v1 =
1 (a + b) + N + μ, 2
v2 =
1 (a + b), 2
v3 =
1 (a + b) + N, 2
v4 =
1 (b − a). 2
(ii) Use a simple change of variables to show (μ)
EN,2 ((t, 1); ξ = 1; xa (1 − x)b ) = tN(a+μ+N)
N DY E (1 − txl )b l=1
JUEN |b →μ
and note too that N (a + μ + N ) = (v3 − v2 )(v1 − v4 ) to conclude from (i) that N E DY (1 − txl )b l=1
JUEN |b →μ
Z t“ = exp
(1)
(v3 − v2 )(v4 − v1 )(t − 1) − e2 t +
0
” dt 1 (2) e2 + VNJ (t; a, b, μ; ξ = 1) , 2 t(t − 1)
where JUEN is defined on 0 < x < 1 as in q.1(iii) above. (iii) Let vˆ = (v3 , v4 , −v2 , −v1 ). Note that the σPVI equation is unchanged by the mapping v → vˆ. Show too that 1 (2) 1 (2) (1) (1) e2 t − e2 − (t − 1)(v3 − v2 )(v1 − v4 ) = eˆ2 t − eˆ2 2 2 and thus conclude that the result of (ii) can be rewritten N E DY (1 − txl )b l=1
Z t“ JUEN |b →μ
= exp
(1)
− eˆ2 t +
0
” dt 1 (2) ˆ eˆ2 + VN (t; a, b, μ; ξ = 1) , 2 t(t − 1)
where VˆN satisfies the same σPVI equation as VN . (iv) As in the derivation of (8.118), use Proposition 3.9.1 and Carlson’s theorem to show N E DY (1 − txl )μ l=1
E MN (0, 0, 1) D Y (a −b )/2 zl |1 + zl |a +b (1 + tzl )μ , MN (a , b , 1) CUEN N
JUEN
=
l=1
363
EIGENVALUE PROBABILITIES — PAINLEVE´ SYSTEMS APPROACH
where a = N + a + μ, b = −(N + a). Use (iii) to conclude N DY
(a −b )/2
zl
|1 + zl |a
l=1
= where here
MN (a , b , 1) exp MN (0, 0, 1)
VNJ
Z
t
(1 + tzl )μ “
E CUEN
(1)
− e2 t +
0
” dt 1 (2) e2 + VNJ (t; −N − b , μ, a + b ; 1) , 2 t(t − 1)
satisfies the σPVI equation with
v1 = a + (1)
+b
N + b + μ , 2
v2 = −
N + b − μ , 2
v3 =
N − b + μ , 2
v4 =
N + b + μ 2
(2)
and e2 , e2 relate to these vs. Note too from Proposition 8.2.6 that this CUEN is a τ -function for the same PVI system. (v) Note that under the mappings μ ↔ N,
a → a + b ,
b → −a
the v parameters in (iv) map to v¯ := (v2 , v1 , v4 , −v3 ), while under this mapping together with the mapping t → 1 − t the σPVI equation is unchanged. Noting too that “ 1 (2) 1 (2) ” (1) (1) e¯2 (1 − t) − e¯2 = − e2 t − e2 , 2 2 argue as in the derivation of (8.119) to conclude that for μ ∈ Z≥0 N DY
(a −b )/2
zl
|1 + zl |a
+b
(1 + tzl )μ
E
l=1
CUEN
∝
μ DY
(a +2b )/2
zl
|1 + zl |a (1 + (1 − t)zl )N
l=1 μ E DY ∝ (1 − (1 − t)xl )N l=1
JUEμ
˛ ˛ ˛ a=b −μ .
E CUEμ
(8.120)
b=a
˜ min = λmin /4N for matrices X† X, with 3. [154] According to (8.98) the p.d.f. for the scaled smallest eigenvalue λ X an N × N square complex Gaussian, N large, takes the simple form 14 e−t/4 . Use (3.12), together with the subsequent discussion, to note that for N → ∞ the scaled condition number κ ˜ (X) := κ(X)/4N is such that κ ˜ (X) = (1/˜ κmin )1/2 .
(8.121)
Hence conclude that the p.d.f. of κ ˜ (X) is equal to 2
e−1/4y , 2y 3
y > 0.
8.4 THE CASES β = 1 AND 4 — CIRCULAR ENSEMBLES AND BULK 8.4.1 Inter-relationships between gap probabilities with unitary symmetry (0)
In general the generating function EN,2 (J; ξ; g), in the case the weight g(x) is even and interval J is symmetrical about the origin, can be written as a product of generating functions with different J, g and N for which the new g have support for x > 0 [197]. In the case of the Cauchy ensemble related to the CUE via a stereographic projection, this leads to a factorization in terms of generating functions for gap probabilities in ensembles of real orthogonal matrices. In the bulk limit this same factorization gives the generating function for the bulk gap probability as the product of two generating functions for the gap probability at the hard edge with parameters a = ± 21 . We will see subsequently that these factorizations are intimately related to
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inter-relationships between gap probabilities in the case of unitary symmetry, with gap probabilities in the cases of orthogonal and symplectic symmetry. P ROPOSITION 8.4.1 Let J{t2i ,t2i+1 } :=
3p−1 i=0
3p−1 (t2i , t2i+1 ) ∪ (−t2i+1 , −t2i )
(8.122)
i=0
with t2i+1 > t2i ≥ 0 and each (t2i , t2i+1 ) disjoint so that J{t2i ,t2i+1 } consists of p disjoint intervals on the positive half line and these same intervals reflected through the origin onto the negative half line. Suppose also that g(x) is even. Then we have (0)
(0)
+ EN,2 (J{t2i ,t2i+1 } ; ξ; g(x)) = E[(N +1)/2],2 (J{t 2
2 2i ,t2i+1 }
(0)
+ ×E[N/2],2 (J{t 2
2 2i ,t2i+1 }
+ where J{t 2
2 2i ,t2i+1 }
; ξ; x−1/2 g(x1/2 )χx>0 )
; ξ; x1/2 g(x1/2 )χx>0 ),
(8.123)
2 2 := ∪p−1 i=0 (t2i , t2i+1 ).
Proof. By definition (0)
EN,2 (J{t2i ,t2i+1 } ; ξ; g(x)) =
1“ C ×
Z
∞
−∞
N Y
Z
g(xl )
l=1
”
−ξ
dx1 · · ·
“Z
−∞
J{t
2i ,t2i+1 }
Y
∞
Z
”
−ξ
dxN J{t
2i ,t2i+1 }
|xk − xj |2 ,
(8.124)
1≤j
(0)
where C is independent of ξ and such that EN,2 (J{t2i ,t2i+1 } ; ξ; g(x))|ξ=0 = 1. Using (5.75) allows this to be rewritten Z h“ Z ∞ i ” N! (0) det EN,2 (J{t2i ,t2i+1 } ; ξ; g(x)) = −ξ . (8.125) g(x)xj+k dx C j,k=0,...,N−1 −∞ J{t ,t } 2i
2i+1
Due to both J{t2i ,t2i+1 } and g(x) being even about the origin, we see the elements in (8.125) vanish whenever j and k have different parity. For p = 1, 2, introducing Z h“ Z ∞ i ” −ξ g(x)x2(j+k) dx Ap := h“ Z
−∞ ∞
= 0
Z −ξ
J{t
2i ,t2i+1 }
J+2
{t ,t2 } 2i 2i+1
j,k=0,...,Np −1
”
g(y 1/2 )y j+k−1/2 dy
i j,k=0,...,Np −1
,
where N1 := [(N + 1)/2], N2 := [N/2], we see by rearranging rows and columns that – » N! N! 0 A1 (0) = EN,2 (J{t2i ,t2i+1 } ; ξ; g(x)) = det det A1 det A2 . 0 A2 C C Use of (8.125) on the r.h.s, and noting that both sides must equal unity when ξ = 0 then implies (8.123).
Let us consider the special case J = (−t, t), g(x) = (1 + x2 )−N of (8.123). We know from (2.51) that the l.h.s. then maps to the CUE upon a stereographic projection, θ θ (0) CUE EN,2 − tan , tan ); ξ; (1 + x2 )−N = EN,2 ((−θ, θ); ξ). (8.126) 2 2
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EIGENVALUE PROBABILITIES — PAINLEVE´ SYSTEMS APPROACH
Applying (8.123) we therefore have [34], [197] θ (0) CUE ; ξ; x−1/2 (1 + x)−N χx>0 ((−θ, θ); ξ) = E[(N +1)/2],2 0, tan2 EN,2 2 θ (0) ×E[N/2],2 0, tan2 ); ξ; x1/2 (1 + x)−N χx>0 2 θ 2 θ J J ;ξ ); ξ) = E[(N 0, sin2 E ((0, sin +1)/2],2 [N/2],2 2 2 a=−1/2,b=b∗ a=1/2,b=−b∗ = EO ∗
1 2
∗
−
(2[(N +1)/2]+1)
((0, θ); ξ)E O
+
(2[N/2]+1)
((0, θ); ξ),
(8.127)
− 21
(N odd), and the second equality follows from the first by a change of where b = (N even), b = variables y = x/(1 + x), and the third equality from (8.80). For definiteness consider the case N → 2N even. Choosing the second and third equalities (8.127) reads θ θ CUE J J E2N,2 ; ξ a=−1/2 EN,2 ; ξ a=1/2 0, sin2 0, sin2 ((−θ, θ); ξ) = EN,2 2 2 b=1/2 b=−1/2 O− (2N +1)
= E2
O+ (2N +1)
((0, θ); ξ)E2
((0, θ); ξ).
(8.128)
(An alternative way to derive this formula is to use the identity (5.95).) We remark too that one can check that this is consistent with the power series expansions (8.79), (8.81). In relation to the first equality in (8.128), we J J know from Proposition 8.3.1 that EN,2 ((0, s); ξ) for general a, b is characterized by the quantity fN (8.75), and furthermore with a = −b the parameters are v1 = v3 = N , v2 = 0, v4 = b. With these parameters we see J that the differential equation (8.76) determining fN is invariant with respect to the mapping b → −b. Thus J the fN characterizing the two terms on the r.h.s. of the first equality in (8.128) satisfies the same differential equation, differing only in the boundary condition. The generating function for the bulk gap probabilities can be obtained from (8.127) by scaling θ → 2πs/N and taking N → ∞. Recalling from Section 7.2 that X → X/2N 2 gives the hard edge scaling in the Jacobi ensemble we see E2bulk ((−s, s); ξ) = E2hard ((0, π 2 s2 ); ξ) E2hard ((0, π 2 s2 ); ξ) a=−1/2
a=1/2
− + = E2O ((0, s); ξ)E2O ((0, s); ξ),
(8.129)
where on the l.h.s. the bulk density has been set to unity.
8.4.2 Gap probabilities with orthogonal symmetry and an evenness symmetry Analogous to the definition (8.7), let us define EN,1 (J; ξ; f ) :=
N (l) (1 − ξχJ ) l=1
OEN (f )
.
(8.130)
For f a classical weight (6.100) and furthermore even (this then excludes the Laguerre case, and restricts the Jacobi case to a = b), and with J = (−t, t), it is possible to relate (8.130) to an average over a matrix ensemble with unitary symmetry [197]. ˜
P ROPOSITION 8.4.2 Let f = e−V1 (x) be as specified above, let e−2V (x) correspond to the different cases ˜ of e−V1 (x) as specified in (5.56) and suppose N is even. We have 1/2 ˜ (0) = EN/2,2 ((0, t2 ); ξ; y −1/2 e−2V (y ) χy>0 ) . (8.131) EN,1 ((−t, t); ξ; e−V1 (x) ) ξ=1
ξ=1
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CHAPTER 8
Proof. According to the method of integration over alternate variables (recall Section 6.3.2) EN,1 ((−t, t); ξ; f ) = where Aj,k =
1“ 2
Z
∞ −∞
Z −ξ
t
N! 1/2 det[Aj,k ]j,k=1....,N , C
“Z ” dx f (x)Rj−1 (x)
−t
∞ −∞
Z −ξ
”
t −t
(8.132)
dy f (y)Rk−1 (y)sgn(y − x)
(8.133)
and C is independent of t. Here {Rj (x)}j=0,1,... is an arbitrary set of monic polynomials. Let us choose Rj (x) even (odd) for j even (odd). Then A2j,2k = A2j−1,2k−1 = 0
for j, k = 1, . . . , N/2
so every alternate element in the matrix [Aj,k ] is zero. Interchanging rows and columns so that the zero elements are all in the top left and bottom right block and noting A2k,2j−1 = −A2j−1,2k shows EN,1 ((−t, t); ξ; f ) =
N! det[A2j−1,2k ]j,k=1,...,N/2 . C
(8.134)
˜
We now make use of the assumption that f is an even classical weight, f = e−V1 (x) . Let {pj (x)}j=0,1,... denote the monic orthogonal polynomials corresponding to the even weight functions e−2V (x) in (5.56). Then pj (x) is even (odd) for j even (odd). Furthermore, from (6.98) and (6.99), ” 1 ˜ d “ −V˜4 (x) ˜ 2j (x) = p2j (x), ˜ 2j+1 (x) = − R R eV1 (x) p2j (x) , (8.135) e γ2j (p2j , p2j )2 dx ˜
where e−V4 (x) is specified by (6.53), have the skew orthogonality property Z ∞ Z 1 ∞ ˜ ˜ ˜ 2j−2 (x) ˜ 2k−1 (y) sgn(y − x) = 1 δj,k . dx e−V1 (x) R dy e−V1 (y) R 2 −∞ γ2k−2 −∞
(8.136)
˜ j (x), j = 0, 1, . . . in (8.133), using (8.136) and integrating by parts shows Choosing Rj (x) = R Z t 1 (2ξ − ξ 2 ) A2j−1,2k = δj,k − e−V2 (x) p2j−2 (x)p2k−2 (x) dx γ2k−2 γ2k−2 (p2k−2 , p2k−2 )2 −t ˜
+
(ξ − ξ 2 )e−V4 (t) p2k−2 (t) γ2k−2 (p2k−2 , p2k−2 )2
Z
t
˜
e−V1 (x) p2j−2 (x)dx.
−t
It follows from this that ˜
EN,1 ((−t, t); 1; e−V1 (x) ) =
h“ 1 det C
Z
∞ −∞
Z −
t −t
”
e−V2 (x) p2j−2 (x)p2k−2 (x) dx
i j,k=1,...,N/2
(8.137)
for some C independent of t. Repeating the workings of the proof of Proposition 5.2.1, but in the reverse order, shows that the r.h.s. of (8.137) is equal to Z t ” Z “Z ∞ Z t ” Y (N/2)! “ ∞ − − (x2k − x2j )2 . dx1 e−V2 (x1 ) · · · dxN/2 e−V2 (xN/2 ) C −∞ −t −∞ −t 1≤j
Changing variables x2j = yj and substituting back in (8.137) gives (8.131).
˜
The most interesting case of (8.131) is when e−V1 (x) = (1 + x2 )−(N +1)/2 . Recalling from (2.51) that this special case of the Cauchy orthogonal ensemble is related to the COE via a stereographic projection, and thus
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EIGENVALUE PROBABILITIES — PAINLEVE´ SYSTEMS APPROACH
in particular
EN,1
θ θ COE − tan , tan ); ξ; (1 + x2 )−(N +1)/2 = EN,1 ()(−θ, θ); ξ) , 2 2 ξ=1 ξ=1
(8.131) then implies
θ (0) COE ; ξ; y −1/2 (1 + y)−2N χy>0 ((−θ, θ); ξ) = EN,2 0, tan2 E2N,1 2 ξ=1 ξ=1 − θ O (2N +1) J 0, 0, sin2 = EN,2 (0; (0, θ)), a=−1/2 = E2 2 b=1/2
(8.138)
where the second and third equalities follow as in (8.127). Thus the probability of there being no eigenvalues in the interval (−θ, θ) for the COE — a β = 1 quantity — is equal to the probability of there being no eigenvalues in (0, θ) for matrices from O− (2N + 1) — a β = 2 quantity. The evaluation of the latter in terms of a σPVI transcendent has been discussed in the last subsection of Section 8.3.1. As with going from (8.127) to (8.129), we can take the bulk limit (bulk density set equal to unity) in (8.138) to obtain − E1bulk (0; (−s, s)) = E2hard (0; (0, π 2 s2 )) = E2O (0; s). (8.139) a=−1/2
This characterization and the results of Section 8.3.4 give the expansion π 4 (ρs)5 π 4 (ρs)6 π 6 (ρs)7 π 6 (ρs)8 π 2 (ρs)3 − + + − + O(s9 ). (8.140) 36 1200 8100 70560 264600 Note that this could have been obtained from the first series expansion in (8.81), by replacing N → N/2, x → ρπs/N , setting ξ = 1 and taking N → ∞, in agreement with (8.138). It follows from (8.140) that E1bulk (0; (0, s)) = 1 − ρs +
1 bulk 1 d2 p1 (0; s) = 2 2 E1bulk (0; (0, s)) ρ ρ ds π 4 (ρs)4 π 6 (ρs)5 π 6 (ρs)6 π 2 (ρs) π 4 (ρs)3 − + + − + O(s7 ). = (8.141) 6 60 270 1680 4725 Substituting the Painlev´e evaluation (8.102) of (8.139) in the first equality of (8.141) gives a Painlev´e evaluation of pbulk (0; s). This can be simplified if we first note from (8.93) and (8.95) that 1 (πt)2 ds (πt)2 d 1 3 ds exp − = − exp − , σIII (s) μ=2 + v s; − ; ξ = 1 dt 2 s 2 s 0 0 a=−1/2 and the following result deduced [229]. P ROPOSITION 8.4.3 We have pbulk (0; s) 1
2˜ u((πs/2)2 ) exp − = s
0
(πs/2)2
u ˜(t) dt , t
where u ˜ satisfies the nonlinear equation 9 2 3 1 u) − u ˜ + s2 (˜ u )2 = (4(˜ u )2 − u ˜ )(s˜ u − u ˜) + (˜ 4 2 4 subject to the boundary condition s s2 8s5/2 − + . 45 135π s→0 3 Although this result is of theoretical interest as an exact form of the Wigner surmise (8.20), for purposes − of numerical computations we can make use of the fact that the moments of pO 2 (s; ξ)|ξ=1 are known from u ˜(s) ∼ +
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CHAPTER 8
p.d.f.
mean
variance
skewness
kurtosis
pbulk (0; s) 1 pbulk (1; s) 1 pbulk (2; s) 1 pbulk (3; s) 1 pbulk (4; s) 1 pbulk (5; s) 1 pbulk (6; s) 1 pbulk (7; s) 1 pbulk (8; s) 1 bulk p1 (9; s)
1 2 3 4 5 6 7 8 9 10
0.28553 06557 0.41639 36889 0.49745 52604 0.55564 24180 0.60091 83521 0.63794 46245 0.66925 53948 0.69637 60657 0.72029 45046 0.74168 65573
0.68718 99889 0.34939 68438 0.22741 44134 0.16645 68639 0.13042 07251 0.10679 47124 0.09018 32871 0.07790 15490 0.06847 07897 0.06101 25387
0.37123 80638 0.02858 27332 −0.01329 56588 −0.01994 68028 −0.02007 29233 −0.01884 07449 −0.01743 19487 −0.01613 54800 −0.01500 75200 −0.01404 07984
Table 8.14 Statistical properties of pbulk (n; s) for various values of n. In the cases n ≥ 4 these are from [79]. 1
Section 8.3.4. Thus we see from (8.139) and the first equality in (8.141) that for p ≥ 1 ∞ ∞ − p−1 sp pbulk (0; s) ds = p2 sp−1 pO 1 2 (0; s) ds, 0
(8.142)
0
where we have set ρ = 1. The results of Table 8.12 then give the results listed in Table 8.14 for the case n = 0. The result of Proposition 8.4.2 leading to (8.138) applies only in the case ξ = 1. Thus at this stage we are COE still to obtain results on the generating function EN,1 ((−θ, θ); ξ) for general ξ. A step in this direction is obtained by considering the implication of the identity (6.154) [146], [396]. P ROPOSITION 8.4.4 We have the inter-relationship n CUE COE COE EN,1 (n; (−θ, θ)) = (2(n − l); (−θ, θ)) + EN,1 (2(n − l) − 1; (−θ, θ)) EN,2 l=0
COE COE (2l; (−θ, θ)) + EN,1 (2l + 1; (−θ, θ)) , × EN,1
(8.143)
COE where EN,1 (−1; (−θ, θ)) := 0, and EN,p (n; (−θ, θ)) = 0 for n > N . Equivalently, in terms of the generating functions CUE EN,2 ((−θ, θ); ξ) :=
± EN,1 ((−θ, θ); ξ) :=
∞
CUE (1 − ξ)n EN,2 (n; (−θ, θ)),
n=0 ∞
COE COE (1 − ξ)n EN,1 (2n; (−θ, θ)) + EN,1 (2n ± 1; (−θ, θ))
(8.144)
n=0
we have − + CUE ((−θ, θ); ξ) = EN,1 ((−θ, θ); ξ)EN,1 ((−θ, θ); ξ). EN,2
(8.145)
Proof. To derive (8.143), we suppose there are n eigenvalues in the interval (0, s) of the CUE, and ask what (6.154) says about the corresponding number of eigenvalues in (−θ, θ) for the individual COEs in the ensemble COE ∪ COE. Let us suppose there are an even number 2(n − l) (0 ≤ l ≤ n) in (−θ, θ) from one of the individual COEs in COE ∪ COE. Because every second eigenvalue is integrated over in the operation alt, this means there must be either 2l or 2l ± 1 (these latter two possibilities occurring with probability 12 ) eigenvalues from the other COE to leave n eigenvalues in (0, s).
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EIGENVALUE PROBABILITIES — PAINLEVE´ SYSTEMS APPROACH
Thus we obtain the probability “ ”” 1 “ COE COE COE COE EN,1 (2l + 1; (−θ, θ)) + EN,1 EN,1 (2(n − l); (−θ, θ)) EN,1 (2l; (−θ, θ)) + (2l − 1; (−θ, θ)) , (8.146) 2 COE where EN,1 (−1; (−θ, θ)) := 0. Similarly, if there is an odd number 2(n − l) − 1 (−1 ≤ l ≤ n) in (−θ, θ) from this same COE, there must be either 2l + 1, or 2l, 2l + 2 (these latter two possibilities both occurring with probability 12 ), so we obtain the probability “ ”” 1 “ COE COE COE COE (2(n − l) − 1; (−θ, θ)) EN,1 (2l + 1; (−θ, θ)) + (2l + 2; (−θ, θ)) . (8.147) EN,1 EN,1 (2l; (−θ, θ)) + EN,1 2
Adding together (8.146) and (8.147) and summing over l gives (8.143). With (8.143) established, multiplying both sides by (1 − ξ)n and summing over n, (8.145) follows immediately.
In addition to the factorization (8.145), we have previously encountered the factorization (8.127). At ξ = 1 we can see that these are in fact the very same factorizations. Thus according to (8.138) we have θ − J COE EN,2 ((0, sin2 ); ξ) a=−b=−1/2 = E2N,1 (0; (−θ, θ)) = E2N,1 ((−θ, θ); ξ) , (8.148) 2 ξ=1 ξ=1 where the second equality follows from the definition (8.144). Equivalently, the equation O± (2N +1)
± ((−θ, θ); ξ) = E2 E2N,1
((0, θ); ξ)
(8.149)
holds for ξ = 1, and is consistent with both factorizations (8.128) and (8.145) for general ξ. In fact methods based on expansions in terms of the eigenvalues and eigenvectors of underlying Fredholm integral operators + (see Section 9.6.1) can be used to establish that (8.149) holds for general ξ in the case of E2N,1 . The factor− ± ization (8.128) then establishes (8.149) for E2N,1 . With E2N,1 thus evaluated, we substitute in (8.144), and use this equation to deduce, with ξ¯ := 2ξ − ξ 2 and thus 1 − ξ¯ = (1 − ξ)2 , that O+ (2N +1)
COE E2N,1 ((−θ, θ); ξ) =
(1 − ξ)E2
¯ + EO ((−θ, θ); ξ) 2 2−ξ
−
(2N +1)
¯ ((−θ, θ); ξ)
.
(8.150)
In the bulk scaling limit (bulk density unity) (8.149) reads O± (2N +1)
E1bulk± ((−s, s); ξ) = E2
((0, s); ξ),
(8.151)
where E1bulk± ((−s, s); ξ) :=
∞
(1 − ξ)n E1bulk (2n; (−s, s)) + E1bulk (2n ± 1; (−s, s)) ,
n=0
while (8.150) reads E1bulk ((−s, s); ξ)
+ ¯ + E O− ((0, s); ξ) ¯ (1 − ξ)E2O ((0, s); ξ) 2 . = 2−ξ
(8.152)
This latter formula is equivalent to the equation E1bulk (n; 2s) = (−1)n
[n/2]
[(n−1)/2]
−
E2O (l; s) −
l=0
l=0
Noting from (8.105) that ±
pO 2 (n; s) = −
d O± E (p; s) ds p=0 2 n
+ E2O (l; s) .
(8.153)
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CHAPTER 8
this implies the simple formula
+ d bulk O− E1 (n; 2s) = (−1)n pO 2 ([(n − 1)/2]; s) − p2 ([n/2]; s) . ds But according to (8.2) (n; s) = pbulk 1
d2 bulk E (n; s) + 2pbulk (n − 1; s) − pbulk (n − 2; s), 1 1 ds2 1
(8.154) ±
(k; s) = 0 for k < 0, so in fact the following recurrence for pbulk (n; s) involving pO where pbulk holds. 2 1 1 P ROPOSITION 8.4.5 We have pbulk (n; s) = 1
− (−1)n d O+ p2 ([(n − 1)/2]; s/2) − pO ([n/2]; s/2) 2 2 ds bulk bulk +2p1 (n − 1; s) − p1 (n − 2; s),
(8.155)
(k; s) := 0 for k < 0. where pbulk 1 ±
Using (8.155) and knowledge of the moments of pO 2 (k; s) from Table 8.12 we then obtain the statistical characterizations listed in Table 8.14 in the cases n = 1, 2, 3. 8.4.3 A relationship between gap probabilities with orthogonal and symplectic symmetry It turns out that knowledge of the COE gap probabilities is sufficient for the determination of the CSE gap probabilities. This is a consequence of (4.32) [397]. P ROPOSITION 8.4.6 We have CSE COE (j; (−θ, θ)) = E2N,1 (2j; (−θ, θ)) + EN,4
1 COE COE E2N,1 (2j − 1; (−θ, θ)) + E2N,1 (2j + 1; (−θ, θ)) (8.156) 2
or equivalently COE pCSE N,4 (n; (−θ, θ)) = p2N,1 (2n + 1; (−θ, θ)).
(8.157)
Proof. Multiplying both sides of (8.156) by n − j + 1, summing over j from 0 to n and making use of (8.17) shows
that (8.156) is equivalent to (8.157). To derive (8.157) we note that with eigenvalues fixed at −θ, θ, the only way these eigenvalues can remain unaffected in the even operation of (4.32), and furthermore this operation leave n eigenvalues inside (−θ, θ), is that there originally be 2n + 1 eigenvalues inside (−θ, θ) (n + 1 of which are integrated over in the even operation). According to (4.32) the eigenvalue p.d.f. for COE2N reduces to that for CSEN upon the even operation, so (8.157) follows.
Recalling the definition (8.149), it follows from (8.156) and (8.149) that 1 O+ (2N +1) O− (2N +1) CSE E2 EN,4 ((−θ, θ); ξ) = ((0, θ); ξ) + E2 ((0, θ); ξ) . 2 In the bulk limit (ρ = 1) this reads − 1 O+ E2 ((0, s); ξ) + E2O ((0, s); ξ) , E4bulk ((0, s); ξ) = 2 while the bulk limit of (8.157) gives (n; s) = 2pbulk (2n + 1; 2s). pbulk 4 1 Note that the latter implies
∞ 0
sp pbulk (n; s) ds = 2−p 4
0
∞
sp pbulk (2n + 1; s) ds. 1
(8.158)
(8.159)
(8.160)
(8.161)
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EIGENVALUE PROBABILITIES — PAINLEVE´ SYSTEMS APPROACH
1. [458] Let U be a N × N unitary matrix, and replace each entry x + iy, x, y ∈ R by its 2 × 2 real matrix representation (1.36) to obtain a 2N × 2N real orthogonal matrix to be denoted Re(U).
E XERCISES 8.4
(i) Show that if the eigenvalues of U are eiθj , (j = 1, . . . , N ), then the eigenvalues of Re(U) are e±iθj , (j = 1, . . . , N ). (ii) With a(eiθ ) = a(e−iθ ) note that N DY j=1
a(eiθj )
E U (N)
=
N DY
a(eiθj )
j=1
E Re(U (N))
,
where on the r.h.s. the average is over one of each of the eigenvalues e±iθj (j = 1, . . . , N ). Substitute (5.95) for the l.h.s. to deduce Ev(Re(U)) = Ev(O+ (N + 1)) ⊕ Ev(O− (N + 1)), where Ev denotes the eigenvalue distribution of the eigenvalues eiθ , 0 < θ < π of the corresponding matrix ensembles. 2
2a −x , x ∈ (−∞, ∞). Under 2. The generalized Hermite √ ensemble is specified by the weight function w2 (x) = |x| e the scaling x → x/ 2N the correlations coincide with those for the spectrum singularity (7.48) with ρ = 1/π [418], and in particular ““ ” 2 t t ” −√ lim EN,2 ,√ ; ξ; |x|2a e−x = E2s.s. ((−t, t); ξ). (8.162) N→∞ 2N 2N
(i) Show that for a = 0 the logarithmic derivative of the generalized Hermite weight has the structure (5.57) with degree f = 2, and so in general this does not define a classical weight. 2
(ii) Apply (8.123) with J{t2i ,t2i+1 } = (−t, t), g(x) = |x|2a e−x and take the scaling limit (8.162) to conclude ˛ ˛ ˛ ˛ ˛ ˛ E2s.s. ((−t, t); ξ)˛ = E2hard ((0, t2 ); ξ)˛ E2hard ((0, t2 ); ξ)˛ . (8.163) ρ=1/π
3.
a→a−1/2
a→a+1/2
(i) Consider the circular β-ensemble. Show from the definitions that for x1 , . . . , xn small ρ(n) (x1 , . . . , xn ) ∼
“ 2π ”βn(n−1)/2 N! n (N − n)!L L
Y
|xk − xj |β
1≤j
MN−n (nβ/2, nβ/2, β/2) , MN (0, 0, β/2)
where Mk is specified by (4.4). Hence deduce that in the bulk scaling limit Y |ρxk − ρxj |β , ρ(n) (x1 , . . . , xn ) ∼ ρn cn (β)(2π)βn(n−1)/2 1≤j
where cn (β) = (β/2)βn(n−1)/2 ((β/2)!)n
n−1 Y k=0
Γ(βk/2 + 1) . Γ(β(n + k)/2 + 1)
(8.164)
(ii) Show from the definitions that for small s Z s Z s 1 (n; s) ∼ dx · · · dxn ρ(n+2) (0, s, x1 , . . . , xn ) pbulk 1 β ρn! 0 0 and use the result of (i) to deduce from this that 1 bulk (n; s) ∼ Sn (β, β, β/2)cn+2 (β)(2π)β(n+2)(n+1)/2 (ρs)n+β(n+2)(n+1)/2 , p ρ β
(8.165)
where Sn is specified by (4.3). Verify that the leading terms in (8.115) and (8.141) exhibit this behavior.
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CHAPTER 8
(iii) [1] Set ρ = 1. In keeping with the Wigner surmise of Exercises 8.1 q.3, and the result of the above exercise, (n; s) of the form make an ansatz for pbulk β 2
γ −(Bs) pW , γ = n + β(n + 2)(n + 1)/2. β (n; s) := As e R∞ W R∞ W Requiring that 0 pβ (n; s) ds = 1, 0 spβ (n; s) ds = n + 1, show that
A=
(Γ(γ/2 + 1))γ+1 2 , γ+1 (n + 1) (Γ((γ + 1)/2)γ+2
B=
Γ(γ/2 + 1) . (n + 1)Γ((γ + 1)/2)
4. [148] Use the identity (8.157) and the second sum rule in (8.18) to deduce N−2 X
pCOE 2N,1 (2n + 1; (0, α)) =
n=0 N−1 X
pCOE 2N,1 (2n; (0, α)) =
n=0
1 CSEN (0, α), ρ 2ρ (2) ” 1 “ COE2N 1 2N (0, α) − ρCSE (0, α) , ρ(2) (2) ρ 2
where ρ = N/π.
8.5 DISCRETE PAINLEVE´ EQUATIONS Amongst the random matrix averages characterizable from the results of this chapter as the solution of certain Painlev´e equations in σ-form are the U (N ) (or equivalently CUEN ) averages
τ III [N ](t; μ) :=
N
1
zlμ e 2
√ t(zl +zl−1 ) U(N )
l=1
,
N τ [N ](t; μ, ν) := (1 + zl )μ (1 + 1/zl )ν etzl
(8.166)
V
l=1
τ V I [N ](t; μ, w1 , w2 ; ξ) :=
U(N )
,
(8.167)
N 1 μ (l) (1 − ξχ(π−φ,π) )ew2 θl |1 + zl |2w1 (1 + tzl )2μ , (8.168) tzl U(N ) l=1
where in (8.168) t = eiφ . It will be shown in this section that these averages have the further property of satisfying a recurrence in N . The recurrences involve auxiliary quantities which themselves satisfy so-called discrete Painlev´e equations. The explicit form of the latter of interest to us are the alternate discrete Painlev´e II equation [422] (0)
(0)
(0)
(0)
(0)
1 v1 + v2 + 2 + 2n 1 v1 + v2 + 2n 1 v +n qn + = + 2 , − 2 qn qn+1 + t 2 qn−1 qn + t qn t t
(8.169)
the discrete Painlev´e IV recurrences [461] (0)
xn + xn−1 =
(0)
t v − v4 − n + 3 , yn 1 − yn (0)
yn yn+1 = t
(0)
(0)
xn − 12 (v1 + v2 ) + 1 + v4 + n (0)
(0)
x2n − 14 (v2 − v1 )2
,
(8.170)
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EIGENVALUE PROBABILITIES — PAINLEVE´ SYSTEMS APPROACH
and the discrete Painlev´e V recurrences [478] (0)
gn+1 gn =
(0)
(0)
t (fn + 1 + n − α2 )(fn + 1 + n − α2 − α4 ) , (0) t−1 fn (fn + α ) 3
(0)
fn + fn−1 = −α3 +
(0) α1
(0) (α0
+n + n)t + . gn − 1 t(gn − 1) − gn
(8.171)
The alternate discrete Painlev´e II equation is satisfied by
n III qnIII := TIII q
v = v (0)
,
(8.172)
the discrete Painlev´e IV recurrences by ∗ ∗ 1 (0) (0) xn = (pVn + t)qnV − (v2 − v1 ), 2 1 yn = V ∗ , qn
where
∗ ∗ pVn := TVn pV
v = v
(8.173)
∗ ∗ qnV := TVn q V
, (0)
v = v (0)
,
and the discrete Painlev´e V recurrences by gn :=
qnV I , qnV I − 1 (0)
(0)
(0)
(0)
fn := qnV I (qnV I − 1)pVn I + (1 + n − α2 − α4 )(qnV I − 1) − α3 qnV I − (α0 + n)
where
pVn I := TVnI pV I
qnV I := TVnI q V I
, (0)
α = α(0)
α = α
qnV I (qnV I − 1) , qnV I − t (8.174)
.
It is simplest to show that (8.172) satisfies (8.169). The required strategy is common to the results relating to (8.170) and (8.171), but avoids some of the more complicated algebraic manipulation required in these latter cases.
P ROPOSITION 8.5.1 The nth shift of the general Painlev´e III transcendent q III |v=v(0) by the Schlesinger operator TIII , qnIII , satisfies the alternate discrete Painlev´e II equation (8.169).
n III := TIII | v= v(0) and write pIII =: Proof. For notational convenience, let us write qnIII =: qn . Also define pIII p n n
pn . Now from the definition of TIII in Table 8.7, the actions in Table 8.3 can be used to express qn+1 = TIII qn , −1 −1 qn−1 = TIII qn , pn+1 = TIII pn , pn−1 = TIII pn each in terms of pn , qn . Explicitly (0)
(0)
(v1 + v2 + 2 + 2n)t t 1 + , qn 2 qn [qn (pn − 1) − (v1(0) − v2(0) )/2] + t qn 1 (0) (0) pn+1 = [qn (pn − 1) − (v1 − v2 )] + 1, t 2 t qn−1 = (0) , (0) (v1 + v2 + 2n)/2pn − qn qn+1 = −
(8.175) (8.176) (8.177)
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CHAPTER 8
F (p, q) = − 12 p
Painlev´e system PII
∂ ∂t HII
PIII
∂ ∂t tHIII ∂ ∂t HIV
PIV PV∗
∂ ∗ ∂t (tHV
PVI = q(1 −
(0)
=p
= −2pq
(0)
(0)
(0)
(0)
(0)
+ (v4 − v2 + n)t) = q(q − 1)p + (v2 − v3 )q + (v4 − v2 + n)
(0) (0) (0) (0) ∂ ∂t (t(t − 1)HV I ) + (v3 − v2 + N )(−v1 − v2 + N ) (0) (0) (0) (0) (0) (0) 2 q)p + 2v3 pq − (v3 + v4 )p + (v3 − v2 + N )(−v2 −
(0)
v3 + N )
Table 8.15 Specification of F (p, q) in (8.179).
while the formula for pn−1 is not needed. The recurrences (8.175) and (8.176) imply qn+1 +
t = qn
(0) 1 (v1 2
(0)
+ v2 + 2 + 2n) . pn+1
(8.178)
Solving (8.177) for pn and substituting in (8.178) gives (8.169).
Next we will relate the variables pn , qn to the particular ratio of τ -functions τ [n + 1]τ [n − 1]/τ 2 [n] for each of the Painlev´e systems PII–PVI. For this we must decide on a normalization of τ [n] — we suppose τ [n] is given by g[n] as in Table 8.10, and thus g[n] satisfies the Toda lattice equation (8.51) (the key point here is that the possible proportionality constant on the r.h.s. of (8.51), permitted from the derivation, is chosen to be unity). Then from the derivation of Proposition 8.2.5 we can read off the following results. P ROPOSITION 8.5.2 With {τ [n]}n=0,1,... for each of the Painlev´e systems normalized as specified, and τ [n] := τ [n](t/4) in the PIII case, τ [n + 1]τ [n − 1] = F (pn , qn ), τ 2 [n]
(8.179)
where F (p, q) is specified by Table 8.15 for each of the Painlev´e systems. In the PIII case the appropriately normalized τ -function sequences is given by (8.68). Making use of Propositions 8.5.2 and 8.5.1 the following results are almost immediate [232].
P ROPOSITION 8.5.3 Let τ III [N ] = τ III [N ](t; μ) as given by (8.166) for μ ∈ Z, and by √ τ III [N ](t; μ) = det[Iμ+j−k ( t)]j,k=1,...,N
(8.180)
for general μ (recall (8.97)), and let pN , qN denote the variables in the corresponding Hamiltonian. The sequences {τ III [N ]}N =0,1,..., {pN }N =0,1,..., {qN }N =0,1,... satisfy the coupled recurrences τ III [N + 1]τ III [N − 1] = pN (N = 1, 2, . . . ), (τ III [N ])2 t →4t q2 μqN +1 (N = 0, 1, . . . ), pN +1 = N (pN − 1) − t t t (1 + N )t qN +1 = − (N = 0, 1, . . . ), (8.181) + qN qN (qN (pN − 1) − μ) + t
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EIGENVALUE PROBABILITIES — PAINLEVE´ SYSTEMS APPROACH
subject to the initial conditions p0 = 0,
√ d log t−μ/2 Iμ ( t), dt √ τ III [1] = Iμ ( t).
q0 = t
τ III [0] = 1,
8.6 ORTHOGONAL POLYNOMIAL APPROACH Another approach to deriving recurrences for the averages (8.166)–(8.168) is via the theory of (bi)orthogonal polynomials on the unit circle [234], [236]. For this some aspects of the latter theory must first be developed. With z = eiθ let π ∞ 1 k wk z , wk = w(z)e−ikθ dθ. (8.182) w(z) = 2π −π k=−∞
Introduce the Toeplitz determinant, or equivalently U (N ) average, 0 [w] := det[wj−k ]j,k=1,...,N = IN
N
w(zl )
l=1
U(N )
(8.183)
0 (for the equality recall (5.77)). If IN [w] is nonzero for each N = 1, 2, . . . , for otherwise general complex w(z) one can construct a system of biorthogonal polynomials {φn (z)}n=0,1,... and {φ¯n (z)}n=0,1,... (here φ¯ does not in general stand for the complex conjugate of φ) such that π 1 w(z) ¯ w(z)φ ˜ w(z) ˜ = . (8.184) m (z)φn (1/z) dθ = δm,n , 2π −π w0
In the case that w(z) is real for |z| = 1 (wk = w−k ) the biorthogonal system reduces to an orthogonal system because then φ¯n = φn , however this will not be the case in general. Some fundamental quantities related to the biorthogonal system are identified in the following definition. D EFINITION 8.6.1 Introduce notation for the various coefficients in φn (z) and φ¯n (z) according to φn (z) = κn z n + ln z n−1 + mn z n−2 + · · · + φn (0), ¯ n z n−2 + · · · + φ¯n (0), φ¯n (z) = κn z n + ¯ln z n−1 + m ¯ n , φ¯n (0) do not in general denote the complex conjugates of ln , mn , φn (0), and κn is real and where ¯ln , m positive. Introduce the reciprocal polynomials by φ∗n (z) = z n φ¯n (1/z),
φ¯∗n (z) = z n φn (1/z)
(cf. the definition of χ ˜k (λ) in (2.88)) and define the so called reflection coefficients rn =
φn (0) , κn
r¯n =
φ¯n (0) . κn
Next we make note of a number of key formulas associated with the biorthogonal system and the quantities of Definition 8.6.1.
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CHAPTER 8
P ROPOSITION 8.6.2 We have N −1 j=0
φ∗ (z)φ¯∗N (ζ) − φN (z)φ¯N (ζ) , φj (z)φ¯j (ζ) = N 1 − zζ κ 2 N −1 = 1 − rN r¯N , κN κn zφn (z) = κn+1 φn+1 (z) − φn+1 (0)φ∗n+1 (z),
κn φn+1 (z) = κn+1 zφn (z) + φn+1 (0)φ∗n (z), κn φn (0)φn+1 (z) + κn−1 φn+1 (0)zφn−1 (z) = (κn φn+1 (0) + κn+1 φn (0)z)φn (z), ln = r¯j rj+1 . κn j=0 n−1
(8.185)
¯ Furthermore the third, fourth and fifth equations remain valid if φ is formally replaced by φ. Proof. Let us refer to the equations (8.185) as (i)–(vi) in order. The summation (i) is analogous to and consistent with the circular form of the Christoffel-Darboux formula (5.82). Like that formula, it can be derived by following the strategy of Exercises 5.1 q.6. Setting z = ζ = 0 in (i) implies that N X
φj (0)φ¯j (0) = κ2N .
j=0
Subtracting this from its counterpart with N → N − 1 and recalling the definitions of rN , r¯N gives (ii). With N = n + 1, multiplication of both sides of (i) with 1 − zζ and comparison of the coefficient of ζ n+1 in (i) implies (iii), while its analogue with φ formally replaced by φ¯ follows by comparison of the coefficient of z n+1 . Replacing z by 1/z and recalling the various definitions this latter equation can be written κn φ∗n (z) = κn+1 φ∗n+1 (z) − φ¯n+1 (0)φn+1 (z). Substituting for φ∗n+1 (z) using (iii) and simplifying using (ii) implies (iv). Replacing n by n − 1 in (iii) and using (iv) to eliminate φ∗n (z) then gives (v). To derive (vi), first equate coefficients of z n in (v). This gives κn φn (0)ln+1 + κ2n−1 φn+1 (0) = κ2n φn+1 (0) + κn+1 φn (0)ln . Making use of (ii) it follows from this that ln+1 ln = + r¯n rn+1 κn+1 κn which is equivalent to (vi).
The second and sixth equations of (8.185) can be used to derive a recurrence system for {τ III [N ]}N =0,1,... which is in fact equivalent to the one of Proposition 8.5.3. First we note that following the strategy of the derivation of (5.74) shows 0 IN [w] = w0N
N −1
κ−2 j ,
j=0
which allows the second equation in (8.185) to be rewritten 0 0 IN +1 [w]IN −1 [w] = 1 − rN r¯N . 0 [w])2 (IN
(8.186)
Now, according to (8.166), with 1
w(z) = z μ e 2
√
t(z+z −1 )
(8.187)
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EIGENVALUE PROBABILITIES — PAINLEVE´ SYSTEMS APPROACH
we have
0 [w] = τ III [N ](t; μ) IN
(8.188)
so the left hand side of (8.186) coincides with the left hand side of (8.179). Thus, whereas in proceeding from (8.179) the strategy was to use the fact that pn and qn satisfy recurrences to determine {τ III [N ]}N =0,1,... by recurrence, our present task is to determine recurrences for rN and r¯N .
P ROPOSITION 8.6.3 For the weight (8.187) the sequences {rN }N =0,1,... , {¯ rN }N =0,1,... satisfy the coupled recurrences 1√ rN r¯N t(rN +1 r¯N + rN r¯N −1 ) + N − μ = 0, 2 1 − rN r¯N rN r¯N 1√ t(¯ rN +1 rN + r¯N rN −1 ) + N + μ=0 2 1 − rN r¯N subject to the initial conditions r0 = r¯0 = 1,
r1 = −
√ Iμ+1 ( t) √ , Iμ ( t)
r¯1 = −
√ Iμ−1 ( t) √ . Iμ ( t)
Proof. To avoid minor complications we suppose here that μ is an integer (if μ is not an integer the contour of integration implied by (8.182) must be modified not to cross the cut along the negative real axis, about which w(z) is discontinuous). Following a method of Laguerre and Freud (see [244]) we consider two different ways to evaluate the integral Z Z π 1 dz ¯ J1 = z 2 w (z)φN (z)φ(1/z) dθ = z 2 w (z)φN (z)φ¯N+1 (1/z) , (8.189) 2π −π 2πiz C where C is the unit circle in the complex plane. Integrating by parts gives Z Z dz dz − z 2 w(z)φN (z)φ¯N+1 (1/z) J1 = − zw(z)φN (z)φ¯N+1 (1/z) 2πiz 2πiz C ZC dz 2 ¯ . + z w(z)φN (z)φN+1 (1/z) 2πiz C But κN N κN φN+1 (z) + πN (z), z 2 φN (z) = φN+1 (z) + πN (z), κN+1 κN+1 (N + 1)κN+1 ¯ φ¯N+1 (1/z) = φN (1/z) κN
zφN (z) =
(8.190)
for some polynomials πN (z) of degree N so the biorthogonality relation (8.184) implies J1 = −
κN κN+1 (N + 1) + (N + 1). κN+1 κN
Alternatively, differentiating (8.187) shows (8.189) can be rewritten Z “μ 1 ”” dz 1√ “ z2 t 1 − 2 w(z)φN (z)φ¯N+1 (1/z) + . J1 = z 2 z 2πiz C Using the first equation in (8.190) together with z 2 φN (z) =
κN 1 “ κN lN+2 ” φN+2 (z) + lN − φN+1 (z) + πN (z), κN+2 κN+1 κN+2
(8.191)
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CHAPTER 8
it follows from the biorthogonality relation (8.184) that J1 = μ
κN 1 √ “ lN lN+2 κN ” + t − . κN+1 2 κN+1 κN+2 κN+1
Equating the two expressions for J1 and eliminating lN using the final equation of (8.185) we deduce the first recurrence. To deduce the second recurrence we apply an analogous strategy to Z dz w (z)φN+1 (z)φ¯N (1/z) . J2 = 2πiz C
In the case μ = 0 the weight (8.187) is real. Then r¯n = rn and the coupled recurrences of Proposition 8.6.3 reduce to the single recurrence 2N rN −√ 2 = rN +1 + rN −1 t 1 − rN subject to the initial conditions r0 = 1,
(8.192)
√ I1 ( t) r1 = − √ . I0 ( t)
Different derivations of this result can be found in [28], [82]. The relationship between rN , r¯N in Proposition 8.6.3 and pN , qN in Proposition 8.5.3 is easily deduced. First note that subtracting the two equations in Proposition 8.5.3 shows 1√ t(LN +1 + LN ) + 2μ = 0, LN := r¯N rN −1 − rN r¯N −1 , (8.193) 2 and the initial conditions therein imply this to be solved subject to the initial condition 2μ L1 = − √ . t By inspection we see that the solution of (8.193) satisfying this initial condition is the constant LN = √ −2μ/ t, N = 1, 2, . . . and thus 1√ t(¯ rN +1 rN − rN +1 r¯N ) + μ = 0. (8.194) 2 P ROPOSITION 8.6.4 We have √ rN +1 qN = − t . pN = 1 − rN r¯N |t →4t , rN t →4t Proof. The equation for pN follows immediately upon comparing the first equation of (8.181) with (8.186) and recalling (8.188). Substituting this equation for pN in the second equation of (8.181) we see that (8.194) results if qN is as specified.
For the PV τ -function sequence relating to (8.167), the formulas of Proposition 8.6.2 again yield coupled recurrences for {rN }N =0,1,..., {¯ rN }N =0,1,..., which in turn can be related to the recurrences obtained from the method of Section 8.5 [234]. In the case of the PVI τ -function sequence relating to (8.168), the formulas (8.185) are inadequate and a more sophisticated formulation based on the Riemann-Hilbert characterization of orthogonal polynomials on the unit circle is called for [236]. E XERCISES 8.6 φn (z) =
1.
(i) Follow the derivation of (5.78) to show
κn det[cj−k z − cj−k−1 ]j,k=1,...,n , In0 [w]
φ¯n (z) =
κn det[cj−k z − cj−k+1 ]j,k=1,...,n . In0 [w]
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EIGENVALUE PROBABILITIES — PAINLEVE´ SYSTEMS APPROACH
(ii) Set z = 0 in the above formulas to deduce rn = (−1)n
In1 [w] , In0 [w]
r¯n = (−1)n
In−1 [w] , In0 [w]
where Inμ [w] := In0 [z μ w]. 2. [306] The objective of this exercise is to show that for w(z) given by (8.187) with μ = 0 √ √ κn−1 “ t t κn−1 φn−1 (0) t φn+1 (0)φn (0) ” n+ φn−1 (z) φn (z) = + − κn 2z 2 κn φn (0) 2 κn+1 κn √ t κn−1 φn−1 (0) φn (z). − 2z κn φn (0)
(8.195)
(i) Let v(z) = − log w(z). Expand φn (z) as a series in {φk (z)} using (8.184), noting that because w(eiθ ) is real φ¯k = φk , to deduce Z “ ” 1 dζ 1 ˜ φn−1 (1/ζ)φk (1/ζ) + 2 φn−1 (1/ζ) φn (ζ)w(ζ) φn (z) = φn−1 (z) ζ ζ 2πiζ C Z n−1 X dζ + φk (z) v (ζ)φk (1/ζ)φn (ζ)w(ζ) ˜ , 2πiζ C k=0
where use is to be made of the fact that for pk (z) a polynomial of degree k and k < n, 0.
R C
φn (ζ)pk (1/ζ) dζ = ζ
(ii) Now make use of the structure of φn (z) in Definition 8.6.1 and the first equation in Proposition 8.6.2 to deduce the differential relation Z v (z) − v (ζ) κn−1 dζ φn−1 (z) + φ∗n (z) ˜ φn (z) = n φn (ζ)φ∗n (1/ζ)w(ζ) κn z−ζ 2πi C Z dζ v (z) − v (ζ) ˜ −φn (z) φn (ζ)φ(1/ζ)w(ζ) . (8.196) z−ζ 2πi C (iii) Note from the explicit form of w(z) that √ “ 1 ” v (z) − v (ζ) t 1 + 2 . =− 2 z−ζ 2 zζ z ζ Substitute this in (8.196) and simplify by noting from Definition 8.6.1 “ κn ∗ ” κn−1 φn−1 (0) ζ φn (ζ) − φn (ζ) = − φn (ζ) + rn−1 (z), φ(0) κn φn (0) φ∗ (ζ) φn+1 (ζ) “ κn ln − κn−1 ln−1 ln+1 ” = ζ n + − φn (ζ) + rn−1 (z), 2 φ(0) κn+1 κn (φn (0)) κn+1 κn where rn−1 (z) is some polynomial of degree n − 1, to deduce the sought equation.
Chapter Nine Eigenvalue probabilities — Fredholm determinant approach The theme of characterizing eigenvalue probabilities as solutions of nonlinear equations is continued, this time using various function theoretic and integrable systems aspects of Fredholm determinants, an approach complimentary to that of Painlev´e systems. The starting point is a Fredholm determinant formula for the generating function of the gap probability in the case of matrix ensembles with unitary symmetry. Fredholm determinants with analytic kernels are well suited to numerical evaluation, and this allows tables of statistical properties of various distributions obtained in the previous chapter using power series to be extended. Function theoretic properties of the Fredholm determinants are used to directly derive Painlev´e transcendent evaluations of such gap probabilities in the scaled limit. In the cases of the soft and hard edges the evaluations involve Painlev´e transcendents distinct from those obtained in the previous chapter. Analysis of the eigenvalues and eigenvectors of the integral operators corresponding to the Fredholm determinants leads to asymptotic formulas for E2scale (n; J)/E2scale (0; J) as |J| → ∞ in the cases that scale equals bulk, soft and hard. The Painlev´e transcendent evaluations can be used to determine the |J| → ∞ behavior of E2scale (0; J), provided the leading form is first deduced from other considerations. In the case of J = [−a, a] and the weight being even, an analysis of the eigenvalues and eigenvectors of the even part of the corresponding ChristoffelDarboux kernel leads to a generating function identity complementary to that obtained for the same quanCOE tity in the previous chapter, which justifies the Painlev´e transcendent evaluation of EN,1 (n; (−θ, θ)) given scale therein. Painlev´e transcendent evaluations of Eβ (n; J) for scale = soft and hard are obtained by making use of superimposed β = 1 ensembles, as considered in Chapter 8, and inter-relations of β = 1 and β = 4 ensembles known from Chapter 6. Integrable systems methods — relating to the Riemann-Hilbert problem and the isomonodromic deformation of linear second order differential equations, and the KP equation and Virasoro constraints — are applied to the β = 2 soft edge gap probability and β = 2 Gaussian ensemble right edge gap probability, respectively.
9.1 FREDHOLM DETERMINANTS Underlying the Fredholm determinant approach to gap probabilities in the case of matrix ensembles with unitary symmetry is a general expression for the generating function of the gap probability as a sum over the corresponding k-point correlation functions. P ROPOSITION 9.1.1 For a continuous one-dimensional statistical mechanical system of N particles a2 N (−ξ)n a2 EN,β ((a1 , a2 ); ξ) = 1 + dx1 · · · dxn ρN (9.1) (n) (x1 , . . . , xn ), n! a1 a1 n=1 where ρN (n) denotes the n-particle correlation function defined by (5.1). Proof. Comparing the definition of EN,β ((sN (a1 ), sN (a2 )); ξ) from Proposition 8.1.2 with the formula for the quantity U [a; z] in Exercises 5.1 q.3(i), we see that they coincide provided we set a(x) = 1, x ∈ (a1 , a2 ), a(x) = 0 otherwise and z = −ξ. Thus the expansion (9.1) follows from the definition of U [a; z] given in Exercises 5.1 q.3. In the general theory of infinite point processes, a relevant concern is conditions on the correlations ρscaled (n)
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FREDHOLM DETERMINANT APPROACH
which are sufficient for the point process to be uniquely determined by its correlations. One such condition is that [372] dx1 · · · dxn ρscaled (9.2) (n) (x1 , . . . , xn ) = o(n!) R
R
for R = j=1 (aj , bj ) any domain which contains a finite number of particles ( R ρ(1) (x) dx < ∞). The point process being uniquely determined by its correlations has the consequence that the joint distribution of the number of particles in R is uniquely determined by its moments, and that the analogue of (9.1) holds for the corresponding generating function. Now, it follows from the definitions that dx1 · · · dxn ρscaled (n) (x1 , . . . , xn ) = #R(#R − 1) · · · (#R − n) , 4l
R
R
where #R denotes the number of particles in the interval R and is thus a so-called factorial moment, from which the usual moments can be calculated by linear combinations. One sees from this that if upon the linear change of scale sN (x) = αN + βN x n dx1 · · · dxn ρN lim |βN | (n) (sN (x1 ), . . . , sN (xn )) N →∞ sN (R) sN (R) = dx1 · · · dxn ρscaled (9.3) (n) (x1 , . . . , xn ), R
R
then the joint distribution of the number of particles in sN (R) for the sequence of point processes specified by {ρN (n) } converges to the joint distribution of the number of particles in R for the process specified by {ρscaled (n) } [496]. In particular, the following result holds.
P ROPOSITION 9.1.2 The convergence (9.3), together with the bound (9.2), implies the convergence Eβ ((a1 , a2 ); ξ) := lim EN,β ((sN (a1 ), sN (a2 )); ξ) N →∞ a2 ∞ (−ξ)n a2 = 1+ dx1 · · · dxn ρscaled (n) (x1 , . . . , xn ). n! a a 1 1 n=1
(9.4)
Before proceeding to study the limit in (9.4), we remark that according to Proposition 8.1.2, the parameter ξ in (9.1) and (9.4) is an auxiliary quantity used to form a generating function. However, a physical interpretation of ξ is also possible [76]. The idea is to suppose that eigenvalues (particles) are independently deleted from samples with probability (1 − ξ). Theoretically, this has the simple effect of multiplying each n-point correlation by ξ n . It thus follows that for 0 < ξ < 1, (9.1) and (9.4) can be interpreted as gap probabilities for the corresponding diluted systems. Returning now to the limit in (9.4), we will first show that for the scaled classical random matrix ensembles with unitary symmetry, the convergence (9.3) can be established. This follows from the asymptotic form of the corresponding Christoffel-Darboux kernels.
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P ROPOSITION 9.1.3 In the notations of Chapters 5 and 7, we have 1 πρ πρy (G) πρx √ KN √ ,√ = K bulk(x, y) + O √ O(1), (9.5) 2N 2N 2N N X Y 1 (G) 1/2 1/2 (2N ) K + ,(2N ) + 21/2 N 1/6 N 21/2 N 1/6 21/2 N 1/6 1 = K soft (X, Y ) + O O(e−X−Y ), (9.6) N 2/3 Y 1 1 (L) X K , = K hard(X, Y ) + O O(1), (9.7) 4N N 4N 4N N 2πi (CJ) 2πix/L 2πiy/L 1 K O(1), (9.8) (e ,e ) = K s.s. (x, y) + O L N N where in the remainder terms the dependence on N is uniform, as is the dependence on x, y in a finite interval for (9.5), X, Y ∈ [w, ∞), w ∈ R, for (9.6), X, Y ∈ [0, w], w ∈ R+ for (9.7) and x, y ∈ [0, L] for (9.8).
Proof. In Chapter 7 we have used asymptotic expansions to show that the pointwise limit of the l.h.s. of the above expressions is given by the first term on the r.h.s. To obtain the correction term to the pointwise limit we make use of the correction term in the asymptotic expansion of the corresponding orthogonal √ polynomials. Consider (9.5). We know from [508] that the correction term in (7.1) also has a uniform dependence on 2nx, in that √ √ Γ(n/2 + 1) −x2 /2 Hn (x) = cos( 2n + 1x − nπ/2) + O(n−1/2 )O(e±i 2nx ). e Γ(n + 1)
(9.9)
√
Of course for x real the term O(e±i 2nx ) can be replaced by a constant; however we also have use for (9.9) in the case of x complex. Using (9.9) to substitute for the denominator in (5.10), and making use too of the normalization (5.48) and Stirling’s formula we immediately deduce (9.5), although at this stage we have to assume |x − y| is bounded away from (G) zero to be sure of the error term. To see that this restriction is not necessary, we take advantage of KN (x, y) being an analytic function of y and write Z 2π “ πρx πρy ” “ πρx πρy ” πρ πρ 1 (G) (G) √ KN √ ,√ = √ ,√ + Reit dt KN √ 2N 2N 2N 2N 2π 0 2N 2N √ for arbitrary R > 0. Choosing R = πρ/ 2N we see from (9.9) that the leading term is an integral representation of (G) K bulk (x, y), while the remainder term is bounded by O(N −1/2 ) coming from the numerator of the definition of KN , times the maximum of the scaled denominator 1 . |y − x + eit | For |y − x| 1 the latter is bounded, thus demonstrating the validity of (9.5) in this domain. The same strategy works in the other cases, once the correction term in the analogue of (9.9) is known. For (9.6) we make use of the asymptotic expansion (7.9) refined [435] so that the correction term O(N −2/3 ) is replaced by the uniform bound O(N −2/3 )O(e−t ) (cf. (7.59)), while for (9.7) we replace the correction term O(N a/2−3/4 ) in (7.29) by [508] j 5/4 x O(N a/2−3/4 ), cN −1 ≤ x ≤ w, 0 < x ≤ cN −1 , xa/2+2 O(N a ), where c, w > 0 are arbitrary. To analyze the remainder term for the Christoffel-Darboux sum in the case of the circular Jacobi ensemble we make use of the integral representation Z 1/2 Γ(a + b + 1) eπix(a−b) |1 + e2πix |a+b (1 + te2πix )r dx = 2 F1 (−r, −b; a + 1; t), (9.10) Γ(a + 1)Γ(b + 1) −1/2 which can be deduced by expanding (1 + te2πix )r according to the binomial theorem and integrating term by term using
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FREDHOLM DETERMINANT APPROACH
(4.4) in the case N = 1. This shows 2 F1 (−n, −b; a
“ “1” ” + 1; t/n) = 1 F1 (−b; a + 1; −t) 1 + O O(t2 ) . n
It remains to establish the bound (9.2). We make use of the fact that ρ(n) is non-negative and, according to Proposition 5.1.2, equal to the determinant of a symmetric matrix. For the determinant of such matrices we have the following inequality, equivalent to a result of Hadamard given in (9.12) below. P ROPOSITION 9.1.4 For A = [aij ]i,j=1,...,n non-negative and symmetric, det A ≤
n
aii .
(9.11)
i=1
Proof. [393] Consider first the case aii = 0 for some i = 1, . . . , n. Because a matrix is non-negative if and only if its principal minors are non-negative, it follows that aii ajj − |aij |2 ≥ 0 for each j = i and thus all elements in the ith row of A must vanish, so det A = 0, in agreement with (9.11). Hence it suffices now to consider the case that A is positive definite. We begin by performing the Laplace expansion – » 0 α † , det A = a11 det A1 − det α A1 where α = [ak1 ]k=2,...,n and A1 = [aij ]i,j=2,...,n . Analogous to a manipulation performed in the proof of Proposition 2.5.1 we have » – – » – » 1 0(N−1)×1 0 α † 0 α † = det A1 det det det 01×(N−1) A−1 α A1 α A1 1 α† A−1 ) > 0, = (det A1 )( 1 α where the inequality follows because A1 is a principal minor of A and thus is itself positive definite, and this implies A−1 1 is positive definite. Hence det A < a11 det A1 , and the result now follows by induction.
Note that writing A = B† B for B an n × n square matrix shows (9.11) is equivalent to the inequality n n (9.12) |bij |2 , | det B|2 ≤ i=1
j=1
which is Hadamard’s result [541]. In light of the determinant structure of ρ(n) noted above Proposition 9.1.4, and the fact that the diagonal entries of the determinant equal the one-body density, the bound (9.11) tells us that ρ(n) (x1 , . . . , xn ) ≤
n
ρ(1) (xi ).
(9.13)
i=1
Assuming the region R is such that the integral of the one-body density over R is finite (i.e., contains a finite number of eigenvalues) we see the integral in (9.2) in fact only grow exponentially fast in n. All criteria of Proposition 9.1.2 are therefore met and so it follows that for the scaled classical random matrix ensemble
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with unitary symmetry E2scaled (J; ξ) = 1 +
∞ 1 2 (−ξ)n dx1 · · · dxn det K scaled(xj , xk ) j,k=1,...,n , n! J J n=1
(9.14)
where scale = bulk, soft, hard or s.s. and J = (−t, t), (t, ∞), (0, t), (−t, t), respectively. We recognise this sum as that occurring in the expansion (5.32) of the determinant of a Fredholm integral operator. Hence E2scaled (J; ξ) = det(1 − ξKJscaled), where
(9.15)
(1 − ξKJscale )f := f (x) − ξ
K scaled(x, y)f (y) dy.
(9.16)
J
With (9.14) as the starting point, and with scaled = bulk, J = (0, t) Jimbo, Miwa, Mˆori and Sato [324] obtained the Painlev´e transcendent evaluation of (9.15) implied by the first equation of (8.111). Furthermore, this work also identified the Fredholm determinant in (9.15) as a τ -function associated with the integrable structure of a particular monodromy preserving deformation of a linear differential equation. In this chapter we will present both the function theoretic properties of (9.15) which lead to its Painlev´e transcendent evaluations, as well as aspects of the underlying integrable systems theory. E XERCISES 9.1
1. [90] Let J(p),I (x1 , . . . , xp ) denote the p-point correlation for the event that there are exactly n particles in the interval I, and these are at the positions x1 , . . . , xp . The quantity J(p),I is referred to as a Janossy density. (i) Consider the p.d.f. (5.139) and let ζj , ζ˜j be linear combinations of {ξl }l=1,...,j , and let ψk , ψ˜k be linear combinations of {ηl }l=1,...,k . With Z ∞ “Z ∞ Z ” w2 (x)ζj (x)ψk (x) dx = δj,k , − w2 (x)ζ˜j (x)ψ˜k (x) dx = δj,k −∞
let KN (x, y) =
−∞
I
PN−1
PN−1 ˜ ˜ ˜ l=0 ζl (x)ψl (y) and KN (x, y) = l=0 ζl (x)ψl (y). Show h i ˜ N (xi , xj ) J(p),I (x1 , . . . , xp ) = det(1 − KN,I ) det K
i,j=1,...,p
,
(9.17)
where KN,I is the integral operator on I with kernel KN (x, y). (ii) Verify that the Laguerre polynomials {L0k (x − t)}k=0,1,... are orthogonal on the interval (t, ∞) with respect to the weight e−x , and use this together with the result of (i) to show that with this choice of w2 and ξl = ηl = xl−1 ˛ ˜ N (x, y) = K (L) (x − t, y − t)˛˛ K , N a=0
(L)
where KN is the Christoffel-Darboux kernel in the Laguerre case with a = 0. Substitute this in (9.17), and relate its hard edge scaled form with p = 2, t = t2 , x1 = t1 , x2 = t2 , to the formula (8.100) with a = 0. 2.
(i) By making use of Propositions 5.2.1 and 5.2.2 with a(x) = 1 − ξχx∈J , deduce that EN,2 (J; ξ; w2 (x)) = det(1 − K), where K is the integral operator on J with kernel ξKN (x, y), as is consistent with the formula (9.15). (ii) Note from (6.27) and (6.76) with a(x) as in (i) that for β = 1, 4 ”2 “ EN,β (J; ξ; e−βV (x) ) = det[12 − ξfβ χJ ], where χJ := diag[χJ , χJ ], and χJ is the indicator function for J.
(9.18)
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FREDHOLM DETERMINANT APPROACH
3. Consider a two-species system, species a and b, with coordinates {aj } and {bj }. Let ρ(n1 ,n2 ) ({aj }j=1,...,n1 ; {bj }j=1,...,n2 ) denote the corresponding (n1 , n2 )-point correlation, and introduce the generating function EN (Ja , Jb ; ξx , ξy ) :=
N DY E (l,a) (l,b) (1 − ξa χJa )(1 − ξb χJb ) l=1
(cf. the case μ = 0 of (8.7)). Show that N X
EN (Ja , Jb ; ξa , ξb ) = 1 + Z
Z
n1 ,n2 =0 (n1 ,n2 )=(0,0)
Ja
Z
Z
da1 · · ·
×
(−ξa )n1 (−ξb )n2 n1 !n2 !
db1 · · ·
dan1 Ja
Jb
dbn2 ρ((n1 ,n2 )) ({aj }j=1,...,n1 ; {bj }j=1,...,n2 ). Jb
In the cases that the correlations have a determinant structure ρ((n1 ,n2 )) ({aj }j=1,...,n1 ; {bj }j=1,...,n2 ) 2 3 [Kaa (aj , ak )]j,k=1,...,n1 [Kab (aj , bk )] j=1,...,n1 k=1,...,n 2 5 = det 4 [Kba (bj , ak )] j=1,...,n2 [Kbb (bj , bk )]j,k=1,...,n2 k=1,...,n1
deduce from this that EN (Ja , Jb ; ξa , ξb ) = det(1 − K),
(9.19)
where K is the 2 × 2 matrix integral operator on R with kernel – » χx∈Ja ξa Kaa (x, y)χy∈Ja χx∈Ja ξa Kab (x, y)χy∈Jb . K(x, y) = χy∈Jb ξb Kba (y, x)χx∈Ja χy∈Jb ξb Kbb (y, y)χy∈Jb
9.2 NUMERICAL COMPUTATIONS USING FREDHOLM DETERMINANTS The sine kernel refers to the kernel K bulk (x, y) :=
sin πρ(x − y) , π(x − y)
(9.20)
and the task is to compute bulk ). E2bulk ((0, s); ξ) = det(1 − ξK(0,s)
(9.21)
Let us first address this problem in a numerical sense. In Section 8.3.5 the Painlev´e transcendent formula (8.111) was used to generate the first 700 terms in its power series, and this in turn was used to compute Table 8.13, giving statistical properties of {pbulk (n; s)} 2 for n up to 4. Due to the computational expense incurred in computing the power series, to get more terms is difficult, and this restriction in turn implies a loss of accuracy in the determination of pbulk (n; s) for higher 2 values of n (recall from (8.165) that the leading term in the power series of pbulk (n; s) is proportional to 2 sn+(n+2)(n+1) ). It turns out that a numerical scheme based on (9.21) can overcome this problem [79]. The first tabulations of spacing distributions were based on Fredholm determinants [259]. Thus the Fred− holm determinant formula (9.81) below for E2O ((0, s); ξ = 1) was used to obtain a graphically accurate determination of the bulk spacing distribution pbulk (0; s) (recall (8.139)). This was done by first developing 1 the theory of Section 9.6.1 below, and thereby determining the eigenvalues of the Fredholm integral oper-
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ator in terms of prolate spheroidal functions. By making use of tables of these functions, E1bulk (0; s) was computed, and a numerical differentiation scheme then used to compute pbulk (0; s) according to (8.154). 1 The same strategy was employed in [392] to tabulate pbulk (0; s) (see [425] for a discussion of the source of 2 inaccuracies in this table). Subsequently, as reproduced in [395], the prolate spheroidal functions were used to plot {Eβbulk (n; s)}, β = 1, 2 for successive values of n up to 10. Recently Bornemann [78], [79] has shown how the Fredholm determinant formulas can be used to compute spacing distributions such as {pbulk (n; s)} for successive values of n up to 14 (and beyond if required), to 2 machine precision of 15-digit accuracy. The strategy relates to the approximation of the integral in definition (9.16) by a sum, according to an m-point quadrature rule of order m (i.e., a rule replacing the integral by a weighted sum of m terms, such that it is exact for a polynomial of degree m − 1) with positive weights wj , m K(x, y)f (y) dy ≈ wj K(x, yk )f (yk ). J
k=1
The integral operator eigenvalue equation ξKJ ψ = λψ is correspondingly replaced by the system of m linear equations ξ
m
wj K(yj , yk )ψ(yk ) = λψ(yj )
(k = 1, . . . , m),
j=1 −1/2
or equivalently (with ψ → wj
ψ)
ξ
m
1/2
1/2
wj K(yj , yk )wk ψ(yk ) = λψ(yj ).
j=1 1/2
1/2
The characteristic polynomial of the symmetric matrix [ξwj K(yj , yk )wk ]j,k=1,...,m at λ = 1 is the Fredholm determinant of 1 − ξKJ in this approximation, and so 1/2
1/2
det(1 − ξKJ ) ≈ det[δj,k − ξwj K(yj , yk )wk ]j,k=1,...,m .
(9.22)
It is shown in [78] that with the kernel K(x, y) analytic in a complex neighbourhood of (a, b), this approximation has error O(ρ−m ) (ρ > 1) and thus converges exponentially fast. In practice this means doubling m will typically double the number of correct digits. The derivative in s, required by (8.115) and analogous formulas, is carried out by converting the numerical tabulation to an interpolation in Chebyshev points. According to the formulas of Proposition 8.1.2, to compute higher order spacing distributions from the generating function, derivatives in ξ of (9.22) are required. This being an entire function of ξ (it is a polynomial), one can use the formula for the k-th derivative 2π k! f (k) (z) = e−ikθ f (z + reiθ ) dθ, r > 0, 2πrk 0 with the trapezoidal rule approximation, which is known to converge exponentially fast for periodic analytic integrands. For the bulk spacings the choice r = 1 was empirically determined to be the most numerically stable in this regard.
9.3 THE SINE KERNEL We now take up the problem of using (9.21) to obtain the characterization in terms of the σPV equation (8.112). Key equations from [324] for this purpose were isolated by Tracy and Widom [517], following on from the works [394], [152]. These equations relate to a number of quantities associated with the integral
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FREDHOLM DETERMINANT APPROACH
operator in (9.15). One of these quantities is the resolvent kernel R(x, y), specified by . (9.23) ξKJ (1 − ξKJ )−1 = R(x, y), . where the notation = denotes that the r.h.s. is the kernel of the integral operator on the l.h.s. Our interest is in the case that J is a single interval, J = [a1 , a2 ] say. Closely related to (9.23), and used in the definition of the other quantities entering the coupled equations, is ρ(x, y) specified by . (1 − ξKJ )−1 = ρ(x, y) (9.24) (note that ρ(x, y) = δ(x − y) + R(x, y)). Both (9.23) and (9.24) have meaning for general kernels; however, the other quantities are special to the Christoffel-Darboux type structure of K scale (x, y). D EFINITION 9.3.1 Let φ(x), ψ(x) be such that ξK scaled (x, y) :=
φ(x)ψ(y) − φ(y)ψ(x) . x−y
(9.25)
In terms of φ(x), ψ(x) and (9.24) introduce the quantities Q(x), qj , P (x), pj according to a2 ρ(x, y)φ(y)dy, qj = Q(aj ) := x→a lim Q(x), Q(x) := (1 − ξKJ )−1 φ := j x∈(a1 ,a2 )
a1
P (x) := (1 − ξKJ )−1 ψ :=
a2
ρ(x, y)ψ(y)dy,
pj = P (aj ) :=
a1
lim
x→aj x∈(a1 ,a2 )
P (x).
The equations of [517] relating to E2bulk ((0, t); ξ) can now be stated. P ROPOSITION 9.3.2 For (9.14) with
K scaled(x, y) = K bulk(x, y)
ρ=1/π
,
and in the notation of Definition 9.3.1 with a1 = −t, a2 = t, we have d 2q 2 p2 qp , (ii) R(t, t) = p2 + q 2 − , (iii) R(t, t) = 2(R(−t, t))2 , t2 t dt 2q p dp 2qp2 dq =p− , (v) = −q + , (iv) dt t dt t where p := p2 and q := q2 . (i) R(−t, t) =
From these coupled equations a single differential equation for σ(t) := −tR(t/2, t/2) can be obtained, and knowledge of σ(t) allows E2 ((0, t); ξ) to be computed. For this we require the general formula ∂ log det(1 − ξKJ ) = (−1)j−1 R(aj , aj ) ∂aj
(j = 1, 2)
(9.26)
(see Exercises 9.3 q.1). It follows from this that −2R(t, t) =
d log E2 ((−t, t); ξ), dt
which in turn implies
E2 ((0, t); ξ) = E2 ((−t/2, t/2); ξ) = exp 0
πρt
(9.27)
σ(u) du. u
(9.28)
On the r.h.s. of (9.28) t has been replaced by πρt to account for a general value of ρ (this is valid because ρ is the only length scale in the problem).
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P ROPOSITION 9.3.3 The quantity σ(t) := −tR(t/2, t/2) satisfies the first of the differential equations in (8.112) (which is the σPV equation of (8.15) with ν0 = ν1 = ν2 = ν3 = 0 and t → −2is) subject to the first of the boundary conditions in (8.113). Proof. From equations (i), (ii), (iv) and (v) of Proposition 9.3.2 it is straightforward to deduce the equations d (tR(−t, t)) = p2 − q 2 dt
and
d (tR(t, t)) = p2 + q 2 . dt
Squaring these equations and subtracting gives „ «2 „ «2 d d (tR(−t, t)) − (tR(t, t)) = −4p2 q 2 = −4t2 (R(−t, t))2 , dt dt where to obtain the last equality equation (i) has again been used. The stated equation now follows by using the equations 4t2 (R(−t, t))2 = 2t
d (tR(t, t)) − 2tR(t, t) and dt
4tR(−t, t)
d d2 (tR(−t, t)) = t 2 (tR(t, t)), dt dt
which are consequences of equation (iii), to eliminate tR(−t, t) and its derivative. The boundary condition follows from the leading small t computation of the n = 1 and n = 2 terms in (9.4) with K(x, y) given by K bulk (x, y) and ρ therein set equal to π1 .
9.3.1 Derivation of the coupled equations Two general operator identities are used throughout the derivation. P ROPOSITION 9.3.4 For any operators L and K, [L, (1 − K)−1 ] = (1 − K)−1 [L, K](1 − K)−1 ,
(9.29)
and for any operator K which can be differentiated with respect to a parameter a, dK d (1 − K)−1 = (1 − K)−1 (1 − K)−1 . da da
(9.30)
Proof. The first identity is verified by letting both sides act on an arbitrary function f , expanding the commutators, and comparing the resulting expressions. The second identity has been derived in Exercises 2.5 q.1.
The following simple lemma is also required. P ROPOSITION 9.3.5 Let M denote multiplication by the independent variable, that is, M f (x) = xf (x). Then for an integral operator LJ with kernel L(x, y) supported on the interval J = [a1 , a2 ], . [M, LJ ] = (x − y)L(x, y). Proof. Since Z (M LJ − LJ M )f := x
a2
Z L(x, y)f (y)dy −
a1
Z
a2
a2
L(x, y)yf (y)dy = a1
(x − y)L(x, y)f (y)dy,
a1
the result follows.
After these preliminaries, we begin the derivation proper with an expression for the resolvent kernel [307]. P ROPOSITION 9.3.6 Let R(x, y) be defined as in (9.23). We have R(x, y) =
Q(x)P (y) − P (x)Q(y) , x−y
(9.31)
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FREDHOLM DETERMINANT APPROACH
where Q and P are specified in Definition 9.3.1, and thus R(x, x) = −Q(x)P (x) + P (x)Q (x).
(9.32)
Proof. We will compute (9.29) with L = M in two different ways. By Proposition 9.3.5 and (9.25), . [M, ξKJ ] = φ(x)ψ(y) − φ(y)ψ(x). . . Thus [M, ξKJ ] = A1 − A2 where A1 = φ(x)ψ(y), A2 = φ(y)ψ(x), so we have [M, ξKJ ](1 − ξKJ )−1 f = A1 (1 − ξKJ )−1 f − A2 (1 − ξKJ )−1 f Z a2 Z a2 Z a2 Z a2 = φ(x) dy ψ(y) ρ(y, y )f (y )dy − ψ(x) dy φ(y) ρ(y, y )f (y )dy a1 a1 a1 a1 Z a2 Z a2 Z a2 Z a2 = φ(x) dy f (y) dy ψ(y )ρ(y, y ) − ψ(x) dy f (y) dy φ(y )ρ(y, y ), a1
a1
a1
a1
where we have used the symmetry ρ(y, y ) = ρ(y , y) which follows since K(x, y) = K(y, x). Hence, using Definition 9.3.1, we have Z Z a2
[M, ξKJ ](1 − ξKJ )−1 f = φ(x)
f (y)P (y) dy − ψ(x)
a1
and so
a2
f (y)Q(y) dy, a1
. [M, ξKJ ](1 − ξKJ )−1 = φ(x)P (y) − ψ(x)Q(y),
which gives
. (1 − ξKJ )−1 [M, ξKJ ](1 − ξKJ )−1 = Q(x)P (y) − P (x)Q(y).
On the other hand, for the l.h.s of (9.29), we have from Proposition 9.3.5 . [M, (1 − ξKJ )−1 ] = (x − y)ρ(x, y) = (x − y)R(x, y). Equating the above two equations gives the stated result.
Equation (i) of Proposition 9.3.2 can now be derived by using (9.31). To see this, we note that with a1 = −t, a2 = t, from the facts that φ(x) is odd, ψ(x) is even and ρ(x, y) = ρ(−x, −y) (which follows since K(x, y) = K(−x, −y)), it follows from Definition 9.3.1 that q2 = −q1 = q,
p2 = p1 = p,
(9.33)
so substitution gives the desired equation. Whereas Proposition 9.3.6 holds for general φ and ψ, the final conclusion of the next result uses a special property of the particular choice ξ ξ sin πρx, ψ(x) = cos πρx, (9.34) φ(x) = π π which gives the kernel (9.20). This result is presented in preparation for the computation of P (x) and Q (x) which occur in the expression (9.32) for R(x, x). P ROPOSITION 9.3.7 Let D denote the operator for differentiation with respect to the independent variable. We have . [D, KJ ] = (Dx + Dy )K(x, y) − δ − (y − a2 )K(x, y) + δ + (y − a1 )K(x, y), where
a2
a1
δ − (y − a2 )f (y)dy := f (a2 )
a2
and a1
δ + (y − a1 )f (y)dy := f (a1 ),
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CHAPTER 9
and consequently, in the cases that (Dx + Dy )K(x, y) = 0, . [D, (1 − ξKJ )−1 ] = R(x, a1 )ρ(a1 , y) − R(x, a2 )ρ(a2 , y). Proof. We have Z a2 Z a2 ∂ ∂ K(x, y)f (y) dy − K(x, y) f (y) dy ∂x a1 ∂y a1 Z a2 ” Z a2 “ ∂ ∂ f (y) K(x, y) dy, := K(x, y)f (y) dy − f (a2 )K(x, a2 ) − f (a1 )K(x, a1 ) + ∂y a1 ∂x a1
[D, KJ ]f :=
which gives the first result. Now suppose (Dx + Dy )K(x, y) = 0. The first result then gives Z a2 Z a2 [D, KJ ](1 − ξKJ )−1 f = dy K(x, y)(δ + (y − a1 ) − δ − (y − a2 )) dy ρ(y, y )f (y ) a1 a1 Z a2 Z a2 = K(x, a1 ) ρ(a1 , y )f (y )dy − K(x, a2 ) ρ(a2 , y )f (y )dy . a1
a1
The second result now follows upon applying (9.29) with L = D and recalling (1 − ξKJ )−1 ξKJ = R.
The derivatives P (x) and Q (x), and consequently R(x, x), can now be computed in terms of quantities in Definition 9.3.1. P ROPOSITION 9.3.8 With K(x, y) given by (9.20) we have Q (x) = πρP (x) + R(x, a1 )q1 − R(x, a2 )q2 , P (x) = −πρQ(x) + R(x, a1 )p1 − R(x, a2 )p2 and consequently R(aj , aj ) = πρ(p2j + qj2 ) −
(q2 p1 − p2 q1 )2 a 2 − a1
(j = 1, 2).
Proof. From Definition 9.3.1 we have Q (x) = D(1 − ξKJ )−1 φ = (1 − ξKJ )−1 Dφ + [D, (1 − ξKJ )−1 ]φ. The stated formula for Q (x) now follows from the fact that Dφ = πρψ, Proposition 9.3.7, (9.23) and Definition 9.3.1. The formula for P (x) is derived similarly, and the formula for R(aj , aj ) is derived by substituting for Q (x) and P (x) in (9.32), and using (9.31) and Definition 9.3.1.
Equation (ii) of Proposition 9.3.2 follows from the formula above for R(aj , aj ) with j = 2, a2 = t and the substitutions (9.33). For the derivation of equation (iii), a preliminary result is needed. ∂ P ROPOSITION 9.3.9 We have ∂a R(x, y) = (−1)j R(x, aj )R(aj , y) and so with (Dx + Dy )K(x, y) = 0, j ∂ ∂ ∂ −R(x, a2 )R(a2 , y), j = 1, + R(x, y) = + R(x, a1 )R(a1 , y), j = 2, ∂aj ∂x ∂y
for x, y ∈ (a1 , a2 ). Proof. We have ∂ ∂ R(x, y) = ρ(x, y) ∂aj ∂aj so we consider (9.30) with a = aj . Now ∂ ∂ KJ f = ∂aj ∂aj
Z
a2 a1
K(x, y)f (y)dy = (−1)j K(x, aj )f (aj )
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FREDHOLM DETERMINANT APPROACH
and so
∂ . KJ = (−1)j K(x, aj )δ ∓ (y − aj ) (j = 1, 2). ∂aj
This gives ∂ ρ(x, y) = (−1)j R(x, aj )ρ(aj , y) ∂aj
(9.35)
and the first result follows since ρ(aj , y) = R(aj , y) for y ∈ (a1 , a2 ). The second result follows from the first result, the fact that for x, y ∈ (a1 , a2 ) « „ ∂ ∂ . R(x, y) [D, (1 − ξKJ )−1 ] = + ∂x ∂y (this can be verified by using integration by parts) and the final equation in Proposition 9.3.7.
Equation (iii) of Proposition 9.3.2 can now be derived by first noting ∂ ∂ ∂ ∂ + R(x, y) R(aj , aj ) = + = (−1)j (R(a1 , a2 ))2 , ∂aj ∂aj ∂x ∂y x=y=aj where the final equality follows from Proposition 9.3.9. This gives d ∂ ∂ R(t, t) = R(a2 , a2 ) − R(a2 , a2 ) = 2(R(−t, t))2 dt ∂a2 ∂a1 a2 =−a1 =t as required. For the derivation of the final two equations in Proposition 9.3.2, the formulas ∂qj ∂ ∂ ∂ ∂pj ∂ + + Q(x) P (x) = and = ∂aj ∂x ∂aj ∂aj ∂x ∂aj x=aj x=aj
(9.36)
(9.37)
(9.38)
are used. The partial derivatives with respect to x are given by Proposition 9.3.8. The derivatives with respect to aj are given by the following result, which is valid for all kernels of the form (9.25). P ROPOSITION 9.3.10 We have ∂ Q(x) = (−1)j R(x, aj )qj ∂aj
and
∂ P (x) = (−1)j R(x, aj )pj . ∂aj
Proof. Now ∂ ∂ Q(x) := (1 − K)−1 φ. ∂aj ∂aj But from (9.35) ∂ . (1 − ξKJ )−1 = (−1)j R(x, aj )ρ(aj , y), ∂aj and the first result follows from Definition 9.3.1. The second result follows similarly.
The final two equations of Proposition 9.3.2 now follow by noting dqj ∂qj ∂qj = − , dt ∂a2 ∂a1 dp
and similarly for dtj , substituting the results of Propositions 9.3.10 and 9.3.8 into (9.38) to compute one of the derivatives, and using Proposition 9.3.10 to compute the other derivative. E XERCISES 9.3
1. The objective of this exercise is toRderive the formula (9.26). For an integral operator KJ on a J = [a1 , a2 ] with kernel K(x, y) we have Tr KJ := a12 K(x, x) dx. Use the formula log det A = Tr log A,
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CHAPTER 9
and the power series expansion of log(1 − ξKJ ) to show that ∂ det log(1 − ξKJ ) ∂aj Z a2 Z ∞ ∂ X ξ n a2 =− dx1 · · · dxn K(x1 , x2 )K(x2 , x3 ) · · · K(xn−1 , xn )K(xn , x1 ) ∂aj n=1 n a1 a1 Z a2 Z a2 ∞ X = (−1)j−1 ξn dx2 · · · dxn K(aj , x2 )K(x2 , x3 ) · · · K(xn−1 , xn )K(xn , aj ). n=1
a1
a1
Identify the final expression with (−1)j−1 R(aj , aj ) as required by (9.26). 2. [324], [517] Here the multiple gap probability will be exhibited as an integrable system. (i) Let E2 (0; J) denote the probability that there are no particles in some collection of intervals J. Show that (9.15) still holds, with the domain of integration of the integral operator now J. (ii) In the case J = ρ = 1/π)
Sm
j=1 (a2j−1 , a2j )
R(ai , ai ) = p2i + qi2 +
2m X
modify the proofs of Propositions 9.3.6–9.3.10 to show that (with
(−1)k
k=1 k=i
(qi pk − pi qk )2 , ai − ak
X ∂qi = pi − (−1)k R(ai , ak )qk ∂ai k=1
2m X ∂pi = −qi − (−1)k R(ai , ak )pk , ∂ai k=1
2m
and
k=i
k=i
∂qj = (−1)k R(aj , ak )qk ∂ak
and
∂pj = (−1)k R(aj , ak )pk ∂ak
with R(aj , ak ) =
qj pk − pj qk qk . aj − ak
q2j+1 = x2j+1 ,
p2j = −iy2j ,
(iii) Introduce the notation q2j = −x2j ,
ω(a) = da log(1 − K) = −
p2j+1 = y2j+1 ,
2m 2m X X (−1)i R(ai , ai )dai = Gi dai , i=1
i=1
2m X (xi yk − yi xk )2 , Gi = x2i + yi2 − ai − ak k=1 k=i
and define the canonical symplectic structure by {xj , xk } = 0,
{yj , yk } = 0,
{xj , yk } =
1 δj,k . 2
Verify that the two pairs of equations in (ii) are equivalent to the Hamilton equations da xj = {xj , ω(a)} and
da yj = {yj , ω(a)}.
Furthermore, verify that the Gi s are in involution so that {Gi , Gj } = 0.
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FREDHOLM DETERMINANT APPROACH
9.4 THE AIRY KERNEL 9.4.1 Painlev´e II equation The Airy kernel refers to the kernel K soft (X, Y ) :=
Ai(X)Ai (Y ) − Ai(Y )Ai (X) , X −Y
(9.39)
and the task is to compute soft E2soft ((s, ∞); ξ) = det(1 − ξK(s,∞) ).
(9.40)
In [518] it is emphasized that a key ingredient in the derivation of differential equations associated with kernels of the form (9.25) is that φ and ψ are related by the coupled first order differential equations m(x)φ (x) = A(x)φ(x) + B(x)ψ(x), m(x)ψ (x) = −C(x)φ(x) − A(x)ψ(x),
(9.41) √ where m, A, B and C are polynomials. For the Airy kernel (9.39), φ(x) = ξAi(x) and ψ(x) = ξAi (x), and since Ai (x) = xAi(x) we see that the equations (9.41) are satisfied with √
m(x) = 1, A(x) = 0, B(x) = 1, C(x) = −x.
(9.42)
Using (9.41), the working of Section 9.3 can be modified to derive coupled nonlinear equations for quantities associated with the Fredholm determinant of the Airy kernel on the interval (a1 , a2 ) = (s, ∞) [519]. The equations involve two quantities in addition to those of Definition 9.3.1. D EFINITION 9.4.1 Let φ, ψ, P and Q be as defined in Definition 9.3.1. In terms of these quantities we write where f |g :=
a2 a1
u := φ|Q
and
v := φ|P = ψ|Q,
f (y)g(y)dy.
P ROPOSITION 9.4.2 For the Airy kernel (9.39), in the notation of Definitions 9.3.1 and 9.4.1 with a1 = s, a2 = ∞, p := p1 , q := q1 and R := R(s, s), we have q = p − qu, p = sq + pu − 2qv, R = pq − qp , R = −q 2 , u = −q 2 , v = −pq. The derivation of these equations will be given in the next subsection. Presently we use of them to derive a single differential equation for R =: R(s; ξ). From this the generating function E2soft ((s, ∞); ξ) that there are no eigenvalues in the interval (s, ∞) can be calculated, since from the general formula (9.26) we have R(s; ξ) = and so E2soft ((s, ∞); ξ)
d log E2 ((s, ∞); ξ) ds = exp −
∞
R(t; ξ)dt .
(9.43)
s
In (8.84) of Proposition 8.3.2 the same formula was obtained, but with R(t; ξ) replaced by us (t; 0; ξ). Indeed it follows from Proposition 9.4.2 that R(t; ξ) and us (t; 0; ξ) are the same quantity. P ROPOSITION 9.4.3 As with us (t; 0; ξ) specified in Proposition 8.3.2, the quantity R of Proposition 9.4.2 satisfies the differential equation (R )2 + 4R (R )2 − sR + R = 0, (9.44)
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CHAPTER 9
and this equation is to be solved subject to the boundary condition R(t; ξ) ∼ ξρsoft (1) (t). t→∞
Proof. Let us refer to the equations of Proposition 9.4.2 according to their order of presentation. Differentiating the first of the stated equations above, and substituting for the first derivatives using the first, second and fifth equations gives q = sq + q 3 + q(u2 − 2v). But from the first, fifth and sixth equations we see qq = uu − v and thus q 2 = u2 − 2v (the constant of integration is zero since all terms vanish as s → ∞). Substituting for u2 − 2v gives q = sq + 2q 3 ,
(9.45)
which is the special case α = 0 of the Painlev´e II differential equation as listed in (8.9). Using the fourth equation it can be verified from this that R = q 2 − sq 2 − q 4 . The stated differential equation for R can now be verified using this equation together with further use of the fourth equation.
The derivation of (9.44) implies a formula for E2soft ((s, ∞); ξ) involving q = q(t; ξ), which according to (9.45) is a Painlev´e II transcendent, in contrast to R(t; ξ) which is a σPII transcendent. Thus, making use of the fourth equation in Proposition 9.4.2 and integrating by parts in (9.43), one obtains [519] ∞ soft E2 ((s, ∞); ξ) = exp − (t − s)q 2 (t; ξ) dt , (9.46) s
where q is the solution of the Painlev´e II equation (9.45) satisfying the condition q(s; ξ) ∼ ξAi(s) as s → ∞.
(9.47)
We remark that it has been proved [294] that (9.45) has a unique solution subject to (9.46) with ξ = 1. Furthermore, this solution exhibits the asymptotic behavior q(s; 1) ∼ −s/2 (9.48) s→−∞
and consequently it follows from (9.46) that [519] E2soft (0; (s, ∞))
∼
s→−∞
3
es
/12
.
(9.49)
9.4.2 Numerical computation using the PII equation The evaluation (9.46) is well suited to the numerical computation of psoft 2 (n; t), provided n is small [451]. The strategy is first to calculate the asymptotic expansion of q(s) := q(s; ξ) in (9.45) about s = ∞, and to use this to obtain an accurate evaluation of q(s0 ) and q (s0 ) for some particular s0 . These values are then used to obtain a power series expansion of q(s) about s = s0 , which in turn provides accurate evaluations of q(s1 ) and q (s1 ) with s1 < s0 , and the procedure continues (cf. Section 8.3.4). P ROPOSITION 9.4.4 Write for the s → ∞ asymptotic expansion of the Airy function Ai(s) ∼
3/2 ∞ e−(2/3)s (−1)n √ 1/4 αn 2 πs ( 2 s3/2 )n n=0 3
(3) so that [435] αn = (6n − 1)(6n − 5)αn−1 /72n, α0 = 1. Let αn = 0≤k≤l≤n αn−l αl−k αk and specify {an }n=0,1,... by the recurrence 1 5 3 1 n− n− an−2 (9.50) an = α(3) n + nan−1 − 4 8 6 6
395
FREDHOLM DETERMINANT APPROACH
subject to the initial conditions a−2 = a−1 = 0. We have q(s; ξ) ∼ Proof. Put q =
√
3/2 ∞ e−2s (−1)k ak ξAi(s) + ξ 3/2 . 3/2 7/4 32π s ( 2 s3/2 )k k=0 3
(9.51)
ξAi(s) + ξ 3/2 Q(s), where |Q(s)| Ai(s) for s → ∞. Then Q ∼ sQ + 2(Ai(s))3 .
Since
(9.52)
3/2 ∞ X (−1)n (3) e−2s αn , 2 8π 3/2 s3/4 n=0 ( 3 s3/2 )n
(Ai(s))3 ∼ we see by substituting
3/2 ∞ X e−2s (−1)k ak 3/2 7/4 32π s ( 2 s3/2 )k k=0 3
Q(s) =
in (9.52) and equating like terms that (9.50) results. 3/2
It is empirically observed in [451] that the sum over k in (9.51) is optimally truncated at k ≈ 43 s0 for given s0 0. This then gives accurate numerical evaluations of q(s0 ; ξ) and q (s0 ; ξ). For a general s0 , knowledge of c0 := q(s0 ; ξ) and c1 := q (s0 ; ξ) allows the power series expansion q(s; ξ) =
∞
cl (s − s0 )l
(9.53)
l=0
to be computed by recurrence. (k)
P ROPOSITION 9.4.5 Let cn k
(q(s; ξ)) about s0 (note
(1) cn
:=
n
(k−1) j=0 cn−j cj
be the coefficients in the power series expansion of
= cn ). We have (3)
cn+2 =
2cn + s0 cn + cn−1 . (n + 2)(n + 1)
(9.54)
Proof. This follows by direct substitution of (9.53) in (9.45), and equating like powers of (s − s0 ). ∂ According to (8.86), (8.3) and (9.46), with u(t) := q(t; 1), v(t) := ∂ξ q(t; ξ)|ξ=1 , we have ∞ d exp − (t − s)u2 (t) dt , psoft 2 (0; s) = ds s ∞ d ∞ soft 2 psoft (1; s) = p (0; s) − 2 (t − s)u(t)v(t) dt exp − (t − s)u (t) dt . 2 2 ds s s
As it is not practical to leave ξ as a variable in the iteration of (9.54) for general c0 = q(s0 ; ξ) and c1 = q (s0 ; ξ), we calculate the power series for v(s) about s0 from knowledge of the power series of u(s) and the initial values v(s0 ), v (s0 ), by noting from (9.45) that v = (s + 6u2 )v. The moments calculated from this procedure [449] are presented in Table 9.1. Numerical computation of the Fredholm determinant (9.40) according to the method of Section 9.2 allows for the high precision computation of {psoft 2 (n; t)} beyond n = 0 and 1 [79], and these too are presented in Table 9.1.
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CHAPTER 9
p.d.f.
mean
variance
skewness
kurtosis
psoft 2 (0; t) psoft 2 (1; t) psoft 2 (2; t) psoft 2 (3; t) psoft 2 (4; t) psoft 2 (5; t)
−1.77108 68074 −3.67543 72971 −5.17132 31745 −6.47453 77733 −7.65724 22912 −8.75452 24419
0.81319 47928 0.54054 50473 0.43348 13326 0.37213 08147 0.33101 06544 0.30094 94654
0.22408 42036 0.12502 70941 0.08880 80227 0.06970 92726 0.05777 55438 0.04955 14791
0.09344 80876 0.02173 96385 0.00509 66000 −0.00114 15160 −0.00405 83706 −0.00559 9855
Table 9.1 Statistical properties of psoft 2 (n; t) for various n.
9.4.3 Further theory In this subsection the theory contained in [517] will be used to derive the equations of Proposition 9.4.2. Now, examination of the theory of Section 9.3.1 shows that for any integral operator on the interval (a1 , a2 ), with kernel KJ of the form (9.25), there are certain inter-relationships between the quantities of Definitions 9.3.1 and 9.4.1 which are always valid. These are referred to as the universal equations. P ROPOSITION 9.4.6 For j, k = 1, 2 we have qj pk − pj qk (j = k), a j − ak
∂ log det(1 − ξKJ ) = (−1)j−1 R(aj , aj ), ∂aj ∂qj ∂pj (iii) = (−1)k R(aj , ak )qk (j = k), (iv) = (−1)k R(aj , ak )pk (j = k), ∂ak ∂ak ∂v ∂u = (−1)k qk2 , (vi) = (−1)k pk qk , (v) ∂ak ∂ak ∂w (vii) = (−1)k p2k , ∂ak
(i) R(aj , ak ) =
(ii)
(9.55)
where j, k = 1,2; R, aj , pj , qj and KJ are as in Definition 9.3.1; u and v are as in Definition 9.4.1, and a w := ψ|P := a12 P (y)ψ(y) dy with P and ψ as defined in Definition 9.3.1. Proof. Equation (i) follows from Proposition 9.3.6, equation (ii) is just (9.26), while (iii) and (iv) follow from Proposition 9.3.10. To derive (v), we note that ∂ ∂u = ∂ak ∂ak
Z
a2
Z Q(y)φ(y) dy = (−1)k φ(ak )qk +
a1
a2
φ(y) a1
∂ Q(y) dy. ∂ak
Substituting the value of ∂Q(y)/∂ak from Proposition 9.3.10 and then noting that Z a2 “ Z a2 ” R(y, ak )φ(y) dy = x→a lim ρ(y, x) − δ(y − x) φ(y) dy = Q(ak ) − φ(ak ) k x∈(a1 ,a2 )
a1
a1
gives the stated result. Equations (vi) and (vii) are derived similarly.
In addition to the universal equations of Proposition 9.4.6 there are equations which depend on the specific form of ψ and φ. Let us take up the task of deriving these equations in the case m(x) = 1,
A(x) = α0 + α1 x,
B(x) = β0 + β1 x,
C(x) = γ0 + γ1 x.
(9.56)
First we seek formulas for P (x) and Q (x) analogous to those given in Proposition 9.3.8 for the sine kernel. For this purpose the result analogous to Proposition 9.3.7 is required.
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FREDHOLM DETERMINANT APPROACH
P ROPOSITION 9.4.7 With D denoting the operator for differentiation with respect to the independent variable, we have [D, (1 − ξKJ )−1 ] 2 . = α1 Q(x)P (y) + P (x)Q(y) + β1 P (x)P (y) + γ1 Q(x)Q(y) − (−1)k R(x, ak )ρ(ak , y). k=1
Proof. From (9.41) and (9.56) it follows that (Dx + Dy )ξK(x, y) ” B(x) − B(y) C(x) − C(y) A(x) − A(y) “ φ(x)ψ(y) + ψ(x)φ(y) + ψ(x)ψ(y) + φ(x)φ(y) = x−y x−y x−y “ ” = α1 φ(x)ψ(y) + ψ(x)φ(y) + β1 ψ(x)ψ(y) + γ1 φ(x)φ(y). The result follows from this according to the proof of Proposition 9.3.7.
The differentiation formulas for P (x) and Q(x) can now be obtained by following the method of the proof of Proposition 9.3.8. P ROPOSITION 9.4.8 We have Q (x) = α0 Q(x) + α1 Q1 (x) + (α1 v + γ1 u)Q(x) + β0 P (x) + β1 P1 (x) +(α1 u + β1 v)P (x) −
2
(−1)k R(x, ak )qk ,
k=1
P (x) = −γ0 Q(x) − γ1 Q1 (x) + (α1 w + γ1 v)Q(x) − α0 P (x) − α1 P1 (x) +(α1 v + β1 w)P (x) − where Q1 (x) :=
a2 a1
2
(−1)k R(x, ak )pk ,
k=1
yρ(x, y)φ(y) dy and P1 (x) :=
a2 a1
yρ(x, y)ψ(y) dy.
The quantities Q1 (x) and P1 (x) can be written in terms of Q(x), P (x), u, v and w. P ROPOSITION 9.4.9 We have Q1 (x) = xQ(x) − vQ(x) − uP (x)
and P1 (x) = xP (x) − wQ(x) − vP (x) .
Proof. From the proof of Proposition 9.3.6 we have . [M, (1 − ξKJ )−1 ] = Q(x)P (y) − P (x)Q(y). Applying this to φ and ψ and rearranging gives the two stated formulas.
Substituting the results of Proposition 9.4.9 in Proposition 9.4.8, then substituting the resulting equations in the formula (9.32) for R(x, x) and taking the limit as x → aj gives the following formula to supplement the equations of Proposition 9.4.6. P ROPOSITION 9.4.10 We have R(aj , aj ) = qj2 (γ0 + γ1 aj − 2γ1 v − 2α1 w) + 2pj qj (γ1 u − α0 + α1 aj − β1 w) +p2j (β0 + β1 aj + 2α1 u + 2β1 v) +
2 k=1 k=j
(−1)k R(aj , ak )(qj pk − pj qk ).
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According to (9.38) and Proposition 9.3.10, the formulas for the derivatives of Q(x) and P (x) give us ∂q ∂p formulas for ∂ajj and ∂ajj . P ROPOSITION 9.4.11 We have ∂qj = qj (α0 + α1 aj + γ1 u − β1 w) + pj (β0 + β1 aj + 2α1 u + 2β1 v) − (−1)k R(aj , ak )qk , ∂aj k=1 2
k=j
∂pj = pj (−α0 − α1 aj − γ1 u + β1 w) + qj (−γ0 − γ1 aj + 2γ1 v + 2α1 w) − (−1)k R(aj , ak )pk . ∂aj k=1 2
k=j
As our final equation we give the analogue of the second equality in (9.36), which follows from the first equality in (9.36), Proposition 9.4.7, the final statement in the proof of Proposition 9.3.9 and the first statement of Proposition 9.3.9. P ROPOSITION 9.4.12 We have 2 2 ∂ R(aj , aj ) = − (−1)k R(aj , ak ) + 2α1 qj pj + β1 p2j + γ1 qj2 . ∂aj k=1 k=j
Derivation of the d.e.’s for the Airy kernel For the Airy kernel, from (9.42) and (9.56), α0 = α1 = 0,
β0 = 1, β1 = 0,
γ0 = 0, γ1 = −1.
(9.57)
Also (a1 , a2 ) = (s, ∞), which means that p2 = q2 = R(a1 , a2 ) = 0 and p1 , q1 depend only on s. The first two equations of Proposition 9.4.2 can thus be seen to follow immediately from Proposition 9.4.11, the third equation from Proposition 9.4.10 and the first two equations, the fourth equation from Proposition 9.4.12, and the fifth and sixth equations from equations (v) and (vi) respectively of Proposition 9.4.7. E XERCISES 9.4
1. [231] The aim of this exercise is to use the two different evaluations of the soft edge gap probability to deduce an identity between Painlev´e II transcendents. (i) Use the first equation in (8.29), the definition of hII in Proposition 8.2.1 and the first of the Hamilton equations in (8.23) to show that 1 “ t ”˛˛ σII (t) = − 1/3 q + q 2 + . ˛ 2 t→−21/3 t 2 Write q = qα (t) where α is the parameter in the PII equation of (8.9) to conclude from Proposition 8.3.3 and (9.43) that ” “ 1 “ t ”˛˛ 2 (t) + q−1/2 (t) + . (9.58) us (t; 0; ξ) = R (t; ξ) = − 1/3 q−1/2 ˛ 2 2 t→−21/3 t (ii) Use the fourth equation in Proposition 9.4.2 and (9.45) to deduce from (i) that 1 “ t ”˛˛ 2 (t) + q−1/2 (t) + . q02 (t) = 1/3 q−1/2 ˛ 2 t→−21/3 t 2 (In fact an identity of Gambier [254] gives −21/3 q02 (−2−1/3 t) =
d 2 (t) − t q/2 (t) − q/2 dt 2
(9.59)
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FREDHOLM DETERMINANT APPROACH
valid for both = ±.)
9.5 BESSEL KERNELS As seen in Chapter 7, there are two kernels which qualify for the title of a Bessel kernel, Ja (X 1/2 )Y 1/2 Ja (Y 1/2 ) − X 1/2 Ja (X 1/2 )Ja (Y 1/2 ) (9.60) 2(X − Y ) (Ja+1/2 (πρx)Ja−1/2 (πρy) − Ja+1/2 (πρy)Ja−1/2 (πρx)) . K s.s. (x, y) := (πρx)1/2 (πρy)1/2 2(x − y) (9.61)
K hard (X, Y ) :=
We will treat each separately. 9.5.1 Spacings at the hard edge Referring to the integral operator on J = (0, t) with kernel K hard (x, y) and resolvent kernel R(x, y), it follows from (9.26) that s E2hard ((0, s); ξ) =: E2hard ((0, s); ξ; a) = exp − R(t; ξ) dt , (9.62) 0
where R(t; ξ) := R(t, t). A theory analogous to that of the sine and Airy kernels can be developed to rederive the characterization of R(t; ξ) as a particular σPIII transcendent implied by (8.88) and Proposition 8.3.3. Furthermore, this development leads to an alternative formula expressing E2hard ((0, s); ξ) in terms of a particular transformed Painlev´e V transcendent √ √[520]. For the Bessel kernel (9.60), φ(x) = ξJa ( x) and ψ(x) = xφ (x). From the d.e. satisfied by the Bessel function it follows that (9.41) holds with m(x) = x, A(x) = 0, B(x) = 1 and C(x) = 14 (x − a2 ). Thus A(x), B(x) and C(x) are of the form assumed in (9.56) with a2 1 , γ1 = (9.63) 4 4 but now m(x) = x. In this case the kernel specific formulas based on Proposition 9.4.7 require modification as the commutator [D, (1 − ξKJ )−1 ] no longer has a simple kernel. Instead it is the commutator [M D, (1 − ξKJ )−1 ] which is used to give the desired formulas. α0 = α1 = 0,
β0 = 1, β1 = 0,
γ0 = −
P ROPOSITION 9.5.1 We have [M D, (1 − ξKJ )−1 ] 2 . = α1 Q(x)P (y) + P (x)Q(y) + β1 P (x)P (y) + γ1 Q(x)Q(y) − (−1)k ak R(x, ak )ρ(ak , y). k=1
Proof. Analogous to the calculation of [D, K] in the proof of Proposition 9.3.7 we find . [M D, K] = (xDx + yDy + 1)K(x, y) − a2 δ − (y − a2 )K(x, y) + a1 δ + (y − a1 )K(x, y). But from (9.41) and (9.56) we see that (xDx + yDy + 1)K(x, y) is given by the r.h.s. of the equation in the proof of Proposition 9.4.7. The stated result now follows from the workings of the proof of the final identity in Proposition 9.3.7.
Due to the similarity of this result with the result of Proposition 9.4.7, by inspecting the role of the commutator [D, K] in the derivations of the formulas for R(aj , aj ) and similar given in the previous section, we
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see that these formulas require only minor modification. P ROPOSITION 9.5.2 In the case m(x) = x the formulas of Propositions 9.4.10 and 9.4.12 for R(aj , aj ) and ∂R(aj , aj )/∂aj hold provided a factor of aj is inserted on the l.h.s. in front of R(aj , aj ) and a factor of ak is inserted in the summand on the r.h.s.. Similarly, the formulas of Proposition 9.4.11 for ∂pj /∂aj and ∂pj /∂aj hold provided a factor of aj is inserted on the l.h.s. and a factor of ak is inserted in the summand on the r.h.s. The analogue of the equations of Proposition 9.4.6 obtained for the Airy kernel can now be obtained. P ROPOSITION 9.5.3 Consider the Bessel kernel (9.60). In the notation of Definitions 9.3.1 and 9.4.1 with a1 = 0, a2 = s, p := p2 , q := q2 and R := R(s, s) we have sq = p + 41 qu, sp = 14 (a2 − s)q + 12 qv − 14 pu, (sR) = 14 q 2 , u = q 2 ,
v = pq.
(9.64)
Proof. The first and second equations follow from Proposition 9.4.11 with j = 2, modified according to Proposition 9.5.2 and with the substitutions (9.63). The third equation follows from Proposition 9.4.12, while the fourth and fifth equations follow immediately from equations (v) and (vi) of Proposition 9.4.6.
Making use of the equations (9.64), a single non-linear equation can be derived for the quantity R. P ROPOSITION 9.5.4 With R(t; ξ), set σ(t; ξ) = −tR. As with uh (t; a, 0; ξ) in (8.88), σ(t) satisfies the differential equation (8.90) with μ = 0, subject to the boundary condition (8.92). d to both Proof. A procedure similar to that used in the derivation of (9.44) can be followed. We begin by applying s ds
sides of the first equation in (9.64), and eliminating the first derivatives by using the second, fourth and fifth equations. This gives 1 1 2 1 s(sq ) = (a2 − s)q + (9.65) (u + 8v)q + sq 3 . 4 16 4 Next, by multiplying the first equation in (9.64) by 2q and using the fourth and fifth equations we see q 2 + 2sqq = 2v +
1 uu + u , 2
which implies
sq 2 = 2v +
1 2 u + u. 4
Substituting for u2 + 8v thus allows (9.65) to be rewritten as s(sq ) =
1 2 1 1 (a − s)q − uq + sq 3 . 4 4 2
(9.66)
Multiplying this equation by q , adding − 21 (sR) to the l.h.s. and − 81 q 2 to the r.h.s. (this is permitted by the third equation of (9.64)) and antidifferentiating shows −sR + (sq )2 =
1 2 1 (a − s − u)q 2 + sq 4 . 4 4
But from the third and fourth equations u = 4sR, so (q 2 − 1)sR =
1 2 1 (a − s)q 2 + sq 4 − (sq )2 . 4 4
(9.67)
Use of this equation, together with the third equation of (9.64), gives the stated differential equation for −tR.
The above working shows that the quantity q can also be characterized as the solution of a non-linear equation. First note substituting sR = u/4 in (9.67) gives an equation expressing (q 2 − 1)u in terms of q. Therefore after multiplying (9.66) by (1 − q 2 ) we can substitute for the term involving (q 2 − 1)u on the
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FREDHOLM DETERMINANT APPROACH
r.h.s. to obtain the differential equation 1 1 s(q 2 − 1)(sq ) = q(sq )2 + (s − a2 )q + sq 3 (q 2 − 2). (9.68) 4 4 The corresponding boundary condition, which follows from the boundary condition for σ(s; ξ) := −sR in Proposition 9.5.4 and the third equation in (9.64), is √ ξ q(s; ξ) ∼ a sa/2 . (9.69) s→0+ 2 Γ(1 + a) The equation (9.68) is related to a Painlev´e equation by a fractional linear transformation. Thus making the transformation 1 + y(x) , s = x2 , q(s) = (9.70) 1 − y(x) one finds that y(x) satisfies the Painlev´e V equation in (8.9) with α = −β = a2 /8, γ = 0 and δ = −2. Furthermore, making use of the third equation in (9.64) we deduce from (9.62) that 1 s s hard E2 ((0, s); ξ; a) = exp − (log )q 2 (t; ξ) dt , (9.71) 4 0 t which shows that q(t; ξ) determines E2hard ((0, s); ξ). 9.5.2 Symmetrical gap about the spectrum singularity In Section 8.3.5 the generating function E2s.s. ((0, t); ξ) for n eigenvalues in the interval (0, t) with a spectrum singularity at the origin was computed. In this section we will compute the generating function E2s.s. ((−t, t); ξ) for E2s.s. (n; ξ), the probability that there are exactly n eigenvalues in the symmetric interval (−t, t) about the spectrum singularity at the origin. Let pbulk 2,n.n. (0; t) denote the probability density function for the spacing between nearest neighbour eigenvalues in the bulk. This is distinct from pbulk (0; t) which represents the p.d.f. for the spacing between consec2 utive eigenvalues in the bulk, and thus the spacing distribution of the average of the spacing between the left neighbor and the right neighbor, only one of which being the nearest neighbor. It follows from the definitions that d s.s. pbulk E2 (0; (−t, t)) . 2,n.n. (0; t) = − dt a=1 We consider the integral operator in (9.15) with K(x, y) = K s.s. (x, y)|ρ=1/π . With R(x, y) denoting the kernel of the corresponding resolvent operator and R(t; ξ) := R(t, t), it follows from (9.26) that πρt E2s.s. (0; (−t, t)) = exp − 2 R(s; ξ) ds . (9.72) 0
Here, in the terminal of the integral t has been replaced by πρt to reinstate the general density, and use has been made of the fact that R(s; ξ) is even in s. According to (9.61), the kernel K s.s. (x, y)|ρ=1/π is of the type (9.25) with ξx ξx Ja+1/2 (x), Ja−1/2 (x), φ(x) = ψ(x) = (9.73) 2 2 and the equations (9.41) hold with m(x) = x,
α0 = −a, α1 = 0,
β0 = 0, β1 = 1,
γ0 = 0, γ1 = 1.
In relation to this kernel, the following coupled equations can be derived [218].
(9.74)
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CHAPTER 9
P ROPOSITION 9.5.5 In the notation of Definitions 9.3.1 and 9.4.1 with a1 = −t, a2 = t, p := p2 , q := q2 , R := R(t, t), relating to the kernel (9.61) with ρ = 1/π and a ∈ Z≥0 , we have (ii) tq = (−a + u − w)q + tp, (iv) (tR) = p2 + q 2 , (vi) w = 2p2 .
(i) tR = 2(−a + u − w)pq + t(p2 + q 2 ) + 2(pq)2 (iii) tp = −tq − (−a + u − w)p, (v) u = 2q 2 , Proof. For a ∈ Z≥0 , we have from (9.73) that φ(−x) = (−1)a−1 φ(x),
ψ(−x) = (−1)a ψ(x),
which gives K s.s. (x, y) = K s.s. (−x, −y) and thus ρ(x, y) = ρ(−x, −y). From Definition 9.3.1 we then see that q1 = (−1)a−1 q2 = (−1)a−1 q,
p1 = (−1)a p2 = (−1)a p,
v = 0.
(9.75)
Equation (i) can now be derived from (9.4.10) with j = 2, modified according to Proposition 9.5.2, by substituting (9.75) and (9.74). Equations (ii) and (iii) follow from Proposition 9.4.11, modified according to Proposition 9.5.2, after noting from equation (i) of Proposition 9.4.6 and (9.75) that R(−t, t) = (−1)a
pq . t
Equation (iv) follows from Proposition 9.4.11, modified according to Proposition 9.5.2, now with use being made of the general formula ˛ “ ∂ ∂ ” d ˛ − . (9.76) f (−t, t) = f (a1 , a2 )˛ dt ∂a2 ∂a1 a2 =−a1 =t Equations (v) and (vi) result from the universal equations (v) and (vii) of Proposition 9.4.6, together with (9.76).
From the equations of Proposition 9.5.5, a procedure similar to that used to derive Proposition 9.3.3 from Proposition 9.3.2 allows σ(s; ξ) = −sR(s/2; ξ) to be specified as the solution of a certain nonlinear equation. P ROPOSITION 9.5.6 The quantity σ(s; ξ) := −sR(s/2; ξ), where R(s; ξ) occurs in (9.72), satisfies the differential equation 2 =0 (9.77) (sσ )2 + 4(−a2 + sσ − σ) (σ )2 − a − (a2 − sσ + σ) subject to the boundary condition σ(s; ξ) ∼ −ξ
2(s/4)2a+1 Γ(1/2 + a)Γ(3/2 + a)
as s → 0.
(9.78)
Proof. Consider the equations of Proposition 9.5.5. Multiply (ii) by p, multiply (iii) by q, add and use (v) to obtain (pq) = p2 − q 2 =
1 (w − u ). 2
Antidifferentiating gives pq =
1 (w − u) 2
which together with (iv) allows (i) to be rewritten as tR = −2a(pq) − 2(pq)2 + t(tR) .
(9.79)
Another equation relating tR to pq is obtained by squaring (iv), squaring the formula (pq) = p2 − q 2 obtained above, and subtracting. This gives ((pq) )2 − ((tR) )2 = −4(pq)2 . (9.80) Solving (9.79) for pq (it follows from a small t expansion that the negative square root is to be taken) and (pq) , substituting in (9.80) and introducing the notation σ(2t) := −2tR gives (9.77). The boundary condition follows from the fact
403
FREDHOLM DETERMINANT APPROACH
˛ ˛ that R(t; ξ) ∼ ξK s.s. (t, t)˛
ρ=1/π
as t → 0, and the corresponding behavior of K s.s. (t, t) deduced from (7.49).
The second order second degree equation (9.77) involves a square root and so is not a σ-form Painlev´e equation (8.15). However (8.15) lists σPIII ; if instead one considers the Hamiltonian theory of PIII rather than PIII , the equation (9.77) can be recognized in this context and σ(s; ξ) identified as an auxiliary Painlev´e III Hamiltonian (see Exercises 9.5 q.1). 1. [548] The Hamiltonian for the PIII system, in contrast to the PIII system (recall Section 8.2.1), is given by [429] “ ” tHIII = 2q 2 p2 − 2tq 2 + (2v1 + 1)q − 2t p + (v1 + v2 )tq,
E XERCISES 9.5
where the parameters v1 , v2 relate to the parameters in the PIII equation of (8.9) by α = −4v2 ,
β = 4(v1 + 1),
(i) With hIII (t) := tHIII +
γ = 4,
δ = −4.
1 (2v1 + 1)2 8
show that hIII = −2q 2 + 2p + (v1 + v2 )q,
8(hIII − thIII ) = (4pq − 2v1 − 1)2 .
Differentiate the latter and use the Hamilton equations (8.23) to show −thIII = (4pq − 2v1 − 1)(2pq 2 + 2p − (v1 + v2 )q). (ii) Use the results of (i) to verify that for = ±1 “ (thIII )2 = [2(hIII − thIII )] 4(hIII )2 + 16[2(hIII − thIII )] q “ 1 ”“ 1 ”” −16(v2 − v1 − 1) 2(hIII − thIII ) − 16 v2 − v1 + ) . 2 2 (iii) For = ±, show that with v1 = −a −
1 , 2
v2 = a +
1 , 2
σ(t) = 2hIII
“t” − a2 4i
the equation in (ii) reduces to (9.77). 2. Make use of (7.53), (8.101), (9.15) and (9.61) to show ˛ “ ± ˛ ch E2O (s; ξ) = det 1 − ξK±1/2 ˛
(0,s)
” ,
(9.81)
ch |(0,s) is the integral operator on (0, s) with kernel (7.54). where K±1/2
9.6 EIGENVALUE EXPANSIONS FOR GAP PROBABILITIES 9.6.1 Commuting differential operators and E2 (n; J) We have seen that the logarithmic derivatives of the Fredholm determinants (9.15) satisfy second order nonlinear differential equations of Painlev´e type. In this section we will show that the eigenvalues of the integral operators KJscale can be obtained from the eigenvalues of certain second order linear differential operators. As a consequence, the asymptotic form of the eigenvalues for large values of the parameter determining J
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can be determined, and this in turn allows the asymptotic form of E2scale (n; J)/E2scale (0; J) — the ratio of the probability that there are n eigenvalues in J to the probability that there are no eigenvalues in J — to be computed [517]. To appreciate this last point, we first remark that for each of the three cases under consideration (no direct analysis of scale = s.s. is considered) KJscale has distinct eigenvalues which can be labeled by a non-negative integer and so ordered E2scale (n; J) =
∞ (1 − ξλl ).
(9.82)
l=0
Use of this formula in (8.1) gives E2scale (n; J) λj1 · · · λjn . = scale (1 − λj1 ) · · · (1 − λjn ) E2 (0; J) 0≤j <···<j 1
(9.83)
n
The utility of this equation lies in the fact that for large values of the parameter specifying J the eigenvalues decrease exponentially fast allowing the corresponding asymptotic form of the l.h.s. to be read off from the first term of the multiple sum, (j1 , . . . , jn ) = (0, 1, . . . , n − 1). Some qualitative features of the λl can readily be deduced. First we note an upper bound. For this we begin by observing that as |J| → 0, the eigenvalues of 1 − ξKJ must tend to 1, and thus each λl must tend to zero. These eigenvalues are a continuous function of the parameter determining J, so the fact that E2scale (J; 1) = E2scale (0; J) is a strictly positive quantity then implies 1 − λl > 0 for each l and for all values of the parameter specifying J. A lower bound on the λl can also be deduced. For this each integral operator KJscale , scale = bulk, soft, hard, is written as the square of another integral operator, thereby implying λl ≥ 0 for each l. Thus the following results hold [259], [519], [520]. ˜ bulk be the integral operator on (−1, 1) defined by P ROPOSITION 9.6.1 Let K (−1,1) 1 sin(t(x − y)) ˜ bulk [f ] = f (y) dy. K (−1,1) π(x − y) −1 We have ˜ bulk = (V bulk )† V bulk , K (−1,1) (−1,1) (−1,1)
bulk V(−1,1) [f ] =
t 1/2 eitxy f (y) dy. −1 2π 1
˜ soft be the integral operator on (0, ∞) defined by Let K (0,∞) ∞ ∞ ˜ soft [f ] = dy du Ai(x + u + t)Ai(y + u + t)f (y). K (0,∞) 0
0
We have
˜ soft = (V soft )2 , K (0,∞) (0,∞)
soft V(0,∞) [f ] =
∞
Ai(x + u + s)f (u) du. 0
˜ hard be the integral operator on (0, 1) defined by Let K (0,1) 1 1 √ √ ˜ hard [f ] = t K dy du Ja ( txu)Ja ( tyu)f (y). (0,1) 4 0 0 We have ˜ hard = (V hard )2 , K (0,1) (0,1)
hard V(0,1) [f ]
√ 1 √ t = Ja ( txu)f (u) du. 2 0
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FREDHOLM DETERMINANT APPROACH
. R
.
Proof. In general, if VJ = V (x, y), then VJ† VJ =
J
V (u, x)V (u, y) du, so the results are immediate.
˜ bulk , K ˜ soft , K ˜ hard result from K bulk |ρ=1/π , K soft , K hard respectively, with the The operators K (−1,1) (0,∞) (0,1) (−t,t) (t,∞) (0,t) kernels written in integral form as given in Exercises 7.1 q.1, after a simple change of variable to scale the t-dependence from the terminals of integration, a transformation which leaves unchanged the eigenvalues. The decompositions of Proposition 9.6.1 lead to the characterizations of the eigenvalues in (9.15) in terms of the eigenvalues of differential operators. This comes about because for each of the integral operators VJ , one can construct a second order linear differential operator of the form L=
d d α(x) + β(x) dx dx
(9.84)
which commutes with VJ . . P ROPOSITION 9.6.2 Let V[a1 ,a2 ] = V (x, y) and let L be given by (9.84). If α(a1 ) = α(a2 ) = 0 and ∂ ∂ ∂ ∂ α(x) + β(x) V (x, y) = α(y) + β(y) V (x, y), (9.85) ∂x ∂x ∂y ∂y then L commutes with V[a1 ,a2 ] . Proof. We have Z
“ ∂ ” ∂ α(x) + β(x) V (x, y)f (y) dy ∂x ∂x a Z 1a2 “ ” ∂ ∂ = α(y) + β(y) V (x, y)f (y) dy ∂y ∂y a1 Z a2 Z a2 “ ” ∂ ∂ β(y)V (x, y)f (y) dy V (x, y) α(y) f (y) dy + =− ∂y ∂y a1 a1 Z a2 “∂ ” ∂ = V (x, y) α(y) + β(y) f (y) dy, ∂y ∂y a1 a2
LV[a1 ,a2 ] [f ] =
where the second equality follows from (9.85), while the third and fourth equalities follow by integration by parts and use of the assumption that α(y) vanishes at y = a1 , a2 .
One can verify that the assumption on α(x) and condition (9.85) are satisfied with αbulk (x) = x2 − 1, αsoft (x) = x, αhard (x) = x(1 − x),
β bulk (x) = t2 x2 , β soft (x) = −(x + t)x, a2 β hard (x) = − 4x − tx 4 .
Consider first the bulk case. Then the corresponding eigenfunctions Sn (x; t) =: Sn (x) say, are known as the prolate spheroidal functions. They form an orthonormal set on (−1, 1) and are entire functions of x, which are real for real x (in the limit t → 0+ they reduce to the Legendre polynomials, up to normalization). For n even Sn (x) is an even function of x, while for n odd, Sn (x) is an odd function of x. bulk Because Lbulk commutes with V(−1,1) it follows that the Sn (x) =: Sn (x; t) defined as the eigenfuncbulk bulk tions of L are also the eigenfunctions of V(−1,1) . Using the parity property of the Sn (x) gives that the corresponding eigenvalue equation can be written t 1/2 1 cos xyt Sn (y; t) dy = μn (t)Sn (x; t), n even, 2π −1 t 1/2 1 i sin xyt Sn (y; t) dy = μn (t)Sn (x; t), n odd, 2π −1
406
CHAPTER 9
where the μn (t) are the eigenvalues. The fact that Sn (x) is real for real x implies that μn (t) is real for n even and is pure imaginary for n odd. It also shows that the Sn (x) are the eigenfunctions of V † , with eigenvalues ˜ bulk = (V bulk )† V bulk with corresponding eigenvalues |μn (t)|2 =: λn (t). μ ¯n (t), and thus of K (−1,1) (−1,1) (−1,1) The computation of the large t asymptotics of the λn (t) makes use of the Hellmann-Feynman formula [296], [174], which gives a relation between λn (t) and the eigenfunction Sn (x; t) evaluated at the special point x = 1. P ROPOSITION 9.6.3 Let K(x, y; t) be a symmetric (in x and y) kernel on (a1 , a2 ) depending smoothly on t. Let gj (x; t) be an eigenfunction, normalized so that a2 (gj (x; t))2 dx = 1, a1
with corresponding eigenvalue λj (t). We have a2 dx gj (x; t) λj (t) = a1
In particular, for the operator μj (t) =
bulk V(−1,1)
a2
dy gj (y; t) a1
∂ K(x, y; t). ∂t
of Proposition 9.6.2 and with a1 = −1, a2 = 1, this gives
μj (t) 2 λj (t) 2 Sj (1; t) or equivalently λj (t) = 2 Sj (1; t). t t
Proof. For the first statement, we follow [519]. Differentiating the eigenvalue equation for gj (x; t) gives Z
a2
a1
∂ K(x, y; t)gj (y; t) dy + ∂t
Z
a2
K(x, y; t) a1
∂ ∂ gj (y; t)dy = λj (t)gj (x; t) + λj (t) gj (x; t). ∂t ∂t
Multiplying both sides by gj (x; t), integrating over x, and using the normalization, the eigenvalue equation and the symmetry in x and y of K gives the desired result. The first equation of the second result follows simply from the first result by noting that ∂ itxy x ∂ itxy e e = , ∂t t ∂x and using integration by parts. The second part follows from the first part and the fact that λj = (−1)j μ2j .
Applying the WKB method of asymptotic analysis to be the eigenvalue equation for the differential oper√ ator L, it was shown by Fuchs [245] (after the identification Sj (x; t) = a−1/2 fj (ax; t), a = t, where fj is the eigenfunction studied in [245]) that the eigenfunctions have the asymptotic behavior ±Sj (1; t) = 2(3j+2)/2 π 1/4 (j!)−1/2 t(j+1/2)/2 e−t μj (t)(1 + o(1)) for j fixed and t → ∞. Substituting in the final equation of Proposition 9.6.3 gives 1 π 1/2 j+1 j+1/2 −2t λ 8 t (t) = e (1 + o(1)), j λ2j (t) j! and this, when integrated from t to ∞, implies π 1/2 j+1 ∞ j+1/2 −2t 4π 1/2 j j+1/2 −2t 1 − 1 ∼ 8 8 t t e dt ∼ e λj (t ) j! j! t (here use has been made of the fact that λj (t) ∼ 1 as t → ∞). Thus one has the following result [245]. ˜ bulk have the asymptotic behavior P ROPOSITION 9.6.4 The eigenvalues λj (t) of the integral operator K (−1,1) 1 − λj (t) ∼ 4π 1/2 8j (j!)−1 tj+1/2 e−2t for j fixed and t → ∞.
(9.86)
407
FREDHOLM DETERMINANT APPROACH
In the soft and hard edge cases the differential operator implied by (9.86) does not lead to special functions previously studied from this viewpoint. Nonetheless an analysis parallel to that sketched above leading to (9.86) can be carried out to obtain the following results [519], [520]. P ROPOSITION 9.6.5 Let i be fixed. One has √ 8 t 3/2 π 5i+3 t 3i/2+3/4 soft 2 − − ∼ exp − 1 − λi t→−∞ i! 2 3 2 √ 2π hard i+(a+1)/2 −2 t 4i+2a+2 1 − λi t ∼ e 2 . t→∞ Γ(a + i + 1)i!
(9.87) (9.88)
Substituting (9.86), (9.87), (9.88) for the first term of (9.83) gives the sought asymptotic forms for the l.h.s. of the latter. P ROPOSITION 9.6.6 Let G(x) denote the Barnes G-function. For n fixed E2bulk (n; t) E2bulk (0; t)
G(n + 1)π −n/2 2−n
2
∼
t→∞
−n/2
(πρt)−n
2
/2 nπρt
e
,
8n t 3/2 E2soft (n; (t, ∞)) G(n + 1) −3n2 /4 − ∼ , (−t/2) exp 2 3 2 E2soft (0; (t, ∞)) t→−∞ π n/2 2(5n +n)/2 E2hard (n; (0, t); a) E2hard (0; (0, t); a)
G(a + n + 1)G(n + 1) −n −n(2n+2a+1) −n2 /2−an/2 2n√t π 2 t e , G(a + 1)
∼
t→∞
(9.89)
where in the bulk case the setting of considering the interval J = (−t, t) and ρ = 1/π has been changed to the interval J = (0, t) with a general ρ by scaling t. 9.6.2 Further asymptotics The results of Proposition 9.6.6 can be supplemented by specifying the asymptotic behavior of E2bulk (0; t), E2soft (0, (t, ∞)) and E2hard (0; (0, t)) as the size of the interval gets large. This in turn is related to the asymptotic form of the Painlev´e transcendents specifying these quantities. A log-gas argument, given in Section 14.6, predicts that for some unspecified case dependent constants c, log E2bulk (0; t) ∼ −ct2 , t→∞
log E2soft (0, (t, ∞))
∼
t→−∞
log E2hard (0, (0, t)) ∼ −ct.
ct3 ,
t→∞
(9.90)
Rigorous determination of the first of these behaviors, which also specifies the proportionality constant as c = π 2 /8 (assuming ρ = 1), can be found in [543], [129]. The second result is just (9.49) and so for this c = 1/12. The behaviors (9.90) imply specific leading behaviors of the corresponding Painlev´e transcendents, and with this established the differential equations generate unique asymptotic expansions. The latter allow (9.90) to be extended [517], [519], [520]. P ROPOSITION 9.6.7 We have e−(πρt) /8 1 1− + O((πρt)−4 ) , 2 1/4 t→∞ 8(πρt) (πρt) csoft −|t|3 /12 3 2025 −9 1 + e + + O(t ) , E2soft (0; (t, ∞)) ∼ t→−∞ (|t|)1/8 26 |t|3 213 t6 √ 1 e−t/4+a t a 9a2 hard + O 3/2 E2 (0; (0, t); a) ∼ chard 1 + 1/2 + , t→∞ 128t ta2 /4 8t t 2
E2bulk (0; t)
∼
cbulk
(9.91)
where cbulk , csoft , chard have the values cbulk = 21/6 e3ζ
(−1)
,
csoft = 21/24 eζ
(−1)
,
chard =
G(1 + a) . (2π)a/2
(9.92)
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CHAPTER 9
Proof. It follows from (8.1) and the first equation in (8.111) that Z
πt
log E2bulk (0; t) = 0
σ(s; 1) ds, s
(9.93)
where σ satisfies the first equation in (8.112). To reproduce the leading behavior (9.90) with c as specified it must be that σ(s; 1) ∼ − s→∞
s2 . 4
In fact the differential equation satisfied by σ has a unique solution of the form σ(s; 1) = a2 s2 + a0 + a−2 s−2 + · · · which from the knowledge that a2 =
− 14
(a2 = 0)
is calculated as
σ(s; 1) = −
“1” s2 5 131 1 1 − − 2 − 4 − 6 +O 8 . 4 4 4s 2s 2s s
Substituting in (9.93) and reinstating the general density ρ gives the first expansion in (9.91). The value of the undetermined constant τbulk can be deduced from a result in the theory of Toeplitz determinants (see Exercise 9.6 q.1). A rigorous determination is also known [363], [163], [127]. An analogous strategy applies in the soft and hard edge cases. Thus according to (9.43) and the first equation in (8.93) Z ∞ Z t ˛ ds ˛ log E2soft (0; (t, ∞)) = − R(s; 1) ds, log E2hard (0; (0, t)) = − σIII (s)˛ μ=0 , (9.94) s t 0 ξ=1 the σPIII equation in (8.15) with v1 = v2 = where R(s; 1) satisfies the differential equation (9.44) and σIII (s) satisfies P 2−j , which calculation shows is a. The first of these differential equations has a unique solution of the form ∞ j=0 a2−j s given by “1” 1 9 s2 − + R(s) ∼ + O , 4 8s 64s4 s7 while the second has a unique solution of the form σ(s) = c1 s + c1/2 s1/2 + c0 + c−1/2 s−1/2 + · · · provided the sign in √ a2 = ±a is fixed. Choosing the minus sign (this makes sense physically from the resulting asymptotic expression for log E2 (0; (0, s))) allows the expansion σ(s) =
a a −1/2 a2 −1 s a2 − s1/2 + + s s + ··· + 4 2 4 16 16
(9.95)
to be generated. Substituting as appropriate in (9.94) gives the final two expansions in (9.91), up to the constants. In the soft edge case the constant follows from the result [126], [30] that for c < 0 Z ∞ Z c “ 1 ” 1 3 1 1 1 |c| + log |c|. R(y; 1) dy + R(y; 1) − y 2 + dy = − log 2 − ζ (−1) + 4 8y 24 12 8 c −∞ For a ∈ Z≥0 , the constant in the hard edge case can be deduced from the integral formula (8.96), as is done for a more general integral formula in (13.52) below. It’s validity for −1 < a < 1 has been proved in [162].
The expansions (9.90) and (9.92) in the hard edge case with a = ± 21 allow us to deduce the large t expansions of Eβbulk (n; t) for β = 1 and 4 [49]. First the case n = 0 will be considered. According to (8.139), E1bulk (0; s) = E2hard (0; (0, (πs/2)2 )|a=−1/2 , where on the l.h.s. the density has been set equal to 1. It thus follows from the final formula in (9.91) that 2 1 e−(πρt) /16−πρt/4 bulk 1/4 . (9.96) 1 + O E1 (0; t) ∼ (2π) G(1/2) t→∞ t (πρt/2)1/8
409
FREDHOLM DETERMINANT APPROACH
Furthermore, it follows from (8.159) that E4bulk (0; t) ∼
t→∞
1 hard E (0; (0, (πt)2 )|a=1/2 . 2 2
(9.97)
The final formula in (9.91) then gives 1 G(3/2) e−(πρt) /4+πρt/2 . 1 + O t→∞ 2(2π)1/4 t (πρt)1/8 2
E4bulk (0; t) ∼
(9.98)
The asymptotic formulas for Eβbulk (0; t), β = 1, 2, 4 are all consistent with the form e−β(πρt) /16+(β/2−1)πρt/2 . (πρt)(1−(2/β)(1−β/2)2 )/4 2
Eβbulk (0; t) ∼ cβ t→∞
(9.99)
Valko and Vir´ag [525] have used the stochastic sine equation (13.179) below to prove this form for general β > 0. In relation to the general n case, the formula (8.153) tells us that hard n even E2 (n/2; (πρt/2)2 )|a=−1/2 , bulk E1 (n; t) ∼ E2hard ((n − 1)/2; (πρt/2)2 )|a=1/2 , n odd. t→∞ Making use of both (9.90) and (9.92) it follows from this that E1bulk (n; t) G(n/2 + 1/2)G(n/2 + 1) −n/2 −n2 /2 π ∼ 2 (πρt/2)−n(n−1)/4 enπρt/2 . G(1/2) E1bulk (0; t) t→∞ Similarly, from (8.159) we have
1 hard E2 (n; (πρt)2 ) . t→∞ 2 a=1/2
E4bulk (n; t) ∼
(9.100)
(9.101)
Recalling (9.97) and making use of (9.90) this gives 2 E4bulk (n; t) G(n + 3/2)G(n + 1) −n −2n(n+1) π 2 ∼ (πρt)−n −n/2 e2nπρt . bulk t→∞ G(3/2) E4 (0; t)
(9.102)
We see from Proposition 9.6.6, (9.100) and (9.102) that the asymptotic formulas for Eβbulk (n; t)/Eβbulk (0; t), β = 1, 2, 4 are all consistent with the form Eβbulk (n; t)
eβnπρt/2
∼ cβ,n , (πρt)βn2 /4+(β/2−1)n/2 Eβbulk (0; t) t→∞
(9.103)
which can in fact be obtained from a log-gas analysis (see Section 14.6.2).
9.6.3 Some gap probabilities at β = 1 with an evenness symmetry In Section 8.4.2 the equation (8.149) relating gap probabilities for the COE (β = 1 quantities) to gap probabilities in the JUE (β = 2 quantities) was shown to hold for ξ = 1. Here (8.149), along with some companion identities, will be derived for general ξ via a calculation which uses evenness symmetry as well as the eigenvalues and eigenvectors of an underlying Fredholm integral operator [395], [197]. First we will derive a determinant formula for the generating function ∞ ∞ 2N 1 (l) E2N,1 ((−t, t); ξ; w1 ) := dx1 · · · dx2N w1 (xl )(1 − ξχ(−t,t) ) |xk − xj | Zˆ2N −∞ −∞ l=1
1≤j
(9.104)
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CHAPTER 9
in the cases that w1 (x) is an even classical weight function, and thus according to (6.100) one of ⎧ 2 Gaussian, ⎨ e−x /2 , w1 (x) = (1 − x2 )(a−1)/2 (−1 < x < 1), symmetric Jacobi, ⎩ Cauchy. (1 + x2 )−(α+1)/2 ,
(9.105)
For this purpose use will be made of the polynomials {pj (x)}j=0,1,... orthogonal with respect to the corresponding classical weight w2 (x) in (5.56), as well as of the eigenvalues and eigenvectors of the integral operator on (−t, t) with kernel K + (x, y) := (w2 (x)w2 (y))1/2
N −1 l=0
p2l (x)p2l (y) . (p2l , p2l )2
According to the proof of Proposition 5.2.2 there are exactly N eigenfunctions. These are of the form (2N ) (2N ) (w2 (x))1/2 q2j (x, t), where each q2j an even polynomial of degree 2N having the structure (2N )
q2j
(x; t) =
N −1 l=0
1 cjl p2l (x), (p2l , p2l )2
(9.106)
with [cjl ]j,l=0,...,N −1 =: C
(9.107)
a real orthogonal matrix. Furthermore t (2N ) (2N ) (2N ) w2 (x)q2j (x; t)q2k (x; t) dx = ν2j (t)δj,k ,
(9.108)
−t
(2N )
(2N )
where {ν2j (t)}j=0,1,...,N −1 are the eigenvalues of the integral operator. The {q2j satisfy the orthonormality condition (2N )
(q2j
(2N )
(x, t), q2k
(x; t))2 = δj,k ,
(x, t)}j=0,...,N −1 also
j, k = 0, . . . , N − 1,
(9.109)
as can be seen from the orthogonality of {p2l (x)}l=0,...,N −1 with respect to this inner product and the structure (9.106). P ROPOSITION 9.6.8 Suppose w1 (x) is given by one of (9.105), and w2 (x) by its companion in (5.56). Let (2N ) (2N ) {ν2j (t)}j=0,1,...,N −1 and {q2j (x; t)}j=0,...,N −1 be specified as in the above text. We have , (9.110) E2N,1 ((−t, t); ξ; w1 ) = det A2j−1,2k j,k=1,...,N
where
(2N ) A2j−1,2k = δj,k 1 − (2ξ − ξ 2 )ν2(j−1) (t) t (2N ) 2 w2 (t) (2N ) q (t) w1 (x)q2j−2 (x; t) dx. +(ξ − ξ ) w1 (t) 2k−2 −t
(9.111)
Proof. Let {Rj (x)}j=0,1,... be the skew orthogonal polynomials with respect to the skew inner product (6.61), setting e−V (x) = w1 (x) therein, and let {pj (x)}j=0,1,... be the monic orthogonal polynomials with respect to w2 (x). According to (6.98), (6.99), (6.97) and (5.65) R2j (x) = p2j (x),
R2j+1 (x) = −
” a2j d “ w2 (x) p2j (x) , w1 (x) dx w1 (x)
rj =
(p2j , p2j )2 , a2j
(9.112)
411
FREDHOLM DETERMINANT APPROACH
8 < 1, ak = k + 1 + a, : 2N + a − 1 − k,
where
Gaussian, symmetric Jacobi, Cauchy.
Substituting (9.112) in the Vandermonde type identity Y (xk − xj ) = det[Rj−1 (xk )]j,k=1,...,2N 1≤j
(cf. (9.71)) we see that 2N Y j=1
=
Y
w1 (xj )
|xk − xj |
1≤j
2
3 w1 (xk )p2j (xk ) ” 5 “ (−a2j ) det 4 d w2 (xk ) p2j (xk ) j=0 dxk w1 (xk ) j=0,...,N −1
N−1 Y
Y
sgn(xk − xj ).
(9.113)
1≤j
k=1,...,2N
This identity holds true with the replacements (N)
a2j → r2j ,
p2j (xk ) → q2j (xk ; t)
(9.114)
on the r.h.s. To see this first rearrange rows in the determinant so they contain in order w1 (xk )p0 (xk ), . . . , w1 (xk )pN−2 (xk ),
” ” d “ w2 (xk ) d “ w2 (xk ) p0 (xk ) , . . . , pN−2 (xk ) . dxk w1 (xk ) dxk w1 (xk )
Then multiplying both sides of (9.113) by
» det
DC 0N
0N DC
– ,
where C is given by (9.107) and “ ” 1 1 D := diag p ,..., p , (p0 , p0 )2 (pN−1 , pN−1 )2 gives (9.113) back but with the substitutions (9.114). Recalling the identity (6.62) we see the r.h.s. of (9.113) with the substitutions (9.114) can be written N−1 Y
rj det[φj−1 (xk )]j,k=1,...,2N Pf[h(xk , xj )]j,k=1,...,2N
j=0
with (2N)
φ2j (x) = w1 (x)q2j
φ2j+1 (x) = −
(x; t),
” d “ w2 (x) (2N) q2j (x; t) , dx w1 (x)
h(x, y) = sgn(y − x).
(9.115)
Substituting this for the integrand of (9.104) and making use of the general integration formula (6.73) and the identity (6.12) shows that N−1 h i1/2 2N 2N ! Y E2N,1 ((−t, t); ξ; w1 ) = rj det Aj,k , (9.116) ˆ j,k=1,...,2N Z2N j=0 where Aj,k =
1“ 2
Z
∞
−∞
Z −ξ
(2N)
Because each of w1 (x), w2 (x), q2j
t
“Z
”
∞
dx −t
−∞
Z −ξ
t
−t
”
dy φj−1 (x)φk−1 (y)sgn(y − x).
(9.117)
(x; t) is even we see from (9.115) that φ2j (x) is even while φ2j+1 (x) is odd. It
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CHAPTER 9
then follows from the definition (9.117) that A2j,2k = A2j−1,2k−1 = 0
for
j, k = 1, . . . ,
N . 2
Interchanging rows and columns so that the zero elements are all in the top left and bottom right blocks, and noting A2k,2j−1 = −A2j−1,2k gives the stated result (9.110) (here Zˆ2N in (9.116) has been eliminated by noting that both sides must equal unity when ξ = 0).
+ The generating function E2N,1 in (8.149) is by definition a generating function for the sum of probabilities COE COE E2N,1 (2n; (−θ, θ)) + E2N,1 (2n − 1; (−θ, θ)). Such probabilities can be computed from the generating func-
tion (9.104) according to the general formula (8.1). Applying this formula to (9.110) allows the following results to be derived. P ROPOSITION 9.6.9 In the setting of Proposition 9.6.8, with qj (x; t) =: qj (x), we have E2N,1 (2n; (−t, t); w1 ) =
N −1
(1 −
ν2j1 (t) · · · ν2jn (t)
0≤j1 <j2 <···≤jn ≤N −1
(1 − ν2j1 (t)) · · · (1 − ν2jn (t))
(2N ) ν2j (t))
j=0
(2N )
(2N )
(2N )
(2N )
(2N ) n w2 (t) q2jα (t) t (2N ) × 1− w (x)q (x) dx , 1 2jα w1 (t) α=1 ν (2N ) (t) −t
(9.118)
2jα
E2N,1 (2n − 1; (−t, t); w1 ) =
N −1
(1 −
ν2j1 (t) · · · ν2jn (t)
0≤j1 <j2 <···≤jn ≤N −1
(1 − ν2j1 (t)) · · · (1 − ν2jn (t))
(2N ) ν2j (t))
j=0
(2N )
(2N )
(2N )
(2N )
(2N ) n w2 (t) q2jα (t) t (2N ) × w1 (x)q2jα (x) dx. w1 (t) α=1 ν (2N ) (t) −t
(9.119)
2jα
Thus E2N,1 (2n; (−t, t); w1 ) + E2N,1 (2n − 1; (−t, t); w1 ) =
N −1
(2N )
(1 − ν2j
j=0
(t))
ν2j1 (t) · · · ν2jn (t)
0≤j1 <j2 <···≤jn ≤N −1
(1 − ν2j1 (t)) · · · (1 − ν2jn (t))
= EN,2 (n; (0, t2 ); x−1/2 w2 (x1/2 )χx>0 ).
(2N )
(2N )
(2N )
(2N )
(9.120)
Proof. The general formula (8.1) tells us that to compute (9.118) we must differentiate (9.116) 2n times with respect to ξ and then set ξ = 1. Now each term in the determinant of (9.116) is quadratic in ξ. If we were to differentiate two distinct rows once with respect to ξ and then set ξ = 1, the two rows would have the form Z t w2 (t) (2N) (2N) q2k−2 (t) − w1 (x)q2j−2 (x) dx w1 (t) −t and thus be proportional to each other, so the determinant will vanish. Hence for a nonzero contribution no two distinct rows are to be differentiated just once. Since a total of 2n differentiations are required, it follows that n distinct rows j1 , . . . , jn say must be differentiated twice. We see immediately from (9.117) that upon setting ξ = 1 the remaining rows are only nonzero on the diagonal. Expanding by these elements, and multiplying by the combinatorial factor (2n − 1)!!
413
FREDHOLM DETERMINANT APPROACH
to account for the number of ways of forming n pairs from 2n rows shows E2N,1 (2n; (−t, t); w1 ) =
N−1 Y
(2N)
(1 − ν2j
X
ν2j1 (t) · · · ν2jn (t)
0≤j1 <j2 <···≤jn ≤N−1
(1 − ν2j1 (t)) · · · (1 − ν2jn (t))
(t))
j=0
(2N)
(2N)
(2N)
(2N)
(2N) Z h i w2 (t) q2jβ (t) t (2N) × det δjα ,jβ − w1 (x)q2jα (x) dx . (2N) w1 (t) ν2j (t) −t α,β=1,...,n β
The determinant in this expression is a special case of the general determinant det[δi,j − xi yj ]i,j=1,...,n = 1 −
n X
x j yj
(9.121)
j=1
(whenever A has rank 1, det(1 − A) = 1 − Tr A), as used in (1.137), so (9.118) results. The only difference in the derivation of (9.119) is that one row is differentiated with respect to ξ exactly once, while n − 1 rows are differentiated twice. This gives E2N,1 (2n − 1; (−t, t); w1 ) = −
N−1 Y
(2N)
(1 − ν2j
j=0
X
(t))
0≤j1 <j2 <···≤jn ≤N−1
(2N)
(2N)
ν2j1 (t) · · · ν2jn (t) (1 −
(2N) ν2j1 (t)) · · · (1
(2N)
− ν2jn (t))
Z ˛ i h d w2 (t) q2jβ (t) t ˛ (2N) w (x)q (x) dx det δjα ,jβ − ˛ . 1 2jα d w1 (t) ν (2N) −t α,β=1,...,n =1 (2N)
×
2jβ
The determinant can again be calculated using (9.121), and (9.119) results. It remains to justify the second equality in (9.120). Now application of the theory of Exercises 9.6 q.3 tells us that N−1 Y
(2N)
(1 − ξν2j
(t)) = EN,2 ((0, t2 ); ξ; x−1/2 w2 (x1/2 )χx>0 )
j=0
so we can identify the r.h.s. of the first equality in (9.120) as ˛ (−1)n ∂ n ˛ 2 −1/2 1/2 E ((0, t ); ξ; x w (x )χ ) ˛ . N,2 2 x>0 n! ∂ξ n ξ=1 Application of (8.1) then gives the second equality in (9.120).
To apply (9.120) to the COE, note that analogous to (8.126) it follows from (2.51) that θ θ (0) COE ((−θ, θ); ξ) = EN,1 − tan , tan ; ξ; (1 + x2 )−(N +1)/2 , EN,1 2 2 or equivalently θ θ COE ; (1 + x2 )−(N +1)/2 . n; (−θ, θ = EN,1 n; − tan , tan EN,1 2 2
(9.122)
Because w1 (x) = (1 + x2 )−(N +1)/2 is an example of the Cauchy weight in (9.105) and the interval (− tan θ2 , tan θ2 ) is symmetric about the origin, for N even (9.120) applies and we conclude COE COE E2N,1 (2n; (−θ, θ)) + E2N,1 (2n − 1; (−θ, θ)) θ ; x−1/2 (1 + x)−(N +1)/2 χx>0 = EN,2 n; 0, tan2 2 2 θ ; x−1/2 (1 − x)1/2 χ0<x<1 = EN,2 n; 0, sin 2
414
CHAPTER 9
where the second equality follows by a change of variables. This equation is equivalent to (8.149) in the case + of E2N,1 . E XERCISES 9.6
1. [219] Consider the one-component log-gas at β = 2 on a circle with circumference length L. Suppose that the circle is divided into M equally spaced lattice points, and that the N particles (N ≤ M ) are constrained to lie on these points. (i) Use the method of the proof of Proposition 5.2.1 and the orthonormality relation M −1 1 X 2πin(k−j)/M e = δk,j M n=0
for |k − j| < M , k, j ∈ Z to show that the n-particle correlation function is given by ρ(n) (l1 , . . . , ln ) = det
h sin πN (l − l )/M i j k . L sin π(lj − lk )/M j,k=1,...,n
(ii) Modify the derivation of the formula (9.1) for Eβ (0; (a1 , a2 )) to show that in general for a one-dimensional lattice gas excluded from the lattice sites l = m1 , m1 + 1, . . . , m2 Eβ (0; [m1 , m2 ]) = 1 +
m2 m2 N X X (−1)n “ L ”n X ··· ρ(n) (l1 , . . . , ln ), n! M n=1 l =m l =m 1
n
1
1
and hence conclude that for the present system h sin πN (j − k)/M i . E2 (0; [m1 , m2 ]) = det δj,k − M sin π(j − k)/M j,k=m1 ,...,m2 With m := m2 − m1 , in the thermodynamic limit N, M, L → ∞, τ := L/M and ρ := N/L fixed, rewrite this result to read h 1 Z 2π−πτ ρ i ei(j−k)θ dθ . E2 (0; m) = det 2π πτ ρ j,k=1,...,m (iii) Widom [542] has proved the following theorem regarding the large behavior of Toeplitz R 2π m asymptotic 1 determinants of the type occurring in (ii): Let Dm [f ] := det[ 2π f (θ)ei(j−k)θ dθ]j,k=1,...,m , where f 0 satisfies f (θ) = f (2π − θ), is supported on a closed arc α ≤ θ ≤ 2π − α and is positive along this arc. Then (assuming some mild restrictions on the derivative of f ), as m → ∞ “ ” “ α 1 α” (9.123) + 3ζ (−1) 21/12 T [F ], Dm [f ] ∼ exp m2 log cos + mG[f ] − log m sin 2 4 2 P∞ ikθ where F (θ) = f (2 cos−1 (cos α/2 cos θ)), and if log g(θ) = , then G[g] := eg0 and k=−∞ gk e 1
T [g] := e 4
P∞
kgk g−k
. Use this theorem in the final formula of (ii) above to show that for large m “ “ ” 1 πρτ ” πρτ 1 − log m sin log 2 + 3ζ (−1) E2 (0; m) ∼ exp m2 log cos + 2 4 2 12 k=1
and thus deduce the first equation in (9.91) by taking the limit τ → 0 with mτ = x. (For a rigorous justification of this argument, see [363].) 2. Use the identity (8.162) and the third asymptotic expansion in (9.91) to deduce that ˛ ˛ E2s.s. (0; (−t, t))˛
” e−t /2+2at “ a + ··· . 1+ 2 +1/4 a 4t t 2
∼ chard |a−1/2 chard |a+1/2
ρ=1/π t→∞
Show that in “ the ” case a = 0 this is consistent with the first asymptotic expansion in (9.91) (this requires the special 1 value of G 2 given above (4.182)).
415
FREDHOLM DETERMINANT APPROACH
3. [197] The objective of this exercise is to interpret the factorization (8.123) in terms of the eigenvalues of the underlying integral operator. In particular we aim to show that if in (8.123) (0)
EN,2 (J{t2i ,t2i+1 } ; ξ; g(x)) =
N−1 Y
(1 − ξλj (J{t2i ,t2i+1 } )),
j=0
where {λj (J{t2i ,t2i+1 } )} are the eigenvalues of the integral operator on J{t2i ,t2i+1 } with kernel KN (x, y) = (g(x)g(y))1/2
N−1 X l=0
pl (x)pl (y) (pl , pl )2
(here {pl (x)} are the monic orthogonal polynomials corresponding to the weight g(x)), then (0)
+ E[(N+1)/2],2 (J{t 2
2 2i ,t2i+1 }
; ξ; x−1/2 g(x1/2 )χx>0 ) =
(0)
+ E[N/2],2 (J{t 2
2 2i ,t2i+1
N−1 Y
(1 − ξλj (J{t2i ,t2i+1 } )),
j=0 j even
; ξ; x1/2 g(x1/2 )χx>0 ) = }
N−1 Y
(1 − ξλj (J{t2i ,t2i+1 } )).
j=0 j odd
(i) Set N1 = [(N + 1)/2]. Deduce from the proof of Proposition 8.4.1 that + EN1 ,2 (0; J{t ; ξ; x−1/2 g(x1/2 )χx>0 ) 2 ,t2 2i 2i+1 } Z Z Z “Z ∞ ” ” 1“ ∞ = −ξ −ξ dx1 w2 (x1 ) · · · dxN1 w2 (xN1 ) C −∞ −∞ J{t ,t J{t ,t 2i 2i+1 } 2i 2i+1 } Y × (x2k − x2j )2 . (0)
1≤j
(ii) Let ρ+ (k) (x1 , . . . , xk ) denote the k-point correlation function associated with the p.d.f. N1 1 Y g(xl ) C l=1
Y
(x2k − x2j )2 .
(9.124)
1≤j
Proceed as in the derivation of Propositions 5.1.1 and 5.1.2 to show h i + (x , . . . , x ) = det K (x , x ) ρ+ 1 α k β N (k)
α,β=1,...,k
,
+ KN (x, y) := (g(x)g(y))1/2
N1 X p2l (x)p2l (y) (p2l , p2l )2 l=0
and use the equality between (9.14) and (9.15) to conclude from this that (0)
+ E[(N+1)/2],2 (J{t 2
2 2i ,t2i+1 }
+ ; ξ; x−1/2 g(x1/2 )χx>0 ) = det(1 − ξKN ),
+ + is the integral operator on J{t2i ,t2i+1 } with kernel KN (x, y). Now use the fact that the eigenfuncwhere KN tions of the latter are the even eigenfunctions of the integral operator on J{t2i ,t2i+1 } with kernel KN (x, y) to deduce the first of the sought results. Apply a similar strategy to derive the second result.
416
CHAPTER 9
9.7 THE PROBABILITIES Eβsoft (n; (s, ∞)) FOR β = 1, 4 In the bulk the probability E1bulk (0; (0, s)) was evaluated via the mapping (8.139) to a gap probability for a β = 2 quantity. For the probability E1soft (0; (s, ∞)) an analogous result is the formula 2 odd(soft)2 E1soft (0; (s, ∞)) = E1 (0; (s, ∞)), (9.125) 2
where E odd(soft) refers to the system with k-point correlation (7.170). Since the latter is a k × k determinant, (9.15) gives 2 E1soft (0; (s, ∞)) = det 1 − (K soft + A ⊗ B) , (9.126) where K soft is the integral operator on (s, ∞) with kernel ∞(9.39) while A is the operator which multiplies by Ai(x), while B is the integral operator with kernel 0 Ai(y − v) dv. (Strictly speaking (9.126) is illdefined due to B not decaying for y → −∞; this can be overcome by redefining A and B in such a way that the corresponding determinant is unchanged. However this technicality does not effect the working formula (9.128) below.) Using (9.126), E1soft (0; (s, ∞)) can be evaluated in terms of the Painlev´e II transcendent q(s) occurring in the expression (9.46) for E2soft (0; (s, ∞)) [521], [195]. P ROPOSITION 9.7.1 For the infinite GOE at the soft edge, scaled as in (7.11), and with q the Painlev´e II transcendent satisfying (9.45) subject to the boundary condition (9.47) with ξ = 1, 2 ∞ soft soft E1 (0; (s, ∞)) = E2 (0; (s, ∞)) exp − q(x) dx . (9.127) s
Proof. Factoring out det(1 − K soft ) from (9.126) and making use of (9.15) shows “
E1soft (0; (s, ∞))
”2
“ ” = E2soft (0; (s, ∞)) det 1 − (1 − K soft )−1 A ⊗ B Z ∞ “ ” (1 − K soft )−1 A[y]B(y) dy , = E2soft (0; (s, ∞)) 1 −
(9.128)
s
where the second equality follows from the fact that (1 − K soft )−1 A[y] is the eigenfunction of the operator (1 − K soft )−1 A ⊗ B, so the eigenvalue is Z ∞ (1 − K soft )−1 A[y]B(y) dy. s
Introducing the notation of Section 9.4, we put φ(x) = A(x) = Ai(x) and Q(x) = (1 − K soft )−1 A[x] so that Z ∞ Z y Z ∞ (1 − K soft )−1 A[y]B(y) dy = dy Q(y) dv φ(v) =: u . (9.129) s
−∞
s
Note from (9.128) that with the notation (9.129) we have ”2 “ E1soft (0; (s, ∞)) = E2soft (0; (0; (s, ∞))(1 − u ).
(9.130)
The strategy now is to derive coupled differential equations for u and the quantity Z ∞ Z y dy ρ(s, y) dv φ(v). q :=
(9.131)
s
−∞
According to Proposition 9.3.10 we have ” “ ∂ Q(y) = −qR(y, s) = −q − δ + (y − s) + ρ(s, y) . ∂s
(9.132)
417
FREDHOLM DETERMINANT APPROACH
p.d.f.
mean
variance
skewness
kurtosis
psoft 1 (0; t) psoft 1 (1; t) psoft 1 (2; t) psoft 1 (3; t) psoft 1 (4; t) psoft 1 (5; t)
−1.20653 35745 −3.26242 79028 −4.82163 02757 −6.16203 99636 −7.37011 47042 −8.48621 83723
1.60778 10345 1.03544 74415 0.82239 01151 0.70315 81054 0.62425 23679 0.56700 71487
0.29346 45240 0.16550 94943 0.11762 14761 0.09232 83954 0.07653 98210 0.06567 07705
0.16524 29384 0.04919 51565 0.01977 46604 0.00816 06305 0.00245 40580 −0.00073 42515
soft Table 9.2 Statistical properties of psoft ˜soft 1 (n; t) for various n. Note from (9.142) that p 4 (n; t) = p1 (2n + 1; t), so the soft second, fourth and sixth lines give the statistical properties of {˜ p4 (n; t)}n=0,1,2 .
Making use of this formula, and recalling q := Q(s), shows Z s Z ∞ “ ”Z y ∂ d φ(v) dv + dy dv φ(v) = −qq . u = −q Q(y) ds ∂s −∞ s −∞
(9.133)
To derive the corresponding formula for the derivative of q we will make use of the formula “∂ ∂ ∂ ” ρ(x, y) = −Q(x)Q(y), + + ∂s ∂x ∂y which is a consequence of Proposition 9.3.7 with the substitutions (9.57) and (9.35). This shows Z ∞ Z ∞ Z y Z y d ∂ dy dv φ(v) − q dy Q(y) dv φ(v) q = − ρ(s, y) ds ∂y s −∞ s −∞ Z ∞ ρ(s, y)φ(y) dy − qu = q(1 − u ). =
(9.134)
(9.135)
s
As q is known, the system of equations (9.133) and (9.135) fully determines u and q once boundary conditions are specified. Now Q(y) is smooth, so we see from (9.129) that u → 0
(9.136)
as s → ∞. On the other hand ρ(s, y) = δ + (s − y) + R(s, y), where R(s, y) is smooth, so for s → ∞ Z ∞ Z ∞ φ(v) dv = Ai(v) dv = 1. q ∼ −∞
(9.137)
−∞
The unique solution of the coupled equations (9.133) and (9.135) satisfying (9.136) and (9.137) is then verified to be u = 1 − e−μ , Z
where μ :=
q = e−μ ,
(9.138)
∞
q(x) dx. s
Substituting the evaluation of us from (9.138) in (9.130) gives (9.127), as required.
Recalling (9.46) and the analogue of (8.86) for β = 1, ζ = 1, it follows from (9.127) that 1 ∞ d 1 ∞ soft 2 exp − (x − t)q (x) dx − q(x) dx , p1 (0; t) = dt 2 t 2 t
(9.139)
where q is as in (9.127). Using this formula, statistical characterizations of psoft 1 (0; t) have been computed [449] as in Table 9.2.
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CHAPTER 9
To compute E1soft (n; (s, ∞)) for n ≥ 1, and to compute ˜ soft (n; (s, ∞)) := lim E4 n; (2N )1/2 + E 4
s −x2 , ∞ ; e ; N/2 , N →∞ 21/2 N 1/6 (this particular definition is made to make connection with (6.124)) we first make note of some inter-relationships between {Eβsoft (n; (s, ∞))}n=0,1,... for β = 1, 2, 4. P ROPOSITION 9.7.2 We have E2soft (n; (s, ∞)) =
2n+1
E1soft (2n + 1 − l; (s, ∞)) E1soft (l; (s, ∞)) + E1soft (l − 1; (s, ∞))
(9.140)
l=0
and ˜ soft (n; (s, ∞)) = E soft (2n; (s, ∞)) + E soft (2n + 1; (s, ∞)). E 4 1 1
(9.141)
The latter equation is equivalent to soft p˜soft 4 (n; (s, ∞)) = p1 (2n + 1; (s, ∞)).
(9.142)
Proof. The identity (9.140) is deduced from (6.138) in an analogous way to how (8.143) is deduced from (6.154). The identity (9.141) follows from (6.142) by an argument similar to that used to deduce Proposition 8.4.6.
We remark that with the notation odd f denoting the odd powers in 1 − ξ of the expansion of f (ξ) about 1−ξ
ξ = 1, and with ξ¯ := 2ξ − ξ 2
(9.143)
so that 1 − ξ¯ = (1 − ξ)2 , it follows from (9.140) that in relation to the corresponding generating functions 2 ¯ = odd E soft ((s, ∞); ξ) (2 − ξ) . (9.144) (1 − ξ)E2soft ((s, ∞); ξ) 1 1−ξ
Now, it follows from (9.140) with n = 0 that 1 E soft (0; (s, ∞)) E1soft (0; (s, ∞)) + 2soft . E1soft (1; (s, ∞)) = 2 E1 (0; (s, ∞)) Making use of the evaluation (9.127) shows 1 ∞ 2 E1soft (1; (s, ∞)) = E2soft (0; (s, ∞)) cosh2 q(x) dx , 2 s while it follows from this, (9.46) and (8.3) that 1 ∞ 1 ∞ d soft soft 2 exp − (x − t)q (x) dx cosh q(x) dx . p1 (1; t) = p2 (0; t) + dt 2 t 2 t
(9.145)
(9.146)
(9.147)
The formula (9.147) was used in the computation of the corresponding statistical data of Table 9.2. However, the equation (9.140) is not sufficient to compute E1soft (n; (s, ∞)) for n ≥ 2. It must be supplemented by the following [198]. P ROPOSITION 9.7.3 We have odd(soft)2
E1
(n; (s, ∞)) =
2n l=0
E1soft (2n − l; (s, ∞)) E1soft (l; (s, ∞)) + E1soft (l − 1; (s, ∞)) ,
(9.148)
419
FREDHOLM DETERMINANT APPROACH
which in terms of generating functions and with ξ¯ specified by (9.143) reads 2 odd(soft)2 ¯ = even E soft ((s, ∞); ξ) (2 − ξ) . ((s, ∞); ξ) E1 1
(9.149)
1−ξ
Furthermore, with q(x; ξ) as in (9.46), odd(soft)2
E1
((s, ∞); ξ) = E2soft ((s, ∞); ξ) cosh
∞
ξ sinh
q(x; ξ) dx −
s
∞
q(x; ξ) dx .
(9.150)
s
Proof. The identity (9.148), which for n = 0 reduces to (9.125), follows from the construction of the scaled ensemble odd(soft)2 as a superposition of two β = 1 soft edge ensembles. In relation to (9.150), because the k-point correlation is given by the determinant formula (7.170), application of (9.15) gives Z ∞ “ ” odd(soft)2 soft ¯ = det(1 − ξK(s,∞) ((s, ∞); ξ) ) 1−ξ (1 − ξK soft )−1 A[y]B(y) dy . E 1
s
But det(1 −
soft ) ξK(s,∞)
Z
=
∞
1−ξ
E2soft ((s, ∞); ξ),
so our task is to show that Z ∞ Z p (1 − ξK soft )−1 A[y]B(y) dy = cosh q(x; ξ) dx − ξ sinh
s
s
∞
q(x; ξ) dx. s
This can be done by modifying the working of the proof of Proposition 9.7.1. First introduce the notation p Qs (x) = (1 − ξK soft )−1 φs (x) φs (x) = ξAi(x), Z
so that
∞
ξ
(1 − ξK soft )−1 A[y]B(y) dy =
Z
s
∞
Z dy Qs (x)
y
−∞
s
φs (v) dv =: us
(cf. (9.129)), and similarly introduce the ξ generalization of (9.131) Z ∞ Z y dy ρ(s, y) dv φ(v), qs := −∞
s
where ρ(s, y) is the kernel of the integral operator (1 − ξK us that us and qs satisfy the coupled equations dus = −q(s; ξ)qs , ds subject to the boundary conditions us → 0, This system has solution Z ∞ Z p qs = ξ cosh q(x; ξ) dx − sinh s
qs →
)
. The workings of the proof of Proposition 9.7.1 tell
dqs = q(s; ξ)(1 − us ) ds p
ξ
as
s → ∞. Z
∞
q(x; ξ) dx, s
soft −1
∞
us = 1 − cosh
q(x; ξ) dx −
p
Z
∞
ξ sinh
s
q(x; ξ) dx. s
The latter is precisely the sought identity.
Adding together (9.140) and (9.148) gives the Painlev´e transcendent evaluation of the generating function E1soft ((s, ∞); ξ) [135], [198]. P ROPOSITION 9.7.4 We have 2 ¯ ¯ + ξ¯ sinh μ(s, ξ) ξ − 1 − cosh μ(s, ξ) soft soft ¯ , E1 ((s, ∞); ξ) = E2 ((s, ∞); ξ) ξ−2
(9.151)
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CHAPTER 9
where ξ¯ is given by (9.143) and
∞
¯ := μ(s, ξ)
¯ dt. q(t; ξ)
(9.152)
s
The expression (9.151) allows {E1soft (n; (s, ∞))} and thus {psoft 1 (n; (s, ∞))} to be computed for small n according to the method detailed in Section 9.4.2, although the need to solve coupled equations makes this a difficult exercise, even for n = 2. Fortunately the generating function E1soft ((s, ∞); ξ) can be expressed in terms of Fredholm determinants, and the method of Section 9.2 employed. Thus it turns out that μ(s, ξ) as soft specified by (9.152), is intimately related to the operator V(0,∞) introduced in Proposition 9.6.1. Explicitly [132] √ det(1 − ξV soft ) (0,∞) √ soft . (9.153) exp − μ(s; ξ) = det(1 + ξV(0,∞) ) This is to be substituted in (9.150), together with the formula soft soft E2soft ((s, ∞); ξ) = det(1 − ξV(0,∞) ) det(1 + ξV(0,∞) ),
(9.154)
which follows from (9.40) and Proposition 9.6.1. Computing appropriate derivates with respect to ξ then gives the l.h.s. of (9.148), and this in turn allows for the recursive computation of {E1soft (n, (s, ∞))}n=0,1,... in terms of Fredholm determinants. The simplest case is n = 0, which shows E1soft (0, (s, ∞)) admits the simple Fredholm form [480], [173] soft E1soft (0, (s, ∞)) = det(1 − V(0,∞) ).
(9.155)
Statistical properties of {psoft 1 (n; (s, ∞))} for n = 2, . . . , 5 computed using this formalism [79] are given in Table 9.2. We now turn our attention to the s → −∞ asymptotic behavior of Eβsoft (0; (s, ∞)) for β = 1 and 4, known from (9.91) in the case β = 2. According to (9.127), (9.141) ∞ and (9.146), in addition to the expansion (9.91) this requires knowledge of the s → −∞ expansion of s q(x) dx. On this point it is known that [30] 1 ∞ √ 3/2 exp − q(x) dx ∼ 2−1/4 e−|s| /3 2 1 + O(1/s4 ) . s→−∞ 2 s ˜4soft (n; (22/3 s, ∞)), the following result then follows Noting from Section 7.6.5 that E4soft (n; (s, ∞)) = E [30]. P ROPOSITION 9.7.5 For β = 1 or 4 one has √
e−β|s| /24+(β/2−1)( 2/3)|s| ∼ τβ s→−∞ |s|(1/8)(1−(2/β)(1−β/2)2 ) 3
Eβsoft (0; (s, ∞))
where τ1 = 2−11/48 e 2 ζ (−1) , τ4 = 2−37/48 e 2 ζ asymptotic form to be valid for β = 2. 1
1
(−1)
3/2
,
(9.156)
. Moreover, comparison with (9.91) shows this same
Theory related to the characterization of the distribution of the largest eigenvalue in terms of (13.186) below allows the leading behavior exhibited in (9.156) to be rigorously established for general β > 0 [463]. E XERCISES 9.7
˜4soft will be identified as τ -functions of Painlev´e systems. 1. [231] In this exercise E1soft and E
(i) Use (9.46) to show that (9.127) can be written E1soft (0; (s, ∞)) = exp
“
−
1 2
Z s
∞
” (t − s)(q 2 (t) − q (t)) dt .
421
FREDHOLM DETERMINANT APPROACH
(ii) With us defined as in Proposition 8.3.2, generalize the derivation of (9.58) to show “ ” 1“ t” 2 . −2−1/3 us (−2−1/3 t; μ; ξ) = − qμ−1/2 (t) + qμ−1/2 (t) + 2 2 (iii) Recalling that q(t) = q0 (t), as well as the definition of σII (= us ) from (8.29) and Proposition 8.2.1, and noting that the differential equation for q (9.45) is unchanged by the negation q → −q, show from results (i) and (ii) that we can write ˛ “ Z ∞“ t2 ” ” ˛ E1soft (0; (s, ∞)) = exp − + HII (t)˛ dt , 8 α=0 s 2
where HII (t)|α=0 + t8 ∼ sponding τ -function by
1 Ai(t) 2
for t → ∞. Thus, defining the auxiliary PII Hamiltonian and corre-
2 ˜ II := HII + t , H 8
τ˜II (t; α) =
d ˜ II , log H dt
conclude + E1soft (0; (s, ∞)) = τ˜II (s; 0), (9.157) 1 ˜ ˜ where the superscript + indicates that HII is subject to the boundary condition HII (t) ∼ 2 Ai(t).
(iv) By writing q → −q in the above working show that “ 1Z ∞ ” − exp − (t − s)(q 2 (t) + q (t)) dt = τ˜II (s; 0), 2 s ˜ II (t) ∼ − 1 Ai(t). Use ˜ II is subject to the boundary condition H where the superscript − indicates that H 2 this result in (9.46) to deduce that + − E2soft (0; (s, ∞)) = τ˜II (s; 0)˜ τII (s; 0).
Substitute (9.157) and (9.158) in (9.141) with n = 0 to show ” “ + − ˜4soft (0; (s, ∞)) = 1 τ˜II E (s; 0) + τ˜II (s; 0) . 2
(9.158)
(9.159)
9.8 THE PROBABILITIES Eβhard (n; (0, s); a) FOR β = 1, 4 Considerations similar to those which yielded the evaluation of Eβsoft (n; (s, ∞)) for β = 1 and 4 suffice for the evaluation of Eβhard [197]. For β = 2 the latter is specified by (8.87) with μ = 0. For β = 1 and 4, so as to relate to (6.124), we define E1hard ((0, s); ξ; a) := lim E1 ((0, s/4N ); ξ; xa e−x/2 ; N ), N →∞
˜ hard ((0, s); ξ; a) := lim E4 ((0, s/4N ); ξ; xa e−x ; N/2). E 4 N →∞
The first step is to make note of the analogue of Proposition 9.7.2, derived as for the results therein using (6.138) and (6.142) with f = x(a−1)/2 e−x/2 , g = xa e−x . P ROPOSITION 9.8.1 We have E2hard (n; (0, s); a) =
2n+1
E1hard (2n + 1 − l; (0, s); (a − 1)/2)
l=0
× E1hard (l; (0, s); (a − 1)/2) + E1hard (l − 1; (0, s); (a − 1)/2)
(9.160)
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CHAPTER 9
and E˜4hard (n; (0, s); a + 1) = E1hard (2n; (0, s); (a − 1)/2) + E1hard (2n + 1; (0, s); (a − 1)/2). The first is equivalent to the generating function identity 2 ¯ a) = odd E hard ((0, s); ξ; (a − 1)/2) (2 − ξ) (1 − ξ)E2hard ((0, s); ξ; 1
(9.161)
(9.162)
1−ξ
and the second to the relation (n; (s, ∞); a + 1) = phard (2n + 1; (s, ∞); (a − 1)/2). p˜hard 4 1
(9.163) 2
The second step is to formulate the analogue of Proposition 9.7.3. For this let odd(hard) refer to the ensemble with k-point correlation (7.172), and let the corresponding generating function for k eigenvalues in 2 (0, s) be denoted E odd(hard) ((0, s); ξ; (a − 1)/2). P ROPOSITION 9.8.2 We have odd(hard)2
E1
(n; (0, s); (a − 1)/2) =
2n
E1hard (2n − l; (0, s); (a − 1)/2)
l=0
× E1hard (l; (0, s); (a − 1)/2) + E1hard (l − 1; (0, s); (a − 1)/2) , which equivalently in terms of generating functions reads 2 odd(hard)2 ¯ (a − 1)/2) = even E hard ((0, s); ξ; (a − 1)/2) (2 − ξ) , ((0, s); ξ; E2 1 1−ξ
(9.164)
(9.165)
where ξ¯ is given by (9.143). Furthermore odd(hard)2
E2
¯ (a − 1)/2) = E hard ((0, s); ξ; ¯ a) cosh μ ¯ a) − ξ sinh μ ¯ a) , ((0, s); ξ; ˜(s; ξ; ˜(s; ξ; 2
where with q(t; ξ) as in (9.71) ¯ := 1 μ ˜ (s; a; ξ) 2
0
s
(9.166)
¯ q(t; ξ) √ dt. t
Proof. The relation (9.164) follows from the definition of the ensemble odd(hard)2 as a superposition, while (9.165)
¯ 2n and summation over n. To deduce (9.166), we note from (7.172) and follows from (9.164) by multiplication by (1 − ξ) (9.15) that “ ” odd(hard)2 E1 (0; (0, s); ξ; (a − 1)/2) = det 1 − ξ(K hard + C ⊗ D) , (9.167) √ hard is the operator on (0, s) with kernel (9.60), C is the operator which multiplies by Ja ( y), and D is the where K integral operator with kernel Z ∞ 1 (9.168) √ √ Ja (t) dt. 2 y y We see from (9.167) that (9.166) is equivalent to the identity Z s p [(1 − ξK hard )−1 C](y)D(y) dy = cosh μ ˜(s; ξ; a) − ξ sinh μ ˜(s; ξ; a). 1−ξ
(9.169)
0
With
p √ Qh (x) = [(1 − ξK hard )−1 φh ](x), φh (x) = ξJa ( x), √ after changing variables t = u in (9.168) we have Z Z ∞ Z s 1 s 1 1 ξ (1 − ξK hard )−1 C[y]D(y) dy = dy Qh (y) √ du √ φh (u) =: uh , 4 y u 0 0 y
(9.170)
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FREDHOLM DETERMINANT APPROACH
and analogous to (9.131) we also introduce Z
∞
qh := 0
Z dy ρh (s, y)
y
1 du √ φh (u), u −∞
(9.171)
where ρh (s, y) denotes the kernel of the integral operator (1 − ξK hard )−1 . Now Proposition 9.3.10 gives “ ” ∂ h Q (y) = q(s; ξ) − δ + (y − s) + ρh (s, y) ∂s (cf. (9.132)). Using this formula and recalling q(s; ξ) := Q(s) we see from (9.170) that d h 1 u = q(s; ξ)qh . ds 4
(9.172)
The corresponding formula for the derivative of qh is derived by making use of the formula x
” 1 ∂ ∂ “ h ∂ h yρ (x, y) + Qh (x)Qh (y) ρ (x, y) + s ρh (x, y) = − ∂x ∂s ∂y 4
(cf. (9.134)) which is a consequence of Proposition 9.5.1, (9.63) and (9.35). This shows Z s “ ” 1 Z ∞ d dq h 1 (yρh (s, y) √ s =− dy du √ φh (u) + q(s; ξ)uh ds dy y u 0 y Z s Z s Z ∞ 1 1 1 =− dy ρh (s, y) √ du √ φh (u) − dy ρh (s, y)φh (y) + q(s; ξ)uh 2 0 y y u 0 1 = − qh − q(s; ξ)(1 − uh ). 2
(9.173)
The coupled equations (9.172) and (9.173) must be solved subject to the s → 0 boundary conditions Z ∞ p Z ∞ p √ h 1 √ φh (u) du = 2 ξ sq ∼ Ja (v) dv = 2 ξ. (9.174) uh ∼ 0, u 0 0 √ √ The occurrence of sq in (9.174) suggests we introduce q˜h := sqh in (9.172) and (9.173). Doing this gives the system of equations √ duh √ h 1 = q(s; ξ)˜ qh , s s(˜ q ) = −q(s; ξ)(1 − uh ). (9.175) ds 4 The solution satisfying (9.174) can be checked to be “p ” uh = 2 ξ cosh μ ˜(s; a; ξ) − sinh μ ˜(s; a; ξ) , p ˜(s; a; ξ) + ξ sinh μ ˜(s; a; ξ), q˜h = 1 − cosh μ thus verifying (9.169).
Adding together (9.165) and (9.162), and using (9.166) gives the analogue of (9.151) for the hard edge, ¯ ¯ + ξ¯ sinh μ ˜(s; a; ξ) a − 1 2 ξ − 1 − cosh μ ˜ (s; a; ξ) hard hard ¯ ) = E2 ((0, s); ξ; a) . (9.176) E1 ((0, s); ξ; 2 ξ−2 Setting ξ = 1 this reads a − 1 2 E1hard 0; (0, s); = E2hard (0; (0, s); a) exp − μ ˜(s; ξ = 1; a) . 2
(9.177)
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CHAPTER 9
Analogous to (9.153), one can show that [132]
√ det(1 − ξV hard ) (0,1) √ hard . exp − μ ˜(s; ξ; a) = det(1 + ξV(0,1) )
(9.178)
This substituted in (9.176) implies
a − 1 hard E1hard 0; (0, s); = det(1 − V(0,1) ), 2 where use is also required of the fact that hard hard ) det(1 + ξV(0,1) ) E2hard (0; (0, s); a) = det(1 − ξV(0,1)
(9.179)
(9.180)
(cf. (9.154)), which in turn follows from (9.15) and Proposition 9.6.1. Substituting (9.178) and (9.179) in (9.164), and computing appropriate derivatives with respect to ξ, we see that the l.h.s. of (9.166) can be expressed in terms of Fredholm determinants. This allows for a recursive computation of {E1hard (n; (0, s); (a − 1)/2)} in terms of Fredholm determinants. We remark that in the case (a − 1)/2 = m ∈ Z+ the probability E1hard (0; (0, s); m) can be written as an m-dimensional integral (see Exercises 9.8 q.1). From this we can deduce the large s asymptotic expansion √
E1hard (0; (0, s); m)
G(3/2) G(2m + 2) e−s/8+m s √ ∼ . G(m + 3/2) G(m + 2) (2 s)m(m+1/2)
(9.181)
We will see in Chapter 12 that an m-dimensional integral evaluation of Eβhard (0; (0, s); m) can also be obtained for m ∈ Z+ and general β > 0. The corresponding asymptotic expansion is given in (13.52). E XERCISES 9.8
1. [233], [231] The objective of this exercise is to derive the identity ˛ D E Pm 2 ˛ E1hard (0; (0, s2 ); m) = e−s /8+ms e−2s j=1 xj ˛ JUEm a=b=1/2 D E P 2 m = e−s /8 es j=1 cos θj ,
(9.182)
Sp(2m)
where the second equality follows from (8.117). (i) Use (9.71) to show that (9.177) can be written “ “ 1 Z s“ q(t) ” ” a − 1” s (log )q 2 (t) + 2 √ dt . E1hard 0; (0, s); = exp − 2 8 0 t t
(9.183)
(ii) Recalling the s → 0+ behavior (9.69) verify that Z s“ Z s ”d √ q(t) √ dt = ( tq(t)) dt log s − log t dt t 0 0 and use this in (i) to show E1hard
“
“ 1 a − 1” 0; (0, s); = exp − 2 4
Z
√
s
“ log
0
” ” s ”“ dq x + q + xq 2 dx , t dx
(iii) Introduce the function y(x) according to (9.70) with s = t. Verify that x
” “ dy dq 1 + q + xq 2 = − y 2 + 4xy + 1 + x. 2x 2 dx (1 − y) dx
(iv) In the Hamiltonian HV of (8.24) make the substitutions q → y, p → z, t → ηx, HV →
1 ˜ HV η
t = x2 .
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FREDHOLM DETERMINANT APPROACH
(cf. (9.145)). Use the relationship between the PV parameters listed below (8.24), and the√fact that the general PV equation with δ = 0 can be reduced to the case with δ = − 12 by the mapping t → −2δt, to show that with y the PV transcendent in (9.70) we must set η = 2,
1 v2 = −v3 = − (a + 1), 4
v1 = −v4 =
1 (a − 1). 4
(9.184)
˜ V so specified, make use of the first of the corresponding Hamilton equations together with the result With H of (iii) to show “a − 1 ” dq x d ˜V ) . x + q + xq 2 = −2 − + (xH dx 2 2 dx (v) Substitute the result of (iv) in (ii), and use the relationship between HV and σV as specified by (8.29) and Proposition 8.2.1 to conclude that σ ˜V := x
“ d a − 1 ” 1 2 (a − 1) a(a − 1) log E1hard 0; (0, x2 ); + x − x+ dx 2 4 2 4
(9.185)
satisfies the σPV equation in (8.15) with parameters (9.184) and t → 2x. (vi) From the results of Exercises 8.3 q.1(i)&(ii) note that x
D E “ Pm 1” d log e−2x j=1 xj +m m+ dx 2 JUEm |a=b=1/2
(9.186)
satisfies the same equation as that satisfied by (9.185) provided (a − 1)/2 = m ∈ Z+ . By verifying that (9.185) and (9.186) have the same small x expansion (for this use the matrix average form in (9.182) together with a result from Exercises 10.7 q.2 below), deduce from this (9.182). 2. [233], [231] The objective of this exercise is to show that 2
˜4hard (0; (0, s2 ); 2l) = e−s E
/8
D E e(s/2)Tr U
U∈O(2l)
(9.187)
(cf. (9.182)). (i) Use (9.160) and (9.161) with n = 0 to show ” “ ”” “ “ E hard (0; (0, s); a) ˜4hard (0; (0, s); a + 1) = 1 E1hard 0; (0, s); a − 1 E . + hard 2 2 2 E1 (0; (0, s); (a − 1)/2) (ii) Use the relationship between the eigenvalue p.d.f. for Sp(2m) and O− (2m+2) noted above (2.69), together with the second formula in (9.182), to note that we can write D E 2 E1hard (0; (0, s); l − 1) = e−s /8 esTr U . U∈O − (2l)
(iii) Note from (9.71) and (9.183) that “ 1 E2hard (0; (0, s); a) = exp − 8 E1hard (0; (0, s); (a − 1)/2)
Z
s
““ log
0
q(t) ” ” s” 2 q (t) − 2 √ dt , t t
and use the workings of q.1 to deduce that the conclusion of q.1(v) holds with E1hard therein replaced by the l.h.s. of this equation, with s → x2 . (iv) Use the results of Exercises 8.3 q.1(i)&(ii) to note that “ E D Pm 1” d +m m+ log e−2x j=1 xj x dx 2 JUEm+1 |a=b=1/2
426
CHAPTER 9
satisfies the same equation as (9.185), provided (a − 1)/2 = m ∈ Z+ . Use (8.117) to rewrite this as x
D E d + m(m + 1/2) log e−(m+1)x e(s/2)Tr U dx U∈O + (2m+2)
and show from this (making use of a result from Exercises 10.7 q.2 below) that the small x expansion is the same as that of (9.185). Conclude that D E 2 E2hard (0; (0, s); 2l − 1) . = e−s /8 e(s/2)Tr U hard E1 (0; (0, s); l − 1) U∈O + (2l) Substitute this, together with the result of (ii), in the result of (i) with a → 2l − 1 to deduce (9.187).
9.9 RIEMANN-HILBERT VIEWPOINT We turn our attention from the function theoretic properties of the Fredholm determinant (9.15) to aspects of the underlying integrable systems theory. A crucial first step [307] is to associate with integral operators having kernels of the form (9.25) a Riemann-Hilbert problem (recall Section 1.4.3). 9.9.1 Integrable kernels Generalize the kernel (9.25) to K(λ, μ) =
f T (λ)g (μ) , λ−μ
f =
f1 f2
, g =
g1 g2
,
(9.188)
where it is required that f T (λ)g (λ) = 0.
(9.189)
Note that this includes the Christoffel-Darboux structure (9.25). The proof of Proposition 9.3.6 again shows (1 − KJ )−1 = 1 + RJ with the kernel of RJ given by R(λ, μ) =
F T (λ)G(μ) , λ−μ
Fj = (1 − KJ )−1 fj , Gj = (1 − KJ )−1 gj (j = 1, 2).
Introduce the 2 × 2 matrix
Y(λ) := 12 −
a2
a1
F (μ)g T (μ) dμ. μ−λ
(9.190)
(9.191)
We see immediately that Y(λ) ∈ Reg(C\[a1 , a2 ]),
(9.192)
where Reg(X) denotes the space of analytic function on the set X, and we see too that Y(∞) = 12 .
(9.193)
In traversing the interval [a1 , a2 ] in the complex plane from a1 to a2 , let the l.h.s. of the interval be denoted +, and the r.h.s. −. For λ ∈ (a1 , a2 ), let Y± (λ) :=
lim
λ →λ λ ∈(±) side
Y(λ ).
(9.194)
427
FREDHOLM DETERMINANT APPROACH
The function Y(λ) is discontinuous across (a1 , a2 ), with Y− (λ) being related to Y+ (λ) by the so-called jump matrix H(λ) specified in the following result. P ROPOSITION 9.9.1 We have Y− (λ) = Y+ (λ)H(λ),
(9.195)
where H(λ) is the 2 × 2 matrix specified by H(λ) = 12 + 2πif(λ)g T (λ).
(9.196)
Also, for λ ∈ (a1 , a2 ) F (λ) = Y+ (λ)f(λ) = Y− (λ)f(λ),
T G(λ) = (Y+ (λ))−1g (λ) = (YT − (λ))−1g (λ).
(9.197)
Proof. We will first derive (9.197). Let λ ∈ (a1 , a2 ) and consider Z Y± (λ)f(λ) := f(λ) −
lim
λ →λ λ ∈(±) side
a2 a1
Z a2 (μ)g T (μ)f(λ) F F (μ)g T (μ)f(λ) (λ) − dμ = f dμ, μ−λ μ−λ a1
where the second equality follows by noting that when μ = λ, the scalar factor g T (μ)f(λ) = f T (λ)g (μ)
(9.198)
in the numerator vanishes (recall (9.189)), and thus the integral is now continuous across [a1 , a2 ]. Use of (9.198) for general λ, μ shows Z Y± (λ)f(λ) = f(λ) +
a2 a1
f T (λ)g (μ) (λ), F (μ) dμ = f(λ) + KJ (F )[λ] = F λ−μ
where the final equality follows by making use of the second equation in (9.190). The derivation of the second equation in (9.197) is analogous. Consider now (9.195). From the definition (9.194) and Cauchy’s residue theorem Z Y+ (λ) − Y− (λ) = −
Cλ
F (μ)g T (μ) (λ)g T (λ), dμ = −2πiF μ−λ
(λ) where Cλ is a circle in the complex μ-plane about the point μ = λ, traversed anticlockwise. Substituting (9.199) for F in this equation gives (9.195).
The function Y(λ), specified only by the analyticity condition (9.192), normalization (9.193) and jump condition (9.195) (and thus jump matrix H) is said to solve the normalized Riemann-Hilbert problem, RHP([a1 , a2 ], H). The solution given by (9.191) is unique (see Exercises 9.9 q.1). An important property relates to its determinant. P ROPOSITION 9.9.2 With Y(λ) specified by (9.191), det Y(λ) = 1.
(9.199)
Proof. According to (9.189), the matrix 2πif(λ)g T (λ) =: iX is nilpotent (X2 = 0) and thus det H(λ) = det[12 + iX] = 1 since the eigenvalues of 12 + iX must equal unity. Consequently, from (9.195) det Y− (λ) = det Y+ (λ),
(9.200)
and so det Y(λ) is analytic for λ ∈ (a1 , a2 ). Furthermore Y(λ) is analytic for λ ∈ C\[a1 , a2 ] and thus det Y(λ) is analytic in this region. At λ = a1 , a2 we see from (9.191) that Y(λ) has a logarithmic singularity. But since det Y(λ) is analytic for all λ = a1 , a2 , this singularity must cancel out of det Y(λ), or else det Y(λ) would have a branch cut.
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CHAPTER 9
Consequently, det Y(λ) is analytic in the finite complex λ-plane. From (9.193), det Y(λ) → 1 as |λ| → ∞. Thus (9.199) follows by Liouville’s theorem.
9.9.2 Schlesinger equations We now specialize to kernels of the form (9.25), and suppose ψ and φ therein furthermore obey the coupled equations (9.41). Note that the latter can be written in the form d ψ(x) ψ(x) a(x) b(x) = A0 (x) , A0 (x) = , (9.201) φ(x) c(x) −a(x) dx φ(x) where a(x), b(x), c(x) are rational functions of x. From the general theory of linear differential equations, (9.201) can be extended to the matrix differential equation ˜ d ψ(x) ψ(x) Φ(x) = A0 (x)Φ(x), Φ(x) = , (9.202) ˜ dx φ(x) φ(x) where the two columns of Φ(x) are linearly independent. We see from the fact that (9.202) has the formal solution x Φ(x) = C0 exp A0 (y) dy, C0 a constant matrix, and the fact that A0 is traceless, that ψ˜ and φ˜ in (9.202) can be chosen so that det Φ(x) = 1.
(9.203)
Following [346], [83], our objective is to show that the matrix product Ψ(x) := Y(x)Φ(x), where Y(x) is specified by (9.191) with ψ(x) , f (x) = φ(x)
g (x) =
(9.204) −φ(x) ψ(x)
(9.205)
satisfies some special partial differential equations. With the choice (9.205), (i6.6e) reduces to ξK scale (x, y) as specified by (9.25). Note that for general f, g, it follows from (9.199) and (9.204) that det Ψ(x) = 1.
(9.206)
P ROPOSITION 9.9.3 With H(x) specified by (9.196), and f, g therein specified by (9.205), we have 1 − 2πiψ(x)φ(x) 2πiφ2 (x) H(x) = −2πiφ2 (x) 1 + 2πiψ(x)φ(x) 1 2πi Φ−1 (x). = Φ(x) (9.207) 0 1 Proof. This can be verified by a direct calculation. It follows from (9.207) and the jump relation (9.195) that the matrix Ψ(x) specified by (9.204) satisfies the jump relation 1 2πi , x ∈ (a1 , a2 ). Ψ− (x) = Ψ+ (x) (9.208) 0 1
429
FREDHOLM DETERMINANT APPROACH
For |x| → ∞, (9.191) and (9.204) show
1 Φ(x). Ψ(x) ∼ 12 + O x In relation to the neighborhood of the endpoints x ∼ ak (k = 1, 2), introduce ◦
Ψ(x) :=
1 0
(−1)k−1 log(x − ak ) 1 ◦ −1
(−1)
2
k−1 4
=: (x − ak )
3 0
1
0
0
5
,
◦ −1
ˆ Ψ(x) := Ψ− (x)Ψ− (x) = Ψ+ (x)Ψ+ (x), ˆ where the final equality follows on use of (9.208) and shows that Ψ(x) is analytic in the neighborhood of ak . Then the facts that Y(x) and thus Ψ(x) has a logarithmic singularity at ak , that det Y(x) = 1, and the jump condition (9.208) imply that in the neighborhood of the endpoints the singular behavior factorizes as ◦
ˆ Ψ(x) = Ψ(x) Ψ(x).
(9.209)
∂Ψ(x) = A(x)Ψ(x), ∂x
(9.210)
P ROPOSITION 9.9.4 We have
where 2
Ak , x − ak k=1 ˆ −1 (ak ) ˆ k) 0 1 Ψ Ak := (−1)k−1 Ψ(a 0 0
ˆ 0 (x) + A(x) = A
(9.211) (9.212)
ˆ 0 (x) is analytic in the neighborhood of ak . Furthermore and the matrix A Ak . x − ak
(9.213)
∂Y −1 ∂Ψ −1 Ψ = Y + Y(x)A0 (x)Y−1 (x). ∂x ∂x
(9.214)
∂Ψ = Uk (x)Ψ, ∂ak
Uk (x) = −
Proof. According to (9.204) and (9.202)
This formula tells us that the possible singularities of A(x) in (9.210) occur at the end points a1 , a2 , along the branch x ∈ (a1 , a2 ) of Y, or at the poles, {bj }, say, of A0 (x). In fact there is no branch cut along x ∈ (a1 , a2 ). To see this note that the fact that the jump matrix in (9.208) is a constant implies ∂Ψ+ −1 ∂Ψ− −1 Ψ+ = Ψ− . ∂x ∂x To study the neighborhood of x = a1 , a2 we note from (9.209) that » – ˆ −1 ∂Ψ −1 (−1)k−1 ˆ ∂Ψ 0 1 ˆ −1 ˆ . Ψ = Ψ Ψ(x) Ψ (x) + 0 0 ∂x x − ak ∂x ˆ is analytic in the neighborhood of each ak , the expansion (9.211) follows. The equation (9.213) is a simple Since Ψ consequence of (9.209).
We remark that it follows from (9.212) that A21 = A22 = 0
(9.215)
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CHAPTER 9
and in particular TrA1 = TrA2 = 0. Also, because according to (9.193) Y is asymptotically equal to the identity, it follows from (9.214) and (9.212) that ˆ 0 (x) ∼ Y(x)A0 (x)Y−1 (x) + O 1 A ∼ A0 (x). (9.216) x |x|→∞ |x|→∞ Together the formulas (9.210) and (9.213) form a Lax pair. Taking the partial derivative with respect to ak of (9.210), and the partial derivative of (9.213) with respect to x, equating and making further use of (9.210) and (9.213) shows ∂Uk ∂A − = [A, Uk ]. ∂x ∂ak
(9.217)
Furthermore, for j = k, it follows by taking the partial derivative of (9.213) with respect to aj that ∂Uk ∂Uj − = [Uj , Uk ]. ∂aj ∂ak
(9.218)
The equations (9.217) and (9.218) themselves imply a pair of partial differential equations for the matrices Ak , known as the Schlesinger equations. P ROPOSITION 9.9.5 We have ∂Ak [Aj , Ak ] = ∂aj a j − ak
(j = k),
∂Aj [Aj , Al ] ˆ 0 (aj )]. = − [Aj , A ∂aj a l − aj l=1 2
(9.219)
l=j
Proof. The first equation follows by substituting the definition of Uk from (9.213) in (9.218) and equating coefficients
of 1/x and 1/x2 on both sides. The second follows by substituting (9.211) and the definition of Uk from (9.213) in (9.217), interchanging k ↔ j, and equating residues at x = aj .
Historically (see, e.g., [310]), the Schlesinger equations arose in the study of the isomonodromy deformation of linear matrix differential equations of the form (9.210). Now, in general the effect of the poles at a1 , a2 in (9.211), and the poles at infinity revealed by the change of variable x → 1/x, is that the solution Ψ(x) at these singularities of A(x) has a branch point. Thus following the solution Ψ(x) about a closed contour encircling any one of these points will not give back Ψ(x) itself but rather some linear combination of the fundamental solutions, Ψ(x) = Ψ(x) Xu , 0 < 1, u = a1 , a2 , ∞. (9.220) 2πi x=u+ e
x=u+
The matrices Xa1 , Xa2 , X∞ are called monodromy matrices. In the special circumstance that the monodromy matrices are independent of a1 and a2 , the differential equation (9.210) is said to be monodromy ˆ 0 (x) = 0 in the definition (9.211) of A(x), it was shown by Schlesinger that preserving. In the case that A (9.210) being monodromy preserving implies (9.213) and thus the equations (9.219). A derivation of this ˆ 0 (x) = 0, the analogous result in given in Exercises 9.9 q.1. The converse is also true. In the case that A result is due to Jimbo, Miwa and Ueno [325]. Associated with the Schlesinger equations is the one-form Tr(A A ) Tr(A A ) 1 2 2 1 ˆ 0 ) ˆ 0 ω= + Tr(A1 A da1 + + Tr(A2 A da2 a 1 − a2 a 2 − a1 x=a1 x=a2 =: f1 (a1 , a2 )da1 + f2 (a1 , a2 )da2 . (9.221)
431
FREDHOLM DETERMINANT APPROACH
Its significance is that with dω :=
∂ ∂ f1 (a1 , a2 )da1 ∧ da2 + f2 (a1 , a2 )da2 ∧ da1 ∂a2 ∂a1
the equations (9.219) imply dω = 0 and thus ω is closed. The general Stokes theorem then implies it is possible to write ω = d log τ
(9.222)
for some τ . It’s explicit form is given by the following result.
P ROPOSITION 9.9.6 With τ defined implicitly in terms of ω by (9.222), which in turn is defined in terms of quantities associated with the kernel (9.188) by (9.221), we have that up to a multiplicative constant τ = det(1 − KJ ).
(9.223)
Proof. Following [83], according to (9.23) and (9.26) “ ∂ ” “ ∂ ” log det(1 − KJ ) da1 + log det(1 − KJ ) da2 ∂a1 ∂a2 = R(a1 , a1 )da1 + R(a2 , a2 )da2 .
d log det(1 − KJ ) :=
Comparing with (9.221) implies we must show R(a1 , a1 ) =
˛ Tr(A1 A2 ) ˆ 0 )˛˛ + Tr(A1 A a1 − a2 x=a1
and the same equation with the indices 1 and 2 reversed. Now we know from (9.190) that R(a1 , a1 ) =
lim
x,y→a1
T (x)G(y) T (y)F (x) F G = lim . x,y→a1 x−y x−y
Making use of (9.197) and (9.204) then shows −1 g T (y)Y+ (y)Y− (x)f(x) x,y→a1 x−y
R(a1 , a1 ) = lim
−1 g T (y)Φ(y)Ψ−1 (x)f(x) + (y)Ψ− (x)Φ . x,y→a1 x−y
= lim From the definitions (9.202), (9.205) Φ−1 f = and so R(a1 , a1 ) =
lim
x,y→a1
»
1 0
– =: eˆ1 ,
g T Φ = [0 1] =: eˆT2 ,
˛ eˆT2 Ψ−1 e1 ˛ + (y)Ψ− (x)ˆ e1 ˛ , = eˆT2 Ψ−1 + (x)Ψ− (x)ˆ x−y x=a1
(9.224)
432
CHAPTER 9
where the final equality follows by l’Hˆopital’s rule. Making use of (9.207) and (9.210) then shows » R(a1 , a1 ) = eˆT2
1 0
−2πi 1
–
2 “ X ˆ Ψ−1 − (x) A0 + k=1
˛ Ak ” ˛ e1 ˛ Ψ− (x)ˆ x − ak x=a1
˛ Ak ˛ e1 ˛ Ψ− (x)ˆ x − a x=a1 k k=1 “» 0 1 – ”˛ ˆ 0 Ψ− (x) ˛˛ = Tr Ψ− (x)−1 A 0 0 x=a1 » – 2 “ ”˛ X Ak 0 1 ˛ + Tr Ψ− (x)−1 Ψ− (x) ˛ , 0 0 x − ak x=a1 2 “ X ˆ = eˆT2 Ψ−1 − (x) A0 +
”
k=1
where the second equality follows on multiplying out the first two terms, while the third equality uses the simple identity “» 0 1 – ” e1 = Tr X eˆT2 Xˆ 0 0 ◦
ˆ is analytic at valid for any 2 × 2 matrix X. Recalling (9.209), and noting that Ψ(x) therein is upper triangular and ψ(x) x = a1 allows this last expression to be rewritten as ”˛ “» 0 1 – ˛ −1 ˆ ˆ ˆ A0 Ψ(x) ˛ Ψ(x) R(a1 , a1 ) = Tr 0 0 x=a1 2 ”˛ “» 0 1 – X ˛ −1 Ak ˆ ˆ Ψ(x) + Tr Ψ(x) . ˛ 0 0 x − ak x=a1 k=1
Recalling the definition (9.212) of A1 we see that ”˛ “» 0 1 – ˛ −1 ˆ ˆ ˆ 0 ), ˆ A0 Ψ(x) ˛ = Tr(A1 A Tr Ψ(x) 0 0 x=a1 » – ”˛ “ 0 1 Tr(A1 A2 ) ˛ −1 A2 ˆ ˆ Ψ(x) = , Tr Ψ(x) ˛ 0 0 x − a2 a1 − a2 x=a1 ˆ making use too of the facts that ψ(x) is analytic in the neighborhood of a1 and so can be expanded in a power series and that Tr A21 = 0 shows ” “» 0 1 – −1 A1 ˆ ˆ Ψ(x) = O((x − a1 )), Tr Ψ(x) 0 0 x − a1 telling us that this vanishes at x = a1 . Thus (9.224) results. The equation for R(a2 , a2 ) follows similarly.
9.9.3 Application to the Airy kernel We know from (9.41) and (9.42) that in the case of the Airy kernel (9.39), and thus ψ(x) = Ai(x), φ(x) = Ai (x) in (9.201), the corresponding matrix A0 (x) has the explicit form 0 x . (9.225) A0 (x) = 1 0 For the Airy kernel we consider J = (s, ∞) and thus a1 = s, a2 = ∞, so in this case (9.211) reads ˆ 0 (x) + A1 . A(x) = A x−s
433
FREDHOLM DETERMINANT APPROACH
We note that it follows from (9.212) that A1 has the structure α −α2 /β . A1 = β −α
(9.226)
Also it follows by substituting (9.225) in the first relation of (9.216) and equating terms proportional to x and proportional to unity, using the fact that Y(x) is to leading order equal to the unit matrix and has unit ˆ 0 (x) has the structure determinant, that A μ x+ν ˆ . (9.227) A0 (x) = 1 −μ In (9.226) and (9.227), α, β, μ, ν depend on s but not x. The compatibility condition (9.217) can be used to derive inter-relations between these quantities. P ROPOSITION 9.9.7 We have dμ dν = β, = −2α, ds ds 2 dα α dβ = + (s + ν)β, = 2(α − μβ). ds β ds
(9.228)
As a consequence α2 + (s + ν)β, β
μ = 2μα −
β = −(ν + μ2 ).
(9.229)
Proof. With (a1 , a2 ) = (s, ∞) the compatibility condition (9.217) takes on the simplified form (x − s)
d ∂ ˆ ˆ 0 (x), A1 ]. A0 (x) + A1 = [A ∂s ds
ˆ 0 (x) given by (9.227) and A1 by (9.226), this expression is linear in x. Equating terms proportional to x gives With A –– » – »» – » d μ ν 0 1 α −α2 /β , = , β −α 0 0 ds 1 −μ which implies the first two differential equations. Equating terms independent of x gives – »» – » –– » – » d d μ ν α −α2 /β μ ν α −α2 /β −s = , , + −α 1 −μ β −α ds 1 −μ ds β and from this the second two differential equations follow. Alternatively these two differential equations can be derived from the second equation in (9.219), appropriately specialized. The stated relations (9.229) can be verified by differentiating and using (9.228).
Using Proposition 9.9.7 we can readily obtain a characterization of the specialization of the resolvent kernel R(s) := R(s, s). P ROPOSITION 9.9.8 We have R(s) = μ,
dR(s) = β, ds
where β satisfies the differential equation d2 β 1 dβ 2 − − 2sβ + 4β 2 = 0, ds2 2β ds which is a particular case of what is referred to in the literature as the Painlev´e XXXIV equation. Setting
434
CHAPTER 9
β = −q 2 the latter reduces to the particular Painlev´e II equation (9.45), and thus the fourth equation of Proposition 9.4.2 is reclaimed. Proof. The relation (9.224), together with the parametrizations (9.226) and (9.227) tell us that for the Airy kernel R(s) = Tr
“» α β
−α2 /β −α
–»
μ 1
–”
s+ν −μ
= 2μα −
α2 + (s + ν)β = μ, β
where the final equality follows from the first equation in (9.229). The stated equation for the derivative of R now follows from the first differential equation (9.228). The differential equation for β can be verified by differentiating the formula for dβ/ds in (9.228), making use of the other differential equations therein, substituting for 2α2 /β − 4αμ in the resulting expression in terms of (dβ/ds)2 , and making use of the second equation in (9.229).
E XERCISES 9.9
1. The objective of this exercise is to show that if the n × n matrix differential equation ∂ Y = AY, ∂x
A :=
n X ν=1
Aν , x − aν
(9.230)
P where the Aν are diagonalizable and A∞ := − n ν=1 Aν is diagonal and has monodromy and connection matrices independent of the aν , then Y also satisfies ∂ Aν Y=− Y. ∂aν x − aν
(9.231) (ν)
For this we require the readily verifiable facts that with the diagonal form of Aν given by A0 the neighborhood of aν the solution of (9.230) permits the expansion
= G−1 ν Aν Gν , in
∞ “ ” X (ν) Yjν (x − aν )j (x − aν )A0 , Y (ν) (x) = Gν 12 +
(9.232)
j=1
while in the neighborhood of infinity the solution is ∞ “ ” X (∞) Yj x−j x−A∞ . Y (∞) (x) = 12 +
(9.233)
j=1
The connection matrices Cν are defined so that Y(x) = Y ∞ (x), (i) With dX := property
Pn
∂ ν=1 ∂aν
˛ ˛ Y˛
Y(x) = Y(ν) (x)Cν .
(9.234)
Xdaν , set Ω(x, a) := dY Y−1 . Defining the monodromy matrices Mν by the
x=u+e2πi
˛ ˛ = Y˛
x=u+
Xu ,
0 < 1, u = a1 , . . . , an , ∞
(cf. (9.220)), use the fact the monodromy matrices are independent of the aν to show that Ω(x, a) is free of branch points. (ii) Use (9.232) and (9.234) to show that for each ν = 1, . . . , n (ν)
dY ∼ Gν A0
(ν) daν (x − aν )A0 Cν , x − aν
and thus
(ν)
Ω = −Gν
(ν)
−A0 Y −1 ∼ C−1 G−1 ν (x − aν ) ν
A0 G−1 ν daν + O(1). x − aν
435
FREDHOLM DETERMINANT APPROACH
Show too that Ω = O
“ ” 1 x
as |x| → ∞. Use Liouville’s theorem to conclude Ω=−
n X ν=1
Aν daν , x − aν
and thus the equations (9.231) must hold.
9.10 NONLINEAR EQUATIONS FROM THE VIRASORO CONSTRAINTS In Section 5.7 special integrability properties of the β = 2 log-gas configuration integral (5.117) were considered. Consideration of such properties of the more general integral n P∞ j 1 Zn [{ti }; R] := dx1 · · · dxn e−V (xl ) e j=1 tj xl (xk − xj )2 , (9.235) n! R R 1≤j
l=1
where R=
3r i=1
[c2i−1 , c2i ] ⊂ support(e−V (x) ),
provides a further method to deduce nonlinear equations for gap probabilities [526]. The key ingredients for this purpose are the partial differential equation (5.136), which remains valid with Zn as specified by (5.117) replaced by Zn as specified by (9.235), and the appropriate generalization of the Virasoro constraints (5.128). It thus remains to consider the latter.
P ROPOSITION 9.10.1 Let Zn be given by (9.235), and suppose for x ∈ support(e−V (x) ) ∞ i g(x) i=0 bi x . V (x) = ∞ =: i f (x) i=0 ai x
(9.236)
With Bp :=
2r
cp+1 f (ci ) i
i=1
∂ ∂ci
and Lp as specified by (5.127) we have
− Bp +
∞ i=0
ai Lp+i − bi
∂ ∂tp+i+1
ZN = 0,
p ≥ −1.
Proof. Proceeding as in the proof of Proposition 5.7.2, we consider N DX ∂ f (xi ) E ∂xi z − xi ZN i=1 Z Z N P∞ Y j 1 := dx1 · · · dxN e−V (xk ) e j=1 tj xk N! E E k=1
Y 1≤j
(xk − xj )2
N X ∂ f (xi ) . ∂x i z − xi i=1
(9.237)
436
CHAPTER 9
Integration by parts shows −
r “ X f (c2k )N k=1
z − c2k
e−V (c2k )+
Z
×
P∞
Z dx1 · · ·
E
dxN−1 E
j=1 tj (c2k )
N−1 Y
j
e−V (xk ) e
Y
P∞
j j=1 tj xk
k=1
E
k=1
Y
N−1 Y
(c2k − xj )2
j=1
1≤j
j f (c2k−1 )N −V (c2k−1 )+P∞ j=1 tj (c2k−1 ) − e z − c2k−1 Z Z N−1 Y −V (x ) P∞ t xj k × dx1 · · · dxN−1 e e j=1 j k
E
(xk − xj )2
(xk − xj )2
1≤j
N−1 Y
(c2k−1 − xj )2
”
j=1
N DX f (xi )V (xi ) E − z − xi ZN i=1
+
N “ N N DX DX ”E X f (xi ) f (xi ) f (xi ) E + 2 + (z − xi )2 (xi − xk )(z − xi ) ZN z − xi ZN k=1 i=1 i=1 k=i
N DX
∞ E X ntn xn−1 i + f (xi ) =0 z − xi ZN i=1 n=1
(9.238)
The sum over k from 1 to r in this expression can be recognized as 2r X f (ck ) ∂ ZN z − ck ∂ck
(9.239)
N N DX DX f (xi )V (xi ) E g(xi ) E = . z − xi z − xi ZN ZN i=1 i=1
(9.240)
k=1
and we see from (9.236) that
Using the simple identity
“ 1 1 1 ” 1 + = (xi − xk )(z − xi ) xi − xk z − xi z − xk
it is easy to see that N “ N N DX DX ”E X f (xi ) f (xi ) f (xi ) E +2 + 2 (z − xi ) (xi − xk )(z − xi ) ZN z − xi ZN k=1 i=1 i=1 k=i
=
N D X i,k=1
N D X E f (xi ) − f (xk ) E f (xi ) + . (xi − xk )(z − xk ) ZN (z − xi )(z − xk ) ZN i,k=1
(9.241)
Substituting (9.239), (9.240), (9.241) in (9.238) shows −
+
2r N N DX D X X f (ck ) ∂ g(xi ) E f (xi ) − f (xk ) E ZN − + z − ck ∂ck z − xi ZN (xi − xk )(z − xk ) ZN i=1 k=1 i,k=1 N D X i,k=1
N ∞ DX E E X ntn xn−1 f (xi ) i + f (xi ) = 0, (z − xi )(z − xk ) ZN z − xi ZN i=1 n=1
(9.242)
437
FREDHOLM DETERMINANT APPROACH
and equating coefficients of z −p for p > 1 gives −
2r X
f (ck )cp−1 k
k=1
+
N N DX E D X ∂ f (xi ) − f (xk ) p−1 E ZN − xp−1 g(xi ) + xk i ∂ck (xi − xk ) ZN ZN i=1 i,k=1
N D X
X p−2
f (xi )
i,j=1
xki xp−2−k j
k=0
E ZN
+
N DX
f (xi )
i=1
∞ X
ntn xn+p−2 i
E
n=1
ZN
= 0.
Substituting the power series expansions (9.236) shows that we can write N DX
xp−1 g(xi ) i
E ZN
i=1 N D X
f (xi )
i,j=1 N DX
f (xi )
i=1
while using too the simple expansion
shows
X p−2
xki xp−2−k j
ZN
n=1
ntn xn+p−2 i
=
bi
∞ X
ai
i=0
E ZN
∞ X i=0
E
k=0 ∞ X
=
=
∞ X i=0
ai
∂ ZN , ∂tp+i−1 p−2 X
∂ ∂ ZN , ∂t ∂t k+i p−2−k k=0 ∞ X
ktk
k=1
∂ ZN , ∂tp+k−2
n−1 X n−1−p p a n − bn = a b a−b p=0
i−1 N ∞ D X X X f (xi ) − f (xk ) p−1 E ∂ ∂ = ai ZN . xk (x − x ) ∂t ∂t Z i p+i−2 k k N i=0 i,k=1 k=0
Noting that p−2 X k=0
p−2+i i−1 X X ∂ ∂ ∂ ∂ ∂ ∂ + = ∂tk+i ∂tp−2−k k=0 ∂tk ∂tp+i−2 ∂t ∂t k p+i−2−k k=0
we obtain (9.237) with p replaced by p − 2, p > 1. Equating coefficients of z −1 in (9.242) gives formally the same results, thus establishing (9.237) in the remaining case, p = −1.
Gap probabilities for the GUE Following [526], we will use the result of Proposition 9.10.1 to specify a partial differential equation for (9.243) F := log Zn 2 t1 =t2 =···=0, V (x)=x
in the variables {c1 , . . . , c2r }. First we note that with V (x) = x2 , (9.236) holds with a0 = 1, a1 = a2 = · · · = 0, b0 = 0, b1 = 2, b2 = b3 = · · · = 0. The first three Virasoro constraints (9.237) can then be written ∞ ∂ ∂ log Zn − nt1 , − iti −B−1 log Zn = 2 ∂t1 i=2 ∂ti−1 ∞ ∂ ∂ −B0 log Zn = 2 log Zn − n2 , − iti ∂t2 i=1 ∂ti
∞ ∂ ∂ ∂ log Zn . − 2n − iti −B1 log Zn = 2 ∂t3 ∂t1 i=1 ∂ti+1
(9.244)
438
CHAPTER 9
Using the equations (9.244) and the KP equation (5.136), the following partial differential equation for (9.243) can be derived. P ROPOSITION 9.10.2 We have 4 2 2 + (8n + 6B−1 F )B−1 + 4(3B02 − 4B−1 B1 + 6B0 ) F = 0. B−1
(9.245)
Proof. We want to substitute for the partial derivatives in (5.136), which all involve t-variables, partial derivatives involving c1 , . . . , c2r . For this purpose we make use of (9.244). It is easy to see from the first equation in (9.244) that ˛ 2 F B−1 ∂2 n ˛ log Z = + , ˛ n ∂t21 4 2 t1 =t2 =···=0
˛ 4 F B−1 ∂4 ˛ log Z = . ˛ n ∂t41 16 t1 =t2 =···=0
(9.246)
The second equation in (9.244) gives ˛ ∂ ˛ log Zn ˛ − n2 , ∂t2 t1 =t2 =···=0 ˛ “ ∂2 ∂ ” ˛ − log Z B02 F = 4 ˛ n ∂t22 ∂t2 t1 =t2 =···=0
−B0 F = 2
and thus we have
˛ “ B2 B0 ” n2 ∂2 ˛ 0 log Zn ˛ = − F+ . 2 ∂t2 4 2 2 t1 =t2 =···=0 A similar calculation using all three equations in (9.244) shows
(9.247)
˛ ∂2 B−1 3B0 F 3n2 ˛ log Zn ˛ = (B1 + nB−1 )F − + . ∂t1 ∂t3 4 4 4 t1 =t2 =···=0
(9.248)
Substituting (9.246)–(9.248) in (5.136) gives the stated result.
Consider the special case r = 1, (c1 , c2 ) = (−∞, x). Then in (9.245) we can make the replacements d d d , B0 → x , B1 → x2 dx dx dx and after simplification deduce the nonlinear differential equation d 2 d3 d F + (−4x2 + 8n) F + 4xF = 0. (9.249) F + 6 3 dx dx dx G (x; μ; ξ) μ=0 (recall Proposition 8.3.1), and thus F must satisfy the On the other hand in this case F = UN B−1 →
ξ=1
σPIV equation in (8.15) with α1 = 0 and α2 = −N . In fact results of Cosgrove [119], [120], reducing a class of third order nonlinear differential equations to the equations (8.15), can be used to give a direct reduction of (9.249) to the sought σPIV equation [526]. E XERCISES 9.10
1. [526] Write Zn,β [{ti }; R] :=
1 n!
Z
Z dx1 · · · R
dxn R
n Y
e−V (xl ) e
Y
P∞
j j=1 tj xl
l=1
|xk − xj |β
1≤j
(cf. (9.235)). The aim of this exercise is to show that (9.237) generalizes to read “
− Bp +
∞ “ X i=0
(β)
ai Lp+i − bi
∂ ∂tp+i+1
”” ZN,β = 0,
p ≥ −1.
439
FREDHOLM DETERMINANT APPROACH
where := L(β) p
p ∞ “ X ∂ ∂ ∂ βX ∂ β” (p + 1) + ktk + 1− . 2 ∂tk ∂tp−k ∂tp+k 2 ∂tp k=0
k=1
(i) Modify the working of the proof of Proposition 9.10.1 to show that (9.241) applies with ZN replaced by ZN,β provided N N D X E D X f (xi ) − f (xk ) E f (xi ) + (xi − xk )(z − xk ) ZN (z − xi )(z − xk ) ZN i,k=1
i,k=1
is replaced by N N D X E ” β “D X f (xi ) − f (xk ) E f (xi ) + 2 (xi − xk )(z − xk ) ZN (z − xi )(z − xk ) ZN i,k=1 i,k=1 N N “ β ”“D X f (xi ) E D X f (xi ) E” + 1− + . 2 (z − xi )2 z − xi i=1 i=1
(ii) Show that the coefficient of z −p , p ≥ 1, in the final line above is equal to (1 − β/2)
∞ “X k=0
ak (p − 1 + k)
∂ ∂tk+p−2
” ZN,β
and use this together with the working below (9.241) to deduce the sought result.
Chapter Ten Lattice paths and growth models Nonintersecting lattice paths are the space-time trajectories of random walkers on a one-dimensional lattice. In some circumstances the probability distribution for the final position of the paths can be written in the form of the Boltzmann factor of a β = 1 or 2 log-gas in which the particles are confined to lattice sites, or as an average over the unitary or symplectic groups. An important role is played by the Schur polynomials, which can be defined combinatorially in terms of weighted lattice paths. Tiling of a hexagon by three species of rhombi is equivalent to some nonintersecting lattice path configurations. By analyzing the vertical rhombi, a multi-species system referred to as a bead process is obtained; the p.d.f. for any one of these species have the form of the JUE, while a further scaled limit gives the joint p.d.f. for the GUE minor process. The Robinson-Schensted-Knuth correspondence from bijective combinatorics leads to a statistical model—the discrete polynuclear growth model—which can be specified in terms of nonintersecting paths. The maximum height can be given as a matrix integral over the unitary group, while imposing certain symmetry constraints on the underlying matrix of nucleation events gives rise to matrix integrals over the symplectic and orthogonal groups. An alternative interpretation of the polynuclear growth model is as a model of directed last passage percolation. This permits a limit to the Hammersley process—a last passage percolation model associated with points distributed in a unit square according to a Poisson rate. The Hammersley process in turn relates to the length of the largest increasing subsequence of a random permutation, and results from random matrix theory relating to spacing distributions are used to compute the corresponding scaled distribution.
10.1 COUNTING FORMULAS FOR DIRECTED NONINTERSECTING PATHS 10.1.1 Nonintersecting ld/rd lattice paths (0)
(0)
(0)
(0)
Consider the square lattice, and mark in points along the x-axis at l1 > l2 > · · · > lp where all lj are required to have the same parity (i.e. be all even or all odd). For each j = 1, . . . , p move a copy of the point (0) (0) (0) (lj , 0) to (lj − 1, 1) or (lj + 1, 1) according to the rule that all the new points are distinct, and draw a left or right sloping segment connecting the point and its copy. Weight each line segment w1∓ according to it sloping to the left (right). We build up p weighted nonintersecting paths each of N segments (also referred to as steps) by repeating this procedure a total of N times, with the weight for going from points along y = j − 1 to points along y = j given by wj∓ for left (right) diagonal segments (abbreviated ld, rd, respectively). An example is given in Figure 10.1. The lattice paths can be considered as the space-time trajectories of random walkers on a one-dimensional lattice. At each time interval the random walkers can move one step to the left or one step to the right subject to the constraint that no two walkers can occupy the same site. In this picture the model under consideration is referred to as the lock step model of vicious walkers [175]. Let us denote the x-coordinates of the final positions by {li }i=1,...,p , l1 > l2 > · · · > lp . We seek ld/rd the total weight GN (l(0) ; l) of all allowed nonintersecting ld/rd lattice paths of N segments starting at (0) (0) (0) (0) {li }i=1,...,p , l1 > l2 > · · · > lp , and finishing at {li }i=1,...,p . This can be determined as a special case of a theorem due to Linstr¨om, Gessel and Viennot (see, e.g., [475]) on computing the generating function for the total weight of a general class of nonintersecting lattice paths.
441
LATTICE PATHS AND GROWTH MODELS
y
y
x
x 1
1
2
4
1
Figure 10.1 A configuration of 3 nonintersecting left diagonal/right diagonal lattice paths of 4 steps starting two units apart (leftmost diagram). An equivalent configuration of up/right diagonal lattice paths starting one unit apart is given in the rightmost diagram, and the paths are also encoded as a semi-standard tableau, with the kth column corresponding to the kth lattice path from the right. The total weight of the right steps in the configuration can be read off from the semi-standard tableau as w1+#1 s · · · wp+#p s .
P ROPOSITION 10.1.1 Let D denote a directed acyclic graph (i.e., no loops), and let u = (u1 , . . . , ur ), v = (v1 , . . . , vr ) be sets of vertices in D so that in connecting {ui } to {vi } by nonintersecting paths along edges of the graph, the only possibility is to connect ui to vi for each i = 1, 2, . . . , r. Let each edge of the graph be weighted, let h(u, v) denote the total weight of all single paths from u to v, and let H({u}, {v}) denote the total weight of all nonintersecting paths starting at {u} and finishing at {v}. Then H({u}, {v}) = det[h(ui , vj )]i,j=1,...,r .
(10.1)
Proof. Following [502], we note that by definition det[h(ui , vj )]i,j=1,...,r =
X
sgn(σ)h(u1 , vσ(1) ) · · · h(ur , vσ(r) )
(10.2)
σ∈Sr
(recall (5.22)). With Pi denoting a path from ui to vσ(i) , and w(Pi ) := h(ui , vσ(i) ) its corresponding weight, this expansion can be viewed as a generating function for (r + 1)-tuples (σ, P1 , . . . , Pr ), assigned weight sgn(σ)w(P1 ) · · · w(Pr ). We want to show that in the sum all such weights of intersecting paths cancel in pairs, while for the nonintersecting paths sgn(σ) = 1. Choose a total ordering of the vertices so that u1 < · · · < ur < v1 < · · · < vr . Consider a (σ, P1 , . . . , Pr ) with at least one pair of intersecting paths, and let v be the smallest vertex in the total ordering such that the paths intersect. Let i and j be the smallest indices such that paths Pi and Pj pass through v, and introduce the notation Pk (→ v) and Pk (v →) to denote the subpaths of Pk from uk to v and v to vσ(k) . Next define (σ , P1 , . . . , Pr ) such that Pl = Pl for l = i, j and Pi = Pi (→ v)Pj (v →), Pj = Pj (→ v)Pi (v →), σ = σ ◦ (i, j) (see Figure 10.2). We note that the set of edges of (P1 , . . . , Pr ) is identical to the set of edges of (P1 , . . . , Pr ) and thus w(P1 ) · · · w(Pr ) = w(P1 ) · · · w(Pr ), while sgn(σ ) = −sgn(σ). Consequently sgn(σ)w(P1 ) · · · w(Pr ) + sgn(σ )w(P1 ) · · · w(Pr ) = 0.
(10.3)
Since the graph D is assumed acyclic, this new set of paths has the same set of intersection points as before (as illustrated in [502], if D has cycles an intersection between two paths may be mapped to a self-intersection, thus violating this property). Hence this construction is an involution and so gives a unique pairing of intersecting paths each with the property (10.3), demonstrating that their contribution cancels in (10.2).
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CHAPTER 10
P1’
P2’
P1’
v
P1
P2
P2’
v
P1
P2
Figure 10.2 The intersecting lattice paths from P1 to P1 and P2 to P2 in the second diagram are constructed from the intersecting lattice paths from P1 to P2 and P2 to P1 according to the prescription in the proof of Proposition 10.1.1. It remains to show that the nonintersecting paths have sgn(σ) = 1. In fact it follows from the assumption on {u} and {v} that the only time nonintersecting paths occur is when σ = I (the identity), so this property is immediate.
To apply Proposition 10.1.1 to nonintersecting ld/rd lattice paths, note that the ld and rd segments form the edges of a directed graph, and that the generating function (total weight) for single lattice paths from l(0) to l is π N (0) 1 ld/rd (wj− e−iθ + wj+ eiθ )e−i(l−l )θ dθ =: gN (l(0) ; l). (10.4) 2π −π j=1 Thus ld/rd
GN
ld/rd (0) (l(0) ; l) = det gN (lj , lk )
j,k=1,...,p
.
(10.5)
The case that the initial and final sites are all odd, l(0) = (2p − 1, 2p − 3, . . . , 1),
l = (2l1 − 1, . . . , 2lp − 1),
(10.6)
or all even, so that the initial points are the minimum allowed distance apart (2 lattice sites), has some special features. First it is easy to see that then each configuration of nonintersecting lattice paths is equivalent to a configuration of nonintersecting lattice paths starting one site apart with segments either vertical or diagonal and to the right (see Figure 10.1). u/rd Let GN denote the total weight of the up/right diagonal (abbreviated u/rd) nonintersecting paths in the case that each rd path segment at step j is weighted qj , and each vertical path segment is weighted unity. From Proposition 10.1.1, for general initial conditions u/rd u/rd (0) GN (m (0) ; m) = det gN (mj , mk ) , (10.7) j,k=1,...,p
where, with ej denoting the elementary symmetric functions (polynomials) as defined in (4.132) (en := 0 for n ∈ Z− ), π N (0) 1 u/rd (0) gN (m ; m) = (1 + qj eiθ )e−i(m−m )θ dθ 2π −π j=1 = em−m(0) (q1 , . . . , qN ).
(10.8)
In the case that the paths start one unit apart, the number of distinct u/rd nonintersecting paths can be written in terms of the components of m in a simple form [365].
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LATTICE PATHS AND GROWTH MODELS
P ROPOSITION 10.1.2 Let m (0) = (p − 1, p − 2, . . . , 0) and set qj = 1 (j = 1, . . . , p). We have u/rd
GN
(m (0) ; m) =
p
(N + p − i)! m !(p + N − 1 − mi )! i i=1
(mj − mk ).
(10.9)
1≤j
Proof. With qj = 1 (j = 1, . . . , p) we see from (10.8) that u/rd
gN
(m(0) ; m) =
“
” N . (0) m−m
(10.10)
Substituting in (10.7) and extracting a common factor from each column shows u/rd
GN
(m (0) ; m) p i h Y N! , (10.11) = det (mk − p + j + 1)p−j (N − mk + p − j + 1)j−1 mi !(p + N − 1 − mi )! j,k=1,...,p i=1
where (a)j is specified in (5.83). But in general [364] h i det (xj + ak+1 ) · · · (xj + an )(xj + b2 ) · · · (xj + bk )
j,k=1,...,n
=
Y
(xk − xj )
1≤j
Y
(ak − bj ). (10.12)
2≤j≤k≤n
This can be seen by subtracting columns j + 1 from columns j (j = 1, . . . , n − 1) in order, and extracting the common factors of (bn − an ), (bn−1 − an−1 ), . . . , (b2 − a2 ) which result in the new columns j = 1, . . . , n − 1. Repeating this procedure starting from column 2 then column 3 etc. reduces the determinant to h i Y (bj − ak ) det (xj + ak+1 ) · · · (xj + an ) . j,k=1,...,n
2≤j≤k≤n
Q The argument used to evaluate the Vandermonde determinant (1.173) shows that the last determinant is equal to n j
xj = mj − p,
ak = k,
bk = k − 1 − N − p
and (10.9) results.
We note that (10.9) has the form of a Boltzmann factor of a one-component log-gas at β = 1 with its domain the integer lattice points −N ≤ l ≤ N + p − 1, subject to a one-body potential with Boltzmann factor w(l) proportional to 1/(N + p − 1 − l)!(N + l)!. Each configuration of these new u/rd nonintersecting paths is equivalent to a semi-standard tableau. D EFINITION 10.1.3 Let λ1 ≥ λ2 ≥ · · · ≥ λn ≥ 0 be non-negative integers. Refer to this ordered set of non-negative integers by λ, which in turn is called a partition. The diagram of a partition is constructed by imagining a blank matrix grid, then for each row j = 1, . . . , n inserting in this grid λj boxes of unit dimension drawn side-by-side starting from column 1. The conjugate partition λ is defined as a diagram by interchanging the rows and columns of the diagram of λ. A semi-standard tableau of shape λ and content N is then a filling of this diagram with the numbers {1, 2, . . . , N } such that the numbers weakly increase along any row, and strictly increase down any column. Given a u/rd nonintersecting lattice path configuration, a semi-standard tableau can be constructed by labeling each right diagonal segment between y = j − 1 and y = j by the number j. The numbers of the rightmost lattice path form the first column of the tableau, the numbers of the second rightmost lattice path form the second column and so on. Here the length of each column k of the tableau (k = 1, . . . , p) is equal to mp−k+1 − (p − k + 1). An example of the construction is given in Figure 10.1. Of fundamental importance is a particular weighted sum over all semi-standard tableaux of a given shape.
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D EFINITION 10.1.4 For semi-standard tableaux of a given shape λ, and content N , define the total weight by #N s sλ (q1 , . . . , qN ) := q1#1 s q2#2 s · · · qN . (10.13) semi standard tableaux shape λ, content N
This is referred to as the Schur polynomial. u/rd
Let GN
be as in (10.7). From the above discussion it follows that u/rd
GN
((p − 1, . . . , 0); (λ1 + p − 1, . . . , λp )) = sλ (q1 , . . . , qN ).
Combining (10.14) and (10.7) and recalling (10.8) shows sλ (q1 , . . . , qN ) = det eλk +j−k
j,k=1,...,p
,
(10.14)
(10.15)
where p = λ1 . Observe from (10.15) that sλ is a symmetric function of q1 , . . . , qN , a property which is not obvious from the definition (10.13). A recurrence implied by the definition (10.13) can be used to verify an expression for the Schur polynomials as the ratio of two determinants. P ROPOSITION 10.1.5 We have sλ (q1 , . . . , qN ) =
det[qjN −k+λk ]j,k=1,...,N det[qjN −k ]j,k=1,...,N
.
(10.16)
Proof. First note from the definition (10.13) that, with ∗ denoting the region λ1 ≥ μ1 ≥ λ2 ≥ μ2 ≥ · · · ≥ λN−1 ≥ μN−1 ≥ λN ,
(10.17)
the Schur polynomials satisfy the recurrence sλ (q1 , . . . , qN ) =
X∗ μ
|λ|−|μ|
sμ (q1 , . . . , qN−1 )qN
.
(10.18)
Subject to the initial condition sλ = 1 for λ = ∅, this uniquely determines the Schur polynomials. Let us denote the r.h.s. of (10.16) by rλ (q1 , . . . , qN ). Following [453], our objective is to show that rλ satisfies the same recurrence and initial condition. The initial condition is immediate. To derive the recurrence, we begin by setting qN = 1 in the determinants and subtracting the last row (which now consists of 1’s in each entry) from all the other rows. Dividing the jth row by qj − 1 (j = 1, . . . , N − 1), and expanding each term as a power series shows » PN−k+λ −1 l – k qj l=0 det 1 j=1,...,N −1 k=1,...,N . (10.19) rλ (q1 , . . . , qN−1 , 1) = » PN−k−1 l – q j l=0 det 1 j=1,...,N −1 k=1,...,N
Next in each determinant, subtract column k from column (k − 1), k = 2, . . . , N in order. The last row then has only its final element nonzero. Expanding by this element shows that (10.19) reduces to the ratio of (N − 1)-dimensional determinants i hP λk l+N−k−1 det l=λk+1 qj X∗ j,k=1,...,N−1 ˆ k˜ = rμ (q1 , . . . , qN−1 ). (10.20) det qj j,k=1,...,N−1 PN Because the determinant in (10.16) is homogeneous of degree |λ| := l=1 λl it follows from the equality between (10.19) and (10.16) that rλ satisfies the recurrence (10.18).
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LATTICE PATHS AND GROWTH MODELS
A fundamental property of the Schur polynomials relates to their orthonormality with respect to averaging over the unitary group, or equivalently over the circular unitary ensemble, sλ (e−iθ1 , . . . , e−iθN )sκ (eiθ1 , . . . , eiθN ) = δλ,κ . (10.21) U(N )
This property is simple to verify from the determinant formula in (10.16). It can also be understood from the fact that the Schur polynomials are eigenfunctions of a certain differential operator which is self-adjoint with respect to the inner product f¯gU (N ) (see Section 12.1.2). A further corollary of (10.16), together with the Vandermonde determinant evaluation (1.173), is the specialization formula PN 1 − q λj −λk +k−j sλ (1, q, . . . , q N −1 ) = q j=1 (j−1)λj . (10.22) 1 − q k−j 1≤j
In the limit q → 1 this yields the evaluation formula sλ (x1 , . . . , xN ) := sλ ((1)N ) = x1 =···=xN =1
1≤j
λj − λk + k − j . k−j
(10.23)
Given a semi-standard tableau, a nonintersecting lattice path configuration can also be constructed where each row (rather than column) corresponds to a lattice path. This will be referred to as the conjugate configuration. Thus if there are p∗ rows, one marks points along y = 1 at x = 0, 1, . . . , p∗ − 1. The numbers in row j correspond to a horizontal right segment of lattice path number p∗ − j at y = j which is weighted by qj , while otherwise the lattice path segments are up. If the length of row j in the tableau is equal to λj , then the final x-coordinate of walker j is λj + p∗ − j (see Figure 10.3). Let the total weight of such paths be denoted u/rh GN . From the definition (10.13) we have u/rh
GN
((p∗ − 1, p∗ − 2, . . . , 0); (λ1 + p∗ − 1, . . . , λp∗ )) = sλ (q1 , . . . , qN ).
(10.24)
u/rh
We can use (10.24) to obtain a different expression for GN and thus the Schur polynomials. For this, introduce the complete symmetric functions (polynomials) hj in the variables q1 , . . . , qn by qk1 · · · qkj = q1l1 q2l2 · · · qnln (j ≥ 1), hj = 1≤k1 ≤···≤kj ≤n
l1 ,...,ln ≥0 l1 +···+ln =j
h0 = 1, h−j = 0. These polynomials are generated by n j=1
∞
1 = hj u j . 1 − qj u j=0
(10.25)
Noting that ∞
1 = 1 + qjl eilθ 1 − qj eiθ l=1
and making use of (10.25) we see that the generating function for a single u/rh path is π N (0) 1 1 u/rh gN (r(0) ; r) = e−i(r−r )θ dθ = hr−r(0) . iθ 2π −π 1 − ql e
(10.26)
l=1
Substituting this in (10.1) and comparing with (10.24) shows sλ (q1 , . . . , qN ) = det hλk +j−k
j,k=1,...,p∗
,
(10.27)
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y 1
1
1
2
2
4
5
5
3
6
6
x
Figure 10.3 A configuration of 3 u/rh nonintersecting lattice paths starting one unit apart with horizontal segments in any of 6 consecutive levels. The level of each right horizontal step of the jth walker from the right is recorded in the jth row to obtain a semi-standard tableau.
where p∗ = μ1 . Another consequence of (10.24), and the evaluation formula (10.23), is a counting formula for the number of u/rh nonintersecting paths starting one unit apart and finishing at arbitrary points l1 , . . . , lp∗ [282]. u/rh
P ROPOSITION 10.1.6 Let l(0) = (p∗ − 1, p∗ − 2, . . . , 0), l = (l1 , . . . , lp∗ ). Let GN number of u/rh paths from l(0) at y = 1 to l at y = N . We have ∗
u/rh GN (l(0) ; l)
=
p i=1
(N − p∗ + li )! li !(N − p∗ + i − 1)!
(lj − lk ).
(l(0) ; l) denote the
(10.28)
1≤j
Proof. With q1 = · · · = qN = 1 in (10.24), substitute (10.23), set λj = lj + j − p∗ (j = 1, . . . , p∗ ), λj = 0 (j = p∗ + 1, . . . , N ), and simplify.
10.1.2 Nonintersecting lattice paths near a wall In the nonintersecting ld/rd lattice path model of the previous section, suppose the paths are conditioned so that their x-coordinate is always positive. The y-axis can then be thought of as an infinitely repelling wall. Consider a single weighted ld/rd lattice path of N segments starting at l(0) and finishing at l with both ld/rd l(0) , l > 0. We know the total weight of such paths is given by gN as specified in (10.4). Consider now w,ld/rd
gN
ld/rd
(l(0) , l) := gN
ld/rd
(l(0) , l) − gN
(l(0) , −l).
(10.29)
To go from l(0) to −l the path must cross the y-axis at least once. If we reflect the portion of the lattice path from the first crossing to −l about the y-axis we obtain a lattice path from l(0) to l which touches or crosses ld/rd the y-axis (see Figure 10.4). This lattice path will be in gN and thus will cancel out of (10.29), leaving only the contribution from lattice paths strictly to the right of the y-axis. The formula (10.1) shows that forming a determinant out of (10.29) gives the total weight of multiple nonintersecting paths that are restricted to the region x > 0 [180]. Furthermore, in the special case that the walkers start at the minimum allowed distance from the wall (1 lattice site) and the minimum distance from each other (2 lattice sites), the determinant can be evaluated [365]. w,ld/rd
P ROPOSITION 10.1.7 Define gN
w,ld/rd
by (10.29) and let GN
(0)
(0)
(l1 , . . . , lp ; l1 , . . . , lp ) denote the total
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LATTICE PATHS AND GROWTH MODELS
y
x
Figure 10.4 In bold is drawn an ld/rd lattice path starting at x = l(0) = 1 and finishing at x = −l = −3 after 6 steps, while the union of this path from x = 1 to its first contact with the y-axis and the reflection of the remaining portion of the path in the y-axis gives a path starting at x = l(0) = 1 and finishing at x = l = 3.
weight of the lattice paths. We have w,ld/rd
GN
(0) w,ld/rd (0) (l1 , . . . , lp(0) ; l1 , . . . , lp ) = det gN (lj , lk )
j,k=1,...,p
and it follows from this that w,ld/rd
G2N
=
(0)
({lj
p i=1
= 2(p − j) + 1}j=1,...,p ; 2l1 − 1, . . . , 2lp − 1)
(2li − 1)(2N + 2i − 2)! (N + li + p − 1)!(N − li + p)!
,
(10.30)
± w =1 k (k=1,...,2N )
(li − lj )(li + lj − 1).
(10.31)
1≤i<j≤p
Proof. As already commented (10.30) follows from (10.1). Regarding (10.31), we make use of (10.10) then proceed as in the derivation of (10.11) to deduce ˛ ˛ w,ld/rd (0) G2N ({lj = 2(p − j) + 1}j=1,...,p ; 2l1 − 1, . . . , 2lp − 1)˛ w± =1 k (k=1,...,2N )
=
p Y i=1
h (2N )! det (N + p − j + 2 − lk )j−1 (N − p + j)2p−j (N + p − li )!(N + p − 1 + li )! i −(N + p − j + 1 + lk )j−1 (N − p + j + 1 − lk )2p−j . j,k=1,...,p
(10.32)
We observe that this determinant with j → p − j + 1 is the special case of the determinant Wp ({xk }k=1,...,p , {aj }j=2,...,p , {bj }j=−p+1,...,p−1 ) i h := det (aj+1 − xk )(aj+2 − xk ) · · · (ap − xk )(b−j+1 + xk )(b−j+2 + xk ) · · · (bp−1 + xk ) − ∗
(10.33)
j,k=1,...,p
where the * denotes the same term but with xk → −xk , obtained by setting x k = lk −
1 , 2
aj = N + j −
1 , 2
bj = N + j +
1 . 2
(10.34)
But in general (see Exercises 10.1 q.3) Wp ({xk }k=1,...,p , {aj }j=2,...,p , {bj }j=−p+1,...,p−1 ) p p p Y Y Y Y = 2p xl (xk − xj )(xk + xj ) (b−j−1+l + aj ) l=1
1≤j
l=2 j=l
Y 0≤j
(bj + bk ).
(10.35)
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Substituting (10.34) we obtain (10.31) after some minor simplification.
The nonintersecting w,ld/rd lattice paths can be mapped to lattice paths restricted to x > 0 in which the allowed steps are u/rd for odd numbered steps and u/ld for even numbered steps. The configurations of the latter are denoted w,u/rd(o)/ld(e), where (e) denotes (e)ven while (o) denotes (o)dd. The steps up are to be weighted unity, the steps rd(o) weighted wj+ for step number 2j − 1, while the steps ld(e) weighted wj− for step number 2j. For the initial conditions, if the original x coordinates in the w,ld/rd case are specified by (0) (0) (0) 2l1 − 1, 2l2 − 1, . . . , 2lp − 1 say, then the original x-coordinates of the case of the w,u/rd(o)/ld(e) lattice (0) (0) (0) paths are given by l1 , l2 , . . . , lp . At odd numbered steps the ld segments then map to up segments with weight 1 and the rd segments remain unaltered, while at even numbered steps the rd segments map to up segments with weight 1 and the ld segments remain unaltered (cf. Exercises 10.1 q.2). It is easy to see that if the final x-coordinates after 2N steps in the w,ld/rd case are 2l1 − 1, . . . , 2lp − 1 then the final x-coordinates in the case of the w,u/rd(o)/ld(e) lattice paths are given by mj = lj (j = 1, . . . , p). Thus in particular w,u/rd(o)/ld(e)
G2N
w,ld/rd
= G2N
(0)
(r1 , . . . , rp(0) ; r1 , . . . , rp )
(0) (2r1 − 1, . . . , 2rp(0) − 1; 2r1 − 1, . . . , 2rp − 1) w−
+ =w =1 2j−1 2j (j=1,...,N )
.
(10.36)
For the initial spacings 1 unit apart, the w,rd(o)/ld(e) lattice paths (or equivalently the w,u/rd/ld lattice paths) can be encoded as oscillating tableaux. The latter are a sequence of semi-standard tableaux, one for each y = 1, 2, . . . , 2N for paths of 2N segments, in which an rd(o) segment of the kth lattice path from the right between y = 2j − 2 and y = 2j − 1 is recorded in column k by a box labeled j, while an ld(e) segment of the same lattice path in going between y = 2j − 1 and y = 2j is recorded by removing the bottom box from column k (see Figure 10.5). Analogous to the situation with semi-standard tableaux noted in Section 10.1.1, by reading the oscillating tableaux along rows rather than down columns we can associate a conjugate set of lattice paths. These lattice paths start one unit from the wall and one unit from each other, and for odd (even) y-coordinates can move an arbitrary number of units to the right (left), or one unit up, subject to the wall constraint that x > 0. Such paths are denoted w,u/rh(o)/lh(e).
1
1
1
1
2
2
2
3 3
2 1 2
2
1
2
2
3
3
3
3
3
1 2 3
Figure 10.5 Illustration of the correspondence between w,u/rd/ld lattice paths, their conjugate w,u/rh(o)/lh(e) lattice paths and oscillating tableaux.
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LATTICE PATHS AND GROWTH MODELS
10.1.3 Single move ld/rd lattice paths As an extension of the ld/rd nonintersecting paths model of Section 10.1.1, allow in addition up steps to get ld/rd/u lattice paths. At step j weight the ld/rd segments by wj− and wj+ , respectively, and the up steps by unity. The generating function for a single path of N segments is π N (0) 1 ld/rd/u (0) gN (l ; l) = (1 + wk+ eiθ + wk− e−iθ )e−iθ(l−l ) dθ. (10.37) 2π −π k=1
Application of Proposition 10.1.1 on the directed lattice formed by left and right sloping diagonal lines as well as vertical lines, gives that the generating function for multiple ld/rd/u paths is equal to ld/rd/u (0) det gN (lj ; lk ) . (10.38) j,k=1,...,l
Suppose now that at each step j exactly one segment is ld or rd, while all others are up. Denote such lattice paths by s,ld/rd (“s” is for single). In the picture of evolution of random walkers on a one-dimensional lattice, this means that at each time step exactly one walker moves to the left or right while all other walkers remain stationary, specifying the so-called random turns model of vicious walkers [175]. We remark that in the version of this model in which only steps in one direction (say to the right) are allowed, and in which the walkers are initially one unit apart, the configurations can be encoded as a standard tableau (see Figure 10.6). A standard tableau is defined as for a semi-standard tableau, except that the content of the tableau must equal the number of boxes, and so each box must be labeled by a different number. y
y
x
x 1
2
3
4
Figure 10.6 Two configurations of 3 nonintersecting single move ld/rd lattice paths of 4 steps. In the second configuration the paths are initially one unit apart and only rd segments are allowed. In this case the paths can be encoded as a standard tableau.
The total weight of p nonintersecting s,ld/rd paths of N steps is the term in (10.38) linear in each of wk+ , (k = 1, . . . , N ). Hence we read off from (10.37) and (10.38) the following result [183].
wk−
P ROPOSITION 10.1.8 We have s,ld/rd
GN
(l(0) ; l) =
1 (2π)p ×
π
−π p N l=1 k=1
dθ1 · · ·
π
−π
dθp
(0) wk+ eiθj + wk− e−iθj det e−i(lα −lβ )θα
α,β=1,...,p
.
(10.39)
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The w,ld/rd lattice paths of Section 10.1.2 can be similarly modified to accommodate the single move condition [41]. s,w,ld/rd
P ROPOSITION 10.1.9 Let GN paths restricted to x > 0. We have
s,w,ld/rd (0) GN (l ; l) =
(l(0) ; l) denote the total weight of p nonintersecting single move ld/rd
1 (2π)p
π
−π
× det e
dθ1 · · ·
π
−π
(0) −i(lα −lβ )θα
dθp
−e
p N k=1
j=1 (0)
−i(lα +lβ )θα
wk+ eiθj + wk− e−iθj
α,β=1,...,p
.
(10.40)
10.1.4 Nonintersecting paths on a triangular lattice Following [330], consider a triangular lattice with sites formed from the integer span of the vectors u1 = (−1, −1) and u2 = (−1, 1), and with edges formed by connecting nearest neighbour points in the directions of u1 , u2 and u1 + u2 (see Figure 10.7). Let us use the basis U = {u1 , u2 } to specify a general site via the coordinate (a, b)U := au1 + bu2 . We define a directed path on this lattice by traversing the edges in their positive direction only. With reference to the xy-coordinate system, such paths consist of left diagonal (ld), negative right diagonal (−rd) and left horizontal steps. Let each step in the direction of u1 or u2 be weighted w, while the steps in the direction of u1 + u2 are to be weighted unity. Let g ld/−rd/lh ((0, 0)U , (n, m)U ) denote the weight of all allowed lattice paths starting at the origin and finishing at (n, m)U .
u2
u1 Figure 10.7 Axes in the directions of the vectors u1 = (−1, −1) and u2 = (−1, 1) provide a convenient coordinate system for the triangular lattice formed by the integer span of these vectors, and connecting points as described in the text. Drawn on this lattice is a family of 3 nonintersecting rd/−ld/rh lattice paths, starting at (1, 0)U , (2, 0)U , (3, 0)U and finishing at (4, hj )U for particular hj .
P ROPOSITION 10.1.10 We have g
ld/−rd/lh
∞ n! wn+m w−2c . ((0, 0)U , (n, m)U ) = (m − c + 1)n n! c=0 (n − c)!c!
(10.41)
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LATTICE PATHS AND GROWTH MODELS
Proof. Let a, b, c denote the number of steps in the direction of u1 , u2 , u1 +u2 , respectively. Then n = a+c, m = b+c, while the number of paths with a given number of steps a, b, c is equal to (n + m − c)! (a + b + c)! = a!b!c! (n − c)!(m − c)!c! each of which is to be weighted by wa+b = wn+m−2c . Multiplying together the right-hand sides of the two displayed equations and summing over c gives (10.41).
Knowledge of (10.41) allows the total weight of p nonintersecting paths starting at (p + 1 − j, 0)U (j = 1, . . . , p) and finishing at (p, hj )U with 0 ≤ h1 < · · · < hp to be computed. P ROPOSITION 10.1.11 Let Gld/−rd/rh ({(p + 1 − j, 0)U }j=1,...,p ; {(p, hj )U }j=1,...,p ) denote the sought total weight. We have Gld/−rd/rh ({(p + 1 − j, 0)U }j=1,...,p ; {(p, hj )U }j=1,...,p ) p (1 + w2 )j−1 = whj j−1 = det g ld/−rd/lh((0, 0)U , (j − 1, hk )U ) w (j − 1)! j,k=1,...,p j=1
(hk − hj ).
1≤j
Proof. The first equality follows immediately from the fact that the paths are directed and Proposition 10.1.1. For the second equality, we note that with (10.41) substituted, the determinant is equal to p Y wj−1+hj (j − 1)! j=1
∞ X c1 ,...,cp
p h i Y (i − ci )ci det (h − c + 1) . j i i−1 ci !w2ci i,j=1,...,p =0 i=1
Elementary row operations reduce the determinant in this expression to the Vandermonde determinant (1.173) in the variables {h1 , . . . , hp } independent of the ci , and so the multiple summation is a product of p independent sums of the form ∞ i−1 “ “ X X 1 ”i−1 i − 1” 1 1 (i − c)c = = 1 + . c!w2c w2c w2 c c=0 c=0
10.1.5 Continuous nonintersecting paths Consider the generating function (10.4) for a single ld/rd lattice path in the case wj− = wj+ = 1/2, and suppose for definiteness that N , l, l(0) are all even, which is accomplished by writing N → 2n, l → 2l, l(0) → 2l(0) . We then have 2N ld/rd , (10.42) g2N (2l(0) , 2l) = 2−2N N + l − l(0) which also have the interpretation as being the probability that a walker reaches site 2l after 2N steps, starting at site l(0) and given that at each step the walker goes to the right or to the left each with probability 12 . The corresponding mean square displacement is N
ld/rd
(2l)2 g2N (0, 2l) = 2N.
l=−N
In the random walk picture, suppose now that the spacing between lattice sites is τ , and consider a continuum limit in which τ → 0, N, l, l(0) → ∞, 2lτ → x, 2l(0) τ → x(0) , 2N τ 2 → Dt.
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CHAPTER 10
Then, using Stirling’s formula, one obtains from (10.42) the limiting p.d.f. 1 1/2 (0) 2 1 ld/rd (0) (B) e−(x−x ) )/2Dt =: gt (x(0) , x), g2N (2l , 2l) → 2τ 2πDt (B)
which has mean square displacement Dt. Note that gt diffusion equation
(10.43)
(B for Brownian) is the unique solution of the
1 ∂u 1 ∂2u = D ∂t 2 ∂x2 subject to the initial condition u|t=0 = δ(x − x(0) ). It follows from (10.43) and (10.5) that the p.d.f. for p nonintersecting Brownian walkers, or equivalently for continuous nonintersecting paths in the xt-plane, is given by the Karlin-MacGregor formula [347] (B)
(B)
(0)
Gt (x(0) ; x) = det[gt (xj , xk )]j,k=1,...,p 1 p/2 Pp (0) 2 (0) 2 = e− j=1 (xj +(xj ) )/2Dt det[exj xk /Dt ]j,k=1,...,p . 2πDt (0)
Setting xj
(10.44)
= (j − 1)a (equally spaced initial condition) and using the Vandermonde formula (1.173) shows
(B) Gt ({(j
− 1)a}; x) has the functional form (2.73) with β = 1. that Suppose now that the positions x(0) at time t are prescribed, and the nonintersecting condition is required up to time T . For a given starting position y define the average over positions time t later as (B) Gt (y ; x) dx. Nt (y ) = xN >···>x1
For 0 < t0 < t < T the p.d.f. for the particles arriving at positions x at time t is then [349] (B)
Gt−t0 (x(0) ; x)NT −t (x) , NT −t0 (x(0) )
(10.45)
as the numerator is the p.d.f. for the event that there is a transition from x(0) to x in time t−t0 , and that following this the particles continue to obey the nonintersecting condition, while the denominator is the p.d.f. for the nonintersecting condition on the prescribed x(0) . The following relationship to the GUE eigenvalue p.d.f. can now be deduced. P ROPOSITION 10.1.12 For p continuous nonintersecting Brownian walkers, starting at the origin and with the nonintersecting condition required for all times, the p.d.f. for the event that they arrive at x after time t is 1 p2 /2 Pp 2 1 e− j=1 xj /2Dt (xk − xj )2 . (10.46) p−1 p/2 Dt (2π) l! l=1 1≤j
Proof. Since the determinant in the final expression of (10.44) vanishes for any of {x(0) j } equal or {xj } equal, it follows that for large t (0)
det[exj
xk /Dt
“ 1 ”p(p−1)/2 1 ]j,k=1,...,p ∼ Qp−1 Dt l=1 l!
Y
(0)
1≤j
Consequently, in the limit T → ∞ (10.45) reduces to (B)
Q
1≤j
x(0) ; x) Gt−t0 ( Q
xj )
(0)
(xk − xj )(xk − xj ).
.
Substituting (10.44), the limits x(0) → 0 can now be taken by further use of (10.47).
(10.47)
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LATTICE PATHS AND GROWTH MODELS
The continuous limit of nonintersecting paths near a wall is considered in Exercises 10.1 q.6. E XERCISES 10.1
1. Consider 2N -step ld/rd lattice paths with − w2j−1 = 1,
+ w2j−1 = aj ,
+ w2j = 1,
− w2j = bj
(j = 1, . . . , N ).
(i) Show that the generating function for a single path is ld/rd
g2N (2l(0) , 2l) =
1 2π
Z
π
N Y
(0)
(1 + aj eiθ )(1 + bj e−iθ )e−i(l−l
)θ
dθ.
−π j=1
(ii) Make use of (10.1) and (5.77) to deduce that for p nonintersecting paths ld/rd
(0)
= 2(p − j)}j=1,...,p ; {lj = 2(p − j) + 2r}j=1,...,p )
N Y
p Y
G2N ({lj D =
e−2iθk r (1 + aj eiθk )(1 + bj e−iθk )
j=1 k=1
E U (p)
.
(10.48)
(iii) Use (3.121) and (4.4) to deduce from (10.48) that when aj = bj = 1 (j = 1, . . . , N ) the number of nonintersecting paths of the type in (ii) is equal to p Y i=1
(2N + p − i)!(i − 1)! (N − r + p − i)!(N + r + i − 1)!
and show that this same formula can be deduced from (10.9). 2.
(i) Make use of q1.(i), (10.30), (10.29) and (10.36) to show that with rd(o) steps weighted a1 , . . . , aN and ld(e) steps weighted b1 , . . . , bN , w,u/rd(o)/ld(e)
(0)
= p − j + 1}j=1,...,p ; r1 , . . . , rp ) Z π 1 dθ · · · dθp = 1 (2π)p −π −π p “ N ” h i Y Y × (1 + aj eiθl )(1 + bj e−iθl ) e−i(p−l+1)θl det eirα θβ − e−irα θβ
G2N
Z
l=1
({rj
π
α,β=1,...,p
j=1
.
(ii) Suppose {aj } = {bj } (in any order), so that p N Y Y
(1 + aj eiθl )(1 + bj e−iθl ) =
l=1 j=1
p N Y Y
|1 + aj eiθl |2 .
(10.49)
l=1 j=1
Note that in general for an integral over an interval symmetric about the origin the replacement θl → −θl leaves the integral unchanged, as does the relabeling θj ↔ θk for a multiple integral. For the integral in (ii) with the substitution (10.49), use the fact that the integrand changes sign under such mappings to deduce w,u/rd(o)/ld(e)
(0)
({rj = p − j + 1}j=1,...,p ; r1 , . . . , rp ) Z π Z π 1 1 = dθ · · · dθp 1 (2π)p 2p p! −π −π p N p “ X ” h i Y Y Y × |1 + aj eiθl |2 ε(w) e−i(p−l+1)w(θl ) det eirα θβ − e−irα θβ
G2N
l=1 j=1
w∈W
l=1
α,β=1,...,p
,
where W denotes the set of all permutations of {θ1 , . . . , θp } together with negations, while ε(w) denotes the parity of the sum of the number of interchanges and the number of negations.
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CHAPTER 10
(iii) Note that X
ε(w)
w∈W
p Y
e−i(p−l+1)w(θl ) =
l=1
X σ∈Sp
ε(σ)
p “ Y
e−i(p−l+1)σ(θl ) − ei(p−l+1)σ(θl )
”
l=1
i h = det e−i(p−α+1)θβ − ei(p−α+1)θβ
α,β=1,...,p
,
and thus conclude w,u/rd(o)/ld(e)
(0)
({rj = p − j + 1}j=1,...,p ; r1 , . . . , rp ) Z π Z π p N Y Y 1 1 dθ · · · dθ |1 + aj eiθl |2 = 1 p (2π)p 2p p! −π −π l=1 j=1 h h i i × det e−i(p−α+1)θβ − ei(p−α+1)θβ det eirα θβ − e−irα θβ
G2N
α,β=1,...,p
α,β=1,...,p
. (10.50)
(iv) Make use of the type C Vandermonde formula in Exercises 5.5 q.4, and the explicit form of the eigenvalue p.d.f. for Sp(2p) (2.69), to deduce from (10.50) that w,u/rd(o)/ld(e)
G2N
(0)
({rj
= p − j + 1}j=1,...,p ; {rj = p − j + 1}j=1,...,p ) =
p N DY Y
|1 + aj eiθl |2
E
l=1 j=1
Sp(2p)
.
(10.51) 3. The objective of this exercise is to prove (10.35). (i) In (10.33) add row j − 1 to row j for j = p, p − 1, . . . , 2 in order to deduce that Wp ({xk }k=1,...,p , {aj }j=2,...,p , {bj }j=−p+1,...,p−1 ) = " × det
p Y
(b−j+1 + aj )
j=2
#
(a2 − xk ) · · · (ap − xk )(b0 + xk ) · · · (bp−1 + xk ) − (xk → −xk )i h (aj+1 − xk ) · · · (ap − xk )(b−j+2 + xk ) · · · (bp−1 + xk ) − (xk → −xk )
,
j=2,...,p
k=1,...,p
and show that repeated use of this strategy yields Wp ({xk }k=1,...,p , {aj }j=2,...,p , {bj }j=−p+1,...,p−1 ) p p Y Y = (b−j−1+l + aj )Xp ({xk }k=1,...,p , {aj }j=2,...,p , {bj }j=0,...,p−1 ), l=2 j=l
where Xp ({xk }k=1,...,p , {aj }j=2,...,p , {bj }j=0,...,p−1 ) i h := det (aj+1 − xk ) · · · (ap − xk )(b0 + xk ) · · · (bp−1 + xk ) − (xk → −xk )
j,k=1,...,p
.(10.52)
(ii) Note that the determinant Xp is homogeneous of degree 32 p2 − 12 p, that it vanishes for xj = xk , −xk (k = j) and for xj = −xj , and is antisymmetric in the xs and symmetric in the bs. Thus conclude that Xp ({xk }k=1,...,p , {aj }j=2,...,p , {bj }j=0,...,p−1 ) p Y Y = xl (xk − xj )(xk + xj )f ({xj }j=1,...,p , {bj }j=0,...,p−1 ), l=1
(10.53)
1≤j
where f is homogeneous of degree 12 p(p − 1) and symmetric in {xj }j=1,...,p , {bj }j=0,...,p−1 , so our re-
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LATTICE PATHS AND GROWTH MODELS
maining task is to show that Y
f ({xj }j=1,...,p , {bj }j=0,...,p−1 ) = 2p
(bj + bk ).
(10.54)
0≤j
(iii) Observe that (10.54) would follow if we could show that the coefficient of x2p−1 x2p−3 · · · xp bp−1 bp−2 · · · b0p−1 1 2 0 1 in Xp was equal to (−1)(p−1)(p−2)/2 2p . For this purpose, note that the only term in (10.52) containing a is the top left entry, and expand by the corresponding coefficient in this entry to show factor of x2p−1 1 [x2p−1 ]Xp ({xk }k=1,...,p , {aj }j=2,...,p , {bj }j=0,...,p−1 ) 1 h i = 2(−1)p−1 det (aj+1 − xk ) · · · (ap − xk )(b0 + xk ) · · · (bp−1 + xk ) − (xk → −xk )
j,k=2,...,p
= 2(−1)p−1 Xp−1 ({xk }k=2,...,p , {aj }j=3,...,p , {bj }j=0,...,p−1 ). Note that here we have a (p − 1) × (p − 1) determinant in which each entry is linear in b0 , so the coefficient is equal to this determinant with (b0 + xk ) and (b0 − xk ) removed from the entries, and thus of bp−1 0 [bp−1 x2p−1 ]Xp ({xk }k=1,...,p , {aj }j=2,...,p , {bj }j=0,...,p−1 ) 0 1 = 2(−1)p−1 Xp−1 ({xk }k=2,...,p , {aj }j=3,...,p , {bj }j=1,...,p−1 ). By iterating this deduce the sought result. 4. Let m (0) = (p − 1, p − 2, . . . , 0) and m = (λ1 + p − 1, . . . , λp ). (i) Note that (1.173) can be written Asym (xn−1 xn−2 · · · x0n ) = 1 2
Y
(xj − xk ),
1≤j
where Asym is specified by (4.135). (ii) Use the result of (i), together with (10.7), the first equality in (10.8), and the formulas (10.16), (10.15) and (5.77) to show sμ (q1 , . . . , qn ) =
p n DY Y
(1 + qj eiθl )sμ (e−iθ1 , . . . , e−iθp )
l=1 j=1
E U (p)
.
(10.55)
(iii) Note from (10.21) that (10.55) is equivalent to the so called dual Cauchy identity p n Y Y
sμ (q1 , . . . , qn )sμ (z1 , . . . , zp ).
(10.56)
μ:μ1 ≤p
l=1 j=1
5.
X
(1 + qj zl ) =
(i) Starting with (10.24), proceed as in the derivation of (10.55) to show sμ (q1 , . . . , qn ) =
p∗ n DY Y l=1 j=1
E 1 −iθ1 −iθp∗ s (e , . . . , e ) , μ 1 − qj eiθl U (p∗ )
(10.57)
where p∗ = μ1 . (ii) Note from (10.21) that (10.57) is equivalent to the so called Cauchy identity p n Y Y l=1 j=1
X 1 = sμ (q1 , . . . , qn )sμ (z1 , . . . , zp ). 1 − qj zl μ
(10.58)
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CHAPTER 10
6. [180], [350] In this exercise the continuous limit of noninteracting paths near a wall is studied. (wB)
( y (0) ; y ). Proceed as in the derivation of (10.44) (i) Let the continuous generating function be denoted Gt to show h (0) “ 1 ”p/2 Pp i (0) 2 (0) 2 (wB) Gt e− j=1 (yj +(yj ) )/2Dt det eyj yk /Dt − e−yj yk /Dt . ( y (0) ; y) = 2πDt j,k=1,...,p (0)
Now set yj = ja (equally spaced initial condition) and use the type C Vandermonde formula from Exercises 5.5 q.4 to show (wB)
Gt
({ja}j=1,...,p ; y) ∝ e−
Pp j=1
yj2 /2Dt
p Y j=1
sinh
ayj Dt
Y
sinh
1≤j
a(yk − yj ) a(yk + yj ) sinh . 2Dt 2Dt
Interpret this as a log-gas system at β = 1 on the half-line y > 0 interacting via the pair potential (2.72) with image charges of like sign in y < 0, and with an appropriate neutralizing background. (ii) For p continuous nonintersecting Brownian walkers, starting at the origin and confined to the half line y > 0 for t > 0, and with the nonintersecting condition required for all times, show the p.d.f. for the event that they arrive at y after time t is proportional to p “ 1 ”p2 +p/2 Pp Y 2 e− j=1 yj /2Dt yl2 Dt l=1
Y
(yk2 − yj2 )2 .
1≤j
Recognize this as the eigenvalue p.d.f. for (2p + 1) × (2p + 1) antisymmetric Gaussian unitary matrices (recall Exercises 1.3 q.5).
10.2 DIMERS AND TILINGS In this section it will be shown how some statistical mechanical models of dimers and tilings relate to nonintersecting paths. 10.2.1 Dimers Consider the brickwork lattice (which is equivalent to the hexagonal lattice) with an odd number of rows and impose periodic boundary conditions in the y-direction by identifying the bottom and top rows. Suppose that on each vertical edge a hard rod (dimer) is placed, thus covering all the lattice sites without overlap. Let us suppose now that along the bottom row a number of dimers are placed across nonconsecutive horizontal edges. For this to be done without overlap, but still cover all lattice sites, for each horizontal dimer a vertical dimer sharing the same site must be removed. This dimer is to be replaced by a horizontal dimer in the next row: this horizontal dimer can in general lie either to the left or to the right of the vacated site, but is constrained not to intersect another horizontal dimer. Because these new horizontal dimers will overlap vertical dimers going between the second and third rows, the procedure of removing vertical dimers and replacing them with non-overlapping horizontal dimers lying either to the left or to the right of the vacated site is repeated. Continuing in this fashion leads to the same number of horizontal dimers being introduced on the final row as were introduced on the first row, which by the assumption of periodic boundary conditions must occur at the same positions as on the first row. Each matching horizontal dimer in the bottom and top rows can be connected by a ld/rd path which starts at an odd numbered column half a lattice spacing below the bottom row and finishes at the same column one half a lattice spacing below the top row (remember the top and bottom rows are identified), with each segment passing through a horizontal dimer. Because dimers do not intersect this gives a family of nonintersecting ld/rd paths. The periodic boundary conditions with an odd number of rows imply the initial positions equal
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LATTICE PATHS AND GROWTH MODELS
to the final positions, and it has already been noted that the former are restricted to odd numbered columns. The paths have the additional constraint of lying between the first and last column inclusive (see Figure 10.8). Conversely, any such family of nonintersecting ld/rd paths determines a dimer configuration on the brickwork lattice.
Figure 10.8 The configuration of all vertical dimers on the brickwork lattice with periodic boundary conditions in the vertical direction (leftmost diagram), and a configuration in which two horizontal dimers are introduced into the bottom row, giving rise to nonintersecting ld/rd paths.
With M columns and 2N + 1 rows, and each horizontal dimer weighted w, the generalized partition function relating to summing over all weighted configurations is given by ∗
Z2N,M (w
2N
)[a] = 1 +
M
w2N p
p
a(2nl − 1)
1≤n1 <···
p=1 ww,ld/rd
×G2N
({2nj − 1}j=1,...,p ; {2nj − 1}j=1,...,p ).
(10.59)
ww,ld/rd (0) (l ; l) denotes the number of ways p nonintersecting paths, conHere M ∗ = [(M − 1)/2] and G2N strained to have x-coordinates in {1, . . . , M − 1} (or equivalently confined to integer sites between infinitely repelling walls at x = 0 and x = M ), and taking N steps each of which is ld/rd. The initial positions along y = 0 are at l(0) , and the final positions along y = 2N are at l. It is possible to write (10.59) as a determinant.
P ROPOSITION 10.2.1 We have ww;ld/rd (2j − 1, 2l − 1)]j,l=1,...,M ∗ , Z2N,M (w2N )[a] = det 1M ∗ + [a(2j − 1)w2N g2N
(10.60)
where ww;ld/rd
g2N
(l(0) , l) =
M−1 πl(0) k πk N 2 πlk sin . 2 cos sin M M M M
(10.61)
k=1
Proof. Applying (10.1) gives ww,ld/rd
G2N ww,ld/rd
where g2N
h i ww,ld/rd (0) (l(0) ; l) = det g2N (lj , lk )
j,k=1,...,N
,
(l(0) , l) is the number of single lattice paths from l(0) to l. Hence ∗
Z2N,M (w
2N
)[a] = 1 +
M X
X
det[anj ,nk ]j,k=1,...,p ,
p=1 1≤n1 <···
where
ww,ld/rd
anj ,nk = a(2nj − 1)w2N g2N
(2nj − 1, 2nk − 1).
(10.62)
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Now, by discretizing (5.32) we see that the r.h.s. of (10.62) can be summed to give “ ” Z2N,M (w2N )[a] = det 1M ∗ + [aj,l ]j,l=1,...,M ∗ which is (10.60). To obtain (10.61) we first generalize the reasoning leading to (10.29) to conclude ww,ld/rd
g2N
∞ “ X
(0)
(lj ; lk ) =
” ld/rd ld/rd g2N (l(0) , l + 2kM ) − g2N (l(0) , −l − 2kM ) .
k=−∞
Substituting (10.4), performing the sum over k using the Poisson summation formula, and simplifying gives (10.61).
The determinant formula (10.60) implies the correlations are given by (5.31) and thus ρ(k) (2n1 − 1, . . . , 2nk − 1) = det[G(2nj − 1, 2nl − 1)]j,l=1,...,k , ww,ld/rd
where, with K = [gN
(2j − 1, 2l − 1)]j,l=1,...,M ∗ , G(2nj − 1, 2nl − 1) = w2N K(1 + w2N K)−1
2nj −1,2nl −1
.
To make this latter formula explicit we note that G := [G(2j − 1, 2l − 1)]j,l=1,...,M ∗ must satisfy G(1 + w2N K) = w2N K and then proceed as in the derivation of (5.105) to deduce that [330] ∗
G(2nj − 1, 2nl − 1) =
M π(2nl − 1)k (2w cos πk/M )2N 2 π(2nj − 1)k sin sin . ∗ M M M 1 + (2w cos πk/M )2N
(10.63)
k=1
In Exercises 10.2 q.1 it is shown that in an appropriate large M, N limit this becomes the sine kernel. 10.2.2 Tiling of a hexagon Nonintersecting ld/rd paths initially equally spaced 2 units apart, and with final points also equally spaced 2 units apart, are equivalent to certain three-species parallelogram tilings of a hexagon, the shape of the latter being determined by the parameters associated with the paths. Thus suppose there are a paths, which each consist of b + c steps, and the x-coordinates of the final points are displaced by c − b lattice sites to the √ right √ of the initial points. Then it is easy to see that the paths are bounded by a hexagon of side lengths 2a, 2b, 2c, √ √ 2a, 2b, 2c, and the paths uniquely determine a tiling of the hexagon (see Figure 10.9). The ld segments √ correspond to the left (right) sloping parallelograms, sloping sides of length 2, horizontal sides√length 2. Changing the angles in the hexagon so that they are all 120◦, and changing the side length 2a → 2a maps this three-species parallelogram tiling into a three-species rhombi tiling. The rhombi tiling can be viewed as the three dimensional diagram of stacked cubes which weakly decrease in height going down columns and across rows of the base, with row 1 counted from the back and column 1 from the left. The heights then specify an array of non-negative integers {aij } :
ai,j ≥ ai+1,j , ai,j ≥ ai,j+1 ,
referred to as plane partitions. Note that plane partitions are a variation on semi-standard tableau in which the entries now weakly decrease along rows and down columns of the diagram of a partition. The number of paths is equal to the maximum number of columns a, while the number of steps to the left in any one path is equal to the maximum number of rows b, and the number of steps to the right determines the maximum height. The correspondence between nonintersecting paths and tilings can be used to enumerate the latter.
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LATTICE PATHS AND GROWTH MODELS
Figure 10.9 Nonintersecting ld/rd paths with initial and final positions 2 units apart are equivalent to a three-species parallelogram tiling of a hexagon involving left-sloping parallelograms (ld segment), right sloping parallelograms (rd segment) and vertical-axis parallelograms. This can be deformed into a rhombi tiling of a hexagon. The rhombi tiling in turn can be viewed as the diagram of a plane partition. Here the rows are counted from the top, and the columns are counted from the right.
P ROPOSITION 10.2.2 Consider a hexagon of the second type in Figure 10.9, with dimensions a × b × c. The number of distinct rhombi tilings is equal to a (a + b + c − i)!(i − 1)! i=1
(a + b − i)!(c + i − 1)!
.
Proof. This follows by setting N = b + c, p = a, mj = p + c − b − j in (10.9). The dimer model of Section 10.2.1 can be formulated as a tiling problem by placing over each vertical dimer a 1 × 2 rectangle, and placing over each horizontal dimer a 2 × 1 rectangle. It is also true that the rhombi tiling of the hexagon is equivalent to a dimer covering of the underlying hexagonal lattice — place a dimer of unit length (a unit being the side length of a rhombus) symmetrically on the major axis of each rhombus. Let us return now to the setting of the first of the diagrams in Figure 10.9. We see that for each level y = 1, 2, . . . , complementary to the x-coordinates of the paths are the x-coordinates of the center point of the vertical-axis parallelograms (see Figure 10.10). Regarding these coordinates as specifying a configuration of a certain lattice gas confined to parallel lines at y = 1, 2, . . . , the number of particles on each line is determined only by the total number of steps in the corresponding lattice paths (b + c), and the displacement c − b of the x-coordinates of the starting positions relative to the final positions; the number of lattice paths (y) plays no role. Thus for y = 1, 2, . . . , b there are 1, 2, . . . , b particles {xi }i=1,...,y ; for y = b + 1, . . . , c (y) there are b particles {xi }i=1,...,b ; and for y = c + 1, . . . , b + c − y there are b − 1, b − 2, . . . , 1 particles (y) {xi }i=1,...,b+c−y . This configuration of particles exhibits the interlacings (y+1)
(y)
(y+1)
x1
> x1 > x2
(y) x1 (y) x1
(y+1) x1 (y+1) x1
> >
(y+1)
> · · · > x(y) y > xy+1
(y+1) > · · · > x(y) y > xy
>
(y) x2
> ··· >
(y+1) xb+c−y−1
(y = 1, . . . , b − 1),
(y = b, . . . , c − 1), (y)
> xb+c−y
(y = c, . . . , b + c − 1).
(10.64)
Furthermore, they must stay within the bounds of the hexagon, and are restricted to odd (even) numbered sites for y odd (even).
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CHAPTER 10
Figure 10.10 Interlacing particle system which results by associating particles with the center point of the vertical-axis parallelograms of a tiling.
Suppose now that the parallelograms are rescaled so that the leftmost extremity (which occurs on the line y = b) is at x = 0, while the rightmost extremity (which occurs on the line y = 0) is at x = 1. By taking the number of lattice paths and thus rhombi to infinity, a continuous multi-species particle system is obtained (the different lines are the different species), which is specified by the interlacings (10.64), supplemented by the requirement that all coordinates are bound between 0 and 1. This multi-species system is referred to as a bead process [94], [177]. One point of interest is the distribution of particles on a given line r. P ROPOSITION 10.2.3 The p.d.f. on line r is proportional to ⎧ r (r) ⎪ (r) (r) (r) ⎪ ⎪ (xj )c−r (1 − xj )b−r (xj − xk )2 , 1 ≤ r ≤ b, ⎪ ⎪ ⎪ j=1 ⎪ 1≤j
(10.65)
1≤j
Proof. We integrate over all particles on lines 1, . . . , r − 1 in order, and further integrate over all particles on lines (1)
(2)
(2)
b + c − 1, . . . , r + 1 in order. The particle coordinate on line 1, x1 is to be integrated between x2 and x1 , giving (2) (3) (3) (2) (3) (2) x1 − x2 . This function now must be integrated over x1 < x1 < x2 < x3 . For this purpose use is made of the Vandermonde determinant expansion (1.173) to write " # (2) 1 x2 (2) (2) x1 − x2 = det . (2) 1 x1 The integration can be done row-by-row to give 2
1 1 (3) 1 6 (3) (3) (3) (3) (3) (x1 − x2 )(x1 − x3 )(x2 − x3 ) = det 4 1 2 2 1
(3)
x3 (3) x2 (3) x1
3 (3) (x3 )2 7 (3) (x2 )2 5 , (3) 2 (x1 )
where the equality follows by further use of (1.173). This must now be integrated over the first region in (10.64) with y = 4, a task which can again be done row-by-row to give 1 3!2!
Y 1≤j
(4)
(4)
(4)
(xj − xk ) = det[(x5−j )k−1 ]j,k=1,...,4 .
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LATTICE PATHS AND GROWTH MODELS
Proceeding in this fashion we see that for r ≤ b, integrating over all particles on lines 1, . . . , r − 1 will give r−1 Y s=1
1 s!
Y
(r)
(r)
(xj − xk ).
(10.66)
1≤j
For r > b, we must continue to integrate (10.66) over the second of the regions in (10.64), noting too that all variables are between 0 and 1. In the case y = b the above method gives an expression proportional to C
b Y
(b+1)
(1 − xj
Y
)
j=1
(b+1)
(xj
(b+1)
− xk
i h (b+1) ) = det (1 − xb+2−j )k
j,k=1,...,b+1
1≤j
.
Continuing this working shows that for b < r ≤ c, integrating over all particles on lines 1, . . . , r − 1 will give b Y
(r)
(1 − xj )r−b
j=1
Y
(r)
(r)
(xj − xk ),
(10.67)
1≤j
up to a proportionality constant. For r > c we must continue to integrate (10.67) over the third of the regions in (10.64), supplemented by the constraint that all variables are between 0 and 1. To do this we can formally continue to integrate over the inequalities of the second (r) (r−1) (r−1) of the regions in (10.64), then take the limit xc → 0, xc−1 → 0 (after first dividing by xc−1 ), then take the limit (r−2) (r−2) 2 xc−2 → 0 (after first dividing by (xc−2 ) ) etc. to so obtain b+c−r Y
(r)
Y
(r)
(1 − xj )r−b (xj )r−c
j=1
(r)
(r)
(xj − xk ),
1≤j
up to a proportionality constant. The results from integrating over all particles on lines b + c − 1, . . . , r + 1 in order are formally the same, after the (r) (b+c−r) identification xj → 1 − xj , and the stated result thus follows.
We note that each of the forms in (10.65) is an example of a Jacobi unitary ensemble with weight defined on [0, 1]. Another point of interest is the joint distribution of the lines 1, . . . , r in the case that c, b → ∞ and the (r) coordinates {xj } are appropriately scaled. (r) (r) √ P ROPOSITION 10.2.4 Consider the bead process with the coordinates scaled xj → 12 (1 + yj / b) and 4r (s) b, c → ∞ with c/b → 1. The joint distribution of s=1 {yj }j=1,...,s is proportional to r
(r) 2
e−(yl
)
(r)
(r)
(yj − yk )
χy(s+1) >y(s) >y(s+1) >···>y(s) >y(s+1) .
s=1
1≤j
l=1
r−1
1
1
s
2
(10.68)
s+1
Proof. The joint p.d.f of the first r lines is obtained by integrating over all lines from y = b + c − 1 down to y = r + 1 inclusive. From the workings of the proof of Proposition 10.2.3 we know this is proportional to r Y j=1
(r)
(r)
(1 − xj )b−r (xj )c−r
Y
(r)
(r)
(xj − xk )
1≤j
r−1 Y s=1
χx(s+1) >x(s) >x(s+1) >···>x(s) >x(s+1) . 1
1
2
s
s+1
Under the stated limiting conditions, this give (10.68).
One recognizes (10.68) as the eigenvalue p.d.f. (5.196) (in the case w(x) = e−x ) of the GUE minor process [333]. 2
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10.2.3 Tiling of the Aztec diamond Another tiling problem which gives rise to nonintersecting lattice paths is domino tiling of the so-called Aztec diamond lattice. The latter can be defined as the union of all vertices and edges which lie on lattice squares [m, m + 1] × [l, l + 1], (m, l ∈ Z) within the diamond shaped region {(x, y) : |x| + |y| ≤ n + 1}. A domino tiling is a covering of this lattice by 2 × 1 and 1 × 2 rectangles whose corners lie on the lattice points. To associate the tiling with lattice paths, color the squares in the Aztec diamond alternating white-black, choosing the left-most square in the top half as white, and the left-most square in the bottom half as black. This gives a checkerboard pattern in which the squares alternate between white-black and black-white down any column or across any row. Then for a horizontal domino which covers a white-black (black-white) pair of squares, reading left to right, no (horizontal) segment of a path is marked; for a vertical domino which covers a white-black (black-white) pair of squares, reading top to bottom, an ld (−rd) segment of a path is marked. As seen in Figure 10.11, the tiling then maps uniquely to certain returning nonintersecting rd/−ld/rh paths (recall Section 10.1.4).
Figure 10.11 Mapping between a domino tiling of the Aztec diamond and nonintersecting rd/−ld/rh paths.
From the correspondence the total number of distinct tilings can be calculated [164]. P ROPOSITION 10.2.5 The number of domino tilings of the Aztec diamond lattice bounded by {(x, y) : |x| + |y| ≤ n + 1} is equal to 2n(n+1)/2 . Proof. This follows by setting hk = k − 1, p = n + 1, w = 1 in the formula of Proposition 10.1.11. E XERCISES 10.2
1. [330] In this exercise a grand canonical ensemble of ww,ld/rd lattice paths is studied.
(i) In (10.63) write M → 2M + 1 and note that then Z 1 lim G(M + x, M + y) = cos πt(x − y) M →∞
(ii) With 2w > 1 and θ0 := kernel
0 2 1 arcos 2w , π
Z
θ0
(2w cos πt/2)2N dt. 1 + (2w cos πt/2)2N
take the N → ∞ limit of the result in (i) to obtain as the limiting
cos πt(x − y) dt =
0
sin πθ0 (x − y) . π(x − y)
(iii) Verify that for j, k = 1, . . . , [(M − 1)/2] [(M −1)/2]
X l=1
ww,ld/rd
gN
(2j − 1, 2l − 1) sin
“ π(2l − 1)k π(2j − 1)k πk ”N sin = 2 cos . M M M
463
LATTICE PATHS AND GROWTH MODELS
Conclude that the eigenvalues of ww,ld/rd
[gN are given by {(2 cos
(2j − 1, 2l − 1)]j,l=1,...,[(M −1)/2]
πk N ) }k=1,...,[(M −1)/2] M
and use this to obtain the evaluation [(M −1)/2] “
ZN,M (ζ)[1] =
Y
k=1
“ πk ”N ” 1 + ζ 2 cos . M
(10.69)
10.3 DISCRETE POLYNUCLEAR GROWTH MODEL 10.3.1 Robinson-Schensted-Knuth correspondence A u/rh and u/lh pair of nonintersecting lattice paths, initially equally spaced, can be put into one-to-one correspondence with a matrix of integers, a result (with a pair of paths recorded as a pair of semi-standard tableaux) usually referred to as the Robinson-Schensted-Knuth (RSK) correspondence [246]. The non-negative integer matrix can itself be thought of as recording events in a discrete space and time growth process [329]. Thus consider an n × n square non-negative integer matrix X = [xi,j ]i,j=1,...,n with rows numbered from the bottom, rotated 45◦ anticlockwise. Label the horizontal rows of the rotated matrix by t = 1, 2, . . . , 2n − 1 and the vertical columns by x = 0, ±1, . . . , ±(n−1), where x = 0 corresponds to the diagonal of the original matrix (see Figure 10.12). The entries xi,j in the matrix for successive t values (t = i + j − 1) are heights of “nucleation events” — columns of unit width and height xi,j centered about the corresponding x-coordinate which are placed on top of the profile formed by earlier nucleation events and their growth. Thus at t = 1 there is a nucleation event at x = 0 which consists of a column of width 1 and height x1,1 marked on the line at y = 0 in the xy-plane. In general, as t → t + 1 the profile of all nucleation events so far recorded is to “grow” one unit in the −x direction and one unit in the +x direction. Thus in going from t = 1 to t = 2 the nucleation event centered at x = 0 of height x1,1 now has width 3 units. On top of this profile, centered at x = −1 and x = 1 nucleation events of unit width and height x2,1 , x1,2 , respectively are then drawn. In now going from t = 2 to t = 3 this new profile is to grow one unit to the left and one unit to the right. In so doing we see that an overlap of width one unit and height min(x2,1 , x1,2 ) will occur. This overlap is ignored in the first diagram (profile on y = 0), and recorded instead as a profile on the line immediately below (here y = −1). The process is repeated with these rules until the nucleation event of height xn,n at t = 2n − 1 has been recorded above x = 0 on the first diagram. The boundary of each profile forms a pair of lattice paths — one to the left and one to the right of x = 0. The lattice paths for x < 0 start at x = −(2n − 32 ) and go either up (in integer amounts with each unit regarded as a step) or to the right (in steps of two units) until they reach x = − 21 , while the lattice paths for x > 0 start at x = (2n − 3/2) and go either up or right until they reach x = 21 . In the y-direction the paths start one unit apart at y = 0, . . . , −(n − 1) (see Figure 10.12). Conversely, each such family of nonintersecting paths corresponds to a unique n × n non-negative integer matrix, which can be constructed by reversing the above procedure, so we thus have a bijection between paths and matrices. The paths in the pair are separately equivalent to the lattice path type of Figure 10.3 and thus to a semi-standard tableau, so the bijection is also between non-negative integer matrices and pairs of semi-standard tableaux of the same shape (the latter constraint comes about because the pairs of paths have the same finishing points). The above correspondence between integer matrices and pairs of nonintersecting lattice paths can be extended to a correspondence between weighted integer matrices and pairs of weighted nonintersecting lattice paths. For this each entry xi,j in the integer matrix is weighted by (1 − ai bj )(ai bj )xi,j , with the factor (1 − ai bj ) representing a normalization allowing the weighting to be interpreted as a probability from a ge-
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CHAPTER 10
b3
0 1
a3 a2
5 4 3 2
1
b1
2 0
2 a1
t=3
t
b2
0 1
a3 a2
4 b1
a12 b2
1
1
y=0
a2
y=−1 0 t=4
x −2 −1 0 1 2
2 b1
3 a2
a3
t=1
b1
a1
4 b1
y=0
a12
0 b2
t=2 b13
b2
y=0
a2
y=−1
a22
0 a12
t=5
y=0
2 b3
a3 a2 3
4 b1 a12 b3
a3 a2
b2 0
y=0 y=−1
Figure 10.12 Mapping from a weighted non-negative integer matrix to a pair of weighted nonintersecting lattice paths.
ometric distribution. Since each entry xi,j corresponds to a nucleation event which adds a unit column of height xi,j to the top profile, the weight can be recorded n on the profile by weighting the left side of the new x x column by bj i,j and the right side by ai i,j (we take i,j=1 (1 − ai bj ) as an overall normalization factor). If columns should overlap as the profiles grow, the resulting column recorded on the profile below is to carry a left side and a right side weight which are exactly equal to those erased in the overlap, thus conserving the total vertical weight in the direction of any side. Crucially the rules of the growth process ensure that at each time step the weight in the direction of any one side is proportional to one particular bj (left sides) or one particular ai (right sides). At the end of the process this means that each vertical step at x = −(2n + 12 − 2j) is weighted by bj , and each vertical step at x = 2n + 12 − 2j by aj (see Figure 10.12). The profile with endpoints at y = −l + 1 (l = 1, . . . , n) will be referred to as the level-l path, and its evolution forms a growth process known as the discrete polynuclear growth model [450], [330]. Of interest is the statistical properties of the profile, in particular its maximum displacement μ1 . To study this quantity the maximum displacements μl of the level-l paths for each l = 1, 2, . . . , n are relevant. The nonintersecting condition implies μ1 ≥ μ2 ≥ · · · ≥ μn so μ = (μ1 , μ2 , . . . , μn ) forms a partition. We know from (10.24) that with each of the vertical steps at x = −(2n + 21 − 2j) weighted by bj , the total weight of all nonintersecting u/rh paths (with horizontal steps two units) initially equally spaced at y = 0, . . . , −(n−1) along x = −(2n− 32 ) and finishing at y = μ1 , μ2 −1, . . . , μn −(n−1) along x = −1/2 is given by the Schur polynomial sμ (b1 , b2 , . . . , bn ). Similarly, with vertical steps at x = (2n + 12 − 2j) each weighted by aj , the total weight of all nonintersecting u/lh (with horizontal steps two units) initially equally spaced at y = 0, . . . , −(n − 1) along x = 2n − 32 and finishing at y = μ1 , μ2 − 1, . . . , μn − (n − 1) along x = 12 is given by the Schur polynomial sμ (a1 , a2 , . . . , an ). It follows that if the non-negative integers xi,j in the matrix X are independent geometric random variables with parameter ai bj and thus Pr(xi,j = k) = (1 − ai bj )(ai bj )k ,
(10.70)
then the probability that such an integer matrix X corresponds to a pair of nonintersecting lattice paths with
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LATTICE PATHS AND GROWTH MODELS
maximum heights μ is given by n
(1 − ai bj )sμ (a1 , a2 , . . . , an )sμ (b1 , b2 , . . . , bn ).
(10.71)
i,j=1
A fundamental statistical quantity relating to the maximum height μ1 =: h of the level-1 path, namely its cumulative probability density, follows from this by summing over μ : μ1 ≤ l. Thus Pr(h ≤ l) =
n
(1 − ai bj )
sμ (a1 , a2 , . . . , an )sμ (b1 , b2 , . . . , bn ).
(10.72)
μ: μ1 ≤l
i,j=1
We remark that since the Schur polynomials are symmetric functions this probability is symmetric in {ai } and {bj }, a feature which is not obvious from the definitions. We note too that the normalization condition for this probability implies the Cauchy identity (10.58). From the combinatorial definition of the Schur polynomial in (10.16), the sum over μ in (10.72) can be regarded as a sum over all pairs of weighted semi-standard tableaux of shape μ, content n and with row length no greater than l. Now we know from Figure 10.1 that such tableaux uniquely code weighted ld/rd nonintersecting lattice paths initially two sites apart consisting of at most l paths and n steps. This can be extended to exactly l paths by appending a suitable number of paths starting immediately to the left of those already present and which move only ld (such paths are not recorded as part of the tableau). A pair of tableaux of the same shape, content n and row length no greater than l, then corresponds to l returning ld/rd nonintersecting lattice paths of 2n steps in which both the initial and final spacings are two sites apart. This viewpoint allows Pr(h ≤ l) to be written in terms of a random matrix average [34]. P ROPOSITION 10.3.1 We have Pr(h ≤ l) =
l n (1 − ai bj ) (1 + aj e−iθk )(1 + bj eiθk )
n i,j=1
j=1 k=1
CUEl
.
(10.73)
Proof. From the returning paths interpretation, Pr(h
≤ l) =
n Y
ld/rd
(1 − ai bj )G2n
(0)
({lj
= 2(l − j) + 1}j=1,...,l ; {lk = 2(l − k) + 1}k=1,...,l ),
(10.74)
i,j=1
where the weights in G2n are specified by ws− = as , ws+ = 1 (s = 1, . . . , n), ws+ = bs , ws− = 1 (s = n + 1, . . . , 2n). The result now follows from (10.48). ld/rd
10.3.2 Joint probabilities Consider the measure on the space of non-negative integer matrices implied by (10.70). Then, as discussed in Section 10.3.1, the probability an n1 × n2 non-negative integer matrix maps to a pair of semi-standard tableaux with shape μ, one of content n1 , the other of content n2 , under the RSK correspondence is equal to n2 n1
(1 − ai bj )sμ (a1 , . . . , an1 )sμ (b1 , . . . , bn2 )
(10.75)
i=1 j=1
(cf. (10.71)). Note that this vanishes for (μ) > min(n1 , n2 ). Next we seek the joint probability that an n1 × (n2 + 1) non-negative integer matrix with measure implied by (10.70) corresponds to a pair of semi-standard tableaux with shape μ, contents n1 and n2 + 1, and that the n1 × n2 bottom left sub-block corresponds to a pair of semi-standard tableaux with shape κ and contents n1 and n2 . According to the remark below (10.75), for the probabilities to be nonzero one requires (μ) ≤ min(n1 , n2 + 1) and (κ) ≤ min(n1 , n2 ).
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CHAPTER 10
The growth model picture of the RSK correspondence tells us that μl equals the height of the level-l path at x = ±1/2, while κl equals the heights of the level-l path at x = − 32 . The inequalities (10.104) imply that μ1 ≥ κ1 ≥ μ2 ≥ κ2 ≥ · · · ≥ κn ≥ μn ,
(10.76)
where n = min(n1 , n2 + 1). The nonintersecting paths corresponding to these heights have total weights sμ (a1 , . . . , an1 )
and
sκ (b1 , . . . , bn2 )
to the right and left, respectively. The path segments connecting the level-l path at x = − 23 to x = − 21 have n1 n2 +1 l weight bμnl2−κ +1 . Noting too that the normalization for the paths is j=1 k=1 (1 − aj bk ), the following result is obtained [222]. P ROPOSITION 10.3.2 Consider an n1 ×(n2 +1) non-negative integer matrix with entries chosen according to (10.70). The joint probability that this matrix maps under the RSK correspondence to a tableau of shape μ, and that the n1 × n2 matrix obtained by deleting the final column maps to a tableaux of shape κ is equal to n1 n 2 +1 Pn (μl −κl ) (1 − aj bk )sμ (a1 , . . . , an1 )sκ (b1 , . . . , bn2 )bn2l=1 , +1 j=1 k=1
where n = min(n1 , n2 + 1), supported on partitions interlaced according to (10.76) and with (μ) ≤ n1 , (κ) ≤ n2 . Suppose now that under the RSK correspondence with weights chosen according to (10.70), the pair of semi-standard tableau corresponding to a particular n1 × n2 matrix (n1 > n2 ) has shape κ. It then follows from Proposition 10.3.2 and (10.71) that the probability an n1 × (n2 + 1) matrix, obtained by adding an extra column to the existing matrix, of having shape μ is P (μ, κ) = χ(μ > κ)
n1 P 2 sμ (a1 , . . . , an1 ) nj=1 (μj −κj )+μn2 +1 (1 − ai bn2 +1 ) , bn2 +1 sκ (a1 , . . . , an1 )
(10.77)
i=1
where χ(μ > κ) = χ(μ1 ≥ κ1 ≥ μ2 ≥ · · · ≥ μn2 ≥ κn2 ≥ μn2 +1 ≥ 0). This in turn allows for an extension of Proposition 10.3.2. P ROPOSITION 10.3.3 Consider an n1 × p (n1 ≥ p) non-negative integer matrix with entries chosen according to (10.70). The joint probability that under the RSK correspondence this matrix is such that the principal n1 × s submatrices (s = 0, . . . , p) corresponds to pairs of semi-standard tableaux with shape μ(s) (s) (note that μi = 0 for i > s) is equal to p n1 i=1 j=1
(1 − ai bj )sμ(p) (a1 , . . . , an1 )
p
Ps−1
bs
(s) (s−1) )+μ(s) p j=1 (μj −μj
χ(μ(s) > μ(s−1) ).
(10.78)
s=1
Proof. Since in the case n = Q 0 (10.71) can be taken as equal to unity, the sought joint probability is obtained from (s+1) , μ(s) ). (10.77) by forming the product p−1 s=0 P (μ 10.3.3 The exponential limit and continuous RSK With ai = e−αi /L , bi = e−βi /L and the height xi,j at each site scaled by xi,j → Lxi,j we see that the discrete geometrical distribution (10.70) becomes the exponential distribution Pr(xi,j ∈ [y, y + dy]) = (αi + βj )e−(αi +βj )y dy,
y ≥ 0.
(10.79)
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This distribution can be used to define a probability measure on n×n matrices with non-negative real numbers as entries. The RSK correspondence gives a bijective mapping with a pair of u/rh nonintersecting lattice paths, each starting two units apart along y = 0 and finishing at y1 > y2 > · · · > yn > 0 along x = 0, in which the up steps are continuous and weighted by an exponential variable proportional to the increment. Explicitly, with the entries of X distributed according to (10.79), the RSK mapping requires that at x = −(2n + 12 − 2j) −βj v (x = 2n + 12 − 2j) the up increment v of each level-l path with (e−αj v ), and nl ≤ j is to be weighted e there is an overall weighting of the paths by the normalization i,j=1 (αi + βj ). 1 The total weight of a single path with vertical increments nof length vj at x = −(2n+ 2 −2j) (j = l, . . . , n), −βj vj weighted by e with prescribed total displacement j=l vj = y is given by ∞ ∞ n n e−βj y n =: ul ({βj }j=l,...,n ; y). (10.80) dvl e−βl vl · · · dvn e−βn vn δ y − vj = μ=l (βj − βμ ) 0 0 j=l
j=l
μ=j
A (semi-)continuous analogue of Proposition 10.1.1 then gives that the corresponding total weight of the set of continuous nonintersecting paths for x < 0 is det[ul ({βj }j=l,...,n ; yk )]k,l=1,...,n . Multiplying this by the weight of the paths for x > 0, and the normalization, we see that the continuous analogue of (10.71) is [222], [87] n
(αi + βj ) det[ul ({αj }j=l,...,n ; yk )]k,l=1,...,n det[ul ({βj }j=l,...,n ; yk )]k,l=1,...,n
i,j=1
n
= n
i,j=1 (αi
i<j (αj
+ βj )
− αi )(βj − βi )
det[eαj yk ]j,k=1,...,n det[eβj yk ]j,k=1,...,n .
(10.81)
In the special case αi = a + (i − 1)c,
βj = a ˜ + (j − 1)˜ c
(10.82)
e−(a+(l−1)c)yk (1 − e−cyk )n−l , cn−l (n − l)!
(10.83)
(10.80) can be simplified to read ul ({αj }j=l,...,n ; yk ) =
and similarly for ul ({βj }j=l,...,n ; yk ). Substituting in (10.81) and making use of the Vandermonde determinant identity (1.173) we obtain the p.d.f. n n 1 (a + a ˜ + (i − 1)c + (j − 1)˜ c) (c˜ c)j−1 Γ2 (j) i,j=1 j=1 Pn ×e−(a+˜a) j=1 yj (e−cyj − e−cyi )(e−˜cyj − e−˜cyi ). (10.84) 1≤i<j≤n
˜) → In the case c = c˜, after the change of variables and replacement of parameters e−cyj → yn+1−j , (a + a c(α + 1), (10.84) reduces to the JUE supported on (0, 1) with a = α, b = 0, while with a + a ˜ = 1 and c = c˜ → 0 it reduces to the LUE with a = 0. Thus, with E(0; J; ME) denoting the probability of having no eigenvalues in the interval J of the matrix ensemble ME, in the case c = c˜ we see that 1 1 Pr l1 ≤ log = E(0; (0, s); JUEn | a=α ), b=0 c s where the JUE is supported on (0, 1) and 0 < s < 1. Similarly, in the case c = c˜ → 0, Pr(l1 ≤ s) = E(0; (s, ∞); LUEn |a=0 ). In Exercises 10.3 q.1 it is shown how a modification of (10.81) leads to more general examples of the JUE
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and LUE. The p.d.f. (10.84) can also be obtained from (10.71), by first specializing the parameters −1 (a1 , . . . , aN ) = (z1 , z1 t1 , z1 t21 , . . . , z1 tN ), 1
−1 (b1 , . . . , bN ) = (z2 , z2 t2 , z2 t22 , . . . , z2 tN ), 2
then simplifying the Schur polynomials according to (10.22). After setting t1 = e−c/L , t2 = e−˜c/L , z1 = e−˜a/L , (μi + N − i)/L = yi and taking the limit as L → ∞ we see that (10.71) multiplied by L2N tends to (10.84). This is to be expected as then the geometric distribution reduces to (10.79) with αi and βj therein equal to (10.82). Continuous RSK limits of joint probabilities can also be taken. In particular, substituting in (10.78) ai = (s) (s) bj = e−1/2L , writing (μi + s − i)/L = xi and taking L → ∞ (note that then Pr(xi,j ∈ [x, x + dx]) = −x e dx) gives, upon use of (10.23), a functional form known from the study of the recursive construction of (s) Wishart matrices in Exercises 4.3 q.4. Explicitly, the joint p.d.f. for {xi }i=1,...,s , s = 1, . . . , p is proportional to p (p) (p) (xl )n1 −p e−xl
(p)
(xi
p−1
χ(x(s+1) > x(s) ),
(10.85)
s=1
1≤i<j≤p
l=1
(p)
− xj )
where χ(x(s+1) > x(s) ) is specified as in (4.100). (s) In the growth model picture the variables xs correspond to the heights λ1 (n1 , s) on the outer profile. It has been shown in Proposition 7.1.4 that with the horizontal distance along the profile O(N 2/3 ), the correlation (s) between the variables xj , appropriately scaled, is given by the dynamical extension of the Airy kernel (7.19). It thus follows that the distribution function for two heights along the profile with this separation have distribution in terms of the dynamical Airy kernel as implied by the result of Exercises 9.1 q.3.
10.3.4 Correlation functions and the Borodin-Okounkov identity The joint p.d.f. (10.71) for the heights in the polynuclear growth model is, in view of the determinant formula (10.16) for the Schur polynomials, of the form (5.139) so the theory of Section 5.8 applies in relation to the calculation of the correlation functions. In fact, by using the determinant formula (10.27) instead of (10.16), a double contour form of the corresponding correlation kernel in the limit n → ∞ is possible [430]. P ROPOSITION 10.3.4 With μ a partition, write μk − (k − 1) =: nk , and consider the joint p.d.f. on n implied by (10.71). For n → ∞ we have ρ(l) (n1 , . . . , nl ) = det[K(nα , nβ )]α,β=1,...,l , where
K(k, l) =
Cr1
dα 2πiαk
Cr2
dβ 1 H(1/β; {bi })H(α; {ai }) . 2πiβ −l+1 α − β H(β; {ai })H(1/α; {bi })
(10.86)
(10.87)
Here r1 > 1 > r2 > 0 and Cr denotes a circle about the origin of radius r, while H(u; {qi }) :=
∞ j=1
1 . 1 − qj u
Proof. In the following some formal manipulations on semi-infinite Toeplitz determinants are carried out; for their justification we refer to [456], [331].
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LATTICE PATHS AND GROWTH MODELS
In terms of {nk }, after use of (10.27) the joint p.d.f. (10.71) reads ∞ Y
h i (1 − ai bj ) det hnk +j−1 ({ai })
i,j=1
j,k=1,...,n
h i det hnk +j−1 ({bi })
j,k=1,...,n
.
As this is a symmetric function in {nk } which vanishes if nj = nk (j = k), we can relax the ordering constraint n1 > n2 > · · · implied by the definition of {nk }. According to the result of Proposition 5.8.1 with ξj (n) = hn+j−1 ({ai }), ηj (n) = hn+j−1 ({bi }), for n → ∞ the l-point correlation is given by (10.86) with ∞ X
K(p, q) =
cjk hp+j−1 ({ai })hq+j−1 ({bi }),
(10.88)
j,k=1
P −1 where [cjk ]j,k=1,2,... = ([ ∞ . l=−∞ ξj (l)ηk (l)]j,k=1,2,... ) In preparation for computing the matrix inverse, introduce the notation for a semi-infinite Toeplitz matrix h i T [f (z)] = [z j−k ]f (z) j,k=1,2,...
(10.89)
for f (z) a Laurent expandable function. We then have ∞ X l=−∞
ξj (l)ηk (l) =
∞ X
hl−j ({ai })hl−k ({bi }) =
„ h i h i« T H(1/z; {ai }) T H(z; {bi }) ,
(10.90)
jk
l=1
where in obtaining the first equality the change of variables l → l − k − j has been performed and the fact that the summation vanishes for l < 1 used, while for the second equality use has been made of (10.25). Because (10.90) is a decomposition into the product of an upper triangular and a lower triangular Toeplitz matrix, and for such matrices in general the inverse is a Toeplitz matrix of the same type but with reciprocal generating function, it follows h [cjk ]j,k=1,2,... = T
i i h i hX 1 1 ej−l ({bi })ek−l ({ai }) , T = H(z; {bi }) H(1/z; {ai }) j,k=1,2,... l=1 ∞
where {ej } denote the elementary symmetric functions (4.132). This result in turn allows us to compute that ∞ X
cjk β k α−j =
j,k=1
assuming 0 < |β| < 1 < |α|. Consider now the generating function
1 β/α , 1 − β/α H(1/α; {bi })H(β; {ai })
∞ X
(10.91)
αk β −l K(k, l).
k,l=−∞
Substituting (10.88) and making use of (10.91) shows ∞ X
αk β −l K(k, l) =
k,l=−∞
1 H(1/β; {bi })H(α; {ai }) . 1 − β/α H(β; {ai })H(1/α; {bi })
(10.92)
This is equivalent to the sought result (10.87).
Suppose the coordinates {nj } are restricted to be less than or equal to l. According to the general theory of Section 9.1 we have that lim Pr(h ≤ l) = det(1 − KJ ),
n→∞
(10.93)
where KJ is the Wiener-Hopf operator on J = {l + 1, l + 2, . . . , } with kernel K(j, k). On the other hand
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(10.73) gives lim Pr(h ≤ l) =
n→∞
∞
l ∞ (1 − ai bj ) (1 + aj e−iθk )(1 + bj eiθk )
i,j=1
j=1 k=1
CUEl
.
(10.94)
Equate (10.93) and (10.94) (with θk → −θk ), and substitute ∞
∞
akl = kck ,
l=1
bkl = kc−k
l=1
for the corresponding power sum symmetric functions. Noting that with these substitutions, and uj = aj , uj = bj , respectively, we have ∞
∞ (1 − uj z) = exp − c∓k z k ,
j=1
(10.95)
k=1
and recalling the general identity (5.32) gives an identity expressing a general Toeplitz determinant in terms of a Fredholm determinant. ∞ P ROPOSITION 10.3.5 Let Dn [f ] be specified as in Proposition 14.4.1 below. With c(θ) := n=−∞ cn einθ we have Dn [ec(θ) ] = enc0 +
P∞
p=1
pc−p cp
det(1 − KJ ),
(10.96)
where KJ is the Wiener-Hopf operator on [n + 1, n + 2, . . . ) with kernel ∞ dα dβ 1 exp ck (β k − αk ) + c−k (α−k − β −k ) . K(k, l) = k −l+1 α−β Cr1 2πiα Cr2 2πiβ k=1
A reformulation of (10.96), and an alternative derivation, are given in Exercises 10.3 q.2. This result was given in the context of random matrix theory by Borodin and Okounkov [86]. Later it was realized that the same identity had occurred in the context of Toeplitz determinant theory in an earlier study by Geronimo and Case [260]. In this context, observe that (10.96) effectively gives higher order corrections to the Szeg¨o formula (14.71) below. E XERCISES 10.3
1. [222] In this exercise a generalization of (10.84) will be derived.
(i) In the case that X is an n × m (n ≥ m) matrix of non-negative real numbers distributed according to (10.79), by taking bm+1 = · · · = bn = 0 in (10.71), and noting that we must then have (μ) ≤ m for this to be nonzero, argue that the continuous RSK correspondence maps the distribution on X to m n Y Y
(αi + βj ) det[ul ({αj }j=l,...,n ; yk )]k,l=1,...,m det[ul ({βj }j=l,...,m ; yk )]k,l=1,...,m ,
i=1 j=1
where ul is specified by (10.80). ˜ + (j − 1)˜ c this (ii) Make use of (10.83) (appropriately modified) to show that with αi = a + (i − 1)c, βj = a simplifies to read m n Y “Y
(a + a ˜ + (i − 1)c + (j − 1)˜ c)
m ”“ Y
i=1 j=1
×
m Y
(1 − e−cyk )n−m e−(a+˜a)
k=1
j=1 Pm
j=1
” 1 1 (c˜ c)j−1 Γ(j)Γ(n − m + j) c(n−m)m
Y
yj
1≤i<j≤m
Relate this to the JUE and the LUE, as done for (10.84).
(e−cyj − e−cyi )(e−˜cyj − e−˜cyi ).
(10.97)
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LATTICE PATHS AND GROWTH MODELS
2. [47], [92] The aim of this exercise is to give a reformulation and alternative derivation of (10.96). (i) Let K = [K(l, k)]k,l=1,... be the transpose of the semi-infinite matrix with entries as specified by (10.87). With φ(z) := 1/(H(1/z; {aj })H(z; {bj })) show that (10.96) can alternatively be written Dn [φ(eiθ )] = D∞ [φ(eiθ )] det(1 − Qn KQn ),
(10.98)
where Qn is the semi-infinite diagonal matrix differing from the identity by the deletion of the first n elements. (ii) With φ− (z) := 1/H(1/z; {aj }), φ+ (z) := 1/H(z; {bj }) so that φ(z) = φ− (z)φ+ (z), verify from (10.92) that ∞ X (φ− /φ+ )p+l−1 (φ+ /φ− )−p−k+1 K(k, l) = p=1
where we have made use of the notation fj = [z j ]f (z). After introducing the further notation H(a) = [aj+k−1 ]j,k=1,2,...
H(˜ a) = [a−j−k+1 ]j,k=1,2,... ,
(10.99)
conclude that K as specified in (i) can be written K = H(φ− /φ+ )H(φ + /φ− ).
(10.100)
(iii) In terms of the notation in (ii), together with T (a) = [aj−k ]j,k=1,2,... , by verifying the formula (ab)l =
∞ X
ap bl−p
p=−∞
show that
T (ab) = T (a)T (b) + H(a)H(˜b).
(10.101)
(iv) Let A be an N × N invertible matrix, let Pn,N be the diagonal matrix with the first n elements 1 and all other entries 0, and set Qn,N = 1N − Pn,N . Then, as already noted in (5.178) but in a different notation, a theorem of Jacobi gives [8] det Pn,N A−1 Pn,N =
det Qn,N AQn,N . det A
(10.102)
(note too that (1.150) corresponds to the case n = 1). Under general conditions this remains valid for N → ∞ and thus semi-infinite matrices. By noting from (10.101) that with A = 1 − K, K as in (10.100), Pn,∞ A−1 Pn,∞ = Pn,∞ T (φ− /φ+ )T (φ+ /φ− )Pn,∞ −1 = Pn,∞ T (φ−1 + )T (φ)T (φ− )Pn,∞ −1 = Tn (φ−1 + )Tn (φ)Tn (φ− ),
where Tn := Pn,∞ T Pn,∞ , deduce (10.98).
10.4 FURTHER INTERPRETATIONS AND VARIANTS OF THE RSK CORRESPONDENCE 10.4.1 Stochastic recurrences The nonintersecting paths underlying the discrete polynuclear growth process can be generated by recurrences which couple together the displacements along level-l and level-(l − 1) at varying positions along the paths. Furthermore, in the case of the top path, this recurrence allows for the derivation of some inter-relations between various growth models, and also for some different interpretations of the process relating to this path.
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Let Xn1 ,n2 denote the first n1 rows and n2 columns of the integer matrix X. Extend this to a square . n1 ,n2 of dimension max(n1 , n2 ) × max(n1 , n2 ) by adjoining rows of zeros to the top (or columns matrix X of zeros to the right as appropriate), then apply the growth algorithm and record λ1 (n1 , n2 ), the maximum displacement of the corresponding level-1 path. Consideration of the rules of the growth process leads to an interpretation of λl (n1 , n2 ) for n1 = n or n2 = n as displacements in the paths relating to X itself. Thus one has that λl (n, j) gives the height of the level-l path at x = −(2n + 12 − 2j), while λl (i, n) gives the height of the level-l path at x = (2n + 12 − 2i). One immediate implication is the summations n n n n λl (n, j) − λl (n, j − 1) = λl (i, n) − λl (i − 1, n) = xi,j , xi,j , (10.103) i=1
l=1
l=1
j=1
where λl (0, j) = λl (i, 0). These equate the increments at a given position to the sum of the nucleation events contributing to those increments. ˜ n1 ,n2 , the same considerations give that λl (n∗ , j) is equal to the height For the paths contributing to X of the level-l paths at x = −(2n∗ + 12 − 2j), while λl (i, n∗ ) is equal to the height of the level-l path at x = (2n+ 12 −2i), where n∗ = max(n1 , n2 ). One notes too that λl (n∗ , j) = λl (n1 , j), λl (i, n∗ ) = λl (i, n2 ). Further relevant points are that the λl (n1 − 1, n2 − 1) occur as the displacement of the level-l path at the origin for times t = 2n∗ − 2 (second-last step), and that λl (i, j) ≥ λl (i, j − 1) ≥ λl+1 (i, j),
λl (i, j) ≥ λl (i − 1, j) ≥ λl+1 (i, j).
(10.104)
These considerations, together with the rules of the growth process, imply recurrences for {λl (i, j)}. Thus the rules of the growth process give that λ1 (n1 , n2 ) is equal to the maximum of the heights in level-1 at x = −1 and x = 1, plus the nucleation height xn1 ,n2 , and so (10.105) λ1 (n1 , n2 ) = max λ1 (n1 , n2 − 1), λ1 (n1 − 1, n2 ) + xn1 ,n2 , where λ1 (0, j) = λ1 (i, 0) = 0. Similarly, the maximum displacement λl (n1 , n2 ) of the level-l path (l = 2, . . . , n) satisfies λl (n1 , n2 ) = max λl (n1 , n2 − 1), λl (n1 − 1, n2 ) + x(l−1) (10.106) n1 ,n2 , (l−1)
where xn1 ,n2 is the height of an overlap event (if any) which occurs in the growth of the nucleation events xn1 ,n2 −1 or xn1 −1,n2 and/or corresponding plateaux. This latter height is equal to the minimum of the height in level-(l − 1) at x = ±1, with the height at the origin in the second-last step subtracted so that x(l−1) n1 ,n2 = min λl−1 (n1 , n2 − 1), λl−1 (n1 − 1, n2 ) − λl−1 (n1 − 1, n2 − 1). Substituting this in (10.106) gives that for l > 1 λl (n1 , n2 ) = max λl (n1 , n2 − 1), λl (n1 − 1, n2 ) + min λl−1 (n1 , n2 − 1), λl−1 (n1 − 1, n2 ) − λl−1 (n1 − 1, n2 − 1).
(10.107)
10.4.2 Last passage percolation and increasing subsequences The process interpretation of the mapping from the non-negative integer matrix X to nonintersecting paths can alternatively be viewed as a certain directed last passage percolation [329]. Let us regard each element xi,j of the matrix X as a waiting time associated with the lattice site (i, j) (1 ≤ i, j ≤ n). The directed last passage percolation is defined by forming a u/rh lattice path from the site (1,1) to the site (n1 , n2 ), such that it maximizes the sum of the waiting times (in general such a path will not be unique). Thus the quantity of
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LATTICE PATHS AND GROWTH MODELS
interest is
(1)
LX (n1 , n2 ) := max
xi,j ,
(10.108)
(1,1) u/rh (n1 ,n2 )
which is referred to as the last passage time. From this definition we see that (1) (1) (1) LX (n1 , n2 ) = max LX (n1 , n2 − 1), LX (n1 − 1, n2 ) + xn1 ,n2 (1)
(10.109)
(1)
with LX (0, k) = LX (j, 0) := 0. This is identical to the recurrence (10.105) with (1)
LX (n1 , n2 ) = λ1 (n1 , n2 )
(10.110)
for all 1 ≤ n1 , n2 ≤ n, so the last passage time is equal to the maximum height in the growth process applied ˜ n1 ,n2 . to X (l) The definition (10.108) of the last passage time can be generalized to a quantity LX (n1 , n2 ) which relates to the level-p heights λp (n1 , n2 ) for each p = 1, . . . , l. Thus let (rd∗ )l denote the set of l disjoint rd∗ lattice paths, the latter defined as either a single point, or points connected by segments formed out of arbitrary positive integer multiples of steps to the right and steps up in the rectangle 1 ≤ i ≤ n1 , 1 ≤ j ≤ n2 (the steps are said to be disjoint if they connect no common lattice sites). In terms of this set generalize (10.108) to (l) LX (n1 , n2 ) = max xi,j . (10.111) (rd∗ )l
These quantities obey a recurrence generalizing (10.109), (l) (l−1) (l−1) (l) LX (n1 , n2 ) + LX (n1 − 1, n2 − 1) = max LX (n1 − 1, n2 ) + LX (n1 , n2 − 1), (l−1) (l) LX (n1 , n2 − 1) + LX (n1 − 1, n2 ) + xn1 ,n2 , (10.112) (0)
(l−1)
where LX (n, m) := 0, which can be proved by consideration of the union of the paths realizing LX (l) LX in each term [457]. Noting that in general
and
max(al−1 + bl , bl−1 + al ) − max(al−2 + bl−1 , bl−2 + al−1 ) = max(bl − bl−1 , al − al−1 ) + min(al−1 − al−2 , bl−1 − bl−2 )
(10.113)
we see by subtracting from the recurrence (10.112) the same recurrence with l → l − 1, that the recurrence (10.107) results with the identification (l)
(l−1)
λl (n1 , n2 ) = LX (n1 , n2 ) − LX
(n1 , n2 ),
or equivalently (l)
LX (n1 , n2 ) =
l
λp (n1 , n2 ).
(10.114)
p=1 (l)
The non-negative matrix X can be presented as a two-line array, and LX (n1 , n2 ) interpreted in terms of weakly increasing subsequences (see Figure 10.13). In fact it is this viewpoint which is most prevalent in the combinatorics literature, being the one taken in the pioneering works of Schensted and Greene [481], [273]. A two-line array is constructed from the n1 × n2 matrix Xn1 ,n2 of non-negative integers by writing down in sequence, reading along the first row of the matrix, then the second row and so on (recall that our convention is to count rows from the bottom), xi,j copies of the ordered pair ji (see Figure 10.13 for an example). By construction the entries in the top row of the two-line array are all weakly increasing, but not so
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0
0
1
3
1
2
1
1
1
2
2
2
2
2
2
3
1
0
2
1
3
3
1
1
1
2
3
3
3
t=0
Q1
Q2
Q3
t=1
Q1
Q2
Q3
t=2
Q1
Q2
Q3
t=3
Q1
Q2
Q3
t=4
Q1
Q2
Q3
t=5
Q1
Q2
Q3
t=6
Q1
Q2
Q3
t=7
Q1
Q2
Q3
t=8
Q1
Q2
Q3
Figure 10.13 A 3 × 3 non-negative integer matrix, with the corresponding queueing “ ” process “ ” “ and ” “two-line ” “ ”array. Note (1) 1 2 2 2 that LX (3, 3) = 8, corresponding to the path through the elements 1 , 1 , 2 , 3 , 33 , while the copies of these ordered pairs forms an increasing subsequence of length 8 in the bottom line of the two-line array.
in general the bottom row. The maximum of the lengths of subsequences in the bottom row which are weakly increasing (i.e., are of the form lj1 ≤ lj2 ≤ · · · ≤ ljm ) is readily seen to be given by the formula (10.108) (1) specifying LX (n1 , n2 ). In this context (10.110), with λ1 (n1 , n2 ) interpreted as the length of the first row of the corresponding semi-standard tableau, is due to Schensted. Furthermore, the maximum of the lengths of (l) l disjoint subsequences in the bottom row which are all weakly increasing is the same thing as LX (n1 , n2 ) and in this context (10.114) is due to Greene.
10.4.3 Queues (1)
Focusing attention on LX (n1 , n2 ) only, a number of further interpretations of the non-negative integer elements of X and the corresponding meaning of the last passage time are possible. One such interpretation is to view the xi,j as service times in a queueing process [28]. Specifically, consider an infinite number of queues Qj (j = 1, 2, . . . ) which initially have n jobs labeled i = 1, 2, . . . in queue Q1 , and zero jobs in all other queues, and suppose that immediately after service in Qj , each job moves to Qj+1 (see Figure 10.13 for an example). Then with xi,j denoting the time it takes server j to process job i (once it reaches the server), we see that the time T (i, j) it takes job i to leave queue Qj satisfies the recurrence (10.105) and is thus equal to λ1 (i, j). Theory related to the queueing problem when all the service times are chosen from the same distribution allows for a characterization of {λ1 (i, N )}i=1,...,n . Thus one has the general result that with the service times all chosen from a distribution of mean μ and variance σ 2 , the limiting scaled exit times Di := lim
N →∞
T (i, N ) − μN √ σ N
(10.115)
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LATTICE PATHS AND GROWTH MODELS
of job i from queue N have joint p.d.f. specified by [267] Di =
i−1
sup 0=t0
(Bl (tl+1 ) − Bl (tl ) ,
i = 1, . . . , n.
(10.116)
l=0
In (10.116) each Bl is an independent standard Brownian motion. This general result is independent of the specific distribution from which the xi,j are being sampled. In the case that the distribution is the exponential with mean unity, we know from (10.85) the joint distribution of the (s) variables {xl }l=1,...,s , s = 1, . . . , p, having the interpretation as the maximum heights in the corresponding sequence of the semicontinuous polynuclear growth models. These heights are such that they relate to the (i) exit times in the queueing model by T (i, N ) = xi , provided we set n1 = N . Thus if in (10.85) with √ (s) (s) n1 = N we introduce the scaled variables dl := (xl − N )/ N as suggested by (10.115) (in particular, (s) (s) then ds = Ds ) we obtain for the joint p.d.f. of {dl }l=1,...,p p 1 −(x(p) )2 /2 e l C l=1
1≤i<j≤p
(p)
(xi
(p)
− xj )
p
χ(x(s) > x(s−1) )
s=2
for some normalization C. We recognize as the joint p.d.f. (4.96) for the eigenvalues of successive minors (s) of a random matrix from GUE∗p . In this interpretation xs corresponds to the largest eigenvalue of the s × s minor and so the following result holds true [45]. P ROPOSITION 10.4.1 The joint distribution of {D1 , . . . , Dn } as specified by (10.116) is equal to the joint distribution of {X1 , . . . , Xn }, where Xj denotes the largest eigenvalue in the top j × j sub-block of a random matrix chosen from GUE∗n .
10.4.4 Corner growth model We consider next a directed growth process (the corner growth model) associated with X, distinct from the polynuclear growth model already considered. For this define a subset of entries of X by , + (xij + 1) ≤ t . (10.117) A(t) := (m, n) : max (1,1)u/rh(m,n) (1)
Comparing with the definition of LX (m, n), we see that (1) ˜ X (m, n). (xij + 1) = LX (m, n) + (m + n − 1) =: L max
(10.118)
(1,1)u/rh(m,n)
Because the sum is over all paths (1, 1)u/rh(m, n) and we must take the maximum, it follows that if (m∗ , n∗ ) ∈ A(t), then (i, j) ∈ A(t) for all 1 ≤ i ≤ m∗ , 1 ≤ j ≤ n∗ , so A(t) forms a staircase-shaped region in the square 1 ≤ m, n ≤ N . The boundary of A(t) can be extended from the x-axis to the y-axis by a single u/lh lattice path enclosing a region referred to as the diagram of A(t). Incrementing t by 1 unit, we see that A(t + 1) is related to A(t) by the addition of at most one lattice site to each column of the latter, and for all columns but the first this site must have its left neighbor in A(t) (the left neighbor will then be referred to as occupied). Thus if ˜ X (m, n) = k + 1 L
(10.119)
then the lattice site (m, n) is added to the diagram of A(k) and so becomes part of the diagram of A(k + 1). We say this event happens at time t = k + 1. Let us now focus attention on the probability of the event (10.119). For m + n > k + 2 we see from
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(10.118) that ˜ X (m, n) = k + 1) = 0. Pr(L For m + n ≤ k + 2, (10.118) and the recurrence (10.109) shows ˜ X (m, n) = k + 1) Pr(L =
k+2−m−n
˜ X (m, n − 1), (L ˜ X (m − 1, n)) = k − j . Pr(xmn = j)Pr max(L
(10.120)
j=0
On the other hand consider a growth process in which A(k + 1) is created from A(k) by first adding to columns according to the rule that for all columns but the first its left neighbour is in A(k), then for each such site (i, j) so added, accepting with probability 1 − qij and rejecting with probability qij . For such a process we see that ˜ X (m, n) = k + 1) Pr(L =
j ˜ X (m, n − 1), L ˜ X (m − 1, n)) = k − j (1 − qmn )qmn Pr max(L
k+2−m−n
(10.121)
j=0
(the significance here of the operation max(·) is that it ensures that both (m, n − 1) and (m − 1, n) are in j A(k), while the factor (1 − qmn)qmn represents the probability that site (m, n) was sampled a total of j times before being accepted). Comparing (10.120) and (10.121) we see that they are identical provided j Pr(xmn = j) = (1 − qmn )qmn .
(10.122)
This condition is fulfilled by the weighting (10.70) of X if we choose qmn = am bn .
(10.123)
4 3 2 1
−4
−3
−2
−1
0
1
2
3
√ Figure 10.14 The square lattice 1 ≤ m, n ≤ N rotated by 45 anticlockwise and stretched by a factor of 2. On the rotated lattice squares are placed in positions stable with respect to motion under gravity according to the rules of the growth process. The top endpoint of each left diagonal segment of the boundary of the accepted placed squares, extended to run along the rotated axes, can be projected onto the x-axis to give a particle configuration. ◦
LATTICE PATHS AND GROWTH MODELS
477
10.4.5 Totally asymmetric simple exclusion process The growth process relating to A(t) is itself related to certain stochastic dynamics of particles on a line [330]. To see this rotate the diagram of A(t) as indicated in Figure 10.14. Instead of thinking of the growth process as adding lattice points, we now think in terms of squares being added. From the shape of the boundary of the accepted squares, a particle configuration on the integer lattice of the x-axis is defined according to the rule specified in Figure 10.14. It follows that in a time interval t → t + 1 of the stochastic dynamics, only particles with their right neighbouring site vacant can move, and they must either move to that lattice site or stay where they are. The probability of these events is the same as the probability of accepting or rejecting the corresponding square in the growth process. According to the prescriptions (10.122) and (10.123) these probabilities are in general lattice site dependent. But in the mapping to the particle configurations all points in the rotated lattice with i − j constant are mapped to the same particle coordinate. For (10.123) then to be constant and also to be less than 1 we must take qij = q so that it is independent of the lattice site. Thus particles with right neighbor vacant move to that site with probability 1 − q, and stay where they are with probability q. This stochastic dynamics is referred to as the totally asymmetric simple exclusion process. ˜ X (m, n) = k + 1 means the particle initially at position i = −(n − 1) has moved m steps at Note that here L time t = k + 1. 10.4.6 The dual RSK correspondence The correspondence between non-negative integer matrices and pairs of nonintersecting u/rd paths has a variant, equivalent to the dual Robinson-Schensted-Knuth correspondence [246], in which a matrix of 0s and 1s is put into one-to-one correspondence with u/rh nonintersecting lattice paths paired with ld/lh nonintersecting paths. Again the matrix itself can be thought of as recording events in a discrete space and time growth process. Let X = [xi,j ] i=1,...,m be an m × n matrix with rows counted from the bottom in which each entry xi,j j=1,...,n is either 0 or 1. Here xi,j = 1 represents a nucleation event (a unit square) which is positioned above the segment x = j − 1 to x = j, on top of earlier nucleation events and their growth. These nucleation events occur at successive time intervals t = 1, 2, . . . , n corresponding to the 1st, 2nd and subsequent rows of the matrix X. Thus at t = 1 there are nucleation events between x = [j1 − 1, j1 ], [j2 − 1, j2 ], . . . , [jp − 1, jp ] where x1j1 , . . . , x1jp are the nonzero entries of the first row of X. These are marked on the line y = 0 as unit squares. In this growth process, as t → t + 1 the profile of all nucleation events so far recorded grows to the right (but not to the left) until they join up with the neighboring nucleation event on the right. If there is no right neighboring nucleation event, and this nucleation event has not yet grown, it is to grow to x = n + 1, while the rightmost profile from the growth of all nucleation events at earlier times is to grow one unit to the right. The shape of the rightmost unit square after growth is to be modified by the removal of the upper triangular half. Furthermore, the meeting of nucleation events in this top profile (which occurs at j2 , . . . , jp in going from t = 1 to t = 2) is to be recorded on the line y = −1 as new nucleation events drawn immediately to the right of these points (thus along the segments [j2 , j2 + 1], . . . , [jp , jp + 1] in going from t = 1 to t = 2), which themselves grow according to the specified rule and give rise to nucleation events on the line y = −2 and so on. After the growth and creation of new nucleation events in lower levels at time t + 1, new nucleation events are placed on the 1st level in correspondence with the nonzero entries of the (t + 1)st row from the bottom of the matrix X, and this completes the process at that time. This procedure is to stop after time m + l in level l (l = 1, . . . , m + 1), this being the maximum time for which new nucleation events can be created and then grow once. The growth profile at each level is of the form of a u/rh path from x = 0 to x = n − 1 and a ld/lh path from x = n + m to x = n + 1 (see Figure 10.15). In level-l these pairs of paths start and finish at y = −(l − 1), have the same maximum height μl units, say, above this line, and do not intersect with the path above, so that μ = (μ1 , μ2 , . . . , μn ) forms a partition.
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t=1 1
1
1
t 3
1
0
0
2
0
1
1
1
x 1
2
y=0 t=2
0
t=5
3 y=0
y=0
y=−1
y=−1
3 t=3
0
y=−2
3 0
3
y=0 t=6
y=−1 t=4
0
y=0
3
y=−1 y=−2
y=0 0
y=−1
3
y=−2 3
0
Figure 10.15 Mapping from a 0,1 matrix to a pair of nonintersecting lattice paths.
It follows that with each entry of X weighted according to the probability Pr(xij = k) =
(ai bj )k , 1 + ai b j
k = 0, 1,
(10.124)
to obtain a one-to-one correspondence between weighted 0,1 matrices and weighted nonintersecting lattice μ(j,l) ν(i,l) and ν(i, l) should be weighted ai . paths, in the latter the displacements μ(j, l) should be weighted bj It follows from (10.24) that the total weight of all nonintersecting u/rh paths initially equally spaced at y = 0, . . . , −(n − 1) along x = 0, finishing at y = μj − (j − 1) (j = 1, . . . , n) along x = n − 1, with up steps at x = j − 1 weighted bj is given by sμ (b1 , b2 , . . . , bn ). Similarly, the total weight of all nonintersecting ld/lh paths initially equally spaced at y = 0, . . . , −(n − 1) along x = n + m, finishing at y = μj − (j − 1) (j = 1, . . . , n) along x = n + 1, with no more than one up step at x = n + m + 1 − i (the latter weighted by ai ) is given by sμ (a1 , a2 , . . . , am ). Here μ denotes the conjugate partition (recall Definition 10.1.3). Hence the probability that a 0,1 matrix with entries distributed according to (10.124) corresponds to a pair of such lattice paths is given by n m
(1 + ai bj )−1 sμ (a1 , . . . , am )sμ (b1 , . . . , bn )
(10.125)
i=1 j=1
(cf. (10.71)). Note that the normalization condition for this probability implies the dual Cauchy identity (10.56). Let Pr(h0,1 ≤ l) denote the probability that the maximum height in the level-1 profile is less than or equal to l. According to (10.125) and the fact that μ1 therein corresponds to the maximum height, we have Pr(h0,1 ≤ l) =
n m i=1 j=1
(1 + ai bj )−1
sμ (a1 , . . . , am )sμ (b1 , . . . , bn ).
(10.126)
μ1 ≤l
We know that the Schur polynomial sμ can be interpreted in terms of nonintersecting u/rh paths by reading along rows (there being μ1 rows in μ and thus μ1 ≤ l paths), while sμ can be interpreted in terms of u/rd paths by reading down columns (in μ there are μ1 columns and thus again μ1 ≤ l paths). The first set of paths consist of m steps, and the second consists of n steps, with the final positions matching. Let
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LATTICE PATHS AND GROWTH MODELS
1
1
1
1
0
0
0
1
1
3 4 2 2 3 4 0 1 2 3 4
0
0 1 3 4 5 2 2 3 4 5 0 1 2 3 4 5
2 2 0 1 2
3 2 2 3 0 1 2 3 Figure 10.16 Oriented digital boiling evolution corresponding to the same 0,1 matrix as in Figure 10.15. u/rh,u/rd (0) (0) (0) Gm,n (l ; l ), l := (l − 1, l − 2, . . . , 0), denote the total weight of this joining of paths, starting and finishing at l(0) , with rh steps weighted by {aj } and rd steps by {bj }. We then have
Pr(h
0,1
≤ l) =
n m
(1 + ai bj )−1 Gu/rh,u/rd (l(0) ; l(0) ). m,n
i=1 j=1
On the other hand, the generating function for a single path of this type is π n m (0) 1 u/rh,u/rd (0) gm,n (l ; l) = (1 + aj eiθ ) (1 − bk e−iθ )−1 e−iθ(l−l ) dθ. 2π −π j=1 k=1
Making use of (10.1) and (5.77) allows [34] Pr(h0,1 ≤ l) =
n m
u/rh,u/rd (0) (0) (l ; l ) Gm,n
(1 + ai bj )−1
i=1 j=1
l m j=1 k=1
to be written as an average over U (l), giving
l n −1 (1 + aj eiθk ) (1 − bj e−iθk ) j=1 k=1
U(l)
.
The top path in the above dual RSK growth process allows for an alternative description, known as oriented digital boiling [272]. To begin, a unit square is placed with its bottom left corner at x = 0, y = −1. At time t = 1 a unit square is added flush right, and a unit square is also added above the existing cube with probability p. Generally, in going from time t to time t + 1, all plateaux grow one unit to the right. Furthermore, above all existing squares still part of the outer profile, a square is added to each independently with probability p. A 0,1 matrix can be associated with the addition of a square, with a 1 (0) placed in column j and row i (rows counted from the bottom) if a square is added (not added) to column j of the growth process at time i − j + 1. If the square is added deterministically, either a 0 or a 1 may be recorded. Conversely, given a 0,1 matrix, a sequence of oriented digital boiling configuration can be constructed (see Figure 10.16 for an example).
E XERCISES 10.4
(i) Argue that “ ” 0,1 0,1 h0,1 1 (n1 , n2 ) = max h1 (n1 , n2 − 1), h1 (n1 − 1, n2 ) + xn1 ,n2 .
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P 0,1 (ii) Show that LX (n1 , n2 ) := xi,j , where the sum is over all u/rd paths from the bottom row (row 1) to the 0,1 (n1 , n2 ). top row (row n1 ), satisfies the same recurrence and thus conclude that h10,1 (n1 , n2 ) = LX
10.5 SYMMETRIZED GROWTH MODELS 10.5.1 Matrices symmetric about the antidiagonal In general the antidiagonal of an n × n matrix X refers to the elements xi,j with i = n + 1 − j. Reflecting the entries about the antidiagonal gives the matrix XR := [xn+1−j,n+1−i ]i,j=1,...,n . Represented geometrically, because we have adopted the convention of numbering rows from the bottom, this reflection is about the top left to bottom right diagonal. Our interest here is in the Robinson-Schensted-Knuth correspondence applied to non-negative integer matrices with the symmetry X = XR . For this use will be made of two theoretical results, which will be stated without proof. First we require a relation between the paths resulting from the RSK mapping applied to XR to those resulting from applying the RSK mapping to X in the generic case that X = XR (see, e.g., [475]). P ROPOSITION 10.5.1 Let (QR , P R ) and (P, Q) denote the pairs of tableaux which result from applying the RSK mapping to XR and X respectively. We have #j s in P = #n + 1 − j s in P R
(10.127)
utzenberger dual of the tableau P and and similarly for Q and QR . The tableau P R is referred to as the Sch¨ can be constructed by the evacuation algorithm (see Exercises 10.5 q.2). Note that (10.127) is a simple consequence of (10.103). It follows from Proposition 10.5.1 that if X = XR and thus X is symmetric about the antidiagonal, the RSK correspondence gives a bijection with a pair of tableaux (P, P R ). Because P R can be constructed from P , we have that in fact there is a bijection between weighted versions of each. For this we choose the weight of P R to equal the weight of P by weighting each occurrence of j in P as well as each occurrence of n+1−j in P R by aj . This weighting of the tableaux is equivalent in the path picture to weighting the up steps at both x = 2n + 12 − 2i and x = −2i + 32 by ai . With the maximum displacement of the level-l path μl , the total weight of such paths is equal to sμ (a21 , . . . , a2n ). Also, from Figure 10.12 such weighting of the paths results by choosing the weights xi,j of the matrix X to be proportional to (ai an+1−j )xi,j . But to obtain a probabilistic setting we should weight only the sites i ≤ n+1−j, with the value of xi,j for i < n+1−j fixed by symmetry. This can be achieved without affecting the weights of the pairs of paths by simply squaring the √ weights at sites i < n + 1 − j and readjusting the normalizations. With ai = qi , we therefore choose Pr(xi,j = k) = (1 − qi qn+1−j )(qi qn+1−j )k , Pr(xi,n+1−i = k) = (1 − qi )qik .
i < n + 1 − j, (10.128)
From the above discussion we then have the following result. P ROPOSITION 10.5.2 Consider a non-negative integer matrix X, symmetric about the antidiagonal with independent entries chosen according to (10.128). The probability that under the RSK correspondence X √ maps to a pair of paths with up steps weighted proportional to qi for x = 2n + 12 − 2i and x = −2i + 32 , and with final displacements μ, is equal to n (1 − qi ) i=1
(1 − qi qj )sμ (q1 , . . . , qn ).
(10.129)
1≤i<j≤n
The second property of the RSK correspondence when [xi,j ] is symmetric about the antidiagonal to be
481
LATTICE PATHS AND GROWTH MODELS
used but not proved is [246] #{xi,n+1−i : xi,n+1−i odd} = #{μj : μj odd} =
n
(−1)j−1 μj .
(10.130)
j=1
Hence if we generalize the second probability in (10.128) to read (1 − qi2 ) kmod 2 k β qi , 1 + βqi
Pr(xi,n+1−i = k) =
(10.131)
then we have the corresponding generalization of (10.129) [34], n (1 − q 2 )
i
i=1
1 + βqi
(1 − qi qj )β
Pn
j=1 (−1)
j−1
μj
sμ (q1 , . . . , qn ).
(10.132)
1≤i<j≤n
Writing h for the maximum height in the growth model picture of this generalized model, it follows from (10.132) that we have Pr(h ≤ l) =
n 1 − qi2 1 + βqi i=1
(1 − qi qj )
β
Pn
j=1 (−1)
j−1
μj
sμ (q1 , . . . , qn ).
(10.133)
μ:μ1 ≤l
1≤i<j≤n
It is possible to write (10.133) as a random matrix average analogous to (10.73) [34]. P ROPOSITION 10.5.3 Let ·Sp(2l) denote an average with respect to the eigenvalue p.d.f. (2.69) for random matrices from the classical group Sp(2l). We have Pr(h ≤ 2l) =
n 1 − qi2 1 + βqi i=1
l (1 − qi qj )
1≤i<j≤n
k=1
n 1 iθk 2 |1 + q e | , j |1 − βe−iθk |2 j=1 Sp(2l)
(10.134) Pr(h ≤ 2l + 1) =
n
(1 − qi2 )
i=1
l n (1 − qi qj ) |1 + qj eiθk |2 k=1 j=1
1≤i<j≤n
Sp(2l)
.
(10.135)
Proof. We follow [224]. According to (10.56) we can write n l Y Y
X
|1 + qj eiθk |2 =
sμ (q1 , . . . , qn )sμ (eiθ1 , e−iθ1 , . . . , eiθl , e−iθl ).
(10.136)
μ:μ1 ≤2l
k=1 j=1
Comparing (10.134) with this substitution to (10.133) shows the former is equivalent to the integration formula l DY k=1
E P2l j−1 1 sρ (eiθ1 , e−iθ1 , . . . , eiθl , e−iθl ) = β j=1 (−1) ρj . −iθ 2 |1 − βe k | Sp(2l)
(10.137)
Recalling now the eigenvalue p.d.f. for Sp(2l) (2.69), noting that the integrand is unchanged by θl → −θl and making use of (10.13) shows the l.h.s. of (10.137) is equal to 1 (2π)l 2l l!
Z
π
−π
Z dθ1 · · ·
π
−π
dθl
– » iθ (ρ l Y (eiθk − e−iθk ) e j 2l−k+1 +k−1) det |1 − βe−iθk |2 e−iθj (ρ2l−k+1 +k−1)
k=1
.
(10.138)
j=1,...,l k=1,...,2l
Proceeding as in the proof of Proposition 6.1.8 reduces this to Z π i 1 h 1 eiθ − e−iθ iθ(ρj −j−ρk +k) −iθ(ρj −j−ρk +k) Pf (e − e )dθ , 2l 2π −π |1 − βe−iθ |2 j,k=1,...,2l
(10.139)
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CHAPTER 10
where as the final step the order of all rows and columns of the Pfaffian have been reversed. Suppose in the integral of (10.139) that k > j so that ρj − ρk + k − j > 0. The integrand is readily computed, showing that (10.139) is equal to h i β −l Pf sgn(k − j)β |ρj −ρk +k−j| . (10.140) j,k=1,...,2l
The above Pfaffian can be evaluated by setting xj = ρj − j, f (xj ) = β Pf
xj
in the general formula
l i h“ f (x ) ”sgn(xj −xk ) Y f (xQ(2j−1) ) j sgn(xj − xk ) = ε(Q), f (xk ) f (xQ(2j) ) j,k=1,...,2l j=1
(10.141)
where the permutation Q is such that xQ(2j−1) > xQ(2j) ,
Q(2j) > Q(2j − 1) (j = 1, . . . , l)
to give the r.h.s. of (10.137). To see the validity of (10.141), note that of the (2l−1)!! permutations in the sum contributing to the Pfaffian, only one term contributes to the r.h.s. with the rest cancelling in pairs. It remains to verify (10.135). Use of an appropriate modification of (10.136) shows that this is equivalent to the integration formula E D sρ (eiθ1 , e−iθ1 , . . . , eiθl , e−iθl , β)
Sp(2l)
=β
P2l+1 j=1
(−1)j−1 ρj
.
(10.142)
Proceeding as in the derivation of (10.138) and (10.139) shows that the l.h.s. is equal to 2 iθj (ρ2l−k+2 +k−1) 3 Z π Z π l e Y (eiθk − e−iθk ) 1 4 e−iθj (ρ2l−k+2 +k−1) 5 dθ · · · dθ det 1 l (2π)l 2l l! −π |1 − βe−iθk |2 −π k=1 β ρ2l−k+2 +k−1 j=1,...,l k=1,...,2l+1 » – 1 [β ρj +2l+1−j ]j=1,...,2l+1 A(2l+1)×(2l+1) = l Pf , [β ρk +2l+1−k ]k=1,...,2l+1 0 2 where
» A(2l+1)×(2l+1) :=
1 2π
Z
π
−π
eiθ − e−iθ (eiθ(ρj −j−ρk +k) − e−iθ(ρj −j−ρk +k) )dθ |1 − βe−iθ |2
The integral can be computed to give for the l.h.s. » [sgn(k − j)β |ρj −ρk +k−j| ]j,k=1,...,2l+1 β −(l+1) Pf [−β ρk +2l+1−k ]k=1,...,2l
– . j,k=1,...,2l+1
[β ρj +2l+1−j ]j=1,...,2l 0
– .
This is precisely the same as (10.140) with l → l + 1, ρ2l+2 = 0, and so reduces to the r.h.s. of (10.142).
We see from the exact results of Proposition 10.5.3 that Pr(h ≤ 2l + 1) = Pr(h ≤ 2l)|β=0 .
(10.143)
To understand this result, note that for matrices symmetric about the antidiagonal, the path (0, 0)u/rh(n, n) which maximizes (0,0)u/rh(n,n) xi,j can likewise be chosen to be symmetric about the antidiagonal. This means that for h to be odd, the value of xi,n+1−i must be odd, as all values of xi,j off the antidiagonal contributing to h occur in pairs. Moreover (10.131) gives , Pr(xi,n+1−i = 2l + 1) + Pr(xi,n+1−i = 2l ) = Pr(xi,n+1−i = 2l ) β=0
whereas Pr(xi,n+1−i = 2l − 1)|β=0 = 0, so pairing the odd values on the diagonal with the even values in this way we reclaim the setting of Pr(h ≤ 2l)|β=0 as required by (10.143).
483
LATTICE PATHS AND GROWTH MODELS
Matrices zero above the antidiagonal Consider a general n × n non-negative integer matrix X = [xi,j ]i,j=1,...,n and suppose xi,j = 0 for i > n + 1 − j. For i ≤ n + 1 − j let xi,j occur with the geometric probability (10.70). Applying the RSK correspondence in the nonintersecting paths formulation but stopping at t = n rather than t = 2n − 1 gives a bijection between X and weighted w,u/rh(o)/lh(e) nonintersecting lattice paths as specified in Section 10.1.2, (1) with the spacing between levels one unit (see Figure 10.17). As usual LX (n, n) is equal to the maximum ˜ say. On the other hand h ˜ is equal to the number of nonintersecting of the heights in the level-1 path, h, w,u/rd(o)/ld(e) lattice paths dual to the w,u/rh(o)/lh(e) lattice paths, where in both cases the paths start and finish on the same level, with the interspacing between levels one unit. Hence w,u/rd(o)/ld(e) (0) ˜ ≤ l) = Pr(h (1 − ai bn+1−j )G2n ({rj = l − j + 1}j=1,...,l ; {rj = l − j + 1}j=1,...,l ). 1≤i≤j≤n
t=1 b1 b1
b2
1
t=2
0
2
2
a2
1
x
2
a2 3
b1
3
b2
y=0
1
−2 −1
0
2
a1
2
1
a1
y=0
0
b3 t
a3
a1
1
0 t=3
2
2
b3
a3
a2
4
b1
2
a1 b2
a2
y=0 y=−1
0
Figure 10.17 Mapping from a weighted non-negative integer matrix with entries zero above the antidiagonal (this portion of the matrix is not shown) and weighted nonintersecting lattice paths. The latter, upon rotation by 90◦ clockwise, can be regarded as w,u/rh(o)/lh(e) type paths.
Let us now set bi = an+1−i , ai = qi (i = 1, . . . , n) so that for sites (i, j) with i < n + 1 − j the probabilities are the same as (10.128) determining h . Making use of (10.51) shows ˜ ≤ l) Pr(h b
i =an+1−i ai =qi
=
l n (1 − qi qj ) |1 + qj eiθk |2 k=1 j=1
1≤i≤j≤n
Sp(2l)
.
Comparing with (10.134) we thus have
˜ ≤ l) = Pr(h ≤ 2l) Pr(h
β=0
To anticipate this, we note from (10.131) that Pr(xi,n+1−i
= k)
β=0
=
.
(1 − qi2 )qik , k even, 0, k odd.
(10.144)
(10.145)
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Also, as remarked in the discussion of (10.143), in the case [xi,j ] is symmetric about the antidiagonal xi,j = max χi,j xi,j , (10.146) max (1,1)u/rh(n,n)
(1,1)u/rh(l,n+1−l)
where (l, n+ 1 − l) is any point on the antidiagonal while χi,j = 1 for points on the antidiagonal and χi,j = 2 otherwise. Because according to (10.145), in the case β = 0 the points on the antidiagonal are restricted to even values, we can redefine χi,j in (10.146) to equal 2 for all points without changing Pr(h ≤ 2l) , β=0
provided we redefine the probability (10.145) so that = (1 − qi2 )qi2k , Pr(xi,n+1−i = k) β=0
k = 0, 1, . . .
˜ ≤ l) in (10.144) are precisely the same as With this redefinition we see that the probabilities making up Pr(h ˜ which together imply (10.144). those specifying Pr(h ≤ 2l)|β=0 , and furthermore h = 2h,
10.5.2 Matrices symmetric about the diagonal The rules of the growth process tell us that if the non-negative integer matrix X = [xi,j ]i,j=1,...,n maps to the pair of tableaux (P, Q), then the transposed matrix XT := [xj,i ]i,j=1,...,n maps to (Q, P ). Hence the Robinson-Schensted-Knuth correspondence when applied to non-negative integer matrices symmetric about the diagonal gives a bijection with a single semi-standard tableau, or equivalently a single set of u/rh lattice paths, since then P = Q. To obtain a bijection between weighted symmetric matrices and a weighted set of u/rh lattice paths the simplest situation is to require that the sets of paths have the same weighting, and then to weight a single set by the square of the original weighting. We see from Figure 10.12 that this will happen if we restrict the weights so that ai = bi (i = 1, . . . , n), and thus weight each entry xi,j by (1 − ai aj )(ai aj )xi,j . But as in (10.70) and (10.128) we would like to relate these weights to probabilities, for which purpose we should weight only the sites i ≤ j, with the value of xi,j for i > j fixed by symmetry. This can be done by squaring the weights at sites i < j and readjusting the normalizations. The following analogue of Proposition 10.5.2 is then obtained [329]. P ROPOSITION 10.5.4 Consider a non-negative integer matrix X, symmetric about the diagonal with independent entries chosen according to Pr(xi,j = k) = (1 − qi qj )(qi qj )k , i < j,
Pr(xi,i = k) = (1 − qi )qik .
(10.147)
The probability that under the RSK correspondence X maps to a pair of paths with up steps weighted pro√ portional to qi for x = ±(2n + 12 − 2i), and with final displacements μ, is equal to
n (1 − qi ) i=1
(1 − qi qj )sμ (q1 , . . . , qn ).
(10.148)
1≤i<j≤n
The probability on the diagonal entries in (10.147) can be generalized [34]. This is possible because in the RSK mapping for a symmetric matrix n j=1
xj,j =
n
(−1)j−1 μj
(10.149)
j=1
(see Exercises 10.5 q.1). Thus if the probability in question is generalized to read Pr(xi,i = k) = (1 − αqi )(αqi )k ,
(10.150)
485
LATTICE PATHS AND GROWTH MODELS
(10.148) should correspondingly be generalized to n
(1 − αqi )
i=1
Pn
(1 − qi qj )α
j=1 (−1)
j−1
μj
sμ (q1 , . . . , qn ).
(10.151)
1≤i<j≤n
With X symmetric about the diagonal, let us denote the maximum height of the level-1 path by h . Noting that n n (−1)j−1 μj = #(columns of odd length in μ) = μk mod 2, (10.152) j=1
k=1
where l = μ1 , we see from (10.151) that Pr(h ≤ l) =
n i=1
(1 − αqi )
(1 − qi qj )
1≤i<j≤n
Pn
α
k=1
μk mod 2
sμ (q1 , . . . , qn ).
(10.153)
μ:μ1 ≤l
As with (10.133) this can be written in terms of an average over a classical group. P ROPOSITION 10.5.5 We have n (1 − αqi ) Pr(h ≤ l) = i=1
n (1 − qi qj ) det(1l + αU) det(1l + qj U) j=1
1≤i<j≤n
U∈O(l)
.
(10.154)
Proof. As with the proof of Proposition 10.5.3, we follow [224]. Use of (10.56) in (10.154) and comparison with (10.153) shows that (10.154) is equivalent to the matrix integral evaluation D E Pl det(1l + αU)sρ (U) = α j=1 ρj mod 2 ,
(10.155)
U∈O(l)
where sρ (U) denotes the Schur polynomial as a function of all the eigenvalues of U. For definiteness, consider the l even case, l → 2l, and consider separately the components O± (2l) of O(2l). Recalling the eigenvalue p.d.f. for O+ (2l) (2.62), and proceeding as in the derivation of (10.140) shows det(1l + αU)sρ (U)U∈O + (2l) = 21−l Pf[ajk ]j,k=1,...,2l where
(10.156)
“ ” ajk = (1 + α2 )δ(ρj −j)−(ρk −k),odd + 2αδ(ρj −j)−(ρk −k),even sgn(k − j) “1 ” 1 = (1 + α)2 − (1 − α)2 (−1)(ρj −j)−(ρk −k) sgn(k − j). 2 2
The task is therefore to compute the Pfaffian of the matrix with these entries. For this one uses the identity [502] P X Pf(A + B) = (−1) j∈S j−|S|/2 Pf S (A)Pf S¯ (B), (10.157) S⊆{1,2,...,2l} |S| even
where Pf S (A) denotes the Pfaffian of A restricted to rows and columns specified by the index set S, and similarly Pf S¯ (B) (S¯ denotes the complement of S). With h1 i h 1 i (1 + α)2 sgn(k − j) A= , B = − (1 − α)2 (−1)(ρj −j)−(ρk −k) sgn(k − j) , 2 2 j,k=1,...,2l j,k=1,...,2l from the simple identities Pf[sgn(k − j)] = 1,
Pf[aj,k (−1)(ρj −j)−(ρk −k) ] = (−1)
P
(ρj −j)
Pf[aj,k ],
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we see that
¯
Pf S A = 2−|S|/2 (1 + α)|S| ,
Pf S¯ B = (−2)−|S|/2 (1 − α)2l−|S| (−1)
P
¯ j∈S
ρj −j
.
Application of (10.157) gives “ 1 + α ”|S| “ 1 − α ”2l−|S|
X
21−l Pf(A + B) = 2
2
S⊆{1,2,...,2l} |S| even
2
P
(−1)
¯ j∈S
ρj
.
(10.158)
But in general X
x|S| y 2l−|S| (−1)
S⊆{1,2,...,2l} |S| even
=
=
1“ 2
¯ j∈S
ρj
x|S| y 2l−|S| (−1)
P
¯ j∈S
ρj
(x + (−1)ρj y) +
j=1
X
+
S⊆{1,2,...,2l}
2l 1“ Y
2
X
P
x|S| (−y)2l−|S| (−1)
P
¯ j∈S
ρj
”
S⊆{1,2,...,2l} 2l Y
” (x − (−1)ρj y) .
j=1
Using this result to evaluate (10.158) and substituting in (10.156) gives the matrix integral evaluation P2l
det(1l + αU)sρ (U)U∈O + (2l) = α
j=1
ρj mod 2
P2l
j=1 (ρj +1)mod 2
+α
.
(10.159)
It remains to compute the corresponding formula for the average over O− (2l). The analogue of (10.156) in this case is
(1 − α2 ) [ζ]Pf[ajk + ζbjk ]j,k=1,...,2l 2l−1 = (−1)ρk −k − (−1)ρj −j . Observing
det(1l + αU)sρ (U)U ∈O −(2l) = (cf. (6.159)), where ajk is as in (10.156) while bjk [bjk ] = uw T − w uT ,
u = [1]j=1,...,2l , w = [(−1)ρj −j ]j=1,...,2l
(10.160)
(10.161)
shows that [bjk ] has rank 2. It follows that the Pfaffian in (10.160) is linear in ζ, and so the r.h.s. of (10.160) can be rewritten i ” (1 − α2 ) 1 “ h (10.162) Pf [ajk ] + ζ[bjk ] − Pf[ajk ] . l−1 2 ζ With γ, ζ1 , ζ2 arbitrary nonzero constants, the structure (10.161), and applying elementary row and column operations, verifies that this in turn can be rewritten 3 2 « „ [ajk ] ζ1 w ζ2 u (1 − α2 ) 1 T 5 − γPf[ajk ] . 4 −ζ1 w 0 γ (10.163) Pf 2l−1 ζ1 ζ2 uT −γ 0 −ζ2 Setting ζ1 = 12 (1 − α)2 , ζ2 = (1 + α)2 , adding one half of the final row/column to the second-last row/column, subtracting the second-last row/column from the final row column, and setting γ = (1 + α2 ) allows (10.163) to be recognised as ˛ ” 21−l “ ˛ 2 ] − (1 + α )Pf[a ] Pf[a . (10.164) ˛ jk jk 2l×2l 2(l+1)×2(l+1) 1 − α2 ρ2l+1 =ρ2l+2 =0 Comparing (10.156) and (10.159) tells us that “ P2l ” P2l Pf[ajk ]2l×2l = 2l−1 α j=1 ρj mod 2 + α j=1 (ρj +1)mod 2 . Substituting in (10.164) and simplifying implies the matrix integral evaluation P2l
det(1l + αU)sρ (U)U∈O −(2l) = α
j=1
ρj mod 2
P2l
−α
j=1 (ρj +1)mod 2
.
(10.165)
487
LATTICE PATHS AND GROWTH MODELS
As the matrix integrals over both components O+ (2l) and O− (2l) have now been evaluated, it only remains to take the arithmetic mean of these to obtain (10.155).
We remark that the Pieri formula (12.190) below enables the general α case of (10.155) to be deduced from the result for α = 0. E XERCISES 10.5
1. [360], [221] The objective of this exercise is to derive (10.149).
(i) Note that for the nonintersecting paths interpretation of the RSK correspondence, [xi,j ]i,j=1,...,n symmetric about i = j implies λl (j, i) = λl (i, j), and use this in (10.105) and (10.107) to conclude λ1 (i, i) = λ1 (i, i − 1) + xi,i , λl (i, i) = λl (i, i − 1) + λl−1 (i, i − 1) − λl−1 (i − 1, i − 1),
l > 1.
(ii) Form appropriate linear combinations of the equations in (i), making use of the fact that λl (l, l − 1) = 0, to deduce the equation xi,i =
i X l=1
(−1)l−1 λl (i, i) −
i−1 X
(−1)l−1 λl (i − 1, i − 1).
l=1
Sum this over i from 1 to n and recall that λl (n, n) = μl to deduce (10.149). 2. Sch¨utzenberger’s evacuation operation on a semi-standard tableau P can be described as follows (see, e.g., [246]). Remove the number n0 , say, of the first square in the first row of P , thus creating a blank square. Interchange the place of the blank square and its neighbor immediately to the right, or immediately below, whichever is smaller (if both neighbors have the same number, choose the one below). Repeat this procedure until the blank square is at the end of a row or the end of a column. Then the square in this position of P R is numbered n + 1 − n0 . The blank square is removed from P and the procedure is repeated to complete the filling of P R . (i) Consider the pair of tableaux (P, Q) equivalent to the pair of lattice paths in Figure 10.12. Note that since the matrix X therein is such that X = XR , we must have P R = Q. (ii) Show that the result of (i) is consistent with the construction of P R from the evacuation operation.
10.6 THE HAMMERSLEY PROCESS Closely related to the last passage percolation model associated with the non-negative matrix X is a last passage percolation model associated with random points in the unit square, known as the Hammersley process. The latter is defined by first marking in the unit square points uniformly at random according to 2 a Poisson rate with intensity z 2 , so that the probability the square contains N points is equal to z 2N e−z /N !. From the points one forms a continuous path by joining points with straight line segments of positive slope, and this path is extended to begin at (0, 0) and finish at (1, 1) by adding an extra segment at both ends. With the length of the extended path defined as the number of points it contains, the stochastic variable l is defined as the maximum of the lengths of all possible extended paths (see Figure 10.18). To relate the Hammersley process so defined to the last passage percolation model of Section 10.4.2, consider an n × n matrix X with entries chosen according to the geometric distribution (10.70). Set ai = bj = z/n. Then to leading order in 1/n2 the probability that xij = 1 is equal to z 2 /n2 while to the same order the probability that xij > 1 is zero. If we now think of the lattice sites (i, j) scaled to the points (i/n, j/n) then in the limit n → ∞ we see that a Poisson process with intensity z 2 in the unit square is generated for the distribution of matrix elements with value unity. This latter point follows from the general fact that the Poisson process in question can be realized as the M → ∞ limit of the discrete process of dividing the unit square up into a regular M × M grid and marking a point randomly within each subsquare with probability z 2 /M 2 .
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CHAPTER 10
1
1 Figure 10.18 Eight points in the unit square, and the extended directed paths of maximum length. Since the number of points in these paths is 3, here ln = 3. (1)
Furthermore, one has that in this limit the last passage time LX (n, n) as defined by (10.108) coincides with the definition of l given above. As a consequence the cumulative probability distribution for l can be deduced from knowledge of the formula for the last passage time (denoted h ) in Proposition 10.3.1. P ROPOSITION 10.6.1 For the Hammersley process Pl 2 Pr(l ≤ l) = e−z e2z j=1 cos θj
U(l)
.
(10.166)
Proof. The relationship between the Hammersley process and the last passage percolation model of Section 10.4.2 tells us that Pr(l
≤ l) = lim Pr(h n→∞
˛ ˛ ≤ l)˛
ai =bj =z/n
.
Computing the limit in (10.73) gives (10.166).
10.6.1 Relationship to random permutations Any particular realization of the Hammersley process containing N points gives a geometrical construction of a random permutation of {1, 2, . . . , N }. To see this label the x-coordinates of the points by 0 < x1 < · · · < xN < 1 and similarly the y-coordinates by 0 < y1 < · · · < yN < 1. Each point will then have a coordinate of the form (xj , yP (j) ), where {P (1), . . . , P (N )} is a permutation of {1, 2, . . . , N }. Because the N points are distributed at random with uniform distribution, the permutation is also random with uniform distribution. In addition, the quantity l has an interpretation in terms of the permutation. Thus the analogue of a continuous path consisting of segments of positive slope is a subsequence 1 ≤ j1 < j2 < · · · < jr ≤ N such that P (j1 ) < P (j2 ) < · · · < P (jr ), which is referred to as an increasing subsequence. The length of an increasing subsequence is defined as the value r. We then see from the definitions that the maximum length of all increasing subsequences of P coincides with l . 10.6.2 Droplet PNG model Associated with the non-negative matrix X, both a last passage percolation model and a discrete polynuclear growth model have been identified. Likewise, associated with random points in the unit square are a last passage percolation model (the Hammersley process) and a particular polynuclear growth model known as the droplet PNG model [450]. To define the latter consider the xt-half-plane t > 0. Let this half plane be filled with points uniformly at random and such that the mean density is unity. Analogous to the entries xij of the matrix X these points are
LATTICE PATHS AND GROWTH MODELS
489
to be thought of as seeds for nucleation events of layered growth of unit height, although now the nucleation event grows continuously to the left and right with unit velocity and is created with zero width. In the droplet model, at (x, t) = (0, 0) a single layer, taken to be at height zero, starts spreading with unit velocity to the left and to the right. All nucleation events and thus subsequent layers are constrained to occur above this initial layer. As the √ initial layer grows √ with unit speed, only nucleation events bounded by the “lightcone” axis v = (t + x)/ 2, u = (t − x)/ 2 are created at a time that their position coordinate makes contact with the ground layer or its growth. The nucleation events (xi , ti ) inside the lightcone create the beginning of a portion of a layer of unit height on top of the ground layer, or existing layers, at position xi . The layers are formed by the growth of the nucleation events with unit velocity to the left and to the right; if two growing portions of a layer collide, then growth at that point ceases and the two portions become one, growing only at the end points of this one portion (see Figure 10.19 for an example). Of interest are the statistical properties of the height at the origin after this growth process has been underway for time t = T .
Figure 10.19 Example of the plateau profile at the time of four successive nucleation events, including the initial event (which is labeled the 0th event and its plateau the 0th level). Note that between the second and third nucleation events, two plateaux on the first level have coalesced.
To √ analyze this quantity, the √ first observation is that only those nucleation events in the region [u = 0, u = T / 2] × [v = 0, v = T / 2] of the lightcone can contribute to the height at x = 0 up to time t = T . Suppose in a realization of the nucleation events there are N points in this region. For a Poisson process of 2 unit density this occurs with probability z 2N e−z /N !, where z 2 = T 2 /2 is the area of the region. One then uses the construction of the previous subsection to associate with the configuration of points a permutation P , and furthermore marks in the world lines of the growth of the nucleation events (see Figure 10.20). The world lines show clearly the layered structure of the growth, and in particular the height at the origin after time T . To relate this height to a property of P , we first note a construction which determines the layer in which each particular nucleation event occurs. This can be done by partitioning the permutation into decreasing subsequences using the leftmost digits at all times. The jth such decreasing subsequence tells us the coordinates of the nucleation events, projected onto the line x = −t, which lie in the jth layer in the growth process. For example, in Figure 10.20 the permutation is 5374162, and the decreasing subsequences formed from the leftmost digits are (531)(742)(6). As is shown in Exercises 10.6 q.1, it is generally true that the number of decreasing subsequences of this type is equal to the length of the longest increasing subsequence of the same permutation. Thus studying the height at the origin in the PNG model after time T is equivalent to studying the maximum path length in the Hammersley process with intensity z 2 = T 2 /2.
10.6.3 Permutation matrices and increasing subsequences We have seen that each configuration of N points in a realization of the Hammersley process is equivalent to a random permutation of {1, 2, . . . , N }, read off from the labels of the points (xj , yP (j) ), j = 1, . . . , N . Also associated with the labels is a permutation matrix defined so that the entry (j, P (j)) of row j is equal to unity, while all other entries in the row are equal to zero. The RSK correspondence applied to the permutation matrix maps to a pair of u/rh and u/lh lattice paths with the constraint that for each allowed position of the up
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CHAPTER 10
t
7
6
T
5
7 6 5
4 3
2
1
12
3
4 x
Figure 10.20 World lines for the endpoints of the plateaux. Only nucleation events inside the square-shaped region including the lines x = ±t, and the lines from t = T to these lines affect the height at the origin. The nucleation points occur at v-shaped configurations, while the inverted v part of the world lines correspond to the joining of the plateaux originating from two different nucleation events. The labeling on the lines x = t and x = −t allow the world lines to be identified uniquely with a permutation [532].
steps (recall Figure 10.12) there is a total of 1 such step only. Thus there is a correspondence with a pair of single move u/rh and u/lh lattice paths initially equally spaced, with a total of n moves each, or equivalently with a pair of standard tableaux having the same shape and each of content n (recall Section 10.1.3). From the discussion of the paragraph below (10.114) we know that if the pair of tableaux have shape κ, l then κ1 is equal to the longest increasing subsequence length of the permutation, while i=1 κi is equal to the maximum of the lengths of l disjoint increasing subsequences. Denoting by ln the longest increasing subsequence length, it follows that for a random permutation with uniform measure 1 Pr(ln ≤ l) = # (of pairs of standard tableaux with content n, κ1 ≤ l) . (10.167) n! This can be written as a random matrix average [262], [455]. P ROPOSITION 10.6.2 We have Pr(ln ≤ l) = Proof. We note that
l l 2n 1 iθj 2n n! e = 2 cos θj . n! j=1 (2n)! U(l) U(l) j=1
fnλ := #(standard tableaux of shape λ and content n)
(10.168)
(10.169)
occurs as a particular coefficient in the monomial expansion of the Schur polynomial sλ (w1 , . . . , wn ). Thus we recall from (10.16) that Q sλ is#jdefined as a weighted sum over semi-standard tableau of shape λ and content n with each tableau weighted by n j=1 wj . Since a standard tableau is a special case of a semi-standard tableau in which each number j = 1, . . . , n occurs exactly once, we see immediately that [w1 w2 · · · wn ]sλ (w1 , . . . , wn ) = fnλ ,
(10.170)
491
LATTICE PATHS AND GROWTH MODELS
where [w1 w2 · · · wn ] denotes the coefficient of w1 w2 · · · wn . Hence Pr(ln ≤ l) =
X 1 sκ (a1 , a2 , . . . , an )sκ (b1 , b2 , . . . , bn ). [a1 · · · an b1 · · · bn ] n! κ:κ ≤l
(10.171)
1
But according to (10.72) and (10.73) X
sκ (a1 , a2 , . . . , an )sκ (b1 , b2 , . . . , bn ) =
κ:κ1 ≤l
l n Y DY
(1 + aj e−iθk )(1 + bj eiθk )
E CUEl
j=1 k=1
.
Reading off the coefficients of a1 · · · an b1 · · · bn in the random matrix average gives the first equality in (10.168). The second equality follows by noting l “X
2 cos θj
”2n
=
j=1
2n “ l X 2n ”“ X j=0
j
eiθk
l ”j “ X
k=1
e−iθk
”2n−j
,
(10.172)
k=1
then observing that only the term j = n contributes to the U (l) average.
We note that the result (10.168) could have been derived from the result (10.166) for the Hammersley process. Thus from the definitions, Pr(lN ≤ l) and Pr(l ≤ l) are related by Pr(l ≤ l) = e−z
2
∞ z 2N Pr(lN ≤ l). N!
(10.173)
N =0
Expanding the random matrix average in (10.166) in a power series in z and using this relation leads to the second equality in (10.168). 10.6.4 Random words A permutation can be regarded as a bijective mapping from {1, . . . , n} to {1, . . . , n}. If we consider instead functions which map from {1, . . . , n} to {1, . . . , k}, then presenting the function as a two-line array specifies a so-called word of length n from an alphabet of k letters. Presenting the two-line array for a word as a nonnegative integer matrix gives a 0, 1 matrix of dimension n× k, constrained so that each row contains exactly a single 1. Applying the mapping of Figure 10.15 with this constraint gives a bijection with u/rh nonintersecting lattice paths making k steps paired with ld/lh nonintersecting lattice paths making n steps (as for general 0,1 matrices), but with the latter lattice paths constrained so that at any one step the number of ld steps is exactly one. In terms of tableaux we thus have a bijection with a semi-standard tableau of content k, and a standard tableau of the same shape also of content n. word Denoting by ln,k the longest increasing subsequence length, it follows that for a random word word Pr(ln,k ≤ l) = k −n #( semi-standard tableaux, shape μ, content k)fnμ . μ:μ1 ≤l
Use of (10.13) and (10.169) then shows word Pr(ln,k ≤ l) = k −n [b1 · · · bn ] sμ (a1 , . . . , ak )sμ (b1 , . . . , bn ) μ:μ1 ≤l
a1 =···=ak =1
l k n = k −n [b1 · · · bn ] (1 + ar eiθp ) (1 + bs e−iθp ) p=1
= k −n
l j=1
(1 + eiθj )k
r=1 l j=1
s=1
e−iθj
U(l) a1 =···=ak =1
n U(l)
,
(10.174)
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where the second equality follows from the equality between (10.72) and (10.73). 1. Note from the discussion of Section 10.1.3 that, with l(0) = (l − 1, l − 2, . . . , 0) and wl− = 0, wl+ = 1 (l = 1, . . . , n), wl+ = 0, wl− = 1 (l = n + 1, . . . , 2n),
E XERCISES 10.6
s,ld/rd
# (of pairs of standard tableaux with content n, κ1 ≤ l)) = G2n
(l(0) ; l(0) ).
Substitute this in (10.167), and make use of (10.39) to reclaim the first equality in (10.168). 2. [133] Consider a permutation π of {1, 2, . . . , N }, and let (π) denote the maximum length of the increasing subsequences. (i) Show that the minimum number of single integer moves required to return π back to the identity is N −(π). (ii) Suppose the integers in the permutation, reading from left to right, are sorted into piles according to the rules that low numbers are put on top of high numbers on the left-most possible pile, and if this is not possible, a new pile is started to the right of existing piles. Show that the number of piles is equal to (π).
10.7 SYMMETRIZED PERMUTATION MATRICES 10.7.1 Longest increasing and decreasing subsequence lengths of a random involution A permutation matrix X is a real orthogonal matrix and thus has the property X−1 = XT (remember that we are counting the rows from the bottom, so the operation T corresponds to reflection about the bottom left to top right diagonal). Thus permutation matrices with the symmetry X = XT have the property X2 = 1 and therefore correspond to involutions. Further, the number of 1s on the diagonal corresponds to the number of fixed points of the involution. Our interest is in the cumulative distribution for ln,k , the longest increasing subsequence length of an involution of {1, . . . , n} consisting of k 2-cycles. One approach to computing the cumulative distribution for ln,k is to consider the symmetrized Hammersley model underlying random involutions. First one notes that a random permutation matrix with the symmetry X = XT and n − 2k points on the diagonal corresponds to n random points in the unit square symmetric about the diagonal y = x and with n − 2k of the n points on this line. A Poissonized version of this setting is to break the unit square into M × M equal subsquares, and to break the diagonal up into M equal segments. In each subsquare below the diagonal (together with its image above the diagonal) a point is marked with probability z 2 /M 2 , while a point is marked on each segment of the diagonal with probability αz/M . The probability that there are a total of n points in the square with n − 2k of the points on the diagonal is then given by the coefficient of wn γ n−2k in 2 αzwγ M αz z2 z 2 w2 M /2 + 1− 1− 2 + . M M M M2 Taking the limit M → ∞ it follows from this that the probability is equal to [wn γ n−2k ]eαzγw+z
2
w 2 /2 −z 2 /2−αz
e
where sn,k =
= e−z
2
/2−αz z
n
n!
αn−2k sn,k ,
n (2k)! 2k 2k k!
is the number of involutions consisting of k 2-cycles and n − 2k fixed points. With l denoting the maximum of the lengths of the paths consisting of positive sloping segments, it then follows that Pr(l ≤ l) = e
−αz−z 2 /2
[n/2] ∞ z n n−2k α sn,k Pr(ln,k ≤ l). n! n=0 k=0
(10.175)
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LATTICE PATHS AND GROWTH MODELS
The significance of the relation (10.175) is that the symmetrized Hammersley model can be realized as a limiting case of the growth process associated with matrices symmetric about the diagonal discussed in Section 10.5.2. We see from the first formula in (10.147), and (10.150), that the required limit is to set qi = z/n for each i = 1, . . . , n and take n → ∞. From the evaluation of the cumulative distribution for h given in Proposition 10.5.5, we can then compute the cumulative distribution for l according to (10.176) Pr(l ≤ l) = lim Pr(h ≤ l) n→∞
qi =z/n
to obtain the following result [34]. P ROPOSITION 10.7.1 We have Pr(l ≤ l) = e−αz e−z
2
/2
det(1l + αU)ezTr U
U∈O(l)
.
(10.177)
Rotating all the elements of a permutation matrix with the symmetry X = XT by 90◦ anticlockwise gives a permutation matrix with the symmetry X = XR . In the two line array presentation of an involution, this is equivalent to reversing the ordering of the bottom line, and so interchanging increasing subsequences with decreasing subsequences. Thus the maximum length of the increasing subsequences in the permutation corresponding to the rotated matrix ln,k say, is equal to the maximum length of the decreasing subsequences in the underlying involution. As with (10.175), this is related to the maximum of the length of paths consisting of positive sloping segments in the corresponding Hammersley process by Pr(l ≤ l) = e−βz−z
2
/2
[n/2] ∞ z n n−2k β sn,k Pr(ln,k ≤ l), n! n=0
(10.178)
k=0
where now the antidiagonal y = 1 − x (0 < x < 1) contains points with a Poisson intensity βz. The analogue of the formula (10.176), together with the results of Proposition 10.5.3, then give the following random matrix form [34]. P ROPOSITION 10.7.2 We have Pl 1 ez j=1 2 cos θj , −iθ 2 |1 − βe k | Sp(2l) k=1 Pl 2 . Pr(l ≤ 2l + 1) = e−z /2 ez j=1 2 cos θj Pr(l ≤ 2l) = e−βz−z
2
/2
l
Sp(2l)
(10.179) (10.180)
10.7.2 Relationship to the flat PNG model The Hammersley process symmetric about the antidiagonal, but with no points thereon, is relevant to the study of the PNG model in which growth is from a flat substrate [450]. Here, in distinction to the case of the droplet PNG model, growth from the nucleation events (which again occur at random with unit mean in space and time) is no longer restricted to happen above an initial plateau, but rather may occur over the length of the whole x-axis. The profile will then on average be flat and the statistical properties of the height fluctuation independent of the position, which can therefore without loss of generality be studied at x = 0. For nucleation √ events√to affect the height at the origin after time 2T they must occur at positions x√ and times t such √ √ that |x| < 2T − t. This is a triangular shaped region, with vertices (x, t) = (− 2T, 0), ( 2T, 0), (0, 2T ) in the xt-plane, in which nucleation events occur uniformly at random with unit density. From the rules of the PNG model one sees that the height at the origin is equal to the maximum number of points in an upward directed √ path formed from the nucleation events, which starts along the line (x, 0) and finishes at the point (0, 2T ). The triangular shaped region of points can be extended to a square √ shaped region √ by reflecting it about the x-axis. The length of the longest upwards-directed path from (0, − 2T ) to (0, 2T ) within the square
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is the same thing as the length of the longest up/right path in a symmetrized Hammersley process. The symmetrization is with respect to the antidiagonal, which itself contains no points. The height h at the origin of the PNG model is equal to half the length of the longest up/right path in the Hammersley process, and thus the cumulative distribution Pr(h ≤ l) is given by (10.179) with z = T and β = 0. E XERCISES 10.7
1. [246] The objective of this exercise is to derive the explicit formula Q 1≤i<j≤p (li − lj ) Qp fnλ = n! , li := λi + p − i, i=1 li !
(10.181)
where p := μ1 and |λ| = n. (i) Argue that
n X
fnλ =
λ
(i) fn−1 ,
(10.182)
i=1 λ
(i) := 0 for λ(i) not a partition, and note that this formula where λ(i) := (λ1 , . . . , λi − 1, . . . , λn ) and fn−1 0n together with the initial condition f0 = 1 specifies fnλ by induction. Q (ii) With Δ := i<j (li − lj ) show that for the r.h.s. of (10.181) to satisfy the formula of (i) we require
nΔ(l1 , . . . , lp ) =
p X
li Δ(l1 , . . . , li − 1, . . . , lp ).
i=1
Verify this formula by establishing the identity “ “p” ” xi Δ(x1 , . . . , xi + t, . . . , xp ) = x1 + · · · + xp + t Δ(x1 , . . . , xp ). 2 i=1
p X
(First observe that the l.h.s. is anti-symmetric in the xi , and hence has as a factor Δ(x1 , . . . , xp ). From this conclude the expression must be a linear function of t.) 2. [7] In this exercise the combinatorial setting will be used to determine the asymptotic behavior of some matrix integrals. (i) Consider (10.178) in the case β = 0. Noting that Pr(l2n,n ≤ 2l) = 1 (n ≤ l), show Pr(l
˛ ˛ ≤ 2l)˛
β=0
1 , s2(l+1),l+1
Pr(l2(l+1),l+1 ≤ 2l) = 1 −
=1−
z 2l+2 + O(z 2l+4 ). (2l + 2)!
Substitute this in (10.179) with β = 0 to conclude E D Pl 2 2l+2 /(2l+2)!+O(z 2l+4 ) = ez /2−z . ez j=1 2 cos θj Sp(2l)
(ii) Repeat the considerations of (i), applied to (10.175) and (10.177) with α = 0, to show ezTr(U) U∈O(2l) = ez
2
/2−z 2l+2 /(2l+2)!+O(z 2l+4 )
.
(iii) Use the fact that the eigenvalues 0 < θ < π for O− (2l + 2) have the same p.d.f. as the eigenvalues in 0 < θ < π for Sp(2l) to deduce from the results of (i) and (ii) that ezTr(U) U∈O± (2l) = ez
2
/2±z 2l /(2l)!+O(z 2l+2 )
.
495
LATTICE PATHS AND GROWTH MODELS
10.8 GAP PROBABILITIES AND SCALED LIMITS 10.8.1 Growth in the square Consider the probability (10.73) relating to the maximum height in the polynuclear growth model with a1 = √ √ · · · = an = q, b1 = · · · = bm = q, bm+1 = · · · = bn = 0. In this case Pr(h ≤ l) = (1 − q)
nm
l
(1 + qeiθk )m (1 + e−iθk )n
k=1
U(l)
.
Let us now write q = e−1/L , l → Lt and take the limit L → ∞ (this is the Laguerre case of the exponential limit discussed in Section 10.3.3). The result of Exercises 10.3 q.1 implies [329] lim Pr(h ≤ Lt)
L→∞
1 Γ(j)Γ(n − m + j) j=1
t
= m
n−m −x
= Em,2 (0; (t, ∞); x
e
dx1 · · ·
0
t
dxm 0
m
xn−m e−xl l
l=1
(xk − xj )2
1≤j
)
(10.183)
where the final equality follows from the definition. In words this says that the limiting probability is equal to the probability of no eigenvalues in the interval (t, ∞) (the soft edge) of the Laguerre unitary ensemble UEm (xn−m e−x ). Consider now the Poisson points limit, in which the cumulative distribution for the maximum height is given by (10.166). The CUEl average in this expression is a special case of (8.96), so we see that Pr(l ≤ l) = E2hard (0; (0, 4z 2); l),
(10.184)
where use is made of (8.87). Hence Pr(l ≤ l) is equal to the probability of no eigenvalues at the hard edge of the LUE with a = l. 10.8.2 Growth symmetric about the antidiagonal For the polynuclear growth model with the underlying integer matrix symmetric about the antidiagonal, and with only even entries allowed on the diagonal, the maximum height h is given by (10.134) with β = 0. In the case q1 = · · · = qn = q this reads . (10.185) Pr(h ≤ 2l) = (1 − q 2 )n(n+1)/2 det(1 + qU)n U∈Sp(2l)
We consider the particular exponential limit obtained by writing q = e−1/2L , 2l → Lt and taking L → ∞. From the appropriate limit of (10.133) we see that then [27] t t n 1 lim Pr(h ≤ Lt) = dx1 · · · dxn e−xl /2 |xk − xj | L→∞ W0,1,n 0 0 l=1
= En,1 (0; (t, ∞); e
−x/2
1≤j
),
(10.186)
and thus the limiting probability is equal to the probability that there are no eigenvalues in the soft edge interval (t, ∞) of LOEn with a = 0. Although this probability is a β = 1 quantity, it can be written as a determinant [233]. P ROPOSITION 10.8.1 In addition to (10.186) we also have lim Pr(h ≤ Lt) =
L→∞
2 (2π)n/2 (t/4)n /2 e−nt/4 det[Ij−k+1/2 (t/4)]j,k=1,...,n . G(n + 1)
(10.187)
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Proof. According to Exercises 8.3 q.3 it follows from (10.185) that Pr(h
≤ 2l) = (1 − q 2 )n(n+1)/2 (4q)nl
l “ ˛ DY ”n E 1 ˛ . (1 − q)2 + λj ˛ 4q JUEl a=b=1/2 j=1
(10.188)
Now, simple manipulation of (8.120) shows that for the JUEl supported on (0, 1) with general a, b l DY j=1
(u + xj )n
E
DY “ u ”l E n!(1 + u)nl (b−a−l)/2 zj |1 + zj |a+b+l , + zj Mn (b + l, a + l, 1) j=1 1+u CUEn n
JUEl
=
(10.189)
where the explicit form of the proportionality follows by consideration of the u → ∞ limit. According to (5.76), with z = eiθ the CUEn average can be written as the Toeplitz determinant h 1 Z π “ u ”l i z (b−a−l)/2+(j−k) |1 + z|a+b+l . det + z dθ 2π −π 1+u j,k=1,...,n The integral in the determinant can be evaluated in terms of the Gauss hypergeometric function (expand (u/(1 + u) + z)l by the binomial theorem and make use of (4.4) in the case N = 1) to give h det
“ u ”i Γ(a + b + l + 1) . 2 F1 − l, −b − l − (j − k); a + 1 − (j − k); Γ(a − (j − k) + 1)Γ(b + l + j − k + 1) 1 + u j,k=1,...,n
With q = e−t/4l and u = (1 − q)2 /4q, for l → ∞ we have u → (t/8l)2 . Making use of the series (5.83) and the identity “ 2 ”μ 2 Iμ (s), (10.190) 0 F1 (− ; 1 + μ; s /4) = Γ(μ + 1) s we see that the limiting form of the general entry of the determinant is (t/8l)(j−k)−a Ia−(j−k) (t/4) and thus to leading order the determinant is equal to (t/8l)−(an) det[Ia−(j−k) (t/4)]j,k=1,...,n .
(10.191)
Also, noting from (4.4) and (4.182) that Mn (a, b, 1) =
G(1 + b) G(n + 1 + a + b) G(1 + a) G(n + 2), G(1 + a + b) G(n + 1 + a) G(n + 1 + b)
we compute using (4.185) the asymptotic expansion 2
Mn (a + l, b + l, 1) ∼ l−n l→∞
/2
2
(2π)−n/2 22nl+n
/2+n(a+b)
G(n + 2).
(10.192)
Substituting (10.191) and (10.192) in (10.189) and substituting the result in (10.188) implies (10.187).
Consider now the Poisson points limit of this model, in which the cumulative distribution of the maximum height is given by (10.179) with β = 0. Recalling (9.182) we see that Pr(l ≤ 2l) = E1hard (0; (0, (2z)2 ); l).
(10.193)
Thus Pr(l ≤ 2l) is equal to the probability of no eigenvalues in the scaled hard edge interval (0, (2z)2 ) of a matrix ensemble with orthogonal symmetry and singularity proportional to xl near the hard edge.
497
LATTICE PATHS AND GROWTH MODELS
10.8.3 Growth symmetric about the diagonal In the case of the polynuclear growth model with the underlying integer matrix symmetric about the diagonal, and all entries on the diagonal itself set equal to zero, the distribution of the maximum height h is specified by (10.154) with α = 0. With q1 = · · · = qn = q this can be written 1 Pr(h ≤ l) = (1 − q 2 )n(n−1)/2 det(1 + qU)n U∈O+ (l) + det(1 + qU)n U∈O− (l) . (10.194) 2 In the RSK mapping between integer matrices symmetric about the diagonal and semi-standard tableaux, it follows from (10.149) that with all the diagonal elements set equal to zero, the tableaux must have row lengths equal in pairs, μ1 = μ2 , μ3 = μ4 , . . . , so only μ1 , μ3 , . . . are independent. Suppose for convenience that n is even. We see from (10.151) and (10.23) that with q1 = · · · = qn = q and α = 0, hj = μj + (n − j) the joint p.d.f. for h := h1 and h3 , . . . , hn−1 is given by (1 − q 2 )n(n−1)/2 Pn/2 q j=1 h2j−1 n−1 l! l=1 × (h2i−1 − h2j−1 )2 (h2i−1 − h2j−1 + 1)(h2i−1 − h2j−1 − 1),
q −(n−1)n/2
1≤i<j≤n/2
where hj = μj + (n − j) and so h1 > h3 > · · · > hn−1 ≥ 0. It follows from this that in the particular exponential limit obtained by writing q = e−1/L , l → Lt and taking L → ∞ [27], lim Pr(h ≤ Lt) =
L→∞
(n/2)!
1 n−1 l=1
l!
t
dx1 · · ·
0
= En/2,4 (0, (t, ∞); e
dxn/2 0
−x
n/2
t
e−xj
j=1
).
(xk − xj )4
1≤j
(10.195)
Hence here the limiting probability is equal to the probability that there are no eigenvalues in the soft edge interval (t, ∞) of the LSEn/2 with a = 0. As with the gap probability in (10.185), manipulation of (10.194) implies that it has a determinantal form [233]. P ROPOSITION 10.8.2 We have 1 (2π)n/2 = lim Pr(h ≤ Lt) −1/L L→∞ 2 G(n + 1) q1 =···=qn =e n2 /2 −nt/4 det[Ij−k+1/2 (t/4)]j,k=1,...,n + det[Ij−k−1/2 (t/4)]j,k=1,...,n . (10.196) e ×(t/4) It remains to consider the Poisson points limit of this model, in which the cumulative distribution of the maximum height is given by (10.177) with α = 0. Recalling (9.187) we see that ˜4hard (0; (0, (2z)2 ); 2l). Pr(l ≤ 2l) = E
(10.197)
Thus Pr(l ≤ 2l) is equal to the probability of no eigenvalues in the scaled hard edge interval (0, (2z)2 ) of a matrix ensemble with symplectic symmetry and singularity proportional to x2l near the hard edge. 10.8.4 Scaled limit In the setup of the Hammersley process there are on average z 2 points in the unit square. As z → ∞ the length of the longest up/right path l will thus diverge. Of interest is the statistical properties of l in this limit. In particular, what is the mean of l and can the fluctuations about the mean be quantified? Some insight into these questions is provided by the identity (10.184). Thus we know from Exercises 7.1 q.1(iii) that the hard edge with singularity proportional to xa scales to the soft edge upon change of variables x → a2 − 2a(a/2)1/3 x. This then suggests that to obtain a well-defined scaled limit we should introduce a
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CHAPTER 10
scaled variable y such that 4z 2 = l2 − 2l(l/2)1/3y or equivalently y = (l − 2z)/(l/2)1/3. Indeed, at a formal level it follows from Exercises 7.1 q.2(iii) that lim E2hard (0; (0, l2 − 2l(l/2)1/3y); l) = E2soft (0; (y, ∞)),
(10.198)
l − 2z lim Pr ≤ y = E2soft (0; (y, ∞)). z→∞ z 1/3
(10.199)
l→∞
and thus
One reads off that for z → ∞ the mean of l is to leading order equal to 2z, and the fluctuations about the mean are proportional to z 1/3 , with the distribution of the scaled quantity (l − 2z)/z 1/3 identical to that of the largest eigenvalue at the soft edge of a scaled matrix ensemble with unitary symmetry. The limit formula (10.199) was first derived and proved by Baik, Deift and Johansson [31]. Here we will present a proof based on the hard to soft edge limiting transition at β = 2 [85]. The key result is a variation of (10.198). P ROPOSITION 10.8.3 Let Ql (y) := (l − (l/2)1/3 y)2 . We have lim E2hard (0; (0, Ql (y)); l) = E2soft (0; (y, ∞)).
l→∞
(10.200)
Proof. Starting with the formula E2hard (0, (0, l2 − 2l(l/2)1/3 y); l) = 1 +
Z y Z ∞ h i X (−1)n y dx1 · · · dxn det Klhard (xj , xk ) , n! j,k=1,...,n 0 0 n=1
which is a consequence of (9.4) and Propositions 5.1.2 and 7.2.1, we adopt a strategy analogous to the proofs of Propositions 9.1.2 and 9.1.3. Thus we focus attention on the l → ∞ form of (Ql (x)Ql (y))1/2 Klhard (Ql (x), Ql (y)). According to (7.31) and the definition of Ql , we require asymptotic estimates of Jl (l − (l/2)1/3 y),
Jl (l − (l/2)1/3 y).
The uniform large ν asymptotic expansion of Jν (νz) valid for all z ∈ C, arg z = π obtained by Olver [435] can be used to show that “ 2 ”1/3 Jν (ν − w(ν/2)1/3 ) ∼ Ai(w) + O(ν −1 )O(e−w ), ν “ 2 ”1/3 “ν ” Ai (w) + O(ν −2/3 )O(e−w ). Jν (ν − w(ν/2)1/3 ) ∼ − 2 ν Substituting in (7.31) shows that for |x − y| bounded away from zero at least (Ql (x)Ql (y))1/2 K2hard (Ql (x), Ql (y)) = K2soft (x, y) + O(l−1/3 )O(e−x−y ). The validity of this estimate for |x − y| → 0, and thus all x, y > 0 can be established by the same technique as used in the proof of Proposition 9.1.3. It follows from this that soft l |Ql (x1 ) · · · Ql (xn )|ρhard (n) (Ql (x1 ), . . . , Ql (xn ))|a=l = ρ(n) (x1 , . . . , xn ) + Rn (x1 , . . . , xn ),
(10.201)
where Rnl → 0 as l → ∞ uniformly on xj ∈ [y, ∞) (j = 1, . . . , n). Thus the analogue of the property (9.3) relating to Proposition 9.1.1 is valid. Hence the conclusion of the latter applies, telling us that lim E2hard (Ql (y); l) = 1 +
l→∞
Z ∞ Z ∞ X (−1)n ∞ dx1 · · · dxn ρsoft (n) (x1 , . . . , xn ), n! y y n=1
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LATTICE PATHS AND GROWTH MODELS
which recalling (9.4) is (10.200).
We can deduce from Proposition 10.8.3 the following limit theorem, which includes (10.198). P ROPOSITION 10.8.4 We have lim E2hard (0; (0, l2 − 2l(l/2)1/3 y + O(l)); l) = E2soft (0; (y, ∞)).
l→∞
(10.202)
Proof. Since Ql (y) = l2 − 2l(l/2)1/3 y + O(l2/3 ), we can rewrite the l.h.s. of (10.202) as lim E2hard (0; (0, Ql (y) + O(l)); l)
l→∞
which in turn can be written as lim E2hard (0; (0, Ql (y)); l) + lim E2hard (0; (Ql (y), Ql (y) + O(l)); l).
l→∞
l→∞
From Proposition 10.8.3 we know that the first term is equal to E2soft (0; (y, ∞)). For the second term, the working of the proof of Proposition 10.8.3 gives that it equals lim E2soft (0; (y − O(l−1/3 ), y))
l→∞
and thus vanishes.
Closely related to the scaled form (10.199) for the cumulative distribution of l is an analogous limit formula for lN , l − 2√N ≤ y = E2soft (0; (y, ∞)). (10.203) lim Pr N 1/6 N →∞ N This result, quantifying the limiting distribution of the largest increasing subsequence length of a large random permutation, was derived in [31] as a corollary of (10.199). Some of the working is given in Exercises 10.8. The analogue of (10.201) can readily be established for β = 1 and 4. This allows us to deduce from (10.193) and (10.197) the limit formulas [35] l − 2z l − 2z soft ˜4soft (0; (y, ∞)), lim Pr ≤ y = E (0; (y, ∞)), lim Pr ≤ y =E 1 z→∞ z→∞ z 1/3 z 1/3 and these in turn tell us that in relation to increasing subsequences √ √ l l 2k,k − 2 2k 2k,k − 2 2k soft ˜4soft (0; (y, ∞)). ≤ y = E (0; (y, ∞)), lim Pr ≤ y =E lim Pr 1 k→∞ k→∞ (2k)1/6 (2k)1/6 1. [328] Let the sequence {qn }n=0,1,... satisfy the bounds 0 ≤ qn ≤ 1 and be monotonically decreasing so that qn ≥ qn+1 . Let ∞ X ξn qn φ(ξ) := e−ξ n! n=0
E XERCISES 10.8
and for given d > 0 write p √ μ(d) n log n, n = n + (2 d + 1 + 1)
p √ νn(d) = n − (2 d + 1 + 1) n log n.
The objective of this exercise is to show that −d φ(μ(d) ≤ qn ≤ φ(νn(d) ) + Cn−d n ) − Cn
for all n ≥ n0 , where C is some positive constant.
(10.204)
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CHAPTER 10
(i) Suppose ξ > 0. With wn (ξ) := e−ξ ξ n /n!, and f (x) = x log x + 1 − x, use Stirling’s formula to show wn (ξ) ≤ C exp(−ξf (n/ξ)). (ii) Note that for 0 ≤ x ≤ 2, f (x) ≥ (x − 1)2 /4 and use this in the result of (i) to deduce √ √ [ξ−2 d+1 ξ log ξ]
X
C , ξd
wn (ξ) ≤
n=0
[2ξ] X √ √ n=[ξ+2 d+1 ξ log ξ]
wn (ξ) ≤
C . ξd
Also, use the fact that f (x) ≥ x/10 for x ≥ 2 to show that for ξ > ξ0 > 0 ∞ X
wn (ξ) ≤ C(ξ0 )e−ξ/5 .
n=[2ξ]
Combine these inequalities to conclude that for ξ large enough ∞ X
wn (ξ) −
n=0
X √ √ n:|n−ξ|≤2 d+1 ξ log ξ
wn (ξ) ≤
C . ξd
(iii) Use the assumption that 0 ≤ qn ≤ 1 to deduce from the final result of (ii) that φ(ξ) −
X √ √ n:|n−ξ|≤2 d+1 ξ log ξ
qn wn (ξ) ≤
C . ξd
P Use the assumption that qn ≥ qn+1 and the fact that ∞ n=0 wn (ξ) = 1 to show X qn wn (ξ) ≤ qξ−2√d+1√ξ log ξ , √ √ n:|n−ξ|≤2 d+1 ξ log ξ
and similarly show X √ √ n:|n−ξ|≤2 d+1 ξ log ξ
qn wn (ξ) ≥ q[ξ+2√d+1√ξ log ξ] −
C . ξd
By combining the above three inequalities, deduce (10.204). 2. [31] In (10.173) replace l by 2z + z 1/3 y so that it reads ∞ ” ” “ l − 2z X z 2N “ lN − 2z −z 2 ≤ y = e ≤ y . Pr Pr z 1/3 N! z 1/3 N=0
Now apply (10.204) with qn → qn (ξ), ξ = z 2 to deduce from (10.199) the limit formula (10.203).
10.9 HAMMERSLEY PROCESS WITH SOURCES ON THE BOUNDARY In the droplet PNG model nucleation events occur above the plateau formed by an initial nucleation event at (x, t) = (0, 0). An extension of this model is to allow for growth of the droplet due to nucleation events forming on the left and right boundaries of the droplet. In particular, suppose nucleation events are created at the rate α− dt on the left boundary, and at the rate α+ dt on the right boundary. An equivalent viewpoint is that surrounding the initial nucleation event at (x, t) = (0, 0) are two staircase structures of growing plateaux (downward sloping for x < 0, upward sloping for x > 0) with vertical increments of one unit having intensity α− for x < 0 and α+ for x > 0. The arrival of a layer of the staircase structure at the boundary
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LATTICE PATHS AND GROWTH MODELS
of the droplet realizes the creation of nucleation events thereon. It turns out that this particular extension of the droplet PNG model is relevant to the study of the KPZ (Kardar-Parisi-Zhang) universality class of critical phenomena [451]. √ To analyze the height at the origin in this model after time t = 2T we note that it is equivalent to the length L(T, α+ , α− ) of the longest up/right path in a Hammersley process with z 2 = T 2 , and furthermore in which there are points on the x-axis (y-axis) of the unit square with Poisson distribution of intensity α+ (α− ). The cumulative distribution of the longest path length L(T, α+ , α− ) can be expressed as a random matrix average by a limiting procedure applied to the discrete PNG model of Section 10.3.1. For this purpose we extend the n × n non-negative integer matrix [xij ]i,j=1,...,n to an (n + 1) × (n + 1) matrix [xij ]i,j=0,...,n , but again with all elements chosen according to the geometric distribution (10.70). As in the discussion at the beginning of Section 10.6, to recover the setting of the Hammersley model in the region of the square away from the boundary we set ai = bj = T /n (i, j = 1, . . . , n) and take n → ∞. The boundary distributions in this limit are obtained by setting a0 = α+ , b0 = α− . However this procedure leaves at the origin a non-negative integer geometric random variable with parameter α+ α− (this distribution will be denoted g(α+ α− )). Denoting by L+ (T, α+ , α− ) the longest up/right path in this process, we see it is related to L(T, α+ , α− ) by the simple relation χ ∈ g(α+ α− ).
L+ (T, α+ , α− ) = L(T, α+ , α− ) + χ,
(10.205)
Furthermore, the limiting procedure applied to (10.73) (appropriately extended to include the variables a0 and b0 ) gives 2 ˜ l, Pr(L+ (T, α+ , α− ) ≤ l) = (1 − α+ α− )e−(α+ +α− )T −T D
(10.206)
where ˜ l := D
l
(1 + α+ eiθj )(1 + α− e−iθj )e2T
Pl j=1
cos θj
U(l)
j=1
.
(10.207)
The formulas (10.205) and (10.206) provide us with a formula for the cumulative distribution of L(T, α+ , α− ). P ROPOSITION 10.9.1 We have 2 ˜ l − α+ α− D ˜ l−1 ). Pr(L(T, α+ , α− ) ≤ l) = e−(α+ +α− )T −T (D
(10.208)
Proof. Introducing the generating functions Q(x) =
∞ X
Pr(L(T, α+ , α− ) ≤ l)xl ,
l=0
Q+ (x) =
∞ X
Pr(L+ (T, α+ , α− ) ≤ l)xl ,
l=0
we see from (10.205) that Q+ (x) =
∞ X l=0
xl
l X
Pr(L(T, α+ , α− ) ≤ l − k)Pr(χ = k)
k=0
= (1 − α+ α− )
∞ X l=0
xl
l X
Pr(L(T, α+ , α− ) ≤ l − k)(α+ α− )k =
k=0
1 − α+ α− Q(x), 1 − xα+ α−
where the final equality follows by writing xl = xl−k xk and summing independently over l − k and k. Multiplying both sides of this identity by 1 − xα+ α− and equating like powers of x gives (10.208).
˜ l in (10.208) with α+ = 1/α− = α can be expressed in terms of The quantity D Pl ˜ l = e2T j=1 cos θj Dl = D α+ =α− =0
U(l)
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familiar from (10.166), and monic orthogonal polynomials {πj (eiθ )}j=0,1,... with respect to the weight e2T cos θ , π 1 1 πj (eiθ )πk (eiθ )e2T cos θ dθ = 2 δj,k . 2π −π κj P ROPOSITION 10.9.2 We have ˜ l = (1 − l)πl (−α)πl (−α−1 ) − απl (−α)πl (−α−1 ) − α−1 πl (−α)πl (−α−1 ) Dl . (10.209) D α+ =1/α− =α
Proof. Let πn∗ (z) := z n πn (1/z) (cf. Definition 8.6.1). It follows from the working of Exercises 5.1 q.6(i) (for the first equality) and the first, third and fourth equations in Proposition 8.6.2 (for the second equality) that E DQ n iθj − x)(e−iθj − y) n j=1 w(θj )(e 1 X 2 π ∗ (x)πn∗ (y) − xyπn (x)πn (y) U (n) DQ E = 2 κk πk (x)πk (y) = n . (10.210) n κn k=0 1 − xy j=1 w(θj ) U (n)
Taking the limit y → 1/x = −1/α in the second equality gives the stated result.
Our interest is in the scaled limit of (10.208) with α+ = 1/α− = α for l = [2T + T 1/3 s],
α = 1 − y/T 1/3,
According to (10.166), (10.199) and (9.46) lim e
T →∞
−T 2
D[2T +T 1/3 s] (t) =
E2soft (0, (s, ∞))
T → ∞.
= exp −
∞
(t − s)q 2 (t) dt ,
(10.211)
s
where q(t) is the Painlev´e II transcendent satisfying (9.45) subject to the boundary condition (9.47) with ξ = 1. The remaining task then is to analyze the polynomials πl (−α) etc. in (10.209). A rigorous treatment of this problem has been given by Baik and Rains [35], [33] using Riemann-Hilbert methods. To avoid technicalities we will proceed instead via a heuristic analysis [451], [196]. First we note from the third and fourth equation in Proposition 8.6.2 that πn+1 (z) = zπn (z) + rn+1 πn∗ (z),
∗ πn+1 (z) = rn+1 zπn (z) + πn∗ (z).
(10.212)
Introduce Rn (T ) := (−1)n−1 rn (T ),
Pn (α) = e−T α πn∗ (−α),
Qn (α) = −e−T α (−1)n πn (−α).
(10.213)
Then we see that the equations (10.212) are consistent with the asymptotic forms R[2T +T 1/3 s] (T ) ∼ T −1/3 u(s),
P[2T +T 1/3 s] (1−y/T 1/3) ∼ a(s, y),
Q[2t+T 1/3 s] (1−y/T 1/3 ) ∼ b(s, y), (10.214)
in which u, a, b are coupled by the partial differential equations ∂a ∂b = ub, = ua − yb. (10.215) ∂s ∂s Moreover, in terms of the limiting functions a and b we see from the first equality in (10.210) that ˜ [2T +T 1/3 s] (T ) /D[2T +T 1/3 s] (T ) g(s, y) := lim e−(α+ +α− )T D T →∞ α+ =1/α− =1−y/T 1/3 s s = a(u, y)a(u, −y) du = b(u, y)b(u, −y) du, (10.216) −∞
−∞
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LATTICE PATHS AND GROWTH MODELS
while (10.209) gives g(s, y) = (s − y 2 )a(s, y)a(s, −y) + a(s, −y)
∂ ∂ a(s, y) − a(s, y) a(s, −y). ∂y ∂y
(10.217)
Knowledge of g(s, y) is sufficient for the scaled form of the cumulative distribution of L(t, α+ , α− ), since from (10.208) and (10.211) it follows that L(T, 1 − y/T 1/3 , 1 + y/T 1/3) − 2√T F˜y (s) := lim Pr ≤s 1/6 T →∞ T ∂ g(s, y)E2soft (0, (s, ∞)) . (10.218) = ∂s From the definition (10.210) of φ∗n and the definitions (10.213) and (10.214), we see that a and b are related by 1
a(s, y) = −b(s, −y)e 3 y
3
−sy
.
(10.219)
The task then is to determine a(s, y) or b(s, y). In fact it is more convenient to obtain a system of equations determining both these quantities. The equations (10.215) give their dependence on s, but involve the unspecified limiting function u(s). Thus it remains to determine u(s), partial differential equations for the dependence of a and b on y, and boundary conditions for the partial differential equations. First we will determine equations for the dependence of a and b on y. P ROPOSITION 10.9.3 We have n r t rn t rn+1 rn t n+1 t + 2− πn (z) + − 2 πn∗ (z), πn (z) = z z z z z r rn+1 rn t ∗ n+1 t ∗ + rn t πn (z) + − t + πn (z). πn (z) = − z z
(10.220) (10.221)
Proof. The first equation follows immediately from (8.196) with t → (2T )2 , after making use of the second and third
equations in Proposition 8.6.2. The second equation is derived similarly, after deriving an equation for πn∗ (z) analogous to (8.196) [232].
Formal substitution of the asymptotic forms (10.214) in the equations of Proposition 10.9.3 gives the partial differential equations ∂a = u2 a − (u + yu)b, ∂y
∂b = (u − yu)a + (y 2 − s − u2 )b. (10.222) ∂y √ To determine u we read off from (8.192) with t → 2T that rn satisfies the second order difference equation −
n rn = rn+1 + rn−1 T 1 − rn2
with
r0 = 1,
r1 = −
I1 (2T ) , I0 (2T )
(10.223)
which we know is a transformed version of a particular discrete PII equation. With Rn (t) as specified in (10.213), and the scaled form of this quantity as specified by (10.214), this difference equation becomes the differential equation d2 u = su + 2u3 . (10.224) ds2 We recognize (10.224) as the differential equation (9.45) satisfied by the Painlev´e II transcendent q in (10.211). Furthermore, to be compatible with the first of the initial conditions in (10.223), we require for
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CHAPTER 10
s = −2T 2/3 and T → ∞ that T −1/3 u(s) → −1, or equivalently u(s) ∼ − −s/2. s→−∞
Up to the minus sign, this is precisely the boundary condition (9.48), and we know from [294] that (10.224) has the unique solution with this property, u(s) = −q(s).
(10.225)
In relation to the boundary condition, we note from (10.219) that a(s, 0) = −b(s, 0). Using this in (10.215) with y = 0 and u given by (10.225) shows ∞ a(s, 0) = A exp − q(t) dt . s
To determine the constant A we note from (10.227) below that a(s, 0) → 1 as s → ∞ and thus ∞ q(t) dt . a(s, 0) = −b(s, 0) = exp −
(10.226)
s
With all quantities now completely determined, the results can be summarized as follows [35], [33]. P ROPOSITION 10.9.4 Denote by −q(t) the Painlev´e II transcendent satisfying (10.211), subject to the boundary condition (9.47) with ξ = 1, and define F˜y (s) as the scaled limit in (10.218). We have ∞ ∂ g(s, y) exp − F˜y (s) = (t − s)q 2 (t) dt , ∂s s where g(s, y) is given in terms of a(s, y) by (10.217). The function a(s, y) in turn is specified as the solution of the coupled partial differential equations ∂a ∂b = qb, = qa − yb, ∂s ∂s ∂b ∂a = q 2 a − (q + yq)b, = (q − yq)a + (y 2 − s − q 2 )b, ∂y ∂y subject to the boundary condition (10.226). E XERCISES 10.9
1.
(i) For a general real weight w(θ) show Q iθj )U (n) n j=1 w(θj )(1 − xe Qn = πn∗ (x). j=1 w(θj )U (n)
(ii) In the case of the weight w(θ) = e2T cos θ , use the result of (i) and (10.206) to show that πn∗ (−x) =
˜ n | α+ =x D α− =0
˜ n |α =α =0 D + −
=
exT Pr(L+ (T, x, 0) ≤ n) . Pr(L+ (T, 0, 0) ≤ n)
From this conclude πn∗ (−x) → exT as n → ∞ and thus a(s, y) → 1
as s → ∞.
(10.227)
Chapter Eleven The Calogero–Sutherland model Consideration of shifted mean parameter-dependent Gaussian random matrices, or equivalently Hermitian matrices with entries undergoing Brownian motion, leads to the Dyson Brownian motion model of the onecomponent log-gas. In the classical cases, a similarity transformation of the corresponding Fokker-Planck operator gives the Schr¨odinger operator for the Calogero-Sutherland model, which is the name given to the quantum many-body system for particles interacting on a line or a circle via the 1/r2 pair potential. By generalizing these Schr¨odinger operators to include exchange terms, decompositions into more elementary operators can be exhibited, and these operators can be used to establish integrability. For the coupling β = 2 the pair potential term in the Schr¨odinger operator is not present, giving rise to a free Fermi system and allowing the corresponding Green function to be expressed as a determinant. In the Gaussian and cases, this determinant form can be related to certain matrix integrals, including that due to Harish-Chandra, and Itzykson and Zuber. The determinant form also allows for the calculation of dynamical correlation functions, by applying formulas worked out in Chapter 5. These are analyzed in various scaled limits.
11.1 SHIFTED MEAN PARAMETER-DEPENDENT GAUSSIAN RANDOM MATRICES In some energy spectra problems for chaotic quantum systems there is a parameter which varies the spectrum continuously. A well-known example, highlighted in [284], is the spectrum of the hydrogen atom as a function of the magnetic field strength. One approach to modeling such systems is to consider a random matrix H with distribution interpolating between a fixed matrix H0 (when τ = 0) and one of the Gaussian ensembles of Chapter 1 (when τ = ∞). D EFINITION 11.1.1 With H real symmetric (β = 1), Hermitian (β = 2), or self-dual quaternion real (β = 4), the parameter-dependent Gaussian ensembles are defined to have the joint p.d.f. for the independent elements N 1 (0) Pτ (H(0) ; H) = exp − β |Hjk − e−τ Hjk |2 /2|1 − e−2τ | C j,k=1 1 = exp − βTr(H − e−τ H(0) )2 /2|1 − e−2τ | , C where C is the normalization and H(0) is a fixed Hermitian matrix. Comparing Definition 11.1.1 with the definitions given in Chapter 1 of the parameter-independent Gaussian ensembles we see that instead of the elements being chosen with mean zero and fixed standard deviation, they now have a mean determined by H(0) (recall Section 1.8) and a parameter-dependent variance. For H(0) fixed, we see by changing variables that H = |1 − e−2τ |1/2 X + e−τ H(0) ,
(11.1)
where X a member of the Gaussian β-ensemble (β = 1, 2, 4), and thus H is equal to the sum of a random and deterministic matrix. This setting is sometimes referred to as a random matrix model with a source [95].
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CHAPTER 11
In applications H(0) may itself be a random matrix. For example, let β = 2 and suppose the joint p.d.f. for the elements at τ = 0 is that of the GOE, P0 (H(0) ) =
N 1 (0) 2 (0) 2 1 1/2 −(Hjj ) /2 e e−(Hjk ) 1/2 2π π j=1 j
(recall Definition 1.1.1). The joint distribution of the elements of H, Pτ (H) say, is obtained by integrating (0) (0) over Hjj and Hjk , Pτ (H) := Pτ (H(0) ; H)P0 (H(0) ) (dH(0) ) =
2 −2τ N 2 −2τ 2 −2τ e−Hjj /(1+e ) 2 √ e−2(Re Hjk ) /(1+e )−2(Im Hjk ) /(1−e ) . −4τ −2τ π(1 + e ) j
This expression shows that H can be decomposed as a sum of a real symmetric matrix X(1) , and a real antisymmetric matrix X(2) , 1 − e−2τ 1/2 1 + e−2τ 1/2 X(1) + i X(2) . (11.2) H= 2 2 The diagonal and off-diagonal elements of X(1) have independent Gaussian distributions with mean zero and variance 1 and 12 respectively, while the off-diagonal elements of X(2) have mean zero and variance 12 . As emphasized in [284], the precise dependence on τ in Definition 11.1.1 has been chosen so that the joint p.d.f. satisfies the multidimensional Fokker-Planck equation of the Ornstein-Uhlenbeck process, or Brownian motion in a harmonic potential, ∂ ∂Pτ 1 ∂ 2 Pτ = . (11.3) (Hμ Pτ ) + Dμ ∂τ ∂Hμ β ∂Hμ2 μ Here the label μ ranges over the independent elements, including both the real and imaginary parts of the off-diagonal elements if they are complex, and Dμ = 1 for the diagonal elements and Dμ = 12 for the offdiagonal elements. This equation can be verified from the factorization property in Definition 11.1.1, the fact that (11.3) is valid for the p.d.f. of each particular μ, and the product rule for differentiation. Analogous to the studies of Chapters 1–3, let us take up the problem of calculating the eigenvalue p.d.f. of H from the distribution of the elements of Pτ . Our task is to compute an integral involving the variables associated with the eigenvectors of H. Recalling (1.27) we see the eigenvalue p.d.f. for a particular H(0) is given by β Pτ (H(0) ; H)(U† dU), |λk − λj | (11.4) 1≤j
where H = ULU and U is real orthogonal, unitary or unitary symplectic for β = 1, 2 or 4, respectively. As written it appears the p.d.f. (11.4) is a function of H(0) and the eigenvalues of H. In fact (11.4) depends only on the eigenvalues of both matrices. This follows by making the change of variables U → U(0) U, where H(0) = U(0) L(0) U(0)† , and it is assumed H(0) is diagonalized by the same set, or a subset, of the unitary matrices which diagonalize H. From the cyclic property of the trace we check that then Pτ (H(0) ; H) → Pτ (L(0) ; H), while it is a fundamental property of the volume form (U† dU) that it is unchanged by such a transformation (recall (2.9)), thus showing that there is indeed no dependence on U(0) . Next we want to change our viewpoint and regard H in (11.4) as fixed, in the sense that the unitary matrices U being integrated over are independent of H. This can be achieved by changing variables U → UV and defining VLV† to be the fixed matrix H. Doing this, noting again that (U† dU) is unchanged by this mapping,
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THE CALOGERO–SUTHERLAND MODEL
and interchanging the names of U and V for consistency with (11.4), we conclude that (11.4) is equal to |λk − λj |β Pτ (H(0) ; VHV† ) (V† dV). (11.5) 1≤j
Finally, with the assumption {V} is sufficient to diagonalise H(0) , suppose the eigenvalue p.d.f. of H(0) is p0 (λ(0) ). Then we have that the eigenvalue p.d.f. of H averaged over p0 , pτ (λ) say, is given by ∞ ∞ (0) (0) |λk − λj |β dλ1 · · · dλN p0 (λ(0) ) pτ (λ) = 1≤j
×
−∞
−∞
Pτ (H(0) ; VHV† ) (V† dV).
(11.6)
The p.d.f. (11.6) can be specified as the solution of a certain transformed Fokker-Planck equation. Essential to the analysis is the fact that since Pτ (H(0) ; H) satisfies the Fokker-Planck equation (11.3), the integral in (11.6) satisfies the same equation, provided we change variables in the Fokker-Planck equation from the elements of H to the eigenvalues and variables dependent on the eigenvectors. This latter task is accomplished starting with the formula (1.12) for the metric form (and its analogue in the complex Hermitian and quaternion real cases) and using formulas from tensor calculus, as we will proceed to demonstrate. For the metric form we have (ds)2 = Tr(dHdH† ) = gμν dHμ dHν , (11.7) μ,ν
where gμν = δμ,ν for the diagonal elements and gμν = 2δμ,ν for the off-diagonal elements. In general the Laplacian associated with a metric form of the type (11.7) is given by (see, e.g., [410]) 1 ∂ ∂ √ ∇2 = det g(g−1 )μ,ν . (11.8) ∂H ∂H det g μ ν μ,ν With g diagonal we see ∇2 =
Dμ
μ
∂2 , ∂Hμ2
(11.9)
which is precisely the second order operator in (11.3). Also ∂ ∂ β (Hμ Pτ ) = Hμ Pτ + N + N (N − 1) Pτ , ∂Hμ ∂Hμ 2 μ μ
(11.10)
so the task is to change variables in (11.9) and (11.10). With H = ULU† , U specified as in (11.4) we have U† dHU = U† dUL + dL − LU† dU, which when substituted in (11.7) gives (ds)2 =
N
(dλj )2 +
j=1
N
|λk − λj |2
β
(s)
(δujk )2 ,
(11.11)
s=1
j,k=1 j=k
(s)
where δujk denotes the real and imaginary components in U† dU := [δuj,k ]. This metric form is of the type (1.13) with yjj = λj ,
hjj = 1,
(s)
(s)
dyjk = δujk ,
(s)
hjk = |λk − λj |.
(11.12)
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CHAPTER 11
Thus if we write (ds)2 =
gμν dyμ dyν
μ,ν
then g := [gμν ] is diagonal with det g = |λk − λj |β =: J
and gμμ = 1 (μ = (jj)).
j
Use of these facts in (11.8) (with {Hν } replaced by {yν }) gives that in terms of {λj } and variables relating to U N ∂ 1 ∂ 2 J + OU , (11.13) ∇ = J j=1 ∂λj ∂λj where the operator OU involves derivatives with respect to variables relating to U. Analogous to (11.8) we have ∂ ∂ Hμ = Hμ (g−1 )μ,ν . ∂H ∂H μ ν μ μ,ν Changing variables as implied by (11.12) gives
∂ ∂ = λj + OU , ∂Hν ∂λ j j=1 N
Hν
ν
(11.14)
involves derivatives with respect to variables relating to U. where OU
P ROPOSITION 11.1.2 We have ∂pτ = Lpτ , ∂τ
L :=
N N N ∂ 1 1 ∂2 λ , + − j β j=1 ∂λ2j j=1 ∂λj λj − λk k=1 k=j
subject to the initial condition that p0 (λ1 , . . . , λN ) is specified.
Proof. To obtain the p.d.e, we note that since Pτ (H(0) ; H) satisfies the Fokker-Planck equation (11.3) it follows immediately that the final integral in (11.6), I := I(H; τ ), say, satisfies the same equation, provided we change variables in the Fokker-Planck equation from the elements of H to the eigenvalues and variables dependent on the eigenvectors. The change of variables is accomplished by using the formulas (11.9), (11.10), (11.13) and (11.14), which give “ ” ∂I = L0 I + βN (N − 1)/2 I, ∂τ
L0 :=
„ « X N N ∂ ∂ 1 X ∂ J + λj . βJ j=1 ∂λj ∂λj ∂λ j j=1
Substituting I = J −1 p and expanding the derivatives using the product rule gives « N „ ∂pτ ∂J ∂J −1 ∂pτ ∂J −1 ∂ 2 J −1 1 X ∂ 2 pτ + pτ +J + pτ J = ∂τ β j=1 ∂λ2j ∂λj ∂λj ∂λj ∂λj ∂λ2j +Jpτ
N X j=1
λj
N “ ” ∂ −1 X ∂ J + λj pτ + βN (N − 1)/2 pτ . ∂λj ∂λj j=1
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THE CALOGERO–SUTHERLAND MODEL
This can be simplified by noting J
∂ ∂J −1 =− log J ∂λj ∂λj
and
N X
λj
j=1
N X ∂ λj log J = β = βN (N − 1)/2, ∂λj λ − λk j j,k=1 j=k
where the last equality follows by interchanging j and k in the sum and taking the arithmetic mean. Thus « N „ ∂pτ ∂ ∂ ∂pτ 1 X ∂ 2 pτ ∂2 − p log J − log J + λ p = τ j τ ∂τ β j=1 ∂λ2j ∂λ2j ∂λj ∂λj ∂λj which, after computing ∂ log J/∂λj , is the stated p.d.e. The initial condition follows from the τ → 0 behavior of Pτ (H(0) ; H): it is nonzero only if the elements and thus the eigenvalues of H and H(0) coincide.
The connection between log-potential systems and the eigenvalue p.d.f. specified by the solution of a p.d.e. in Proposition 11.1.2 comes from the classification of the p.d.e. [469]. It is the Fokker-Planck equation for the log-potential system of Proposition 1.4.4. To see this we recall that for an interacting N -particle system with a general potential energy W , executing overdamped Brownian motion in a fictitious viscous fluid with friction coefficient γ at inverse temperature β, the evolution of the p.d.f. pτ (λ1 , . . . , λN ) for the location of the N particles at the points λ1 , . . . , λN is given by the solution of the Fokker-Planck equation N ∂ ∂pτ ∂W −1 ∂ = Lpτ γ . (11.15) where L = +β ∂τ ∂λj ∂λj ∂λj j=1 For the system of Proposition 1.4.4 W = W (H) := −
1 2 λ 2 j=1 j N
log |λj − λk | +
1≤j
and so
N 1 ∂W =− + λj , ∂λj λ − λk j k=1
(11.16)
k=j
and (11.15) becomes the evolution equation of Proposition 11.1.2. (0) (0) We remark that in small time intervals δτ := τ − τ0 the displacements λj − λj := λj (τ ) − λj (τ0 ) as described by (11.15) are Gaussian random variables, with mean and variance (0)
(λj − λj ) = −
∂W δτ, ∂λj
(0)
(λj − λj )2 =
2 δτ, β
(0)
(11.17)
(0)
respectively, while to leading order in δτ the correlation (λj − λj )(λk − λk ) vanishes, as do all correlations involving three or more displacements. In fact these criteria uniquely specify the equation (11.15), and have been used by Dyson [145] to provide a different method of establishing Proposition 11.1.2. We will present this method in Exercises 11.1 q.3. The equations (11.17) can be used to recast the Fokker-Planck equation (11.15) as the set of coupled Langevin equations γ
dλj (τ ) ∂W =− + Fj (τ ) dτ ∂λj
(j = 1, . . . , N ).
Here Fj (τ ) denotes a random force related to standard Brownian motion Bj (τ ) by 2γ B (τ ). Fj (τ ) = β j
(11.18)
(11.19)
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CHAPTER 11
tau 1 0.8 0.6 0.4 0.2
-2
-3
-1
1
2
x
Figure 11.1 Brownian paths of the eigenvalue motion of a parameter-dependent Gaussian random matrix.
Thus Fj (τ ) is a Gaussian random variable with zero mean and variance given by Fi (τ )Fj (τ ) =
2γ δi,j δ(τ − τ ). β
Another general property of the Fokker-Planck equation (11.15) is that as τ → ∞, pτ becomes proportional to e−βW (see Exercises 11.1 q.1 (iii)). Now with W as in (11.16), e−βW is proportional to the p.d.f. for the eigenvalues of the Gaussian β-ensemble of Chapter 1, and so the eigenvalue p.d.f. of the parameter-dependent Gaussian random matrices converges to the Gaussian β-ensemble in the limit τ → ∞. For β = 1, 2 and 4 this of course must be the case, as for τ → ∞ the joint p.d.f. of the elements of the parameter-dependent Gaussian random matrices coincides with the Gaussian ensembles of Chapter 1. Using Definition 11.1.1, a realization of the Brownian motion paths described by the Fokker-Planck equation in Proposition 11.1.2 can be constructed. Thus we choose a matrix H(0) from the GUE, say. The eigenvalues of this matrix give the initial positions. Then, for a particular increment of the parameter τ , δτ say (δτ 1), we construct a matrix according to Definition 11.1.1 (take β = 2 for definiteness): the diagonal and upper triangular off-diagonal elements (the latter have an independent real and imaginary part) are chosen with p.d.f.’s 1/2 (0) 2 (0) 2 −2δτ −2δτ 1 2 ) ) e−2|Hjk −Hjk | /(1−e e−(Hjj −Hjj ) /(1−e and . π(1 − e−2δτ ) π(1 − e−2δτ ) The eigenvalues of this matrix give the new positions. This procedure is continually repeated, with the matrix H(0) given by the matrix computed in the previous step. Non-intersecting Brownian paths corresponding to the eigenvalues computed in this way are shown in Figure 11.1. This numerical procedure draws attention to the alternative viewpoint on parameter-dependent random matrices implied by the working of Exercises 11.1 q.3, namely, that the entries themselves of the matrix H can be regarded as Brownian paths.
E XERCISES 11.1
1. [469] Consider the Fokker-Planck operator (11.15).
(i) Show that as an operator identity Le−βW =
« N N „ X 1 X ∂ −βW ∂ ∂W ∂ 1 ∂2 e = e−βW − , β j=1 ∂λj ∂λj β ∂λ2j ∂λj ∂λj j=1
511
THE CALOGERO–SUTHERLAND MODEL
and hence conclude p = e−βW is, up to normalization, the steady state solution of the Fokker-Planck equation. (ii) Use the first equality in (i) to show that eβW/2 Le−βW/2 is a Hermitian operator, and use the second equality to obtain the explicit formula eβW/2 Le−βW/2 =
« N „ X 1 ∂2 β “ ∂W ”2 1 ∂ 2 W . − + 2 2 β ∂λj 4 ∂λj 2 ∂λj j=1
(iii) Use (ii) to check the identity eβW/2 Le−βW/2 = −β −1
N X
Π†j Πj
where
Πj :=
j=1
1 ∂ iβ ∂W − , i ∂λj 2 ∂λj
show from this that the eigenvalues of L are ≤ 0, and thus conclude that as τ → ∞ p is proportional to the steady state solution e−βW . Also check that Πj e−βW/2 = 0. 2. Let {|n}n=0,1,... be the complete set of eigenfunctions of a Hermitian operator A with corresponding eigenvalues {λn }n=0,1,... , all assumed distinct. Let {|n() }n=0,1,... be the complete set of eigenfunctions of the perturbed () operator A + B with corresponding eigenvalues {λn }n=0,1,... .Write |n() = |n + |n1 + 2 |n2 + · · · ,
2 λ() n = λn + λn,1 + λn,2 + · · · ,
and choose the normalization of |n() so that () n|n() = 1 for all . (i) Use the normalization condition to deduce j n|nj = 0 (j = 1, 2, . . . ). (ii) Equate coefficients of and 2 in the eigenvalue equation for |n() and use (i) to show that λn,1 = n|B|n and λn,2 = 1 n|B|n1 . (iii) In the equation (A − λn )|n1 + (B − λn,1 )|n = 0 obtained P by equating coefficients of in the eigenvalue equation for |n() , note from (ii) that (B − λn,1 )|n = j =n j|B|n|j and operate on both sides with (A − λn )−1 to express |n1 in terms of {|j} to obtain the formula λ() n − λn = n|B|n −
X n|B|jj|B|n + O(3 ). λj − λn j =n
(11.20)
3. [145] The following results were used by Dyson to derive Proposition 11.1.2. To begin, replace τ by δτ := τ − τ0 in Pτ as given by Definition 11.1.1 and write Z f := f Pδτ (H(0) ; H) (dH). (i) Show that as δτ → 0
j
(Hμ − Hμ(0)) ) = −δτ Hμ(0) , (0)
2/β, μ = (jj), 1 otherwise,
|Hμ − Hμ(0) |2 = δτ
(0)
¯ν − H ¯ ν ) vanishes at order δτ as do all higher order correlations. while (Hμ − Hμ )(H (0)
(0)
(0)
(ii) Suppose that with τ = τ0 , H is diagonal and thus Hjj = λj , Hjk = 0 (j = k). Deduce from the formula in q.2(iii) that to second order in δτ (0)
λj − λj
(0)
= Hjj − Hjj +
(0) N X |(Hjk − Hjk |2 (0)
k =j
(0)
λj − λk
.
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CHAPTER 11
Use the results of (i) and the perturbation expansion to show that as τ → τ0 N “ X (0) λj − λj = −δτ λj − k=1 k=j
” ∂W (H) 1 = −δτ λj − λk ∂λj
(0)
(λj − λj )2 = δτ
and
2 , β
which are the equations (11.17). 4. [534], [251] Consider the parameter-dependent Gaussian ensemble (11.1) with H(0) also a member of the Gaussian β-ensemble and set e−τ = cos φ. Note that upon an appropriate modification, (11.20) implies Kn :=
(φ) ˛ X |n|X|j|2 ∂ 2 λn ˛ = −λ + 2 . ˛ n ∂φ2 φ=0 λn − λj j =n
The quantity Kn is referred to as the level curvature. (i) Show that for the Gaussian β-ensembles (β = 1, 2, 4), averaging over X allows |n|X|j|2 to be replaced by 12 (i.e. the sum of the variances of the independent real numbers making up each of the off-diagonal elements equals 12 in each case). As in the proof of Proposition 1.8.2 note that for large N X j =n
1 ∼ λn − λj
Z
√
2N
√ − 2N
where the equality follows from (1.130) with J =
√
ρb (y) dy = λn , λn − y
2N . Hence conclude Kn = 0.
(ii) The distribution of level curvatures at λn = 0 is defined as P(K) =
1
N DX
N ρ(1) (0)
δ(K − Kn )δ(λn )
n=1
E X,H(0)
.
Use the Fourier integral form of δ(K −Kn ) to perform the average over X and so obtain (after implementing too the effect of δ(λn )) 1
1 P(K) = N ρ(1) (0) 2π
Z
∞ −∞
e−iωK
N DY j=2
E |λj |β dω. 2 β/2 (1 + 2iω/βλj ) Eβ,N −1 (e−βλ /2 )
Remark: In the large N limit this has been shown to be proportional to 1/(K 2 + 2N )β/2+1 .
11.2 OTHER PARAMETER-DEPENDENT ENSEMBLES 11.2.1 Circular ensembles A parameter-dependent extension of the circular ensembles presented in Chapter 2 has been given by Dyson [145]. First it is necessary to extend the defining equation in Definitions 2.2.2 and 2.2.3 so that dS is unitary at second order, 1 S + dS = V 1 + i dM − (dM)2 U (11.21) 2 (V = UT for β = 1, V = UD for β = 4). A parameter δτ is introduced into the entries of the matrix of differentials dM by requiring 1 1 2/β, μ = (jj), s s 2 δMμ = 0, (δMμ ) = (11.22) 1/β, otherwise, δτ δτ
513
THE CALOGERO–SUTHERLAND MODEL
where δMμs denotes the parameter-dependent version of dMμs , while all other moments vanish to the same order. Proceeding as in Exercises 11.1 q.3 (ii) we now suppose that for a particular parameter τ , U = V = diag[eiθj /2 ], so that S is diagonal. Then the perturbation expansion of Exercises 11.1 q.2(iii) applied to the s r.h.s. of (11.21) gives that up to second order in dMjk (s labels the independent real component(s) of the entries) s 2 eiθk s (dMjk ) 1 i(θj +δθj ) iθj s 2 1 + i dMjj − e =e (dMjk ) + iθ iθ j k 2 e −e k,s k=j 1 i s 2 (dMjk ) cot(θk − θj )/2 . (11.23) = eiθj 1 + i dMjj − (dMjj )2 − 2 2 s k=j
Expanding the l.h.s. to second order in δθj , equating real and imaginary parts on both sides and averaging s over dMjj and dMjk , we see using (11.22) that δθj = −
∂W δτ ∂θj
where W = W (C) := −
and (δθj )2 =
2 δτ, β
log |eiθk − eiθj |,
(11.24)
j
which are the equations (11.17) and thus equivalent to the Fokker-Planck equation (11.15) (with λj replaced by θj and W = W (C) ). The eigenvalues for matrices from the parameter-dependent circular ensemble can therefore be regarded as particles in a one-component log-gas on a circle undergoing overdamped Brownian motion. An alternative approach is to begin with the equation (2.12) (for β = 2) and (2.15), (2.17) (for β = 1, 4) with the matrix elements of dM therein, dMμs , say, replaced by δMμs as specified by (11.22). The distribution of the elements of M then satisfy the Fokker-Planck equation ∂P ∂2P 1 = Dμ (11.25) ∂τ β μ,s ∂(Mμs )2 (cf. (11.3)). According to the theory given by (11.7)–(11.9) the r.h.s. is equal to β1 ∇2 , and ∇2 can in turn be calculated by changing variables in (11.8), using the metric form expansion (2.22) as in going from (11.11) to (11.13). This reclaims the Fokker-Planck equation (11.15) with W = W (C) , λj = θj . Notice that the perturbative expansion (11.23) can be interpreted as originating from the equation S(τ + dτ ) = S(τ )ei dM .
(11.26)
This can be used to approximately compute members of the parameter-dependent circular ensembles.
11.2.2 Wishart matrices Parameter-dependent Wishart matrices are constructed from parameter-dependent Gaussian random matrices in an analogous fashion to their parameter independent counterparts. Thus if X is an n × m parameter-
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CHAPTER 11
dependent Gaussian random matrix with joint p.d.f. for the elements given by m n 1 (0) |Xjk − e−τ Xjk |2 /2|1 − e−2τ | Pτ (X(0) ; X) := exp − β C j=1 k=1 1 = exp − βTr((X − e−τ X(0) )† (X − e−τ X(0) ))/2|1 − e−2τ | , C then A = X† X defines a parameter-dependent Wishart matrix. As with parameter-dependent Gaussian random matrices, the corresponding eigenvalue p.d.f. can be computed in at least two different ways: via the method of the proof of Proposition 11.1.2 [311] or via Dyson’s method of Exercises 11.1 q.3. Here the former method will be outlined (for the latter method see Exercises 11.2 q.1). The starting point is to decompose X according to its singular value decomposition X = ULV† (L = diag(x1 , . . . , xm )), where {x2j } are the eigenvalues of A. We want to substitute this in Pτ and integrate out the dependence on U and V to obtain the p.d.f. of {x2j }. Recalling from (3.4) the eigenvalue portion of the Jacobian, analogous to (11.4) this operation is equivalently given by J (U† dU) (V† dV)Pτ (X(0) ; UXV† ), (11.27) m 2 2 β where J ∝ j=1 xβa+1 j 1≤j
{(xj )2 }, the p.d.f. for {x2j } corresponding to the eigenvalues of A = X† X averaged over X(0) is given by ∞ ∞ (0) 2 (0) 2 2 2 2 (0) 2 pτ (x1 , . . . , xm ) = J d(x1 ) . . . d(x(0) m ) p0 ((x1 ) , . . . , (xm ) ) 0 0 × (U† dU) (V† dV)Pτ (X(0) ; UXV† ). (11.28) By repeating the proof of Proposition 11.1.2 with J now given as below (11.27), we find that (11.28) can be characterized as the solution of the Fokker-Planck equation describing the overdamped Brownian dynamics of the log-gas with potential energy W = W (L) :=
N N 1 2 a xj − log x2j − 2 j=1 2 j=1
log |x2k − x2j |,
(11.29)
1≤j
where a = a + 1/β and N = m. 11.2.3 Jacobi ensembles We have seen in Section 3.7 that the Jacobi ensemble can be realized by the p.d.f. for the singular values which form the matrix Λt in the decomposition (3.99). As S in (3.99) is unitary (with the additional constraint of being symmetric for β = 1, and self-dual quaternion for β = 4), a parameter-dependent extension is possible via the evolution equation (11.26) [238]. The corresponding evolution equation for the singular values can be calculated via the method used for the parameter-dependent circular ensembles, as given in the paragraph including (11.25). This in turn relies on the validity of (11.13), for which (up to a proportionality constant) one requires that hjj = c, c a constant, in the analogue of (1.13), as holds true in (11.12). Now, in terms of the variables λj , we see from the expression for T5 in the proof of Proposition 3.8.1 that hjj = 2/(1 − λ2j ). However, if we write λj = sin φj and consider {φj } as the variables we then obtain hjj = 2. The theory of Section 11.1 then tells us that {φj } evolves according to the Fokker-Planck equation (11.15) with λj replaced
515
THE CALOGERO–SUTHERLAND MODEL
by φj and W such that e−βW = J, J being the Jacobian specified in Proposition 3.8.1, with the change of variables λj = sin φj . Specifically we have W = W (J) := −
N N a b log sin2 φj − log cos2 φj − 2 j=1 2 j=1
log | sin2 φj − sin2 φk |
(11.30)
1≤j
for certain a , b . 11.2.4 DMPK equation Important in the theory of conductance in disordered wires is a parameter-dependent extension of the trans˜ †I M ˜ = fer matrix decomposition (3.114). As these matrices have the property (3.94), one sees that dM n,n † † ˜ I dM ˜ and consequently M ˜ I dM ˜ = i dM for some 2n × 2n Hermitian matrix M. This tells us −M n,n n,n that the appropriate metric form is Tr((dM)2 ), which together with the decomposition (3.114) can be used in the evolution equation (11.26) to obtain an evolution equation for the variables λi in Λ = diag(λ1 , . . . , λN ) of (3.114). As in the above discussion of Jacobi ensembles, we must first make a change of variables in order that hjj = c (constant) in the analogue of (1.13). This is achieved by setting λj = sinh2 xj , and considering {xj } as the independent variables. We conclude that {xj } evolves according to the Fokker-Planck equation (11.15) with λj replaced by xj and e−βW = J. The Jacobian is first computed in the variable λj according to the calculation of Proposition 3.8.1, and then the change of variables to {xj } is made. All three values β = 1, 2, 4 can be considered by imposing the constraints (3.100), (3.101). One finds (see e.g. [53]) W = W (D) := −
N 1 log | sinh 2xj | − 2β j=1
log | sinh2 xk − sinh2 xj |.
(11.31)
1≤j
In the variables {λi } this particular Fokker-Planck equation is named the Dorokhov-Mello-Pereyra-Kumar (DMPK) equation. E XERCISES 11.2
1. [11] In this exercise the Fokker-Planck equation for Wishart matrices will be rederived us(0) (0) ing the method of Dyson. The eigenvalues of X† X (X(0)† X(0) ) are denoted by x21 , . . . , x2m ((x1 )2 , . . . , (xm )2 ) (0) (0) (0) (0) = [xjk ], δxj := xj − xj , and δxjk := xjk − xjk . Furthermore, we suppose that and X = [xjk ], X X(0)† X(0) is diagonal, and consider a small increment in parameter δτ . (0)
(i) From the p.d.f. for the elements of X show that δxjk = −xjk δτ and (δxsjk )2 = β2 δτ , while all other moments vanish to the same order in δτ . Also, note from the assumption regarding X(0)† X(0) that P Pn Pβ (0) (0) 2 (0)s 2 (xj )2 = n k=1 |xkj | := k=1 s=1 (xkj ) . (ii) Use the perturbation formula of Exercises 11.1 q.2 with B = X† X − X(0)† X(0) to show that to second order in P (0) (0) n “ n ” X X | m ¯lj δxlk + δ x ¯lj xlk + x ¯lj δxlk )|2 l=1 (δ x (0) (0) (0) x2j −(xj )2 = ¯kj xkj +δxkj x ¯kj + . |δxkj |2 +δ x x2j − x2k k=1 k=1
=j
(iii) Average over {δxjk } in (ii) using the formulas of (i) to deduce that to order δτ (δxj )2 + 2xj δxj =
“
− 2x2j + 2n + 2
m X x2j + x2k ” δτ. x2j − x2k k=1 =j
(0)
(iv) Add (xj )2 to both sides of the equation in (ii), take the square root, expand the r.h.s, and average over
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CHAPTER 11
{δxjk } to deduce that X x2j + x2k ” 1 “ 2 − 2x2j + 2n − + 2 δτ. 2xj β x2j − x2k k=1 m
δxj =
=j
(v) From the results of (iii) and (iv) deduce the equations (11.17) with W given by (11.29). 2. [106] In the theory of radial stochastic Loewner evolution (radial SLE), the conformal mapping associated with N (N) such processes, Gt (z) satisfies the differential equation (N)
dGt dt where
(N)
= −Gt
N (N) X Gt + eiθj (t) (N)
j=1
Gt
− eiθj (t)
,
N X dθj (t) θj (t) − θk (t) √ cot = + κBj (t). dt 2 k=1
(11.32)
k=j
By comparing with (11.18), identify (11.32) as equivalent to the Fokker-Planck equation (11.15) in the case W is given by (11.24), γ = 12 and κ = 4/β.
11.3 THE CALOGERO-SUTHERLAND QUANTUM SYSTEMS 11.3.1 The Schr¨odinger operators In Exercises 11.1 q.1(i) we have seen that the Fokker-Planck operator of the type (11.15) can always be transformed into a Hermitian operator. When W corresponds to the parameter-dependent random matrix ensembles, the transformation can be written −eβW/2 Le−βW/2 = (H − E0 )/β,
(11.33)
where H is the Schr¨odinger operator for a quantum many-body system in which the particles interact via a 1/r2 pair potential (all three-body terms which occur in general cancel). In the Gaussian and circular case, these quantum many-body systems were first studied by Calogero [105] and Sutherland [506]. The name Calogero-Sutherland model has now been given to a class of quantum many-body systems, including those originally studied by Calogero and Sutherland, which are related to root systems for classical reflection groups [434] (see Section 11.5.5). We will consider a subclass of these models for which the Schr¨odinger operator has a potential energy consisting of one- and two-body terms only, H=−
N N ∂2 + V1 (xj ) + ∂x2j j=1 j=1
V2 (xj , xk ),
(11.34)
1≤j
and for which the ground state ψ0 factorizes into the form ψ0 =
N j=1
f1 (xj )
f2 (xj , xk ).
(11.35)
1≤j
Note that all Schr¨odinger operators derived from (11.33) with W consisting of one- and two-body potentials only must have the property (11.35) since, as follows from Exercises 11.1 q.1, ψ0 is proportional to e−βW/2 . P ROPOSITION 11.3.1 Consider the Fokker-Planck operator (11.15) with λj replaced by xj and W given
517
THE CALOGERO–SUTHERLAND MODEL
by (11.16). The operator identity (11.33) holds with H =H and E0 =
Nβ 2
+
N N ∂2 β2 2 := − + x + β(β/2 − 1) ∂x2j 4 j=1 j j=1
(H)
β2 4 N (N
1≤j
1 (xj − xk )2
− 1).
Proof. We use the general formula of Exercises 11.1 q.1(ii). For the given W N N N X X X ∂2W 1 =N+ 2 ∂x (x − x k )2 j j j=1 j=1 k=1 k=j
and
«2 N „ X ∂W j=1
∂xj
=
N X
N X
x2j +
j=1
j,k1 ,k2 =1 k1 ,k2 =j
N X 1 1 xj . −2 (xj − xk1 ) (xj − xk2 ) xj − xk j,k=1 k=j
The last sum occurred in the proof of Proposition 11.1.2. It is equal to N (N − 1)/2. Also, in the triple sum all terms with k1 = k2 cancel (these are the apparent three-body terms), due to the identity 1 1 1 + + = 0. (a − b)(a − c) (b − a)(b − c) (c − a)(c − b) Thus
«2 N „ X ∂W j=1
∂xj
=
N X
x2j +
j=1
N X j,k=1 k=j
(11.36)
1 − N (N − 1). (xk − xj )2
Substituting in the general formula gives the stated Schr¨odinger operator.
For W corresponding to the other parameter-dependent random matrix ensembles, a similar result holds. P ROPOSITION 11.3.2 With W given by (11.24), (11.29), (11.30) and (11.31), the operator identity (11.33) holds with N π 2 ∂2 1 (C) , H = H := − + β(β/2 − 1) 2 2 ∂x L sin (π(x j − xk )/L) j j=1 1≤j
H=H
(J)
1 1 a β a β b β b β −1 −1 := − + 2 2 2 2 cos2 φj sin2 φj j=1 1 1 +β(β/2 − 1) + , sin2 (φj − φk ) sin2 (φj + φk ) 1≤j
∂ + 2 ∂φj j=1 2
N
N 2 ∂ 1 + ∂x2j sinh2 2xj j=1 1 1 +β(β/2 − 1) + , sinh2 (xj − xk ) sinh2 (xj + xk ) 1≤j
H = H (D) := −
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CHAPTER 11
where 0 ≤ xj ≤ L, xj > 0, 0 < φj < π, xj > 0 and π 2 N (N 2 − 1) (C) E = E0 := β 2 , L 12 β2N βN (L) (a + N − 1) + , E = E0 := 2 2 β 2 N 2 (J) a + b2 + (N − 1)(a + b ) + 2a b + 4(N − 1)(N − 2) + βN (N − 1), E = E0 := 4 β2 (D) N (N − 1)(N − 2), E = E0 := −N − 2βN (N − 1) − 3 respectively. Proof. The derivation follows the same procedure as the proof of Proposition 11.3.1, with (11.36) being replaced by the identities cot(a − b) cot(a − c) + cot(b − a) cot(b − c) + cot(c − a) cot(c − b) = −1, a b c + + = 0, (a − b)(a − c) (b − a)(b − c) (c − a)(c − b) a(1 − a) b(1 − b) c(1 − c) + + = −1 (a − b)(a − c) (b − a)(b − c) (c − a)(c − b)
(11.37)
(the latter applies to both W (J) and W (D) ).
11.3.2 The Green function and eigenfunctions Note that in general the substitutions (11.33) and pτ = eE0 τ /β e−βW/2 ψτ
(11.38)
in the Fokker-Planck equation (11.15) with γ = 1 give the imaginary time Schr¨odinger equation ∂ ψτ = Hψτ . (11.39) ∂τ Since (11.39) is linear, to calculate its solution for general initial conditions it suffices to obtain the solution when the precise particle positions are known initially, −β
ψτ (x)
τ =0
=
N
(0)
δ(xl − xl )
(0)
(0)
(x1 < · · · < xN ).
(11.40)
l=1
With this initial condition we say that ψτ =: GSτ (x(0) ; x) is the Green function solution of the equation (11.39). Because of the ordering constraint in (11.40), GSτ has the τ = 0 normalization dx1 · · · dxN GSτ (x(0) ; x)|τ =0 = 1, (11.41) A
where A is the region −∞ < x1 < x2 < · · · < xN < ∞. Note that it follows from (11.38) that the Green S function solution of the Fokker-Planck equation (11.15), GFP τ say, is given in terms of Gτ by GFP x(0) ; x) = eE0 τ /β τ (
e−βW (x1 ,...,xN )/2
GSτ (x(0) ; x). (11.42) (0) (0) e−βW (x1 ,...,xN )/2 The Green function solution of (11.39) can be expressed in terms of the eigenfunctions and eigenvalues of the Schr¨odinger operator H. Thus, according to the method of separation of variables, if H has a complete
519
THE CALOGERO–SUTHERLAND MODEL
set of normalized eigenfunctions {ψκ } with corresponding eigenvalues {Eκ } (in general the label κ will be an N -tuple of integers), then GSτ (x(0) ; x) = ψκ (x(0) )ψκ (x)e−Eκ τ /β . (11.43) κ (D)
is distinguished from the other Hamiltonians in that it does not support At this stage the Hamiltonian H bound states. This means the complete set of eigenfunctions are labeled by continuous quantum numbers. For the other Hamiltonians there is a complete set of bound states, and the label κ is an N -tuple of integers. Only these cases will be considered in the following. It turns out that the study of the eigenfunctions has its natural setting in generalizations of H which involve exchange terms. These generalizations also play a crucial role in the study of the integrability of H. D EFINITION 11.3.3 A Schr¨odinger operator H of the form (11.34) is said to be integrable if there exists a set of N algebraically independent commuting operators {Xi }i=1,...,N such that [H, Xi ] = 0,
i = 1, . . . , N.
The theme of integrability will be addressed in the next two sections. The commuting operators which emerge will be used to study the eigenfunctions of the Schr¨odinger operators in subsequent chapters. (C)
1. [505] The objective of this exercise is to generalize the Schr¨odinger equation H (C) ψ0 In particular, we seek W (x) even and periodic of period L such that the Schr¨odinger operator
E XERCISES 11.3 (C) (C) E0 ψ0 .
H=−
N X ∂2 + ∂x2j j=1
has a ground state of the form
X
(11.44)
1≤j
Y
ψ0 =
W (xk − xj )
=
u(xk − xj )
(11.45)
1≤j
for u(x) even or odd. (i) Substitute (11.45) in (11.44) to deduce that Hψ0 =
X
“
N “X N ” ”2 X φ(xj − xk ) ψ0 , W (xk − xj ) − 2φ (xk − xj ) ψ0 − j=1
1≤j
k=1 k=j
where φ(x) := u (x)/u(x), and thus φ is odd. (ii) Note from the equation of (i) that if there is an even function f such that N “X N X j=1
k=1 k=j
φ(xj − xk )
”2
=
N “ N X ” X φ2 (xj − xk ) − (N − 2)f (xj − xk ) , j=1 k=1 k=j
then ψ0 as specified by (11.45) is the ground state of (11.44) with “ ” W (x) = 2 φ (x) + φ2 (x) − (N − 2)f (x) +
2E0 . N (N − 1)
(iii) Show that (11.46) holds if, for x + y + z = 0, φ(x)φ(y) + φ(y)φ(z) + φ(z)φ(x) = f (x) + f (y) + f (z). (iv) With φ(x) = αζ(x),
ζ(x) :=
π θ1 (πx/L; q) , L θ1 (πx/L; q)
(11.46)
520
CHAPTER 11
where θ1 (z; q) is specified by (2.79), use the identity [541] (ζ(x) + ζ(y) + ζ(z))2 = −(ζ (x) + ζ (y) + ζ (z)) valid for x + y + z = 0 to show that with W (x) = (2α − N + 2)ζ (x) + (2α2 − N + 2)ζ 2 (x) +
2E0 N (N − 1)
the ground state of (11.44) is given by (11.45) with u(x) = (θ1 (πx/L; q))α . (With α = β/2, the absolute value squared of (11.45) is therefore proportional to (2.83).) (C)
(v) Reclaim the equation H (C) ψ0
(C)
(C)
by taking the q → 0 limit.
= E0 ψ0
2. [77] With a1 < a2 < · · · < an and gj > 0 (j = 1, . . . , n) consider the function of x specified by y =x−
n X l=1
gl x − al
(cf. (4.54)). (i) By sketching a graph, show that for each value of y there are n + 1 values of x, one in each of the intervals (al−1 , al ), (l = 1, . . . , n) with a0 := −∞, an+1 := ∞. Let these values of x be denoted x(l) , l = 0, . . . , n. (ii) Deduce from (i) that
Z
Z
∞
∞
f (y) dx = −∞
f (y)
n “X
−∞
” 1/(dy/dx(l) ) dy,
l=0
valid or general f such that the integral is well defined. (iii) Let CR denote the circle of radius R, centered at 0. By evaluating limR→∞ (1/2πi) integral, then by the residue theorem, deduce that n X
1/(dy/dx(l) ) = 1
R CR
dx/ y
first as a contour
(11.47)
l=0
Z
and hence conclude
Z
∞
∞
f (y) dx = −∞
f (y) dy.
(11.48)
−∞
(iv) Use the result of (iii) in the case n = 1, a1 = 0 to show Z ∞ √ 2 2 2 e−(x +g /x ) dx = πe−2g ,
g > 0.
(11.49)
−∞
3. [21] In this exercise a derivation of the integral evaluation Z
∞ −∞
Z dx1 · · ·
N “ “1 X dxN exp − β x2j + 2 j=1 −∞ ∞
X 1≤j
”” “ 2π ”N/2 g2 e−βgN(N−1)/2 , (11.50) = (xk − xj )2 β
valid for g > 0, which can be regarded as a multidimensional generalization of the working of q.2 will be outlined. This integral evaluation was first derived by taking the classical limit of the quantum partition functions associated with H (H) [253]. (i) Consider the integral in (11.50) for N = 2. Show that after an appropriate change of variables, (11.49) can be used to obtain its evaluation.
521
THE CALOGERO–SUTHERLAND MODEL
(ii) With
N X
wj = xj − g
k=1,k =j
show that
N X
wj2 =
j=1
N X
x2j + 2g 2
j=1
N X j
1 xj − xk
(11.51)
1 − N (N − 1)g. (xk − xj )2
(iii) The denominators in (11.51), when set equal to zero, form a hyperplane arrangement, which partitions space into chambers Δ. Analogous to the result of q.2(i) there is exactly one solution of (11.51) for each chamber Δ, denoted xΔ . Further, analogous to (11.47) it can be proved that h ∂w i˛ X 1 j ˛ . = 1, J(Δ) := det ˛ J(Δ) ∂x x= xΔ k chambers Δ Use this to deduce that Z Z ∞ dx1 · · · −∞
Z
∞
∞
dxN f (w1 , . . . , wN ) = −∞
−∞
Z dx1 · · ·
∞
dxN f (x1 , . . . , xN ) −∞
(cf. (11.48)), valid for any f such that the integral is well defined. With an appropriate choice of f , use this result and the result of (ii) to reclaim (11.50).
¨ 11.4 THE SCHRODINGER OPERATORS WITH EXCHANGE TERMS 11.4.1 Definitions Let sjk denote the operator which acts on the coordinates x1 , . . . , xN of a function f by interchanging xj and xk . In investigating the integrability of H (H) and H (C) as defined in Propositions 11.3.1 and 11.3.2 respectively, Polychronakos [446] introduced more general Schr¨odinger operators H (H,Ex) and H (C,Ex) in which the factor (β/2 − 1) outside the sum over pairs is replaced by (β/2 − sjk ) and placed inside the sum over pairs. Explicitly H (C,Ex) = − H (H,Ex) := −
N π 2 ∂2 + β ∂x2j L j=1
1≤j
N N ∂2 β2 2 + x +β ∂x2j 4 j=1 j j=1
β/2 − sjk , sin2 (π(xj − xk )/L)
1≤j
0 ≤ xj ≤ L,
β/2 − sjk (xj − xk )2
(11.52)
(11.53)
(here and below in similar terms written as fractions, the operation in the numerator (β/2 − sjk ) is to be carried out first). The Schr¨odinger operators H (L) and H (J) can be similarly generalized to the operators H (L,Ex) and H (J,Ex) with exchange terms, although the procedure is complicated by the underlying root system being of B and BC type rather than the simpler A type which corresponds to H (H) and H (C) . The complication means that in addition to the exchange operator sjk it is also necessary to introduce the change of sign operator, σj say, which replaces the coordinate xj by −xj . Note that the operators sjk and σj make up the type B Weyl group (recall Section 4.7.2). In terms of these operators one defines N N N ∂2 β 2 2 βa βa /2 − σj + x + ∂x2j 4 j=1 j 2 j=1 x2j j=1 β/2 − sjk β/2 − σj σk sjk , +β + 2 (xj − xk ) (xj + xk )2
H (L,Ex) := −
1≤j
xj > 0,
(11.54)
522
CHAPTER 11
N N N ∂2 βa βa /2 − σj βb βb /2 − σj + + ∂φ2j 2 j=1 sin2 φj 2 j=1 cos2 φj j=1 β/2 − sjk β/2 − σj σk sjk + , 0 < φj < π. +β sin2 (φj − φk ) sin2 (φj + φk ) 1≤j
H (J,Ex) := −
(11.55)
From the definition we see that the operator sjk acts like the identity on symmetric functions. Thus when restricted to act on symmetric functions (which must also be even in the cases of (11.54) and (11.55)) the (·) operators H (·,Ex) reduce to their original form H (·) . One consequence is that e−βW /2 , which we know (·) from Exercises 11.1 q.1 is the symmetric ground state wave function of H , remains the symmetric ground state wave function of H (·,Ex) . Another is that a decomposition of H (·,Ex) implies a decomposition of H (·) after restricting the former to act on symmetric functions. In subsequent sections, for each H (·,Ex) , a set of N commuting operators will be identified which establishes integrability and provides a simple decomposition of the Schr¨odinger operator; by the above remarks these properties are established for H (·) after restricting the operators to act on symmetric functions. 11.4.2 The type A Dunkl operator A few years before Polychronakos introduced the operators H (·,Ex) , Dunkl [141] introduced a class of differential operators with exchange terms. These came about in the context of studying multivariable polynomials and integral transforms associated with weight functions of the form of the integrand in (4.152). It turns out that these same operators play an essential role in establishing the integrability of H (·,Ex) . D EFINITION 11.4.1 The type A Dunkl operators are defined by dj :=
N ∂ 1 1 − sjk + , ∂zj α k=1 zj − zk
j = 1, . . . , N.
k=j
When acting on homogeneous polynomials, the Dunkl operators lower the degree by 1. Another fundamental property is that they mutually commute. P ROPOSITION 11.4.2 For all 1 ≤ j, k ≤ N we have [dj , dk ] = 0. Proof. From the definitions of the dj and sjk we see [dj , dk ] =
« N „ 1 X sjl skj − skj sjl sjk skl − skl sjk sjl skl − skl sjl + + . α2 l=1 (zj − zl )(zk − zl ) (zk − zj )(zj − zl ) (zj − zk )(zk − zl ) l=j,k
But for distinct indices sjk skl = slj sjk = skl slj
(11.56)
(the indices are cycles of (jkl) with the middle index repeated twice), so we can write « N „ 1 1 1 1 X + + (sjl slk − skl slj ) − [dj , dk ] = 2 α l=1 (zj − zl )(zk − zl ) (zk − zj )(zj − zl ) (zj − zk )(zk − zl ) l=j,k
and according to the identity (11.36) the terms in the large brackets cancel.
The Dunkl operators can be used to define a set of commuting degree preserving operators ξi , and these operators in turn can be used to decompose the operator H (C,Ex) [58].
523
THE CALOGERO–SUTHERLAND MODEL
D EFINITION 11.4.3 The Cherednik operators ξi are defined in terms of the Dunkl operators di by ξi := αzi di + 1 − N +
N
sip
p=i+1
= αzi
zi zp ∂ + (1 − sip ) + (1 − sip ) + 1 − i, ∂zi pi i
(11.57)
where i = 1, . . . , N . P ROPOSITION 11.4.4 For all 1 ≤ i, j ≤ N we have [ξi , ξj ] = 0. Proof. From Proposition 11.4.2 and the second and third members of the readily verified relations P (i) [di , zi ] = 1 + α1 p =i sip (iii) [di , zj ] = − α1 sij (i = j)
(ii) di sip = sip dp , (iv) [di , sjp ] = 0 (i = j, p),
(11.58)
we see that [zi di , zj dj ] = −α(zi di − zj dj )sij . Also, the second and fourth relations give N N h i h X i X z i di , α sjp + sip , αzj dj = α(zi di − zj dj )sij p=j+1
p=i+1
while for i > j (and similarly i < j) N h X p=i+1
N i “ ” X [sip , sjp ] + [sip , sji ] = 0, sjp =
N X
sip ,
p=j+1
p=i+1
where to obtain the last equality (11.56) has been used.
P ROPOSITION 11.4.5 With α = 2/β and zj := e2πixj /L we have H (C,Ex) =
N 2π 2 1 N − 1 βW (C) /2 2 −βW (C) /2 e e ξ + . j L α2 j=1 2
Proof. Writing (11.52) in terms of α and the zj gives H (C,Ex) =
N “ “ 2π ”2 „ X
L
j=1
zj
« N N ∂ ”2 zj zl zj zl 1 X 1 X . − 2 + s jl ∂zj α j,l=1 (zj − zl )2 α j,l=1 (zj − zl )2 j=l
On the other hand, we see from (11.24) with θj = 2πxj /L that e−βW N Y j=1
−(N−1)/2α
zj
Y
(11.59)
j=l (C)
/2
is proportional to
(zk − zj )1/α ,
(11.60)
j
so Definition 11.4.3 gives e−βW
(C)
/2
“ X X N − 1 ” βW (C) /2 ∂ sjp sjp = αzj − zj − zp . e ξj + 2 ∂zj p<j zj − zp p>j zj − zp
(11.61)
524
CHAPTER 11
Summing over j, dividing by α and squaring we see that the term independent of α in the resulting expression is N “X j=1
zj
∂ ”2 , ∂zj
and, after some straightforward manipulation, that the term proportional to 1/α is equal to the term required by (11.59). Furthermore, use of (11.36) and (11.56) shows that the term proportional to 1/α2 is also as required by (11.59).
11.4.3 Integrability of H (C,Ex) and H (C) Consider the operator X (C) (u) :=
N j=1
(1 + uξj ) =
N
up Xp(C) ,
Xp(C) :=
ξi1 · · · ξip .
(11.62)
1≤i1 <···
p=0
According to Propositions 11.4.4 and 11.4.5, e−βW /2 X (C) (u)eβW /2 commutes with H (C,Ex) . Thus (C) (C) (C) the set of N algebraically independent operators e−βW /2 Xp (u)eβW /2 , p = 1, . . . , N , are mutually (C) (C) commuting, and commute with H (C,Ex) which is itself a linear combination of X2 , X1 and a constant. This establishes the integrability of H (C,Ex) . Also, we know that when restricted to act on symmetric functions H (C,Ex) reduces to H (C) . It follows that the integrability property of H (C,Ex) carries over to H (C) with the set of algebraically independent, mutually (C) (C) (C) commuting operators e−βW /2 Xp (u)eβW /2 restricted to act on symmetric functions. (C)
(C)
11.5 THE OPERATORS H (H,Ex) , H (L,Ex) AND H (J,Ex) 11.5.1 Definitions of the transformed operators An alternative way to view Proposition 11.4.5 and the results of Section 11.4.3 is as a decomposition of the transformed Schr¨odinger operator 2 ˜ (C,Ex) := L eβW (C) /2 (H (C,Ex) − E (C) )e−βW (C) /2 H 0 2π N N ∂ 2 N − 1 ∂ zj = + zj ∂z α ∂z j j j=1 j=1 zj zk ∂ 2 ∂ 1 − sjk − , (11.63) + − α zj − zk ∂zj ∂zk zj − zk 1≤j
2πixj /L
where zj := e and α := 2/β, in terms of a set of mutually commuting operators. The analogous transformation in the Hermite, Laguerre and Jacobi cases gives ˜ (H,Ex) := − 2 eβW (H) /2 (H (H,Ex) − E (H) )e−βW (H) /2 H 0 β N 2 ∂ 1 − sjk ∂ 2 1 ∂ ∂ + − , = − 2yj − 2 ∂yj ∂yj α yj − yk ∂yj ∂yk yj − yk j=1 j
(11.64)
525
THE CALOGERO–SUTHERLAND MODEL
˜ (L,Ex) := − 1 eβW (L) /2 (H (L,Ex) − E (L) )e−βW (L) /2 H 0 2β N 1 ∂2 1 ∂ ∂ = + −2yj + (2a + 1) 2 4 j=1 ∂yj ∂yj yj ∂yj +
1 ∂ ∂ yj2 + yk2 1 y − − y (1 − s ) , j k jk α yj2 − yk2 ∂yj ∂yk yj2 − yk2
(11.65)
j
˜ (J,Ex) := 1 eβW (J) /2 (H (J,Ex) − E (J) )e−βW (J) /2 H 0 4 N ∂ 2 1 zj − 1 N − 1 ∂ 1 zj + 1 + b+ + zj zj = + a+ ∂zj 2 zj − 1 2 zj + 1 α ∂zj j=1 zj zk ∂ 1 − sjk 2 ∂ − + − α zj − zk ∂zj ∂zk zj − zk 1≤j
(11.66)
1≤j
where a := (a β − 1)/2,
b := (b β − 1)/2. To obtain (11.64) and (11.65) the change of variables yj = β/2xj has been made, and for (11.65) the restriction to functions even in yj has been made, while for (11.66) the change of variables zj = e2iφj has been made along with the restriction to eigenfunctions unchanged by the mapping zj → 1/zj .
11.5.2 The operators hi Associated with the type A Dunkl operators di is the generalized Laplacian ΔA :=
N
d2i .
(11.67)
i=1
The explicit expansion of ΔA is closely related to (11.64) [141]. P ROPOSITION 11.5.1 With the type A Dunkl operators di defined in terms of the variables {yj } we have ΔA =
N 2 ∂ 1 − sjk ∂2 1 ∂ − . + − 2 ∂yj α yj − yk ∂yj ∂yk yj − yk j=1
(11.68)
j
Proof. This follows from Definition 11.4.1 of the di , the relations (11.56) and the identity (11.36) (the latter two results are used to show that the apparent term proportional to 1/α2 vanishes).
Comparison between (11.64) and (11.68) shows ˜ (H,Ex) = ΔA − 2 H
N j=1
2 1 ∂ = ΔA − ξj − N (N − 1) ∂yj α α N
yj
j=1
(11.69)
526
CHAPTER 11
where the second equality follows from Definition 11.4.3 of ξj (written in terms of {yj }), or equivalently ˜ (H,Ex) = − 2 H α
N
hi −
i=1
1 N (N − 1), α
hi := ξi −
α 2 d . 2 i
(11.70)
Like the result of Proposition 11.4.5 this gives a decomposition of the corresponding Schr¨odinger operator in terms of simpler operators, although the simpler operators occur only to the first power in (11.70) whereas they are squared in Proposition 11.4.5. From the results of Section 11.4.3 we see that the operator X (H) (u) :=
N
(1 + uhj ) =
N
up Xp(H) ,
hi1 · · · hip
(11.71)
1≤i1 <···
p=0
j=1
Xp(H) :=
can be used to establish integrability, provided we can show that the {hj } mutually commute. This latter task can be readily accomplished, once note is taken of some commutation formulas. P ROPOSITION 11.5.2 We have
⎧ ⎨ di sij , dj sij , [ξj , di ] = ⎩ −αdj − p<j sjp dj − p>j dj sjp
i < j, i > j, i = j.
Proof. From the definition of ξj in terms of dj we see that [ξj , di ] =
N X 1 [sjp , di ]. [yj dj , di ] + α p=j+1
(11.72)
But in general [AB, C] = A[B, C] + [A, C]B,
(11.73)
which together with Proposition 11.4.2 implies [yj dj , di ] = [yj , di ]dj . Using this equation and (11.58) in (11.72) gives the stated result.
P ROPOSITION 11.5.3 The operators hi , (i = 1, . . . , N ) as defined by (11.70) mutually commute. Proof. We have α α α2 2 2 [ξi , d2j ] − [d2i , ξj ] + [d , d ]. 2 2 4 i j According to Propositions 11.4.5 and 11.4.2 the first and last commutators vanish, while making use of (11.73) and Proposition 11.5.2 shows that the sum of the second and third commutators vanishes too. [hi , hj ] = [ξi , ξj ] −
Another way of showing that {hi } commute is first to note from the definition (11.70) and Definition 11.4.3 that we can write N di di + 1 − N + hi = α y i − sip , (11.74) 2 p=i+1 and then note that the algebra (11.58), supplemented by the relation [yi , yj ] = 0, is unchanged by the mapping φ(A) (yi ) = yi −
di , 2
φ(A) (di ) = di ,
φ(A) (sij ) = sij ,
(11.75)
where φ(A) is multiplicative (i.e., φ(A) (uv) = φ(A) (u)φ(A) (v)). Since by these rules φ(A) (ξi ) = hi ,
(11.76)
527
THE CALOGERO–SUTHERLAND MODEL
we see by applying φ(A) to the commutation relation [ξi , ξj ] = 0 that [hi , hj ] = 0 as required. (H) It follows immediately from Proposition 11.5.3 that {Xp }p=1,...,N as defined by (11.71) mutually com˜ (H,Ex) , thus establishing the integrability of the latter operator. These operators mute and commute with H restricted to act on symmetric functions provide a mutually commuting set of algebraically independent op˜ (H) , and establish integrability in this case as well. erators which commute with H 11.5.3 The operators li ˜ (L,Ex) requires the Dunkl operator of type B. The study of the Schr¨odinger operator H D EFINITION 11.5.4 For i = 1, . . . , N the Dunkl operators for the root system of type B are defined by ∂ 1 1 − sip 1 − σi σp sip a + 1/2 (B) + + + (1 − σi ). di := ∂yi α yi − yp yi + yp yi p=i
(B)
A direct calculation using (11.56) and the second of the identities (11.37) shows that the di (B)
[di
commute,
(B)
, dj ] = 0.
(11.77)
Analogous to the result of Proposition 11.5.1, the corresponding generalized Laplacian is closely related ˜ (L,Ex) . Thus a direct calculation, making use of the same identities required to establish to the operator H (11.77), shows ΔB :=
N N ∂ (B) ˜ (L,Ex) + 1 , (di )2 = 4 H yj 2 j=1 ∂yj i=1
(11.78)
N provided ΔB is restricted to act on functions even in each yj . Rewriting j=1 yj ∂y∂ j in terms of {ξˆi }, where ξˆi denotes the Cherednik operators of Definition 11.4.3 with the change of variables zj = yj2 (the operators ξˆi rather than ξi are needed for latter purposes), we see from this that 1 ˜ (L,Ex) = − 1 N (N − 1), H li − 2α i=1 2α N
α (B) li := ξˆi − (di )2 . 4
(11.79)
The algebra isomorphism (11.75) can be extended to the type B case, and from this the fact that the {li } commute can be deduced. First we note from the definitions that N α 1 (B) yi di + 1 − N + ξˆi = (sip + σi σp sip ) even , 2 2 p=i+1 functions which in turn shows that li as specified by (11.79) can be rewritten li =
α 4
)di
(B)
, sij , σi } is noted in (11.88) below. We see that the mapping
Now the algebra satisfied by {yi , di φ(B) (yi ) = yi −
dB i , 2
(B)
N 1 (sip + σi σp sip ) even . 2 p=i+1 functions
(B)
(2yi − di
+1−N +
B φ(B) (dB i ) = di ,
φ(B) (sij ) = sij ,
φ(B) (σi ) = σi ,
(11.80)
(11.81)
with φ(B) multiplicative is an algebra isomorphism of (11.88), and also φ(B) (ξˆi ) = li .
(11.82)
528
CHAPTER 11
Thus it follows by applying φ(B) to the commutation relation [ξˆi , ξˆj ] = 0 that [li , lj ] = 0, which is the sought result. ˜ (L,Ex) in (11.79), and the commutivity of {li }, we see that the operator From the decomposition of H X (L) (u) :=
N
(1 + ulj ) =
N
up Xp(L) ,
Xp(L) :=
li1 · · · lip ,
(11.83)
1≤i1 <···
p=0
j=1
˜ (L,Ex) (which itself was defined with when restricted to act on functions even in y1 , . . . , yN commutes with H the restriction that it acts on functions even in y1 , . . . , yN ). By also restricting X (L) (u) to act on symmetric ˜ (L) shows functions, the fact that the decomposition (11.79) when restricted to symmetric functions gives H ˜ (L) , thus establishing integrability in both cases. that X (L) (u) commutes with H (BC)
11.5.4 The operators ξj
The analogue of the Cherednik operator of Definition 11.4.3 for the BC root system is [60], [297] (BC)
ξj
=
N 1 1 1 − σj σk sjk 1 1 − σj 1 1 − σj 1 ξj − − a+ − b+ + (a + b + 1). α α k=1 1 − zj zk 2 1 − zj 2 1 + zj 2
(11.84)
=j
A direct calculation using the final identity of (11.37), and (11.56), shows that ˜ (J,Ex) = H
N
(BC)
2
ξj
j=1
1 (J) − E0 , 4
(11.85)
(BC)
)2 is restricted to functions which are even in the zi and unchanged by the provided the action of each (ξj mapping zi → 1/zi (i = 1, . . . , N ). This latter point indicates an essential difference with the decomposition (BC) 2 of Proposition 11.4.5 which is otherwise very similar: the operator (ξj ) must be considered as a single (BC) (BC) ξj .
entity which cannot be further decomposed as the product ξj commuting operator (11.62) is X (J) (u) :=
N
(BC) 2
(1+u(ξj
) )=
j=1
N p=0
up Xp(J) ,
Consequently, the analogue of the
Xp(J) :=
(BC) 2
(ξi1
(BC) 2
) · · · (ξip
) , (11.86)
1≤i1 <···
where the action is restricted to functions which are even in the zi and unchanged by the mapping zi → 1/zi (i = 1, . . . , N ). By further restricting the action to functions which are symmetric in the zi we see that (J) ˜ (J) and thus establish its integrability. {Xp }p=1,...,N commute with the operator H 11.5.5 Root system structure The Schr¨odinger operators (11.52) and (11.55) both describe quantum many-body systems on a circle. They can be cast in the common form N π 2 k (k − s ) ∂2 α α α α·α H (·,Ex) = − + , (11.87) 2 α/2 −α/2 2 ∂xj L (z −z ) j=1 α ∈Δ+
αN for α equal to the N -component vector (α1 , . . . , αN ). In where zj := e2πixj /L and z α := z1α1 · · · zN (11.87) Δ+ denotes the positive roots of the underlying root system (A type for H (C,Ex) and BC type for H (J,Ex) ), which are listed in Table 4.1. The operator sα denotes the reflection in the subspaces perpendicular to α. These operators are specified in terms of the exchange and negation operators for the root systems AN −1
529
THE CALOGERO–SUTHERLAND MODEL
and BCN and thus by sjk and sjk , σj σk sjk , σj , respectively. Finally, the kα are required to have the property that kα = kw(α) for all w ∈ W (the Weyl group generated by the reflections sα ). In the A type case this means kα is constant. Choosing kα = β/2, we reclaim (11.52). (B)
E XERCISES 11.5
1. [39] Show that the analogue of the equations (11.58) for the operators di
are the equa-
tions (B)
[di
“ ” 1 1X , yj ] = δij 1 + (sik + σi σj sij ) + (2a + 1)σj − (1 − δij )(sij − σi σk sik ), α k =i α (B)
(B)
sij dj
= di
(B) σj d j
(B) = −di σj ,
(B)
,
sij dk
(B) σj d k
(B)
= dk sij =
(B) d k σj
(k = i, j), (k = j).
(11.88)
2. [39] This exercise relates to some algebraic properties of {ξi }. (i) Use Definition 11.4.3 of the ξi and the formulas (11.58) to verify that ξi si − si ξi+1 = 1,
ξi+1 si − si ξi = −1,
[ξi , sj ] = 0 (j = i − 1, i).
(11.89)
Remark: These relations, together with [ξi , ξj ] = 0 and s2j = 1,
sj sj+1 sj = sj+1 sj sj+1 ,
[si , sj ] = 0 (|i − j| ≥ 2)
(11.90)
define the degenerate type A Hecke algebra. Note that the second relation in (11.89) follows from the first by preand post-multiplying by si . (i) Use Definition 11.4.3 of the ξi , and the commutator relations (11.58) to show 8 i < j, < −zj sij , −zi sijP , i > j, [ξi , zj ] = P : αzi + p>i zp sip + p
[si , ξi ξi+1 ] = 0
and from these equations, together with the last equation of (i), conclude that [si , f (ξ1 , . . . , ξN )] = 0 for any symmetric function f . (iii) Use the algebra isomorphism (11.75) and (11.76) to verify that the equations of (i) remain valid with the ξi replaced by the hi . ˜ (L) and H ˜ (J) are noted. 3. [36] Here some changes of variables for H ˜ (L) refer to (11.54) when acting on symmetric functions, so that sjk = 1. Replace yj in favor of uj (i) Let H according to uj = yj2 to show that in terms of uj ˜ (L) = H
N “ X j=1
uj
N ∂2 uj ∂ ” ∂ 2 X + (a − u + 1) + . j 2 ∂uj ∂uj α k=1 uj − uk ∂uj =j
˜ (J ) refer to (11.55) when acting on symmetric functions, so that sjk = 1. Replace zj in favor of yj (ii) Let H
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CHAPTER 11
according to yj = sin2 φj to show that in terms of yj ˜ (J) = H
N “ X
yj (1 − yj )
j=1
N ∂2 ∂ 2 X yj (1 − yj ) ∂ ” + . 2 + (a + 1 − yj (a + b + 2)) ∂yj ∂yj α k=1 yj − yk ∂yj
(11.91)
=j
11.6 DYNAMICAL CORRELATIONS FOR β = 2 11.6.1 Formulation The Schr¨odinger operators of Proposition 11.3.1 and 11.3.2, related to the corresponding Fokker-Planck operators by the transformation (11.33), have the property that at β = 2 the two-body term vanishes identically. The Schr¨odinger operators then describe free particles in an external potential, permitting computations of dynamical correlation functions not possible for the other couplings. In relation to the dynamical correlations, we recall from Section 11.3.2 that the Green functions GFP is τ the solution of the Fokker-Planck equation (11.15) which satisfies the delta function initial condition as in (11.40). The p.d.f. for the event that the particles are at positions x(1) for parameter value τ1 , and at positions x(2) for parameter value τ2 , is calculated from the Green functions according to ∞ ∞ 1 (0) (0) (1) (2) dx · · · dxN p0 (x(0) )GFP x(0) ; x(1) )GFP x(1) ; x(2) ),(11.92) pτ1 ,τ2 (x ; x ) = τ1 ( τ2 −τ1 ( N ! −∞ 1 −∞ where p0 is the prescribed initial p.d.f. in terms of the variables x(0) . The corresponding dynamical correlation (1) (2) ρ(m1 ,m2 ) ({xj }j=1,...,m1 ; τ1 ; {xj }j=1,...,m2 ; τ2 ) is then calculated from the formula N! N! (1) (2) ρ(m1 ,m2 ) ({xj }j=1,...,m1 ; τ1 ; {xj }j=1,...,m2 ; τ2 ) := (N − m1 )! (N − m2 )! ∞ ∞ ∞ ∞ (1) (1) (2) (2) dxm1 +1 · · · dxN dxm2 +1 · · · dxN pτ1 ,τ2 (x(1) ; x(2) ). × −∞
−∞
−∞
(11.93)
−∞
At β = 2 we read off from Propositions 11.3.1 and 11.3.2 that H (H) = −
N N ∂2 + x2j , 2 ∂x j j=1 j=1
H (L) = −
N N ∂2 a (a − 1) + + x2j , 2 2 ∂xj j=1 xj j=1
H (J) = −
N N ∂2 a (a − 1) b (b − 1) , + + ∂x2j j=1 cos2 xj sin2 xj j=1
(11.94)
which all describe independent particles in an external one-body potential. The same is true for H (D) , although as already remarked that case does not fit into the scheme below since there are no bound states. However, the fact that in the original Fokker-Planck description the particles cannot overlap due to the repulsive two-body interaction means the particles are to be regarded as fermions and thus an antisymmetric Green function is sought. Now in general for Schr¨odinger operators of the form (11.94), the free fermion Green function solution GSτ is given in terms of the Green function solution with N = 1, gτS (x1 , x2 ), say, by GSτ (x; x ) = det[gτS (xj , xk )]j,k=1,...,N .
(11.95)
Furthermore the method of separation of variables gives that for N = 1 the Green function solution of
531
THE CALOGERO–SUTHERLAND MODEL
the imaginary time Schr¨odinger equation (11.39) with β = 2 can be written in terms of the eigenfunctions {Φn }n=0,1,··· and corresponding eigenvalues {νn }n=0,1,··· according to gτS (u(0) , u) =
∞ Φj (u(0) )Φj (u)
Φj |Φj
j=0
e−γj τ ,
γj =
νj 2
where f |g := I f (u)g(u) du. For the particular operators (11.94) the eigenfunctions can be expressed in terms of the classical Hermite, Laguerre and Jacobi polynomials. A unified presentation of the three cases is possible by introducing the new variable yj = xj in H (H) , yj = x2j in H (L) and 12 (1 − yj ) = sin2 xj in H (J) , and defining the transformed single-particle Green function gτ by gτ (y, y )dy = gτS (x, x )dx. Recalling (11.42) we then find that x; x )dx1 · · · dxN = Gτ (y ; y )dy1 · · · dyN , GFP τ ( where
Gτ (y , y ) = e
E0 τ /2
N j=1
#
w2 (yj ) w2 (yj )
1≤j
(yk − yj ) det[gτ (yj , yk )]j,k=1,...,N (yk − yj )
(11.96)
(11.97)
with gτ (y, y ) =
w2 (y)w2 (y )
∞ pj (y)pj (y ) j=0
(pj , pj )2
e−γj τ .
(11.98)
Here the weight function is one of the classical forms of Section 5.4.1, the corresponding monic polynomials are given by (5.46), while ⎧ 1 Hermite, ⎪ ⎨ (n + 2 ), 2(n + (a + 1)/2), Laguerre, γn = (11.99) ⎪ ⎩ 2 2(n + (a + b + 1)/2) , Jacobi, where a := a − 12 , b := b − 12 . To proceed further the initial condition p0 in (11.92) must be specified. In the example of the parameterdependent random matrix given by (11.2), the appropriate choice would be the GOE eigenvalue p.d.f. This leads to a quaternion determinant formula for (11.93) [216], but the details will not be pursued. Instead we consider the simpler situation that P0 (H) is the joint p.d.f. for elements of the GUE, x(1) ; x(2) ). pτ1 ,τ2 (x(1) ; x(2) ) = GFP τ2 −τ1 ( The latter in turn is the special case w ˜2 (x) = w2 (x) of choosing in (11.92) p0 (x(0) ) =
N 1 w ˜2 (xl ) C l=1
1≤j
(xk − xj )2 .
(11.100)
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CHAPTER 11
11.6.2 Relationship to matrix integrals Consider the Fokker-Planck equation (11.15) with W given by (11.16). For β = 1, 2, 4 the Green function solution is given by (11.5) and thus P 2 † (0) ˜ 2 PN (λ(0) )2 ˜ β −β˜ N λ(0) ; λ) ∝ j j=1 λj −βt j=1 e2βtTr(VHV H ) (V† dV), (11.101) GFP ( |λ − λ | e k j τ 1≤j
where β˜ := β/2|1 − e−2τ |, t := e−τ and V is real orthogonal, unitary or unitary symplectic for β = 1, 2 or 4, respectively. On the other hand, for β = 2, this Green function is also given by the Hermite case of (11.96). Comparing the two expressions, and making use of the classical summation formula [508] ∞ 2 2 2 2 2 Hk (w)Hk (z) k 1 √ t = √ (1 − t2 )−1/2 e−t (z +w )/(1−t ) e2wzt/(1−t ) , k k!2 π π
|t| < 1,
(11.102)
k=0
gives the following matrix integral formula, due to Harish-Chandra [292] and Itzykson and Zuber [309]. † P ROPOSITION † 11.6.1 Let (U dU) denote the Haar volume form for N × N unitary matrices, normalized so that (U dU) = 1. Let A and B be N × N Hermitian matrices with eigenvalues {aj } and {bj }, respectively. We have N −1 † det[eaj bk ]j,k=1,...,N eTr(UAU B) (U† dU) = . (11.103) j! 1≤j
Proof. Summing (11.98) in the Gaussian case according to (11.102) and substituting in (11.97) shows (0) ; λ) ∝ GFP τ (λ
Y
(λk − λj )
1≤j
(λk − λj )
(0)
(0)
e−
PN
j=1
˜ 2 λ2 j −βt
e
PN
(0) 2 2 j=1 (λj +(λj ) )
h ˜ (0) i det e2βtλj λk
j,k=1,...,N
.
(11.104)
˜ j = aj , λ(0) = bj , gives (11.103) up to the proportionality constant. To Comparing this with (11.101), and setting 2βtλ j verify the latter is given correctly, take the limit a1 , . . . , aN → 0 and make use of the Vandermonde determinant formula (1.173) to show that both sides are equal to unity.
Consider next the Fokker-Planck equation (11.15) with W given by (11.29). Then the Green function solution is given by (11.27) and thus GFP x(0) ; x) ∝ τ (
N j=1
×
xβa+1 j
˜ PN
|x2k − x2j |β e−β
j=1
˜ 2 x2j −βt
PN
(0) 2 j=1 (xj )
1≤j
(U† dU)
˜
†
(V† dV)eβtTr(VX
U† X(0) +X(0)† UXV† )
,
(11.105)
˜ t are as in (11.101) where β, For β = 2 this Green function is also given by the Laguerre case of the l.h.s. of (11.96). Making use of the classical summation formula [508] ∞ n! xyt t (x + y) 0 F1 a + 1; (11.106) Lan (x)Lan (y)tn = (1 − t)−a−1 exp − (a + 1)n 1−t (1 − t)2 n=0
533
THE CALOGERO–SUTHERLAND MODEL
we obtain the alternative formula x(0) ; x) ∝ GFP τ (
N
x2a+1 e−xj j
2
j=1
(x2k − x2j )
(0) 2 1≤j
(0) ˜ 2 × det 0 F1 (a + 1; (xj xk tβ) )
−
˜
(0) (xj )2 )
j,k=1,...,N
e−βt
2
PN
(0) 2 2 j=1 (xj +(xj ) )
.
(11.107)
Comparing (11.105) in the case β = 2 with (11.107), and interchanging the role of U and V for convenience, the following analogue of (11.103) is obtained [277], [311]. P ROPOSITION 11.6.2 Let (U† dU), (V† dV) denote the normalized Haar volume for N ×N , n×n unitary matrices with n ≥ N . Let X and Y be n × N complex matrices, with the eigenvalues of X† X, Y† Y given 2 by x21 , . . . , x2N and y12 , . . . , yN , respectively. We have † † † † † (U dU) (V† dV)eTr(UX V Y+Y VXU ) =
N −1
(n − N + 1)j j!
j=0
det[0 F1 (n − N + 1; (xj yk )2 ]j,k=1,...,N . 2 2 2 2 1≤j
(11.108)
Here the normalization has been fixed by requiring that both sides equal unity in the limit x1 , . . . , xN → 0. We note that the 0 F1 function can be replaced in favor of the Bessel function of pure imaginary argument by using the identity (10.190). 11.6.3 Character expansions A major aspect of the theory of the classical group U (N ), and consequently of group integrals over the corresponding Haar measure, is that of its representations. This theory is detailed in many texts (see e.g. [540]). Following [487] we will present, without further explanation, a number of key notions and formulas which together allow for the computation of matrix integrals of the type (11.103) and (11.108). A key role is played by the Schur polynomials in their determinant form (10.16). First, irreducible representations of U (N ) and more generally Gl(N ) (the group of N × N invertible complex matrices) are labeled by partitions μ, and they have dimension expressible in terms of Schur polynomials by sμ ((1)N ) =
N 1 Γ(N − j + 1 + κj ) , hμ j=1 Γ(N − j + 1)
(11.109)
where the equality follows from (12.105) below and hμ is specified by (12.58) below with α = 1. Second, if X ∈ Gl(N ) has eigenvalues λ1 , . . . , λN , then its representations X(μ) have the property χμ (X) := Tr(X(μ) ) = sμ (λ1 , . . . , λN )
(11.110)
independent of the basis. The quantity χμ (X) is a group character. Third, for U ∈ U (N ), the elements of U(μ) have the integration property 1 (μ) ¯ (ρ) † Uij U δμ,ρ δi,k δj,l (11.111) kl (U dU) = sμ ((1)N ) (cf. the first result in Proposition 2.3.3), where (U† dU) denotes the normalized Haar volume form. We will make use of these results to deduce the following matrix integral, which generalizes (11.103) [438]. P ROPOSITION 11.6.3 Let A, B be N ×N complex matrices with eigenvalues {a1 , . . . , aN }, {b1 , . . . , bN }.
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CHAPTER 11
We have N −1 det(1N − AU† BU)−r (U† dU) = k=1
k! det[(1 − aj bk )N −1−r ]j,k=1,...,N . (r − N + 1)k 1≤j
(11.112)
Proof. Denote the integral by I(A, B). According to the generalized binomial theorem, (12.133) below in the case α = 1,
X Y Γ(r − j + 1 + κj ) sκ (X) 1 = . (det(1 − X))r Γ(r − j + 1) hκ κ j=1 N
Substituting this in the integrand and using (11.110) shows I(A, B) =
Z N X 1 Y Γ(r − j + 1 + κj ) Tr(A(κ) U(κ)† B(κ) U(κ) )(U† dU). hκ j=1 Γ(r − j + 1) κ
The integral is quadratic in the entries of the unitary matrix representation and so can be computed using (11.111) to give I(A, B) =
N X 1 Y 1 Γ(r − j + 1 + κj ) χκ (A)χκ (B). N h Γ(r − j + 1) s κ κ ((1) ) κ j=1
(11.113)
Next, substitute for sκ ((1)N ) using (11.109) and substitute the determinant form of the Schur polynomials (10.16). Finally, write kj := κj − j + N so that 0 ≤ kN < kN−1 < · · · < k1 and apply the general formula (5.170) to obtain I(A, B) =
N ∞ i hX Y Γ(N − j + 1) Γ(r + 1 − N + p) 1 , det (aj bk )p Δ(a)Δ(b) j=1 Γ(r − j + 1) Γ(1 + p) j,k=1,...,N p=0
Q where here Δ(a) := 1≤j
11.6.4 Matrix integrals and biorthogonal ensembles The matrix integrals (11.103), (11.108) and (11.112) all lead to biorthogonal ensembles. For definiteness, consider the Harish-Chandra/Itzykson-Zuber integral (11.103). As already noted, it implies for the Green function solution of the Fokker-Planck equation (11.15) with W given by (11.16) and β = 2 the explicit form (11.104). This Green function is equivalent to the eigenvalue p.d.f. of the complex parameter-dependent random matrix specified by Definition 11.1.1. Its functional form is the biorthogonal ensemble (5.148) with 2 w2 (x) = e−x . An alternative way to state this result is in terms of the shifted mean GUE, with distribution of elements proportional to exp(−(H − H(0) )2 ). Thus we have that the p.d.f. for the eigenvalues {xj } of H, given the 2 eigenvalues {yj } of H(0) , is equal to (5.148) with w2 (x) = e−x and aj = −2yj . In the case that aj = 0 (j = r + 1, . . . , N ) the matrix H(0) has rank r, and the corresponding correlation kernel is given by (5.172). We know that the correlation kernel exhibits a soft edge√scaling limit on introducing the scaled variables (7.16), and in particular setting yk = N/2 − (N 1/6 / 2)sk for the nonzero eigenvalues of H(0) . The significance of this latter value can be seen from Proposition 1.8.2, which tells us that in the caser = 1 a single eigenvalue will separate from the Wigner semicircle for y1 > N/2, thereby giving y1 = N/2 as the critical value of this effect. Another application [256] of (11.103) leading to the biorthogonal ensemble (5.148) is the computation of the eigenvalue p.d.f. of complex correlated Wishart matrices X† X, where X is a n × N (n ≥ N ) complex Gaussian matrix with joint p.d.f. 1 −Tr(X† XΣ−1 ) e C
535
THE CALOGERO–SUTHERLAND MODEL
(cf. (3.71)). Writing Σ−1 = 1N + B, we see that the joint p.d.f. of the elements of A = X† X is proportional to (det A)n−N e−Tr(A) e−Tr(AB) , where use has been made of (3.23). Now changing variables to the eigenvalues and eigenvectors of A according to (1.27) with β = 2 shows that the eigenvalue p.d.f. of A is proportional to N † n−N −λl 2 e−Tr(U AUB) (U† dU). λl e (λk − λj ) (11.114) 1≤j
l=1
Applying (11.103) gives (5.148) with w2 (x) = xn−N e−x . Let the eigenvalues of B be denoted {bi }, and suppose b1 = · · · = bN −r = 0 (in this situation the covariance matrix is said to be spiked [334], [29]). The corresponding correlation kernel is then given by (5.169) with a := n − N , and the nonzero eigenvalues of B denoted a1 , . . . , ar . We know from (7.40), (7.41) that this correlation kernel permits a soft edge scaling, where in particular it is required that aj = − 21 + sj /2N 1/3 , or equivalently that the eigenvalues of the covariance matrix Σ different from unity are of the form 2 + 2sj /N 1/3 . To understand the significance of this value, specialize to the case n = N , r = 1. For Y an n×n matrix of standard complex Gaussians, the matrix X† X = Σ1/2 Y† YΣ1/2 has the first column of X having variance unity. Noting that the eigenvalues of X† X are the same as those for XX† , from this latter matrix we see the problem can be regarded as a rank 1 perturbation of the type (3.72) with b = 2 + 2sj /N 1/3 . According to Proposition 3.5.1 the value b = 2 is critical for the largest eigenvalue to separate from the bulk.
11.6.5 Continuous nonintersecting paths In the setting of nonintersecting paths (recall Section 10.1.5) it is natural to prescribe not just the initial but also the final configuration. Take x(0) to be the initial configuration at time t = 0, and x(m+1) to be the (B) final configuration at time t = 2T . With Gt given by (10.44), the p.d.f. for arriving at x(s) in time t = ts , s = 1, . . . , m, where 0 < t1 < · · · tm < 2T , is then m+1 (B) x(l−1) ; x(l) ) l=1 Gtl −tl−1 ( . (11.115) (B) G2T (x(0) ; x(m+1) ) The simplest situation is that when x(0) = x(m+1) = 0. According to (10.47), for x and/or y approaching 0, 1 p2 /2 Pp 1 (B) − j=1 (x2j +yj2 )/2t Gt (x; y ) ∼ e (xj − xk )(yj − yk ). p−1 (2π)p/2 l=1 l! t 1≤j
1≤j
(1)
(1)
(m)
(xj − xk )(xj
(m)
− xk )
m−1
(B)
Gtl+1 −tl (x(l) ; x(l+1) ).
(11.116)
l=1
Prescribed initial and final positions can also be realized in the setting of parameter-dependent random matrices. Thus it was remarked at the end of Section 11.1 that the latter can be regarded as random matrices in which the entries are Brownian paths. More explicitly, the Brownian paths are those determined by the Ornstein-Uhlenbeck process corresponding to the Fokker-Planck equation (11.3). If instead we define the
536
CHAPTER 11
parameter-dependent ensemble to have joint p.d.f. for the elements, Pt say, to be proportional to exp − βTr (H − H(0) )2 /4t ,
(11.117)
then Pt satisfies the multidimensional heat equation ∂Pt ∂ 2 Pt 1 = Dμ , ∂t β μ ∂Hμ2 where Dμ is as in (11.3). The entries are now thus free Brownian motion, beginning at the entries of H(0) , and so this latter matrix determines the eigenvalues of H at t = 0. In particular, if we choose H(0) = 0, then the Brownian motions corresponding to the entries begin at the origin, and the eigenvalues of H are all initially zero. If the Brownian paths specifying the elements are also constrained to return to the origin at t = 2T (and thus be Brownian bridges), then we have a situation in which both the initial and final positions of the eigenvalues are prescribed as equaling 0. In the case β = 2, when the matrices in (11.117) are complex Hermitian, the corresponding p.d.f. for the eigenvalues equaling λ(s) at times ts (s = 1, . . . , m) with 0 < t1 < · · · < tm < 2T is in fact precisely (11.116). To see this, we read off from the limit λ, τ → 0, √ (B) λ/ τ fixed of (11.104) that the Green function Gt in this case is related to the Green function Gt (10.44) for the nonintersecting paths problem by (λk − λj ) (B) Gt (λ(0) ; λ) = Gt (λ(0) ; λ). (0) (0) 1≤j
11.6.6 Correlations and determinant formulas The first task in evaluating the dynamical correlation (11.93) is to compute (11.92) in a convenient form. Throughout we will work with the variables yj used in (11.97) and consider the initial condition (11.100). P ROPOSITION 11.6.4 Let {uj (z; τ )}j=0,1,... , {vj (z; τ )}j=0,1,... be a set of monic polynomials with uj , vj of degree j. For the p.d.f. (11.92) with Green function (11.97) and initial condition (11.100) we have N 1 (2) (2) (1) (2) (1) (2) pτ1 ,τ2 (y ; y ) = w2 (yj ) det vj−1 (yk ; τ2 ) det[gτ2 −τ1 (yl , yj )]j,l=1,...,N C j=1 j,k=1,...,N N (1) (1) × w2 (yj ) det u ˜j−1 (yk ; τ1 ) j=1
where 1 u ˜j−1 (y; τ ) := w2 (y)
I
j,k=1,...,N
,
w ˜ (z) 2 uj−1 (z; τ )gτ (z, y) dz. w2 (z)
(11.118)
(11.119)
Proof. From the definition y (1) ; y (2) ) = pτ1 ,τ2 (
N q N Y 1 Y (2) (2) (2) (1) (2) w2 (yj ) (yj − yl ) det[gτ2 −τ1 (yl , yj )]j,l=1,...,N C j=1 j>l
Z
Z (0)
×
(0)
dy1 · · · I
I
dyN
N (0) Y w ˜2 (yl ) Y (0) (0) (0) (1) q (yk − yj ) det[gτ1 (yk , yj )]j,k=1,...,N (11.120) (0) l=1 w2 (yl ) j
537
THE CALOGERO–SUTHERLAND MODEL
for some normalization C. Write Y
(0)
(0)
(0)
(yk − yj ) = det[uj−1 (yk ; τ1 )]j,k=1,...,N ,
(11.121)
j
so that there are two determinant factors in the integrand. Since both determinants are antisymmetric in {yj }j=0,...,N−1 Q (0) we can replace (11.121) by N ! N j=1 uj−1 (yj ; τ1 ) in the integrand. Integrating row-by-row in the remaining determinant shows that the last line in (11.120) can be written N q “Y ” h i (1) (1) N! w2 (yl ) det u ˜j−1 (yk ; τ1 )
j,k=1,...,N
l=1
,
and the result follows.
We recognize the structure (11.118) as an example of the p.d.f. (5.175) with k = 2. Consequently the correlations are given by Proposition 5.9.6. The double summations therein reduces to a single summation if we choose the polynomials {uj (x; τ )} and {vj (x; τ )} so that they have the biorthogonality property w ˜2 (z) dy w2 (y) dz (11.122) uj−1 (z; τ1 )gτ1 (z; y) vk−1 (y; τ2 ) = hj−1 (τ2 )δj,k . w2 (z) I I At τ1 = τ2 = 0 the l.h.s. of (11.122) reduces to w ˜2 (y)uj−1 (y; 0)vk−1 (y; 0) dy. I (0) {pl (y)}l=0,1,...
denotes the monic orthogonal polynomials with respect to the weight function Thus if w ˜2 (y), then we must have (0)
uj−1 (y; 0) = pj−1 (y),
(0)
vk−1 (y; 0) = pk−1 (y).
Now let {pl (y)}l=0,1,... denote the orthogonal polynomials with respect to the weight function w2 (y) as in (11.98), and write (0)
pk−1 (y) =
k−1
α ˜k−1 l pl (y),
α ˜ k−1 k−1 = 1.
(11.123)
l=0
A direct calculation verifies the following result. P ROPOSITION 11.6.5 The monic polynomials vk−1 (y; τ ) = e−γk−1 τ
(0)
uj−1 (z; τ ) = pj−1 (z),
k−1
α ˜ k−1 l pl (y)eγl τ
l=0 (0)
(0)
with normalization hj−1 (τ ) = e−γj−1 τ (pj−1 , pj−1 )w˜2 have the biorthogonality property (11.122). Note that in the special case w ˜2 (y) = w2 (y) Proposition 11.6.5 gives uj (z; τ ) = pj (z),
vk (z; τ ) = pk (z),
(11.124)
with normalization hj (τ ) = e−γj τ (pj , pj )2 . Application of Proposition 5.9.6 gives the following formula for the dynamical correlations. P ROPOSITION 11.6.6 For the p.d.f. (11.118) (1) (2) (1) (2) ρ(m1 ,m2 ) (y1 , . . . , ym ; τ1 ; y1 , . . . , ym ; τ2 ) 1 2
= det
11 ]j,k=1,...,m1 [σjk
12 j=1,...,m [σjk ] 1
21 j=1,...,m [˜ σjk ] 2
22 [σjk ]j,k=1,...,m2
k=1,...,m1
k=1,...,m2
,
(11.125)
538
CHAPTER 11
where mn = σjk mn σ ˜jk
(m) (n) N u ˜l−1 (yj ; τm )vl−1 (yk ; τn ) (m) (n) , w2 (yj )w2 (yk ) hl−1 (τn ) l=1 (m) (n) gτm −τn (yj |yk )
− (τm > τn ) (m) (n) ∞ u ˜l (yj ; τm )vl (yk ; τn ) (m) (n) . = − w2 (yj )w2 (yk ) hl (τn )
mn = σjk
(11.126)
l=N
Proof. It follows from the biorthogonality (11.122) and the definition (11.119) that an arbitrary function f (x) can be expanded ∞ R X w2 (t)f (t)˜ ul (t; τ ) dt I f (x) = vl (x; τ ). h l (τ ) l=0 p Choosing f (x) = gτ (x, x )/ w2 (x) and noting from Proposition 11.6.5 that uj−1 (z; τ ) in (11.119) is independent of τ , we see from this that (m)
gτ (yj
(n)
, yk ) =
q
(m)
w2 (yj
(n)
)w2 (yk )
(m) (n) ∞ X u ˜l (yj ; τ + τ )vl (yk ; τ ) . hl (τ ) l=0
Choosing τ = τn , τ = τm − τn (τm > τn ), verifies the final equality in (11.126), and allows the integrals in the mn formulas of Proposition 5.9.6 for σjk to be evaluated to give the first equality.
We remark that pτ1 ,τ2 (x; y ) effectively specifies a two-species system. As such the formula (9.19) applies for the corresponding gap probability generating function. E XERCISES 11.6
1.
(i) [508] Consider the second order differential equation
d2 pn (z) dpn (z) + β(z) = λn pn (z). dz 2 dz p Let x = x(z) be such that x (z) = 1/ α(z), and put α(z)
ψ0 (x) = e−w(z(x))
with w =
(11.127)
α − 2β . 4α
Show that the functions ψn (x) = ψ0 (z)pn (z(x)) then solve the Schr¨odinger equation “
−
” d2 + V (x) ψn (x) = −λn ψn (x) dx2
where
1 (2β − α )(2β − 3α ) 1 − α + β . 16α 4 2 (ii) [285] The Hermite, Laguerre and Jacobi polynomials satisfy (11.127) with 8 Hermite, < (1, −2z, −2n), 4(z, (a + 1 − z), −n), Laguerre, (α(z), β(z), γn ) = : 4(1 − z 2 , b − a − (a + b + 2)z, −n(n + a + b + 1)), Jacobi V (x) = v(z(x)),
v=
(the factors of 4 in the Laguerre and Jacobi cases are chosen for convenience). Show that the theory of (i) is consistent with the results stated in the paragraph including (11.96). 2. [409] The following p.d.f. can be interpreted as specifying a preferred basis. (i) Let U be a member of the CUE, and define a probability density function for Hermitian matrices H by
539
THE CALOGERO–SUTHERLAND MODEL
requiring that
†
2
Pr(H) ∝ e−Tr H e−bTr([U,H][U,H] ) .
(11.128)
Integrate over U according to (11.103) to deduce that the corresponding eigenvalue p.d.f. is proportional to h i 2 2 det e−(b+1/2)(xj +xk )+2bxj xk . j,k=1,...,N
(ii) Regard (11.128) as specifying a probability distribution on unitary matrices U rather than Hermitian matrices H. By integrating over H show that the corresponding eigenvalue p.d.f. is proportional to Y 1≤j
3.
(i) Set c =
p
sin2 (θk − θj )/2 . 1 + 4b sin2 (θk − θj )/2 (1)
t(2T − t)/T . In the case m = 1, note that (11.116) with xj
=: xj can be written
2 Pp Y “ xk 2p /2 c−p x j ”2 − j=1 (xj /c)2 − e Q c c (2π)p/2 p−1 l=1 l! 1≤j
and check from Proposition 4.7.1 that it is correctly normalized with respect to the domain x1 > · · · > xp . (ii) Make use of (11.102) to show that for general p, (11.116) can be written m ”p(p−1)/2 Y T 2−p(p−1)/2 “ (c1 cm )p(p−1)/2 c−p Q l p−1 π p/2 l=1 l! t1 (2T − tm ) l=1
×
m Y s=1
× det where cj :=
p
e−
Pp
(s) 2 j=1 (xj /cs )
(1)
(m)
det[Hj−1 (xk /c1 ]j,k=1,...,p det[Hj−1 (xk /cm ]j,k=1,...,p
∞ hX i e−n(τs+1 −τs ) (s+1) (s) √ n /cs+1 )Hn (xk /cs ) , Hn (xj π2 n! j,k=1,...,p n=0
tj (2T − tj )/T , τs := − 21 log(2T − ts )/ts .
(iii) Use the continuum version of Proposition 5.9.5 to obtain from the result of (ii) the dynamical correlation function ρ(r) ((x1 , τ1 ), . . . , (xr , τr )) =
r Y
− c−1 j e
Pr
j=1 (xj /cj )
2
˜ det[K((x j , τj ), (xk , τk ))]j,k=1,...,r
j=1
where
8 p−1 X ek(t−s) > > > √ k Hk (y/cy )Hk (x/cx ), > < π2 k! k=0 ˜ K((x, s), (y, t)) = ∞ X > ek(t−s) > > √ k Hk (y/cy )Hk (x/cx ), > : − π2 k! k=p
t ≥ s, t < s.
4. Let (U† dU) denote the normalized Haar volume form for N × N unitary matrices, and let A, B be general N × N complex matrices. The method of Section 11.6.3 has been used to derive the matrix integral evaluation [482] Z † (det U)ν e(1/2)(AU+BU ) (U† dU) = 2N(N−1)/2
N−1 Y k=1
k!
“ det B ”ν/2 det[μj−1 I ν+j−1 (μi )]i,j=1,...,N i , QN 2 2 det A i<j (μi − μj )
540
CHAPTER 11
where ν ∈ Z and {μ2i } are the eigenvalues of AB. By taking the limits μ2 → μ1 = succession deduce from this (8.180).
√
t, μ3 → μ2 , etc., in
11.7 SCALED LIMITS With the β = 2 initial state (11.100) assumed, let us suppose further that w ˜2 (y) = w2 (y) so that this state corresponds to the equilibrium state. In this case (11.124) gives that the in general biorthogonal, parameterdependent polynomials {uj (x; τ )}, {vj (x; τ )} reduce to orthogonal, parameter-independent polynomials, and we see from (11.126) that (m) (n) N pl−1 (yj )pl−1 (yk ) γ (τn −τm ) (m) (n) mn σjk = w2 (yj )w2 (yk ) e l−1 , (pl−1 , pl−1 )2 mn σ ˜jk =−
l=1
(m)
w2 (yj
(n)
)w2 (yk )
(m) (n) ∞ pl−1 (yj )pl−1 (yk ) γl−1 (τn −τm ) e . (pl−1 , pl−1 )2
(11.129)
l=N +1
The scaled form of these quantities can be computed in the bulk and at the hard and soft edges [441], [378], [216].
11.7.1 Bulk Consider the Gaussian weight w2 (x) = e−x . The monic orthogonal polynomials are given by the polyno(G) mials pn (x) in (5.46), their corresponding normalization is given in (5.48) and the constant γn is given (m) (m) mn in (11.99). We see from (11.129) that in the case m = n, σjk is precisely the function KN (yj , yk ) of Proposition 5.1.3 and is thus summed by the Christoffel-Darboux formula (5.10). Its bulk scaled limit is (m) (n) mn given in Proposition 7.1.1. Thus with σjk := σ(yj , yk ; τm − τn ) we have 2
(m) sin πρ(y − yk ) πρy πρ πρyj j lim √ σ √ , √ k ;0 = . (m) (m) N →∞ 2N 2N 2N π(yj − yk ) (m)
(m)
(m)
In the case m = n the Christoffel-Darboux formula cannot be used. Instead we proceed as in the asymptotic analysis of the summations in (6.152), which involves substituting the asymptotic form (7.1). One finds (m) (n) πρy πρ πρyj (πρ)2 lim √ (τm − τn ) σ √ , √ k ; N →∞ 2N 2N 2N 2N 1 2 (m) (n) bulk (m) (n) e(τn −τm )(πρu) /2 cos πuρ(yj − yk ) du. (11.130) =: S2 (yj , yk ; τm − τn ) = ρ 0
Note that τ has been scaled by 1/N to obtain a well-defined result. Similarly, now assuming τn < τm , we have (m) (n) πρy (πρ)2 πρ πρyj lim √ σ ˜ √ , √ k ; (τm − τn ) N →∞ 2N 2N 2N 2N ∞ 2 (m) (n) (m) (n) bulk ˜ =: S2 (yj , yk ; τm − τn ) = −ρ e(τn −τm )(πρu) /2 cos πuρ(yj − yk ) du. (11.131) 1
541
THE CALOGERO–SUTHERLAND MODEL
Recalling (11.125) we therefore have (1) (2) (2) πρ m1 +m2 πρy (1) πρy πρym πρym √ ρ(m1 ,m2 ) √ 1 , . . . , √ 1 ; τ1 ; √ 1 , . . . , √ 2 ; τ2 τ1 →(πρ)2 τ1 /2N N →∞ 2N 2N 2N 2N 2N τ2 →(πρ)2 τ2 /2N
lim
(1)
(2)
(1) (2) =: ρbulk (m1 ,m2 ) (y1 , . . . , ym1 ; τ1 ; y1 , . . . , ym2 ; τ2 ) (1) (1) (1) (2) [S2bulk (yj , yk ; τ1 − τ2 )]m1 ×m2 [S2bulk (yj , yk ; 0)]m1 ×m1 = det (2) (1) (2) (2) [S˜2bulk (yj , yk ; τ2 − τ1 )]m2 ×m1 [S2bulk (yj , yk ; 0)]m2 ×m2
.
(11.132)
11.7.2 Soft edge At the soft edge the appropriate scaling is √ y → 2N +
y τ , τ → 1/3 , 21/2 N 1/6 N with the scaling of y thus being the same as in the static theory (recall (7.11)). Using (7.9) one finds, for τn ≥ τm 2/3 (m) (n) √ yj yk e−N (τm −τn ) √ 1 σ 2N + , 2N + ; 1/3 (τm − τn ) 1/2 1/6 1/2 1/6 1/2 1/6 N →∞ 2 N 2 N 2 N N ∞ (m) (n) (m) (n) Ai (yj + v)Ai(yk + v)e−v(τn −τm ) dv, =: S2soft (yj , yk ; τm − τn ) =
lim
(11.133)
0
while for τn < τm 2/3 (m) (n) √ yj yk e−N (τm −τn ) √ 1 σ ˜ 2N + , 2N + ; (τ − τ ) m n N →∞ 21/2 N 1/6 21/2 N 1/6 21/2 N 1/6 N 1/3 0 (m) (n) (m) (n) =: S˜2soft (yj , yk ; τm − τn ) = − Ai (yj + v)Ai(yk + v)e−v(τn −τm ) dv. (11.134)
lim
−∞
Note that in the case τn = τm , (11.133) must reduce to the soft edge scaled form of the Gaussian kernel (7.12), a fact which is seen from the first formula of Exercises 7.1 q.1(ii). Setting (1)
(2)
(1) (2) ρsoft (m1 ,m2 ) (y1 , . . . , ym1 ; τ1 ; y1 , . . . , ym2 ; τ2 ) (1) (1) m1 +m2 √ √ 1 y1 y m1 τ1 ρ 2N + , . . . , 2N + ; 1/3 ; := lim (m ,m ) 1 2 1/2 1/6 1/2 1/6 1/2 1/6 N →∞ 2 N 2 N 2 N N (2) (2) √ √ y ym τ2 (11.135) 2N + 1/21 1/6 , . . . , 2N + 1/2 21/6 ; 1/3 2 N 2 N N bulk and recalling (11.125) it follows that ρsoft and (m1 ,m2 ) is given by the determinant in (11.132), but with S2 bulk soft soft ˜ ˜ replaced by S and S respectively. S 2
2
2
11.7.3 Hard edge To study the hard edge we use the Laguerre weight w2 (x) = xa e−x , x > 0, and the corresponding monic orthogonal polynomials from (5.46). The appropriate scaling is τ y , τ → , y → 4N 2N where again the scaling of y is chosen to coincide with the static theory (recall (7.30)).
542
CHAPTER 11
Use of (7.29) gives y 1 yj τm − τn lim σ , k ; N →∞ 4N 4N 4N 2N (m)
(m)
:= S2hard (yj
(n)
(n)
, yk ; τm − τn ) =
1 4
1
Ja 0
(m) (n) Ja uyj uyk e−u(τm −τn ) du (11.136)
and, for τn < τm , y 1 yj τm − τn σ ˜ , k ; N →∞ 4N 4N 4N 2N (m)
(n)
lim
(m)
:= S˜2hard (yj
(n)
, yk ; τm − τn ) = −
1 4
∞
Ja 1
(m) (n) Ja uyj uyk e−u(τm −τn ) du. (11.137)
Setting (1)
(2)
(1) (2) ρhard (m1 ,m2 ) (y1 , . . . , ym1 ; τ1 ; y1 , . . . , ym2 ; τ2 ) (1) (2) (2) 1 m1 +m2 y (1) ym τ1 y1 ym τ2 ,... 2; ρ(m1 ,m2 ) 1 , . . . , 1 ; = lim N →∞ 4N 4N 4N 2N 4N 4N 2N
(11.138)
bulk and and recalling (11.125) we thus have that ρhard (m1 ,m2 ) is given by the determinant in (11.132) with S2 bulk hard hard ˜ ˜ replaced by S and S , respectively. S 2
2
E XERCISES 11.7
2
1.
(i) With mn mn fjk := σjk ((m, n) = (2, 1)),
21 21 fjk =σ ˜jk ,
mn 21 where σjk and σ ˜jk are specified by (11.126), use (11.122) to check that Z Z m1 1p mp 1 m2 2p mp 2 fjl flk dyl = fjk ξmp , fjl flk dyl = fjk ξmp , I
(11.139)
I
where m, p = 1 or 2 and
» ξ1 =
1 0
1 0
–
»
0 0
ξ2 =
,
1 1
– .
(ii) Using the orthogonality relations (11.139), adopt the method of integration of Proposition 5.1.2 to show that Z det I
» [σ 11 ] jk j,k=1,...,ν 21 [˜ σjk ] j=1,...,μ k=1,...,ν
= (N − μ + 1) det
–
12 [σjk ] j=1,...,ν
dyμ(2)
k=1,...,μ
22 [σjk ]j,k=1,...,μ
» [σ 11 ] jk j,k=1,...,ν
12 [σjk ]
21 [˜ σjk ] j=1,...,μ−1
j=1,...,ν k=1,...,μ−1
–
22 [σjk ]j,k=1,...,μ−1
k=1,...,ν
and Z det I
» [σ 11 ] jk j,k=1,...,ν 21 [˜ σjk ] j=1,...,μ k=1,...,ν
= (N − ν + 1) det
12 [σjk ] j=1,...,ν
–
k=1,...,μ
22 [σjk ]j,k=1,...,μ
dyν(1)
» [σ 11 ] jk j,k=1,...,ν−1 21 [˜ σjk ]
j=1,...,μ k=1,...,ν−1
Explain how these results are consistent with (11.125).
12 [σjk ] j=1,...,ν−1 – k=1,...,μ
22 [σjk ]j,k=1,...,μ
.
Chapter Twelve Jack polynomials We have seen that the calculation of dynamical correlation functions for the Brownian evolution of the loggas requires knowledge of the Green function solution of the Fokker-Planck equation. It has also been noted that the Green function can be expressed in terms of the eigenvalues and eigenfunctions of the corresponding Schr¨odinger operator. In the cases of interest, these eigenfunctions factorize into a product of the ground state wave function times a multivariable polynomial. The most fundamental case is the Schr¨odinger operator H (C,Ex) . The polynomial part of the eigenfunctions are then termed the nonsymmetric Jack polynomials, and they form the natural starting point from which to investigate the properties of their symmetric counterparts, which are the polynomial part of the eigenfunctions of H (C) . Here a self-contained development of the theory of Jack polynomials is undertaken. We discuss orthogonality properties, with respect to both a pairing inner product and an integral inner product, and compute the associated normalization constants. The pairing inner product relates to the Cauchy product expansion, and this combined with the integral inner product allows a generalization of the Selberg integral, in which the Jack polynomial appears as a factor in the integrand, to be evaluated. Interpolation Jack polynomials, which can be defined by their vanishing properties, are introduced for their use in the derivation of the branching coefficients in the expansion of the product of a symmetric Jack polynomial and an elementary symmetric function (Pieri formulas), and their relation to generalized binomial coefficients.
12.1 NONSYMMETRIC JACK POLYNOMIALS 12.1.1 Construction as eigenfunctions ˜ (C,Ex) (11.63). According to Proposition 11.4.5 this operator has the Consider the transformed operator H decomposition N N − 1 2 L 2 (C) ˜ (C,Ex) = 1 H ξ + − E0 . j α2 j=1 2 2π
The fact that the ξj all commute suggests that we seek simultaneous eigenfunctions of the ξj . Such eigen˜ (C,Ex) . We will show not only that such eigenfunctions (if they exist) will then be eigenfunctions of H functions exist, but that they form a complete set, being in one-to-one correspondence with the monomials ηN z1η1 z2η2 · · · zN =: z η (ηj ∈ Z≥0 ; η is said to form a composition of non-negative integers, and the ηj form the parts). The mechanism behind this result is that the ξj have a triangular action on the monomial z η . To be more precise, we must first introduce a partial ordering for compositions. D EFINITION 12.1.1 Let η + denote the particular reordering of the parts of the composition (η1 , η2 , . . . , ηN ) → (ηP (1) , ηP (2) , . . . , ηP (N ) ) such that ηP (1) ≥ ηP (2) ≥ · · · ≥ ηP (N ) (note that the associated permutation P need not be unique). We know from Definition 10.1.3 that in general an N -tuple of non-negative integers so ordered is referred to as
544
CHAPTER 12
a partition. The modulus of a composition η, or its associated partition η + , is defined as |η| :=
N
ηj .
j=1
Suppose |η| = |ν| for compositions η = ν. In such cases the partial ordering <, known as the dominance ordering, is defined by the statement that ν < η if p j=1
νj ≤
p
ηj
(12.1)
j=1
for each p = 1, . . . , N . A further partial ordering ≺, referred to as the Bruhat ordering, r is defined on compositions by the statement that ν ≺ η if ν + < η + , or in the case ν + = η + , if ν = l=1 sil jl η, where ηil > ηjl , il < jl . We emphasize that both < and ≺ are partial orderings only. For example, the partitions (4, 1, 1) and (3, 3, 0) are incomparable according to the requirement (12.1). We will see that ξj z η gives monomials for which the corresponding exponents are all comparable with η in the partial ordering ≺, and are in fact smaller than η in this ordering. P ROPOSITION 12.1.2 Let (η1 , . . . , ηj + r, . . . , ηk − r, . . . , ηN ), 1 ≤ r ≤ ηk − ηj − 1, ηk > ηj , η(j, k, r) = (η1 , . . . , ηj − r, . . . , ηk + r, . . . , ηN ), 1 ≤ r ≤ ηj − ηk , ηj > ηk , and η¯j = αηj − #{k < j|ηk ≥ ηj } − #{k > j|ηk > ηj }.
(12.2)
For each i = 1, . . . , N we have ξi z η = η¯i z η + sgn(ηi − ηp ) z η(p,i,r) + sgn(ηi − ηp ) z η(i,p,r) , r
p
(12.3)
r
p>i
where the first sum over r is from 1 ≤ r ≤ ηi − ηp − 1 (ηi > ηp ) and 1 ≤ r ≤ ηp − ηi (ηp > ηi ), while the second sum over r has the same constraints with ηi and ηp interchanged. The exponents η(j, k, r) occurring on the r.h.s. of (12.3) are smaller in the partial ordering ≺ than η. Proof. The identity a n bm − b n a m = an−1 bm + an−2 bm+1 + · · · + am bn−1 , a−b shows that zi η (1 − sip )ziηi zp p = zi − zp zp η (1 − sip )ziηi zp p = zi − zp
(
(
n > m,
(ziηi zp p + ziηi −1 zp p + · · · + zi p zpηi −1 ), η −1 η −2 η −(zp p ziηi +1 + zp p ziηi +2 + · · · + zpηi zi p ), η
η
η +1
η −1
η +1
η −1
−(zp p ziηi + zp p ziηi +1 + · · · + zpηi +1 zi p ), η +1 η +2 η (ziηi −1 zp p + ziηi −2 zp p + · · · + zi p zpηi ),
(12.4)
ηi > ηp , ηp > ηi , ηp > ηi , ηi > ηp .
According to Definition 11.4.3 of the ξi , these identities suffice to give the action of ξi on z η , and they indeed imply the stated formulas (to obtain η¯i in the form stated requires using the identity #{k < j|ηj > ηk } = j − 1 − #{k < j|ηk ≥ ηj }). Finally, note that in the definition of η(j, k, r) for r = ηj − ηk we have η(j, k, r) < η, while for r = ηj − ηk , η(j, k, r) ≺ η (the latter follows because then η(j, k, r) = sjk η with ηj > ηk , j < k).
In general a triangular action on an ordered set of basis vectors φ0 < φ1 < φ2 < · · · , Aφj = αj φj +
545
JACK POLYNOMIALS
j−1 l=0
c˜jl φl , allows for a systematic construction of eigenfunctions of A, ψj = φj +
j−1
cjl φl
l=0
with eigenvalue αj , provided {αj } are distinct. For each individual ξi , although there is a triangular action, the eigenvalues are not distinct. Thus instead of considering a particular ξi , we consider their generating function as specified by the operator X
(C)
N ({ul }) := (1 + ul ξl )
(12.5)
l=1
which generalizes (11.62). Proposition 12.1.2 gives that this operator also has a triangular structure when acting on z η , showing N (1 + ul η¯l )z η + lower order terms,
(12.6)
l=1
where the lower order terms are with respect to the partial ordering ≺. Thus X (C) ({ul }) has eigenfunctions of the form Eη (z) := Eη (z; α) := Eη (z1 , . . . , zN ; α) = z η + aην z ν , (12.7) ν≺η
N
with corresponding eigenvalues j=1 (1 + uj η¯j ), which according to (12.2) are distinct for distinct compositions. Due to this latter fact the construction of the eigenfunctions from their triangular structure is now unique. However, it is not immediately clear that (12.7) is independent of {ul }. To see that indeed this must be the case, we can use the fact that the ξj commute to act on the eigenvalue equation X (C) ({ul })Eη (z) =
N (1 + ul η¯l )Eη (z)
(12.8)
l=1
with the operator X (C) ({vl }), {vl } = {ul }. This leads to the conclusion that X (C) ({vl })Eη (z) is an eigenN function of X (C) ({ul }) with eigenvalue l=1 (1 + ul η¯l ). But the unique (up to normalization) polynomial with this property is Eη (z), so X (C) ({vl })Eη (z) must be proportional to Eη (z). Of course eigenfunctions of X (C) ({vl }) cannot depend on {ul } so Eη (z) must in fact be independent of {ul }. Equating coefficients of uj on both sides of the eigenvalue equation (12.8), we see that ξj Eη (z) = η¯j Eη (z)
(12.9)
for j = 1, . . . , N so that the Eη are simultaneous eigenfunctions of each ξj . The fact that X (C) ({ul }) has a unique eigenfunction of the form (12.7) implies that Eη is the unique polynomial of the form (12.7) which is a simultaneous eigenfunction of {ξj }. The Eη are referred to as the nonsymmetric Jack polynomials [436], [58]. In the limit α → ∞, we see by including only the terms in ξj and η¯j proportional to α that the eigenvalue equation (12.9) reduces to zj
∂ Eη (z) = ηj Eη (z). ∂zj
Thus we see that Eη (z; ∞) = z η .
(12.10)
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CHAPTER 12
12.1.2 Orthogonality and a multidimensional integral The Schr¨odinger operator H (C,Ex) as specified by (11.52) is self-adjoint with respect to the inner product L/2 L/2 dx1 · · · dxN f (z1 , . . . , zN )g(z1 , . . . , zN ), (12.11) f |g := −L/2
−L/2
where zj := e2πixj /L , so that f |H (C,Ex) g = H (C,Ex) f |g. ˜ (C,Ex) , it follows that H ˜ (C,Ex) is self-adjoint with From the relationship (11.63) between H (C,Ex) and H respect to the inner product L/2 L/2 f |g(C) := dx1 · · · dxN |zk − zj |2/α f (z1 , . . . , zN )g(z1 , . . . , zN ) (12.12) −L/2
−L/2
1≤j
(recall that 2/α := β). In light of the decomposition of Proposition 11.4.5 we might also suspect that the Cherednik operator ξi is self-adjoint with respect to (12.12). In fact this is indeed the case. P ROPOSITION 12.1.3 For each i = 1, . . . , N we have f |ξi g(C) = ξi f |g(C) . Proof. By writing F = e−βW that
(C)
/2
f and G = e−βW
F |(e−βW
(C)
/2
ξi eβW
(C)
/2
(C)
/2
g we see that the stated formula is equivalent to the statement
)G = (e−βW
(C)
/2
ξi eβW
(C)
/2
)F |G,
where ·|· refers to the inner product (12.11). The validity of this latter formula is seen from the explicit expression (C) (C) (11.61) for e−βW /2 ξi eβW /2 .
An immediate consequence of Proposition 12.1.3 is that the operator X (C) ({ul }) is self-adjoint with respect to (12.12). Now we know that the nonsymmetric Jack polynomials Eη are the polynomial eigenfunctions of X (C) ({ul }), and that the corresponding eigenvalue is distinct. Thus we can apply the elementary theorem, stating that eigenfunctions of a Hermitian (self-adjoint) operator with distinct eigenvalues are orthogonal, to conclude the following result. P ROPOSITION 12.1.4 For compositions η = ν, Eη |Eν (C) = 0. We note that Proposition 12.1.4 together with the structure (12.7) implies that it is possible to construct {Eη } via the Gram-Schmidt orthogonalization procedure from {z η } with the partial order ≺. Because {z η } is a basis for analytic functions in N variables, it follows that so is {Eη }. 12.1.3 Orthogonality and a product expansion The pairing inner product Eη , Eν q := uη δη,ν ,
(12.13)
for suitable normalization uη , is of importance in the subsequent development of the general theory. Motivating the definition of (12.13) and in particular the choice of uη is the expansion of the function Ω(x, y) :=
N
N 1 1 1/α (1 − x y ) (1 − x i i j yk ) i=1 j,k=1
(12.14)
547
JACK POLYNOMIALS
in a basis of {Eη (x)} [476]. P ROPOSITION 12.1.5 For some coefficients uη = 0 we have 1 Eη (x)Eη (y). Ω(x, y) = uη η
(12.15)
Proof. Since {Eη } is a basis for analytic functions of N variables and Ω(αx, y/α) = Ω(x, y) we must have Ω=
X
Eη (x)Uη (y)
(12.16)
η (x)
(y)
for some homogeneous polynomials Uη (y) of degree |η|. With ξi (ξi ) denoting that ξi is written in terms of x1 , . . . , xN (y1 , . . . , yN ), we will show (x) (y) ξi Ω = ξi Ω (i = 1, . . . , N ). (12.17) (y)
Now, using (12.17) in (12.16) gives ξi Uη (y) = η¯i Uη (y) for each i = 1, . . . , N , so we conclude that Uη (y) is a multiple of Eη (y), and thus (12.15) holds. (x) It remains to derive (12.17). Since Ω is symmetric under interchange of xi and yi it suffices to show that Ω1 ξi Ω is symmetric under interchange of xi and yi . Now, from the definitions X 1 ∂ 1 αxi yi αxi Ω= −N + , Ω ∂xi 1 − x i yi 1 − x i yj j=1 N
which allows the explicit form of symmetry property.
1 (x) ξ Ω Ω i
1 1 − sij yi − yj , Ω= Ω xi − xj (1 − xi yj )(1 − xj yi )
to be written down. Straightforward manipulation then exhibits the sought
With the uη occurring in (12.15) being chosen as the uη in (12.13), another basis fundamental to the nonsymmetric Jack polynomials can be identified [139]. D EFINITION 12.1.6 With Ω(x, y) as specified in (12.14), for each composition η define a polynomial qη homogeneous of degree |η| by qη (x)y η . (12.18) Ω(x, y) = η ηN The monomial y η by definition has the factorization y1η1 · · · yN . Similarly, defining qk (xi ; x) by the expansion N ∞ 1 1 = qk (xi ; x)yik , (1 − xi yi ) j=1 (1 − xj yi )1/α
(12.19)
qη (x) = qη1 (x1 ; x)qη2 (x2 ; x) · · · qηN (xN ; x).
(12.20)
k=0
we see that
Note from (12.19) that the simplest nontrivial example of the elementary polynomials qk (xi ; x) is 1 xj . α j=1 N
q1 (xi ; x) = xi + (x)
(12.21) (x)
We saw in Proposition 12.1.2 that ξj xη gives a series in {xν } with ν η. It turns out that ξj qη (x) gives a series in {qν (x)} with η ν, implying an upper triangular structure, whereas the former structure is
548
CHAPTER 12
lower triangular. To demonstrate this, we note from the definition (12.18) that (x) (x) ξi Ω(x, y) = (ξi qη (x))y η .
(12.22)
η
But making use of (12.17) in (12.18) we also have (x)
ξi Ω(x, y) =
(y)
qη (x)ξi y η .
η
Substituting (12.3) in this and rearranging the sum over η to be of the form in (12.22) we deduce the following analogue of Proposition 12.1.2, which exhibits the claimed triangular structure. P ROPOSITION 12.1.7 We have (x) sgn(ηi − ηp ) qη∗ (p,i,r) + sgn(ηi − ηp ) qη∗ (i,p,r) , ξi qη (x) = η¯i qη (x) + r
p
where η ∗ (j, k, r) =
(12.23)
r
p>i
(η1 , . . . , ηj − r, . . . , ηk + r, . . . , ηN ), 1 ≤ r ≤ ηk − ηj − 1, ηk > ηj , (η1 , . . . , ηj + r, . . . , ηk − r, . . . , ηN ), 1 ≤ r ≤ ηj − ηk , ηj > ηk ,
and qη∗ (j,k,r) = 0 if η ∗ contains a negative part. The largest composition with respect to the ordering ≺ and such that |η| = p is η # := (p, 0, . . . , 0). (x) For this composition (12.23) gives ξi qη# (x) = η¯i qη# (x) for each i = 1, . . . , N so qη# (x) ∝ Eη# (x). For general η we can make use of the upper triangular structure and the operator (12.5) to systematically (x) construct simultaneous eigenfunctions of the ξi , which must have the same upper triangular structure. On the other hand we know these eigenfunctions to be the Jack polynomials. Thus we obtain the analogue of (12.7) expressing the Jack polynomials as a series in {qν (x)}, a ˜ην qν (x), (12.24) Eη (x) = cη qη (x) + η≺ν
for some cη . We will show subsequently that cη is precisely the normalization uη in (12.13). Let us now proceed to relate {qη } to the pairing inner product (12.13). P ROPOSITION 12.1.8 Let {vη } and {wη } be sets of homogeneous multivariable polynomials such that vη (x)wη (y) (12.25) Ω(x, y) = η
with Ω(x, y) specified by (12.14) and suppose that {wη (y)} is a basis. Then {vη (x)} is a basis and furthermore vη , wν q = δη,ν .
(12.26)
Proof. Since {Eγ } is a basis we can write vη =
X
aηγ Eγ /uγ
and
wν =
γ
X
bνγ Eγ .
γ
Then from the definition (12.13) we have vη , wν q =
X
aηγ bνγ .
(12.27)
γ
On the other hand, equating (12.15) and (12.25) with the above expansions substituted in the latter implies X aγη bγν = δη,ν . γ
(12.28)
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JACK POLYNOMIALS
Consider compositions of modulus p, and with a total order on such compositions define A = [aγη ], B = [bγν ]. Then (12.28) is equivalent to the matrix equation AT B = 1. Since {vη } is a basis the matrix A is invertible so it follows from this that B = (AT )−1 ; (12.29) this in turn tells us that B is invertible and thus {wν } is a basis. Also, in this notation the r.h.s. of (12.27) reads [ABT ]ην . According to (12.29) this is equal to [1]ην and (12.26) follows.
With vη (x) = qη (x) and wη (y) = y η all the assumptions of Proposition 12.1.8 are met, so we conclude qη , xν q = δη,ν .
(12.30)
This relation may well be taken as the definition of ·, ·q , with (12.13) then a corollary, although then one has to work harder to first establish that {qη } form a basis [139]. As a consequence of (12.30) the quantity cη in (12.24) can be determined. Thus substituting (12.24) for Eη and (12.7) for Eν in (12.13) it follows from (12.30) that cη = uη . Hence (12.24) can be written a ˜ην qν (x). (12.31) Eη (x) = uη qη (x) + η≺ν
We would like to express the quantity uη occurring in (12.13), (12.15) and now (12.31) as an explicit function of the parts of η. However, this is not possible until some further theory, introduced in the next section, is developed. p 1. Show that z p Eη (z) = Eη+pN (z), where z p := z1p · · · zN and
E XERCISES 12.1
η + pN := (η1 + p, η2 + p, . . . , ηN + p). (It is assumed that p ≥ − minj ηj ; however, more generally this equation can be used to give meaning to Eη in the case that η has negative parts.) 2. Show that Eη
“1” z
= E−ηR (z R ),
where z R := (zN , zN−1 , . . . , z1 ),
η R := (ηN , ηN−1 , . . . , η1 )
(12.32)
and E−ηR is interpreted as remarked in q.1. (Hint: Make the change of variables zi → 1/zi in the definition of ξi , and relate the resulting operator to ξ |z→z R .) (N)
(N−1)
(n)
, where ξi = ξi 3. Show from Definition 11.4.3 that ξi |zN =0 = ξi definition of ξi are z1 , . . . , zn . Use this to deduce that for ηN = 0,
indicates that the variables in the
Eη (z1 , . . . , zN−1 , 0) = Eη (z1 , . . . , zN−1 ). 4. Let κ denote a partition, and let κ denote its conjugate (recall Definition 10.1.3)κ Show that N X
(i − 1)κi =
i=1
5.
N X
κi (κi − 1)/2.
(12.33)
i=1
(i) Verify that j (si η)i =
η¯i+1 , η¯i ,
ηi =
ηi+1 , ηi = ηi+1 ,
j (si η)i+1 =
η¯i , η¯i+1 ,
ηi =
ηi+1 , ηi = ηi+1 ,
(si η)j = η¯j
(j = i, i + 1)
to deduce that {¯ ηj }j=1,...,N = {(P η)j }j=1,...,N , where P η denotes some permutation of η. (ii) Deduce from (i) that the eigenvalue of X (C) (u) := X (C) ({uj })|uj =u corresponding to the eigenfunction
550
CHAPTER 12
Eη (z) is the same for all compositions η giving rise to the same partition η + , and has the explicit form N Y
(1 + u(η + )j ) =
j=1
N Y
(1 + u(αηj+ − (j − 1))).
j=1
˜ (C,Ex) , with the eigenvalue corresponding to Eη (z) (iii) Show that the same features hold true of the operator H being given by e(η + ; α) := 2b((η + ) ) +
“N − 1 ” 2 + 1 |η| − b(η + ), α α
b(κ) :=
N X
(j − 1)κj ,
(12.34)
j=1
where use has been made of the formula of (12.33).
12.2 RECURRENCE RELATIONS 12.2.1 Elementary recurrences It turns out that all the nonsymmetric Jack polynomials can, in principle, be calculated recursively by just two elementary operations. The first is the elementary transposition si := si i+1 , which interchanges the ith and (i + 1)th coordinates [436]. P ROPOSITION 12.2.1 Let δ¯i,η := η¯i − η¯i+1 . We have ⎧ 1 1 ⎪ Esi η (z), ηi > ηi+1 E (z) + 1 − 2 ⎪ δ¯i,η ⎨ δ¯i,η η si Eη (z) = Eη (z), ηi = ηi+1 ⎪ ⎪ ⎩ 1 E (z) + Esi η (z), ηi < ηi+1 δ¯i,η η where si acts on compositions by interchanging the ith and (i + 1)th part. Proof. The derivation requires the three relations (11.89). Let E := δ¯i,η si Eη − Eη . Then making use of the first of the relations in (11.89) shows ξi E = η¯i+1 E, the second relation gives ξi+1 E = η¯i E and the final relation gives ξj E = η¯j E (j = i, i + 1). Hence E must be proportional to Esi η . For ηi < ηi+1 it follows from the structure (12.7) that the −1 , thus implying the stated formula in these coefficient of z si η in E is δ¯i,η , while for ηi = ηi+1 the coefficient is −δ¯i,η cases. The stated formula for ηi > ηi+1 follows from the formula for ηi < ηi+1 by replacing η by si η, acting on both sides with si and then once again using the formula in its original form.
The second elementary operation is specified by a raising-type operator Φ [359]. D EFINITION 12.2.2 When acting on functions define Φ := zN sN −1 sN −2 · · · s1 so that Φf (z1 , . . . , zN ) = zN f (zN , z1 , . . . , zN −1 ). Also define Φ to act on compositions by Φη := (η2 , . . . , ηN , η1 + 1). P ROPOSITION 12.2.3 The operator Φ obeys the relations ξi Φ = Φξi+1
(i = 1, . . . , N − 1),
ξN Φ = Φ(ξ1 + α)
551
JACK POLYNOMIALS
η 0N
Eη (z) 1
0N −1 1 0N −2 10 0N −3 102 .. . N −1 10 0N −2 12 0N −3 101 0N −4 102 1 .. . 10N −2 1 0N −1 2
zN 1 zN −1 + α+N −1 zN 1 zN −2 + α+N −2 (zN −1 + zN ) .. . 1 z1 + α+1 (z2 + z3 + · · · + zN ) zN zN −1 1 zN (zN −2 + α+N −1 zN −1 ) 1 zN (zN −3 + α+N −2 (zN −2 + zN −1 )) .. . 1 zN (z1 + α+2 (z2 + · · · + zN −1 )) 1 zN (zN + α+1 (z1 + · · · + zN −1 ))
Table 12.1 Recursive generation of some nonsymmetric Jack polynomials
and consequently ΦEη (z) = EΦη (z). Proof. The first relation follows from the facts that ∂ ∂ Φ = Φzi+1 , ∂zi ∂zi+1 zi zi+1 (1 − sip )Φ = Φ (1 − si+1 p+1 ), zi − zp zi+1 − zp+1 zp zp+1 (1 − sip )Φ = Φ (1 − si+1 p+1 ), zi − zp zp+1 − zi+1 zN z1 (1 − siN )Φ = Φ (1 − si+1 1 ) zi − zN zi+1 − z1 zi
p = N,
and the definition of ξi , while the second relation is a consequence of the same equations with the first replaced by zN
∂ ∂ Φ = Φz1 + Φ. ∂zN ∂z1
To derive the final statement, we note from the first two relations that ξi (ΦEη ) = η¯i+1 (ΦEη ) (i = 1, . . . , N − 1),
ξN (ΦEη ) = (¯ η1 + α)(ΦEη ),
which imply ΦEη is proportional to EΦη . That the proportionality constant is unity follows from comparison of the term proportional to z Φη on both sides.
The recursive generation of some particular Eη obtained using the above formulas is given in Table 12.1. Note that only the case ηi < ηi+1 of the action of si is required. ˆ which is the counterpart to the raising-type operator Φ Finally we will discuss a lowering-type operator Φ [39].
552
CHAPTER 12
ˆ when acting on functions of z1 , . . . , zN by D EFINITION 12.2.4 Define Φ ˆ = d1 s1 s2 · · · sN −1 = s1 · · · si−1 di si · · · sN −1 , Φ where di denotes the type A Dunkl operator, and when acting on compositions by ˆ = (ηN − 1, η1 , . . . , ηN −1 ). Φη ˆ A result analogous to Proposition 12.2.3 for Φ holds for Φ. ˆ obeys the relations P ROPOSITION 12.2.5 The operator Φ ˆ = Φξ ˆ i−1 (i = 2, . . . , N ) ξi Φ
ˆ = Φ(ξ ˆ N − α) ξ1 Φ
and consequently d ˆ η = η¯N + N − 1 E ˆ = 1 η E ˆ . ΦE Φη Φη α α dφη ˆ Proof. Using the formulas of (11.89) we see that ξj s1 s2 · · · sN−1 = s1 s2 · · · sj−2 ξj sj−1 sj · · · sN−1 = s1 s2 · · · sj−2 (sj−1 ξj−1 − 1) sj · · · sN−1 = s1 s2 · · · sN−1 ξj−1 − s1 s2 · · · sj−1 sj · · · sN−1 , which together with Proposition 11.5.2 gives ˆ = ξj d1 s1 s2 · · · sN−1 = d1 (ξj + s1j )s1 s2 · · · sN−1 ξj Φ = d1 (s1 s2 · · · sN−1 ξj−1 − s1 s2 · · · sj−2 sj · · · sN−1 + s1j s1 s2 · · · sN−1 ) . Since the last two terms cancel, the first relation follows. For the second relation, use of the formulas of (11.89) shows ξ1 s1 s2 · · · sN−1 = s1 s2 · · · sN−1 ξN +
N−1 X
s1 · · · sj−1 sj+1 · · · sN−1 ,
j=1
and this together with Proposition 11.5.2 gives ! X ˆ s1p s1 s2 · · · sN−1 ξ 1 Φ = d1 ξ 1 − α − p>1
= d1
s1 s2 · · · sN−1 (ξN − α) +
N−1 X
s1 · · · sj−1 sj+1 · · · sN−1 −
j=1
X
! s1p s1 s2 · · · sN−1
p>1
ˆ N − α), = Φ(ξ where the last equality follows after noting that the last two terms in the line above cancel. ˆ η is a constant multiple of E ˆ . To determine this constant, note that the The two relations together imply that ΦE Φη ˆ Φη ˆ η , writing Φ ˆ = s1 · · · sN−1 dN shows that the leading term in E ˆ is z . To determine the coefficient of this term in ΦE Φη
N −1 ηN −1 zN in dN Eη . Recalling that coefficient is equal to that of z1η1 · · · zN−1
η
ξN = αzN dN + 1 − N
(12.35)
then gives the coefficient as (¯ ηN + N − 1)/α as required.
As remarked, the operators si and Φ generate the Eη recursively. For some scalar operators A, it is possible to compute A(Esi η ) and A(EΦη ) and so then to compute A(Eη ) recursively. In the next section three
553
JACK POLYNOMIALS
examples of this procedure will be given: the first will provide the evaluation of Eη ((1)N ) (the notation (1)N means z1 = · · · = zN = 1) [476]; in the second the normalization of the Eη with respect to the inner product (12.12) will be established [39]; in the third the normalization uη occurring in (12.13), (12.15) and (12.31) will be evaluated [144].
12.3 APPLICATION OF THE RECURRENCES 12.3.1 The specialization formula The evaluation of Eη ((1)N ) is in terms of the quantities α(a(i, j) + 1) + l(i, j) + 1 , eη := α(a (i, j) + 1) + N − l (i, j) . (12.36) dη := (i,j)∈η
(i,j)∈η
For completeness we also list at this stage similar quantities which are required subsequently: α(a(i, j) + 1) + l(i, j) , eη := α(a (i, j) + 1) + N − 1 − l (i, j) . dη := (i,j)∈η
(12.37)
(i,j)∈η
Here the notation (i, j) ∈ η refers to the diagram of the composition η, in which each part ηi becomes the nodes (i, j), 1 ≤ j ≤ ηi on a square lattice labeled as is conventional for a matrix (cf. Definition 10.1.3) The quantities a(i, j), a (i, j), l(i, j) and l (i, j) are specified by the following definition [476]. D EFINITION 12.3.1 Referring to the diagram of a composition η, let a(i, j) := ηi − j denote its arm length at node (i, j), and let a (i, j) := j − 1 denote its coarm length. Also define the leg length l(i, j) = ul(i, j) + ll(i, j) and coleg length l (i, j) = ul (i, j) + ll (i, j), by setting ll(i, j) = #{k > i|j ≤ ηk ≤ ηi }, ul(i, j) = #{k < i|j ≤ ηk + 1 ≤ ηi } ll (i, j) = #{k > i|ηk > ηi }, ul (i, j) = #{k < i|ηk ≥ ηi } (note that ll (i, j) and ul (i, j) are independent of j). Use will be made of the quantities (12.36) and (12.37) with η replaced by si η and Φη. In particular, we require the readily verified formulas [476] ⎧ ¯ ⎧ ¯ +1 ⎨ δi,η ⎨ ¯ δi,η , ηi > ηi+1 , ¯i,η , ηi > ηi+1 , d d δ δi,η −1 s s η η i i esi η = eη , esi η = eη , = = δ¯i,η +1 ⎩ δ¯i,η , η < η , ⎩ dη d η , ηi < ηi+1 , i i+1 δ¯i,η −1 δ¯i,η dΦη eΦη = = η¯1 + α + N, dη eη
dΦη eΦη = = η¯1 + α + N − 1, dη eη
dη = η¯N + N − 1, dΦη ˆ
dη = η¯N + N. dΦη ˆ
(12.38)
eη . dη
(12.39)
P ROPOSITION 12.3.2 We have Eη ((1)N ) =
Proof. Starting with η = 0N , all η can be generated from the action of Φ and si on η, and furthermore the latter can be restricted to the case ηi < ηi+1 . Thus it suffices to show EΦη ((1)N ) and Esi η ((1)N ) (the latter with ηi < ηi+1 ) are given correctly by (12.39), which is certainly true for η = 0N . Now according to Proposition 12.2.5, EΦη ((1)N ) =
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CHAPTER 12
ΦEη ((1)N ) = Eη ((1)N ). On the other hand, from (12.38) we also have eΦη /dΦη = eη /dη , which is in agreement with (12.39). Furthermore, according to Proposition 12.2.1, Esi η ((1)N ) = while (12.38) gives
δ¯i,η − 1 Eη ((1)N ), δ¯i,η
ηi < ηi+1
δ¯i,η − 1 eη esi η = ¯ , dsi η δi,η dη
again in agreement with (12.39).
12.3.2 The normalization integral Next we will turn our attention to the evaluation of the normalization integral L/2 L/2 dx1 · · · dxN |zk − zj |2/α Eη (¯ z )Eη (z), Nη(C) := Eη |Eη (C) := −L/2
−L/2
(12.40)
1≤j
where zj = e2πixj /L . P ROPOSITION 12.3.3 We have (C)
Nη
(C)
N0N
=
dη eη . dη eη
(12.41)
Proof. As in the proof of Proposition 12.3.2, it suffices to show EΦη |EΦη (C) and Esi η |Esi η (C) are correctly given by (12.41). Since the transpositions si are self-adjoint with respect to the inner product (12.12), we see from Definition 12.2.2 that in general Φf |Φg(C) = f |g(C) , so Φ is an isometry. Using the final equation of Proposition 12.2.5 we thus have EΦη |EΦη (C) = ΦEη |ΦEη (C) = Eη |Eη (C) . (C)
Using (12.38) shows that the r.h.s. of (12.41) satisfies the same recurrence. Consider next Nη −1 Eη , so we have In the case ηi < ηi+1 , Proposition 12.2.1 gives Esi η = si Eη − δ¯i,η
with η replaced by si η.
−1 −1 )Eη |(si − δ¯i,η )Eη (C) Esi η |Esi η (C) = (si − δ¯i,η −2 −1 = (1 + δ¯i,η )Eη |Eη (C) − 2δ¯i,η Eη |si Eη (C) −2 = (1 − δ¯i,η )Eη |Eη (C) ,
(12.42)
where the final equality follows by substituting for si Eη according to Proposition 12.2.1 in the previous line, and using the fact that for ηi = ηi+1 , Eη and Esi η are orthogonal. On the other hand (12.38) shows that the r.h.s. of (12.41) satisfies the same recurrences under the action of si .
12.3.3 The normalization uη To calculate the normalization uη in (12.13), a strategy similar to that used in the proof of Proposition 12.3.3 ˆ rather than the raising type operator Φ can be adopted, except that direct use of the lowering type operator Φ is made. P ROPOSITION 12.3.4 We have uη =
dη . dη
(12.43)
Proof. Again we seek to express Esi η , Esi η q and EΦη , EΦη q in terms of Eη , Eη q and show that the equations are consistent with (12.43). Equation (12.42) makes use of the orthogonality of {Eν } and self-adjointness of si , but no
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JACK POLYNOMIALS
other detail of the inner product, so again we have −2 Esi η , Esi η q = (1 − δ¯i,η )Eη , Eη q .
(12.44)
This is indeed consistent with (12.43). For the second sought recurrence, following [144], note from (12.30) and (12.20) that in general q1 (xi ; x)f, xi gq = f, gq = σf, σgq , ˆ we thus where q1 (xi ; x) is given by (12.21) and σ is any permutation of N symbols. Recalling Definition 12.2.4 of Φ have 1 (¯ ηN − 1 + N )q1 (xN ; x)dN Eη , Eη q , α (12.45) where the final equality follows by making use of (12.35). To proceed further, we note from Definition 11.4.1 of dN and the explicit form (12.21) of q1 (xN ; x) that EΦη ˆ , EΦη ˆ q = dN Eη , dN Eη q = q1 (xN ; x)dN Eη , xN dN Eη q =
dN q1 (xN ; x) = dN xN +
1 . α
From this and (12.35) we see that dN q1 (xN ; x)dN Eη = Thus q1 (xN ; x)dN Eη −
1 (¯ ηN α
1 (¯ ηN + N )dN Eη . α
+ N )Eη is annihilated by dN , and consequently from (12.23)
qN (xN ; x)dN Eη =
1 (¯ ηN + N )Eη + α
X
cην qν .
ν:|ν|=|η|, νN =0
Substituting this in (12.45) and using the structure formula (12.24), together with the orthogonality (12.30), show that for ηN > 0 1 dη dη ˆ η q = 1 (¯ ˆ η , ΦE ηN + N )(¯ ηN − 1 + N )Eη , Eη q = 2 Eη , Eη q , ΦE 2 α α dΦη ˆ ˆ dΦη ˆ η , then replacing where the second equality follows from (12.38). Substituting the result of Proposition 12.2.5 for the ΦE η by Φη (this is always possible since we are requiring ηN > 0) shows EΦη , EΦη q =
dη dΦη Eη , Eη q , dΦη dη
which like (12.44) is consistent with (12.43).
12.4 A GENERALIZED BINOMIAL THEOREM AND AN INTEGRATION FORMULA 12.4.1 The generalized binomial theorem The classical binomial theorem refers to the power series expansion ∞ 1 (r)n n x , = r (1 − x) n! n=0
|x| < 1,
N where (r)n is specified as in (5.83). Here the binomial theorem will be generalized so as to express j=1 (1 − xj )−r as a series in {Eη }. This will be done by making use of the expansion formula (12.15) as well as the
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specialization formula (12.39) [39]. Replacing the quantity (r)n will be the generalized Pochhammer symbol (α)
[u]η+ :=
N Γ(u − α1 (j − 1) + ηj+ )
Γ(u − α1 (j − 1))
j=1
.
(12.46)
Note from the definitions (12.36) and (12.37) that eη and eη are expressible in terms of the generalized Pochhammer symbol. Thus eη = α|η| [1 + N/α]η+ ,
eη = α|η| [1 + (N − 1)/α]η+ .
(α)
(α)
(12.47)
P ROPOSITION 12.4.1 With uη as in (12.15), and thus given explicitly by (12.43), we have (α)
(α)
α|η| [r]η+ α|η| [r]η+ 1 = E (x) = Eη (x). η (1 − xj )r uη dη dη η η
N j=1
(12.48)
Proof. We begin by replacing N by kN , kN ∈ Z+ in (12.15), then setting y1 , . . . , ykN equal to 1 and xN+1 , . . . , xkN equal to 0. Noting that uη as defined below (12.15) and dη as defined by (12.36) are independent of N , and making use of the formula (12.47) for eη gives the formula N Y j=1
˛ X eη ˛ X α|η| [1 + kN/α](α) 1 η+ N→kN = Eη (x) = Eη (x). u u (1 − xj )kN/α+1 η dη η dη η η
Now, from the definition (12.46) we see that each coefficient of Eη (x) is a polynomial in kN/α. Also, expanding the l.h.s. as a power series gives that the same must be true here too. It follows that since both sides are equal for each kN ∈ Z+ , they must in fact be equal for all (complex) values kN/α =: r − 1.
12.4.2 A generalization of the Morris constant term identity and the Selberg integral Since {Eη } is an orthogonal basis for analytic functions f with respect to the scalar product ·|·(C) , it follows that a general analytic function f has the expansion f (x1 , . . . , xN ) =
Eη |f (C) (C)
η
Nη
Eη (x).
(12.49)
N Setting f = j=1 (1 − xj )−r and comparing with (12.48) shows that we can evaluate the inner product Eη |f (C) in this case. In fact manipulation of the inner product provides us with a generalization of the Morris constant term identity (4.4) [39]. . P ROPOSITION 12.4.2 Let Δ(x) := j=k (1 − xj /xk )1/α . For a, b, 1/α ∈ Z≥0 we have N 1 b a (α) . (1 − x ) (1 − ) E (x) Δ(x) CT i η [−b]η+ i=1 xi N = Eη ((1) ) . (12.50) (α) N 1 b. a [1 + a + (N − 1)/α]η+ CT i=1 (1 − xi ) (1 − x ) Δ(x) i
Proof. First note that for (Laurent) polynomials f and g and 1/α ∈ Z≥0 we can write (12.12) as „ “ « 1 1 ” e f |g(C) = LN CT f ,..., . g(x1 , . . . , xN )Δ(x) x1 xN Thus for r ∈ Z≤0 , comparison of (12.49) with f = CT
N Y i=1
QN
j=1 (1
! (1 −
−r e x−1 Eη (x)Δ(x) i )
− xj )−r and (12.48) give α|η| [r]η+
(α)
=
dη
“ ” e CT Eη (x−1 )Eη (x)Δ(x) .
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JACK POLYNOMIALS
Now make the replacement η → η + a and rewrite Eη+a according to the formula of Exercises 12.1 q.1. Also, set r = −a − b and note “ “ 1 ”a+b 1 ”b = (−1)a (1 − xi )a 1 − , xai 1 − xi xi to conclude that ! N “ “ ” Y 1 ”b 1 (α) a e e CT (1 − xi ) 1 − Eη (x)Δ(x) = (−1)aN α|η| [−a − b]η+ +aN CT Eη (x−1 )Eη (x)Δ(x) . x d i η+aN i=1 (12.51) Expressing the l.h.s. of this in the form of the l.h.s. of (12.50) we have “Q ” N a 1 b (α) e CT α|η| [−a − b]η+ +aN daN Nη(C) i=1 (1 − xi ) (1 − xi ) Eη (x)Δ(x) ” “Q , (12.52) = (α) (C) N 1 be a [−a − b]aN dη+aN N0N CT i=1 (1 − xi ) (1 − x ) Δ(x) i
(C) Nη
is defined by (12.40) and the ratio where we first note from the definition (12.46) that
(C) (C) Nη /N0N
evaluated by (12.41). To simplify the r.h.s. of this expression,
(α)
[−a − b]η+ +aN (α)
[−a − b]aN
(α)
= [−b]η+ .
(12.53)
To deduce the dependence of dη+aN on a we note from the formula of Exercises 12.1 q.1 and (12.39) that eη+aN dη+aN = . dη eη (C)
(C)
The formula of Exercises 12.1 q.1 also tells us that Nη+aN /Nη dη+aN dη
=
eη+aN eη
and thus
= αaN
= 1, so making use of (12.54) in (12.41) shows
N [1 + a + (N − 1)/α]η+ Y Γ(1 + a + (j − 1)/α) [1 + (N − 1)/α]η+ j=1 Γ(1 + (j − 1)/α)
dη+aN daN
(12.54)
=
[1 + a + (N − 1)/α]η+ dη . [1 + (N − 1)/α]η+
(12.55)
Substituting (12.53) and (12.55) in (12.52) and making further use of (12.47) and (12.41) give the sought result.
With the notation (4.4), (12.50) can be rewritten as the trigonometric integral 1/2 1/2 N 1 dx1 · · · dxN eπixl (a−b) |1 + e2πixl |a+b Eη (−e2πix1 , . . . , −e2πixN ) MN (a, b; 1/α) −1/2 −1/2 l=1
×
1≤j
(α)
|e
2πixk
−e
|
2πixj 2/α
N
= Eη ((1) )
[−b]η+
(α)
[1 + a + (N − 1)/α]η+
.
(12.56)
In this form the restriction to a, b, 1/α ∈ Z≥0 required in (12.50) can be lifted. Furthermore, Proposition
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CHAPTER 12
3.9.1 can be used to convert (12.56) to the generalized Selberg integral 1 1 N 1 dt1 · · · dtN tλl 1 (1 − tl )λ2 Eη (t1 , . . . , tN ) SN (λ1 , λ2 , 1/α) 0 0 l=1
= Eη ((1)N )
[λ1 + (N − 1)/α +
(α) 1]η+ (α)
[λ1 + λ2 + 2(N − 1)/α + 2]η+
2/α
|tj − tk |
1≤j
.
(12.57)
1. Argue that Eη ((−1)N ) = (−1)|η| Eη ((1)N ) is the limiting value of the l.h.s. of (12.56) for a = b and a → ∞. Show that this is consistent with the asymptotic form of the r.h.s.
E XERCISES 12.4
2. Let κ denote the partition conjugate to κ (recall Definition 10.1.3). Show that = (−α)−|κ| [−αa]κ [a](α) κ
(1/α)
.
3. [342] Let κ denote a partition. Define dκ as in (12.37) and set ” Y “ hκ := αa(i, j) + l(i, j) + 1 .
(12.58)
(i,j)∈κ
In this exercise the formulas α|κ| [N/α]κ
(α)
hκ =
dκ =
1/α fN (κ)
α|κ| [(N − 1)/α + 1]κ , 1/α f¯ (κ)
fnλ (κ) :=
,
Y 1≤i<j≤n
(α)
f¯nλ (κ) :=
N
Y 1≤i<j≤n
((j − i)λ + κi − κj )λ , ((j − i)λ)λ
(1 − λ + (j − i)λ + κi − κj )λ , (1 − λ + (j − i)λ)λ
(12.59)
(12.60)
will be established by induction. Consider for definiteness the first identity. Note that it is true for κ = 0N . (i) Assume the identity is true for a particular partition κ = (κ1 , . . . , κN ). Verify that the identity is then true for the partition κ(i) := (κ1 , . . . , κi−1 , κi + 1, κi+1 , . . . , κN ) , provided κi N Y κj − i + 1 + (κi + 1 − j)α Y (j − i)/α + κi − κj + 1/α = N − (i − 1) + κi α. κj − i + 1 + (κi − j)α j=i+1 (j − i)/α + κi − κj j=1
(ii) Verify the identity in (i) by defining i ≤ j1 < j2 · · · < ji ≤ N such that κi+1 , . . . , κj1 = κi ,
κj1 +1 , . . . , κj2 = κi − 1,
...
, κji +1 , . . . , κN = 0,
and expressing κj in terms of j1 , . . . , ji .
12.5 INTERPOLATION NONSYMMETRIC JACK POLYNOMIALS 12.5.1 Definitions and fundamental properties Of importance in the development of aspects of the theory of nonsymmetric Jack polynomials which will arise later, in particular generalized binomial coefficients, and also in the derivation of Pieri type formulas, is the notion of an interpolation nonsymmetric Jack polynomial (also referred to as a shifted nonsymmetric Jack polynomial) [432], [358], [477]. In one variable the Jack polynomials are simply the monomials {xp }. The interpolation Jack polynomials in one variable generalize the monomial xp to the polynomial (−1)p (−x)p = x(x − 1) · · · (x − p + 1).
(12.61)
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JACK POLYNOMIALS
This can be characterized by being the unique (up to normalization) polynomial of degree at most p which vanishes at x = 0, 1, . . . , p − 1 and is nonzero at x = p. Likewise, in the N -variable case the interpolation nonsymmetric Jack polynomials are defined by a vanishing condition, which is based on the following result [358], [477]. P ROPOSITION 12.5.1 For a given composition ν, define ν¯j by (12.2), and let {F (¯ ν )} be a given set of complex numbers indexed by ν¯. There exists a unique polynomial f in N variables of degree at most d such that for all |ν| ≤ d f (¯ ν1 /α, . . . , ν¯N /α) = F (¯ ν1 , . . . , ν¯N ).
(12.62)
Proof. Since for each distinct ν there is a distinct value of ν¯, (12.62) gives the same number of linear equations for the coefficients aν in the expansion X aν xν f (x1 , . . . , xN ) = ν:|ν|≤d
as there are distinct coefficients. Hence if a solution to the equations exists, then it must be unique. To show existence, we will construct f of the form “ N − 1” (12.63) f (x1 , . . . , xN ) = g(x1 , . . . , xN−1 ) + xN + h(xN − 1, x1 , . . . , xN−1 ) α by induction on N and d. The validity for N = 1, d arbitrary follows from the fundamental theorem of algebra, while in the case d = 0, N arbitrary, there is nothing to prove. Consider first the set of compositions such that |ν| ≤ d and with νN = 0. From (12.2) we then have ν¯N = −(N − 1) and so f (x1 , . . . , xN−1 , ν¯N /α) = g(x1 , . . . , xN−1 ).
(12.64)
Since g depends only on the variables x1 , . . . , xN−1 , the induction hypothesis gives that with xj = ν¯j /α, f can take on prescribed values. Consider next the cases that νN = 0. We want to find a function h such that ν1 /α, . . . , ν¯N−1 /α) = f (¯ ν1 /α, . . . , ν¯N /α) − g(¯
1 νN /α − 1, ν¯1 /α, . . . , ν¯N−1 /α) (¯ νN + N − 1)h(¯ α
ˆ (recall Definition and h can take on prescribed values at these points. The essential point here is that the composition Φν ˆ 1 = ν¯N − α, (Φν) ˆ j = ν¯j−1 , j ≥ 1. Since ν¯N + (N − 1) = 0 for 12.2.2) which has degree |ν| − 1 is such that (Φν) ˆ νN = 0, it follows by induction in d that there is a polynomial h which takes on prescribed values at Φν/α.
It follows from Proposition 12.5.1 that there is a unique polynomial (up to normalization), Eη∗ (x), say, of degree ≤ |η|, which has the vanishing property Eη∗ (¯ ν /α) := Eη∗ (¯ ν1 /α, . . . , ν¯N /α) = 0,
|ν| ≤ |η| (ν = η),
(12.65)
while η /α) = 0. Eη∗ (¯
(12.66)
In the next proposition it will be shown that the coefficient of xη in Eη∗ (x) is nonzero. Choosing the coefficient to be unity gives Eη∗ (x) = xη + aην xν . (12.67) |ν|≤|η| ν=η
Note in particular that {Eη∗ (x)} forms a basis for analytic functions. With this normalization, the polynomial Eη∗ (x) is referred to as the interpolation nonsymmetric Jack polynomial (its relationship to the nonsymmetric Jack polynomial will become evident as the theory is developed).
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P ROPOSITION 12.5.2 The polynomial Eη∗ contains the monomial xη with a nonzero coefficient. Furthermore, with Δ(N ) f (x1 , . . . , xN ) := f (xN − 1, x1 , . . . , xN −1 ) we have
N − 1 (N ) ∗ ∗ Δ Eη (x) = EΦη (x), xN + α where Φη := (η2 , . . . , ηN , η1 + 1) as in Definition 12.2.2.
(12.68)
Proof. Analogous to the decomposition (12.63) write “ N − 1” Eη∗ (x1 , . . . , xN ) = g(x1 , . . . , xN−1 ) + xN + h(xN − 1, x1 , . . . , xN−1 ). α
(12.69)
For the first statement we will proceed by induction on N and |η|. In the case ηN = 0, (12.64) gives g(x1 , . . . , xN−1 ) = Eη∗# (x1 , . . . , xN−1 , η¯N /α) ∝ Eη∗# (x1 , . . . , xN−1 ),
(12.70)
where η# := (η1 , . . . , ηN−1 ) and the validity of the final proportionality follows from (12.65) and (12.66). Thus g contains xη# by the induction hypothesis in N . For ηN = 0, from the proof of Proposition 12.5.1 we can take g = 0 and ∗ h(x1 , . . . , xN ) = EΦη ˆ (x1 , . . . , xN ).
(12.71)
ˆ
∗ Φη by the induction hypothesis in |η|, and so it follows from (12.69) that Eη∗ contains xη . But EΦη ˆ contains x To derive (12.68), substitute (12.71) in (12.69) with g = 0 and replace η by Φη.
A consequence of (12.68) is an evaluation formula for Eη∗ (x) at the special point x = η¯/α. P ROPOSITION 12.5.3 We have η /α) = α−|η| dη . Eη∗ (¯
(12.72)
Proof. Noting from (12.67) that the proportionality constant in (12.70) is unity, we see that if ηN = 0, then Eη∗ (¯ η /α) = Eη∗# (¯ η# /α), where η# := (η1 , . . . , ηN−1 ). The r.h.s. of (12.72) has this property, which provides us with an induction in N . For induction in |η|, we note from (12.68) that “ η¯ + α + N − 1 ” 1 ∗ η /α) = EΦη (Φη/α), Eη∗ (¯ α where we have used the facts (Φη)j = ηj+1 (j = 1, . . . , N − 1), (Φη)N = η¯1 + α. We see from (12.38) that the r.h.s. of (12.72) satisfies this same recurrence.
To proceed further we require the fact that the Eη∗ are eigenfunctions of the family of operators [358] ˜ i := xi − σi · · · σN −1 xN + N − 1 Δ(N ) σ1 · · · σi−1 , Ξ (12.73) α where 1 1 − si (i = 1, . . . , N − 1). (12.74) σi := si + α xi − xi+1 Note that in the case N = 1 we have ˜ 1 = x1 (1 − Δ(1) ), Ξ
(12.75)
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JACK POLYNOMIALS
where Δ(1) f (x1 ) := f (x1 − 1), which is indeed an eigenoperator for (12.61). A feature of (12.75), deduced by noting that Δ(1) =
∞ (−1)j dj , j! dxj1 j=0
(12.76)
is that it consists of a degree preserving term equivalent to x1 dxd 1 , plus degree lowering terms. Similarly, inspection of (12.73) using (12.76) and the fact that Δ(N ) = sN −1 sN −2 · · · s1 Δ(1) ,
Δ(1) f (x1 , x2 , . . . , xN ) := f (x1 − 1, x2 , . . . , xN )
(12.77)
for general N shows ˜ i = 1 ξi + degree-lowering terms. Ξ α ˜ i } are eigenoperators, this means we can replace (12.67) by the more refined statement If {Ξ a ˜ην Eν (x). Eη∗ (x) = Eη (x) +
(12.78)
(12.79)
|ν|<|η|
To establish that (12.73) is indeed an eigenoperator, it is first necessary to specify the structure of σi f (x)|x=¯ν /α . From the definition (12.74) we see 1 ν¯i − ν¯i+1 − 1 σi f (x) = f (¯ ν /α) + f (si ν/α). ν¯i − ν¯i+1 ν¯i − ν¯i+1 x=¯ ν /α Now, from Exercises 12.1 q.5(i) we have that for νi = νi+1 , si ν¯ = si ν, while for νi = νi+1 , ν¯i − ν¯i+1 = 1, so we conclude that in all cases σi f (¯ ν /α) = af (¯ ν /α) + bf (si ν/α)
(12.80)
˜ i Eη∗ = 1 η¯i Eη∗ . Ξ α
(12.81)
for a and b independent of f . P ROPOSITION 12.5.4 We have
Proof. The Eη∗ are uniquely characterized, up to normalization, by the vanishing condition (12.65), together with the property (12.66). Now, we see from (12.80) and the action of Δ(N) that ˛ “ N − 1 ” (N) ˛ σi · · · σN−1 xN + Δ σ1 · · · σi−1 Eη∗ (x)˛ α x=¯ ν /α consists of a linear combination of Eη∗ (¯ ρ/α) for compositions ρ such that |ρ| = |ν| − 1. But for such compositions ˜ i Eη∗ (¯ Eη∗ (¯ ρ/α) = 0. Recalling the definition (12.73), this means Ξ ν /α) = 0 for |ν| ≤ |η|, ν = η, while the xi term in ∗ ∗ 1 ˜ (12.73) gives Ξi Eη (¯ η /α) = α η¯i Eη (¯ η/α), thus establishing (12.81).
12.5.2 Algebra isomorphism The eigenvalue equations (12.81) and the fact that {Eη∗ (x)} form a basis for analytic functions imply ˜ j ] = 0. ˜ i, Ξ [Ξ
(12.82)
A direct calculation using the definition (12.74) shows that like {si }, the {σi } satisfy the algebra σj2 = 1,
σj σj+1 σj = σj+1 σj σj+1 ,
[σi , σj ] = 0 (|i − j| ≥ 2).
(12.83)
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˜ j } form a representation of the degenerate type A Hecke algebra specified in Exercises 11.5 In fact {σi , αΞ q.2. To see this, first note that xi σi = σi xi+1 +
1 , α
[xi , σj ] = 0 (j = i − 1, i).
(12.84)
Now introduce the operator [358]
˜ i = σi · · · σN −1 xN + N − 1 Δ(N ) σ1 · · · σi−1 . Zi := xi − Ξ α We see immediately that Zi σi = σi Zi+1 .
(12.85)
(12.86)
Also, noting that Δ(N ) σj = σj−1 Δ(N ) and using the second relation in (12.84) we see Zi σj = σj Zi
(j = i − 1, i).
(12.87)
Combining (12.87), (12.86) and (12.84), we have from the definition of Zi that ˜ i+1 ) = 1, ˜ i )σi − σi (αΞ (αΞ
˜ i , σj ] = 0 (j = i − 1, i), [αΞ
(12.88)
which is precisely the relations (11.89), thus completing (with (12.82) and (12.83)) the defining relations of the degenerate type A Hecke algebra. An immediate consequence is the analogue of Proposition 12.2.1.
P ROPOSITION 12.5.5 We have σi Eη∗ =
⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩
∗ E + 1− δ¯i,η η 1
Eη∗ , 1 E∗ δ¯i,η η
1
2 δ¯i,η
Es∗i η , ηi > ηi+1 , ηi = ηi+1 ,
+ Es∗i η ,
ηi < ηi+1 .
Let ψ be the operator which has the property ψEη = Eη∗ ψ of mapping the nonsymmetric Jack polynomials to the interpolation nonsymmetric Jack polynomials. It follows immediately from (12.81) that ˜ i ψEη = ψ −1 Ξ
1 η¯i Eη α
and thus ˜ i ψ. ψξi = Ξ Also, Propositions 12.2.1 and 12.5.5 together imply ψsi = σi ψ. It is also possible to determine explicitly the action of ψ on xi [358], which turns out to involve the operator Zi (12.85). Now, from its definition, Zi acts on Eη∗ by Zi Eη∗ = (xi − η¯i /α)Eη∗ .
(12.89)
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JACK POLYNOMIALS
Because the r.h.s. vanishes for all x = ν/α, |ν| ≤ |η|, and has degree |η| + 1 we must have aην Eν∗ . Zi Eη∗ =
(12.90)
ν:|ν|=|η|+1
Equating the leading degree terms on the r.h.s.’s of (12.89) and (12.90) gives xi Eη = aην Eν , ν:|ν|=|η|+1
and this equation together with (12.90) implies ψxi = Zi ψ
(12.91)
which is the sought relation. Since {xi } commute, it follows immediately from this that [Zi , Zj ] = 0, while another consequence is the Rodrigues type formula Eη (Z1 , . . . , ZN ) · 1 = Eη∗ (x1 , . . . , xN ).
(12.92)
12.5.3 Extra vanishing condition ν /α) = 0 for |ν| ≤ |η|, ν = η. In fact the eigen-equation (12.81) implies there are By definition Eη∗ (¯ further conditions under which Eη∗ must vanish [358]. This requires introducing a further partial order on compositions. D EFINITION 12.5.6 Let ν, η be compositions. We write ν η if there exists a permutation π such that νi < ηπ(i) for i < π(i) and νi ≤ ηπ(i) for i ≥ π(i). Note that for ν and η partitions the statement ν η is equivalent to ν ⊆ η (inclusion of diagrams). However for compositions, although ν ⊆ η implies ν η, the converse is not true in general. The significance of the partial order lies with the following result. P ROPOSITION 12.5.7 For a composition η let IS be the set of functions which vanish at S¯η := {¯ ν /α : η ˜ i (IS ) ⊆ IS . ν}. Then Ξ ˜ i f (¯ ˜ i f (¯ Proof. For ν¯/α ∈ S¯η we have to show that Ξ ν /α) = 0. We see from (12.73) and (12.74) that Ξ ν /α) is a linear combination of f (¯ ν /α) and ˛ ˛ σ ˜i · · · σ ˜N−1 Δ(N) σ ˜1 · · · σ ˜i−1 f (x)˛ , x=¯ ν /α
where each σ ˜i is either 1 or si . This is equal to f (¯ ρ/α), where for some subset J = {j1 , . . . , jr } of {1, . . . , N } with j1 < · · · < jr νji = ρji+1 (i = 1, . . . , r − 1), νjr = ρj1 + 1, νj = ρj (j ∈
J). ρ/α) = 0, which in From Definition 12.5.6 we see that ρ ν. Since ν η we have ρ η. Thus ρ¯/α ∈ S¯η and so f (¯ ˜ i f (¯ ν /α) = 0. turn implies Ξ
The extra vanishing property can now be established. ν /α) = 0. P ROPOSITION 12.5.8 For compositions η, ν such that η ν we have Eη∗ (¯ Proof. Consider the set IS of Proposition 12.5.7, and choose a member of this set, hη say, which does not vanish at η. Writing hη (x) =
X ρ
aρ Eρ∗ (x)
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CHAPTER 12
˜ i hη ∈ IS we must have that and noting from Proposition 12.5.7 that Ξ Eρ∗ (¯ ν /α) = 0 for all ν¯/α ∈ S¯η .
(12.93)
The nonvanishing condition gives that for some ρ (ρ say) Eρ∗ (¯ η /α) = 0 and so from (12.65) |η| ≥ |ρ |. But Eρ∗ (ρ¯ /α) = 0 by (12.66) and so ρ¯ /α ∈ S¯η , which means η ρ . The conditions |η| ≥ |ρ | and η ρ together imply ρ = η, so we ν /α) = 0 for all η ν. conclude from (12.93) that Eη∗ (¯
12.6 THE SYMMETRIC JACK POLYNOMIALS 12.6.1 The symmetrization operator We know that when restricted to act on symmetric functions, the Schr¨odinger operator H (C,Ex) (11.52) reduces to the Schr¨odinger operator H (C) of Proposition 11.3.2. Furthermore, transforming the operator H (C) by factoring out the ground state gives the operator 2 ˜ (C) := L eβW (C) /2 (H (C) − E (C) )e−βW (C) /2 H 0 2π N N ∂ 2 N − 1 ∂ zj zk ∂ 2 ∂ , (12.94) zj = + zj + − ∂zj α j=1 ∂zj α zj − zk ∂zj ∂zk j=1 1≤j
2πixj /L
where zj := e . Comparison with (11.63) shows that the transformed Schr¨odinger operator H (C,Ex) when restricted to act on symmetric functions reduces to (12.94). Now, as seen in (12.34), the eigenvalue of H (C,Ex) for the eigenfunction Eη depends only on the underlying partition η + . Thus the linear combination a ˜η Eη (z), (12.95) η:η+ =κ
where κ denotes a partition, is also an eigenfunction of H (C,Ex) . In fact it is possible to choose the coefficients a ˜η such that the linear combination is a symmetric function of z, thereby constructing a symmetric polynomial ˜ (C) for each partition κ. eigenfunction of H P ROPOSITION 12.6.1 The polynomial
Pκ (z) = Pκ (z; α) := dκ
η:η+ =κ
1 Eη (z) dη
(12.96)
is symmetric in z1 , . . . , zN , and an eigenfunction of (12.94), and is referred to as the symmetric Jack polynomial. Furthermore, it has the structure Pκ (z) = mκ + bκσ mσ , (12.97) σ<κ
where mκ denotes the monomial symmetric function in the variables z1 , . . . , zN associated with the partition κ (for example, with κ = 21 and N = 2, m21 = z12 z2 + z1 z22 ), < denotes the dominance ordering for partitions, and the coefficients bκσ are independent of N . Proof. The special structure is a consequence of the special structure of the nonsymmetric Jack polynomials exhibited by (12.7). The fact that the coefficient of mκ in (12.96) is unity is seen by noting that the only polynomial in {Eη }η+ =κ containing the monomial z κ is Eκ itself (a consequence of (12.7)) and this occurs in (12.96) with coefficient unity. The coefficients bκσ are independent of N as a consequence of the stability property of Exercises 12.1 q.3. It remains to check that the r.h.s. of (12.96) is symmetric, which is equivalent to showing that it is unchanged by the action of the elementary
565
JACK POLYNOMIALS
transposition si (i = 1, . . . , N − 1). This can be done by writing ” “1 X 1 X 1 Eη (z) = χηi ηi+1 Eη (z) + Esi η (z) dη dη dsi η + + η:η =κ
η:η =κ ηi ≤ηi+1
with χηi ηi+1 = 1/2 (ηi = ηi+1 ), χηi ηi+1 = 1 otherwise, then using Proposition 12.2.1 to compute the action of si on the r.h.s, which shows that in fact the individual terms in this grouping remain unchanged.
We remark that in the case N = 2, Pκ (z; α) can be expressed in terms of the Jacobi polynomials (γ,γ) Pκ1 −κ2 (x), γ = (α − 1)/2 and x related to z1 , z2 (see Exercises 12.6 q.6). ˜ (C,Ex) ), the eigenvalue correIn Exercises 12.1 q.5(ii), it was noted for the operator X (C) (u) (as with H + sponding to the eigenfunction Eη depends only on η . Hence any linear combination of the form (12.95) is also an eigenfunction of X (C) (u). In particular Pκ is an eigenfunction satisfying the eigenvalue equation X
(C)
(u)Pκ (z) =
N
(1 + u(ακj − (N − j))) Pκ (z)
j=1
and is in fact the unique eigenfunction of X (C) (u) with the structure (12.97). To see this latter point, consider X (C) (u)z η , as implied by (12.6). The action of X (C) (u) on mκ can be deduced by symmetrizing both sides, as specified by the operation Sym f (x1 , . . . , xN ) := f (xP (1) , . . . , xP (N ) ) P ∈SN
(cf. (4.135)). Since Sym can be constructed out of elementary transpositions si , and X (C) (u) is symmetric in the {ξi }, it follows from Exercises 11.5 q.2(ii) that Sym commutes with X (C) (u). This shows X (C) (u)mκ =
N
(1 + u(ακj − (N − j))) mκ + ακσ mσ
(12.98)
σ<κ
j=1
N for some coefficients ακσ . Because of this triangular structure, and the fact that the eigenvalue j=1 (1 + u(ακj − (N − j))) is distinct for distinct partitions κ, the uniqueness follows. The Sym operator can be used to give a different formula to (12.96) relating Pκ (z) to {Eη (z)}. Now, from (12.8) and Exercises 12.1 q.5(ii) we have X (C) (u)Eη (z) =
N
(1 + u(ακj − (N − j))) Eη (z).
j=1
By applying Sym to both sides, and using the fact that Sym commutes with X (C) (u), we thus have X (C) (u)Sym Eη (z) =
N
(1 + u(ακj − (N − j))) Sym Eη (z).
(12.99)
j=1
Since, from the structure formula (12.7) Sym Eη (z) must, up to normalization, have the structure (12.97), we conclude from this that Sym Eη (z) = aη Pη+ (z) N
for some constant aη . Substituting z = (1)
(12.100)
shows
aη =
N !Eη ((1)N ) . Pη+ ((1)N )
(12.101)
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The explicit value of Eη ((1)N ) is known from (12.39), while the explicit value of Pη+ ((1)N ) can in fact be deduced from (12.101) by making use of the additional fact that aη can be independently computed for the composition η +R (recall (12.32)). Thus the structure (12.7) shows the only term in Eη+R (z) which when +R symmetrized contributes to mη+ in Pη+ (z) is z η , and has coefficient unity. It follows from (12.100) that Sym z η
+R
= aη R m η + ,
and so +
aη+R =
η1
fj !,
(12.102)
j=0
where fj denotes the frequency of the integer j in η + (for example, if η + = 211100 then f0 = 2, f1 = 3, f2 = 1). Equating (12.101) with (12.102) in the case η = κ for κ a partition shows eκ , dκR j=0 fj !
N! Pκ ((1)N ) = + η1
(12.103)
where use has also been made of (12.39) and the fact, which follows from (12.47), that eκR = eκ . Simplification of the r.h.s. of (12.103) gives the sought analogue of (12.39) in the case of the symmetric Jack polynomial [500]. P ROPOSITION 12.6.2 Let hκ be given by (12.58) and let αa (i, j) + N − l (i, j) = α|κ| [N/α](α) bκ := κ .
(12.104)
(i,j)∈κ
We have Pκ ((1)N ) =
bκ . hκ
(12.105)
Proof. From the formulas (12.47) and (12.104) we see that N eκ 1 Y = (ακj + N − j + 1), bκ N ! j=1
so it suffices to show
h Qκ1 κ
j=0 fj !
dκR . j=1 (ακj + N − j + 1)
= QN
(12.106)
This is done in Exercises 12.6 q.1.
12.6.2 Orthogonality with respect to ·|·(C) The orthogonality of {Eη } with respect to (12.12), together with the expansion formula (12.96) implies that {Pκ } are also orthogonal with respect to (12.12). This can also be deduced from the fact that the eigenoperator X (C) (u) for {Pκ } is self-adjoint with respect to (12.12) and has distinct eigenvalues (recall the argument leading to Proposition 12.1.4). Note also that the orthogonality together with the structure (12.97) implies that {Pκ } can be constructed from {mκ } with the partial ordering < according to the Gram-Schmidt procedure. The normalization of Pκ with respect to (12.12) can be deduced from the knowledge of the normalization of Eη (12.41), together with the formulas (12.96), (12.100) and (12.102) [376], [40].
567
JACK POLYNOMIALS
(C)
P ROPOSITION 12.6.3 Let Nκ
:= Pκ |Pκ (C) . We have (C)
Nκ
(C)
N0N
dκ eκ bκ dκ = , eκ hκ j=0 fj ! dκR eκ
N! = κ1
where dκ , dκ , eκ , eκ , hκ , bκ are as defined by (12.36), (12.37), (12.58), (12.104). Proof. According to (12.96), the orthogonality of {Eη } with respect to (12.12) and the normalization formula (12.41) we see that d d eκ (C) Pκ |EκR (C) = κ EκR |EκR (C) = κ N0 . dκR dκR eκ On the other hand, from the fact that the weight functions in ·|·(C) is symmetric, as is Pκ , we have Qκ1 1 j=0 fj ! (C) (C) = = Pκ |EκR Pκ |Sym EκR Nκ(C) , N! N! (C)
(C)
where the final equality follows from (12.100) Q 1 and (12.102). The first stated formula for Nκ /N0N follows. For the second equality we substitute for N !eκ / κj=0 fj !dκR using the equality between (12.103) and (12.105).
12.6.3 Orthogonality, a Cauchy-type product and the antisymmetric Jack polynomials In Section 12.1.3 the product Ω(x, y) was introduced, as were the associated polynomials {qη } and the inner product ·, ·q for which {Eη } forms an orthogonal set. An analogous theory can be developed for the symmetric Jack polynomials [376], although the path we will take is to first study the antisymmetric Jack polynomials [40]. In general the antisymmetrization operation Asym is specified as in (4.135). Since it is always possible to write Asym f (z1 , . . . , zN ) = Δ(z)g(z1 , . . . , zN ), Δ(z) := (zj − zk ), (12.107) 1≤j
where g is symmetric, we see that for f analytic it is possible to expand Asym f in terms of {Δmκ }. Now with δ := (N − 1, N − 2, . . . , 0)
(12.108)
and κ a partition we have Asym z δ+κ = Δ(z)(mκ + terms smaller with respect to <), so proceeding as in the derivation of (12.98) we can conclude X (C) (u)Δ(z)mκ =
N
(1 + u(α(κj + N − j) − (N − j))) Δ(z)mκ + ακσ Δ(z)mσ σ<κ
j=1
for some constants ακσ . By the argument used below (12.98), this implies that X (C) (u) possesses unique antisymmetric eigenfunctions of the form Sκ+δ (z) = Sκ+δ (z; α) := Δ(z) mκ + (12.109) aκσ mσ , σ<κ
and furthermore, if ρ has distinct parts, Asym Eρ (z) = cρ Sρ+ (z; α),
ρ+ = κ + δ
(12.110)
for some proportionality constant cρ (in the cases that ρ has repeated parts, Asym Eρ = 0; see Exercises 12.6
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CHAPTER 12
q.2). In preparation for computing the explicit value of cρ , we first make note of the expansion of the antisymmetric Jack polynomial Sρ+ (z) in terms of {Eη }. P ROPOSITION 12.6.4 Let the partition μ have all parts distinct. We have 1 Sμ (z) = ε(σ)dσ(μ) Eσ(μ) (z). dμ
(12.111)
σ∈SN
Proof. As noted in the proof of Proposition 12.6.1, the only polynomial in {Eσ(μ) }σ∈SN containing the monomial z μ
is Eμ itself. This corresponds to the term σ = id (the identity), and gives that the coefficient of z μ is unity as required by (12.109). It thus remains to check that the r.h.s. of (12.111) is antisymmetric. For this we write (12.111) as “ ” X 1 Sκ (z) = ε(σ) dσ(μ) Eσ(μ) (z) − dsi σ(μ) Esi σ(μ) (z) dμ σ∈S N σ(μ)i ≤σ(μ)i+1
for a given i, (1 ≤ i ≤ N − 1), and proceed as in the proof of Proposition 12.6.1.
Also required is the analogue of (12.102), whereby we independently determine cρ in the special case ρ = (κ + δ)R . From the structure (12.7) we have that the only term in E(κ+δ)R (z) which contributes to the R term Δ(z)mκ in the expansion (12.109) of Sκ+δ (z) is z (κ+δ) . We therefore have that R
Asym z (κ+δ) = c(κ+δ)R Δ(z)mκ and so c(κ+δ)R = (−1)N .
(12.112)
Using (12.112) and (12.111) the value of cρ for general ρ can be determined. P ROPOSITION 12.6.5 We have cρ = ε(σ)
dρ dρ+R
,
(12.113)
where σ is the permutation such that ρ = σ(ρ+ ). Proof. Following [211], we introduce the polynomial G(x, y) =
X dρ Eρ (x)Eρ (y). dρ +
(12.114)
ρ:ρ =κ
We can check from Proposition 12.2.1 and the relations (12.38) that (x)
(y)
si G(x, y) = si G(x, y),
i = 1, . . . , N − 1
and consequently Asym(x) G(x, y) = Asym(y) G(x, y).
(12.115)
Applying (12.115)–(12.114) and using (12.110) shows Sρ+ (x)
X dρ X dρ c E (y) = Sρ+ (y) cρ Eρ (x). ρ ρ d dρ ρ + +
ρ:ρ =κ
It follows from this that Sρ+ (x) = c˜ρ+
ρ:ρ =κ
X dρ cρ Eρ (x) dρ +
ρ:ρ =κ
(12.116)
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JACK POLYNOMIALS
for some proportionality constant c˜ρ+ . Comparing with (12.111) shows c˜ρ+
dρ dρ cρ = ε(σ) , dρ dρ+
(12.117)
where σ is such that ρ = σ(ρ+ ). Setting ρ = ρ+R and making use of (12.112) evaluates c˜ρ+ as c˜ρ+ =
dρ+R dρ+
,
and substituting this in turn in (12.117) we deduce (12.113).
An immediate consequence of the {Sκ+δ } being eigenfunctions of the operator X (C) (u) with distinct eigenvalues is that the antisymmetric Jack polynomials are orthogonal with respect to the inner product (12.12). Noting that |zk − zj |2/α |Δ(z)|2 = |zk − zj |2(α+1)/α , j
j
we have that the polynomials Sκ+δ (z)/Δ(z) can be constructed by orthogonalizing {mκ } in the order implied by the partial order <, with respect to the inner product ·|·(C) . α →α/(1+α)
But we already know that the unique symmetric polynomials with this property are the symmetric Jack (α/(1+α)) (z), so we have polynomials Pκ Sκ+δ (z; α) = Δ(z)Pκ (z; α/(1 + α)),
(12.118)
and consequently with ρ+ = κ + δ Asym Eρ (z) = cρ Δ(z)Pκ (z; α/(1 + α)).
(12.119)
In the limit α → ∞, substituting (12.10) in (12.119) with ρ = (δ+κ)R and making use of the Vandermonde determinant formula (1.173) shows k−1+κN +1−k
Pκ (z; 1) =
]j,k=1,...,N k−1 det[zj ]j,k=1,...,N
det[zj
.
(12.120)
This is the ratio of alternants formula (10.16) for the Schur polynomials in the N variables z1 , . . . , zN , and thus Pκ (z; 1) = sκ (z).
(12.121)
The formula (12.119) can be used to obtain the analogue of (12.15) for the symmetric Jack polynomials, referred to as a Cauchy type expansion [500] ((10.58) is the special case α = 1). P ROPOSITION 12.6.6 Let Π(α) (x, y) :=
N i,j=1
We have Π(α) (x, y) =
1 . (1 − xi yj )1/α
hκ κ
dκ
Pκ (x)Pκ (y).
(12.122)
(12.123)
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CHAPTER 12
Proof. Suppose we can show that Π(α) (x, y) =
X 1 Pκ (x)Pκ (y) vκ κ
(12.124)
for some nonzero vκ . Then to determine vκ , we proceed as in the derivation of (12.48) to deduce from (12.124), (12.105) and (12.104) that N Y X α|κ| [r](α) 1 κ = Pκ (x). (12.125) r (1 − x v j) κ hκ j=1 κ Now substituting (12.96) for Pκ and comparing with (12.48) gives vκ =
dκ hκ
(12.126)
as required. It thus remains to establish (12.124). For this we begin by applying Asym(x) to both sides of (12.14). On the l.h.s. we make use of the Cauchy double alternant formula (4.34) to note that 1 Δ(x)Δ(y) = QN , (1 − x y ) j j j=1 j,k=1 (1 − xj yk )
Asym(x) QN and so obtain
Δ(x)Δ(y) , (1 − xj yk )(α+1)/α j,k=1
Asym(x) Ω(x, y) = QN
(12.127)
(12.128)
while on the r.h.s. we use (12.119). This gives Δ(x)Δ(y)Π(α/(1+α))(x, y) = Δ(x)
X ∗ cρ ρ
uρ
Pκ (x; α/(α + 1))Eρ (y; α),
(12.129)
where the ∗ denotes that the sum is restricted to ρ with distinct parts and ρ+ = κ + δ. Applying Asym(y) to both sides of this identity we see that the l.h.s. is simply multiplied by N !, while on the r.h.s. we again use (12.119). Cancelling Δ(x)Δ(y) from both sides and summing over permutations of ρ which give the same κ, and finally replacing α/(1 + α) by α gives the sought formula.
The structure of (12.123) together with the result of Proposition 12.1.8 suggests we define a pairing inner product for symmetric functions by Pκ , Pμ g :=
dκ δκ,μ . hκ
Introducing the symmetric homogeneous polynomials gκ by Π(α) (x, y) = gκ (x)mκ (y)
(12.130)
(12.131)
κ
allows us to deduce that {gκ } are a basis for symmetric functions and furthermore gκ , mμ g = δκ,μ .
(12.132)
We remark that there is an alternative way to expand the product (12.122), in terms of the power sum basis for symmetric functions. This is covered in Exercises 12.6 q.5, and leads to an alternative characterization of the scalar product ·, ·g . The formula N j=1
α|κ| [r]κ 1 = Pκ (x), (1 − xj )r dκ κ (α)
(12.133)
obtained by substituting (12.126) in (12.125), is a more compact way to write the generalized binomial
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JACK POLYNOMIALS
theorem (12.48). A useful corollary of (12.133) follows by making the replacements xj → xj /r and taking the limit r → ∞. Thus, since lim r−|κ| [r](α) κ = 1,
lim (1 − xj /r)−r = exj
r→∞
r→∞
we obtain e(x1 +···+xN ) =
α|κ| dκ
κ
Pκ (x; α).
(12.134)
Equating terms homogeneous of degree p on both sides shows as a further corollary, (x1 + · · · + xN )p = p!
α|κ| Pκ (x; α). dκ
(12.135)
|κ|=p
The result (12.135) together with the expansion formula (12.123) allows a property of Pκ first noted in the work of Stanley [500] to be deduced. P ROPOSITION 12.6.7 Suppose |κ| ≤ N , and denote by [x1 · · · xk ]f (x) the coefficient of x1 · · · xk in f . One has |κ|! [x1 · · · x|κ| ]Pκ (x) = . hκ Proof. Since Π(α) (x, y) =
N “ Y
1+
i,j=1
we see that [x1 · · · xk ]Π(α) (x, y) =
” 1 xi yj + O((xi yj )2 ) α
X 1 1 (y1 + · · · + yN )k = |κ|! Pκ (y; α), k α dκ |κ|=k
where to obtain the second equality (12.135) has been used. But according to (12.123) [x1 · · · xk ]Π(α) (x, y) = Equating coefficients of
(α) 1 P (y) dκ κ
X hκ [x1 · · · xk ]Pκ (x; α) Pκ (y; α). dκ κ
in the above two equations gives the stated result.
12.6.4 Shift operators The Cherednik operators (11.57) can be used to define operators G+ , G− with an action on symmetric Jack polynomials which shifts the value of 1/α by unity, G− Pκ+δ (z; α) = c− κ Pκ (z; α/(1 + α)),
G+ Pκ (z; α/(1 + α)) = c+ κ Pκ+δ (z; α),
(12.136)
for some proportionality constants c± κ [437], [344]. Use of (12.118) shows that equivalently −1 (z)Sκ+δ (z; α), G− Pκ+δ (z; α) = c− κΔ
G+ (Δ−1 (z)Sκ+δ (z; α)) = c+ κ Pκ+δ (z; α),
thereby exhibiting that the G± map between the symmetric and antisymmetric Jack polynomials (the latter divided by Δ(z)) of the same index and parameter. We will see that the operators G± can be constructed out of the operators ˜ ± := G (ξk − ξj ± 1). 1≤j
572
CHAPTER 12
To see the significance of these operators, we note from the results of Exercises 11.5 q.2(ii) that (ξi+1 − ξi ∓ 1)si = −si (ξi+1 − ξi ± 1 − 2si ),
(ξj − ξi+1 − 1)si = −si (ξj − ξi + 1),
(12.137)
˜ + maps antisymmetric where the latter equation is valid for j = i, i + 1. It follows immediately that G ˜ functions to symmetric functions, while G− maps symmetric functions to antisymmetric functions. Also of ˜ − on antisymmetric functions. In this regard, the relations (12.137) allow us to importance is the action of G establish, via an induction in N , that N −1 l ˜ − A(z) = 2G ˜ + A(z) 1+ sj G
(12.138)
l=1 j=1
for general antisymmetric functions A. ˜ ± and relating inner products (12.12) with 1/α shifted From (12.138) an adjoint type property involving G by unity can be established. (C)
P ROPOSITION 12.6.8 Write f |g(C) = f |gα , and let Δ(z) be given by (12.107). For f, g symmetric functions (C) ˜ − f |g(C) ˜ Δ−1 (¯ z )G α/(α+1) = f |G+ (Δ(z)g)α .
(12.139)
Proof. From the definition of f |g(C) , (C) ˜ − f |g ˜ ˜ − (Δ(z)g)(C) Δ−1 (¯ z )G = f |G α , α/(α+1) = G− f |Δ(z)gα (C)
(12.140)
˜ − is self-adjoint with respect to ·|·(C) where the second equality follows from the fact that G α . Changing the names of the integration variables zj → zN+1−j (j = 1, . . . , N ) in the integral specifying the inner product does not change its value. Doing this to the final inner product in (12.140) and then taking the arithmetic mean with the final inner product unaltered shows l N−1 E(C) ” Y Y 1 D ˛˛“ ˜ − (Δ(z)g)(C) ˜ − (Δ(z)g) f |G = sj G . f˛ 1 + α 2 α j=1 l=1
The sought identity (12.139) now follows upon use of (12.138).
The property (12.139) allows the operators G± (12.136) to be made explicit. P ROPOSITION 12.6.9 Choosing ˜ + Δ(z), G+ = α−N (N −1)/2 G
˜− G− = α−N (N −1)/2 (Δ(z))−1 G
the actions (12.136) hold with ± c± κ = aκ ((α + 1)/α),
a± κ (β) :=
(λj − λk ∓ 1 + (k − j ± 1)β).
j
Proof. Only the case of G− will be considered; similar working establishes the results for G+ . The fact that Δ(z)Pκ (z; α) = Pκ+δ (z; α) +
X
aκμ Pμ (z; α)
μ<κ+δ
˜ + being an eigenoperator of Pκ+δ (z; α) show that together with G Y −N(N−1)/2 (κ + δ j − κ + δ k + 1). c+ κ = α 1≤j
Recalling that for a partition ρ, ρ¯j = αρj − (j − 1), the stated formula for c+ κ follows.
(12.141)
573
JACK POLYNOMIALS
To establish that G− has the sought action, we note that in terms of G± (12.139) reads (C)
G− f |gα/(α+1) = f |G+ g(C) α . Choose g = Pκ (z; α/(1 + α)) and f = mμ with μ < κ + δ. Noting that then X aμρ Pρ (z; α/(α + 1)) G− f = Pμ−δ (z; α/(α + 1)) + ρ<μ−δ (C)
for μ − δ a partition, G− f = 0 otherwise, it follows that mμ |G+ gα proportional to Pκ+δ (z; α).
= 0 for all μ < κ + δ, and hence G+ g is
The actions (12.136) together with the adjoint type property (12.141) can be used to deduce a formula for (C) the normalization Pκ |Pκ α in the case that 1/α is a positive integer. Thus we see that (C)
Pκ (·, 1 + 1/α)|Pκ (·, 1 + 1/α)1/(1+1/α) =
a− κ (1 + 1/α) Pκ+δ (·, 1/α)|Pκ+δ (·, 1/α)(C) α a+ κ (1 + 1/α)
which upon iteration gives (C)
Pκ (·, k + 1/α)|Pκ (·, k + 1/α)1/(k+1/α) =
k−1
a+ κ+jk ((k − j) + 1/α)
j=0
a− κ+jk ((k − j) + 1/α)
Pκ+kδ (·, 1/α)|Pκ+kδ (·, 1/α)(C) α .
Now set 1/α = 0. Since Pμ (z; 0) = mμ , we see that the inner product on the r.h.s. is equal to LN N ! and (C) thus an explicit formula for Pκ (·, k)|Pκ (·, k)k is obtained. This can be checked to agree with the formula in Proposition 12.6.3 upon making use of (12.59) and (12.60). 12.6.5 Integration formulas By summing (12.56) over η, η + = κ, according to (12.96) we obtain the trigonometric integral [192] 1/2 1/2 N 1 dx1 · · · dxN eπixl (a−b) |1 + e2πixl |a+b Pκ (−e2πix1 , . . . , −e2πixN ) MN (a, b; 1/α) −1/2 −1/2 l=1
×
(α)
|e2πixk − e2πixj |2/α = Pκ ((1)N )
1≤j
[−b]κ
(α)
[1 + a + (N − 1)/α]κ
.
(12.142)
Similarly, we obtain from (12.57) the generalized Selberg integral [386], [338], [342] 1 1 N 1 2/α dt1 · · · dtN tλl 1 (1 − tl )λ2 Pκ (t1 , . . . , tN ) |tj − tk | SN (λ1 , λ2 ; 1/α) 0 0 l=1
= Pκ ((1)N )
[λ1 + (N − 1)/α +
j
(α) 1]κ (α)
[λ1 + λ2 + 2(N − 1)/α + 2]κ
.
(12.143)
We recall that MN and SN are given explicitly in terms of gamma functions by (4.4) and (4.3) respectively. On the other hand, substituting for Pκ using (12.100) shows that (12.142) implies (12.56) and (12.143) implies (12.57), so the nonsymmetric and symmetric versions are equivalent. A corollary of (12.142), which is of particular use in the calculation of dynamical correlation functions, can be deduced as a consequence of the following result.
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CHAPTER 12
P ROPOSITION 12.6.10 Let f (z1 , . . . , zN ) be symmetric in z1 , . . . , zN (zj := e2πixj ), periodic in xj of period 1 and homogeneous of integer order k (k = 0). Let u (xl ) have the small expansion u (xl ) = 1 + a(xl ) + o(). Then we have 1 →0 lim
1/2
−1/2
dx1 · · ·
1/2
−1/2
dxN
1/2
=N −1/2
Proof. For small ,
z1k a(x1 )dx1
N Y
N
u (xl )f (z1 , . . . , zN )
l=1 1
dx2 · · ·
0
1
dxN f (1, z2 , . . . , zN ). 0
u (xl ) ∼ 1 +
l=1
N X
a(xl )
l=1
and thus, since f is assumed homogeneous of nonzero integer order, Z
1/2 −1/2
Z dx1 · · ·
1/2
dxN −1/2
N Y
Z u (xl )f (z1 , . . . , zN ) ∼
l=1
1/2 −1/2
Z dx1 · · ·
1/2
dxN −1/2
N X
a(xl )f (z1 , . . . , zN ).
l=1
P The stated result now follows by using the assumption that f is symmetric to replace N l=1 a(xl ) in the integrand by N a(x1 ), then using the assumption that f is periodic to replace zj by z1 zj (j = 2, . . . , N ) and finally the fact that f is homogeneous of order k to write f (z1 , z1 z2 , . . . , z1 zN ) = z1k f (1, z2 , . . . , zN ).
P ROPOSITION 12.6.11 We have 1 1 N dx2 · · · dxN |1 − e2πixl |2/α Pκ (1, e2πix2 , . . . , e2πixN ) N 0
×
0
l=2
|e2πixk − e2πixj |2/α = Pκ ((1)N )|κ|(κ1 − 1)!
2≤j
[0]κ (α) (N/α)! , (1/α)!N [1 + (N − 1)/α](α) κ (12.144)
where the dash on [0]κ (α) means that the j = 1 term in its definition (12.46) is to be omitted. Proof. This is deduced from (12.142) by setting a = −b = and taking the limit → 0. To do this on the l.h.s. we use Proposition 12.6.10 with u (x) = z and Y |zk − zj |2/α , f (z1 , . . . , zN ) = Pκ (−z1 , . . . , −zN ) 1≤j
and use the fact that
Z
1/2
e2πix|κ| x dx =
−1/2
(−1)|κ| . 2πi|κ|
On the r.h.s. this limit is taken directly, using the formula lim
→0
1 (α) = (κ1 − 1)![0]κ (α) . [] κ
575
JACK POLYNOMIALS
We remark that it follows from Proposition 12.6.11 that one has the expansion [287] βκ k = Pκ (z1 , . . . , zN ) pk := z1k + · · · + zN Pκ |Pκ (C)
(12.145)
|κ|=k
where βκ is equal to the r.h.s. of (12.144) (see Exercises 12.6 q.7). Another class of integration formulas involving Jack polynomials generalize the vanishing implied by (10.137) in the limit β → 0, (10.155) and (10.159) in the limit α → 0, and the counterparts of these latter two integrals relating to O± (2l + 1). Recalling the eigenvalue p.d.f.’s for Sp(2l) and O± (l) from Section 2.6, these vanishings can be seen to be the case α = 1 of the following integrals [460], which we state without proof. P ROPOSITION 12.6.12 Let Δ± :=
(eiθk − eiθj )(1 − eiθj eiθk ),
1≤j
write (z ± , w± , . . . ) as an abbreviation for (z, z −1 , w, w−1 , . . . ), and denote by (κ) the number of nonzero parts of κ. We have the generalized Sp(2l) vanishing integral π π l dθ1 · · · dθl Pκ (e±iθ1 , . . . , e±iθl ) |1 − eiθj |2/α |1 + eiθj |2/α |Δ± |2/α = 0 0
0
j=1
unless (κ) ≤ 2l and each κ1 = κ2 , κ3 = κ4 , . . . ; the generalized O+ (2l) vanishing integral π π l dθ1 · · · dθl Pκ (e±iθ1 , . . . , e±iθl ) |1 − eiθj |1/α−1 |1 + eiθj |1/α−1 |Δ± |2/α = 0 0
0
j=1
unless (κ) ≤ 2l and each κi has the same parity; the generalized O∓ (2l + 1) vanishing integral π π l dθ1 · · · dθl Pκ (e±iθ1 , . . . , e±iθl , 1) |1 − eiθj |3/α−1 |1 + eiθj |1/α−1 |Δ± |2/α = 0 0
0
j=1
unless (κ) ≤ 2l + 1 and each κi has the same parity; the generalized O− (2l) vanishing integral π π l−1 dθ1 · · · dθl−1 Pκ (e±iθ1 , . . . , e±iθl−1 , 1, −1) |1 − eiθj |3/α−1 |1 + eiθj |3/α−1 |Δ± |2/α = 0 0
0
j=1
unless (κ) ≤ 2l and each κi has the same parity. Moreover, with κ2i−1 = μi (i = 1, . . . , l) the normalized value of non-zero cases of the first integral equals (j − 1 + (2l + 2 − 2i)/α)(μi − j + 1 + (2μj − 2i + 1)/α) , (j + (2l + 1 − 2i)/α)(μi − j + (2μj − 2i + 2)/α) (i,j)∈μ
while with κi = 2μi (i = 1, . . . , 2l) the arithmetic mean of the normalized value of second and fourth integral in the nonzero cases equals (2j − 2 + (2l + 1 − i)/α)(2μi − 2j + 1 + (μj − i)/α) . (2j − 1 + (2l − i)/α)(2μi − 2j + (μj − i + 1)/α) (i,j)∈μ
E XERCISES 12.6
1. [40] The objective of this exercise is to verify the formula (12.106). Q (i) Let (κ) be as in Proposition 12.6.12. Check that f0 ! = N j=(κ)+1 (N − j + 1).
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CHAPTER 12
Q 1 (ii) Show from the definitions that the nodes at the end of each row of the diagram of κ contribute κi=1 fi ! to Q (ακ + N − j + 1) to d . hκ , while the nodes at the beginning of each row of κR contribute (κ) R j κ j=1 (iii) Check that the node (i , j ) ∈ κ (not at the end of a row) in the ith row of the jth block (the jth block contains parts equal to j) has the same leg length as the node (i , j ) ∈ κR in the ith row of the jth block, shifted one column to the right. As these nodes have arm lengths differing by one, conclude they give a contribution to hκ and dκR , respectively, which is equal. 2.
(i) For ρ with repeated parts, use the structure (12.7) to deduce that the coefficient of Δ(z)mκ (z) in Asym Eρ , (α) κ = ρ − δ must vanish, and thus Asym Eρ is proportional to Sμ (z) with μ < κ. Note that this is inconsistent with (12.111) unless the proportionality constant is zero, and thus Asym Eρ = 0. (ii) Substitute the r.h.s. of (12.123) with α → α/(α + 1) in (12.129), rewrite Δ(y)Pκ (y; α/(α + 1)) therein using (12.118), and equate coefficients of Pη+ (x; α/(α + 1))Eρ (y; α) on both sides to deduce that cρ = ε(σ)
dρ (α) hη+ (α/(α + 1)) . dρ+ (α) dη+ (α/(α + 1))
Remark: This can be further simplified (see [40]) to reclaim the formula (12.113). 3. [376] The power sum symmetric functions are given by fκ
pκ = pκ1 pκ2 · · · pκN = pf11 pf22 · · · pκ11 ,
pj = pj (x) =
N X
xjm
(12.146)
m=1
(fj is used in the same sense as in (12.102)). They form a basis for symmetric functions in an infinite number of variables, i.e. in the limit N → ∞ (for fixed N , {pκ } are not linearly independent). (i) Show from the definition of pj in an infinite number of variables x = {xi }i=1,2,... that ∞ X
pj (x)tj−1 =
j=1
and deduce from this that
∞ Y k=1
∞ X 1 d log , dt 1 − xk t k=1
“ X p (x)tj ” 1 j = exp . 1/α (1 − xk t) αj j=1 ∞
(ii) Replace t by yl in the above identity and take the product over l to conclude ∞ Y k,l=1
X α−(κ) 1 = pκ (x)pκ (y), zκ (1 − xk yl )1/α κ
zκ :=
κ1 Y
lfl fl !.
l=1
(iii) Use the result of (ii) together with Proposition 12.1.8 to deduce pκ , pσ g = α(κ) zκ δκ,σ , where ·, ·g is specified by (12.130). 4. [376], [500] This exercise relates to expansion formulas for Pκ (x). (i) Check from the definition (12.34) that e(μ; α) < e(κ; α) for μ < κ. (ii) Use (12.94) together with (12.4) to show that ˜ (C) mκ = e(κ; α)mκ + 1 H α
X 1≤j
[(κj −κk )/2]
(κj − κk )
X r=1
mκ(j,k,r) ,
577
JACK POLYNOMIALS
where κ(j, k, r) is the partition obtained by rearranging the composition (κ1 , . . . , κj − r, . . . , κk + r, . . . , κN ). ˜ (C) has unique eigenfunctions of the form (12.97), and thus the (iii) Use the results of (i) and (ii) to show that H ˜ (C) which are of the form (12.97). Pκ can be characterized uniquely as the eigenfunctions of H (iv) Use these results to show P21k 0N −k−1 (z) = m21k 0N −k−1 (z) +
(k + 1)(k + 2) m1k+2 0N −k−2 (z). k+1+α
(12.147)
(v) Suppose N > (κ). Introduce the notation κ∗ := (κ2 , . . . , κN , 0),
x∗ := (x2 , . . . , xN ),
and write ˜ (C) mκ (x) = H
X
X
˜ (C)∗ mκ∗ (x∗ ) = H
bκμ mμ (x),
bκ∗ μ∗ mμ∗ (x∗ ).
(12.148)
μ∗ ≤κ∗
μ≤κ
Use the result of (ii) to show that bκμ = bκ∗ μ∗
for
bκκ − bμμ = bκ∗ κ∗ − bμ∗ μ∗ .
μ < κ,
From this deduce that if Pκ (x) =
X
aκμ mμ
then Pκ∗ (x∗ ) =
where the notation
aκ∗ μ∗ mμ∗ .
μ∗ ≤κ∗
μ≤κ
(vi) Use (v) to show
X
[xκ1 1 ] Pκ (x) = Pκ∗ (x∗ ), [xκ1 1 ]
denotes the coefficient of
(12.149)
xκ1 1 .
(vii) Let
mλ(1) mλ(2) .. . mλ(n−1) mλ
e(λ(1) ) − e(λ) bλ(2) λ(1) .. . bλ(n−1) λ(1) bλλ(1)
e(λ
(2)
0 ) − e(λ)
bλ(n−1) λ(2) bλλ(2)
0 0 .. .
··· ··· ··· ···
3 0 7 0 7 7 .. 7. . 7 (n−1) e(λ ) − e(λ) 5 bλλ(n−1) (12.150)
Use this to rederive (12.147). 5. [437] Here a factorization property of Pκ+2αδ (z; 1/(−α + 1/2)), α ∈ Z+ is deduced. (i) Recalling the notation (12.34) and (12.107), start from the eigenvalue equation ” ” “ “ 2π ”2 “ (C) e(κ; α) |Δ(z)|α Pκ (z; 1/α), H (C) |Δ(z)|α Pκ (z; 1/α) = E0 + L then replace α by 1 − α and make use of the fact that H (C) |α→1−α = H (C) and (12.94) to deduce ” “ ˜ (C) |Δ(z)|1−2α Pκ (z; 1/(1 − α)) = e(κ; 1 − α)|Δ(z)|1−2α Pκ (z; 1/(1 − α)). H
578
CHAPTER 12
(ii) Next replace α by −α + 12 , κ by κ + α(N − 1), and use the fact that z p Pκ (z) = Pκ+p (z) (recall Exercises 12.1 q.1) to show that for α a non-negative integer ˛ ” “ “ 1” ˜ (C) ˛˛ H Δ(z))2α Pκ (z; 1/(α+1/2)) = e κ+α(N −1); α+ (Δ(z))2α Pκ (z; 1/(α+1/2)). 2 α→−α+1/2 With δ as in (12.108), deduce from this that (1/(−α+1/2))
(Δ(z))2α Pκ(1/(α+1/2)) (z) = Pκ+2αδ
(z).
(12.151)
(iii) Let α = −(k + 1)/(r − 1) with k + 1, r − 1 positive, relatively prime integers. For κ a partition of N parts and s ∈ Z+ , let N − (κ) = (k + 1)s − 1 =: n0 and let the smallest nonzero part of κ equal (r − 1)s + 1. Specifically, κ is specified in terms of its frequencies [f0 f1 f2 · · · ] according to κ := κ(k, r, s) = [n0 0s(r−1) k0r−1 k0r−1 k · · · ]. A conjecture of [61] (consistent in the case s = 1 with properties of Jack polynomials with negative, rational α known from [171]) gives ˛ ˛ Pκ(k,r,s) (z1 , . . . , zN )˛
z1 =···=z(k+1)s−1 =z
=
N Y
(z−zi )(r−1)s+1 Pκ(k,r,1) (zs(k+1) , zs(k+1)+1 , . . . , zN ).
i=s(k+1)
Show that with k = s = 1 this is equivalent to the κ = 0N case of (12.151). 6. [550] The objective of this exercise is to derive the formula Pκ1 κ2 (z1 , z2 ; 2/α) = (α,β)
where Pn
“1 ” 2κ1 −κ2 (γ,γ) (z1 + z2 )(z1 z2 )−1/2 , (z1 z2 )(κ1 +κ2 )/2 Pκ1 −κ2 aκ1 −κ2 2
(12.152)
(x) denotes the Jacobi polynomial (5.47), γ := (α − 1)/2 and an :=
“ n + γ ” (n + 2γ + 1) n
(γ + 1)n
n
2−n .
(i) Let κ1 ,κ2 = κ21 + κ22 − ακ2 and “ ∂ ”2 “ ∂ ”2 z1 z2 “ ∂ ∂ ” . + z2 +α − L := z1 ∂z1 ∂z2 z1 − z2 ∂z1 ∂z2 Use the result of q.4 (iii) above to show that (L − κ1 ,κ2 )Pκ1 κ2 (z1 , z2 ; 2/α) = 0. (ii) Show that with the change of variables u = 12 (z1 + z2 )(z1 z2 )−1/2 , v = (z1 z2 )1/2 , the operator L reads L=
∂2 ∂2 ∂ 1“ 2 ∂ ” α“ ∂ ∂ ” (u − 1) 2 + v 2 2 + u +v + u −v . 2 ∂u ∂v ∂u ∂v 2 ∂u ∂v
(iii) With the ansatz Pκ1 κ2 (z1 , z2 ; 2/α) = v κ1 +κ2 f (u) in the equation of (i), use the result of (ii) to show that f satisfies the equation “
(u2 − 1)
” ∂2 ∂ − (κ1 − κ2 )(κ1 − κ2 − α) f (u) = 0. + (1 + α)u 2 ∂u ∂u (γ,γ)
Since this is the differential equation satisfied by the Jacobi polynomial Pκ1 −κ2 (u), γ = (α − 1)/2, conclude that (12.152) is correct up to the proportionality constant. Use the expansion (5.47) to choose the proportionality constant so the coefficient of z1κ1 z2κ2 on the r.h.s. of (12.152) is unity.
579
JACK POLYNOMIALS
(iv) Note from (12.118) and (12.152) that in the case N = 2 the antisymmetric Jack polynomial has the explicit form “ ” 2κ1 −κ2 α+1 (γ ,γ ) 1 (z1 z2 )(κ1 +κ2 )/2 Pκ1 −κ2 Sκ1 κ2 (z1 , z2 ; 2/α) = (z1 −z2 ) (z1 +z2 )(z1 z2 )−1/2 , γ := , aκ1 −κ2 2 2 where an is defined as in (12.152) except that γ is replaced by γ . 7. In this exercise the expansion (12.145) will be derived. (i) Note from the orthonormality and completeness of {Pκ } that Z
1/2
βκ = −1/2
Z dx1 · · ·
1/2 −1/2
−k dxN (z1−k + · · · + zN )Pκ (z1 , . . . , zN )
N Y
|zk − zj |2/α .
j
PN
(ii) Use the symmetry of the integrand to replace p=1 zp−k by N z1−k . Now change variables zj → z1 zj (j = 2, . . . , N ) to deduce that βk is equal to the l.h.s. of (12.144). 8. Apply the limiting procedure used to deduce Proposition 4.7.3 to show from (12.143) that 1
Z
˜ λ ,α,N W 1
∞ 0
Z
∞
dt1 · · ·
dtN 0
N Y
tλl 1 e−tl Pκ (t1 , . . . , tN )
l=1
Y
|tk −tj |2/α = Pκ ((1)N )[λ1 +(N −1)/α+1](α) κ ,
j
(12.153) where
Z
∞
˜ λ ,α,N = W 1
Z
∞
dt1 · · ·
0
dtN 0
N Y
tλl 1 e−tl
l=1
Y
|tk − tj |2/α .
j
12.7 INTERPOLATION SYMMETRIC JACK POLYNOMIALS With aη as in (12.100), let us introduce the interpolation symmetric Jack polynomials Pκ∗ (x) according to Sym Eη∗ (x) = aη Pκ∗ (x),
κ = η+ .
(12.154)
Suppose η itself is a partition, and let ν be a composition such that η ν + or equivalently η ⊆ ν + . It follows from Definition 12.5.6 that then η ρ(ν + ) for any permutation ρ, and the extra vanishing condition Proposition 12.5.8 gives Eη∗ (ρ(ν + )/α) = 0. Thus with ν + = μ we have from (12.154) that μ/α) = 0 Pκ∗ (¯
for κ ⊆ μ
(12.155)
for |μ| ≤ |κ|, (μ = κ).
(12.156)
and in particular μ/α) = 0 Pκ∗ (¯
The proof of Proposition 12.5.1 shows there is a unique (up to normalization) polynomial with the property (12.156) together with the requirement that Pκ∗ (¯ κ/α) = 0, and this can be used to characterize the interpolation symmetric Jack polynomials, with (12.155) an extra vanishing condition. It follows from (12.79) that Pκ∗ (x) = Pκ (x) + cκμ Pμ (x) (12.157) |μ|<|κ|
for some coefficients {cκμ }. Also, proceeding as in the proof of Proposition 12.6.1, making use of Proposition 12.5.5 rather than Proposition 12.2.1, we see that analogous to (12.96) 1 Pκ∗ (x) = dκ Eη∗ (x). d η + η:η =κ
580
CHAPTER 12
From this and the vanishing property (12.65) we see that κ/α) = Eκ∗ (¯ κ/α) = α−|κ| dκ , Pκ∗ (¯
(12.158)
where the final equality follows from (12.72). Next we turn our attention to characterizing {Pκ∗ (x)} as eigenfunctions and deriving a special form of the eigenoperator. Regarding the former, because of the algebra isomorphism of Section 12.5.2, the working leading to the result of Exercises 11.5 q.3(iii) allows us to conclude ˆ 1, . . . , Ξ ˆ N )] = 0. [si , f (Ξ Consequently, as in the derivation of (12.99) we conclude N
ˆ j )P ∗ (x) = (1 + uΞ κ
N
(1 + u(κj − (N − j)/α))Pκ∗ (x).
(12.159)
j=1
j=1
Next we seek a special form of this eigenoperator, which has significance in the development of the theory of the symmetric Jack polynomials. This is the determinant formula [359] N
ˆ j ) (1 + uΞ = symmetric functions
j=1
1 −j+1 det (xi u + 1)(xi + 1/α)N −j − uxN Δi , i Δ(x) i,j=1,...,N
(12.160)
where Δ(x) is specified by (12.107) and Δi f (x1 , . . . , xN ) = f (x1 , . . . , xi−1 , xi − 1, xi+1 , . . . , xN ) =: f (x − εi ).
(12.161)
To establish this formula, we first show that the r.h.s. of (12.160) maps Sym(xκ ) for κ a partition to a polynomial of degree less than or equal to |κ|. P ROPOSITION 12.7.1 Let −j+1 Δi . Δij := (xi u + 1)(xi + 1/α)N −j − uxN i
With sκ denoting the Schur polynomial, we have 1 det[Δij ]i,j=1,...,N Sym(xκ ) Δ(x) =
N
(1 + u(κj − (N − j)/α))sκ (x1 , . . . , xN ) +
cκμ sμ (x1 , . . . , xN ).
(12.162)
|μ|≤|κ| μ=κ
j=1
Proof. Making use of (10.16) shows (12.162) can be rewritten det[Δij ]i,j=1,...,N Sym(xκ ) =
N Y
(1 + u(κj − (N − j)/α))Asym(xκ+δN ) +
j=1
X
cκμ Asym(xμ+δN ), (12.163)
|μ|≤|κ| μ=κ
where δ =: δN is specified by (12.108). Since in general an operator of the form det[Aj [xi ]]i,j=1,...,N where Aj [xi ] acts only on the xi variable maps symmetric functions to antisymmetric functions, it suffices to check that the action of det[Δij ]i,j=1,...,N on the monomial xη (η + = κ) is consistent with (12.163). For this task, expanding (xi + 1/α)N−j by the binomial theorem shows Δij = ((N − j)u/α + 1)xN−j + uxN−j+1 (1 − Δi ) + O(xN−j−1 ), i i i and thus Δij xηi i = (1 + u(ηi + (N − j)/α))xηi i +N−j + O(xηi i +N−j−1 ). Taking the determinant indeed gives a structure consistent with (12.163).
(12.164)
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JACK POLYNOMIALS
With the property (12.162) established, we verify (12.160) by checking that for the r.h.s. is an eigenoperator the {Pκ∗ }. For this, let I ⊆ {1, . . . , N }, and with εi as in (12.161) put εI := i∈I εi , ΔI f := ( i∈I Δi )f = f (x − εI ). Also, introduce the functions φI (x) := det[cIij (x)]i,j=1,...,N where N −j+1 for i ∈ I, xi cIij (x) = (12.165) (xi + 1/α)N −j for i ∈ I. P ROPOSITION 12.7.2 Let μ be a partition and δN be as in (12.144). If μ − εI is not a partition then φI (μ + δN /α) = 0. Proof. Consider first the case that μN = 0 and N ∈ I. Then with x = μ + δN /α, xN = 0 and thus the final row of
φI (μ + δN /α) vanishes. The only other instance for which μ − εI is not a partition is when there is an i < N such that i ∈ I, i + 1 ∈ / I and μi = μi+1 . Then with x = μ + δN /α, xi = xi+1 + 1/α and according to the definition (12.165), [cIij (μ + δN /α] has two rows the same.
P ROPOSITION 12.7.3 For each partition κ, N 1 (1 + u(κj − (N − j)/α))Pκ∗ (x). det[Δij ]i,j=1,...,N Pκ∗ (x) = Δ(x) j=1
Proof. The value of the eigenvalue follows from (12.162). To check that Pκ∗ (x) is the corresponding eigenfunction we use the characterization (12.156). Put D := det[Δij ]i,j=1,...,N and expand X d I ΔI D=
(12.166)
I
with
j dI = det[dIij ],
dIij =
−uxN−j+1 i (xi u + 1)(xi + 1/α)N−j
for i ∈ I, for i ∈ / I.
Since dI has φI as a factor, Proposition 12.7.2 holds. To make use of this fact, we note (12.166) gives X DPκ∗ (μ + δN /α) = dI |x=μ+δN /α Pκ∗ (μ − I + δN /α).
(12.167)
I
For μ − εI a partition the factor Pκ∗ (μ − εI + δN /α) vanishes according to (12.156), while for μ − εI not a partition the factor dI |x=μ+δN /α vanishes according to Proposition 12.7.2. Hence all terms in (12.167) vanish so we conclude that (Δ(x))−1 DPκ∗ (x) is proportional to Pκ∗ (x) as required.
12.7.1 Isomorphism between Pκ and Pκ∗ Let Ψ be the invertible linear map with the action ΨPκ = Pκ∗ for all partitions κ. Closely related to Ψ is the operator [359] 1 −j+1 det (xi + 1/α)N −j + uxN E := Δi i Δ(x) i,j=1,...,N (cf. definition of Δij in Proposition 12.7.1). Expanding E = 1 + uE1 + · · · + uN EN we can show that Ek maps Pκ∗ to the space spanned by {Pλ∗ } with |λ| = |κ| + k.
(12.168)
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CHAPTER 12
P ROPOSITION 12.7.4 Let ek (x) denote the elementary symmetric functions (4.132) in x1 , . . . , xN of degree k. We have Ek Pκ∗ (x) = cκλ Pλ∗ (x), (12.169) λ:|λ|=|κ|+k
where the cκλ are such that
cκλ Pλ (x) = ek (x)Pκ (x).
(12.170)
λ:|λ|=|κ|+k
Proof. For (12.169) it suffices to check that Ek Pκ∗ (x) vanishes at x = μ + δN /α for all |μ| < |κ| + k. With φI defined above (12.165), we see 1 X Ek = φ I ΔI . (12.171) Δ(x) |I|=k
Now arguing as in the proof of Proposition 12.7.3 gives (12.169). The result (12.170) follows by recalling the structure (12.157), and noting that the degree preserving action of ΔI is just the identity while the highest degree terms in (Δ(x))−1 φI are equal to (Δ(x))−1 det[˜ cIij (x)] where j c˜Iij (x) = This in turn simplifies to
Q i∈I
xN−j+1 i xN−j i
for i ∈ I, for i ∈ / I.
xi , and tells us that the highest degree action of Ek is simply multiplication by ek (x).
It follows immediately from (12.170) and (12.169) that Ψ(ek Pκ ) = Ek Ψ(Pκ ),
(12.172)
and this in turn implies n n m m n n m Ψ(em j ek ) = Ψ(ek ej ) = Ej Ek · 1 = Ek Ej · 1,
n, m ∈ Z≥0 ,
or equivalently for analytic f symmetric in x1 , . . . , xN Ψf = f (E1 , . . . , EN ) · 1.
(12.173)
12.7.2 Binomial coefficients The classical binomial coefficients are related to the polynomials (12.61), fp (x), say, by l fp (l) = . p fp (p)
(12.174)
They furthermore occur in the expansion (1 + x)l =
l l p x , p p=0
(12.175)
which may alternatively be used to define the binomial coefficients, with (12.174) then a corollary. From the latter viewpoint, generalized symmetric binomial coefficients can then be defined by κ Pμ (x) Pκ (1 + x) = . (12.176) Pκ ((1)N ) μ Pμ ((1)N ) μ:|μ|≤|κ|
Then, in keeping with (12.174), the generalized symmetric binomial coefficients can be expressed in terms of the interpolation symmetric Jack polynomials [431].
583
JACK POLYNOMIALS
P ROPOSITION 12.7.5 We have
Pμ∗ (¯ κ/α) κ . = ∗ Pμ (¯ μ/α) μ
(12.177)
Proof. Let Dμ ({ξi }) be a symmetric polynomial of degree ≤ |μ| in the Cherednik operators {ξi }. Then, according to Taylor’s theorem, and making use too of the structure (12.97), we have that for some particular polynomials Dμ X Pκ (1 + x) = Dμ ({ξi })Pκ |x=(1)N Pμ (x). μ:|μ|≤|κ|
Since Dμ ({ξi })Pκ |x=(1)N = Dμ ({¯ κi })Pκ ((1)N ), this reads X
Pκ (1 + x) = Pκ ((1)N )
Dμ ({¯ κi })Pμ (x).
(12.178)
μ:|μ|≤|κ|
κi }) is a polynomial in {κi } of degree ≤ |μ| with the additional features that Now Dμ ({¯ j 0, |κ| ≤ |μ| (κ = μ), κi }) = Dμ ({¯ 1/Pμ ((1)N ), κ = μ. κ/α)/(Pμ∗ (μ/α)Pμ ((1)N )). Substituting in (12.178) and comparing But the only polynomial with these properties is Pμ∗ (¯ with (12.176) gives the result.
It follows from (12.155) that (12.177) is nonzero only if μ ⊆ κ. Hence (12.176) can be refined to read κ Pμ (x) Pκ (1 + x) = . (12.179) Pκ ((1)N ) μ Pμ ((1)N ) μ⊆κ
1. Use the fact that for κ = pN , Pκ (x) = xp1 · · · xpN , and the binomial theorem (12.133) with r = −p to show from (12.179) that for κ = pN ,
E XERCISES 12.7
“κ” μ
α|μ| [−p]μ . dμ (α)
=
(12.180)
12.8 PIERI FORMULAS 12.8.1 The Macdonald automorphism ωα Macdonald [376] introduced the operator ωα according to its action on the power sum basis, ωα pκ := α(κ) pκ ,
(12.181)
where (κ) denotes the length of κ as in Exercises 12.6 q.1. The action of this operator on the Cauchy-type product (12.122) and the symmetric Jack polynomials turns out to be very simple, leading to a useful variant of the identity (12.123). Consider first the product (12.122). From the identity of Exercises 12.6 q.3(ii) we see that (y)
ω−α
∞
∞ (−1)(κ) 1 = p (x)p (y) = (1 − xk yl ). κ κ zκ (1 − xk yl )1/α κ k,l=1 k,l=1
This result can be used to deduce the action of ω−α on Pκ . Setting yl = 0 for l > N we can substitute
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CHAPTER 12
(12.123) in the l.h.s to conclude hκ κ
dκ
N Pκ (x; α) ω−α Pκ (y; α) = (1 − xk yl ).
(12.182)
k,l=1
˜ (C) on Further progress relies on the following identity for the action of H
N
k,l=1 (1
− xk yl ).
˜1 given by P ROPOSITION 12.8.1 With E ˜1 = E
N
xj
j=1
∂ ∂xj
(12.183)
we have N N 1 ˜ (C)(y) N −1 (y) ˜ (x) ˜ ˜ (C)(x) − ( N − 1 + 1)E H + 1) E H (1 − x y ) = − − ( (1 − xk yl ). k l 1 1 α α α α →1/α k,l=1
k,l=1
(12.184) ˜ (C) , we see that there are terms on both sides independent of α, and terms Proof. From the definition (12.94) of H Q proportional to 1/α. The equality of these terms can be checked directly. For example, with Λ := terms independent of α on the l.h.s. are Λ
N N X X j=1 l,l =1
x2j yl yl , (1 − xj yl )(1 − xj yl )
N k,l=1 (1 − xk yl ),
the
(12.185)
l=l
while the terms independent of α on the r.h.s. are 2Λ
N X j,k=1 j=k
N yj yk X xl . yj − yk l=1 1 − yj xl
The final multiple summation can be rewritten by interchanging the summation labels j and k, then taking the arithmetic mean with the original form. This gives agreement with the term (12.185). Similar manipulation gives equality between the terms proportional to 1/α.
N If we substitute for k,l=1 (1 − xk yl ) in the l.h.s. of (12.184) according to the formula of (12.182), the ˜ (x) can be computed from the fact that it is an eigenoperator for ˜ (C)(x) − ((N − 1)/α + 1)E action of H 1 the symmetric Jack polynomials Pκ (x; α) with eigenvalue 2b(κ ) − α2 b(κ) (recall (12.34)). Making the same (α) (α) substitution on the r.h.s. and equating coefficients of Pκ (x) we see that ω−α Pκ (y) satisfies the eigenvalue equation 2 ˜ (C)(y) − N − 1 + 1 E ˜ (y) H b(κ) ω−α Pκ (y; α). ω P (y; α) = 2b(κ ) − −α κ 1 α α α →1/α Using the fact that Pκ (x; α) has leading term mκ (x), we see from (12.182) that (hκ /dκ )ω−α Pκ (y; α) must have leading term (−1)|κ| mκ (y). But the unique polynomial satisfying the eigenvalue equation with this property is (−1)|κ| Pκ (y; α), so we have the following result. P ROPOSITION 12.8.2 The Macdonald automorphism (12.181) acts on the symmetric Jack polynomials according to ω−α Pκ (x; α) = (−1)|κ|
dκ Pκ (x; 1/α) hκ
(12.186)
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JACK POLYNOMIALS
and consequently N
(1 − xk yl ) =
(−1)|κ| Pκ (x; α)Pκ (y; 1/α).
(12.187)
κ
k,l=1
The identity (12.187) is referred to as the dual Cauchy product type expansion ((10.56) is the case α = 1). 12.8.2 The Pieri and combinatorial formulas According to (12.121) the Schur polynomials are the special case α = 1 of the symmetric Jack polynomials Pκ (x; 1/α). Earlier, in (10.16) we defined the Schur polynomials in terms of semi-standard tableaux. Here the analogue of this combinatorial formula will be derived in the Jack case. This will be done by first providing the Jack analogue of the Pieri formula for Schur polynomials. The latter expresses em (x)sκ (x), where em (x) denotes the mth elementary symmetric function (recall (4.132)), in terms of {sμ (x)}. Its Jack analogue is the expansion of em (x)Pκ (x; α) in terms of {Pμ (x; α)}. The Pieri formula for Schur polynomials is easy to deduce by making use of the ratio of alternants formula (12.120), but before doing so it is necessary to introduce a definition relating to diagrams of partitions. D EFINITION 12.8.3 Let κ and λ be partitions described by their diagrams, and suppose κ ⊂ λ. The skew diagram λ/κ consists of those boxes of λ which are not in κ. A skew diagram is said to be a vertical (horizontal) m-strip if λ/κ consists of m boxes, all of which are in distinct rows (columns). P ROPOSITION 12.8.4 We have em (x)sκ (x) =
sλ (x).
(12.188)
λ λ/κ a vertical m strip
Proof. From (12.120) and the mth elementary function written in the form em (x) =
X
Y
M ∈{1,...,N } |M |=m
i∈M
xi
we have “ ” 1 Asym em (x)xδ+κ Δ(x) X 1 = Asym(xδ+κ+χM ) = Δ(x) M ⊆{1,...,N }
em (x)sκ (x) =
|M |=m
X
sκ+χM (x),
(12.189)
M ⊆{1,...,N } |M |=m
where χM = (χ(i ∈ M ))i=1,...,N with χ(i ∈ M ) = 1 if i ∈ M and χ(i ∈ M ) = 0 otherwise. In other words κ + χM is obtained from κ by adding 1 to κi for each i ∈ M . Now if κj = κj+1 and χ(j ∈ M ) = 0, χ(j + 1 ∈ M ) = 1 then κ + χM is not a partition. However, we see from (12.120) that then two columns in the definition of sκ+χM (x) are equal and so in such cases sκ+χM (x) = 0. Hence all nonzero contributions to (12.189) are of the form λ = κ + χM for λ/κ a vertical m-strip and (12.188) results.
The general structure of (12.188) persists in the Jack case [500], [376]. P ROPOSITION 12.8.5 For suitable coefficients ψλ/κ , em (x)Pκ (x) = λ λ/κ a vertical m strip
ψλ/κ Pλ (x).
(12.190)
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CHAPTER 12
Proof. We must have
X
em (x)Pκ (x) =
cκ,λ Pλ (x).
(12.191)
λ:|λ|=|κ|+m
By considering the leading term in the monomial basis on the l.h.s. of (12.191) we must have λ ≤ κ + 1m 0N−m . Applying (12.186) to both sides of (12.191), then replacing α by 1/α gives X c˜κ,λ Pλ (x) (12.192) P(m) (x)Pκ (x) = λ:|λ|=|κ|+m
for suitable c˜κ,λ proportional to cκ,λ . Consideration of the leading term on the l.h.s. of (12.192) shows λ ≤ κ +m1 0N−1 . This combined with the inequality λ ≤ κ + 1m 0N−m just derived implies (λ) = (κ) or
(λ) = (κ) + 1.
(12.193)
We will now show that these two classes of permitted terms in (12.191) have the further requirement that λ/κ is a vertical m-strip. Setting all terms but x1 , . . . , x(κ) in (12.192) equal to zero eliminates the terms with (λ) = (κ) + 1 and gives X c˜κ,λ Pλ (x1 , . . . , x(κ) ). (12.194) Pm1 0N −1 (x1 , . . . , x(κ) )Pκ (x1 , . . . , x(κ) ) = λ:|λ|=|κ|+m (λ)=(κ)
We now proceed by induction. Suppose (12.194), with the sum restricted to λ/κ a vertical m-strip, is true for partitions κ ˆ with |ˆ κ| < |κ| and (ˆ κ) ≤ (κ) (this is certainly the case for κ ˆ = 0(κ) ). Noting that Pκ (x1 , . . . , x(κ) ) = x1 · · · x(κ) Pκ −1 (x1 , . . . , x(κ) ) and applying the induction hypothesis to P(m) Pκ −1 gives that the sum in (12.194) can be restricted to λ/κ a vertical m-strip for all κ. It remains to obtain the same result for partitions λ such that (λ) = (κ) + 1 in (12.191) or (12.192). Such partitions can be isolated in (12.191) by equating coefficients of xκ1 1 +1 on both sides. Making use of (12.149) with x∗ = (x2 , . . . , xN ), N ≥ (κ) + 1, shows that (12.191) then reduces to X cκ,λ Pλ∗ (x∗ ). em−1 (x∗ )Pκ∗ (x∗ ) = λ:|λ|=|κ|+m (λ)=(κ)+1
After application of (12.186) this reads Pm−1 0N −1 (x∗ )Pκ −1 (x∗ ) =
X
c˜κ,λ Pλ −1 (x∗ ).
λ:|λ|=|κ|+m (λ)=(κ)+1
The same argument as used below (12.194) now gives that λ can be restricted to λ/κ a vertical m-strip, and (12.190) follows.
The Jack polynomial generalization of (12.188) can be determined by using the theory of the interpolation symmetric Jack polynomials [359]. P ROPOSITION 12.8.6 For each lattice point (i, j) of the diagram of a partition λ, define cλ (i, j) := α(λi − j) + (λj − i + 1) = αa(i, j) + (i, j) + 1, cλ (i, j) := α(λi − j + 1) + (λj − i) = α(a(i, j) + 1) + (i, j). In terms of these quantities, for κ ⊂ λ define = ψλ/κ
cλ (i, j)/c (i, j) λ , cκ (i, j)/cκ (i, j)
(i,j)
(12.195)
587
JACK POLYNOMIALS
where the product is taken over all pairs (i, j) ∈ λ such that κi = λi and κj < λj (i.e., rows such that λ/κ is empty, and columns such that λ/κ is non-empty). We have ψλ/κ Pλ (x). (12.196) em (x)Pκ (x) = λ λ/κ a vertical m strip
Proof. We seek the coefficients aκλ , λ/κ a vertical m-strip, such that em (x)Pκ (x) =
X
aκλ Pλ (x).
λ:|λ|=|κ|+m
From (12.172) this can equivalently be written Em Pκ∗ (x) =
X
aκλ Pλ∗ (x).
λ:|λ|=|κ|+m
From the vanishing property (12.156) a particular coefficient aκλ in this equation can be isolated by evaluation at x = ¯ λ/α. Thus, with I ⊂ {1, . . . , N }, ˛ ”˛ “ 1 X 1 1 ˛ ˛ Em Pκ∗ (x)˛ = ∗ ¯ φI (x)ΔI Pκ∗ (x) ˛ aκλ = ∗ ¯ ¯ ¯ x=λ/α x=λ/α Pλ (λ/α) Pλ (λ/α) Δ(x) |I|=m =
1 ∗ ¯ κ/α), φχM (λ/α)P κ (¯ ¯ ¯ Pλ∗ (λ/α)Δ( λ/α)
(12.197)
where the second equality follows from (12.171), and the third equality (in which M is such that λ = κ + χM ) makes further use of the vanishing property, together with the definition of ΔI given above (12.165). According to (12.158) and the definition (12.37) Q κ/α) Pκ∗ (¯ (i,j)∈κ cκ (i, j) m dκ m Q = α = α . (12.198) ¯ dλ Pλ∗ (λ/α) (i,j)∈λ cλ (i, j) ¯ := {i ∈ / M }. From the definitions we see that unless i ∈ M , j ∈ / J, We write M / M }, J := {λi |i ∈ M }, J¯ = {λi |i ∈ ¯, j ∈ ¯ cλ (i, j) = cκ (i, j). Thus (12.198) can be written cλ (i, j + 1) = cκ (i, j) while unless i ∈ M / J, Y Pκ∗ (¯ κ/α) 1 = αm ∗ ¯ c (i, 1) Pλ (λ/α) λ i∈M
Y
Y cκ (i, j) c (i, j + 1) ¯ i∈M,j∈J¯ λ i∈M ,j∈J
cκ (i, j) . cλ (i, j)
(12.199)
Also, from the definitions, ¯ Δ(λ/α) =
Y“
(λi + (N − i)/α) − (λj + (N − j)/α)
”
i<j
while it follows from the Vandermonde formula (1.173) and (12.165) that ” Y Y“ ¯ φχM (λ/α) = (λi + (N − i)/α) (λi + (N − i + χi∈M¯ )/α) − (λk + (N − k + χk∈M¯ )/α) . i∈M
i
To relate these products to those in (12.195), we note that for j ∈ J¯, ¯ |λk = j} = {λj+1 + 1, λj+1 + 2, . . . , κj }, {k ∈ M while if j = 0 this same formula holds with κ0 := N . Similarly, for j ∈ J {i ∈ M |λi = j} = {κj + 1, κj + 2, . . . , λj }.
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CHAPTER 12
Using these formulas we can rewrite the products to obtain Y ¯ φχM (λ/α) = α−m cλ (i, 1) ¯ Δ(λ/α) i∈M
Y
cλ (i, j + 1) cκ (i, j) ¯
i∈M,j∈J
Y ¯ ,j∈J i∈M
cλ (i, j) . cκ (i, j)
(12.200)
Multiplying together (12.199) and (12.200), and substituting in (12.197) gives (12.195).
Combinatorial formula A consequence of the dual Cauchy expansion (12.187), together with the Pieri formula (12.190), is a formula expressing Pκ (x) as a series in {xnN }n=0,...,κ1 [343], [376]. P ROPOSITION 12.8.7 With ψλ/κ = ψλ/κ (α) specified by (12.195), define
ψκ/μ =: ψκ/μ (α) = ψκ /μ (1/α).
(12.201)
We have Pκ (x1 , . . . , xN ) =
κ1
xrN
ψκ/μ Pμ (x1 , . . . , xN −1 ).
(12.202)
μ κ/μ a horizontal r strip
r=0
Proof. We must have Pκ (x1 , . . . , xN ; α) =
κ1 X
X
xrN
aκμ Pμ (x1 , . . . , xN−1 ; α)
μ |κ|=|μ|+r
r=0
for some aκμ . Substituting in the r.h.s. of (12.187) gives X
(−1)|κ| Pκ (y; 1/α)
κ
κ1 X
X
xrN
aκμ Pμ (x1 , . . . , xN−1 ; α).
(12.203)
μ |κ|=|μ|+r
r=0
On the l.h.s. of (12.187), we make use of (12.196) and lower dimensional cases of the r.h.s. by writing N Y
(1 − xk yl ) = N “X j=1
=
(1 − xN yj )
j=1
k,l=1
=
N Y
X μ
(−xN )j ej (y)
N Y
(1 − xk yl )
k=2,l=1
”X (−1)|μ| Pμ (y; 1/α)Pμ (x1 , . . . , xN−1 ; α) μ
(−1)|μ| Pμ (x1 , . . . , xN−1 ; α)
N X
X
(−xN )j
ψκ /μ (1/α)Pκ (y; 1/α).
κ κ /μ a vertical j strip
j=1
(12.204) Equating coefficients of Pκ (y; 1/α)Pμ (x1 , . . . , xN−1 ; α) in (12.203) and (12.204) gives aκμ = ψκ /μ (1/α) for κ /μ a vertical r-strip, or equivalently κ/μ a horizontal r-strip, and aκμ = 0 otherwise, as required by (12.202).
The combinatorial significance of (12.202) comes about because it can be expressed as a sum over semistandard tableaux. To see this let x = {x1 , . . . , xN } be elementary weights and in terms of these quantities define the weight W1 (T ) of a semi-standard tableau T of content N as
s . Wx (T ) = x1#1 s x2#2 s · · · x#N N
(12.205)
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JACK POLYNOMIALS
Furthermore, let pi,j denote the number entered in the diagram of κ at site (i, j). Define the partitions κ(h) in terms of their diagram by κ(h) = {(i, j) ∈ κ | pi,j ≤ h}. With αh := #h’s we have that κ(h) /κ(h−1) is a horizontal αh -strip, and T can be uniquely specified in terms of the sequence of partitions κ(0) , κ(1) , . . . , κ(N ) (note κ(N ) = κ). Consider now the general structure N
gκ (x1 , . . . xN ) :=
T shape κ
c(κ(h) /κ(h−1) )Wx (T ).
h=1
This satisfies the recurrence relation gκ (x1 , . . . , xN ) =
N
xrN
N
c(κ/μ)
μ T κ/μ a horizontal r strip shape μ
r=0
=
xrN
N −1
c(κ(h) /κ(h−1) )Wx1 ,...,xN −1 (T )
h=1
c(κ/μ)gμ (x1 , . . . , xN −1 ),
(12.206)
μ κ/μ a horizontal r strip
r=0
where in obtaining the final line the fact that κ(h) = μ(h) for h = 0, 1, . . . , N − 1 has been used. Comparing (12.206) with (12.202) gives the sought combinatorial formula for the Jack polynomials [500]. P ROPOSITION 12.8.8 With ψκ/μ = ψκ/μ (α) specified by (12.201) and Wx (T ) by (12.205), we have Pκ (x1 , . . . , xN ; α) =
N T shape κ
ψκ(h) /κ(h−1) (α)Wx (T ).
(12.207)
h=1
Note that when α = 1, (12.207) reduces to the combinatorial definition of the Schur polynomial in (10.16), and furthermore (12.202) reduces to the recurrence (10.18). 12.8.3 An average over the Dixon-Anderson density The binomial expansion (12.176) can be used in the recurrence (12.202) to deduce an identity which in turn can be used to compute the average of a Jack polynomial with respect to the Dixon-Anderson density (4.11) [431], [369], [361]. P ROPOSITION 12.8.9 We have ¯ = Pμ∗ (λ/α)
∗
(α)
[N/α]μ
(α)
[(N − 1)/α]μ
ψλ/ν
Pν ((1)N −1 ) ∗ P (¯ ν /α), Pλ ((1)N ) μ
(12.208)
where the sum * is over partitions ν of N − 1 parts, such that λ1 ≥ ν1 ≥ λ2 ≥ ν2 ≥ · · · ≥ νN −1 ≥ λN .
(12.209)
Proof. Setting xN = 1 in (12.202) shows Pλ (x1 , . . . , xN−1 , 1) =
X∗
ψλ/ν Pν (x1 , . . . , xN−1 ).
Next set xj = 1 + yj (j = 1, . . . , N − 1), make use of the binomial expansion (12.176) on both sides, and equate
590
CHAPTER 12
coefficients of Pμ (y1 , . . . , yN−1 ) to deduce Pλ ((1)N )
“λ” μ
=
“ν ” Pμ ((1)N ) X∗ ψλ/ν Pν ((1)N−1 ) . N−1 μ Pμ ((1) )
The result (12.208) follows from this upon use of (12.177) and (12.105).
Suppose λ in (12.208) is replaced by ax, where a is a scalar such that each component of ax is a nonnegative integer. Taking the limit a → ∞ allows the component of x to take on continuous values, and we ¯ have Pμ∗ (λ/α) ∼ Pμ∗ (ax), while (12.157) gives lima→∞ Pμ∗ (ax)/a|μ| = Pμ (x). Applying the same scaling to the r.h.s. of (12.208), replacing ν → ay, and making use of (12.105), (12.59) and (12.211) below gives the evaluation of the average of a Jack polynomial with respect to the Dixon-Anderson density (4.11). P ROPOSITION 12.8.10 Let DAN (1/α) refer the Dixon-Anderson density (4.11) with n → N , s1 , . . . , sn = 1/α, {λj } → {yj }, {aj } → {xj }. For (μ) ≤ N − 1 we have (α)
Pμ (x) =
[N/α]μ
(α)
[(N − 1)/α]μ
Pμ (y)DAN (1/α) .
(12.210)
Note that with xN = 0 (12.210) can be interpreted as an integral eigenoperator equation for Jack polynomials in N − 1 variables. E XERCISES 12.8
1. [376] Let gnk (λ, ν) =
Y 1≤i≤j≤n
(νi − λj+1 + k(j − i) + 1)k−1 (λi − νj + k(j − i) + 1)k−1 . (λi − λj+1 + k(j − i) + 1)k−1 (νi − νj + k(j − i) + 1)k−1
Use an induction procedure similar to that used in Exercises 12.4 q.3 to show that with ψκ/μ specified according to (12.201) and (12.195), 1/α (12.211) ψκ/μ = gN (κ, μ). (C)
2. [343] In this exercise a recursion for Nκ
is established.
(i) Show from the definition (4.132) that em
“1 t1
,...,
N 1 ” Y −1 = ti eN−m (t1 , . . . , tN ), tN i=1
and show that analogous to the formula of Exercises 12.1 q.1, z p Pκ (z) = Pκ+pN (z). (ii) From the results of (i) deduce that em (t)Pκ−χM (t)Pκ (1/t) = Pκ+χM¯ (t)eN−m (1/t)Pκ (1/t), where χM is as in (12.189). (iii) Expand em (t)Pκ−χM (t) and eN−m (1/t)Pκ (1/t) according to (12.196), set tj = e2πixj and take the inner product (12.12) of both sides to deduce Nκ(C) =
ψ(κ+χ ¯ )/κ M
ψκ/(κ−χ M)
(C)
Nκ−χM .
Check that this is consistent with the result of Proposition 12.6.3. 3. [537] Let K(λ1 , λ2 , λ)[Pμ (y)] be defined as in (4.17) but with Pμ (y) included in the integrand. By making use of (12.210) obtain the analogue of the first formula for K in Proposition 4.2.2, and use the same argument as that
591
JACK POLYNOMIALS
used in the proof of Proposition 4.2.2 to derive the second formula for K therein. Hence conclude (α)
SN+1 (λ1 , λ2 , 1/α)[Pν ] [(N + 1)/α]μ = (α) SN+1 (λ1 , λ2 , 1/α) [N/α]μ
SN (λ1 + 1/α, λ2 + 1/α, 1/α)[Pμ ] , SN (λ1 + 1/α, λ2 + 1/α, 1/α)
(μ) ≤ N.
Now use the symmetric Jack analogue of Exercises 12.1 q.1 to show that with (μ) ≤ N + 1 this implies (α)
[(N + 1)/α]μ−μN +1 SN (λ1 + 1/α, λ2 + 1/α, 1/α)[Pμ−μN +1 ] SN+1 (λ1 − μN+1 , λ2 , 1/α)[Pν−μN +1 ] = . (α) SN+1 (λ1 , λ2 , 1/α) SN (λ1 + 1/α, λ2 + 1/α, 1/α) [N/α]μ−μN +1 Making use of (12.105), (12.59), show that this can be iterated to reclaim (12.143). 4. Replace yl by 1/yl and make use of the results of Exercises 12.1 q.1 and q.2 (as they apply to symmetric Jack polynomials) to rewrite (12.187) to read N Y
(xk − yl ) =
k,l=1
where μ = N N − κR .
X κ
(−1)|μ| Pκ (x; α)Pμ (y; 1/α),
(12.212)
Chapter Thirteen Correlations for general β Our ability to calculate correlations for log-gas systems beyond the special random matrix couplings β = 1, 2 and 4 relies on Jack polynomial theory. One application of the theory is to particular generalizations of the Selberg integral—the so-called Selberg correlation integrals—which allow the exact calculation of the two particle correlation for the log-gas on a circle at even β. Another is to the exact calculation of the densitydensity correlation in the case of two different initial conditions: the first corresponding to a perfect gas and the second, which is restricted to β rational, to the equilibrium state. In the latter case, with β = p/q in reduced form, the correlation is expressed as a (p + q)-dimensional integral, with p integrals on [0, 1] and q integrals on [1, ∞), and is closely related to the Dotsenko-Fateev integral from Chapter 4. A consequence is a functional equation relating the static structure functions at β and 4/β. These applied studies motivate further development of Jack polynomial theory, in particular to generalized hypergeometric functions, generalized classical polynomials and zonal polynomials. Two other topics relating to general β are also given in this chapter. One is the inter-relations between spacing distributions, linking β = 2(r + 1) to β = 2/(r + 1) (r ∈ Z+ ), which follow as a consequence of the generalization of the Dixon-Anderson integral given in Chapter 4; the other is the stochastic differential equations specifying the bulk and edge distributions.
13.1 HYPERGEOMETRIC FUNCTIONS AND SELBERG CORRELATION INTEGRALS 13.1.1 Generalized hypergeometric functions The Cauchy-type identities (12.123) and (12.187), used in conjunction with the generalized Selberg integral (12.143), give rise to a further class of generalized Selberg integrals (the Selberg correlation integrals [19], [342], [185]) of direct relevance to the calculation of correlation functions in the log-gas, and displays their relationship to generalized hypergeometric functions. First the latter quantities will be specified [342], [373], [550]. (α)
D EFINITION 13.1.1 Define the renormalized symmetric Jack polynomial Cκ (x) by Cκ(α) (x) :=
α|κ| |κ|! Pκ (x; α) dκ
(13.1)
(α)
and let [u]κ denote the generalized Pochhammer symbol as defined by (12.46). In terms of these quantities, (α) the generalized hypergeometric functions p Fq are specified by (α) p Fq (a1 , . . . , ap ; b1 , . . . , bq ; x1 , . . . , xm ) :=
(α) 1 [a1 ](α) κ · · · [ap ]κ C (α) (x1 , . . . , xm ). (α) (α) κ |κ|! [b1 ]κ · · · [bq ]κ κ (α)
(α)
(13.2)
Since in the one-variable case we have κ = k, Ck (x) = xk and [u]k = (u)k , we see that with m = 1 (α) the generalized hypergeometric function p Fq reduces to the classical hypergeometric function, specified in the case p = 2, q = 1 by (5.83). We note that for general number of variables, in the cases p = q = 0 and
593
CORRELATIONS FOR GENERAL β
p = 1, q = 0, (13.2) can be expressed in terms of elementary functions which are independent of α. Thus it follows from (12.134) that (α) 0 F0 (x1 , . . . , xm )
= ex1 +···+xm
(13.3)
while the generalized binomial theorem (12.133) implies (α)
1 F0 (a; x1 , . . . , xm ) =
m
(1 − xj )−a .
(13.4)
j=1
In fact these results are not independent. Indeed (13.3) can be deduced as a limiting case of (13.4) by making use of the general formula lim F (α) (a1 , . . . , ap ; b1 , . . . , bq ; x1 /ap , . . . , xm /ap ) ap →∞ p q = p−1 Fq(α) (a1 , . . . , ap−1 ; b1 , . . . , bq ; x1 , . . . , xm ), (α)
(13.5)
(α)
which follows from the explicit form (12.46) of [ap ]κ and the fact that Cκ (x) is homogeneous of degree |κ|. (α) Next we will show that it is the generalized hypergeometric function 2 F1 that is related to Selberg correlation integrals. P ROPOSITION 13.1.2 We have 1/2 1/2 N m 1 eπixl (a−b) |1 + e2πixl |a+b dx1 · · · dxN (1 + tl e2πixl ) MN (a, b; 1/α) −1/2 −1/2 l=1 l =1 (1/α) 2πixk 2πixj 2/α × |e −e | = 2 F1 (−N, αb; −(N − 1) − α(1 + a); t1 , . . . , tm ), (13.6) j
1 SN (λ1 , λ2 ; 1/α) =
1
1
dx1 · · ·
0
(1/α) (−N, −(N 2 F1
dxN 0
m N 2/α λ1 λ2 xl (1 − xl ) (1 − tl xl ) |xj − xk | l =1
l=1
j
− 1) − α(λ1 + 1); −2(N − 1) − α(λ1 + λ2 + 2); t1 , . . . , tm ).
(13.7)
Proof. We will give only the derivation of (13.6); (13.7) follows from similar reasoning. Consider the r.h.s. of (12.142). According to (12.105), bκ α|κ| [N/α]κ = , (13.8) hκ hκ where the second equality follows form (12.104). Next we write all the terms on the r.h.s. in terms of the conjugate partition κ . From the definitions (12.58) and (12.37) and the facts that for κ → κ , a(i, j) ↔ l(i, j), we see that hκ = α|κ| dκ |α→1/α . This together with the formula of Exercises 12.4 q.2 shows that the r.h.s. can be written (α)
Pκ ((1)N ) =
(−1)|κ| α−|κ| [−N ]κ
(1/α)
(1/α)
[αb]κ
(1/α)
dκ |α→1/α [−(N − 1) − α(1 + a)]κ
.
With this substitution, now multiply both sides of (12.142) by (−1)|κ| Pκ (t1 , . . . , tm ; 1/α) and make use of (12.187) on the l.h.s. and the formulas (13.1) and (13.2) (the latter with p = 2, q = 1 and κ replaced by κ ) on the r.h.s.
Also of interest are Selberg correlation integrals based on the Cauchy type formula (12.123).
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CHAPTER 13
P ROPOSITION 13.1.3 We have 1/2 1/2 N m 1 eπixl (a−b) |1 + e2πixl |a+b dx1 · · · dxN (1 + tl e2πixl )−1/α MN (a, b, 1/α) −1/2 −1/2 l=1 l =1 N 1 (α) , −b; (N − 1) + a + 1; t1 , . . . , tm , (13.9) |e2πixk − e2πixj |2/α = 2 F1 × α α j
1 1 m N 1 2/α xλl 1 (1 − xl )λ2 dx1 · · · dxN (1 − tl xl )−1/α |xj − xk | SN (λ1 , λ2 , 1/α) 0 0 l=1 l =1 j
Finally we will note the analogous multidimensional integral formulas which result from the generalized binomial theorem (12.133). P ROPOSITION 13.1.4 We have 1/2 1/2 N 1 dx1 · · · dxN eπixl (a−b) |1 + e2πixl |a+b (1 + te2πixl )−r MN (a, b, 1/α) −1/2 −1/2 l=1 1 (α) 2πixk 2πixj 2/α r, −b; (N − 1) + a + 1; (t)N , × |e −e | = 2 F1 α
(13.11)
1≤j
1 1 N 1 2/α dx1 · · · dxN xλl 1 (1 − xl )λ2 (1 − txl )−r |xj − xk | SN (λ1 , λ2 , 1/α) 0 0 l=1 j
(13.12)
Proof. These formulas follow immediately from (12.142) and (12.143) upon multiplying by α|κ| [r](α) κ /dκ and summing
over κ using (12.133) on the l.h.s. and (13.8), (13.1), (13.2) on the r.h.s.
An immediate consequence of Proposition 13.1.4 is that it provides a multidimensional generalization of the classical Gauss summation formula for 2 F1 at t = 1. Thus by observing that with t = 1 the integral in the second formula becomes an example of the Selberg integral (4.1) we see that [550] 1 S (λ , λ − r, 1/α) 2 N 1 2 (α) , (13.13) r, (N − 1) + λ1 + 1; (N − 1) + λ1 + λ2 + 2; (1)N = 2 F1 α α SN (λ1 , λ2 , 1/α) or equivalently, after making use of (4.3), (α)
N 2 F1 (a, b; c; (1) ) =
N −1 j=0
Γ(c − j/α)Γ(c − a − b − j/α) . Γ(c − a − j/α)Γ(c − b − j/α)
(13.14)
595
CORRELATIONS FOR GENERAL β
13.1.2 Holonomic partial differential equations In general a differential equation (or set of differential equations) is said to be holonomic if it generates a unique (up to normalization) solution of a given type. A classical example is the differential equation for the hypergeometric function 2 F1 (a, b; c; t), dF d2 F − abF = 0. t(1 − t) 2 + c − (a + b + 1)t dt dt Seeking a power series solution of this equation gives a recurrence relation for the coefficients, the solution to which gives (5.83). It turns out [342] (see also [550], [373]) that this property extends to (13.2) with p = 2, q = 1 and general m. P ROPOSITION 13.1.5 For c − α1 (i − 1) ∈ / Z≤0 , i = 1, . . . , m, each of the partial differential equations ∂F 1 ∂2F 1 − abF tp (1 − tp ) 2 + c − (m − 1) − (a + b + 1 − (m − 1))tp ∂tp α α ∂tp m ∂F ∂F 1 1 tp (1 − tp ) = 0, (13.15) − tj (1 − tj ) + α j=1 tp − tj ∂tp ∂tj j=p
p = 1, . . . , m has the same symmetric formal power series solution equal to unity at the origin. This solution can be calculated by recurrence and is given by the series (13.2) with p = 2, q = 1, which is analytic in the neighbourhood of the origin for |tj | < 1, j = 1, . . . , m. The proof of Proposition 13.1.5, which is a generalization of the proof given by Muirhead [410] in the case α = 2, first establishes the uniqueness property by transforming the equations in terms of elementary symmetric functions and establishing a recurrence relation for the coefficient. Next it is shown that there is a unique series solution of the equations (13.15) summed over p from 1 to m of the form F = ακ Cκ(α) (t1 , . . . , tm ), κ
provided the coefficients ακ are independent of m. This is done by establishing a recurrence equation for the ακ . Finally the recurrence equation is solved to give (13.2) in the case p = 2, q = 1. In Exercises 13.1 q.1 it will be shown directly that the Selberg correlation integral (13.7) satisfies the set of partial differential equations (13.15). In the remainder of this subsection, some consequences of Proposition 13.1.5 will be discussed by way of (α) transformation formulas for the generalized hypergeometric function 2 F1 . First we note the analogue of the classical Kummer relations [550]. P ROPOSITION 13.1.6 We have (α)
2 F1 (a, b; c; t1 , . . . , tm ) =
=
m j=1 m
(1 − tj )−a 2 F1
(α)
t1 tm a, c − b; c; − ,...,− 1 − t1 1 − tm (α)
(1 − tj )c−a−b 2 F1 (c − a, c − b; c; t1 , . . . , tm ).
j=1
Proof. These formulas are established by verifying that the functions on the r.h.s. satisfy the p.d.e.’s (13.15) for which (α)
use is made of the particular form of the p.d.e.’s satisfied by the 2 F1 functions on the r.h.s. Because the functions of the r.h.s. are analytic at the origin, according to Proposition 13.1.5 the fact that they satisfy the p.d.e.’s (13.15) implies that (α) they are equal to the function 2 F1 (a, b; c; t1 , . . . , tm ).
Replacing tj by tj /b (j = 1, . . . , m) in the second equality of Proposition 13.1.6, taking b → −∞ and
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CHAPTER 13
using (13.5) gives a generalization of the second Kummer formula, (α)
1 F1 (a; c; t1 , . . . , tm ) =
m
(α)
etj 1 F1 (c − a; c; −t1 , . . . , −tm ).
(13.16)
j=1 (α)
When a or b ∈ Z≤0 for some 1 ≤ i ≤ m, the series (13.2) defining 2 F1 terminates. In such cases the method of proof of Proposition 13.1.6 shows that a further transformation formula holds. P ROPOSITION 13.1.7 Subject to the above noted condition for the termination of the series, (α)
2 F1 (a, b; c; t1 , . . . , tm ) =
(α) 2 F1 (a, b; a + b + 1 + (m − 1)/α − c; 1 − t1 , . . . , 1 (α) m 2 F1 (a, b; a + b + 1 + (m − 1)/α − c; (1) )
− tm )
where the denominator on the r.h.s. can be evaluated according to (13.13). 13.1.3 Hypergeometric functions of two sets of variables The generalized hypergeometric function (13.2) can be obtained as a special case of a more general class of generalized hypergeometric functions based on two sets of variables. D EFINITION 13.1.8 With Eη (x) := Eη (x1 , . . . , xn ; α) denoting the nonsymmetric Jack polynomial of n variables, let (α) |η| [a1 ](α) Eη (x)Eη (y) η + · · · [ap ]η + α . p Kq (a1 , . . . , ap ; b1 , . . . , bq ; x1 , . . . , xn ; y1 , . . . , yn ) := (α) (α) d Eη (1n ) η η [b1 ] + · · · [bq ] + η
(13.17)
η
Setting y1 = · · · = yn = 1 in (13.17), using (12.96) and comparing with the definition (13.2) we see that n p Kq (a1 , . . . , ap ; b1 , . . . , bq ; x1 , . . . , xn ; (1) )
= p Fq(α) (a1 , . . . , ap ; b1 , . . . , bq ; x1 , . . . , xn ).
(13.18)
Furthermore, by symmetrizing the function p Kq in the variables x1 , . . . , xn (or y1 , . . . , yn ) a hypergeometric function of two sets of variables results which contains the symmetric Jack polynomials in its definition. To show this requires first establishing a transformation property of (13.17) under the action of the elementary (x) transposition operator si [39]. (x) (y) P ROPOSITION 13.1.9 Let F = η Aη Eη (x)Eη (y). Then si F = si F if and only if the coefficients Aη satisfy 1 Asi η = 1 − ¯2 Aη , ηi > ηi+1 , δi,η or equivalently
1 −1 Aη , Asi η = 1 − ¯2 δi,η
ηi < ηi+1 .
Proof. This follows from the formulas for si Eη given in Proposition 12.2.1. Making use of (12.38) we see that the conditions of Proposition 13.1.9 are met by the definition (13.17) of so we have
p Kq ,
(x)
si
p Kq (a1 , . . . , ap ; b1 , . . . , bq ; x1 , . . . , xn ; y1 , . . . , yn ) (y)
= si
p Kq (a1 , . . . , ap ; b1 , . . . , bq ; x1 , . . . , xn ; y1 , . . . , yn ).
This result in turn can be used to determine the form of p Kq under the action of Sym(x) .
(13.19)
597
CORRELATIONS FOR GENERAL β
P ROPOSITION 13.1.10 Let (α) p Fq (a1 , . . . , ap ; b1 , . . . , bq ; x1 , . . . , xn ; y1 , . . . , yn )
:=
(α) (α) (α) 1 [a1 ](α) κ · · · [ap ]κ Cκ (x1 , . . . , xn )Cκ (y1 , . . . , yn ) . (α) (α) |κ|! [b1 ](α) Cκ ((1)n ) κ · · · [bq ]κ κ
(13.20)
We have Sym(x) p Kq (a1 , . . . , ap ; b1 , . . . , bq ; x1 , . . . , xn ; y1 , . . . , yn ) = n! p Fq (a1 , . . . , ap ; b1 , . . . , bq ; x1 , . . . , xn ; y1 , . . . , yn ).
(13.21)
Proof. For convenience we first abbreviate (13.17) by writing p Kq
=
X
Aη Eη (x)Eη (y).
η
Making use of (12.100) gives Sym(x) p Kq =
X
Aη aη Pη+ (x)Eη (y).
(13.22)
η
Suppose we now make a further application of Sym(x) . On the l.h.s. of the above formula a factor of n! results. However, on the r.h.s. we can use the formula (13.19) to replace Sym(x) with Sym(y) and compute its action again using (12.100) to conclude 1 X Aη a2η Pη+ (x)Pη+ (y). Sym(x) p Kq = n! η Recalling (12.101) and making use of (12.96) in the case z = (1)n gives the sought result.
The hypergeometric function of two sets of variables can be used to provide an integral representation of (α) 2 F1 (a, b; c; t1 , . . . , tN ) [550]. P ROPOSITION 13.1.11 We have 1 1 2 (α) a, (N − 1) + λ1 + 1; (N − 1) + λ1 + λ2 + 2; t1 , . . . , tN = 2 F1 α α SN (λ1 , λ2 , 1/α ×
1
dx1 · · ·
0
0
1
(α)
dxN 1 F0 (a; t1 , . . . , tN ; x1 , . . . , xN )
N
xλl 1 (1 − xl )λ2
|xk − xj |2/α .
1≤j
l=1
(13.23) Proof. This follows by multiplying both sides of (12.143) by (α)
(α)
[a]κ Cκ (t) |κ|! Pκ ((1)N ) and summing over N .
Since, from the definition (13.20), together with the formula (13.4), we have (α) N 1 F0 (a; (t) ; x1 , . . . , xN )
(α)
= 1 F0 (a; tx1 , . . . , txN ) =
N j=1
and we see that (13.12) follows as a special case of the formula (13.23).
(1 − txj )−a ,
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CHAPTER 13
E XERCISES 13.1
1. [342] The objective of this exercise is to show that the Selberg correlation integral in (13.7) satisfies the partial differential equations (13.15) for suitable values of the parameters. For this purpose let Z SN,m [f ] :=
dx1 . . . dxN f [0,1]N
N “ Y
xλj 1 (1 − xj )λ2
j=1
m Y
(xj − tk )
k=1
”
Y
|xk − xj |2λ ,
1≤j
where f may be an operator acting on all terms to the right.
(i) Use the three equations 0 = SN,m
N N N hX hX i hX ∂ i ∂ ∂ 1 i xj , 0 = SN,m , 0 = SN,m ∂xj ∂xj ∂xj xj − tp j=1 j=1 j=1
(p = 1, . . . , m) (cf. (4.128) and (4.129)), which follow from the fundamental theorem of calculus, to show that for λ1 , λ2 > 0 0 = λ1 SN,m
m N N N X hX hX hX 1i 1 i 1 i − λ2 SN,m + SN,m , xj 1 − xj x j − tk j=1 j=1 j=1 k=1
0 = N (1 + λ1 + λ2 + m + (N − 1)λ)SN,m [1] − λ2 SN,m
N hX j=1
hXX t i 1 i k + SN,m , 1 − xj x j − tk j=1 m
N
k=1
and 0 = −2λSN,m
h
X
hX i λ 1 1 1 i 1 SN,m ( − ) + (xj − tp )(xk − tp ) tp x j − tp xj j=1 N
1≤j
hX“ 1 hX“ 1 1 ”i X 1 1 ”i λ2 SN,m + SN,m − + . 1 − tp x j − tp 1 − xj tp − tl x j − tp x j − tl l=1 j=1 j=1 N
−
m
N
l=p
(ii) By regarding λ1 SN,m
N hX 1i xj j=1
and
λ2 SN,m
N hX j=1
1 i 1 − xj
as unknowns in the first two equations of (i) obtain the equations λ1 SN,m
m N N X hX hX 1i 1 − tk i = N (1 + λ1 + λ2 + m + (N − 1)λ)SN,m [1] − SN,m , xj x j − tk j=1 j=1 k=1
λ2 SN,m
N hX j=1
hXX t i 1 i k = N (1 + λ1 + λ2 + m + (N − 1)λ)SN,m [1] + SN,m , 1 − xj x j − tk j=1 N
m
k=1
599
CORRELATIONS FOR GENERAL β
and substitute these equations in the third equation of (i) to show i h X 1 0 = −2λSN,m (xj − tp )(xk − tp ) „ +
1≤j
λ1 λ2 − tp 1 − tp
−SN,m
m N X hX
SN,m
N hX j=1
1“ 1 i − N (1 + λ1 + λ2 + m + (N − 1)λ)SN,m [1] x j − tp tp
1 “ 1 − tk i” − N (1 + λ1 + λ2 + m + (N − 1)λ)SN,m [1] x j − tk 1 − tp
j=1 k=1
+SN,m
«
m N X hX
N m hX tk i” 1 X 1 1 i − + SN,m . x j − tk tp − tl j=1 xj − tp x j − tl l=1
j=1 k=1
l=p
(iii) Show from the definition of SN,m := SN,m [1] that hX 1 i ∂SN,m = −SN,m , ∂tp x j − tp j=1 N
h ∂ 2 SN,m = 2SN,m 2 ∂tp
X 1≤j
i 1 (xj − tp )(xk − tp )
and substitute in the final equation of (ii) to obtain the p.d.e. (13.15) with α → λ,
a = −N,
b = (λ1 + λ2 + m + 1)/λ + N − 1,
c = (λ1 + m)/λ,
F = SN,m . (13.24)
(iv) Write the general solution F of (13.15) as F =
m Y
˜ t−a j F (1/t1 , . . . , 1/tm ),
j=1
and show that F˜ (t1 , . . . , tm ) satisfies the same p.d.e.’s with b → a − c + 1 + (m − 1)/α,
c → a − b + 1 + (m − 1)/α.
(v) Use the results of (iii) and (iv) to show that the Selberg correlation integral in (13.7) satisfies the system of equations (13.15) with α → 2.
1 , α
a = −N,
b = −(N − 1) − α(λ1 + 1),
c = −2(N − 1) − α(λ1 + λ2 + 2).
(i) Use the result of q.1(iii) and Proposition 13.1.5 to show 1 SN (λ1 + m, λ2 , λ) =
Z dx1 · · · dxN
[0,1]N
(λ) 2 F1 (−N, (λ1
N “ Y
xλj 1 (1 − xj )λ2
j=1
m Y
(xj − tk )
”
k=1
Y 1≤j
+ λ2 + m + 1)/λ + N − 1, (λ1 + m)/λ; t1 , . . . , tm ).
Use this in the case m = 1 together with (5.92) to show N DY
(t − xj )
j=1
E
(α,β)
ME2λ,N (xλ1 (1−x)λ2 )
with α = (λ1 + 1)/λ − 1, β = (λ2 + 1)/λ − 1.
∝ PN
|xk − xj |2λ
(1 − 2t)
(13.25)
600
CHAPTER 13
(j)
(ii) Replace the wi , (i = 0, 1, 2) in (4.49) by their mean values and verify that the resulting recurrence is ˆ (α, ˆ β) (1 − 2x) where α ˆ = (α(j) + 1)/λ − 1, βˆ = (β (j) + 1)/λ − 1 and cj is satisfied by A# j (x) = cj Pj # chosen such that Aj (x) is monic. Note that this is consistent with the result of (i). (iii) Show that in the case λ = 1, the result of (i) is consistent with the first formula in Proposition 5.1.4. (iv) Use the first formula in Exercises 6.4 q.1 to show that the result of (i) with λ = formula in (6.98).
1 2
is consistent with the first
(α)
3. Set tp → tp /b in (13.15) and take the limit b → ∞ to conclude that 1 F1 (a; c; t) satisfies each of the p.d.e.’s tp
m “ ” ∂F 1 1 X ∂2F 1 “ ∂F ∂F ” + c − − aF + − t (m − 1) − t t = 0, p p j ∂t2p α ∂tp α j=1 tp − tj ∂tp ∂tj
(13.26)
j=p
and that these uniquely determine this function. 4.
(i) By making the replacement t → t/r and taking the limit r → ∞, show from (13.11) that 1 MN (a, b, 1/α)
Z
1/2 −1/2
Z dx1 · · ·
N Y
1/2
dxN −1/2
2πixl
eπixl (a−b) |1+e2πixl |a+b e−te
l=1
Y
|e2πixk −e2πixj |2/α
j
(α)
= 1 F1 (−b; a + 1 + (N − 1)/α; (t)N ). (ii) Consider the case a, b, 2/α ∈ Z≥0 when the integrand in (i) is analytic. Argue that in this case it is possible to replace e2πixj by Re2πixj , R > 0. Now make the replacement t → −t/b, put R = bt−1/2 and take the limit b → ∞ to show that [187] “ 1 ”(c−1)N/2 (α) N 0 F1 (c + (N − 1)/α; (t) ) = BN (c, α) t N “ 1 ”N Z Y Y 1/2 × e2t cos θj ei(c−1)θj |eiθk − eiθj |2/α dθ1 · · · dθN , (13.27) 2π [−π,π]N j=1 1≤j
where c ∈ Z≥1 and BN (c, α) =
N Y Γ(1 + 1/α)Γ(c + (j − 1)/α) . Γ(1 + j/α) j=1
(13.28)
Argue that this is valid for general α. 5. [38] The objective of this exercise is to deduce the identity ep1 (x) 0 K0 (x; y) = 0 K0 (x; y + 1),
(13.29)
where p1 (x) is specified by (12.146). (y)
(y+1)
(y)
(y)
(y)
(i) Let ξi refer to (11.57) with z → y. By noting ξi = ξi +αdi , and using the fact that ξi (x) ξi 0 K0 (x; y) together with the second identity in (13.86) below, deduce that ” “ (y+1) p1 (x) (x) e ξi ep1 (x) 0 K0 (x; y) . 0 K0 (x; y) = ξi (ii) Note from the definition (13.17) that ep1 (x) 0 K0 (x; y) =
X η
α|η|
dη Uη (y)Eη (x) dη eη
for some polynomial Uη (y) with leading term Eη (y). Now use the result from (i) to deduce (y+1)
ξi
Uη (y) = η¯i Uη (y),
0 K0 (x; y)
=
601
CORRELATIONS FOR GENERAL β
and thus conclude Uη (y) is equal to Eη (y + 1), as required by (13.29). (iii) Analogous to (12.176) the nonsymmetric binomial coefficients can be defined by X „ η « Eν (z) Eη (1 + z) = . Eη ((1)N ) ν Eν ((1)N )
(13.30)
ν:|ν|≤|η|
Use this to substitute for the Eη (y + 1) which occur on the r.h.s. of (13.29), and equate coefficients of Eη (y) on both sides to obtain X |ν| 1 „ ν « 1 Eν (x). ep1 (x) Eη (x)α|η| = α (13.31) dη dν η ν 6. [342], [38] Here the symmetric analogues of q.5 are considered. (i) Symmetrize (13.29) and so obtain (α)
(α)
ep1 (x) 0 F0 (x; y) = 0 F0 (x; y + 1). (ii) Substitute for Pκ (y + 1) in the series expansion of the r.h.s. of the identity in (i) using (12.179) and equate coefficients of Pκ (y) to deduce the symmetric analogue of (13.31) ep1 (x) Cκ(α) (x) =
X “ μ ” |κ|! μ
κ |μ|!
Cκ(α) (x).
(iii) By equating terms of degree |κ| + 1 on both sides of the identity of (ii) note that X
p1 (x)Pκ (x) = α
μ:|μ|=|κ|+1
“ μ ” d
κ
κ dμ
Pμ (x).
By comparing this with (12.196) in the case m = 1 read off that for |μ| = |κ| + 1, “μ” κ
=
1 dμ ψμ/κ . α dκ
13.2 CORRELATIONS AT EVEN β 13.2.1 Circular β-ensembles The generalizations of the Selberg integral presented in Section 13.1.1 allow the general n-point correlation for the log-gas on a circle at even β, or equivalently the circular β-ensemble, to be expressed as a particular (β/2) generalized hypergeometric function 2 F1 in β(n − 1)/2 variables. Suppose for convenience that there are N + n particles in the system. According to the results of Proposition 1.4.1 for the Boltzmann factor, and Proposition 4.7.2 for the normalization, the general formula (5.1) gives (N + 1)n ((β/2)!)N +n ρ(n) (r1 , . . . , rn ) = |e2πirk /L −e2πirj /L |β IN,n (β; r1 , . . . , rn ), (13.32) n L (β(N + n)/2)! 1≤j
where
IN,n (β; r1 , . . . , rn ) := [0,1]N
dx1 · · · dxN
n N j=1 k=1
|1 − e2πi(xj −rk /L) |β
|e2πixk − e2πixj |β ,
1≤j
(13.33) and we have scaled the angles θj → 2πxj /L. In the case β even, the Selberg correlation integral (13.6) can be shown to include (13.33), which thus implies a formula for ρ(n) in terms of the generalized hypergeometric
602
CHAPTER 13
(β/2)
function 2 F1
[185].
P ROPOSITION 13.2.1 For the n-point correlation (13.32) with β even we have (N + 1)n ((β/2)!)N +n ρ(n) (r1 , . . . , rn ) = |e2πirk /L − e2πirj /L |β MN (nβ/2, nβ/2, β/2) n L (β(N + n)/2)! 1≤j
×
n
eπiN β(rk −r1 )/L 2 F1
(β/2)
(−N, n; 2n; 1 − t1 , . . . , 1 − t(n−1)β ),
(13.34)
k=2
where MN is specified by (4.4), tk := e−2πi(rj −r1 )/L ,
k = 1 + (j − 2)β, . . . , (j − 1)β (j = 2, . . . , n)
(i.e., e−2πi(rj −r1 )/L for each j = 2, . . . , n is repeated β times). Proof. First change variables xj → xj + r1 /L +
1 2
in (13.33) to obtain
Z IN,n (β; r1 , . . . , rn ) = [−1/2,1/2]N
×
Y
|e
dx1 · · · dxN
N “ Y
|1 + e2πixj |β
j=1
2πixk
−e
n Y
|1 + e2πi(xj +(r1 −rk )/L) |β
”
k=2
2πixj β
| .
j
Now, for β even, |1 + e2πi(xj +(r1 −rk )/L) |β = e−πiβ(xj +(r1 −rk )/L) (1 + e2πi(xj +(r1 −rk )/L) )β , which shows IN,n (β; r1 , . . . , rn ) =
n Y
e−πiβN(r1 −rk )/L
[−1/2,1/2]N
k=2
×
n Y
Z
(1 + e2πi(xj +(r1 −rk )/L) )β
k=2
dx1 · · · dxN
N “ Y
e−πiβ(n−1)xj |1 + e2πixj |β
j=1
”Y
|e2πixk − e2πixj |β .
j
Comparison with (13.6) shows that this integral is of the type therein with m = (n − 1)β,
a − b = −β(n − 1),
a + b = β,
2/α = β
and the tk as specified in the statement of the proposition. Thus (13.6) shows that IN,n is proportional to (β/2) (−N, n; n 2 F1
− 2/β − N − 1; t1 , . . . , t(n−1)β ). (β/2)
Use of Proposition 13.1.7 gives the form stated in the proposition; the proportionality constant then follows since 2 F1 at the origin is unity, while at the origin IN,n = MN (nβ/2, nβ/2, β/2).
According to (13.23) the generalized hypergeometric function in m variables has an m-dimensional integral representation. However, the kernel of this integral representation is itself a generalized hypergeometric function of two sets of variables, which in general does not further simplify. An exception is the case in which all the m variables are equal. Then the particular generalized hypergeometric function of two sets of variables can be summed using the generalized binomial theorem, and the integral representation (13.12) involving only elementary functions holds. Proposition 13.2.1 shows this is precisely the case of the two-point correlation function. Thus we have the following β-dimensional integral representation [190].
603
CORRELATIONS FOR GENERAL β
P ROPOSITION 13.2.2 For the two-point correlation, (13.32) with n = 2, and β even, ρ(2) (r1 , r2 ) =
(N + 2)(N + 1) (βN/2)!((β/2)!)N +2 MN (nβ/2, nβ/2, β/2) L2 (β(N + 2)/2)! Sβ (1 − 2/β, 1 − 2/β, 2/β) β −πiβN (r1 −r2 )/L du1 · · · duβ ×(2 sin π(r1 − r2 )/L) e [0,1]β
×
β
−1+2/β
(1 − (1 − e2πi(r1 −r2 )/L )uj )N uj
(1 − uj )−1+2/β
j=1
|uk − uj |4/β .
j
13.2.2 The bulk limit By using Proposition 13.2.1, the evaluation of ρ(n) in the bulk limit follows immediately from (13.5) and the formulas (4.183), (4.185) and (4.180). P ROPOSITION 13.2.3 For the n-point correlation (13.32) with β even we have ρbulk (n) (r1 , . . . , rn ) := lim ρ(n) (r1 , . . . , rn ) N,L→∞ N/L=ρ
= ρn cn (β)
|2πρ(rk − rj )|β
1≤j
(β/2)
eπiρβ(rk −r1 )
k=2
(β/2) ×1 F1 (n, 2n; −2πiρ(r2
where in the argument of 1 F1
n
− r1 ), . . . , −2πiρ(rn − r1 )),
each −2πiρ(rj − r1 ) (j = 2, . . . , n) occurs β times, and
cn (β) = (β/2)βn(n−1)/2 ((β/2)!)n
n−1 k=0
Γ(βk/2 + 1) . Γ(β(n + k)/2 + 1)
In the case of the two-point function we have, from Proposition 13.2.2, the integral representation 3 −πiβρ(r1 −r2 ) (2πρ(r1 − r2 ))β bulk 2 β ((β/2)!) e ρ(2) (r1 , r2 ) = ρ (β/2) du1 · · · duβ β!(3β/2)! Sβ (−1 + 2/β, −1 + 2/β, 2/β) [0,1]β ×
β
−1+2/β
e2πiρ(r1 −r2 )uj uj
j=1
(1 − uj )−1+2/β
|uk − uj |4/β .
j
(13.35)
13.2.3 Asymptotics The integral representation (13.35) is well suited to study the large |r1 − r2 | =: r asymptotics [188]. Being a Fourier transform on a finite (multidimensional) interval, its major contributions will come from the neighborhood of the end points. We partition the integration variables into two sets, and expand the integrand about zero in the first set and about unity in the second set of variables. To leading order this procedure gives for the expansion of the integral eπiβρr β cos 2πlρr 2 Jβ/2−l,β (−i)Jβ/2+l,β (i), (2πρr)β β/2 − l (2πρr)4l2 /β β/2
l=0
(13.36)
604
CHAPTER 13
where
n
Jn,β (z) :=
(0,∞)n j=1
−1+2/β −ztj
tj
e
|tk − tj |4/β dt1 · · · dtn .
(13.37)
1≤j
The integral (13.37) is convergent for Re(z) > 0, but the formula (13.36) requires its value with Re(z) = 0. However, a simple change of variables ztj → sj shows that 1 Jn,β (1), z 2n2 /β which is an analytic function for all z = 0. Furthermore, according to Proposition 4.7.3 we have the explicit evaluation n Γ(1 + 2j/β)Γ(2j/β) Jn,β (1) = , Γ(1 + 2/β) j=1 Jn,β (z) =
so all terms in (13.36) are known explicitly. The next order oscillatory terms are calculated by including in the expansion of the integrand about the endpoints the leading order correction terms. For each l = 0, . . . , β/2, writing tj = 1 − sj (j = β/2 − l + 1, . . . , β), and noting
β/2−l
(1 − tj )−1+2/β
β
(1 − sj )−1+2/β
j =β/2−l+1
j=1
2 4 ∼1− 1+ l+ β β shows that we must evaluate n tj := Jn,β (z) j=1
(0,∞)n
β/2−l j=1
β/2−l
β
a=1
l =β/2−l+1
2 4 tj − 1 − l + β β
n n −1+2/β −ztj tj tj e j=1
(1 − ta − sl )4/β
j=1
β
sj ,
(13.38)
j=β/2−l+1
|tk − tj |4/β dt1 · · · dtn .
1≤j
This can be evaluated explicitly for we see immediately that n d (2n2 /β) tj = − Jn,β (z) = 2n2 /β+1 Jn,β (1). Jn,β (z) dz z j=1
(13.39)
Consequently, the first order correction to (13.36) is given by eπiβρr β sin 2πlρr 2 bl Jβ/2−l,β (−i)Jβ/2+l,β (i), (2πρr)β β/2 − l (2πρr)4l2 /β β/2
l=0
bl = −
2l2 8l 1− . β β
(13.40)
The l = 0 term in (13.40), which in (13.36) gives the leading order nonoscillatory term (and thus must equal ρ2 ), vanishes. To get the next order non-oscillatory term we must therefore expand the case l = 0 (equal numbers of variables expanded about the end points) to second order. This is done in Exercises 13.2 q.2. Putting together the above results we thus obtain the desired asymptotic expansion. P ROPOSITION 13.2.4 The two-point correlation function (13.35) for the log-gas at even β has the large r asymptotic expansion β/2 cos 2πpρr 1 1 1 ap sin 2πpρr +O cos 2πpρr +b +2 , ρ (r, 0) ∼ 1− +O p (2) ρ2 β(πρr)2 r4 2πρr r2 (2πρr)4p2 /β p=1
605
CORRELATIONS FOR GENERAL β
where
p ap =
j=1 Γ(2j/β + 1) p−1 j=1 Γ(−2j/β + 1)
2
and bp is as in (13.40). For β = 2 this agrees with (7.65), and for β = 4 it agrees with (7.94).
13.2.4 Jacobi β-ensemble Here we will consider the Jacobi β-ensemble on the interval [0, 1], which is obtained from (3.74) by the change of variables xj = (1 − yj )/2, and furthermore we will set aβ/2 = λ1 and bβ/2 = λ2 . It follows from (13.25) that for the Jacobi β-ensemble in question, with N + n particles and β even, the n-point correlation is given by ρ(n) (r1 , . . . , rn ) :=
n n (N + n)n rkλ1 (1 − rk )λ2 |rk − rj |β SN +n (λ1 , λ2 , β/2) k=1
×
dx1 . . . dxN [0,1]N
N
j
xλj 1 (1 − xj )λ2
j=1
n
(xj − rk )β
k=1
|xk − xj |β
1≤j
n n SN (λ1 + nβ, λ2 , β/2) λ1 = (N + n)n tk (1 − tk )λ2 |tk − tj |β SN +n (λ1 , λ2 , β/2) k=1
(β/2)
×2 F1
j
(−N, 2(λ1 + λ2 + m + 1)/β + N − 1; 2(λ1 + m)/β; t1 , . . . , tβn )
{tj } →{rj }
, (13.41)
where m = βn and tk = rj
for k = 1 + (j − 1)β, . . . , jβ (j = 1, . . . , n).
(13.42)
In the special case n = 1 the arguments of the generalized hypergeometric function in (13.41) are all equal, and we have available β-dimensional integral representations by way of (13.11) and (13.12). However the parameters in the hypergeometric function are such that these integrals are singular. This can be overcome by using the first of the Kummer type relations in Proposition 13.1.6 to note that the hypergeometric function can be transformed to read t1 tβ (β/2) (1 − r)βN 2 F1 − N, 2(λ1 + 1)/β − N + 1; 2λ1 /β + 2; − ,...,− . 1 − t1 1 − tβ t1 =···=tβ =r Now we can make use of (13.11) to conclude ρ(1) (r) = (N + 1) ×
1/2
−1/2
rλ1 (1 − r)λ2 +βN SN (λ1 + β, λ2 , β/2) SN +1 (λ1 , λ2 , β/2) Mβ (2(λ1 + 1)/β − 1, 2(λ2 + 1)/β + N − 1; 2/β) 1/2 β dx1 · · · dxβ eπixl (2(λ1 −λ2 )/β) |1 + e2πixl |2(λ1 +λ2 +2)/β+N −2
× e−πixl −
−1/2
r πixl N e 1−r
l=1
1≤j
|e2πixk − e2πixj |4/β .
(13.43)
606
CHAPTER 13
13.2.5 Laguerre β-ensemble The limiting procedure specified in Section 4.7.1 converting the Selberg integral into the normalization for the Laguerre ensemble allows the Selberg correlation integral to be similarly transformed. P ROPOSITION 13.2.5 We have ∞ ∞ N m 1 aβ/2 xj e−βxj /2 dx1 · · · dxN (xj − tk ) Wa+2m/β,β,N 0 0 j=1 k=1
=
(β/2) (−N ; a 1 F1
Proof. In (13.25) set
|xk − xj |β
1≤j
+ 2m/β; t1 , . . . , tm ).
(13.44)
xj → xj /L, tk → tk /L, λ2 → βL/2, λ1 = aβ/2, λ = β/2
and taking the limit L → ∞, using (13.5) on the r.h.s, gives (13.44) up to a proportionality constant. The latter is fixed by requiring that both sides agree at t1 = · · · = tm = 0.
We can use this result to express the general n-point correlation function for the Laguerre ensemble, β (β/2) even, in terms of the generalized hypergeometric function 1 F1 . In a system of N + n particles, this correlation is defined by the N -dimensional integral n (N + n)n aβ/2 −βrj /2 ρ(n) (r1 , . . . , rn ) = r e |rk − rj |β Wa,β,N +n j=1 j 1≤j
×
(0,∞)N
dx1 · · · dxN
N
n
aβ/2 −βxj /2
xj
e
j=1
|rk − xj |β
k=1
|xk − xj |β .
j
For β even this is a special case of the integral in (13.44) with m = βn and {tk } related to {rj } by (13.42), and thus n (N + n)n aβ/2 ρ(n) (r1 , . . . , rn ) = Wa,β,N rj e−βrj /2 |rk − rj |β Wa+2n/β,β,N +n j=1 1≤j
(β/2)
We remark that in the case n = 1 the function 1 F1 Exercises 13.1 q.4(i), and thus ρ(1) (r) = (N + 1) ×
β l=1
has the β-dimensional integral representation of
raβ/2 e−βr/2 Wa+2/β,β,N +1 Mβ (2/β − 1 + a, N, β/2) Wa,β,N
1/2
−1/2
eπixl (2/β−1+a−N ) |1 + e2πixl |N +2/β−1+a e−re
2πixl
dx1 · · ·
β
1/2
−1/2
dxβ
|e2πixk − e2πixj |4/β . (13.46)
j
According to (7.30) hard edge scaling requires we compute the limit 1 n X Xn 1 ,..., . ρhard ρ(n) (n) (X1 , . . . , Xn ) = lim N →∞ 4N 4N 4N Substituting (13.46) and using the confluence (13.5) and the explicit evaluation of Wa,β,N given in Proposi-
607
CORRELATIONS FOR GENERAL β
tion 4.7.3 shows that for β even n
ρhard (n) (X1 , . . . , Xn ) = An (β)
βa/2
Xj
j=1 (β/2) ×0 F1 (a
|Xk − Xj |β
1≤j
+ 2n; Y1 , . . . , Ynβ )
{Yj } →{−Xj /4}
,
(13.47)
where (Γ(1 + β/2))n
An (β) = 2−n(2+aβ+β(n−1)) (β/2)n(1+aβ+β(n−1)) 2n j=1
Γ(1 + aβ/2 + β(j − 1)/2)
.
For the density (n = 1), and with a = c − 2/β, c a positive integer, the generalized hypergeometric function (β/2) can be written as a β-dimensional integral according to (13.27) to give 0 F1 β 1/2 eiX cos θj ei(c−1)θj |eiθk − eiθj |4/β dθ1 · · · dθβ , (13.48) ρ(1) (X) = a(c, β)X β/2−1 [−π,π]β j=1
1≤j
where a(c, β) = (−1)(c−1)β/2 (2π)−β
1 β β (Γ(1 + 2/β))β . 2 4 Γ(β)
The correlation integrals (13.44) can also be used to give expressions for some spacing distributions at the hard edge in the Laguerre ensemble. Now, by definition ∞ ∞ N 1 dx1 · · · dxN e−βxj /2 xaj |xk − xj |β EN,β (0; (0, s); xa e−βx/2 ) = W2a/β,β,N s s j=1 =
e−N βs/2 W2a/β,β,N
∞
0
dx1 · · ·
j
∞
dxN 0
N j=1
e−βxj /2 (xj + s)a
|xk − xj |β ,
j
(13.49) where the second equality follows by the change of variables xj → xj + s. From this, we can calculate the distribution of the smallest eigenvalue according to (N )
d EN,β (0; (0, s); xa e−βx/2 ) ds ∞ N −1 N e−N βs/2 a ∞ s dx1 · · · dxN −1 xβj e−βxj /2 (xj + s)a |xk − xj |β , = W2a/β,β,N 0 0 j=1
pβ (0; s; a) = −
j
(13.50) where the second equality follows by differentiating the first equality in (13.49) and then changing variables xj → xj + s. We see that for a ∈ Z≥0 , the final integrals in both (13.49) and (13.50) are examples of the correlation (β/2) integral (13.44) and thus can be expressed in terms of the hypergeometric function 1 F1 . P ROPOSITION 13.2.6 For a ∈ Z≥0 we have (−N ; 2a/β; (−s)a ), EN,β (0; (0, s); xa e−βx/2 ) = e−βN s/2 1 F1 W2a/β+2,β,N −1 (β/2) (N ) (−N + 1; 2a/β + 2; (−s)a ). pβ (0; s; a) = N sa e−βN s/2 1 F1 Wa,β,N (β/2)
608
CHAPTER 13
From the above formulas, and with the aid of the confluent limit (13.5), we can calculate the scaled limits Eβhard (0; (0, s); a) := lim EN,β (0; (0, s/4N ); xa e−βx/2 ), N →∞
phard (0; s; a) := β
lim
N →∞
1 (N ) p (0; (0, s/4N ); a). 4N β
P ROPOSITION 13.2.7 For a ∈ Z≥0 we have Eβhard (0; (0, s); a) = e−βs/8 0 F1
(β/2)
(2a/β; (s/4)a ),
(0; s; a) = Aa,β sa e−βs/8 0 F1 phard β
(β/2)
(2a/β + 2; ; (s/4)a ),
where Aa,β = 4−(a+1) (β/2)2a+1
Γ(1 + β/2) . Γ(1 + a)Γ(1 + a + β/2)
(β/2)
(β/2)
We know from Exercises 13.1 q.4 that both 1 F1 and 0 F1 occurring in Propositions 13.2.6 and 13.2.7 have a dimensional integral representations. In particular, this gives for the probability that there are no eigenvalues in the interval (0, s) in the scaled limit the formula 1 (−1+2/β)a/2 1 a Eβhard (0; (0, s); a) = e−βs/8 Ba (2/β, β/2) s 2π a 1/2 × es cos θj ei(−1+2/β)θj |eiθk − eiθj |4/β dθ1 · · · dθa , [−π,π]a j=1
1≤j
(13.51) where Ba is specified by (13.28). An immediate application of this formula is the calculation of the large s asymptotic expansion. Now, in 1/2 this limit the maximum of the factors es cos θj in the integrand occur at θj = 0. Expanding to leading order about these points gives 1 a(a+1)/2β−a/4 1 1/2 Eβhard (0; (0, s); a) ∼ e−βs/8+as (13.52) τa,β 1 + O 1/2 , s s where a 1 a 2 Ba (2/β, β/2) e−tj /2 |tk − tj |4/β dt1 · · · dta . τa,β = 2(2/β−1)a 2π a (−∞,∞) j=1 1≤j
The latter integral is evaluated explicitly in Proposition 4.7.1, which together with the explicit form (13.28) for Bm gives the simplification a 1 a/2 Γ(2j/β). τa,β = 2(2/β−1)a 2π j=1 In the case β = 2 this gives τa,β = (2π)−a/2 G(a + 1), which verifies the conjectured asymptotic form in (9.92) for a ∈ Z≥0 . 13.2.6 Gaussian β-ensemble We know from Section 4.7.1 that changing variables and replacing according to β xj β tj 1 1 1− , tj → 1− , λ1 = λ2 = L2 , xj → 2 2 L 2 2L
(13.53)
609
CORRELATIONS FOR GENERAL β
and taking the limit L → ∞, the Selberg correlation integral of Exercises 13.1 q.1 becomes proportional to N m −βx2j /2 Im,N (t1 , . . . , tm ; β) := dx1 · · · dxN e (tk − xj ) |xk − xj |β . (13.54) (−∞,∞)N
j=1
k=1
j
However, unlike in (13.44) the corresponding generalized hypergeometric function does not reduce to another generalized hypergeometric function. Nonetheless (13.54) satisfies analogous p.d.e.’s. To see this note that the substitutions (13.53) in the p.d.e.’s (13.15) with parameters (13.24) satisfied by the Selberg correlation integral tell us that for each p = 1, . . . , m, 2 N ∂ ∂ 2 1 ∂ ∂ Im,N = 0, − 2t + 2N + − (13.55) p ∂t2p ∂tp α l=1 tp − tl ∂tp ∂tl l=p
where α = β/2. Summing up the equations (13.55) and comparing with (11.53) we see that the resulting equation can be written ˜ (H) Im = −2N mIm,N H m
(13.56)
(H)
˜ (H,Ex) with m coordinates and restricted to symmetric functions. ˜ m equal to the operator H with H ˜ (H,Ex) . The logical development of the theory at this stage is to consider the polynomial eigenfunctions of H This will be carried out in the next section. Here we list some results as relevant for present purposes. ˜ (H,Ex) are the Hermite polynomials 2−k Hk (x), In the case N = 1 the monic polynomial eigenfunctions of H which have eigenvalue −2k, and these polynomials have the integral representation (5.156). Likewise, for ˜ (H,Ex) permits polynomial eigenfunctions Eη(H) (x), referred to as nonsymmetric Hermite polygeneral N , H nomials with the structure Eη(H) (x) = Eη (x) + aην Eν (x), (13.57) |ν|<|η|
having eigenvalue −2|η| and admitting the integral representation 2 2 ex1 +···+xN (H) (H) (y), Eη (x) = 0 K0 (2y; −ix)Eη (iy) dμ (H) (−∞,∞)N N0 where dμ(H) (y) := e−(y1 +···+yN ) 2
(H) N0
2
(13.58)
|yk − yj |2/α dy1 · · · dyN ,
1≤j
dμ(H) (y)
:=
(13.59)
(−∞,∞)N
(H)
(note that N0 is essentially the Mehta integral (4.140); it is given in terms of gamma functions by (13.80) below) and 0 K0 is specified by (13.17). Now (13.54) has the structure Im,N (t1 , . . . , tm ; β) = (t1 · · · tm )N + lower terms.
(13.60)
But as (t1 · · · tm )N = EN m (t1 , . . . , tm ), (H)
and as this is symmetric, it follows from (13.19) and (13.58) that EN m is symmetric and has the structure
610
CHAPTER 13
(H) (H) ˜m (13.60). Furthermore EN m is an eigenfunction of H with eigenvalue −2|η| = −2N m, so we conclude (H) Im,N (t1 , . . . , tm ; β) = EN m (t1 , . . . , tm ) . (13.61) α=β/2
Analogous to (13.41) we have that for the Gaussian β-ensemble with N + n particles and β even the (H) n-point correlation function is given in terms of Im,N and thus EN m by Gβ,N Iβn,N ({tj } → {rk }; β) Gβ,N +n Gβ,N (H) = (N + n)n EN βn ({rk }) , Gβ,N +n α=β/2
ρ(n) (r1 , . . . , rn ) = (N + n)n
(13.62)
where Gβ,N is the normalization in (1.28). In the special case n = 1, according to (13.42) all the arguments (H) in EN βn ({rk }) are equal. Noting from (13.18) and (13.3) that m 0 K0 (2y1 , . . . , 2ym ; (−it) )
= 0 F0 (−2iy1t, . . . , −2iymt) = e−2it (α)
Pm
j=1
yj
(13.63)
one can then deduce from (13.58) the integral representation e−βr /2 Gβ,N ρ(1) (r) = (N + 1) Gβ,N +1 N (H) | N →β 2
0
×
(−∞,∞)β
α →β/2
du1 · · · duβ
β
(iuj + r)N e−uj
2
j=1
|uk − uj |4/β .
(13.64)
1≤j
The form (13.64) is suitable for asymptotic analysis [132]. One finds for the global density √ 2 1 2 Γ(1 + 2/β) ρ(1) ( 2Nx) ∼ ρW (x) − N π (πρW (x))6/β−1 N 2/β 1 1 , 8/β , (13.65) × cos 2πN P (x) + (1 − 2/β)Arcsin x + O min N N where ρW (x) and P (x) are as in (7.26), while at the soft edge √ Γ(1 + β/2) 4π β/2 Γ(1 + 2/β) x 1 √ Kβ,β (x) + O(N −1/3 ), ∼ ρ(1) 2N + √ 2π β Γ(1 + 2j/β) 2N 1/6 2N 1/6 j=1 (13.66) where i∞ i∞ n 3 1 Kn,β (x) := − dv · · · dv evj /3−xvj |vk − vl |4/β . (13.67) 1 n n (2πi) −i∞ −i∞ j=1 β
1≤k
We remark that the expansion (13.65) is consistent with the previously computed global expansions of the Gaussian ensemble density at β = 2 and 4 given by (7.25) and (7.98) respectively. Let us denote the N → ∞ limit of (13.66) by ρsoft,β (1) (x). Asymptotic analysis of the multiple integral
611
CORRELATIONS FOR GENERAL β
(13.67) shows [132] 3/2 1 Γ(1 + β/2) e−2βx /3 1 , + O x→∞ 2π (4β)β/2 x3β/4−1/2 x3β/4+1 1 4 |x| Γ(1 + β/2) 2 π 1 soft,β − 6/β−1 3/β−1/2 cos |x|3/2 − 1− +O ρ(1) (x) ∼ . , x→−∞ π 3 2 β 2 |x| |x|5/2 |x|6/β−1/2 (13.68)
ρsoft,β (1) (x) ∼
The latter result is consistent with the expansions (7.69), (7.100) given previously for β = 2, 4, while the former is consistent with the expansion (7.72) obtained previously for β = 2. E XERCISES 13.2
1. Use the formula (13.16) to verify that the expression for ρ(n) in Proposition 13.2.3 is real. Similarly, use the first transformation formula in Proposition 13.1.6 to verify that the expression for ρ(n) in Proposition 13.2.1 is real.
2.
(i) Suppose that the integrand in (13.35) has β/2 variables expanded about tj = 0 and β/2 variables expanded about tj = 1. Show that the next order term in the expansion (13.38) is then “
4 ”“ X ”“ tj β2 j=1 β/2
1+
β X j=β/2+1
−
β/2 ” 1“ 2 ”2 “ X ”2 1 “ 2 ”2 “ sj + tj + 1+ 1+ 2 β 2 β j=1
β/2 (1 + 2/β) X 2 (1 + 2/β) tj − 2 2 j=1
β X
β X
sj
”2
j=β/2+1
s2j .
j=β/2+1
P Pn 2 (ii) Use the result (13.39), and its derivative, to evaluate Jn,β (z)[ n j=1 tj ] and Jn,β (z)[( j=1 tj ) ]. Also, note from (12.147) that n X
t2j := m20n−1 = P20n−1 (t1 , . . . , tn ) −
j=1
2 P n−2 (t1 , . . . , tn ), 1 + α 110
and use this together with (12.153) to deduce that Jβ/2,β (z)
n hX j=1
t2j
i
1 = 2 z
„
« −2/β + 2 + 3β/2 Jn,β (z). 1 + 2/β
(iii) Use the results of (ii) to show that the total contribution of the second order terms in (i) to the asymptotic expansion of (13.35) is ρ2 − . β(πρr)2 (H)
3. [39] The nonsymmetric Hermite polynomials Eη can be specified as the polynomials with the structure (13.57) (H) (H) which satisfy the eigenfunction equation hi Eη = η¯i Eη for each i = 1, . . . , N . The nonsymmetric Laguerre (L) (L) polynomials Eη are even functions of the variables y = (y1 , . . . , yN ), and this is indicated by writing Eη (y 2 ). 2 They can be specified as the polynomials with the structure (13.57) (with x → y ) which satisfy the eigenfunction (L) (L) equation li Eη (y 2 ) = η¯i Eη (y 2 ) for each i = 1, . . . , N . In this exercise Rodrigues-type formulas for these polynomials will be derived. (i) Apply the map φ(A) specified by (11.75), (11.76) to the eigenvalue equation (12.9), to show “ “ 1 ” 1 ” hi Eη y − d · 1 = η¯i Eη y − d · 1 2 2 where Eη (y − 12 d) denotes the polynomial Eη (y) with each variable yi replaced by yi − 12 di . Deduce from
612
CHAPTER 13
this that
“ 1 ” Eη(H) (y) = Eη y − d · 1 2
(13.69)
(cf. (12.92)). (ii) Use the map φ(B) specified by (11.81), (11.82) to show ““ ”2 ” 1 Eη(L) (y 2 ) = Eη y − d(B) · 1. 2
(13.70)
4. [187] Use the fact that the integrand in (13.48) is periodic to translate the domain of integration to [−π + , π + ], 0 < 1. By then expanding β/2 of the variables about θ = 0 and β/2 variables about θ = π, deduce that for X→∞ ˛2 “ β ”˛ Z Y 2 +···+θ 2 ˛ ˛ iX 1/2 (θ1 β/2 )/2 e |θk −θj |4/β dθ1 · · · dθβ/2 ˛ . ρ(1) (X) ∼ |a(c, β)|X β/2−1 2β ˛ β/2 (−∞,∞)β/2 1≤j
Change variables iX 1/2 θk2 → −λ2k and use the evaluation (1.160) to deduce from this the explicit asymptotic form 1 ρ(1) (X) ∼ . 2πX 1/2 Note that this is consistent with (7.74) and (7.114). 5.
(i) By an appropriate change of variables show that EN,β (0; (0, t); xa (1 − x)b χ0<x<1 ) Z 1 Z N “ Y t ”a 1 1 = (1 − t)b+N+βN(N−1)/2 dx1 · · · dxN (1 − xl )b xl + C 0 1−t 0 l=1
Y
|xk − xj |β ,
1≤j
where C is such that the r.h.s. equals unity at t = 0. With the hard edge scaling specified by t = X/4N 2 , as is consistent with the scale identified in (7.45) after a change of variables appropriate for the Jacobi ensemble defined on (0, 1), use this result with a = 0 to reclaim the result of Proposition 13.2.7 in the case a = 0. (ii) Make use of (13.10) in the case m = 1, λ1 = λ2 = 0, α = 2 to deduce that Z
1
Z
1
dx1 · · ·
0
dxN 0
“1
N “ Y
xl +
l=1
t ”−1/2 1−t
Y
|xk − xj |
1≤j
“N N + 1 ”−N/2 1” −1 , ; N + 1; 1 − 2 F1 t 2 2 t ” “N N + 1 −N/2 , ; N + 1; 1 − t = (1 − t) 2 F1 2 2 ∝
where the equality follows upon use of the first of the Kummer relations in Proposition 13.1.6. (iii) By use of the connection formula [541] 2 F1 (a, b; c; z)
= A 2 F1 (a, b; a + b − c; 1 − z) + B 2 F1 (c − a, c − b; c − a − b + 1; 1 − z), A :=
Γ(c)Γ(c − a − b) , Γ(c − a)Γ(c − b)
B :=
Γ(c)Γ(a + b − c) Γ(a)Γ(b)
deduce that lim
N→∞
2 F1
“N N + 1 “ 1 X ” X 1/2 “3 X ” √ X ” ∝ 0 F1 ; , ; N + 1; − − ; = e X/4 , 0 F1 2 2 2 4N 2 16 4 2 16
where the equality follows from (10.190) and (7.50).
613
CORRELATIONS FOR GENERAL β
(iv) [154] Use the results of (i)–(iii) to show E1hard (0; (0, X); a = −1/2) = e−X/4+
√ X/4
and hence by following the reasoning of Exercises 8.3 q.4 conclude that the p.d.f. for κ ˜ (A) in the case of real Gaussian matrices is equal to y + 2 −1/(2y)−1/(8y 2 ) e 8y 3 (use must be made too of Proposition 3.2.2). 6.
(i) Suppose μ ∈ Z+ . By manipulating the integrands as implied by the average, making use of Proposition 13.1.4, and applying Proposition 13.1.7, show that N DY
|z − eiθl |2μ
E
l=1
(2/β)
CE0 β,N
∝ 2 F1
(−μ, −μ; −2μ; (1 − |z|2 )N ).
Now apply Proposition 13.1.2 to conclude N DY
|z − eiθl |2μ
l=1
E CE0 β,N
∝
μ “ DY
1 − (1 − |z|2 )xl
”N E
l=1
ME4/β,μ (xα (1−x)α )
,
(13.71)
where α = 2/β − 1. (ii) Suppose μ ∈ Z+ . By making use of (13.12), then transforming according to the theory of Exercises 13.1 q.1 (iv), and finally applying the first of the Kummer type transformations of Proposition 13.1.6, show that N E DY (x−xl )2μ l=1
(2/β)
MEβ,N (xa (1−x)b )
∝ (1−x)2μN 2 F1
(−2μ, 1+(N −1)β/2+b; −2μ−a; (x/(x−1))N ).
Now make use of (13.6) to conclude this average is proportional to 2μ DY
eiθl (a−b)/β |1 + eiθl |2(a+b+2)/β+N−2 ((1 − x)e−iθl /2 − xeiθl /2 )N
l=1
E CE0 4/β,2μ
.
(13.72)
13.3 GENERALIZED CLASSICAL POLYNOMIALS (H)
The nonsymmetric Hermite polynomials Eη have been introduced in Section 13.2.6, while the nonsymmetric Laguerre polynomials have appeared in Exercises 13.2 q.3. Here, following [38], the properties of both will be further developed. A brief development of the theory of the generalized Jacobi polynomials is also given. 13.3.1 Generalized Hermite polynomials (H)
˜ (H,Ex) In Section 13.2.6 the nonsymmetric Hermite polynomials Eη were defined as the eigenfunctions of H with the structure (13.57). In Exercises 13.2 q.3 it was remarked that they can alternatively be specified as simultaneous eigenfunctions of {hi } with corresponding eigenvalues η¯i . To see the consistency between these two specifications, note from the definition (11.70) that hi consists of ξi , a degree-preserving operator, and d2i which lowers the degree by 2. Thus the hi have the action ˜bην Eν (y). hi Eη (y) = η¯i Eη (y) + |ν|<|η|
614
CHAPTER 13
Proceeding now as in the working which leads to (12.9), we can use this to conclude the existence of a simultaneous set of polynomial eigenfunctions of the operator X
(H)
N ({u}) := (1 + ul hl )
(13.73)
l=1 (H)
of the form (13.57), with {aη ν } therein independent of {ul }. The relation (11.70) then gives that the Eη are also polynomial eigenfunctions of H (H,Ex) with eigenvalue −2|η|.
Orthogonality (H)
The polynomials {Eη } are orthogonal with respect to the inner product f (y)g(y) dμ(H) (y), f |g(H) :=
(13.74)
(−∞,∞)N
where dμ(H) (y) is given by (13.59), so that Eν(H) |Eη(H) (H) = Nη(H) δν,η .
(13.75)
This fact relies on the eigenoperator (13.73) separating the eigenvalues and being self-adjoint with respect to (13.74). The self-adjointness property of X (H) (u) in turn follows from the fact that each hi is self-adjoint with respect to (13.74). P ROPOSITION 13.3.1 We have f |di g(H) = (2yi − di )f |g(H) and thus f |hi g(H) = hi f |g(H) . Proof. The first equation is verified by direct substitution of Definition 11.4.1 and integration by parts. Denote the adjoint just determined by d∗i so that d∗i = 2yi − di . Then according to (11.76) we can write hi =
N X α ∗ sip di di + 1 − N + 2 p=i+1
which immediately implies the second result.
Recurrences We know that all the nonsymmetric Jack polynomials can be generated from the constant polynomial E0N (z) := 1 by application of just two fundamental operators: the transposition si and the raising type operator Φ. Also ˆ was identified. The analogous operators for the Eη(H) (y) can be the explicit form of the lowering operator Φ deduced by simply applying the map φ(A) as specified by (11.75) to the corresponding formulas for Eη (z) in Propositions 12.2.1, 12.2.3 and 12.2.5, and making use of (11.76) and (13.69).
615
CORRELATIONS FOR GENERAL β
P ROPOSITION 13.3.2 We have ⎧ (H) (H) 1 1 ⎪ Esi η (y), ηi > ηi+1 , E η (y) + 1 − δ¯2 ⎪ ¯ δ ⎪ i,η ⎨ i,η (H) si Eη(H) (y) = Eη (y), ηi = ηi+1 , ⎪ ⎪ ⎪ ⎩ 1 E (H) (y) + E (H) (y), ηi < ηi+1 , si η δ¯i,η η ˆ ∗ Eη(H) (y) = 2E (H) (y), Φ Φη ˆ (H) (y) = 1 ΦE η α
(13.76)
ˆ ∗ := sN −1 · · · s1 d∗1 = sN −1 · · · s1 (2y1 − d1 ) Φ
dη (H) EΦη ˆ (y), dΦη ˆ
(13.77) (13.78)
ˆ ∗ is the adjoint of Φ ˆ with respect to the inner product (13.74). where Φ (C)
Analogous to the determination of the normalization Nη using the operators si and Φ given in the proof (H) of Proposition 12.3.3, the normalization of Eη with respect to the inner product (13.74) can be determined ˆ ∗. using the operators si and Φ P ROPOSITION 13.3.3 We have Nη(H) := Eη(H) |Eη(H) (H) =
dη eη (H) 1 N0 , (2α)|η| dη
(13.79)
where (H)
N0
:= N0N = 1|1(H) = 2−N (N −1)/2α π N/2 (H)
N −1 j=0
Γ(1 + (j + 1)/α) . Γ(1 + 1/α)
(13.80)
Proof. Because the formula (13.76) is formally the same as in Proposition 12.2.1, we have analogous to (12.42) −2 Es(H) |Es(H) (H) = (1 − δ¯i,η ) Eη(H) |Eη(H) (H) . iη iη
(13.81)
(H)
Furthermore, taking the inner product of both sides of (13.77) with EΦη and dividing by 2 gives (H)
(H)
EΦη |EΦη (H) =
1 (H) ˆ ∗ (H) (H) 1 ˆ (H) (H) (H) = ΦE . E |Φ Eη Φη |Eη 2 Φη 2
(13.82)
But a simple change of variables in (13.78) gives ˆ (H) = 1 dΦη Eη(H) , ΦE Φη α dη
so we have (H)
(H)
EΦη |EΦη (H) =
1 dΦη (H) (H) (H) Eη |Eη . 2α dη (H)
(H)
(13.83) (H)
The formulas (13.83) and (13.82) completely determine Eη , |Eη (H) starting from N0N = 1|1(H) , the value of which is given by Proposition 4.7.1. It follows immediately from (12.38) that the r.h.s. of (13.79) obeys the same recurrences, implying the result.
Generating function The classical Hermite polynomials can be defined by the generating function ∞ Hk (x)y k k=0
k!
= e2xy e−y . 2
(13.84)
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CHAPTER 13
A similar result holds for the nonsymmetric Hermite polynomials, with the role of the exponential e2xy played by the type A Dunkl kernel KA (2x; y), which is specified in terms of the hypergeometric function (13.17) by dη α|η| Eη (x)Eη (y) (13.85) KA (x; y) = 0 K0 (x; y) = d η eη η (here (12.39) has been substituted for Eη ((1)N )). We first require some additional properties of KA (2x; y). P ROPOSITION 13.3.4 We have ˆ (x) KA (x; y) = Φ(y) KA (x; y), Φ
(x)
di
KA (x; y) = yi KA (x; y).
(13.86)
ˆ (x) term by term to the series definition (13.17) using Proposition 12.2.5, Proof. The first equality follows by applying Φ
ˆ in the summation, then using the final formula of Proposition 12.2.3 to rewrite EΦν (y) in terms of writing ν = Φη (x) ˆ (x) , respectively, and using the first Φ(y) Eν (y). The second result follows by writing yi and di in terms of Φ(y) and Φ equality.
˜1 specified by (12.183) and ΔA by (11.67), we see from (11.68) that the Hamiltonian (11.64) can With E be written as ˜ (H,Ex) = ΔA − 2E ˜1 . H
(13.87)
Now from the definition (11.67) of ΔA , it follows immediately from the second equality in (13.86) that (x)
ΔA KA (2x; y) = 4p2 (y)KA (2x; y)
(13.88)
(recall the meaning of p2 (y) as specified in (12.146)). Also, the fact that the Eη are homogeneous of degree |η| implies ˜ (y) Eη (y) = |η|Eη (y), E 1 (y)
˜ where E 1
(13.89)
is given by (12.183) with {x} → {y}. The generating function can now be established.
P ROPOSITION 13.3.5 We have (2α)|η| dη η
eη dη
Eη(H) (x)Eη (y) = KA (2x; y)e−p2 (y) .
(13.90)
Proof. From the definition (13.85) we see that KA (2x; y)e−p2 (y) =
X (2α)|η| dη Qη (x)Eη (y), eη dη η
˜1(y) to both sides using (13.89) and the formula where Qη (x) is a polynomial with leading term Eη (x). Applying E “ ” ˜ (y) KA (2x; y)e−p2 (y) = e−p2 (y) E ˜ (y) KA (2x; y) − 2p2 (y)KA (2x; y)e−p2 (y) E 1 1 ˜1 KA (2x; y) − = e−p2 (y) E (x)
1 (x) Δ KA (2x; y), 2 A
where use has been made of (13.88), shows “
” X (2α)|η| dη ˜ (x) − 1 Δ(x) e−p2 (y) KA (2x; y) = |η|Qη (x)Eη (y). E 1 A 2 eη dη η
Substituting (13.85) on the l.h.s, recalling (13.87) and equating Eη (y) on both sides shows ˜ (H,Ex) Qη (x) = −2|η|Qη (x). H
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CORRELATIONS FOR GENERAL β
(H)
The unique polynomial eigenfunction solution of this equation with leading term Eη is Eη , and the stated formula follows.
By combining the generating function with a corollary of (13.88), an exponential operator can be identified which maps the Jack polynomials to the nonsymmetric Hermite polynomials. First, note that (13.88) implies 1 k k (x) ΔA KA (2x; y) = p2 (y) KA (2x; y), 4 which after multiplication by (−1)k /k! and summing over k gives 1 (x) exp − ΔA KA (2x; y) = e−p2 (y) KA (2x; y). 4 Using the generating function (13.90) on the r.h.s. and the series formula (13.85) on the l.h.s, and equating coefficients of Eη (y) on both sides shows that exp(− 14 ΔA ) has the desired action, 1 (13.91) exp − ΔA Eη (x) = Eη(H) (x). 4 (H)
Now Eη (x) is an eigenfunction of ξj with eigenvalue η¯j , while Eη (x) is an eigenfunction of hj with the (H) same eigenvalue. Hence (13.91) remains valid with the replacements Eη (x) → ξj Eη (x) and Eη (x) → 1 hj exp(− 4 ΔA )Eη (x). This shows that we have the intertwining relation 1 1 (13.92) exp − ΔA ξj = hj exp − ΔA . 4 4 Integration formulas The Dunkl kernel KA (x, y) occurs as the kernel in an integral transform that relates the nonsymmetric Jack and Hermite polynomials. One of these is (13.58), while its inverse and the transform underlying these results are given by the following. (H)
P ROPOSITION 13.3.6 Let dμ(H) (y) be given by (13.59), and let N0N be as in (13.79). Then we have (H) KA (2y; z)KA (2y; w) dμ(H) (y) = N0N ep2 (w)+p2 (z) KA (2z; w), (13.93) (−∞,∞)N (H) KA (2y; z)Eη(H) (y) dμ(H) (y) = N0N ep2 (z) Eη (z). (13.94) (−∞,∞)N
Proof. To derive the first formula, multiply both sides by e−p2 (w)−p2 (z) , use the generating function (13.90) twice on (H)
the l.h.s., and then use the orthogonality of {Eη (y)} with respect to (13.74) to compute the integral. The resulting sum is identified as proportional to KA (2z; w). The second formula follows from the first after multiplying by e−p2 (w) , using the generating function on the l.h.s., and equating coefficients of Eη (w) on both sides. We remark that (13.58) is derived from (13.93) similarly, by replacing z by iz and w by −iw, then using the generating function on the r.h.s., and equating coefficients of Eη (w).
We remark that in the case N = 1 (13.94) reduces to the classical identity (5.153). Symmetric Hermite polynomials (H)
(H)
The symmetric Hermite polynomials Pκ (y; α) := Pκ (y) can be defined as the eigenfunctions of the operator (11.71) with the structure Pκ(H) (y) = Pκ (y) + aκμ Pμ (y). (13.95) |μ|<|κ|
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CHAPTER 13
Analogous to the case of the symmetric Jack polynomials (recall Section 12.6), the linear combination a ˜η Eη(H) (y) η:η+ =κ
˜η = dη+ /dη gives is an eigenfunction of X (H) (u). Furthermore, according to (12.96) and (13.57), choosing a the leading term equal to Pκ (y) as required by (13.95). Because the leading term is symmetric, the polynomial itself must be symmetric (X (H) (u) maps symmetric functions to symmetric functions), so we must have 1 Eη(H) (y). (13.96) Pκ(H) (y) = dη+ d η + |η|:η =κ
Alternatively, analogous to (12.100), the symmetric Hermite polynomials can be constructed from their nonsymmetric counterparts by symmetrization. Thus Pκ(H) (y) =
1 Sym Eη(H) (y), aη
(13.97)
where κ = η + and aη is given by (12.101). Applying Sym to both sides of the generating function (13.90), making use of Proposition 13.1.10, (12.100) and (13.96), we obtain the generating function for the symmetric Hermite polynomials, (2α)|κ| Pκ(H) (x)Pκ (y) |κ|
dκ
Pκ ((1)N )
= 0 F0 (2x; y)e−p2 (y) . (α)
(13.98)
Furthermore, applying Sym to both sides of (13.91) shows that 1 exp − D0 Pκ (y) = Pκ(H) (y). (13.99) 4 Here D0 is the generalized Laplacian ΔA restricted to symmetric functions, which from (11.68) has the explicit form N ∂2 1 ∂ 2 ∂ D0 = . + − 2 ∂yj α yj − yk ∂yj ∂yk j=1 j
The integration formulas of Proposition 13.3.6 can also be symmetrized. Consider in particular the formula (13.58). Applying Sym(z) to both sides, making use of (13.97) and Proposition 13.1.10 gives (α) (H) (H) (H) (y) = N0N e−p2 (z) aη Pη+ (z). N! 0 F0 (2y; −iz)Eη (iy) dμ (−∞,∞)N
(α)
But 0 F0 (2y; −iz) is symmetric in {yi } so (α) (H) F (2y; −iz)E (iy) dμ (y) = N! η 0 0 (−∞,∞)N
(−∞,∞)N
(α) (H) (y) 0 F0 (2y; −iz)SymEη (iy) dμ
= aη (−∞,∞)N
(α) (H) (y). 0 F0 (2y; −iz)Pη + (iy) dμ
Hence the symmetric analogue of (13.58) is (α) (H) (H) (y) = N0N e−p2 (z) Pκ(H) (z). 0 F0 (2y; −iz)Pκ (iy) dμ
(13.100)
(−∞,∞)N
(H)
The formulas (13.96) and (13.97) together allow the norm of the Pκ
with respect to (13.74) to be com-
619
CORRELATIONS FOR GENERAL β
puted from knowledge of the nonsymmetric norm (13.79). P ROPOSITION 13.3.7 We have Pκ(H) |Pκ(H) (H) = (2α)|κ|
Pκ ((1)N ) (H) N0N . dκ
Proof. This is derived in a similar fashion to the norm of the symmetric Jack polynomials as given in the proof of Proposition 12.3.3. An alternative derivation of this result using the generating function (13.98) is given in Exercises 13.3 q.1.
13.3.2 Generalized Laguerre polynomials (L)
The nonsymmetric generalized Laguerre polynomials, to be denoted Eη , are simultaneous eigenfunctions of the {li }. Analogous to the theory of the nonsymmetric Hermite polynomials, the fact that li consists of ξˆi plus a degree lowering operator, together with the fact that the {li } commute, implies the simultaneous eigenfunctions have the structure Eη(L) (y 2 ) = Eη (y 2 ) + cην Eν (y 2 ), (13.101) |ν|<|η| (L)
and the corresponding eigenvalue is η¯i . The relationship (11.79) shows that each Eη (y 2 ) is also a polyno˜ (L,Ex) with corresponding eigenvalue −|η|. mial eigenfunction of H Orthogonality ˜ (L,Ex) together with the Since the Schr¨odinger operator H (L,Ex) (11.54) is Hermitian, the definition of H above facts tells us immediately that for compositions ν and η such that |ν| = |η| (at least), Eν(L) |Eη(L) (L) = 0 where
(13.102)
f |g(L) := dμ(L) (y 2 ) :=
(−∞,∞)N N l=1
2 2 f (y12 , . . . , yN )g(y12 , . . . , yN ) dμ(L) (y 2 ),
|yl |2a+1 e−yl
2
(13.103)
|yk2 − yj2 |2/α dy1 · · · dyN .
(13.104)
1≤j
In fact the orthogonality (13.102) holds for general ν = η. This in turn follows from the fact that the li , as (L) well as being eigenvalues of the {Eη (y 2 )} which separate the eigenvalues, are self-adjoint with respect to (13.103). P ROPOSITION 13.3.8 We have a f |dˆi g(L) = (1 − 2 − dˆi )f |g(L) , yi
(13.105)
f |li g(L) = li f |g(L) .
(13.106)
Proof. The result (13.105) follows from (13.103) and Definition 11.5.4 after integrating by parts (the fact that f and g (B)∗
are even in (13.103) is not used in the derivation). Hence with di
(B)
denoting the adjoint of di
with respect to (13.103)
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CHAPTER 13
(B)∗
we have di
(B)
= 2yi − di
. Substituting this in (11.80) shows
li =
“α 4
(B)∗ (B) di
di
+1−N +
N ”˛ 1 X ˛ (sip + σi σp sip ) ˛ even , 2 p=i+1 functions
and thus li is self-adjoint with respect to (13.103), which is the statement (13.106).
Recurrences If we make the Eη in Proposition 12.2.1 a function of yi2 and then apply the map φ(B) specified by (11.81) and make use of (13.70) we obtain the recurrences ⎧ (L) 2 (L) 1 1 ⎪ Esi η (y 2 ), ηi > ηi+1 , E (y ) + 1 − 2 ⎪ δ¯i,η ⎪ δ¯i,η η ⎨ (L) (13.107) si Eη(L) (y 2 ) = Eη (y 2 ), ηi = ηi+1 , ⎪ ⎪ ⎪ ⎩ 1 E (H) (y 2 ) + E (L) (y 2 ), ηi < ηi+1 . si η δ¯i,η η As in the theory of the nonsymmetric Jack and Hermite polynomials, the operator si is one of the two fun(L) damental operators needed to generate the Eη by recurrence. The other is a raising type operator. In preparation for defining the raising type operator, and its lowering type companion, we introduce the raising type 2 operator for Eη (y 2 ), Ψ := yN sN −1 · · · s2 s1 so that Ψ Eη (y 2 ) = EΦη (y 2 )
(13.108)
ˆ of (cf. the final equation in Proposition 12.2.3). One might also suspect we should consider the operator Φ 2 Definition 12.2.4 with the change of variables zj = yj . However this is not the case; instead the lowering type operator is defined by ˆ := B1 s1 s2 · · · sN −1 , Ψ (B) 2
where Bi := 14 (di
) .
P ROPOSITION 13.3.9 With q := 1 + (N − 1)/α we have ˆ =Ψ ˆ ξˆj−1 , ξˆj Ψ ˆ =Ψ ˆ (ξˆN − α), ξˆ1 Ψ
2 ≤ j ≤ n,
dη 2 ˆ Eη (y 2 ) = 1 [a + q]η Ψ EΦη ˆ (y ). α [a + q]Φη dΦη ˆ ˆ
(13.109)
Proof. Let dˆi denote the type A Dunkl operator of Definition 11.4.1 with the change of variables zj = yj2 (j = 1, . . . , N ) so that
1 ∂ 1 X 1 − sip dˆi = + . 2yi ∂yi α yi2 − yp2 p =i
Similarly define the Cherednik operator with the same change of variables X sjp . ξˆj := αyj2 dˆj + (1 − N ) + p>j
One can verify directly that 8 < Bi sij , Bj sij , P [ξˆj , Bi ] = P : −αBj − p<j sjp Bj − p>j Bj sjp ,
i < j, i > j, i = j.
621
CORRELATIONS FOR GENERAL β
The first two results of (13.109) follow from this commutator formula in a manner analogous to the deduction of the first two results of Proposition 12.2.5 from Proposition 11.5.2. The final result, up to the proportionality constant cη , say, follows from the first two results. Examination of the leading term gives “ η¯ + N − 1 ” N cη = (a + q − 1 + η¯N /α) , α which can be identified with the stated form.
ˆ From the definitions we see Consider now the action of φ(B) on Ψ and Ψ. ˆ = Ψ, ˆ φ(B) Ψ 1 (B) 2 1 (B) 2 ˆ ∗, φ(B) Ψ = yN − di sN −1 · · · s2 s1 = sN −1 · · · s2 s1 yN − di = Ψ 2 2 ˆ with respect to (13.103). Thus, applying the map φ(B) to (13.108) and ˆ ∗ denotes the adjoint of Ψ where Ψ (L) (13.109) we obtain the sought raising and lowering formulas for the Eη . P ROPOSITION 13.3.10 We have dη (L) 2 ˆ E (L) (y 2 ) = 1 [a + q]η Ψ EΦη η ˆ (y ), α [a + q]Φη d ˆ ˆ Φη ˆ ∗ E (L) (y 2 ) = E (L) (y 2 ). Ψ η Φη ˆ ∗ given by the above result allows E (L) , |E (L) (L) to be related to Eη(L) , |Eη(L) (L) as The action of Ψ Φη Φη (L)
in the derivation of (13.83). Furthermore, the formulas (13.107) and the orthogonality of the {Eη } with respect to (13.103) implies the formula corresponding to (13.81) holds in the Laguerre case. Proceeding as in the proof of Proposition 13.3.3 allows the normalization of the nonsymmetric Laguerre polynomials to be computed. P ROPOSITION 13.3.11 With q = 1 + (N − 1)/α as in Proposition 13.3.9, we have Nη(L) := Eη(L) |Eη(L) (L) =
[a + q]η dη eη (L) N0 , dη α|η|
where (L)
N0
:= N0N = 1|1(L) = α(1−N −(N −1) (L)
2
/α)
N −1 j=0
Γ(1 + (j + 1)/α)Γ(a + 1 + j/α) . Γ(1 + 1/α)
Generating function The classical Laguerre polynomials can be specified by the generating function 2
ez 0 F1 (a + 1; −(xz)2 ) =
∞ Lak (x2 )z 2k . (a + 1)k
(13.110)
k=0 (L)
In the case of the Eη , an analogous formula holds, with the hypergeometric function 0 F1 (a + 1; −(xz)2 ) replaced by KB (x2 ; z 2 ) := 0 K1 (a + q; x2 ; z 2 ) :=
η
α|η| dη Eη (x2 )Eη (z 2 ). [a + q]η dη eη
(13.111)
Before presenting the generating function, we make note of some fundamental properties of KB , which are analogous to the properties of KA presented in Proposition 13.3.4.
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CHAPTER 13
P ROPOSITION 13.3.12 We have ˆ (x) KB (x2 ; y 2 ) = Ψ(y) KB (x2 ; y 2 ), Ψ (x)
Bi
KB (x2 ; y 2 ) = yi2 KB (x2 ; y 2 ).
(13.112)
ˆ (x) term-by-term to the series definition (13.111) using the final equation Proof. The first equality follows by applying Ψ
ˆ in the summation, then using (13.108) to rewrite EΦν (y 2 ) in terms of Ψ(y) Eν (y 2 ). in Proposition 13.3.9, writing ν = Φη (x) ˆ (x) , respectively, and using the first result. The second result follows by writing yi and Bi in terms of Ψ(y) and Ψ
In further preparation for deriving the generating function, we note with the notation (11.78) the operator ˜ (L,Ex) can be rewritten to read H ˜ (L,Ex) = ΔB − 2 H
N l=1
xl
∂ . ∂xl
(13.113)
Furthermore, we note the second equality in Proposition 13.3.12 implies ΔB KB (x2 ; y 2 ) = 4p1 (x2 )KB (x2 ; y 2 )
(13.114)
(p1 is as specified in (12.146)). These facts allow us to proceed as in the derivation of (13.90) to establish the following formula. P ROPOSITION 13.3.13 We have (−α)|η| dη Eη(L) (x)Eη (z) = KB (x; −z)ep1 (z) . e [a + q] d η η η η
(13.115)
In Exercises 13.3 q.6, the generating function (13.115) is used to give the explicit form of the coefficients cην in (13.101) in terms of the generalized binomial coefficients. Analogous to the exponential operator formula (13.91) we can use (13.114) to deduce from the generating (L) function (13.115) an exponential operator formula expressing Eη (x2 ) in terms of Eη (x2 ). Thus we have 1 (13.116) exp − ΔB Eη (x2 ) = Eη(L) (x2 ). 4 Similarly, analogous to (13.92) this in turn implies the intertwining relation 1 1 (13.117) exp − ΔB ξˆj = lj exp − ΔB . 4 4 Integration formulas The strategy used to derive the integration formulas of Proposition 13.3.6, with the generating function (13.115) taking the role of the generating function (13.90), implies analogous integration formulas in the Laguerre case. P ROPOSITION 13.3.14 We have (L) KB (x; −za )KB (x; −zb ) dμ(L) (x2 ) = N0N e−p1 (za ) e−p1 (zb ) KB (za ; zb ), [0,∞)N (L) KB (x; −za )Eη(L) (x) dμ(L) (x2 ) = N0N e−p1 (za ) Eη (−za ), [0,∞)N (L) KB (x; −za )Eη (−x) dμ(L) (x2 ) = N0N e−p1 (za ) Eη(L) (za ). [0,∞)N
(13.118)
623
CORRELATIONS FOR GENERAL β
Symmetric Laguerre polynomials The theory of symmetric Laguerre polynomials can be developed from the nonsymmetric Laguerre polynomials in an analogous fashion to the development of the symmetric Hermite polynomials. Thus the symmetric Laguerre polynomials can be defined as the eigenfunctions of (11.54) with the structure Pκ(L) (y 2 ) = Pκ (y 2 ) + a ˜κμ Pμ (y 2 ). (13.119) |μ|<|κ|
Furthermore, they can be expressed in terms of the nonsymmetric Laguerre polynomials by either of the formulas 1 E (L) (y) (13.120) Pκ(L) (y) = dη+ dη η + η:η =κ
or Pκ(L) (y) =
1 SymEη(L) (y), aη
where aη is specified by (12.101). Applying Sym to both sides of (13.116) shows 1 exp − ΔB symmetric Pκ (y 2 ) = Pκ(L) (y 2 ). 4 functions
(13.121)
(13.122)
Changing variables y 2 → x, and noting that N ∂ 1 ΔB symmetric = D1 + (a + 1) , 4 ∂x j functions j=1
where D1 :=
N j=1
xj
∂2 ∂ 2 xj + , 2 ∂xj α xj − xk ∂xj j=k
we see that (13.122) can equivalently be written N ∂ exp − D1 + (a + 1) Pκ (x) = Pκ(L) (x). ∂xl
(13.123)
l=1
The Sym operator can also be applied to both sides of the generating function (13.115) to give that (−α)|κ| Pκ(L) (x)Pκ (z) = 0 F1 (a + q; x; −z)ep1 (z) . dκ N) [a + q] P ((1) κ κ κ
(13.124)
Also, similar to the derivation of the norm of the symmetric Hermite polynomials, Proposition 13.3.7, the formulas (13.120) and (13.121) allow the norm of the symmetric Laguerre polynomials to be computed from knowledge of the nonsymmetric norm Proposition 13.3.11.
P ROPOSITION 13.3.15 We have (L)
Pκ(L) |Pκ(L) (L) = |κ|!N0 Cκ(α) ((1)N )[a + q](α) κ , (α)
where Cκ
is specified by (13.1).
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CHAPTER 13
13.3.3 Generalized Jacobi polynomials Let η be an N -tuple of non-negative integers, let ε be an N -tuple with each entry +1 or −1, and define εη as the N -tuple formed from ε and η by multiplication of their respective parts. A direct calculation using (11.84) (J) shows that for suitable ej,εη , cεη,ε η (J) ξjBC z εη = ej,εη z εη + cεη,ε η z ε η . η <η
˜ (J,Ex) permits a complete set of From this action we can make use of the operator (11.86) to conclude that H nonsymmetric eigenfunctions Eη(J) (sin2 φ) = Eη (sin2 φ) + aνη Eν (sin2 φ). (13.125) |ν|<|η| (J)
In the variable y = sin2 φ, the Eη (y) are polynomials, referred to as the nonsymmetric Jacobi polynomials. (BC) Another direct calculation shows ξj is self-adjoint with respect to the inner product π/2 π/2 (J) (J) f |gφ := dφ1 · · · dφN |ψ0 |2 f (¯ z )g(z), (13.126) 0
where z := e (J)
ψ0 =
2iφ
N
0
and
−(N −1)/α−(a+b+1)/2
zj
(zj − 1)a+1/2 (zj + 1)b+1/2
j=1
(zk − zj )1/α (1 − zj zk )1/α .
(13.127)
j
To see this, one makes use of the operator identity zj ∂ 1 zl (J) (BC) (J) ψ0 ξj (ψ0 )−1 = zj − slj + slj ∂zj α zj − zl zj − zl l<j
+
1 α
N k=1 =j
l>j
σj σk sjk 1 σj 1 σj + a+ + b+ . 1 − zj zk 2 1 − zj 2 1 + zj
(J)
As a consequence, {Eη (sin2 φ)} is orthogonal with respect to the inner product (13.126). Changing vari(J) ables yj = sin2 φj shows the nonsymmetric Jacobi polynomials {Eη (y)} are orthogonal with respect to the inner product 1 1 N N (J) f |gy := dy1 · · · dyN yla (1 − yl )b |yk − yj |2/α f (y)g(y). (13.128) 0
Symmetrizing
(J) Eη (y)
0
l=1
j
gives the corresponding symmetric Jacobi polynomial Pκ(J) (y) =
1 Sym Eη(J) (y), aη
where κ = η + and aη is given by (12.101). The symmetric Jacobi polynomials form a complete set of ˜ (J) (11.91), with corresponding eigenvalues symmetric polynomial eigenfunctions of H e(J) κ
:=
N
κj (κj + 2(N − j)/α + a + b + 1).
j=1
They appear in a generalization of the dual Cauchy product (12.212) [484] ,[404].
(13.129)
625
CORRELATIONS FOR GENERAL β
P ROPOSITION 13.3.16 With μ := N N − κ one has N
(xk − yl ) =
(−1)|μ| Pκ(J) (x; α)Pμ(J) (y; 1/α).
(13.130)
κ
k,l=1
Proof. First, by a direct calculation, we can check that ˜ (J)x −H
N Y
(xk − yl ) =
k,l=1
N ” Y 1 “ ˜ (J)y (J) (xk − yl ), H∗ − e(N)N ∗ α
(J)y
˜∗ ˜ (J)x refers to (11.91) with y → x, while H where H α → 1/α, Expanding
N Y
(13.131)
k,l=1
(J)
, eκ∗ refer to (11.91), (13.129) with parameters
a → aα + α − 1,
(xk − yl ) =
k,l=1
X
b → bα + α − 1.
fκ (y)Pκ(J) (x),
(13.132)
κ⊆(N)N
we have from (12.212) that fκ is of the form ” “ X cκρ Pρ (y; 1/α) , fκ (y) = (−1)|μ| Pκ (y; 1/α) +
(13.133)
ρ<μ
where μ := N N − κ . Now, substituting (13.132) in (13.131) we obtain X “ (J) 1 (J) ” 1 X ˜ (J)y H∗ fκ (y)Pκ(J) (x). − eκ − eN N ∗ fκ (y)Pκ(J) (x) = α α N N κ⊆N
κ⊆N
But from the explicit formula (13.129) we can make use of (12.33) to check that “ 1 (J) ” e −α e(J) = e(J) κ − μ∗ . α NN ∗ ˜ ∗(J)y with the structure (13.133), eigenvalue e(J) As the unique eigenfunction of H μ∗ , is fκ (y) = (−1)|μ| Pκ(J) (y; 1/α) (13.130) follows.
E XERCISES 13.3
1. Following the method of [36] an alternative derivation of Proposition 13.3.7 will be given. (H)
(i) By multiplying both sides of (13.98) by Pκ (y) and integrating with respect to the measure dμ(H) (y) (H) (H) (H) deduce that with Nκ := Pκ |Pκ (H) , Nκ(H)
(2α)|κ| Pκ (z) = e−p2 (z) dκ Pκ ((1)N )
Z (−∞,∞)N
(α) (H) (H) (y). 0 F0 (2y; z)Pκ (y) dμ
(ii) Set z1 = · · · = zN = c in the formula of (i), make use of (13.3) and complete the square to show Z Y (2α)|κ| N P Nκ(H) ((c) ) = e−p2 (y) |yk − yj |2/α Pκ(H) (y + c) dy1 · · · dyN . κ N dκ Pκ ((1) ) (−∞,∞)N j
Now take the limit c → ∞, noting from (13.95) that limc→∞ Pκ (y +c)/Pκ (c) = 1 to deduce the formula (H) for Nκ given in Proposition 13.3.7.
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CHAPTER 13
2. Use the integral representation (13.58), the definition (13.85) and the integration formula (13.93) for |t| < 1 to show that X 1 E (H) (w)Eη(H) (z)t|η| (H) η η Nη ” “ ” “ 2wt 1 t2 z , (13.134) = (H) (1 − t2 )−Nq/2 exp − ; (p2 (w) + p2 (z)) KA 2 2 1/2 2 1/2 (1 − t ) (1 − t ) (1 − t ) N0N where q := 1 + (N − 1)/α. Similarly, make use of the first and third formulas in (13.118) and (13.111) to provide a verification of the formula X 1 E (L) (x)Eη(L)(y)t|η| (L) η N η η “ ” “ y 1 t tx ” , (13.135) = (L) (1 − t)−N(a+q) exp − (p1 (x) + p1 (y)) KB ; 1−t 1−t 1−t N N 0
valid for |t| < 1. 3.
(i) Use the formula (13.91) in (13.58) to deduce that for f analytic such that all terms converge Z (H) KA (2y; −iz)f (iy) dμ(H) (y) = N0N e−p2 (z) e−ΔA /4 f (z).
(13.136)
(−∞,∞)N
(ii) Use this result to show that if we define the integral transform (a generalized Fourier transform, or Dunkl transform) by Z ep2 (z) F (z) = KA (2y; −iz)f (iy) dμ(H) (y) (H) N0N (−∞,∞)N then the inverse transform is f (z) =
e−p2 (z) (H) N0N
Z (−∞,∞)N
KA (2y; z)F (y) dμ(H) (y).
4. [142] With di denoting the type A Dunkl operator, and p and q homogeneous polynomials of N variables, define the pairing ˛ ˛ [p, q]A := p(d)q ˛ , (13.137) x=0
where p(d) is the operator obtained from p(x) by replacing each xi by di . (i) Note that for p and q of different degrees, [p, q]A = 0. (ii) By forming an appropriate linear combination of the second formula in (13.86) note that Eν (d(x) )KA (x; y) = Eν (y)KA (x; y). Now set x = 0 and equate coefficients of Eν (y) to deduce that [39] [Eν , Eη ]A =
1 dη eη δν,η . α|η| dη
(iii) Compare (13.138) and (13.75) to show [Eν , Eη ]A =
2|η| (H)
N0N
Eν(H) |Eη(H) (H) .
(13.138)
627
CORRELATIONS FOR GENERAL β
Now make use of (13.91) to deduce [p, q]A =
2|η| (H)
N0N
e−ΔA /4 p|e−ΔA /4 q(H) .
5. With the symmetric Hermite polynomials characterized by the structure (13.95) and orthogonality Pκ(H) |Pμ(H) (H) ∝ δκ,μ , and the symmetric Laguerre polynomials characterized by the structure (13.119) and orthogonality Pκ(L) |Pμ(L) (L) ∝ δκ,μ , use the characterization of the symmetric Jacobi polynomials as having the structure (13.125) and orthogonality (J) (J) (J) Pκ |Pμ y ∝ δκ,μ to deduce the limit relations ˛ ”˛ “1 ˛ ˛ lim L|κ| Pκ(J) (x/L)˛ = Pκ(L) (x), lim (−2L)|κ| Pκ(J) (1 − x/L) ˛ a →L2 = Pκ(H) (x). L→∞ L→∞ 2 b→L b →L2 6. The objective of this exercise is to derive the formula Eη(L) (x) =
„ « (−1)|η| [a + q]η eη X (−1)|ν| dν η Eν (x), dη [a + q]ν eν ν
(13.139)
νη
giving the explicit expansion of the nonsymmetric Laguerre polynomials in terms of the nonsymmetric Jack polynomials. (i) Use (13.31) to obtain the formula KB (x; −z)ep1 (z) =
X (−1)|η| dη X α|ν| „ ν « Eν (z). Eη (x) [a + q]η eη dν η η ν
(ii) Substitute the result of (i) in (13.115) and equate coefficients of Eη (z) to deduce (13.139). (iii) Symmetrize (13.139) to show Pκ(L) (x) = (−1)|κ| Pκ(α) ((1)N )[a + q](α) κ
X „ κ « (−1)|σ| Pσ (x) . N σ [a + q](α) σ Pσ ((1) )
σ⊆|κ|
(iv) Use (12.180) together with (12.105) to deduce from (iii) that in the case κ = pN Pκ(L) (x) = (−1)|κ| Pκ ((1)N )[a + q](α) κ 1 F1 (−N ; a + q; x). (α)
(L)
(L)
(L)
(α)
Note also that with η = pN , Eη (x) = Pη (x), so Eη (x) is also thus related to 1 F1 .
13.4 GREEN FUNCTIONS AND ZONAL POLYNOMIALS 13.4.1 Hermite case (H)
The symmetric Hermite polynomials Pκ (13.97) are a complete set of polynomial eigenfunctions of H (H,Ex) with eigenvalue −2|κ|. Recalling the transformation and change of variables which gave rise to the eigenop(H) erator (11.64) for {Pκ }, we see that the complete set of symmetric eigenfunctions of H = H (H) is , + (H) (13.140) e−βW (x)/2 Pκ(H) ( β/2x) , κ
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CHAPTER 13
(H)
(H)
where here and in the remainder of the subsection Pκ (u) = Pκ (u; 2/β), with corresponding eigen(H) (H) (H) values E0 + β|κ|. Denoting Nκ |α=2/β := Pκ |Pκ (H) |α=2/β , a simple change of variables gives (H) ( β/2)−N −βN (N −1)/2 Nκ |α=2/β for the normalization of (13.140). The formula (11.43) shows the corresponding Green function is specified by (H) (0) (H) (x(0) ; x) = ( β/2)N +βN (N −1)/2 e−E0 τ /β e−βW (x )/2 e−βW (x)/2 GS(H) τ Pκ(H) ( β/2x(0) )Pκ(H) ( β/2x) × e−|κ|τ . (13.141) (H) N | κ α=2/β κ S(H)
With Gτ known, the general formula (11.42) gives the Green function for the corresponding Fokker-Planck operator as Pκ(H) ( β/2x(0) )Pκ(H) ( β/2x) FP(H) (0) N +βN (N −1)/2 −βW (H) (x) Gτ (x ; x) = ( β/2) e e−|κ|τ . (H) Nκ |α=2/β κ (13.142) Symmetrizing (13.134) allows the multidimensional analogue of the classical summation formula (11.102) to be deduced, Pκ(H) (x)Pκ(H) (y)t|κ| (H)
κ
Nκ |α=2/β
=
1 (H)
N0
|α=2/β
(2/β)
× 0 F0
(1 − t2 )−N (1+β(N −1)/2)/2 exp −
2xt y , ; (1 − t2 )1/2 (1 − t2 )1/2
N t2 2 2 (x + y ) j 1 − t2 j=1 j
|t| < 1.
(13.143)
(H)
Recalling (13.20), substituting (13.143) in (13.142) gives GFP as a sum over Jack polynomials. 13.4.2 Laguerre case (L)
The symmetric Laguerre polynomials Pκ (y 2 ) (13.119) are a complete set of symmetric polynomial eigen˜ (L,Ex) with eigenvalue −|κ|, and so the complete set of symmetric eigenfunctions of H (L) functions of H is , + (L) e−βW (x)/2 Pκ(L) (βx2 /2) a →βa /2−1/2 α →2/β
(L)
κ
(L)
with eigenvalue Eκ = E0 + 2β|κ|. Denoting NκL |α=2/β := PκL |PκL (L) |α=2/β , and proceeding as in the derivation of (13.142) shows
GFP(L) (x(0) ; x) = (β/2)N (1+β(N +a −1))/2 e−βW (x) τ Pκ(L) (β(x(0) )2 /2)Pκ(L) (βx2 /2) −2|κ|τ × e a →βa /2−1/2 . (L) Nκ α=2/β κ (L)
(13.144)
In the classical case N = 1, this can be summed according to (11.106). By symmetrizing (13.135) one obtains the multivariable generalization [36] N Pκ(L) (x)Pκ(L) (y)t|κ| 1 t −N (a+q) = (1 − t) exp − (x + y ) j j (L) (L) 1 − t j=1 α=2/β Nκ N0N |α=2/β κ y xt (2/β) ; , |t| < 1, (13.145) a + q; × 0 F1 1−t 1−t
where q = 1 + β(N − 1)/2, thus giving an alternative form for the r.h.s. of (13.144).
629
CORRELATIONS FOR GENERAL β
13.4.3 Zonal polynomials FP(H)
is given in terms of a matrix integral. We know from (11.101) that for β = 1, 2, 4 the Green function Gτ FP(L) Similarly (11.105) gives Gτ for β = 1, 2, 4 in terms of a matrix integral. Comparing with the expressions for the Green functions as implied by (13.142), (13.143) and (13.144), (13.145) the following matrix integral evaluations result. P ROPOSITION 13.4.1 Let (U† dU), (V† dV) denote the normalized Haar volume form for N × N , n × n matrices with n ≥ N . Let H, H(0) be as in (11.101) and let X, Y be n × N matrices with real (β = 1), complex (β = 2) or real quaternion (β = 4) elements. We have (0) † (2/β) (0) eTr(H U HU) (U† dU) = 0 F0 (λ ; λ), (13.146) † † † † (2/β) (βn/2; x2 ; y 2 ), (13.147) (U† dU) (V† dV) eTr(UX V Y+Y VXU ) = 0 F1 (0)
where {λj }, {λj } are the eigenvalues of H(0) , H and {x2j }, {yj2 } are the eigenvalues of X† X, Y† Y. In the case β = 2 these are the same matrix integrals evaluated by (11.103) and (11.108), respectively. Underlying the integration formulas of Proposition 13.4.1 are further integration formulas directly involv(2/β) (2/β) ing {Cκ } in the integrand. These relate to a fundamental theory for the {Cκ } as so-called zonal polynomials associated with the symmetric spaces gl(N, R)/O(N ) (β = 1), gl(N, C)/U (N ) (β = 2) and u∗ (2N )/Sp(2N ) (β = 4) [376]. However a development of this theory is outside the scope of the present work. Instead we will be content with noting some key integration formulas from the theory. P ROPOSITION 13.4.2 Let U be as in (13.146). Let A, B be N × N matrices with real, complex and real quaternion entries for β = 1, 2, 4, respectively. We have (2/β) (2/β) (A)Cκ (B) Cκ Cκ(2/β) (AU† BU) (U† dU) = , (13.148) (2/β) Cκ ((1)N ) (2/β)
(2/β)
(AU† BU) is defined as Cκ (y) with y1 , . . . , yN denoting the eigenvalues of AU† BU, and where Cκ (2/β) (2/β) similarly the meaning of Cκ (A), Cκ (B). In fact this integration formula can be taken to be the defining property of the corresponding zonal polynomials, up to normalization. Note that summing (13.148) over κ using (13.2), (13.3), (13.20) reclaims (13.146). Consider now (13.147) in the case n = N , Y = 1. Noting that the integrand is then independent of V, and recalling from Section 1.3.2 that in the case β = 4 Tr in (13.147) refers to the matrices regarded as having real quaternion rather than scalar elements, we obtain (2)
eTr(XO) O∈O(N ) = 0 F1 (N/2; XXT /4), †
†
(1) eTr(XU) eTr(U X ) U∈U (N ) = 0 F1 (N ; XX† ), (1/2) (2N ; XX† ). eTr(XS) S∈Sp(2N ) = 0 F1
(13.149) (13.150) (13.151)
In (13.151) Tr now refers to the matrices regarded as having complex entries and use has been made of the fact that for X real quaternion Tr(XS) = Tr(XS)† . It turns out that underlying (13.149)–(13.151) are group integrals transforming Schur polynomials to zonal polynomials [312], [376], [454].
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CHAPTER 13
P ROPOSITION 13.4.3 One has sλ (AO)O∈O(N ) =
⎧ (2) ⎪ ⎨ Cκ (AAT ) (2)
C ((1)N ) ⎪ ⎩ 0 κ
, λ = 2κ,
(13.152)
otherwise,
(1) Cκ (AA† ) , sλ (AU)sκ (U† A† )U∈U (N ) = δλ,κ (1) Cκ ((1)N )
sλ (AS)S∈Sp(2N ) =
⎧ (1/2) ⎪ ⎨ Cκ (AA† ) (1/2)
C ⎪ ⎩ 0κ
((1)N )
(13.153) , λ = κ2 ,
(13.154)
otherwise,
where in (13.152) the partition 2κ is the partition obtained by doubling each part of κ, while in (13.154), κ2 is the partition obtained by repeating each part of κ twice. Note that with A = 1 (13.154) reduces to (10.137) with β = 0, while (13.152) reduces to (10.155) with α = 0. To deduce (13.149)–(13.151) from the results of Proposition 13.4.3 first note from (13.1) and (13.8) that for general α (α)
Cκ(α) ((1)N ) = |κ|!
α2|κ| [N/α]κ . hκ dκ
(13.155)
In (13.152) and (13.154) multiply both sides by 1/dλ |α=1 and use (13.3) on the l.h.s. On the r.h.s. use (13.155), (13.2) and the identities (hκ dκ )|α=2 = 1, d2κ |α=1
22|κ| (hκ dκ )|α=1/2 =1 dκ2 |α=1
(13.156)
to obtain (13.149) and (13.151). For the identity (13.150), first multiply both sides of (13.153) by 1/dκ |α=1 , and simplify by noting that for α = 1, dκ = hκ . 13.4.4 Duality formulas The calculations of Section 13.2 are based on expressions for the correlations in terms of multidimensional integrals whose dimension is independent of N . As the correlations themselves are multidimensional integrals whose dimension is proportional to N , a feature of Jack polynomial theory which gives identities between multidimensional integrals is being put to use (see also Exercises 13.2 q.6). Here, following [131], such a duality type formula will be developed for states in the Dyson Brownian motion model of the log-gas in a harmonic well with general initial conditions. P ROPOSITION 13.4.4 We have N n √ (α) (sj − αyk ) ep2 (x) 0 F0 (2y; −ix) j=1 k=1
= (−i)nN ep2 (s)
(1/α) (2y; −is) 0 F0
ME2/α,N (e−y2 )
N n
(yj +
√
αxk )
j=1 k=1
ME2/α,n (e−y2 )
Proof. In the case that f is a symmetric function, (13.136) reads Z (−∞,∞)N
(α) (H) (y) 0 F0 (2y; −iz)f (iy) dμ
(H) −p2 (z) −ΔA /4
= N0
e
e
f (z),
.
(13.157)
631
CORRELATIONS FOR GENERAL β
or equivalently ep2 (x) (p,s)
Let ΔA
D
(α) 0 F0 (2y; −ix)f (iy)
E 2
ME2/α,N (e−y )
= e−ΔA /4 f (x).
(13.158)
be specified by (11.68) with N → p, y → s and 1 − sjk replaced by zero. A direct calculation shows (N,y)
ΔA
˛ √ (n,s) ˛ (sj − i αyk ) = ΔA ˛
n Y N Y
j=1 k=1
N n Y Y α→1/α
√ (sj − i αyk )
j=1 k=1
and consequently (N,y)
e−ΔA
/4
N n Y Y
N n Y Y (n,s) √ √ (sj − i αyk ) = e−ΔA |α →1/α /4 (sj − i αyk ).
j=1 k=1
(13.159)
j=1 k=1
Suppose now that we choose f (y) =
N n Y Y
√ (sj − i αyk )
(13.160)
j=1 k=1
in (13.158). Making use of (13.159) shows ep2 (x)
D
(α) 0 F0 (2y; −ix)f (iy)
E
(n,s)
−y2
ME2/α,N (e
)
= e−ΔA
|α →1/α /4
N n Y Y
√ (sj − i αxk ).
(13.161)
j=1 k=1
To deduce (13.157) we make a further application of (13.158), this time with N → n, α → 1/α, x = s, y = x and now f (u) given by the r.h.s. of (13.160) with y → x. Substituting for the r.h.s. of (13.161) the l.h.s. of (13.158) so modified gives (13.157). (α)
Consider (13.157) with s = (c)n , x = (0)N . Making use of (13.63) with 0 K0 replaced by 0 F0 we see that (13.157) simplifies to involve only elementary functions in the average. Completing the square gives N
√ (c − αyj )n
j=1
−y2
ME2/α,N (e
)
=
n
(c − iyj )N
j=1
ME2α,n (e−y2 )
.
(13.162)
This is a generalization of the duality formula used to derive (13.64). In the special cases 2α = 1, 2 and 4 we can substitute (13.146) in (13.157) to obtain the matrix integrals duality [96], [97] n √ 2 eTr X e−2iTr(XY) det(sj − αY) −y2 Y∈ME2/α,N (e
j=1
= (−i)nN eTr S
2
e−2iTr(SY)
N k=1
det(Y +
√ αxk )
)
Y∈ME2α,n (e−y2 )
,
where now MEβ,m refers to the appropriate ensemble of Gaussian random matrices. 1. [509], [287], [454] For β = 1, 2, 4, let X, A, B be N × N random matrices with standard Gaussian real, complex, real quaternion entries respectively. Define Z † 1 e−Tr(XX ) f (X)(dX), f (X)X := C X
E XERCISES 13.4
where C is such that the r.h.s. equals unity for f = 1. (i) By noting that the distribution of X is invariant under the mapping X → UX for U ∈ O(N ), U (N ), Sp(2N ),
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CHAPTER 13
respectively, conclude
f (AXBX† )X = f (AUXBX† U† )U X .
(13.163)
Similarly, use the invariance under X → XU to conclude f (AXBX† )X = f (AXUBU† X† )U X . (2/β)
(ii) Choose f = Cκ
(13.164)
in (13.163) to deduce from (13.148) that (2/β)
Cκ
Cκ(2/β) (AXBX† )X =
(A)
(2/β) Cκ ((1)N )
Cκ(2/β) (XBX† )X .
(iii) Use (13.164) and (13.148) to evaluate the average on the r.h.s. of the identity in (ii) and obtain Cκ(2/β) (AXBX† )X =
(2/β)
Cκ
(2/β)
(A)Cκ
(B)
Cκ(2/β) (XX† )X .
(2/β) (Cκ ((1)N ))2
(13.165)
(iv) Note from (13.152) that with X an N × N random matrix of standard real Gaussians s2κ (AX)X = s2κ (AXO)O X =
1 (2) Cκ ((1)N )
Cκ(2) (AT AXXT )X .
Now use (13.165) in the case β = 1 to deduce from this that (2)
s2κ (AX)X =
Cκ (AAT ) (2)
(Cκ ((1)N ))2
Cκ(2) (XXT )X .
(v) With X an N × N random matrix of standard complex Gaussians, show from (13.153) and (13.165) in the case β = 2 that (1) Cκ (AA† ) sλ (AX)sκ (X† A† )X = δλ,κ (1) Cκ(1) (XX† )X . (Cκ ((1)N ))2 (vi) With X an N × N random matrix of standard real quaternions, show from (13.154) and (13.165) in the case β = 4 that (1/2) Cκ (AA† ) sκ2 (AX)X = Cκ(1/2) (XX† )X . (1/2) (Cκ ((1)N ))2 2. [249] In this exercise, the simplest case (p = 1) of the matrix integral duality formula Z
∞
| det(z1N − AU)|2p U∈U (N) ∝ 0
Z
∞
dt1 · · ·
dtp 0
p Y det(|z|2 1N + tl AA† ) (1 + tl )N+2p l=1
Y
|tk − tj |2
1≤j
(13.166) will be established. (i) With sκ (X) denoting the Schur polynomial in the eigenvalues of X, note from the dual Cauchy identity (10.56) that N X det(1N + qX) = q r s1r (X). (13.167) r=0
(ii) Use (13.167) together with the integration formula (13.153), to show | det(z1N − AU)|2 U∈U (N) = |z|2N
N X r=0
|z|−2r
C1r (AA† ) (1)
(1)
C1r ((1)N )
.
633
CORRELATIONS FOR GENERAL β
(iii) Note from the definition (13.1) and (12.105) that C1r (AA† ) (1)
(1) C1r ((1)N )
=
“ N ”−1 r
s1r (AA† ),
and change variables x → t/(1 − t) in the Euler beta integral (4.2) to show Z ∞ “ N ”−1 tr = (N + 1) dt. r (1 + t)N+2 0 Substitute this in the previous formula, substitute the result in the formula of (ii), and make a further use of (13.167) to obtain (13.166) in the case p = 1.
13.5 INTER-RELATIONS FOR SPACING DISTRIBUTIONS A generalization of the Dixon-Anderson integral was used in Section 4.4 to show that the p.d.f. of every (r + 1)th eigenvalue in certain β-ensembles with β = 2/(r + 1) is equal to the p.d.f. of another β-ensemble with β = 2(r+1). In the case r = 1 this has been used in Sections 8.4.3, 9.7 and 9.8 to relate gap probabilities in the bulk, and at the soft and hard edges, respectively, of matrix ensembles with orthogonal and symplectic symmetry. In this section the inter-relations of Section 4.4 will be used for general r ∈ Z+ to deduce the corresponding inter-relations between spacing distributions, both in the case of finite N ensembles and in scaled limits [199]. First some notation is required. D EFINITION 13.5.1 Suppose the eigenvalues in the matrix ensemble are ordered. Let the p.d.f. for the (k + 1)th largest eigenvalue be denoted pmax (k; s; MEβ,N ). In the case that the support of the spectrum is restricted to x > 0 let the p.d.f. for the (k + 1)th smallest eigenvalue be denoted pmin (k; s; MEβ,N ). In the circular β-ensemble CEβ,N let pspacing (k; s; CEβ,N ) denote the p.d.f. for eigenvalues which are (k + 1)th neighbors. With these definitions, we read off from (4.112)–(4.115) the following inter-relations. P ROPOSITION 13.5.2 One has pmax ((r + 1)k + r; s; ME2/(r+1),(r+1)N +r (e−x )) = pmax (k; s; ME2(r+1),N (e−(r+1)x )), pmin ((r + 1)k + r; s; ME2/(r+1),(r+1)N +r (xa e−x )) = pmin (k; s; ME2(r+1),N (x(r+1)a+2r e−(r+1)x )), (r+1)b+2r
pmin (((r + 1)k + r; s; CEb2/(r+1),(r+1)N +r ) = pmin (k; s; CE2(r+1),N ), pspacing ((r + 1)k + r; s; CE02/(r+1),(r+1)N ) = pspacing (k; s; CE02(r+1),N ). We know from Section 7.2 that the soft and hard edge scalings of MEβ,N (xa e−cx ) are given by x →
β (4N + 2(2N )1/3 sβ x), 2c
x →
β x , 2c 4N s˜β
(13.168)
respectively, where sβ , s˜β are arbitrary length scales. Thus we can calculate the soft and hard edge distributions according to β (2N )1/3 sβ pmax (k; s; MEβ,N (e−cx )), N →∞ c β 1 phard (k; s; a) := lim pmin (k; s; MEβ,N (xa e−cx )). β N →∞ 2c 4N s ˜β psoft β (k; s) := lim
In the ensemble CEbβ,N the mean spacing between eigenvalues is 2π/N and so the appropriate scaled distri-
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CHAPTER 13
butions are 2π min p (k; 2πs/N ; CEbβ,N ), N 2π spacing p pbulk,sp. (k; s) := lim (k; 2πs/N ; CE0β,N ). β N →∞ N These limits can be taken in the results of Proposition 13.5.2 to obtain generalizations of the inter-relations (9.141), (9.163) and (8.160). pβbulk,s.s.(k; s; b) := lim
N →∞
P ROPOSITION 13.5.3 Let the scales in (13.168) be chosen so that s2/(r+1) = (r + 1)2/3 s2(r+1) and s˜2/(r+1) (r + 1)2 = s˜2(r+1) . One has soft psoft 2/(r+1) ((r + 1)k + r; s) = p2(r+1) (k; s), hard phard 2/(r+1) ((r + 1)k + r; s; a) = p2(r+1) (k; s; (r + 1)a + 2r), bulk,s.s. bulk,s.s. ((r + 1)k + r; (r + 1)s; b) = p2(r+1) (k; s; (r + 1)b + 2r), (r + 1)p2/(r+1) bulk,sp. (r + 1)pbulk,sp. 2/(r+1) ((r + 1)k + r; (r + 1)s) = p2(r+1) (k; s).
E XERCISES 13.5
1. [199] According to (8.17) pβbulk,sp. (k; s) =
k d2 X (k − j + 1)Eβbulk (j; s). ds2 j=0
Argue that for s large the term j = k on the r.h.s. will dominate, and then use the final equation in Proposition 13.5.3 to deduce bulk bulk ((r + 1)k; (r + 1)s) ∼ E2(r+1) (k; s). (r + 1)E2/(r+1) Show that this is consistent with (9.99) and (9.103).
13.6 STOCHASTIC DIFFERENTIAL EQUATIONS 13.6.1 The bulk The random recurrences obtained in Chapters 1–3 for characteristic polynomials of the various β-ensembles are well suited to obtaining a characterization of the number of eigenvalues in a certain interval, and the spacing between eigenvalues, in terms of stochastic differential equations. We consider first the circular βensemble, which for finite N has the property that the eigenvalues are equal to the zeros of χN (λ) as specified by the coupled system (2.89). For the purpose of studying the N → ∞ bulk scaling limit of the circular βensemble, this suggests seeking the scaled, limiting form of these equations. To do this we don’t work with χk (λ) itself, since for |λ| = 1 it takes values in a two-dimensional region of the complex plane. Instead, following [357] we work with Bk (λ) :=
λχk−1 (λ) χ ˜k−1 (λ)
(13.169)
(cf. (3.138)) which has the property that |Bk (λ)| = 1 for |λ| = 1. The fact that the zeros of χ ˜k−1 (λ), k = 2, . . . , N − 1, are all outside the unit circle, together with the relationship between χk−1 and χ ˜k−1 , shows Bk (λ) is a finite Blanske product and thus has the further property |Bk (λ)| < 1 for |λ| < 1. The (relative) Pr¨ufer phase (recall Section 1.9.4) can be used to parametrize Bk (λ) for |λ| = 1.
635
CORRELATIONS FOR GENERAL β
P ROPOSITION 13.6.1 Write Bk (eiθ ) = eiψk (θ) Bk (1)
(13.170)
so that ψk (θ) denotes the relative Pr¨ufer phase. For k = 0, . . . , N − 1 we have that ψk (θ) is an increasing function of θ, and satisfies the recurrence 1 − γk ψk+1 (θ) = ψk (θ) + θ + 2Im log , ψ0 = θ, (13.171) 1 − γk eiψk (θ) where γk := Bk (1)αk and the logarithm is defined by its power series. Proof. We see from the coupled recurrences (2.89) that {Bk } satisfies the first order recurrence Bk+1 (λ) = λBk (λ)
1−α ¯ k Bk (λ) , 1 − αk Bk (λ)
B0 (λ) = λ.
(13.172)
It follows from this and (13.170) that eiψk+1 (θ) = ei(θ+ψk (θ))
1 − γk 1 − γ¯k e−iψk (θ) . 1 − γ¯k 1 − γk e−iψk (θ)
The recurrence (13.171) now follows by noting that for general z = 0, z/¯ z = exp(2iIm log z). To show ψk is an increasing function of θ we use the fact that |Bk (z)| is less than one, equal to one and greater than 1 for |z| < 1, |z| = 1, |z| > 1, respectively, to conclude ∂|Bk (z)| >0 ∂r
(13.173)
for r := |z| = 1. Writing Bk (z) = |Bk (z)| exp iψk ((r, θ)) (and in particular ψk ((1, θ)) = ψk (θ)), we see from the radial Cauchy-Riemann equation that ”˛ ∂ “ ∂ ˛ = −i eiψk (θ) . |Bk (z)|eiψk ((r,θ)) ˛ ∂r ∂θ r=1 Minor manipulation and use of (13.173) shows ∂ψk (θ)/∂θ > 0 as required.
According to (13.172), BN (λ) = 0 when BN −1 (λ) = 1/α ¯ N −1 , with the latter a variable uniform on the unit circle. Thus, with α ¯ N −1 = e−iη , the eigenvalues can be characterized as the values of the relative Pr¨ufer phases ψN −1 (θ) = 2πk + η,
k ∈ Z≥0 .
(13.174)
We seek the bulk scaling of (13.171), in which θ → 2πx/N so that in the variable x the mean spacing is unity. It may be expected that ψk (2πx/N ) ∼ Φ(t; x) where t = k/N , and consequently ∂Φ(t; x) = 2πx + 2Im (eiΦ(t;x) − 1)A(t) , Φ(0; x) = 0 (13.175) ∂t (for this type of scaling of the difference equations for polynomials orthogonal on the unit circle the resulting differential equations are referred to as a Krein system). Here A(t) is the process obtained from the N → ∞ form of N γk (or equivalently N αk , due to the rotational invariance of the latter), as seen by expanding the logarithm to first order. Recalling Proposition 2.8.4, and noting from Definition 2.8.3 that r2 Θν = 2/(ν+1), shows that the process is consistent with A(t) =
1 (B (t) + iB2 (t)) (βt)1/2 1
(13.176)
for B1 , B2 two independent standard Brownian paths. A rigorous demonstration of (13.175), with A(t) given
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CHAPTER 13
by (13.176), as the scaling limit of (13.171) is given in [357]. The criteria (13.174) for an eigenvalue before scaling implies the criteria in relation to (13.175) that Φ(1; x) = 2πk + η,
k ∈ Z≥0 ,
where 0 ≤ η ≤ 2π is chosen uniformly at random and then fixed. At β = 1, 2 and 4, the study of the scaled correlation functions undertaken in Chapter 7 reveals that the process defined by the scaled N → ∞ eigenvalues in the circular β-ensemble is the same as that defined by the bulk scaled eigenvalues of the Gaussian β-ensemble. The characterization in terms of a stochastic differential equation allows for this coincidence to be verified for general β > 0. Moreover, study of the Gaussian β-ensemble from this perspective leads to further insights into the stochastic differential equation [524]. Such a study is based on the matrix recurrence (1.167), and the parametrization in terms of Pr¨ufer phases (1.169). The analysis is complicated by the corresponding recurrence (1.170) being nonlinear. This complication is overcome by viewing the matrix recurrence as specifying a distance preserving mapping in hyperbolic geometry. This in turn leads to a construction of the corresponding point process in terms of the Brownian carousel. D EFINITION 13.6.2 Consider the Poincar´e disk model of the hyperbolic plane (see Section 15.7.2 below). Let b = b(t) be a path in the disk, z a point on the boundary and f : R+ → R+ an integrable function. Rotate z about center b(t) at angular speed λf , and count the integer-valued total windings N (λ) of z about b(t). The hyperbolic carousel point process is the point process corresponding to N (λ). The Brownian carousel point process is the hyperbolic carousel point process with b equal to hyperbolic Brownian motion BH . Workings from [524] give that with the point written as z = eiγλ (t) , the angle γλ satisfies the differential equation dγλ |eiγλ − b|2 = λf dt 1 − |b|2
(13.177)
(the final factor can be recognized as proportional to the reciprocal of the Poisson kernel ; recall (2.39)). Consider the hyperbolic angle αλ (t) determined by z0 := eiγλ (0) , b(t) and eiγλ (t) . With the M¨obius transformation of the unit disk taking z0 to 1, and w to 0 denoted T (z0 , w; z), so that T (z0 , w; z) =
z − w 1 − wz ¯ , 1 − wz ¯ z0 − w
this angle is specified by αλ (0) = 0,
eiαλ (t) = T (eiγλ (0) , b(t); eiγλ (t) ).
(13.178)
It is shown in [524] (by applying Ito’s formula ) that in the Brownian carousel (b = BH ) the equations (13.177) and (13.178) together imply αλ (t) satisfies the stochastic sine equation dαλ = λf dt + Re((e−iαλ (t) − 1)(dB1 + i dB2 )),
αλ (0) = 0
(13.179)
(a coupled one-parameter family of stochastic differential equations — the same B1 , B2 must be used for each λ), where B1 and B2 are as in (13.176). For fixed λ the simplification Im(e−iαλ (t)/2 (dB1 + i dB2 )) = dB, where B is one-dimensional Brownian motion, is valid so it follows from (13.179) that in this circumstance dαλ = λf dt + 2 sin(αλ /2)dB.
(13.180)
The term involving Brownian motion in (13.180) vanishes when αλ = 2πk, k ∈ Z, and this together with the fact that αλ (t) has a limit for t → ∞ implies αλ (∞) is an integer multiple of 2π (in the case of (13.179) this statement holds almost surely: as a function of λ, αλ (∞) jumps between successive integer values). Suppose that in (13.179) f (t) = (β/4)e−βt/4 .
(13.181)
637
CORRELATIONS FOR GENERAL β
Changing variables t = −(4/β) log s (0 < s ≤ 1), αλ (t) = Φλ (s) shows 4 Φλ (1) = 0, (13.182) dΦλ (s) = λds + √ sin Φλ (s) Im e−iΦλ (s)/2 (dB1 (s) + i dB2 (s)) , βs √ where to change variables in B(t) we have used the realization dB(t) = G dt with G a standard normal (see, e.g., [157]). Compare now (13.182) with (13.175). Minor manipulation shows the two equations to be identical, with s ↔ t, Φλ (s) ↔ Φ(t; x), λ ↔ 2πx. However the boundary equations are different, with (13.175) prescribing Φ at t = 0, while (13.182) prescribes Φ at s = 1. Despite this difference in boundary conditions, it is indeed the case that with the choice of f (13.181) the Brownian carousel point process, measured in units of 2π, coincides with the point process for eigenvalues in the bulk of the Gaussian βensemble (mean density unity). Moreover αλ can then be constructed as a relative Pr¨ufer phase, and the distribution of αλ (∞)/2π as specified by (13.180) is equal to the distribution of the number of eigenvalues in [0, λ] of the limiting matrix ensemble. 13.6.2 Soft edge We turn our attention now to the characterization of the scaled eigenvalues at the soft edge of the Gaussian β-ensemble in terms of a stochastic operator [158], [463]. We begin by studying the limit β → ∞. P ROPOSITION 13.6.3 Let Tβ = [tij ]i,j=1,...,N be the random tridiagonal matrix specified in (1.159). Let ΔN be the N × N symmetric tridiagonal matrix with −1s above and below the diagonal, and 2s on the ˜ N be the N × N matrix with nonzero entries on the subdiagonal below the diagonal only, diagonal, and let J these being 1, 2, . . . , N − 1 in order. With i−1 1 √ tk,k+1 D = diag (N/2)−(i−1)/2 i=1,...,N β k=1
one has
√ 1 1 ˜N , N 2/3 ΔN + N −1/3 J lim √ DTβ D−1 − 2N 1N = − √ β→∞ β 2N 1/6
(13.183)
where the equality is in values of the distribution. Furthermore, the eigenvectors of (13.183) are given by Dx where x = [xj ]j=1,...,N with 1/2 2 1 xN −n = √ e−λ /2 Hn (λ). n πn!2 Here Hn (λ) denotes the Hermite polynomial of degree n, and λ is required to be such that HN (λ) = 0. Proof. Since limβ→∞ see that for large β
√1 tk,k+1 β
= limβ→∞ 2
√1 χ ˜ β (n−k)β
√ 0 6 N −1 6 6 1 1 6 lim √ Tβ = √ 6 β→∞ β 26 6 4
√
=
√1 β
N −1 √ 0 N −2
√
n − k is an equality in values of the distribution, we
√
3 N −2 0 .. .
√
N −3 .. . √ 2
..
. 0 1
7 7 7 7 7 7 7 1 5 0
and the result (13.183) now follows by direct calculation. The statement in regards to the eigenvectors can be verified from the three-term recurrence of Exercises 5.1 q.1(vii).
The significance of the r.h.s. of (13.183) is that it is a discretization, lattice spacing h = N −1/3 , of the Airy
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operator d2 + x. (13.184) dx2 With the boundary condition that the eigenfunctions vanish at x = 0 and decay to zero as x → ∞, so as to be consistent with the form of the eigenfunctions for finite N , the Airy operator has eigenvalues given by the zeros of the Airy function. This leads to the conclusion that for β → ∞, and in the soft edge scaling, the eigenvalues of the Gaussian β-ensemble are given by the zeros of the Airy function. For finite N we know from Exercises 1.4 q.8 that for β → ∞ the eigenvalues of the Gaussian β-ensemble crystallize at the zeros of the Hermite polynomial. The asymptotic formula (7.9) tells us that at the soft edge these scale to the zeros of the Airy function, in agreement with this conclusion. These findings suggest studying the l.h.s. of (13.183) for fixed β. Then one has [158] √ 1 1 ˜ N + √2 W , √ DTβ D−1 − 2N IN = − √ (13.185) N 2/3 ΔN + N −1/3 J β β 2N 1/6 −
where W is the bidiagonal random matrix ⎡ N [0, 1] ⎢ b(N −1)β N 1/6 ⎢ ⎢ W=− √ ⎢ 2 ⎢ ⎣
⎤ N [0, 1] b(N −2)β
N [0, 1] .. .
..
. bβ
⎥ ⎥ ⎥ ⎥, ⎥ ⎦
N [0, 1] √ with b(N −j)β = (2χ ˜2(N −j)β − (N − j)β)/ 2βN . We can check that b(N −j)β has mean zero and variance 1 − j/N , so that each element of W has mean zero, and to leading order for j fixed and N large has standard −1/3 deviation N 1/6 . This is consistent with a discretization, , of a Brownian motion √ lattice spacing h = N process which has mean zero and standard deviation h over an interval (x, x + h]. Recalling that (13.183) is a discretization of (13.184), we are therefore led to suspect that (13.185) is a discretization of the stochastic Airy operator −
d2 2 + x + √ B (x), 2 dx β
(13.186)
where B(x) denotes standard Brownian motion. Furthermore, this suggests the distribution of the eigenvalues, calculated when the eigenfunctions are required to vanish at x = 0 and decay to zero as x → ∞, will coincide with the distribution of the scaled eigenvalues at the soft edge of the Gaussian β-ensemble. In the work [463] these statements have been made rigorous.
13.6.3 Hard edge The eigenvalues at the hard edge of the Laguerre β-ensemble also allow for a description in terms of a ˜ β to be the random bidiagonal matrix (3.127) with n − stochastic operator [158], [462]. To begin, define B i m =: a ˜ and m → n. Let A = [(−1) δi,n+1−j ]i,j=1,...,n , and then form ⎡ ⎤ χ(˜a+1)β −χβ ⎢ ⎥ χ(˜a+2)β −χ2β ⎢ ⎥ 1 1 ⎢ ⎥ . . ˜ βA = √ ⎢ .. .. Lβ := √ AB ⎥. ⎥ β β⎢ ⎣ χ(˜a+n−1)β −χ(n−2)β ⎦ χ(˜a+n)β
639
CORRELATIONS FOR GENERAL β
In the limit β → ∞ this reduces to the fixed matrix √ ⎡√ a ˜ + 1 √− 1 √ ⎢ a ˜+2 − 2 ⎢ ⎢ .. L∞ = ⎢ . ⎢ ⎣
⎤ ..
. √ a ˜+n−1
⎥ ⎥ ⎥ ⎥, ⎥ √ −√ n − 1⎦ a ˜+n
(13.187)
which can readily be decomposed as the finite difference scheme for a differential operator [158]. P ROPOSITION 13.6.4 Let h = 1/(2n + a ˜ + 1), xi = h(˜ a + i); ∇ equal the n × n upper biadiagonal matrix with −1s down the diagonal and 1s along the next diagonal; S equal half the upper bidiagonal matrix with all entries on the diagonal and next diagonal equal to 1; and E equal the n × n upper biadiagonal matrix specified by 1 1√ a ˜ −1/2 1 1√ a ˜ −1/2 (E)i,i = x2i + h˜ a− x2i − x2i , (E)i,i+1 = − x2i − h˜ a+ x2i − x2i . h h 2 h h 2 We have 1 1 √ √ √ 2 1 1 L∞ = −2 diag( x2 , x4 , . . . , x2n ) ∇ +a ˜ diag √ , √ , . . . , √ S + E. (13.188) h 2h x2 x4 x2n A Taylor expansion shows that the entries of E are O(h), and so E can be ignored in the limit n → ∞. The other matrices can be viewed as a discretization of the operator √ d a ˜ Ja˜∞ := −2 x +√ , dx x
x ∈ [0, 1]
(13.189)
acting on square integrable functions with the properties that v(1) = 0 (this accounts for the final row of (13.187) having only one entry) and (Ja˜∞ v)(0) = 0 (a consequence of the entries in the first row of (13.187) being O(1)). Now matrices realizing the Laguerre β-ensemble specified by the p.d.f. (3.17) with N → n are given the product β1 Bβ BTβ , where Bβ is specified by (3.127) with a = n − m + 1 − 2/β = a ˜ + 1 − 2/β and m → n. To obtain the hard edge scaling, this product must be multiplied by 4n ∼ 2(2n + a ˜ + 1), giving a matrix product similar to h2 Lβ LTβ provided a = a ˜ + 1 − 2/β. Hence, the hard edge scaled eigenvalues of the β → ∞ limit of the Laguerre β-ensemble must be given by the eigenvalues of the operator Ja˜∞ (Ja˜∞ )∗ , where √ d a ˜+1 + √ (Ja˜∞ )∗ = 2 x dx x is the adjoint of Ja˜∞ and the boundary conditions noted below (13.189) √ are enforced. One can check that the eigenfunctions of this operator are the Bessel functions Ja˜+1 (σk x) where σk denotes the kth zero of Ja˜ (x), and the corresponding eigenvalues are σk2 . This is in keeping with the result of Exercises 4.6 q.1 and the asympototic formula (7.29). The study of (13.188) for fixed β [158] introduces a noise term into the discretization, and suggests that 2 h Lβ
is a discretization of the stochastic Bessel operator √ d a ˜ 2 + √ + √ B (x), Ja˜β = −2 x dx x β
(13.190)
where B(x) denotes standard Brownian motion. The corresponding eigenfunctions must vanish at x = 1, and so too must the action of (13.190) on the eigenfunctions at x = 0. Thus the hard edge scaled eigenvalues in the Laguerre β-ensemble specified by the p.d.f. (3.17) with N → n must therefore be given by the eigenvalues of the operator Ja˜β (Ja˜β )∗ , a fact proved rigorously in [462].
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E XERCISES 13.6
1. [357] Consider the coupled differential equations (Dirac-Krein system) " # !» – d p(x; E) b(x) − − a(x) =0 −E dx d q(x; E) − a(x) −b(x) dx
subject to the initial conditions p(0; E) = 0, q(0; E) = 1.
(i) By writing
»
show that
p(x; E) q(x; E)
–
» =
r(x; E) sin θ(x; E) r(x; E) cos θ(x; E)
– ,
θ(0; E) = 0
“ ” d θ(x; E) = E + Im e2iθ(x;E) (a(x) + ib(x)) . dx
(ii) Suppose a(x) + ib(x) is a random complex function, unchanged by complex rotations so that e2iθ(x;E0 ) (a(x) + ib(x)) = a(x) + ib(x). Under this circumstance, deduce from the differential equation in (i) that ” “ d Ψ(x; E) = 2E + Im (eiΨ(x;E) − 1)(a(x) + ib(x)) , dx where Ψ(x; E) = 2(θ(x; E) − θ(x; 0)), and after an appropriate choice of a(x), b(x) and identification of Ψ, x and E recognize this as the stochastic equation (13.175).
13.7 DYNAMICAL CORRELATIONS IN THE CIRCULAR β ENSEMBLE 13.7.1 Eigenfunction expansion of the density-density correlation Consider the setting of dynamical correlations relating the initial state to a state with parameter value τ . Our interest is in the truncation of the two-point function ρ(1,1) (x; 0; y; τ ) =: ρ(1,1) (x, y; τ ), denoted ρT(1,1) , obtained by subtracting the product of dynamical one-point functions ρ(1) (x; 0)ρ(1) (y; τ ). Making use of (11.92) and (11.93) we therefore have ∞ ∞ ∞ ∞ (0) (0) (0) (0) (1) (1) ρT(1,1) (x, y; τ ) = N 2 dx2 · · · dxN p0 (x, x2 , . . . , xN ) dx2 · · · dxN −∞
−∞ (1)
(1)
−∞
(0)
−∞
(0)
×GFP τ (y, x2 , . . . , xN |x, x2 , . . . , xN ) − ρ(1) (x; 0)ρ(1) (y; τ ) ∞ ∞ N (0) (0) (0) dx1 · · · dxN δ(xj − x) p0 (x(0) ) = −∞
−∞
×
∞
−∞
(1)
dx1 · · ·
×GFP x(0) ; x(1) ) τ (
∞ −∞
j=1 (1)
dxN
N
(1) δ(xj − y)
j=1
− ρ(1) (x; 0)ρ(1) (y; τ ).
(13.191)
641
CORRELATIONS FOR GENERAL β
S For GFP odinger τ we substitute (11.42) with Gτ given in terms of the eigenfunctions ψκ of the corresponding Schr¨ operator by (11.43). This gives
ρT(1,1) (x, y; τ ) βW/2 N (1) (1) N ψ δ(x − y) e δ(x − x) p ψ e−βW/2 κ κ 0 j j j=1 j=1 e−(Eκ −E0 )τ /β = ψ |ψ κ κ κ −ρ(1) (x; 0)ρ(1) (y; τ ), where
(13.192)
f |g :=
∞
−∞
dx1 · · ·
∞ −∞
dxN f (x1 , . . . , xN )g(x1 , . . . , xN ).
(13.193)
The formula (11.43) for the Green function, and consequently the formula (13.192) for ρT(1,1) , assumed we are choosing a real basis of eigenfunctions. We know from Proposition 1.1.6 that this is always possible for a real Hamiltonian operator. However, in studying the Dyson Brownian motion model on the circle the ˜ (C) (12.94) involving the corresponding Schr¨odinger operator H (C) has been transformed to the operator H 2πixj /L complex coordinates zj = e . In this situation the appropriate inner product is that specified by (12.11) (cf. (13.193)). Correspondingly (11.43) must be modified to read (0) (0) GSτ (x; x(0) ) = ψκ (z1 , . . . , zN )ψκ (z1 , . . . , zN )e−Eκ τ /β . (13.194) κ
Now we know from Proposition 12.6.1 that the eigenfunctions of H (C) , which after factorization of the ground state are analytic in the {zj }, are given by 1 −βW (C) (x1 ,...,xN )/2 e Pκ (z1 , . . . , zN ; 2/β), (13.195) C where Pκ denotes the symmetric Jack polynomial and C the normalization. We remark that in terms of the inner product (12.11) the normalization is given by ψκ (z1 , . . . , zN ) =
C 2 = e−βW
(C)
/2
Pκ |e−βW
(C)
/2
Pκ .
(13.196)
According to the result of Exercises 12.1 q.5(iii) the corresponding eigenvalue is 2π 2 e(κ; 2/β). (13.197) Eκ = E0 + L However, the functions (13.195) are not a complete set with respect to the space of Laurent expandable functions in the {zj } as required in the derivation of (13.194). To extend the set we note that, analogous to the result of Exercises 12.1 q.1, −l z1−l · · · zN Pκ (z1 , . . . , zN ; 2/β) = Pκ−lN (z1 , . . . , zN ; 2/β),
(13.198)
κN − l ≥ 0, and moreover the l.h.s. of (13.198) is an eigenfunction of (12.94) with eigenvalue e(κ − l; α) for all l ∈ Z. Hence a complete set of eigenfunctions is given by N
{e−βW
(C)
/2
Pκ (z; 2/β)}κ ∪ {e−βW
(C)
/2 −l
z Pκ (z; 2/β)} κ:κN =0, l=1,2,...
(13.199)
where the κ are partitions. Equivalently, if we replace κ by κ± , where κ± denotes an ordered set of integers (not necessarily non-negative), and use (13.198) to define Pκ± (z) in this setting, we see that {e−βW suffices for the complete set of eigenfunctions.
(C)
/2
Pκ± (z; 2/β)}κ±
(13.200)
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Let us adopt the latter viewpoint. We then have GSτ (x(0) ; x) = e−E0 τ /β
Pκ± (z (0) ; 2/β)Pκ± (z; 2/β) (C) Nκ± |α=2/β
κ±
e−e(κ± ;2/β)τ /β .
Using this formula in place of (11.43), and making use of the inner product (12.12) we see from (13.191) that ρT(1,1) (x, y; τ ) (C) N (1) (0) (1) (C) N ; 2/β)(C) Pκ± (z (0) ; 2/β)|p0 eβW j=1 δ(xj − y)|Pκ± (z j=1 δ(xj − x) = (C) Nκ± κ± =0N
=
×e−τ e(κ± ;2/β)/β N (C) N (1) (0) (C) j=1 δ(xj )|Pκ± (z (1) ; 2/β)(C) Pκ± (z (0) ; 2/β)|p0 eβW j=1 δ(xj ) (C)
Nκ±
κ± :κ± =0N
×e2πi|κ± |(y−x)/L e−τ e(κ± ;2/β)/β .
(13.201)
In obtaining the first equality use has been made of the fact that ρ(1) (y; τ ) = N/L for all τ , while the second equality follows from changing variables in the inner product and using the fact that Pκ is homogeneous of degree |κ|. (C) The normalization Nκ for a partition is given by Proposition 12.6.3. Making use of the formulas (12.105) and (12.47) we see that this can be written (2/β) |κ| ((1)N ) (C) (β/2) dκ Pκ (2/β) [β(N − 1)/2 + 1]κ
Nκ(C) = N0N
(13.202)
while Proposition 4.6.2 gives (C)
N0N = LN
Γ(βN/2 + 1) . (Γ(β/2 + 1))N
(13.203) (C)
(C)
Suppose κ± := κ − lN is such that κN − l < 0. We then note from (13.198) that Nκ−lN = Nκ and make use of (13.202). N (1) Consider next the inner product j=1 δ(xj )|Pκ± (z (1) ; 2/β)(C) in (13.201). For κ a partition the integration formula (12.144) gives l(κ) β N (C) (1) (C) |κ|(κ1 − 1)! (2/β) j=2 (− 2 (j − 1))κj (1) N δ(xj ) Pκ (z ; 2/β) = N0N ((1) ) . (13.204) Pκ (2/β) L [β(N − 1)/2 + 1]κ j=1 In the case that Pκ is replaced by Pκ± , it turns out that this inner product vanishes unless κ± has either all positive parts (which is the case (13.204)) or all negative parts (which we will see can be related to (13.204)).
P ROPOSITION 13.7.1 Suppose κ is a partition with κN = 0 and l ∈ Z>0 , and for η a composition, let η +
643
CORRELATIONS FOR GENERAL β
denote the associated partition, as in Definition 12.1.1. We have N
(C) δ(xj ) z −l Pκ (z; 2/β)
j=1
= LN −1 N ×
N
1/2
−1/2
dx2 e−2πilx2 · · ·
j=2
=
|1 − e2πixj |β
1/2 −1/2
dxN e−2πilxN Pκ (1, e2πix2 , . . . , e2πixN )
|e2πixk − e2πixj |β
2≤j
N j=1 δ(xj )|P(l−κ)+ (1/z; 2/β)(C) , κ1 ≤ l, 0 otherwise.
Proof. The symmetric analogue of the formula of Exercises 12.1 q.2 reads z −l Pκ (z; 2/β) = Pκ−lN (z; 2/β) = P(lN −κ)+ (1/z; 2/β), which immediately implies the stated result for κ1 ≤ l. To show that the inner product vanishes for κ1 > l, we proceed as in the proof of Proposition 12.3.2, this time setting a = −b = −l + in (12.142). According to Proposition 12.6.10, (4.4) and (12.46) we then have N DX
˛ E(C) ˛ δ(xj )˛z −l Pκ (z; 2/β) = LN−1 N (|κ| − N l)Pκ ((1)N ; 2/β)
j=1
× lim
→0
N 1Y Γ(1 + β(j − 1)/2)Γ(βj/2 + 1) . j=1 Γ(1 − l + + β(N − j)/2 + κj )Γ(1 + l − + β(N − j)/2 − κj )Γ(1 + λ)
The product contains the terms
1 . Γ(1 − l + + κN )Γ(1 + l − − κ1 )
With κN = 0 and l ∈ Z>0 , 1/Γ(1−l++κN ) is proportional to for small , and similarly for κ1 > l, 1/Γ(1+l−−κ1 ) is also proportional to for small , thus implying the result.
The result of Proposition 13.7.1 implies the sum in (13.201) can be reduced to a sum over partitions only, provided twice the real part is taken. Thus ρT(1,1) (x, y; τ ) = 2Re
κ κ=0N
N 1 (C)
Nκ
(C) (1) δ(xj ) Pκ (z (1) ; 2/β)
j=1
N (C) (C) (0) × Pκ (z (0) ; 2/β) p0 eβW δ(xj ) e2πi|κ|(y−x)/L e−τ e(κ;2/β)/β . (13.205) j=1
It remains to evaluate the inner product Pκ (z (0) ; 2/β)|p0 eβW p0 = e−βW
(C)
(C)
N j=1
(0)
δ(xj )(C) in (13.205). In the case
(C)
/N0N (initial state equals equilibrium state),
N (C) (C) (0) δ(xj ) = Pκ (z (0) ; 2/β) p0 eβW j=1
N 1 (C) N0N
j=1
(C) (0) δ(xj ) Pκ (1/z (0); 2/β)
(13.206)
644
CHAPTER 13
and thus is evaluated according to (13.204). In the case p0 = 1/LN (initial state a perfect gas) 1/2 N (C) N 1/2 βW (C) (0) (0) (0) (0) Pκ (z ; 2/β) p0 e δ(xj ) = dx2 · · · dxN Pκ (1/z (0) ; 2/β) L −1/2 −1/2 j=1 N (0) Pκ (1/z1 , 0, . . . , 0) L (0) −κ1 N , κ = (κ1 , 0, . . . , 0), L (z1 ) = 0 otherwise. =
(13.207)
Consideration of these facts gives the following formulas for ρT(1,1) in the finite system [212]. P ROPOSITION 13.7.2 For the perfect gas initial condition ρT(1,1) (x, y; τ ) =
∞ 2 2 2 2N Re e2πiκ1 (y−x)/Le−(2π) τ (κ1 +β(N −1)κ1 /2)/βL , 2 L κ =1
(13.208)
1
while for the case that the initial state is equal to the equilibrium state ρT(1,1) (x, y; τ ) =
N (Γ(1 + β(j − 1)/2))4 2 2/β 2 2 ¯2/β Re |κ| (κ − 1)! (κ)f (κ) f 1 N N L2 (Γ(1 + β(N − j)/2 + κj ))2 j=1 κ=0
×
1 exp 2πi(y − x)|κ|/L − e(κ; 2/β)τ /β (.13.209) (Γ(1 + β(j − 1)/2 − κj ))2 j=2 N
Proof. In deriving (13.208) we use the result (13.207) in (13.205), together with the facts that for (κ1 , 0, . . . , 0),
(β/2)|κ| dκ = 1, which allows the ratio of (13.204) and (13.202) to be simplified. In deriving (13.209), we substitute (13.204), (13.206) and (13.202) in (13.205), and use the results „ l(κ) Y“ j=2
−
” β (j − 1) 2 κj
«2 =
Pκ(2/β) ((1)N ) =
N “Y j=2
Γ(1 + β(j − 1)/2) ”2 , Γ(1 + β(j − 1)/2 − κj )
˛ bκ ˛ β/2 = fN (κ), ˛ hκ α→2/β
(13.210)
where fN (κ) is specified by (12.59), as well as the formula (12.60) for dκ . β/2
13.7.2 Bulk limit The formula (13.208) for ρT(1,1) in the case of perfect gas initial conditions is in the form of a Riemann sum approximation to an integral, and the following result is immediate. P ROPOSITION 13.7.3 In the bulk limit N, L → ∞, N/L = ρ (constant), ∞ 2 2 bulk (x, y; τ ) = 2ρ2 e−(2πρ) (u +βu/2)τ /β cos(2πρ(x − y)u) du. ρT(1,1)
(13.211)
0
In contrast, the formula (13.209) for ρT(1,1) in the case that the initial state is equal to the equilibrium state requires further analysis before its limiting value can be computed. One essential difficulty is that in (13.209) the sum is over all partitions κ, and thus N independent quantities, whereas in (13.208) only the largest part κ1 enters into the summation. This feature would appear to make the problem of computing the thermodynamic limit intractable. However, if we restrict attention to the case β rational a significant simplification takes place.
645
CORRELATIONS FOR GENERAL β
P ROPOSITION 13.7.4 Let β be rational and write β/2 = p/q where p and q are relatively prime. Then the summand in (13.209) is nonzero if and only if κ is of the form κ = (α1 , . . . , αq , p, . . . , p, . . . , 1, . . . , 1, 0, . . . , 0), 7 89 : 7 89 : where α1 ≥ α2 ≥ · · · ≥ αq and q +
β1 p s
p j=1
(13.212)
βp 1 s
βj ≤ N .
QN
Γ(1+β(j−1)/2−κj ))2 in (13.209) is non-zero if and only if β(j−1)/2−κj ∈ {−1, −2, . . . } for each j = 1, 2, . . . , N . With β/2 = p/q and j = q + 1 this means p − κq+1 ∈ {−1, −2, . . . } and thus κq+1 ≤ p as required.
Proof. The factor (
j=2
As a consequence of (13.7.4), for β rational the summation in (13.209) can be taken over the p + q coordinates αi (i = 1, . . . , q) and βj (j = 1, . . . , p). In fact it is more convenient to replace the coordinates βj by γj (j = 1, . . . , p) defined so that κq+γa +k = p − a, k = 1, . . . , γa+1 − γa (γ0 := 0, γp+1 := N − q). β/2
P ROPOSITION 13.7.5 For β/2 = p/q and κ as specified above and large N, {αj }, {γj } we have fN (κ) = ABC/D, where A := (j − i)β/2 + αi − αj ∼ (αi − αj )β/2 ; β/2
1≤i<j≤q
B :=
q N
(j − i)β/2 + αi − κj
i=1 j=q+1
1≤i<j≤q
∼ b1 b2
β/2
p q
1 γ β/2 + αi a i=1 a=1
with b1 =
q
1 ; Γ((q − i + 1)β/2 + αi − p) i=1
b2 = q
Γ((N − i + 1)β/2 + αi ),
i=1
and C :=
(j −i)β/2+κi −κj
q+1≤i<j≤N
β/2
p p−1 −N +q Pp ∼ Γ(β/2) (β/2)− j=1 γj c1 c2 (γa −γa )2/β a=1 a =a+1
with c1 :=
p
q+γa+1
Γ((N + 1 − i)β/2 + p − a),
c2 :=
a =1
a=0 i=q+γa +1
D :=
p
(j − i)β/2
1≤i<j≤N
β/2
=
1 ; Γ(γ + 1 + 2(a − 1)/β) a
N 1 N Γ(jβ/2). Γ(β/2) j=1
Proof. The decomposition fNβ/2 (κ) = ABC/D is immediate from the definitions (12.59) and (13.212). For the asymptotic form of A we have ” Y “ (j − i)β/2 + αi − αj 1≤i<j≤q
β/2
Y
Γ((j − i)β/2 + αi − αj + β/2) Γ((j − i)β/2 + αi − αj ) 1≤i<j≤q “ ”β/2 Y Y ∼ (j − i)β/2 + αi − αj ∼
=
1≤i<j≤q
where the first asymptotic expression follows upon using (4.126).
1≤i<j≤q
(αi − αj )β/2 ,
646
CHAPTER 13
For the asymptotic form of B we write N Y j=q+1
=
p Y
Y
γa+1 +q
a=0 j=1+q+γa
and note p Y
Y
γa+1 +q
a=0 j=1+q+γa
Γ((j − i)β/2 + αi − (p − a) + β/2) Γ((j − i)β/2 + αi − (p − a))
=
p Y Γ((γa+1 + q − i)β/2 + αi − (p − a) + β/2) Γ((γa + 1 + q − i)β/2 + αi − (p − a)) a=0
=
p Γ((N − i)β/2 + αi + β/2) Y Γ((γa + q − i)β/2 + αi − (p − a) − 1 + β/2) Γ((1 + q − i)β/2 + αi − p) a=1 Γ((γa + 1 + q − i)β/2 + αi − (p − a))
=
p Γ((N − i)β/2 + αi + β/2) Y 1 , Γ((1 + q − i)β/2 + αi − p) a=1 (γa + q − i)β/2 + αi − (p − a) − 1 − β/2
where in obtaining the second equality the facts γ0 := 0, γp+1 := N − q have been used. The result now follows since γa , αi 1 while the remaining quantities in the product are fixed. For the asymptotic form of C note Y
=
“ p−1 Y
Y
p Y
q+γa+1
q+γa +1
Y
p min(q+γa+1 ,N−1) q+γa+1 ” ”“ Y Y Y .
a=0 i=q+γa +1 a =a+1 j=q+γa +a
q+1≤i<j≤N
a=0
i=q+γa +1
j=i+1
Proceeding as in the simplification of B we see that for the range of indices implied by the first grouping of products Y
p−1
Y
p Y
q+γa+1
Y
q+γa +1
“
(j − i)β/2 + κi − κj
”
a=0 i=q+γa +1 a =a+1 j=q+γa +1
Y
p−1
=
Y
q+γa+1
a=0 i=q+γa +1
Γ((N + 1 − i)β/2 + p − a) Γ((q + γa+1 + 1 − i)β/2)
p Y a =a+1
β/2
1 . (q + γa + 1 − i)β/2 + a − a − 1
Now Y
q+γa+1
i=q+γa +1
(q + γa
“ 2 ”γa+1 −γa Γ(γ − γ 1 a+1 + 1 + 2(a − a − 1)/β) a = , + 1 − i)β/2 + a − a − 1 β Γ(γa − γa + 1 + 2(a − a − 1)/β)
while Y
p−1
p “ 2 ”γa+1 −γa Γ(γ − γ Y a+1 + 1 + 2(a − a − 1)/β) a β Γ(γa − γa + 1 + 2(a − a − 1)/β)
a=0 a =a+1
=
“ p−1 Y “ 2 ”(γa+1 −γa )(p−a) ” a=0
Y
p−1
×
β p Y
a=1 a =a+1
∼
p ” Γ(γa − γa + 1 + 2(a − a)/β) “ Y 1 Γ(γa − γa + 1 + 2(a − a − 1)/β) Γ(γa + 1 + 2(a − 1)/β)
p “ 2 ”N−q “ Y
β
a =1
a =1
1 Γ(γa + 1 + 2(a − 1)/β)
p ” p−1 Y Y
(γa − γa )2/β .
a=0 a =a+1
647
CORRELATIONS FOR GENERAL β
Finally, for the range of indices implied by the second grouping of products p min(q+γa+1 ,N−1) q+γa+1 Y Y Y Γ((j − i + 1)β/2 + κi − κj ) Γ((j − i)β/2 + κi − κj ) a=0 i=q+γ +1 j=i+1 a
Y p
=
Y
a=0
“ =
Y
min(q+γa+1 ,N−1) q+γa+1
i=q+γa +1
j=i+1
Γ((j − i + 1)β/2) Γ((j − i)β/2)
min(q+γa+1 ,N−1) Y 1 ”N−1−q Y Γ((q + γa+1 − i + 1)β/2) Γ(β/2) a=0 i=q+γ +1 p
a
and Y
Y
p−1 q+γa+1
Γ((N + 1 − i)β/2 + p − a) “ Y Γ((q + γa+1 + 1 − i)β/2) a=0 p
a=0 i=q+γa +1
Y
min(q+γa+1 ,N−1)
Γ((q + γa+1 − i + 1)β/2)
”
i=q+γa +1
Y Y 1 Γ((N + 1 − i)β/2 + p − a). Γ(β/2) a=0 i=q+γ +1 q+γa+1
p
=
a
A similar calculation gives the following result. P ROPOSITION 13.7.6 For β/2 = p/q and κ as specified by (13.212), and large N , {αj }, {γj } we have β/2 f¯N (κ) = A B C /D where (j − i)β/2 + αi − αj + 1 − β/2 A := ∼ (αi − αj )β/2 ; β/2
1≤i<j≤q
B :=
q N (j − i)β/2 + αi − κj + 1 − β/2
β/2
i=1 j=q+1
1≤i<j≤q
∼ b1 b2
q p
1 γ β/2 + αi a i=1 a=1
with b1 =
q
Γ((N − i + 1)β/2 + αi + 1 − β/2),
1 ; Γ((q − i + 1)β/2 + αi − p + 1 − β/2) i=1
b2 = q
i=1
C :=
(j − i)β/2 + κi − κj + 1 − β/2
q+1≤i<j≤N
β/2
p−1 p p Pp ∼ Γ(2/β) (β/2)− j=1 γj c1 c2 (γa − γa )2/β a=1 a =a+1
with c1 :=
p
q+γa+1
Γ((N + 1 − i)β/2 + p − a + 1 − β/2),
c2 :=
p a =1
a=0 i=q+γa +1
1 ; Γ(γa + 2a /β)
and D :=
i≤i<j≤N
(j − i)β/2 + 1 − β/2
β/2
=
N j=1
Γ(β(j − 1)/2 + 1).
648
CHAPTER 13
Next we combine terms and products involving N from Propositions 13.7.5 and 13.7.6 with the products in (13.209). P ROPOSITION 13.7.7 With the notation introduced in Propositions 13.7.5 and 13.7.6 we have b1 b1 c1 c1 Γ2 (κ1 )
N −2 Γ(1 + (N − j)β/2 + κj j=1
q
∼ N q(β/2−1)
β/2 +
j=1 N
b2 b2 c2 c2
p Γ((N − q)β/2 + p) αj β/2−1 (β/2)−N +1+q−p (N − γa )2/β−1 , N Γ(N − q + 2p/β) a=1
N 2 −2 Γ(1 + (j − 1)β/2 − κj ) Γ(1 + (j − 1)β/2)
j=2
∼
q
j=2
1 Γ(−(j − 1)β/2)
q 2
β/2−1
αj
j=1 p
p −2 Pp Γ(2a/β) (β/2)2 j=1 γj γa2/β−1
a=1
j=1
a=1
and N Γ(N ) 1 . (Γ(1 + β(j − 1)/2)2 = (Γ(β/2))N (β/2)N −1 DD j=1 Γ(βN/2)
Proof. To derive the first formula we begin by noting b1 b1
q “ Y
Γ(1 + (N − j)β/2 + κj )
j=1
”−2
=
q “ q Y Y |Γ((N − j + 1)β/2 + αj ) αj ”β/2−1 ∼ N q(β/2−1) , β/2 + Γ(1 + (Nj )β/2 + αj ) N j=1 j=1
where in the final line use has been made of (4.126). Also, using the fact that N “ Y
Γ(1 + (N − j)β/2 + κj )
”−2
=
j=q+1
p Y
Y
q+γa+1
“
Γ(1 + (N − j)β/2 + p − a)
”−2
a=0 i=q+γa +1
we have N ”−2 “ Y Γ(1 + (N − j)β/2 + κj )
c1 c1
j=q+1
=
p Y
Y
q+γa+1
a=0 i=q+γa
= lim
→0
p Y
Γ((N + 1 − i)β/2 + p − a) Γ(1 + (N − i)β/2 + p − a) +1 Y
q+γa+1
a=0 i=q+γa +1
Γ((N + 1 − i)β/2 + p − a) ((N − i)β/2 + p − a + )Γ(1 + (N − i)β/2 + p − a + )
p Y Γ((N − q − γa ) + 2(p − a)/β) Γ((N − q − γa+1 ) + 2(p − a + )/β) →0 Γ((N − q − γ Γ((N − q − γa+1 ) + 2(p − a + )/β) a ) + 2(p − a)/β) a=0
= (β/2)−(N−q) lim ∼ (β/2)−N+1+q−p
p Γ((N − q)β/2 + p) Y (N − γa )2/β−1 , Γ((N − q) + 2p/β) a=1
where in the second equality the quantity is introduced because (N − i)β/2 + p − a = 0 for a = p, i = q + γp+1 := N . These results imply the first stated result.
649
CORRELATIONS FOR GENERAL β
To derive the second result we begin by noting q “ q “ q ”−2 Y ”2 Y sin π((j − 1)β/2) ”2 Y “ = Γ(1 + (j − 1)β/2 − κj ) Γ(κj − (j − 1)β/2) . π j=2 j=2 j=2
Hence b2 b2 Γ2 (κ1 )
q “ q “ ”2 ”−2 Y Y Γ(1 + (j − 1)β/2 − κj ) Γ(1 + (j − 1)β/2) j=2
j=1
q “ q q ”2 Y sin π((j − 1)β/2) ”2 Y Γ(αj − (j − 1)β/2) Y “ Γ(1 + (j − 1)β/2) = π Γ(αj + 1 − jβ/2) j=1 j=2 j=1
∼
q “ q Y sin π((j − 1)β/2) ”2 Y
π
j=2
β/2−1
αj
j=1
q “ Y
Γ(1 + (j − 1)β/2)
j=1
”2
=
q Y j=1
β/2−1
αj
q “ Y j=1
”2 1 . Γ(−(j − 1)β/2)
Next we note N Y j=q+1
=
Γ(1 + (j − 1)β/2) Γ(1 + (j − 1)β/2 − κj ) p Y
Y
q+γa+1
a=0 j=q+γa +1
p p Pp Y Y Γ(1 + (j − 1)β/2) 1 Γ(γa + 2a/β), = (β/2) j=1 γj Γ(1 + (j − 1)β/2 − (p − a)) Γ(2i/β) a=1 i=1
which shows N “ Y
c2 c2
j=q+1
p p ”2 Y “Y Pp Γ(1 + (j − 1)β/2) ”2 1 ∼ (β/2)2 j=1 γj γ 2/β−1 . Γ(1 + (j − 1)β/2 − κj ) Γ(2i/β) a=1 a i=1
It remains to specify eκ and |κ| in terms of the coordinates {αj }, {γa }. P ROPOSITION 13.7.8 For the partitions (13.212) we have q p 2π 2 (α2j + (β/2)N αj ) + (β/2)(N γa − γa2 ) βeκ ∼ L a=1 j=1 p q and |κ| = j=1 αj + a=1 γa . Proof. From the definition of {αj } and {γa } we have |κ| =
q X j=1
αj +
p X
(p − a)(γa+1 − γa )
a=0
with γ0 := 0. Simplification of the last series gives the stated result for |κ|. For βeκ , from (12.34) we have βeκ =
q “ “ 2π ”2 X
L +
α2j + (β/2)αj (N + 1 − 2q)
”
j=1
p “ “ 2π ”2 X
L
a=0
” (p − a)2 (γa+1 − γa ) + (β/2)(p − a)(γa+1 − γa )(N − γa+1 − γa − 2q) .
650
CHAPTER 13
We see that the leading order terms in the summand are proportional to N 2 . Ignoring terms of a lesser order gives βeκ ∼
q “ 2π ”2 X
L
(α2j + (β/2)αj N ) +
p “ 2π ”2 X
L
j=1
(β/2)(p − a)(γa+1 − γa )(N − γa+1 − γa ),
a=0
which we identify with the stated result.
Substituting the results of Propositions 13.7.5 to 13.7.8 in (13.209) shows that for large L and N , ρT(1,1) (x, y; τ ) is the Riemann approximation to a multidimensional integral in the variables xj := αj /N (xj ≥ 0) and yj := γj /N (1 ≥ yj ≥ 0) [283]. P ROPOSITION 13.7.9 For β/2 := λ = p/q (p and q relatively prime) we have bulk (x, y; τ ) ρT(1,1)
= Cp,q (λ)
q
i=1
∞
dxi
0
p j=1
1
0
dyj Q2p,q F (p, q, λ|{xi , yj }) cos(Qp,q (x − y)) exp(−Ep,q τ /2λ), (13.213)
where the momentum Q and the energy E variables are given by q p q p xi + yj , Ep,q := (2πρ)2 P (xi ) + H (yj ) Qp,q := 2πρ i=1
j=1
i=1
j=1
with P (x) = x(x + λ) and H (y) = λy(1 − y), the so called form factor F is given by 2λ 2/λ i
(13.214)
and the normalization is given by Cp,q (λ) =
Γq (λ)Γp (1/λ) λ2p(q−1) Γ2 (p) p . q 2 2 2π 2 p!q! i=1 Γ (p − λ(i − 1)) j=1 Γ (1 − (j − 1)/λ)
(13.215)
Proof. Substituting the results of Propositions 13.7.5 – 13.7.8 in (13.209) and minor manipulation gives bulk ρT(1,1) (x, y; τ ) q p ”2 “ Y ”2 1 1 1 “Y (Γ(2/β))p ((Γ(β/2))q ∼ 2π 2 j=2 Γ(−(j − 1)β/2) Γ(2a/β) a=1
×N −(p+q) ×
Y 1≤i<j≤q
X∗
Q2p,q
κ
(uj − ui )β
p Y
(1 − va )2/β−1 (βva /2)2/β−1
a=1
Y
q Y
β/2−1
ui
(β/2 + ui )β/2−1
i=1
(va − va )4/β cos((x − y)Qp,q )e−Ep,q τ /β ,
(13.216)
1≤a
where va := γa /N , ui := αi /N , while Qp,q and Ep,q are as in (13.213) and the asterisk denotes the summation is over the partitions (13.212). Because the summand depends only on the scaled variables {va }, {ui } and further is symmetrical in these sets of variables, we have that for large N Z ∞ Z 1 Z ∞ Z 1 X∗ 1 ∼ du1 · · · duq dv1 · · · dvp . (13.217) N −(p+q) p!q! 0 0 0 0 κ The remaining task is to show the constant prefactor in (13.216), multiplied by 1/p!q! from (13.217), can be written in
651
CORRELATIONS FOR GENERAL β
the form (13.215). Now q Y
Γ(p − β(i − 1)/2)
i=2
=
q Y
q “ ”“ ” “ ”Y (β/2)p 2(p − 1)/β − (i − 1) 2(p − 2)/β − (i − 1) · · · − (i − 1) Γ(−β(i − 1)/2)
i=2
i=2
p q Y Γ(2(j − 1)/β − 1) Y = (β/2)p(q−1) (−1)q−1 (q − 1)! Γ(−β(i − 1)/2), Γ(2(j − 1)/β − q) i=2 j=2
while
p p p−1 Y Y Y Γ(2(j − 1)/β − 1) 1 Γ(2(j − 1)/β) = . Γ(2(j − 1)/β − q) Γ(1 − 2j/β) j=2 j=2 j=1
Qp (q − 1)! 1 j=2 Γ(2(j − 1)/β) p(q−1) Qq , = (β/2) Qp−1 Γ(−β(j − 1)/2) Γ(p − β(i − 1)/2) j=2 j=1 Γ(1 − 2j/β)
Thus
which when substituted in the prefactor of (13.216) gives the form (13.215).
13.7.3 Asymptotic expansions bulk (x, y; τ ) in the case of perfect gas initial conditions. Rewriting Consider first the formula (13.211) for ρT(1,1) the cosine term as a complex exponential and linearizing the exponent about s = 0 shows that for large x and τ 1 bulk ρT(1,1) . (x, 0; τ ) ∼ 2ρ2 Re τ (2πρ)2 /2 − 2πiρx bulk Consider next the formula (13.213) for ρT(1,1) (x, y; τ ) in the case that the initial state equals the final state. bulk Here, analogous to the procedure of Section 13.2.3, the large x and τ asymptotic expansion of ρT(1,1) (x, y; τ ) is obtained by expanding the integrand in (13.213) about xi = 0 for all i = 1, . . . , q and either yj = 0 or yj = 1. Choosing to expand about yj = 0 for j = 1, . . . , p − m and yj = 1 for j = p − m + 1, . . . , m (this
is one of
p m
equivalent choices of partitioning {yj } into p − m variables to be expanded about 1 and m
variables to be expanded about 0), and writing wj = 1 − yj−(p−m) (j = p − m + 1, . . . , p) gives Qp,q = 2πρm + 2πρ
q
xi +
i=1
Ep,q ∼ (2πρ)2
q
λxi +
i=1
p−m
yj −
j=1 p−m
m
wj ,
j=1
λyj +
m
j=1
λwj ,
j=1
q p−m m 2λ 2/λ 2/λ p i
i
j=1
j
k=1
k
Extending the integration interval in the variables yj (j = 1, . . . , p − m) and wj (j = 1, . . . , m) to (0, ∞), and changing variables 2πρ(−ix + πρτ )xi → xi ,
2πρ(−ix + πρτ )yj → yj ,
2πρ(ix + πρτ )wj → wj ,
652
CHAPTER 13
shows that to leading order in terms proportional to cos 2πmρx (m = 0, . . . , p), bulk (x, 0; τ ) ρT(1,1)
1 1 1 A(q, p, λ) + ∼ Cp,q (λ)(2πρ) 2 (−i(2πρ)x + (2πρ)2 τ /2)2 (i(2πρ)x + (2πρ)2 τ /2)2 p m2 /λ p −2mq 2 1 + λ m cos 2πmρx B(q, p − m, λ)C(m, λ) , |i(2πρ)x + (2πρ)2 τ /2|2 m m=1 2
(13.218) where A(q, p, λ) ∞ λ−1 −x1 dx1 x1 e ··· = 0
×
0
∞
0
dyp yp1/λ−1 e−yp
xi +
p
i=1
0
dxq xλ−1 e−xq q
q
B(q, p − m, λ) ∞ λ−1 −x1 dx1 x1 e ··· = ×
∞
∞
0
∞
0
dyp yp−m e−yp−m 1/λ−1
C(m, λ) = 0
∞
0
∞
1/λ−1 −y1
··· p 2λ 2/λ i
e
2 q yj
j=1
dxq xλ−1 e−xq q
∞
1/λ−1 −y1
dy1 y1
e
···
0 p−m 2λ 2/λ |x − x | i i i
q
1/λ−1 −w1 dw1 w1 e
∞
··· 0
1/λ−1 −wm dwm wm e
m
|wj − wj |2/λ .
(13.220) (13.221)
j<j
All the above integrals can be evaluated in terms of the following limiting form of the Dotsenko-Fateev integral from Exercises 4.5 q.1 [237]. P ROPOSITION 13.7.10 Let n m ∞ 2ρ 2/ρ 1 ∞ j<j |ηj − ηj | −β/ρ −zri i
m n−1 m−1 n Γ(l/ρ) Γ(jρ − n) Γ(1 − β/ρ + l/ρ) Γ(1 − n + β + jρ), Γ(1/ρ) j=1 Γ(ρ) j=0 l=1
l=0
(13.223) where cnm (β, ρ) = m + n + mβ − nβ/ρ + m(m − 1)ρ + n(n − 1)/ρ − 2mn. Proof. Consider J(0,n)(m,0) (α, β, ρ) as defined in Exercises 4.5 q.1. Setting ti = exp(si /α) and τj = exp(−ξj /α) after eliminating the parameters α , β , ρ in favor of α, β, ρ, and then considering the limit α → ∞ shows fnm ({esi /ρ }, {e−ξj /ρ }; α, β, ρ) ∼
α→∞
n Y
(si /α)−β/ρ e−si /ρ
i=1
Q
×
i
m Y j=1
Q (si − si )2/ρ j<j α−2ρ (ξj − ξj )2ρ Qn Qm . −2mn (s + ξ )2 i j i=1 j=1 α
−2/ρ
α
(ξj /α)β e−ξj
653
CORRELATIONS FOR GENERAL β
Noting also that
Z
∞
α→∞
1
Z
1 α
dti ∼
Z
∞
1
dsi , 0
dτj ∼
α→∞
0
1 α
Z
∞
dξj , 0
we see that J(0,n)(m,0) (α, β, ρ) ∼ α−cnm (β,ρ) α→∞
i=1
Q
∞ 0
i=1
−β/ρ −si /ρ
dsi si
|si − si |2/ρ Qn Qm
i
×
n Z Y
Q
e
∞ Y
dξj ξ β e−ξj
j=1
|ξj − ξj |2ρ . 2 j=1 (si + ξj ) j<j
(13.224)
On the other hand, using the relation (4.126), we see from the evaluation formula in Exercises 4.5 q.1 that J(0,n)(m,0) (α, β, ρ) ∼ α−cnm (β,ρ) ρ(1−β/ρ+(n−1)/ρ)n α→∞
×
n−1 Y
Γ(1 − β/ρ + l/ρ)
m−1 Y
m n Y Γ(l/ρ) Y Γ(jρ − n) Γ(1/ρ) j=1 Γ(ρ) l=1
Γ(1 − n + β + jρ).
(13.225)
j=0
l=0
After equating (13.224) with (13.225), and introducing the common scaling factor z via the change of integration variables si /ρ = zri , ξj = zηj , the integration formula (13.223) results.
Comparison of (13.219), (13.220) and (13.221) with (13.222) shows ∂2 I (z; (p + )/q − 1, (p + )/q) p,q →0 ∂z 2 z=1 p q−1 2 Γ (l/λ) q!p! = q Γ2 (jλ − p), λΓ (λ) Γ(1/λ) j=1
A(q, p, λ) = q!p! lim
l=1
B(q, p − m, λ) = q!(p − m)!Ip−m,q (1; λ − 1; λ) = q!(p − m)!
p−m l=1
q Γ2 (l/λ) Γ2 (jλ − (p − m)) , Γ(1/λ) j=1 Γ(λ)
C(m, λ) = m!I0,m (1; 1/λ − 1, 1/λ), where in the formula for A(q, p, λ) the auxiliary variable is introduced because Ip,q (z; p/q − 1, p/q) before the double differentiation is singular. Substituting the above results in (13.218), and manipulating (13.215) to write it in the form Γq (λ)Γp (1/λ) 1 Cp,q (λ) = 2 2π 2 p!q! pj=1 Γ2 (j/λ) q−1 i=1 Γ (iλ − p) shows that
1 ρ2 1 + λ (2πρx + i(2πρ)2 τ /2)2 (2πρx − i(2πρ)2 τ /2)2 p cos 2πmρx am , + |2πρx + i(2πρ)2 τ /2|4m2 /β m=1
bulk ρT(1,1) (x, 0; τ ) ∼ −
m 2 −2mq
am = 2m λ
p
j=1
Γ2 (j/λ)
l=p−m+1
Γ2 (l/λ)
q
2 j=1 Γ (jλ − q−1 2 i=1 Γ (iλ
p + m) − p)
(13.226)
,
where as in Proposition 13.7.9 λ = β/2 = p/q. It is straightforward to check that in the special case
654
CHAPTER 13
β/2 ∈ Z>0 , so that for q = 1 this expansion agrees with the asymptotic expansion of Proposition 13.2.4. We turn our attention now to asymptotics associated with the dynamical structure function, defined by ∞ bulk ˆ S(k; τ ) := ρT(1,1) (x, 0; τ )eikx dx. (13.227) −∞
For the case that the initial state equals the equilibrium state, the non-oscillatory term of the asymptotic behavior (13.226) implies that for k small ˆ τ ) ∼ |k| e−πρ|k|τ . S(k; πβ
(13.228)
This can be derived directly from the integral formula (13.213) substituted in (13.227) (see Exercises 13.7 q.1). An analogous calculation [206] also allows the next leading term of the small k expansion for τ = 0 to be computed, giving the result 1 β − 2 ˆ 0) ∼ |k| + 1 S(k; k2 . (13.229) πβ 2πρ πβ β In (14.15) below the same expansion will be deduced from physical principles. 13.7.4 Analytic properties of the static structure function The formula (13.213) in the case τ = 0 allows the structure function for the static log-gas to be computed. Thus, in the notation of that formula, ∞ bulk ˆ ˆ β) = ρT(1,1) (x, 0; 0)eikx dx S(k) =: S(k; −∞
= πCp,q (λ)
q i=1
∞
dxi
0
p j=1
0
1
dyj Q2p,q F (q, p, λ|{xi , yj }) δ(k − Qp,q ).(13.230)
ˆ β) to an analytic function [207]. For values of |k| small enough, (13.230) can be used to relate S(k; P ROPOSITION 13.7.11 Set ρ = 1 and suppose |k| < 2π. Then ˆ β) = |k| f (|k|; β), S(k; πβ where f (k; β) is analytic in the interval |k| < min(2π, πβ), and is given explicitly by p ∞ q ∞ f (k; β) = 2π 2 λCp,q (λ) dxi dyj Q2p,q Fˆ (q, p, λ|{xi , yj }; k) δ(1 − Qp,q |ρ=1 ), i=1
0
j=1
(13.231)
(13.232)
0
with λ = β/2 and 1 p 1−λ 1−1/λ (x (1 + kx /λ)) i i i=1 j=1 (yj (1 − kyj )) 2λ 2/λ i
Fˆ (q, p, λ|{xi , yj }; k) = q
(13.233)
Proof. Note that in (13.230) the integration variables are all positive, and because of the delta function are restricted to the hyperplane q p X X |k| xi + yj = . 2π i=1 j=1
655
CORRELATIONS FOR GENERAL β
We see immediately from these constraints that the restriction yj < 1 in the domain of integration is redundant for |k| < 2π. Assuming this condition the integration over yj can be extended to the region (0, ∞). The change of variables xi → |k|xi and yj → |k|yj then shows that for |k| < 2π the formula (13.231) is valid. The analytic properties of f (k; β) are evident by inspection.
An immediate consequence of (13.232) is a functional equation satisfied by f (k; β). ˆ β) by (13.231), satisfies the functional equaP ROPOSITION 13.7.12 The function f (k; β), related to S(k; tion (7.139). Proof. The integral in (13.232) is unchanged by the mapping λ → 1/λ (and thus p ↔ q) followed by k → −k/λ. The precise functional equation (7.139) then follows provided we can show that Cp,q (λ) = λ2pq−2 Cq,p (1/λ), which indeed readily follows from the definition of Cp,q (λ) in (13.215).
As noted in (7.152), the functional equation (7.139) can be illustrated using the evaluations (7.136) and ˆ β) for β = 1 and 4, respectively in terms of elementary functions. Furthermore, in relation (7.95) of S(k; ˆ β), in the notation of (7.141) the functional equation (7.139) implies the to the small |k| expansion of S(k; functional relation (7.141). The simplest structure of the coefficients Aj (x) allowing (7.141) is Aj (x) = pj (x), where pj (x) is a polynomial of degree j. The functional relation (7.141) can be stated as requiring pj (x) =
j
aj,l xl ,
aj,l = aj,j−l
(j even),
(13.234)
l=0
pj (x) = (x − 1)
j−1
a ˜j,l xl ,
a ˜j,l = a ˜j,j−1−l
(j odd).
(13.235)
l=0
In fact the structure of (13.234) can be refined by noting from the exact result (7.4) that f (k; 2) = 1.
(13.236)
This implies pj (x) = (x − 1)2
j−2
bj,l xl ,
bj,l = bj,j−2−l
(j even).
(13.237)
l=0
Assuming the validity of the assumption that Aj (x) is a polynomial of degree j, the exact results giving S.β (k) as an elementary function of k for various β allow the polynomials pj (x) for j up to nine to be completely specified [207], thus extending the expansion (13.229) up to order k 10 . E XERCISES 13.7
1. [206] The aim of this exercise is to derive (13.228).
ˆ τ ) provided a factor exp(−E ˆp,q,k τ /2λ) is (i) Suppose ρ = 1. Show that the formula (13.231) holds for S(k; included in the integrand of the formula (13.232) defining f (k; β), where ˆp,q,k = (2π)2 k2 E
q “X i=1
x2i − λ
p X
” yj2 .
j=1
(ii) Note that in the limit k → 0 the quantity f (k; β) thus modified is independent of k and thus ˆ τ ) ∼ π|k|Cp,q (λ)I(λ)e−|k|πρτ , S(k;
(13.238)
656
CHAPTER 13
where I(λ) =
q Z Y i=1
∞
dxi 0
p Z Y j=1
∞
1 1−λ Qp
G(q, p, λ|{xi , yj }; k) = Qq i=1
xi
dyj Q2p,q G(q, p, λ|{xi , yj }; k) δ(1 − Qp,q ),
0
Q
i
1−1/λ
j=1
λyj
|xi − xi |2λ Qq Qp
Q
j<j
j=1 (xi
i=1
|yj − yj |2/λ
+ λyj )2
.
(iii) To evaluate I(λ) rewrite it as 1 I(λ) = lim →0+ 2π
Z
∞
du e
iu
−∞
q Z Y
∞
dxi 0
i=1
p Z Y
∞
dyj Q2p,q G(q, p, λ|{xi , yj }; k)e−iQp,q u e−Qp,q ,
0
j=1
and change variables xi → xi /( + iu), yj → yj /( + iu) to obtain “ I(λ) =
lim
→0+
1 2π
Z
∞ −∞
p Z ∞ q Z ∞ ”Y Y eiu du dx dyj Q2p,q G(q, p, λ|{xi , yj }; k)e−Qp,q . i ( + iu)2 0 0 i=1 j=1
Note that the first integral on the r.h.s. is unity, while the second integral is (2πρ)2 times the integral A(q, p, λ) defined by (13.219). Use the fact that (2πρ)2 A(q, p, λ) = 1/(Cp,q (λ)π 2 β), which follows from the workings leading to obtaining the first term in (13.226) from (13.218), to deduce (13.228) from (13.238). 2.
(i) Note from (13.230) that for p = q = 1 and thus β = 2 Z 1 Z ∞ ˆ dx dy δ(|k|/2πρ − (x + y)) S(k) =ρ 0
0
and from this reclaim the result (7.4). (ii) Consider (13.230) with p = 1, q = 2 so that β = 1. By changing variables y=
1 (1 − Y ), 2
x1 =
1 (X1 − 1), 4
x2 =
1 (X2 − 1) 4
deduce that ρ ˆ S(k) = 4
Z
Z
∞ 1
Z
∞
dX1
1
dX2
dY ( −1
1
X1 + X2 |X1 − X2 |(1 − Y 2 ) − Y )2 2 (X12 − 1)1/2 (X22 − 1)1/2
δ(|k|/πρ − ((X1 + X2 )/2 − Y )) × . (X1 − Y )2 (X2 − Y )2
(13.239)
(iii) Now introduce the change of variables Y = v,
X1 + X2 = 2u1 u2 ,
X1 X2 = u21 + u22 − 1.
Show that, for X1 > X2 , the final equation can be replaced by X1 − X2 = (4u21 u22 − 2u21 − 2u22 + 2)1/2 , and hence deduce that the Jacobian for the change of variables from {X1 , X2 } to {u1 , u2 } is equal to 2 2 (u21 − u22 ) = (X12 − 1)1/2 (X22 − 1)1/2 . X1 − X2 (X1 − X2 ) Use this to rewrite (13.239) as Z ∞ Z ∞ Z ˆ S(k) =ρ du1 du2 1
1
1
−1
dv (u1 u2 − v)2 (1 − v 2 )
δ(|k|/πρ − u1 u2 + v) . (u21 + u22 − 1 − 2vu1 u2 + v 2 )2
ˆ (This form of S(k) for β = 1 first appeared in [160], where it was further evaluated to reclaim (7.136).)
CORRELATIONS FOR GENERAL β
657
(iv) Let Ep,q be as specified in Proposition 13.7.9. Show that after the change of variables introduced in (ii) and (iii) π 2 ρ2 (2u21 u22 − u21 − u22 + 1 − v 2 ), E1,2 = 2 and hence deduce from (13.213) that for β = 1 [490] Z ∞ Z ∞ Z 1 (u1 u2 − v)2 (1 − v 2 ) bulk ρT(1,1) (x, y; τ ) = ρ2 du1 du2 dv 2 (u1 + u22 − 1 − 2vu1 u2 + v 2 )2 1 1 −1 “ ” × cos(πρ(x − y)(u1 u2 − v)) exp − (τ /2)(πρ)2 (2u21 u22 − u21 − u22 + 1 − v 2 ) .
Chapter Fourteen Fluctuation formulas and universal behavior of correlations In previous chapters, asymptotic properties of log-gas systems relating to the two-point correlation, density profiles at the edge and spacing distributions were calculated exactly at the random matrix couplings β = 1, 2 and 4. Here macroscopic physical characterizations of the log-gas will be used to predict extensions of the asymptotic forms to the general β case, and furthermore to study fluctuation formulas for certain statistics. As an example, the log-gas has the physical property that it will perfectly screen an external charge density in the long wavelength limit, and this can be used to predict universal asymptotic forms for the two-point correlation. These universal forms can in turn be used to predict that the variance of a linear statistic will be of order unity, and furthermore to give closed forms for this quantity. Macroscopic electrostatics combined with a linear response relation implies that the p.d.f. for such linear statistics will be Gaussian. It is shown that this latter fact ties in with the Szeg¨o and Fisher-Hartwig asymptotic formulas from the theory of Toeplitz determinants. In the second to last section, macroscopic arguments are used to predict asymptotic properties of the gap and spacing probabilities Eβ (n; s) and Pβ (n; s), while in the last section the current correlation is introduced and is used to study the dynamical structure function.
14.1 PERFECT SCREENING In Section 1.4 a macroscopic physical characterization of one-component log-gas systems was used to predict the leading behavior of the particle density. Thus it was hypothesized that, to leading order in the number of particles N , the charge density of the mobile particles will exactly cancel the fixed background density, which is of opposite sign. The reasoning for this is that the absence of cancellation would imply the total charge density in some macroscopic region to be nonzero. An electric field would then be created, and the state would no longer be in equilibrium. A refinement of this physical reasoning is to consider the response of a log-gas system to a periodic perturbation in the background density. If the wavelength of this perturbation is large compared to the inter-particle spacing, the above argument says that for an equilibrium state to be obtained the mobile particles must adjust their density to exactly cancel the charge density of the perturbation. A charged system with this property is said to be in a conductive phase. We will show how it can be used together with a linear response argument to ˆ deduce the small |k| behavior of the structure function S(k), or equivalently the long distance non-oscillatory behavior of the truncated two-particle correlation, for the log-gas on a line [314]. First the linear response formula must be introduced. It states that for any observable A in a statistical mechanical system, the change in its mean value due to a perturbation δU in the total energy U is to leading order in δU given by A − A0 = −βAδU T0 ,
(14.1)
where the subscripts and 0 denote the presence and absence, respectively, of the perturbation in the average. It is simple to derive (14.1) in the canonical ensemble from the meaning of the averages, 1 −β(U+δU) dx1 · · · dxN Ae with Z := dx1 · · · dxN e−β(U +δU) A := Z Ω Ω Ω Ω
FLUCTUATION FORMULAS AND UNIVERSAL BEHAVIOR OF CORRELATIONS
659
and AδU T0 := AδU 0 − A0 δU 0 . Now we will apply (14.1) to the characterization of a conductive phase. For this suppose the system is perturbed by an external charge density eikx (physically this can be thought of as a pair of external charge densities ( cos kx, sin kx)). The total energy of the system is then perturbed by an amount ∞ N ∞ N π ikx δU = − ˆ (1) (k), n log |x − xj | e dx = n ˆ (1) (k) := n(1) (x)eikx dx = eikxj , |k| −∞ −∞ j=1 j=1 where n(1) (x) =
(14.2)
N
j=1 δ(x − xj ) denotes the microscopic particle density and the result ∞ π log |x| eikx dx = − |k| −∞
(14.3)
from the theory of generalized functions has been used. Take for the observable A the charge density at the point x, which for a one-component system with unit charges is equal to the microscopic particle density n(1) (x). (Strictly speaking the microscopic charge density c(1) (x) is equal to n(1) (x) − 1 for a system consisting of unit positive charges and a unit density neutralizing negative background, but the constant −1 plays no role in the subsequent analysis.) Then (14.1) reads π n(1) (x) − n(1) (x)0 = −β n(1) (x)ˆ n(1) (k)T0 . (14.4) |k| Consider first the l.h.s. of (14.4). According to the characterization of a conductive phase, as |k| → 0 the system responds so that its charge density is equal and opposite to that of the external charge density, and thus n(1) (x) − n(1) (x)0 ∼ −eikx . |k|→0
(14.5)
Consider now the r.h.s. of (14.4). Using the translational invariance of the system in the bulk we have ˆ n(1) (k)T0 = eikx S(k), n(1) (x)ˆ
(14.6)
ˆ ˜ S(k) := ρS(k/ρ),
(14.7)
where ˜ with S(k) denoting the dimensionless structure function (7.3). Hence (14.4) implies ˆ S(k) ∼
|k|→0
|k| , πβ
(14.8)
or equivalently, by taking the inverse transformation, ρT(2) (x, x )
∼
|x−x |→∞
−
1 , − x )2
βπ 2 (x
(14.9)
for the leading non-oscillatory term. The exact results (7.136), (7.4), (7.95) and (7.135), (7.65), (7.94) for ˜ S(k) and ρT(2) (x, 0) at β = 1, 2 and 4 in the bulk exhibit the behaviors (14.8) and (14.9) respectively, as do the exact results (13.228) and (13.226) for general rational β. Also, the perturbation results (7.28) and (7.97) are consistent with (14.8). ˆ Setting k = 0 in (14.8) gives S(k) = 0 or equivalently ∞ ρT(2) (x, 0) dx = −ρ, (14.10) −∞
which can be interpreted as saying that the screening cloud about a fixed internal charge has total charge equal and opposite to that of the internal charge. It is known as a charge sum rule, and is expected to hold
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true independent of the phase of the Coulomb system, being necessary for thermodynamic stability [386]. ˆ In this regard, note that the formula S(k) = 0 can be deduced from (14.4) without the assumption (14.5), by simply multiplying both sides by |k| and setting k = 0. Internal screening should also be valid in a semi-infinite system, where it implies ∞ ρT(2) (y1 , y) dy = −ρ(1) (y1 ), (14.11) 0
and in the case that a multiple number n of internal charges are fixed, which requires ρT(n+1) (y1 , y2 , . . . , yn , y) dy = −nρT(n) (y1 , . . . , yn ),
(14.12)
I
valid for an infinite or semi-infinite interval I. A derivation of (14.12) for I equal to the real line is given in Exercises 14.1 q.1, while the case of I semi-infinite is the subject of Exercises 14.1 q.2. The exact results for ρT(n) in Chapter 7 all satisfy (14.12). At β = 2 this follows from the fact ρT(n) is of the form ρT(n) (x1 , . . . , xn ) = (−1)n−1 K(xi1 , xi2 ) · · · K(xin , xi1 ), cycles length n
where K(x, y) has the property
K(x, y)K(y, z) dy = K(x, z), I
these being limiting forms of the second equation in Proposition 5.1.2 and the first equation in (5.7) respectively. Similarly, at β = 1 and 4 the validity of (14.12) follows from the structure of ρT(n) in Propositions 6.1.7 and 6.3.3, together with the fact that the second equation in (6.17) and the first equation in (6.70) remain valid in the thermodynamic limit.
14.1.1 Second order correction A refinement of the linear response argument leading to (14.8) can be given [315] which predicts the term of ˆ order k 2 in the small |k| expansion of S(k). This refinement requires introducing into the r.h.s. of (14.5) a wave number-dependent factor A(k), where A(k) → 1 as |k| → 0. Thus, for |k| small, we have that the net charge in the system after the perturbation is eikx (1 − A(k)). This creates an electric potential ∞ π (1 − A(k))eikx log |x − y|eiky dy = φ(x) = −(1 − A(k)) |k| −∞ and thus a corresponding electric field ∂φ(x) = −iπ sgn(k) (1 − A(k))eikx . (14.13) ∂x Now the existence of this electric field means that there is a force density ρ(1) (x)Ex at point x in the system. Physically, this force density must be balanced for the system to be stable. The balancing force is provided by the gradient of the pressure fluctuation, ∂ ∂ ∂P ∂P ∂ ∂P δP (x) = δρ(1) (x) = − A(k)eikx = −ik A(k)eikx , (14.14) ∂x ∂x ∂ρ ∂x ∂ρ ∂ρ Ex = −
so equating ρ times (14.13) with (14.14) gives |k| ∂P −1 |k| ∂P . A(k) = 1 + ∼1− πρ ∂ρ πρ ∂ρ
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FLUCTUATION FORMULAS AND UNIVERSAL BEHAVIOR OF CORRELATIONS
Note that in deriving (14.14) the fact that for a one-component system the charge and particle densities are proportional has been used. Since an extra factor A(k) on the r.h.s. of (14.5) carries through to an extra factor A(k) on the r.h.s. of (14.8), after recalling (4.169) we thus have |k| 1 |k| 2 ˆ S(k) ∼ − (1 − β/2). (14.15) ρ πβ |k|→0 πβ Inspection of the exact results (7.4), (7.96), and (7.137), and recalling (14.7), gives agreement with (14.15). Similarly the perturbation results (7.28) and (7.97) agree with (14.15).
14.1.2 Multicomponent systems In general for a multicomponent Coulomb system, the charge-charge correlation C(2) (r, r ) is defined by C(2) (r, r ) = c(1) (r)c(1) (r ) − c(1) (r)c(1) (r ),
(14.16)
where c(1) (r) denotes the microscopic charge density c(1) (r) :=
N
qj δ(r − rj )
(14.17)
j=1
with qj denoting the charge of the jth particle. For a p-component system this can be written in terms of the truncated two-particle correlations and the particle densities as p p C(2) (r, r ) = qα qγ ρT(2) (r (α) , r (γ) ) r(α) = r + qα2 δ(r (α) − r )ρ(1) (r (α) ) (α) , (14.18) r (γ) = r
α,γ=1
r
α=1
= r
where here the labels α and γ specify the species. For a multicomponent log-gas on a line, the perfect screening argument of Section 14.1 can be used to predict the leading order non-oscillatory decay of C(2) . Thus, repeating that argument with the observable A chosen to be the microscopic charge density (14.17), we conclude that C(2) (x, x )
∼
|x−x |→∞
−
1 βπ 2 (x − x )2
(14.19)
(cf. (14.9)). Now, for the two-component log-gas with charge ratio 1:2, (14.18) gives C(2) (x, 0) = ρT+1,+1 (x, 0) + 4ρT+1,+2 (x, 0) + 4ρT+2,+2 (x, 0) + δ(x)(ρ+1 (x) + 4ρ+2 (x)). Use of (7.186) shows that for β = 1 C(2) (x, 0) ∼ − x→∞
1 π 2 x2
,
in agreement with the prediction (14.19). In a p-component system we can also consider the consequence of perfect screening of internal charges. Analogous to the equations of Exercises 14.1 q.1 we have (α ) n(1) (y1 1 ) · · · n(1) (yn(αn ) )c(1) (x)T0 dx = 0, I
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where (α ) n(1) (y1 1 ) · · · n(1) (yn(αn ) )c(1) (x)T0
+
n
=
p
(α ) qα ρ(n+1) (y1 1 , . . . , yn(αn ) ; x(α) )
α=1
(αj )
δα,αj δ(x − yj
(α ) ) ρ(n) (y1 1 , . . . , yn(αn ) ) − ρ(1) (x(α) )
j=1
x(α) =x (α1 )
x(α) =x
ρ(n) (y1
, . . . , yn(αn ) ) ,
and as in (14.18) the label α specifies the species. Inductive use of this equation gives p n (α1 ) (α ) T (αn ) (α) (α) qα ρ(n+1) (y1 , . . . , yn , x ) dx = − qαj ρT(n) (y1 1 , . . . , yn(αn ) ) α=1
I
(14.20)
j=1
(cf. (14.12)). The exact results (7.185) satisfy this requirement in the case n = 1. 1. (i) Suppose the observable n(1) (x) in (14.4) is replaced by n(1) (y1 ) · · · n(1) (yn ). By taking the limit k → 0 in the resulting equation, deduce that Z n(1) (y1 ) · · · n(1) (yn )n(1) (y)T0 dy = 0, (14.21)
E XERCISES 14.1
I
where I denotes the real line and n(1) (y1 ) · · · n(1) (yn )n(1) (y)T0 := n(1) (y1 ) · · · n(1) (yn )n(1) (y)0 − n(1) (y1 ) · · · n(1) (yn )0 n(1) (y)0 n “X ” = ρ(n+1) (y1 , . . . , yn , y) + δ(y − yj ) ρ(n) (y1 , . . . , yn ) − ρ(n) (y1 , . . . , yn )ρ(1) (y). j=1
Use this equation inductively to deduce (14.12). (ii) Show from the definitions that (14.21) holds in the finite system, irrespective of the system being a log-gas. 2. For the log-gas on a half-line, let the system be perturbed Rby an external charge of strength at the point x, so that the energy is perturbed by an amount δU = − I log |x − y|n(1) (y) dy. Take for the observable n(1) (y1 ) · · · n(1) (yn ), so the linear response relation (14.1) then reads Z n(1) (y1 ) · · · n(1) (yn ) − n(1) (y1 ) · · · n(1) (yn )0 = −β log |x − y|n(1) (y1 ) · · · n(1) (yn )n(1) (y)T0 dy. I
(i) By taking the limit x → ∞, deduce (14.21). (ii) In the case n = 1 (one-component system), the observable corresponds to the charge density. Perfect screening of the external charge density requires that Z “ ” n(1) (y1 ) − n(1) (y1 )0 dy1 = −. I
Substitute this in the linear response relation to deduce that Z Z 1 = β dy1 dy log |y1 − y|C(2) (y1 , y), I
(14.22)
I
where C(2) (y1 , y) corresponds to the one-component case of the charge-charge correlation (14.18). ˜ 3. Let S(k) denote the dimensionless structure function (7.3), and let v˜(k) denote the dimensionless Fourier transform of the pair potential v(x), Z ∞
v˜(k) = ρ −∞
v(x)eikρx dx.
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FLUCTUATION FORMULAS AND UNIVERSAL BEHAVIOR OF CORRELATIONS
For long-range potentials v(x) such that v˜(k) → ∞ as k → 0, the weak coupling (β → 0) approximation from the theory of classical fluids [289] says that ˜ S(k) −1 = − and thus ρT(2) (x, 0) = −
ρ2 2π
Z
∞ −∞
β˜ v (k) 1 + β˜ v (k)
β˜ v (k) e−iρxk dk. 1 + β˜ v (k)
˜ (i) By rearranging and taking the inverse Fourier transform, show that the approximation for S(k) is equivalent to the integral equation Z ∞ “ ” 1 T v(|y − x|) ρT(2) (x, 0) + ρδ(x) dx. ρ(2) (y, 0) = −β ρ −∞ Show that this equation can be derived from the linear response relation (14.1) with δU the energy due to an external charge placed at the origin, A chosen as the microscopic density at the point y and A − A0 set equal to ρT(2) (y, 0)/ρ. ˜ (ii) For the log-gas show that the formula for S(k) reproduces (14.8), and use the formula for ρT(2) (x, 0) to derive (4.179). (iii) Use the formula for ρT(2) (x, 0) and (4.178) to deduce that the mean energy per particle is given by u=−
” “ 1 1 lim βv(x) + 2 ρT(2) (x, 0) . 2β x→0+ ρ
14.2 MACROSCOPIC BALANCE AND DENSITY In this section the global asymptotic expansions of the Gaussian and Laguerre densities (7.162) and (7.161) will be considered from a macroscopic viewpoint. Consider first the Gaussian case. In Section 1.4 the macroscopic balance equation (1.48) was used to deduce the Wigner semicircle form for the density. Here we want to refine (1.48) to allow a correction to the Wigner semicircle form to be computed. The basic hypothesis is that the electrostatic energy on the r.h.s. should be corrected by subtracting the free energy. According to (4.166), with ρ therein interpreted as ρ(1) (x), this modifies (1.48) so it reads a 1 x2 1 +C = − log ρ(1) (x), ρ(1) (y) log |x − y| dy + x ∈ (−a, a) (14.23) 2 2 β −a √ √ for some √ constant C. We know that to leading order the density is supported on (− 2N , 2N ) so that a = 2N. √ If we introduce the global density σ(X) := 2/Nρ(1) ( 2NX), normalized so that 1 σ(X) dX = 1, (14.24) −1
the balance equation (14.23) reads
1
x2 + C = −1
σ(y) log |x − y| dy +
1 1 1 − log σ(x), N 2 β
(14.25)
again for some constant C. This integral equation can be solved to give the sought correction to ρW (y).
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P ROPOSITION 14.2.1 The integral equation (14.25) permits the solution σ(y) = ρW (y) +
μG (y) , N
(14.26)
where ρW (y) is the Wigner semicircle form as specified in (7.26) while 1 1 1 1 1 − (δ(y − 1) + δ(y + 1)) − . μG (y) = β 2 2 π 1 − y2
(14.27)
Note that this is consistent with (7.162). Proof. Substituting (14.26) in (14.25) and equating terms O(1/N ) gives Z
1
C= −1
μG (y) log |x − y| dy +
“1 2
−
1” log ρW (x), β
x ∈ (−1, 1),
(14.28)
which must be solved subject to the constraint Z
1
μG (y) dy = 0,
(14.29)
−1
deduced by substituting (14.26) in (14.24). Differentiating (14.28) gives
Z
1
0 = PV −1
“1 1” x μG (y) , dy − − x−y 2 β 1 − x2
where PV denotes the principal value. The validity of (14.27) can be verified by making use of the fact that [448] Z 1 1 1 p PV dy = 0, x ∈ (−1, 1) 1 − y2 −1 x − y (cf. (1.66)) and recalling the requirement (14.29).
To study the expansion of the global density in the Laguerre case (7.161) we begin by recalling the result of Proposition 3.1.4. This allows the general chiral ensemble p.d.f. N 1 −βV (xl ) β/2 e xl C l=1
|x2k − x2j |β
(xl > 0)
(14.30)
1≤j
to be interpreted as a log-gas confined to x > 0, with image charges of the same sign in x < 0, and subject to a one-body potential V (x). It has been noted in Exercises 3.1 q.9 that to leading order in N the background density satisfies the same integral equation (1.55) as the background density for the log-gas on a line with confining potential V (x). Analogous reasoning gives that with V (x) =
x2 − a log |x| 2
(14.31)
1 the global density σ(X) := 4ρ(4N X), which is normalized so that 0 σ(X) dX = 1 satisfies (14.25) with a x2 on the l.h.s. replaced by 2x2 − 2N log |x|. Seeking a solution of the form (14.26) with ρW (y) replaced by 2ρW (y) and μG (y) by μch (y) gives (14.27) with μG (y) → μch (y) − a2 δ(y). Taking into consideration the normalization requirement we thus obtain 1 1 1 1 a a μch (y) = − δ(y − 1) − . (14.32) − δ(y) + 2 β 2 π 1 − y2 π 1 − y2 On the other hand, changing variables xl → x2l in the Laguerre β-ensemble MEβ,N (xβ(a+(1−1/β)/2 e−βx/2 ) gives (14.30) with V (x) specified by (14.31). It thus follows that with μL (X) the O(1/N ) correction to the
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FLUCTUATION FORMULAS AND UNIVERSAL BEHAVIOR OF CORRELATIONS
density in the Laguerre β-ensemble with one-body terms xβa e−βx/2 one has 2XμL (X 2 )|a →a+(1−1/β)/2 = μch (X).
(14.33)
This is indeed consistent with (7.161). E XERCISES 14.2
1. Appropriately refine the argument of the proof of Proposition 3.6.3 to conclude that for the Jacobi β-ensemble specified therein
“b − 1 1 ” 1 1 ““ a − 1 1 ” (2/β − 1 + (a + b)/2) ” + + − . δ(y −1)+ δ(y +1)− π(1 − y 2 )1/2 N 2 β 2 β π(1 − y 2 )1/2 (14.34) By changing variables y = cos θ deduce that for the β-ensemble on the half-circle specified by the r.h.s. of (3.76), N −1 ρ(1) (y) ∼
N −1 ρ(1) (θ) ∼
“b − 1 (2/β − 1 + (a + b)/2) ” 1 1 ““ a − 1 1” 1” − + δ(θ) + + δ(θ − π) − . (14.35) π N 2 β 2 β π
14.3 VARIANCE OF A LINEAR STATISTIC One application of the formula (14.9) relates to the calculation of the variance of a linear statistic in the infinite density limit for a log-gas on the line. Before discussing this and similar applications, some preliminary remarks and theory relating to linear statistics and their variance are required. D EFINITION 14.3.1 A linear statistic of a one-component system is any function A that can be written in N the form A = j=1 a(rj ) where rj denotes the particle coordinates. In random matrix theory, an example of a linear statistic is the conductance G/G0 =
N
λj ,
(14.36)
j=1
where {λj } are the eigenvalues of t† t (recall Section 3.8). An observable quantity associated with (14.36) is the mean square fluctuation (i.e., variance) Var(G/G0 ) := (G/G0 − G/G0 )2 = (G/G0 )2 − G/G0 2 .
(14.37)
For a general linear statistic the variance can be expressed in terms of the truncated two-particle correlation function. P ROPOSITION 14.3.2 For a one-component system of N particles confined to an interval I Var(A) = dx dx a(x)a(x ) ρT(2) (x, x ) + ρ(1) (x)δ(x − x ) . I
I
Proof. We have Z
Z
A2 =
dx a(x) I
I
Z
Z dx a(x)
= I
dx a(x )
N D X
N E Z DX E δ(x − xj )δ(x − xj ) + a2 (x) δ(x − xj ) dx I
j,j =1 j=j
` ´ dx a(x ) ρ(2) (x, x ) + δ(x − x )ρ(1) (x)
I
Z
and A =
a(x)ρ(1) (x) dx, I
so the result follows from the definition of Var(A) and ρT(2) .
j=1
(14.38)
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Experiments (see e.g., [53]) have shown that (14.37) is of order unity, independent of N . An important feature in explaining this effect is that the density of the eigenvalues of t† t, which take values between 0 and 1, is infinite in the large N limit. In fact in the infinite density limit similar O(1) fluctuation formulas hold for the one-component log-gas confined to an interval I finite or otherwise. In particular, due to the analogy between the eigenvalues of the Wishart matrix and the log-gas confined to the half-line, the log-gas results can be used to calculate (14.37). Other linear statistics also show themselves in the theory of quantum conductance [53]. One is the shot noise power P , which characterizes time-dependent fluctuations in the current due to the discreteness of the electron charge. With λj , G0 as in (14.36), and V the voltage difference between two electron reservoirs, P = P0
N
λj (1 − λj ),
P0 = 2eV G0 .
(14.39)
j=1
Another relates to the conductance GNS of a normal metal-superconductor junction (recall Section 3.3.2), which is given by GNS
N λ2j 4e2 . = h j=1 (2 − λj )2
(14.40)
From Proposition 14.3.2, to evaluate Var(A) in the infinite density limit we require the infinite density limit of ρT(2) (x, x ) smoothed over the microscopic interparticle spacing. The smoothing kills all terms with period proportional to the inter-particle spacing — these terms do not contribute to Var(A) in the infinite density limit as they integrate to zero. Since in a one-component system of point particles the only length scale is the interparticle spacing 1/ρ, we see that the asymptotic formula (14.9) must also apply for |x − x | fixed and ρ → ∞ and thus the sought asymptotic form is already known in the case that the domain is the real line. Also of interest in the study of Var(A) is the infinite density limit of ρT(2) (x, x ) for the log-gas confined to different one-dimensional domains I, in particular the circle, finite interval and semi-infinite interval. The linear response argument of the previous section cannot readily be adapted to these cases. However, Jancovici [316] has shown how a different linear response argument can be formulated which is also applicable in the more general settings. In the method of Jancovici, the one-dimensional domain I is regarded as embedded in two dimensions, and it is hypothesized that in the infinite density limit a log-gas confined to the domain I behaves as a conductor obeying the laws of two-dimensional electrostatics. The system is perturbed by adding an external charge δq at the point r . The total energy of the system is then perturbed by an amount δU = δqΦ(r ), where Φ(r ) is the electrostatic potential at r due to the charge density induced on I. The observable A is chosen to be N the potential Φ(r) at another point r due to the charges in I (Φ(r) = − j=1 log |r − xj |) so that the linear response formula reads Φ(r) − Φ(r)0 = −βδqΦ(r)Φ(r )T0 . But the assumption of the applicability of two-dimensional electrostatics gives that in the infinite density limit the response of the system is such that Φ(r) − Φ(r )0 = φ(r ), where φ(r) denotes the macroscopic potential at r due to the charge induced on I by the charge at r (φ(r) does not include the potential due to the charge at r itself). Thus φ(r) = −βδqΦ(r )Φ(r )T0 .
(14.41)
Suppose for definiteness that the domain I is some segment (finite or infinite) of the real line. To relate the r.h.s of (14.41) to ρT(2) (x, x ) we note that for a one-component system with unit charges, ρT(2) (x, x ) =
FLUCTUATION FORMULAS AND UNIVERSAL BEHAVIOR OF CORRELATIONS
667
σ(x)σ(x )T0 , where σ(x) denotes the charge density, and it is assumed x = x . But from 2d electrostatics E + (x) − E − (x) = 2πσ(x), where E + (x) − E − (x) is the discontinuity of the electric field perpendicular to the point x on I (cf. Section 1.4.3), so we have T 1 + − + − ρT(2) (x, x ) = . (14.42) 2 (E (x) − E (x))(E (x ) − E (x )) 0 (2π) Now the electric fields and their corresponding averages can be calculated from the potential Φ(r) occurring in (14.41) according to E ± (x) = lim± y→0
∂ Φ(r), ∂y
E r (x)E s (x )T0 = lim± lim± y→0
y →0
∂2 Φ(r)Φ(r )T0 , ∂y∂y
(14.43)
where r, s = ±. Substituting (14.41) in this last equation, and making use of the symmetry relation E r (x)E s (x )T0 = E −r (x)E −s (x )T0 allows (14.42) to be rewritten as 2 ∂ 2 φ(r ) ∂ 2 φ(r ) − lim+ lim− . lim+ lim+ − − ρT(2) (x, x ) = 2 βδq βδq y→0 y →0 ∂y∂y (2π) y→0 y →0 ∂y∂y
(14.44)
At this stage the details of the calculation depend on the particular interval I. To illustrate the details, suppose I is the whole real line, and let us use (14.43) to rederive (14.9). P ROPOSITION 14.3.3 For the one-component log-gas on a line x+1/ρ 1 1 ∂2 ρT(2) (y, x )dy = − =− 2 log |x − x | lim ρ 2 ρ→∞ βπ ∂x∂x βπ 2 (x − x ) x
(14.45)
or equivalently, for a fixed density ρ, the leading non-oscillatory term in the large |x − x | expansion of ρT(2) (x, x ) is given by the r.h.s of (14.45). Derivation. In using macroscopic electrostatics we are implicitly assuming the quantities such as σ(x) and ρT(2) (x, x ) are averaged on a microscopic scale so (14.44) will yield the average quantity in (14.45). To calculate φ(r) suppose for definiteness the perturbing charge is in the upper half plane y > 0. Then for y > 0, from the method of images φ(r) − δq log |z − z | = −δq(log |z − z | − log |z − z¯ |)
(14.46)
(recall that φ(r) does not include the potential due to the charge at r , which is the second term on the l.h.s.), where z = x + iy, z = x + iy , while for y < 0, φ(r) − δq log |z − z | = 0.
(14.47)
An elementary calculation according to (14.44) yields the first equality in (14.45).
The prediction of Proposition 14.3.3 that the leading non-oscillatory term in the large |x − x | expansion of ρT(2) (x, x ) is equal to −1/(βπ 2 (x − x )2 ) has already been obtained in (14.9), and checked against exact results. To apply this method to other domains I we see the main task is to specify the potential φ(r ). There are two cases to consider: I infinite or I finite. For I infinite the system can support a charge excess by redistributing charges at infinity, while for I finite the situation is that of a grounded conductor which carries no nett charge. From the viewpoint of two-dimensional electrostatics, in the former situation we require φ(r)− δq log |z − z | to vanish for r ∈ I independent of the value of r , while in the latter situation we only require this quantity to vanish for both r, r ∈ I. Note that in the latter situation we also require φ(r ) → 0 as |r| → ∞ since the total charge on the conductor is zero. In all cases φ(r) must satisfy Laplace’s equation and be symmetric in r and r unless these points are separated by a closed domain I.
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We will consider the different domains of particular interest separately. Circle For |z | > 1, we can verify that
φ(r) − δq log |z − z | =
-
z − z |) + log |z | , −δq log (|z − z |/|1/¯ −δq log |z |,
|z| > 1, |z| < 1,
(14.48)
is the required potential, as it satisfies all the above requirements. In particular, the “constant” −δq log |z | for |z| < 1 is required by continuity of the two solutions at |z| = 1. For |z | < 1 a similar verification shows that −δq log |z|, |z| > 1, φ(r ) − δq log |z − z | = (14.49) −δq log (|z − z |/|1 − z¯z |) , |z| < 1 is the required potential. Finite Interval (−1, 1) Here φ(r) is obtained from the solution for φ(r) above for the circle geometry in the case |z |, |z| > 1 via the conformal mapping 2w = z + 1/z, which maps the exterior of the unit disk to the plane with a cut from 1/2
1/2
−1 to 1. We thus simply substitute z = w + (w2 − 1) , z = w + (w − 1) (w = reiθ , w = r eiθ ), where the principal branch of the square root is to be taken, in (14.48). The resulting expression is valid for all r and r . 2
Semi-infinite Interval (0, ∞) Here we begin with the infinite line potential (14.46) and map the infinite line to the half-line x > 0 by the conformal mapping w = z 2 . Thus the required potential is obtained by substituting z = w1/2 = r1/2 eiθ/2 1/2 1/2 and z = w = r eiθ /2 in (14.46). The resulting expression is valid for all r and r . From these explicit evaluations of φ(r) we can use (14.44) (appropriately modified in the case of a circle) to evaluate ρT(2) (x, x ) in the infinite density limit. One finds [52], [192] N lim N →∞ 2π
N lim N →∞ 2
θ +2π/N
1
, β(2π) sin2 (θ − θ )/2 ∂2 1 log | sin(θ − θ )/2|, =− 2 βπ ∂θ∂θ
θ
x +2/N
ρT(2) (y, x) dy = −
2
(14.50)
1 − xx
1
, βπ 2 (x − x )2 [(1 − x2 )(1 − x 2 )]1/2 1 ∂2 1 2 1/2 (1 − x ) log |x − x | , =− 2 βπ (1 − x2 )1/2 ∂x∂x
x
(14.51) x+1/ρ
lim ρ
ρ→∞
ρT(2) (φ, θ) dφ = −
x
x+x 1 √ 2πβ xx (x − x )2 √ x − √x ∂2 1 √ , =− 2 log √ x + x βπ ∂x∂x
ρT(2) (y, x ) dy = −
(14.52)
FLUCTUATION FORMULAS AND UNIVERSAL BEHAVIOR OF CORRELATIONS
669
for the unit circle, interval (−1, 1) and semi-infinite line respectively. Note that (14.52) will also give the leading non-oscillatory decay of ρT(2) (x, x ) for ρ fixed and x, x → ∞ (cf. the statement of Proposition 14.3.3). Note too that the soft and hard edge asymptotic formulas obtained in Chapter 7 for ρT(2) (x, y) are all in agreement with the first equality in (14.52). Now, denoting the partial derivative representations in Proposition 14.3.3 and (14.50)–(14.52) by K2 (x, x ) we see that in each case K2 (x, x )dx = 0 (14.53) I
provided we take the differentiation with respect to x outside the integral. On the other hand, without this interchange of limits, there is a non-integrable singularity at x = x in K2 (x, x ). Since (14.53) is precisely the condition (14.10) for the perfect screening of an internal charge, this suggests we substitute K2 (x, x ) for ρT(2) (x, x ) + ρ(1) (x)δ(x − x ) in the formula of Proposition 14.3.2 and interchange the order of integration with respect to x and differentiation with respect to x . This gives ∞ ∞ 1 dx a (x) dy a (y) log |x − y| Var(A) = − 2 βπ −∞ −∞ ∞ 1 2 |k||˜ a(k)| dk, (14.54) = β2π 2 −∞ π π 1 dθ a (θ) dφ a (φ) log | sin(θ − φ)/2| Var(A) = − 2 βπ −π −π ∞ ∞ 4 = nan a−n , a(θ) = an einθ , (14.55) β n=1 n=−∞ 1 1 1/2 1 a(x) a (y)(1 − y 2 ) Var(A) = 2 dx dy βπ −1 (1 − x2 )1/2 −1 x−y ∞ ∞ 2 2 = ncn , a(cos θ) = c0 + 2 cn cos nθ, (14.56) β n=1 n=1 √ ∞ ∞ x − √y 1 dx a (x) dx a (y) log √ Var(A) = − 2 √ βπ 0 x + y 0 ∞ ∞ 1 2 = |ˆ a(k)| k tanh(πk) dk, a ˆ(k) = eikx a(ex ) dx (14.57) β2π 2 −∞ −∞ for the infinite line, unit circle, interval (−1, 1) and semi-infinite line, respectively, where a (u) denotes the derivative of a(u). In deriving the second equality in (14.54) the Fourier transform (14.3) has been used, while the second equality in (14.55) makes use of the Fourier expansion π ∞ 1 1 log | sin(θ−θ )/2| = (p = 0). (14.58) αp eip(θ−θ ) , αp = log | sin(θ/2)|e−ipθ dθ = − 2π −π 2|p| p=−∞ The second equality in (14.56) follows after the change of variables x = cos θ, y = cos φ and use of the cosine expansion (1.73). Also, the second equality in (14.57) makes use of the Fourier transform [165] πy 1 ∞ tanh ax ixy − log tanh e dx = 4a 2 −∞ x in the case a = π. From (14.36) and (14.37) the formula (14.56) with a(x) = (1 + x)/2 (this is the linear function on (−1, 1)
670
CHAPTER 14
that takes on values 0 and 1 at the endpoints), and thus c1 = 1/4, cn = 0 (n > 1), gives for the variance of the dimensionless conductance in the quantum transport problem [51] Var(G/G0 ) =
1 . 8β
14.3.1 Alternative calculation of the smoothed two-point density correlation The linear response formula (14.1) can be used in a different way [51] to that detailed in the previous section to give an alternative derivation of the formulas (14.45) N and (14.50)–(14.52). The idea is to perturb the system by an arbitrary one-body potential so that δU = j=1 u(xj ). The observable A is taken to be the microscopic density at a point x in the system so the linear response relation (14.1) then reads (14.59) n(1) (x ) − n(1) (x )0 = −β u(x) ρT(2) (x, x ) + ρ(1) (x )δ(x − x ) dx, I
ρT(2)
where we have used the definition of to rewrite the r.h.s. of (14.1). Our task is to compute the l.h.s. of (14.59) for arbitrary u and then to calculate ρT(2) by functional differentiation, δ n(1) (x ) − n(1) (x )0 = −β ρT(2) (x, x ) + ρ(1) (x )δ(x − x ) . (14.60) δu(x) Note that (14.60) is exact if we set u(x) = 0 after the functional differentiation and thus is valid with this qualification independent of the applicability of the linear response relation. We can calculate the l.h.s. of (14.59) by viewing u(x) as the potential energy of an external charge density. As in Section 1.4 it is hypothesized that the density difference is determined by macroscopic electrostatics and thus satisfies − log |x − x | n(1) (x ) − n(1) (x )0 dx = u(x) + C, (14.61) I
where the constant C is determined by the particle conservation condition ! n(1) (x ) − n(1) (x )0 dx = 0. I
The equation (14.61) for I an infinite line is solved using Fourier integrals, for I a circle it is solved using Fourier sums, and for I a finite or semi-infinite interval it can be solved using the eigenfunction method of Proposition 1.4.2. We will illustrate the details of the method in the case of a circle (the finite and semi-infinite intervals are covered in Exercises 14.3). P ROPOSITION 14.3.4 The solution of the integral equation π log | sin(θ − θ )/2|Δρ(θ ) dθ = u(θ) + C, where C is chosen so that
π −π
−π
Δρ(θ ) dθ = 0 and u(θ) = Δρ(θ) = −
∞ p=−∞
up eipθ is given by the Fourier series
∞ 1 |p|up eipθ . π p=−∞
Proof. Substituting for log | sin(θ − θ )/2| in the integral equation using (14.58) gives ∞ X p=−∞
Z αp eipθ
π −π
Δρ(θ )e−ipθ dθ =
∞ X
up eipθ + C,
p=−∞
where αp is defined in (14.58). Equating coefficients of eipθ gives the value of the Fourier coefficients of Δρ(θ), and the
671
FLUCTUATION FORMULAS AND UNIVERSAL BEHAVIOR OF CORRELATIONS
stated result follows.
Since up =
1 2π
π −π
u(θ)e−ipθ dθ we see from Proposition 14.3.4 that
∞ δ 1 1 ∂2 Δρ(θ ) = − 2 |p|eip(θ −θ) = 2 log | sin(θ − θ )/2|, δu(θ) 2π p=−∞ π ∂θ∂θ
(14.62)
where to obtain the last equality we have used (14.58). Substituting (14.62) in (14.60) reclaims the infinite density formula (14.50) for the two point correlation on a circle. E XERCISES 14.3
1. The objective of this exercise is to solve the integral equation Z 1 log |x − x |Δρ(x ) dx = u(x) + C, −1
where C is chosen so that interval (−1, 1).
R1 −1
Δρ(x ) dx = 0, and to use this solution to rederive (14.51) for the case of the
(i) Substitute x = cos θ, x = cos σ, then use the eigenfunction method of Section 1.5 to show that Δρ(cos θ) = −
∞ “Z π ” X 2 p u(cos σ) cos pσ dσ cos pθ. 2 π sin θ p=1 0
(ii) From this result and the final expression of Exercises 1.5 q.1 show that “ 1/2 ∂ ”2 1 δΔρ(x) 2 = 2 ) log |x − x | (1 − x 2 δu(x ) ∂x π (1 − x2 )1/2 (1 − x )1/2 and thus reclaim (14.51). 2. In this exercise the integral equation Z
with C chosen so that
R∞ 0
∞
log |x − x |Δρ(x ) dx = u(x) + C,
0
Δρ(x ) dx = 0, is solved and the solution is used to rederive (14.52).
(i) Use the formal identity
Z
∞
log |x2 − t2 | = −2
cos kx cos kt 0
to show that
1 Δρ(y ) = − 2 π y
Z
„Z
∞
2
«
∞
dk k 0
dk k
2
dx u(x ) cos kx cos ky. 0
(ii) Use the formulas in (i) to show that 1 ∂ “ 1/2 ∂ ” δΔρ(x) = − x log |x − x | δu(x ) ∂x π 2 x 1/2 ∂x and thus reclaim (14.50) in the case of the semi-infinite interval. 3. For two statistics A and B, the covariance is defined by Cov(A, B) := (A − A)(B − B) = AB − AB, while the correlation coefficient is defined by Cor(A, B) :=
Cov(A, B) . (Var(A)Var(B))1/2
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CHAPTER 14
Use the Schwarz inequality to show |Cor(A, B)| ≤ 1. 4. Use (14.39) and (14.56) with a(x) = (1 − x2 )/4 to show that Var (P/P0 ) =
1 . 64β
14.4 GAUSSIAN FLUCTUATIONS OF A LINEAR STATISTIC 14.4.1 The infinite density limit After having evaluated the variance of the linear statistic A, it is natural to seek the full distribution PA (t) of A. For a general statistic A this is defined as 1 dx1 · · · dxN δ(t − A)e−βU (14.63) PA (t) = Zˆ I I (cf. the equation of Exercises 3.2 q.3(ii)) and represents the p.d.f. for the event that A = t. The evaluation of P (t) for a log-gas in the infinite density limit can be deduced by a linear response argument [445], [316]. N To do this, suppose δU = j=1 a(xj ) in (14.1), and take for the observable A therein (not to be confused with the linear statistic A) the microscopic density n(1) (x ). This gives n(1) (x ) − ρ(x )0 = −βn(1) (x )AT0 .
(14.64)
Now terms ignored on the r.h.s. of (14.64) are quadratic and higher in a(x). But on the l.h.s, the formula (14.61) with u(x) replaced by a(x) says that in the infinite density limit ρ(1) (x ) − ρ(1) (x )0 is linearly related to a(x). This implies (14.64) must be exact in the infinite density limit. Multiplying by a(x ) and integrating over x then shows, at the same level of physical argument, that the relation A − A0 = −βA2 T0
(14.65)
is exact in the infinite density limit. Replacing A by −ikA/β and introducing the infinite density ρ → ∞ (∞) limiting form of PA (t), PA (t) say, we see that (14.65) can be rewritten as ∞ d (∞) log eik(t−A0 ) PA (t) dt = −kA2 T0 . (14.66) dk −∞ After integrating with respect to k and exponentiating this implies ∞ N 2 2 T (∞) eik(t−A0 ) PA (t) dt := lim eik(a(xl )−a) = e−k A 0 /2 . −∞
ρ→∞
(14.67)
l=1
(∞)
(t − A0 ) is thus exactly given by the Gaussian 1/2 2 2 1 (∞) PA (t − A0 ) = e−t /2σ , σ 2 = Var(A) = A2 T0 . 2πσ 2
The distribution PA
(14.68)
Note that for the particular choices of I discussed in the previous two sections Var(A) is order one and known (∞) explicitly, so PA (t − A0 ) is completely specified. The exactness of (14.65) also has consequence for the higher order correlations ρT(n) (x1 , . . . , xn ) implying they must vanish in the infinite density limit for x1 , . . . , xn distinct [445]. This can be seen from the cumulant
FLUCTUATION FORMULAS AND UNIVERSAL BEHAVIOR OF CORRELATIONS
673
expansion
∞
log −∞
where
eikt PA (t) dt =
∞ (ik)n Mn (a), n! n=1
(14.69)
dx1 a(x1 ) · · ·
Mn (a) := I
I
dxn a(xn ) ρT(n) (x1 |x2 | . . . |xn ).
(14.70)
In (14.70) ρT(n) (x1 |x2 | . . . |xn ) is the truncated n-particle correlation function which includes the contribution of coincident points, ρT(2) (x1 |x2 ) = ρT(2) (x1 , x2 ) + δ(x1 − x2 )ρ(1) (x1 ), ρT(3) (x1 |x2 |x3 ) = ρT(3) (x1 , x2 , x3 ) + δ(x1 − x2 )ρT(2) (x2 , x3 ) + δ(x1 − x3 )ρT(2) (x1 , x2 ) +δ(x2 − x3 )ρT(2) (x1 , x3 ) + δ(x1 − x2 )δ(x2 − x3 )ρ(1) (x1 ), and so on. Substituting (14.69) in (14.66) we see that Mn (A) = 0 for n ≥ 3. Since A is arbitrary, this implies ρT(n) (x1 |x2 | . . . |xn ) = 0. But for x1 , . . . , xn distinct, ρT(n) (x1 |x2 | . . . |xn ) = ρT(n) (x1 , . . . , xn ), so indeed ρT(n) (x1 , . . . , xn ), n ≥ 3, must vanish for noncoincident points. 14.4.2 Szeg¨o’s asymptotic formula The result (14.68) can be used both to anticipate Sz¨ego’s asymptotic formula from the theory of Toeplitz determinants, and to interpret it as a fluctuation formula. We recall that an N × N Toeplitz determinant is characterized by each entry in row j, column k depending on the difference j −k. Thus all entries are constant along each diagonal parallel to the main diagonal. Szeg¨o’s asymptotic formula gives the large N asymptotic behavior of a general class of Toeplitz determinants [327]. P ROPOSITION 14.4.1 Let ea(θ) and a(θ) have Fourier decompositions ea(θ) =
∞
αp eipθ ,
a(θ) =
p=−∞
respectively, where it is required DN [e
a(θ)
∞
−π ≤ θ ≤ π,
ap eipθ ,
p=−∞
∞ p=−∞
|p||ap |2 < ∞. One has
] := det[αj−k ]j,k=1,...,N
∞ ∼ exp N a0 + pap a−p + o(1) .
N →∞
(14.71)
p=1
We remark that the Borodin-Okounkov identity (10.96) shows that the correction terms to (14.71) are in general exponentially small. To anticipate (14.71), we note from (5.77) that DN [ea(θ) ] =
N l=1
ea(θl )
U(N )
,
(14.72)
where ·U (N ) can be interpreted as the canonical average for the one-component log-gas on a circle at β = 2. Hence DN [eika(θ) ] = PˆA (k), where PˆA (k) is the Fourier transform of the probability distribution PA (t) for the linear statistic A =
674 N j=1
CHAPTER 14
a(θj ) in the log-gas. According to (14.67) and (14.55) ∞ PˆA (k) = exp N ika0 − k 2 pap a−p + o(1) ,
(14.73)
p=1
which is the sought result. A consequence of (14.73) is that Szeg¨o’s asymptotic formula can be interpreted as a Gaussian fluctuation formula [327]. In Exercises 14.4 q.3 below, it will be shown that (14.73) can be derived using the continuous approximation applied directly to the potential energy of the Boltzmann factor in (14.72). This procedure has been adapted by Johansson [328] to rigorously prove (14.73), and more generally (14.68) in the case of the log-gas on the unit circle for all general β > 0. The general formula (14.67) can be used to give the analogue of (14.71), interpreted as an asymptotic formula for the U (N ) average (14.72), for averages over the classical groups O(N ) and Sp(2N ). With a(θ) → a(cos θ) the variance is given by (14.56). This follows from Proposition 3.7.1 which relates the eigenvalue p.d.f.’s for the classical groups to particular Jacobi unitary ensembles upon changing variables yj = cos θj . ˜ denote the eigenvalue p.d.f. of the eigenLet G be one of the classical groups O± (N ), Sp(2N ), and let G values with angles θ ∈ (0, π). The formulas (14.67), (14.56) and (14.35) give that N∗ l=1
ea(cos θl )
∼
˜ N →∞ G
∞ ∞ ∞ 1 2 exp N ∗ c0 + a cn + b (−1)n cn + pc , 2 p=1 p n=1 n=1
(14.74)
∞ where a(cos θ) = c0 + 2 n=1 cn cos nθ and N ∗ , a, b are as in Proposition 3.7.1. This is in agreement with known rigorous results (see, e.g., citeJo97, [46]).
14.4.3 Asymptotics of a class of Hankel determinants The first identity in Exercises 5.1 q.3(iii) tells us that ˜ N (e−N V (x)+a(x) ) D ∞ ∞ N 1 −N V (x1 ) := dx1 e ··· dxN e−N V (xN ) ea(xl ) (xk − xj )2 N ! −∞ −∞ l=1 1≤j
(14.75)
−∞
∞ where ap = −∞ xp ea(x)−N V (x) dx, thus relating the multiple integral to a Hankel determinant (by definition the latter have entries which depend on the sum of the row index j and the column index k). The following asymptotic formula for these integrals is known [328].
P ROPOSITION 14.4.2 Let V (x) be an even degree polynomial which is independent of N with positive leading coefficient and no real zeros. Let ρ(1) (x) be the scaled density in the log-gas system corresponding ˜ N (e−N V (x) ), supported on [c1 , c2 ] and normalized so that to D c2 ρ(1) (x) dx = 1. c1
FLUCTUATION FORMULAS AND UNIVERSAL BEHAVIOR OF CORRELATIONS
675
For sufficiently well behaved functions a(x) one has c2 ˜ N (e−N V (x)+a(x) ) D a(x)ρ(1) (x) dx ∼ exp N ˜ N (e−N V (x) ) N →∞ D c1 c2 1 c2 a (y) (y − c1 )(c2 − y) a(x) . dx dy × exp 4π 2 c1 x−y (x − c1 )(c2 − x) c1 (14.76) A key identity used in the derivation of (14.76) is given in Exercises 14.4 q.4(iii). We recognize (14.76) as an example of (14.67) with β = 2, a(x) → −ia(x) and A2 T0 = Var(A) given by (14.56) (in the latter the kernel must be translated so that the endpoints are at x = c1 , c2 ). The result of Proposition 14.4.2 holds with the product of differences squared in (14.75) raised to the power of β/2, provided a factor of 2/β multiplies the argument of the second exponential in (14.76), and furthermore the d r.h.s. is multiplied by the additional factor exp( c a(y)μ(y) dy), where μ(y) is the O(1/N ) correction to the density (recall (14.26)). 14.4.4 Gaussian fluctuations of the spacing distribution Consider a continuous one-dimensional statistical mechanical system in the bulk in a fluid state with unit density. Let Fβ (s) denote the probability that the spacing, to the right say, of a given particle is greater that s. Analogous to the formula (9.4) we can write s ∞ (−1)l−1 s Fβ (s) = dx2 · · · dxl ρ(l) (0, x2 , . . . , xl ). (14.77) (l − 1)! 0 0 l=1
Let η(L, s) denote the number of particles in an interval of length L which have their right-neighbor spacing greater than s. Clearly the expected value of η(L, s) is LFβ (s). For the log-gas at β = 2, Soshnikov [494] has considered the distribution of η(L, s), and proved that it has a Gaussian form as L → ∞. The analysis of η(L, s) is complicated by the fact that it is not a linear statistic. Consequently the formulas (14.65) and (14.69) do not hold, so other methods of analysis are necessary. The approach taken in [494] is to prove p (p − 1)!!(b(s))p/2 + o(1), p even 1/2 (η(L, s) − η(L, s))/L (14.78) = o(1), p odd where b(s) is the variance of the scaled statistic (η(L, s) − η(L, s))/L1/2 and (2n − 1)!! := (2n − 1)(2n − 3) · · · 3 · 1. These moments uniquely characterize a Gaussian distribution. Note that the variance of η(L, s) itself is proportional to L, which is the same behavior exhibited by a gas with short-range interactions. We will not attempt to give details of the working which leads to (14.78) for general p, but we will consider the details of the case p = 2 and thus calculate the variance explicitly. First, some notation is required. Denote by ξ(xi , dxi , s) the indicator for the event that there is an eigenvalue in [xi , xi + dxi ] and no eigenvalue in [xi + dxi , xi + s]. Then L ξ(x, dx, s) and ξ(xi , dxi , s) = Fβ (s)dxi (14.79) η(L, s) = 0
while for |x1 − x2 | < s we must have ξ(x1 , dx1 , s)ξ(x2 , dx2 , s) = 0.
(14.80)
For |x1 − x2 | > s the following result holds. P ROPOSITION 14.4.3 Consider a fluid state in the thermodynamic limit. Let x1 , . . . , xm+2 be particle
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CHAPTER 14
coordinates with x3 , . . . , xm+2 restricted to the intervals [x1 , x1 + s] ∪ [x2 , x2 + s]. Suppose in a particular configuration of the particles xi1 , . . . , xik ∈ [x1 , x1 + s] and xj1 , . . . , xjm+2−k ∈ [x2 , x2 + s], and define the correlation function ρm+2,2 by ρm+2,2 (x1 , . . . , xm+2 ) = ρm+2 (x1 , . . . , xm+2 ) − ρk (x1 , xi1 , . . . , xik )ρ2+m−k (x2 , xj1 , . . . , xj2 +m−k ). Then for |x1 − x2 | > s we have Cov ξ(x1 , dx1 , s)ξ(x2 , dx2 , s) = g(x1 , x2 )dx1 dx2 , where
(14.81)
∞ (−1)m ρm+2,2 (0, |x1 − x2 |, x3 , . . . , xm+2 )dx3 · · · dxm+2 g(x1 , x2 ) = m! Im m=0
with I = [0, s] ∪ [|x1 − x2 |, |x1 − x2 | + s]. Proof. Analogous to (14.77), for |x1 − x2 | > s we can write ξ(x1 , dx1 , s)ξ(x2 , dx2 , s) =
Z ∞ ” “X (−1)m ρm+2 (x1 , x2 , x3 , . . . , xm+2 )dx3 · · · dxm+2 dx1 dx2 , m! (I )m m=0
where I = [x1 , x1 + s] ∪ [x2 , x2 + s] for |x1 − x2 | < s. Introducing the correlation function ρm+2,2 this reads ξ(x1 , dx1 , s)ξ(x2 , dx2 , s) Z ∞ “X ” (−1)m = ρm+2,2 (x1 , x2 , x3 , . . . , xm+2 )dx3 · · · dxm+2 dx1 dx2 m! (I )m m=0 ∞ “X ” l Z (−1) + ρl+1 (x1 , x3 , . . . , xl+2 )dx3 · · · dxl+2 l! [x1 ,x1 +s]l l=0 ∞ ” “ X (−1)l Z ρl+1 (x2 , x3 , . . . , xl+2 )dx3 · · · dxl+2 dx1 dx2 . × l! [x2 ,x2 +s]l l=0 The final term is precisely ξ(x1 , dx1 , s)ξ(x2 , dx2 , s), so subtracting this term from both sides gives the l.h.s. of (14.81) (recall the definition of covariance given in Exercises 14.3 q.3). The r.h.s. is obtained after using translation invariance to shift the interval I to I.
The necessary ingredients are now available to compute the variance. P ROPOSITION 14.4.4 For L → ∞ we have Var(η(L, s)) ∼ Lb(s),
b(s) = |x|>s
g(s, x) dx − 2sFβ2 (s) + Fβ (s).
Proof. By definition, and making use of (14.79), DZ
L
“
Var(η(L, s)) = 0
DZ +
L
ξ(x1 , dx1 , s) − Fβ (s)dx1 “
”Z
L
“ ”E ξ(x2 , dx2 , s) − Fβ (s)dx2
0
ξ(x1 , dx1 , s) − Fβ (s)dx1
”2 E
x1 =x2
.
0
In the integrand of the final average, since (ξ(x1 , dx1 , s))2 = ξ(x1 , dx1 , s) we see that this term of the integrand is of order dx1 , while all other terms are of order (dx1 )2 and so do not contribute. Computing the average of ξ(x1 , dx1 , s)
677
FLUCTUATION FORMULAS AND UNIVERSAL BEHAVIOR OF CORRELATIONS
using (14.79) shows that this term equals LFβ (s). To compute the first average, we need to consider separately the cases |x1 − x2 | > s and |x1 − x2 | < s. For |x1 − x2 | > s the average of the terms in the integrand is precisely equal to Cov(ξ(x1 , dx1 , s)ξ(x2 , dx2 , s)) and so is given by (14.81). For |x1 − x2 | < s, (14.80) applies, so after using the second formula in (14.79), the average of the integrand of the first term in this region reduces to −(Fβ (s))2 dx1 dx2 . Thus we have Z Z Z Z L
L
dx1
Var(η(L, s)) = 0
L
dx2 g(x1 , x2 ) −
0 |x1 −x2 |>s
L
dx1 0
dx2 (Fβ (s))2 + LFβ (s),
0 |x1 −x2 |<s
from which the stated result follows.
One notes that no specific properties of the log-gas at β = 2 were used in the derivation of Proposition 14.4.4, rather, this result for Var(η(L, s)) is universal for general statistical mechanical systems in onedimension. 1. [37] Suppose e−βU in (14.67) is given by the eigenvalue√p.d.f. of Proposition 1.3.4 for the Gaussian ensemble and a(x) = x. Also, make the change of variables xj → 2N xj so that the support of the density is (−1, 1).
E XERCISES 14.4
(i) For β even complete the square to show that independent of N , Pˆ (k) = e−k P (u) =
“ β ”1/2 π
2
/4β
, and so
2
e−βu .
Use Carlson’s theorem (recall Proposition 4.1.4) to argue that this result must remain true for all β > 0. (ii) Show that the formula for P (u) in (i) is in agreement with (14.68) by evaluating (14.56) with a(x) = x. 2. Consider the one-component log-gas at β = 2 subject to a one-body potential with Boltzmann factor eu(θ) . Make the replacement a(θ) → u(θ) + ika(θ) in the Szeg¨o formula (14.71) to deduce that (14.73) again holds true. 3.
(i) Let e−βW [a] := eβ
P j
P iθ log |e j −eiθk |+ik N j=1 a(θj )
Show that for large N the maximum of the exponent occurs when Z β π cos((φ − θ)/2) Δρ(θ) dθ + ika (φ) = 0, 2 −π sin((φ − θ)/2)
.
φ ∈ [−π, π],
where ρ(1) (θ) = 1 + Δρ(θ)/N is the density of points {θj } and solve this equation for Δρ(θ) using the result of Proposition 14.3.4. (ii) Note that for large N „
Z Z ikN π ik π a(θ)dθ + Δρ(θ)a(θ)dθ 2π −π 2π −π « Z π Z π β + 2 dθ Δρ(θ) dφ Δρ(φ) log |eiθ − eiφ | . 8π −π −π
e−β(U [a]−U [0]) ∼ exp
Substitute for Δρ(θ) using the result of (i) and simplify the integrals making use of the second equality in (14.55) to reclaim the r.h.s. of (14.73) in the case β = 2. 4. [328] Generalize the definition (14.75) to read ˜ N (e−NV (x)+a(x) )[f ] := 1 D N!
Z
∞ −∞
dx1 e−NV (x1 ) · · ·
Z
∞ −∞
dxN e−NV (xN )
N Y j=1
ea(xj )
Y
(xk − xj )2 f.
1≤j
(i) Let φ(x) be differentiable for all x ∈ R and suppose φ (x) is bounded at infinity. Make the change of ˜ N with f = 1 and expand to leading variables xj = yj + φ(yj ), 0 < 1, in the above definition of D
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order in to conclude ˜ N (e−NV (x)+a(x) ) −N D
N hX
N i hX i ˜ N (e−NV (x)+a(x)) φ(xj )V (xj ) + D φ(xj )a (xj )
j=1
˜ N (e−NV (x)+a(x) ) +2D
h
j=1
X 1≤j
N hX i φ(xk ) − φ(xj ) i ˜ φ (xj ) = 0. + DN (e−NV (x)+a(x) ) xk − xj j=1
(ii) Show that this same equation can be obtained by integrating by parts ˜ N (e−NV (x)+a(x) ) D
N hX i d φ(xj ) dxj j=1
as in Aomoto’s method of Section 4.6. (iii) Choose φ(x) =
1 z−x
to deduce from the equation in (i) that
˜ N (e−NV (x)+a(x) ) ND
N N hX hX V (z) − V (xj ) i 1 i ˜ N (e−NV (x)+a(x) ) − N V (z)D z − xj z − xj j=1 j=1
N hX “ ˜ N (e−NV (x)+a(x) ) + D j=1
1 i”2 + R = 0, z − xj
where ˜ N (e−NV (x)+a(x) ) R = 2D
h
X 1≤j
h X 1 i”2 i “ 1 ˜ N (e−NV (x)+a(x)) − D (z − xk )(z − xj ) z − xj j=1 N
N N i hX hX a (xj ) i ˜ 1 ˜ N (e−NV (x)+a(x) ) + DN (e−NV (x)+a(x) ) . +D z − xj (z − xj )2 j=1 j=1
(iv) In general it is known that 1 N2
P „˜ ˜ N (e−NV (x)+a(x) )[PN u(xj )] ”2 « DN (e−NV (x)+a(x))[2 1≤j
as N → ∞ while lim
N→∞
˜ N (e−NV (x)+a(x) )[PN u(xj )] Z ∞ 1 D j=1 u(x)ρ(1) (x) dx, = ˜ N (e−NV (x)+a(x))[1] N D −∞
where ρ(1) (x) is the limiting normalized particle density for the statistical mechanical system with Boltz˜ N (e−NV (x)+a(x) )[1]. Use these facts to deduce from the equation in (iii) that mann factor implied by D Z ∞ Z ∞ “Z ∞ 1 ”2 V (z) − V (x) 1 ρ(1) (x) dx − V (z) ρ(1) (x) dx + ρ(1) (x) dx = 0, z−x −∞ −∞ z − x −∞ z − x and hence conclude from this that for z → support ρ(x), Z ∞ 1 ρ(1) (x) dx ∼ V (z). −∞ z − x Identify this as the fundamental equation in the complex electric field approach of Section 1.4.3 to the calculation of ρb (x).
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FLUCTUATION FORMULAS AND UNIVERSAL BEHAVIOR OF CORRELATIONS
5.
(i) Let ρW (x) be as in (7.26). Verify that for s > 1 Z 1 p p 1 log |s − x|ρW (x) dx = s2 − s s2 − 1 − log(2(s − s2 − 1)) − 2 −1 by first showing that both sides tend to log s + O(1/s) as s → ∞, and then verifying that the derivative is consistent with (1.132). (ii) By making use of the leading asymptotics in (14.76), which remains valid for general β > 0 (the leading asymptotics is independent of β), deduce that for s > 1 and large N [248] 2
e−Ns
N DY
E |s − xl |
l=1
OEN
2 (e−N x )
∼ exp
“
“ p ” p 1” − N s s2 − 1 + log(2(s − s2 − 1)) + + O(1) . 2
Conclude from this that for the GOE “ “ p ” ” p √ ρ(1),N+1 ( 2N s) ∼ exp − N s s2 − 1 + log(s − s2 − 1) + O(1) ,
(14.82)
valid for s > 1. 6. Let ρ˜(1) (x) = ρb (x)/N , where ρb (x) is the density (3.61) for the scaled chiral ensemble. Furthermore let √ 1 ˜(1) ( y) ρL (1) (y) = √ ρ 2 y and thus ρL (1) (y) is the normalized density in the Laguerre ensemble scaled so that m = N , λ → N λ, a = N α with N large. (i) Verify from (3.69) that for x ∈ / [c2 , d2 ] Z
d2
2 c2
ρL 2(α + 2) α “ α2 ”1/2 (1) (y) , dy = 1 − − 1 − + 2 x−y x x x
and deduce from this that Z d2 ˛u + x − 2 − α˛ ˛ α(α + u − x) − 2x ˛ ˛ ˛ ˛ ˛ + (2 + α) log ρL 2 ˛ ˛ ˛, (1) (y) log |x − y| dy = x − α − u + α log ˛ 2x2 2 c2 where u := ((x − α)2 − 4x)1/2 . (ii) Assuming the validity of the large N formula N DY
|x − λl |β
E
l=1
MEβ,N (λβN α/2 e−βN λ/2 )
Z “ ∼ exp βN
d2 c2
log |x − y|ρL (1) (y) dy
”
for x ∈ / [c2 , d2 ], read off from (i) that xβNα/2 e−βNx/2
N DY
|x − λl |β
l=1
∼ exp
“ βN “ 2
E MEβ,N (λβN α/2 e−βN λ/2 )
˛ ˛ ˛ ˛”” ˛ α(α + u − x) − 2x ˛ ˛u + x − 2 − α˛ − u + α log ˛ ˛ + (2 + α) log ˛ ˛ . 2x 2
Hence conclude that for x ∈ / [c2 , d2 ] ρL (1),N+1 (x) ∼ exp
“ βN “ 2
˛ α(α + u − x) − 2x ˛ ˛ u + x − 2 − α ˛”” ˛ ˛ ˛ ˛ −u+α log ˛ log ˛ . (14.83) ˛+(2+α) ˛ 2(α + 1)1/2 x 2(α + 1)1/2
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14.5 CHARGE AND POTENTIAL FLUCTUATIONS 14.5.1 Charge fluctuations In the bulk of the one-component log-gas, by setting a(x) = χ[−l/2,l/2] , where χ[−l/2,l/2] = 1 if x ∈ N [−l/2, l/2] and zero otherwise, we have that the linear statistic A := j=1 (a(xj ) − a) represents the fluctuation in the number of particles, or equivalently charge, in the interval [−l/2, l/2]. Since ∂a(x)/∂x = δ(x − l), (14.54) gives that the variance diverges in the infinite density limit. In the log-gas on a line with fixed density (which we take equal to unity, for convenience), it follows that the variance of A diverges as l → ∞. Here we want to determine the asymptotic form of this divergence. Rewriting the formula (14.38) for Var(A) by use of the Fourier transform gives ∞ 1 4l ∞ sin2 (y/2) ˆ ˆ Var(A) = |ˆ a(k)|2 S(k)dk = S(y/l)dy, (14.84) 2π −∞ π 0 y2 ˆ where S(k) denotes the structure function (14.7). The asymptotics can be determined by breaking the range of integration into the intervals [0,1], [1, l] and [l, ∞), and using (14.8) in the first two intervals. One finds [146] Var(A) ∼ where Bβ =
2 2 C+ π2 β π
0
1
2 log l + Bβ , π2 β
1 ˆ 2 y dy + S(y) − 2 y πβ π
(14.85) 1
∞
1 ˆ S(y)dy, y2
(14.86)
with C denoting Euler’s constant. In deriving (14.86) use has been made of the identity 1 ∞ (1 − cos y) cos y dy − dy = C. y y 0 1 ˆ For β = 1, 2 and 4, S(k) is known in terms of elementary functions according to (7.136), (7.4) and (7.95), ˆ ˜ respectively (from (14.7), for ρ = 1, S(k) = S(k)). The integrals in (14.86) can then be computed explicitly to give 1 2 2 B1 = 2 C + 2 1 + log 2π − , π π 4 1 1 B2 = 2 C + 2 1 + log 2π , π π 1 1 1 (14.87) B4 = 2 C + 2 1 + log 4π + . 2π 2π 4 We remark that the leading behavior exhibited by (14.85) is consistent with that obtained by substituting the r.h.s. of (14.45) for ρT(2) (x, x ) + ρ(1) (x)δ(x − x ) in (14.38) with a(x) = χ[−l/2,l/2] and ignoring the singular terms in the ensuing evaluation of the integral. This same heuristic procedure predicts the leading behavior 1 Var(A) ∼ log l (14.88) 2π 2 β for the charge fluctuation in the interval [0, l] of the log-gas on the half-line, a result which can be proved rigorously for the hard edge at β = 2 [498]. The fact that the fluctuations in the number of particles in an interval of size l is proportional to log l contrasts to the situation with an integrable (at infinity) pair potential ˆ V (x). In such systems S(0) = 0 and the formula (14.84) gives that the variance is proportional to l. Also of interest is the probability distribution (14.63) associated with charge fluctuations. We expect the
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FLUCTUATION FORMULAS AND UNIVERSAL BEHAVIOR OF CORRELATIONS
linear response relation (14.1) to again be exact in the “macroscopic” limit, which here is l → ∞, provided 2 N the ratio (a(x ) − a) /Var(A) is finite. For the infinite system with a finite density this implies j j=1 1/2 2 1/2 2 2 1 log l log l P u ∼ e−u /2 2 2 1/2 π β π β (2π)
as → ∞.
(14.89)
For β = 1, 2 and 4 this has been rigorously established by Costin and Lebowitz [121]. The proof in the case β = 2 has been extended in [495], [465] to apply to all determinantal point processes (recall Section 5.8). N P ROPOSITION 14.5.1 Let I be an interval in a determinantal point process such that A := j=1 χxj ∈I → √ ∞ as N → ∞. Then for N → ∞, (A − A)/ Var A has Gaussian distribution with zero mean and unit variance. Proof. Setting a(x) = χx∈I in the first result of Exercises 5.1 q.3(iii) allows (14.69) to be rewritten Z
∞
log
eikt P (t) dt =
−∞
∞ X 1 ik ¯n , (e − 1)n U n! n=1
Z ¯n := U
Z dxn ρT(n) (x1 , . . . , xn ).
dx1 · · · I
I
Putting ik = log(1 + x) and equating with the r.h.s. of (14.69) shows that for k > 2, ¯k = Mk − U
k−1 X
bkj Mj − (−1)k (k − 1)!M1
(14.90)
j=2
for some coefficients bkj . On the other hand, because we are considering a determinantal point process, ρT(n) has the form given in Proposition 5.1.2 and consequently Z Z ¯k = (−1)k−1 (k − 1)! dx1 · · · dxk KN (x1 − x2 )KN (x2 − x3 ) · · · KN (xk − x1 ) U I
I
k = (−1)k−1 (k − 1)!Tr KN ,
where KN is the integral operator on I with kernel KN (x, y) (cf. Exercises 9.3 q.1). Since M1 = Tr K this allows (14.90) to be rewritten k−1 X k )+ bkj Mj . (14.91) Mk = (−1)k (k − 1)!Tr(KN − KN j=2
But k Tr(KN − KN )=
k−1 X
j j+1 2 Tr(KN − KN ) < (k − 1)Tr(KN − KN ) = (k − 1)Var A,
(14.92)
j=1
where the inequality follows from the fact that the eigenvalues of KN must satisfy the inequality 0 ≤ λl < 1 (recall Section 9.6.1). Substituting (14.92) in (14.91) and proceeding inductively gives Mk = O(Var A) for k > 2. Since by assumption Var A → ∞ as N → ∞, it follows that 1 Mk → 0, (Var A)k
k ≥ 3,
which implies the stated result.
A proof of (14.89) for general β has been given in [355], by making use of the theory of Section 2.8. 14.5.2 Potential fluctuations Another linear statistic giving rise to a logarithmic divergence of the variance is the potential at the origin N a(x) = − log |x|. First, note from (14.54) and (14.3) that the variance of A = − j=1 log xj diverges in
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the infinite density limit. In the case of the semi-infinite or infinite line this divergence must persist at finite density due to the equivalence with the infinite density limit for the logarithmic potential. We should therefore consider the asymptotic form of the divergence in the variance of A in a finite system with fixed density as a function of the size of the system. Suppose in particular N the log-gas is confined to a circle of circumference length L, and consider the linear statistic A = − j=1 log |2 sin(πxj /L)|, −L/2 < xj ≤ L/2, which corresponds to the potential at the origin. Using (14.38) and (14.58) we have Var(A) =
∞ L 1 rn , 4 n=−∞ n2 n=0
∞ where the Fourier coefficients rn are specified by ρT(2) (x − x ) + δ(x − x ) = L1 n=−∞ rn e2πi(x−x )n/L . But analogous to (14.8) we expect rn ∼ (2π/L)(|n|/πβ) for 0 ≤ |n| ≤ O(L), which gives Var(A) ∼
1 log L β
as L → ∞.
(14.93)
Furthermore, analogous to (14.89) we expect the corresponding probability distribution to be asymptotically Gaussian, 1 1/2 1 1/2 2 1 P u ∼ e−u /2 as L → ∞. (14.94) log L log L β β (2π)1/2 A statistic closely related to the potential at the origin is the fluctuation of the average (angular) displaceN π ment from its mean, A = L l=1 xl , or equivalently the fluctuation in the scaled dipole moment. Thus comparing the Fourier expansions − log |2 sin
∞ cos 2πmx/L πx |= L m m=1
∞ sin 2πmx/L πx = L m m=1
and
(14.95)
we see that the variance will again exhibit the behavior (14.93), and the probability distribution the behavior (14.94). Whereas the potential fluctuations are infinite in the infinite system at finite density, the fluctuation of the potential difference A(x) := −
N
log |x − xj | −
j=1
N
log |xj |
(14.96)
j=1
is a well-behaved quantity. Thus by use of the formula (14.3) in (14.38) we have ∞ sin2 (kx/2) ˆ Var(A(x)) = 2π S(k)dk, k2 −∞
(14.97)
which, on identifying ρx with ρl, is precisely π 2 times (14.84). This quantity therefore diverges in the infinite density limit, or equivalently as x → ∞, exhibiting the behavior Var(A(x)) ∼
x→∞
2 log x + π 2 Bβ , β
and analogous to (14.89) we expect 2 1/2 2 1/2 2 1 P u ∼ e−u /2 β β (2π)1/2
(14.98)
as x → ∞.
(14.99)
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FLUCTUATION FORMULAS AND UNIVERSAL BEHAVIOR OF CORRELATIONS
14.5.3 Fisher-Hartwig asymptotics The prediction (14.94) for the probability distribution of the linear statistics in the log-gas on a circle with π a(x) = − log 2| sin πx/L| and a(x) = L x can be proved [37] at β = 2 using a generalization of the Szeg¨o asymptotic formula (14.71). This generalization was conjectured by Fisher and Hartwig [176], and has subsequently been proved to varying degrees of generality in relation to the number of singularities R and constraints on {ar , br }; see, e.g., [43]. P ROPOSITION 14.5.2 Consider the Toeplitz determinant in Proposition 14.4.1 with a(θ) = g(θ) − i
R
br arg ei(θr +π−θ) +
r=1
= g(θ) +
R
R
ar log |2 − 2 cos(θ − θr )|
r=1
(ar + br ) log(1 + ei(θ−θr −π) ) + (ar − br ) log(1 + ei(θr +π−θ) ) ,
r=1
∞ 2 ipθ , where ∞ where −π < arg z ≤ π, and assume g(θ) = p=−∞ gp e p=−∞ |p||gp | < ∞. Then for |Re(ar )| < 12 , |Re(br )| < 12 or Re(ar ) > − 12 , Re(br ) = 0 (and possibly in other regions; although see Exercises 14.5 q.2) DN [ea(θ) ] ∼ eg0 N e
PR
2 2 r=1 (ar −br ) log N
E,
(14.100) ∞ where E is independent of N . To specify E, write g(θ) − g0 = g+ (θ) + g− (θ), where g+ (θ) = p=1 gp eipθ −1 and g− (θ) = p=−∞ gp eipθ . Then E=e
P∞
k=1
kgk g−k
R
e−(ar +br )g− (θr ) e−(ar −br )g+ (θr )
r=1
×
1≤r=s≤R
R −(ar +br )(as −bs ) G(1 + ar + br )G(1 + ar − br ) , 1 − ei(θs −θr ) G(1 + 2ar ) r=1
where G is the Barnes G-function. The functional form of (14.100), together with the structure of the constant E, can be anticipated by suitably truncating the logarithms in the definition of a(θ), and applying Szeg¨o’s asymptotic formula (14.71). The necessary working is given in Exercises 14.5 q.3. N Consider now the application of Proposition 14.5.2 to the p.d.f. of A = − j=1 log 2| sin πxj /L| with β = 2. From the first line of (14.67) and the definition of DN in (14.71) we have ik/2 . Pˆ (k) = DN (2 − 2 cos θ) The Fisher-Hartwig formula (14.100) with g(θ) = 0, R = 1, b1 = 0, a1 = ik/2 and θ1 = 0 gives that 2 Pˆ (k) ∼ e−(k /4) log N ,
(14.101)
N →∞
which is equivalent to (14.94) with β = 2, L ∼ N . Similarly, with A :=
π L
N l=1
xl we have that at β = 2
Pˆ (k) = DN [eikθ/2 ]. The Fisher-Hartwig formula (14.100), this time with g(θ) = 0, R = 1, a1 = 0, b1 = ik/2 and θ1 = −π, again implies the asymptotic behavior (14.101), as expected. For the log-gas on a circle at general rational β, to prove the Gaussian behavior (14.101) for the linear statistics a(x) = − log 2| sin πx/L| and a(x) = πx/L it suffices to compute the asymptotic form of the
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average N
eibθl |2 + 2 cos θl |a
CβEN
l=1
,
(14.102)
where CβEN refers to the circular β-ensemble as specified by the p.d.f. (2.20). In fact (14.102) is given in terms of gamma functions according to the Morris integral (4.4). In terms of the quantity fn (α, c) defined by (4.187) we see that N
eibθl |2 + 2 cos θl |a
CβEN
l=1
=
fN (2a, c) , fN (a + b, c)fN (a − b, c)
c := β/2.
(14.103)
Proceeding now as in the derivation of (4.188) provides the sought asymptotic form for β rational. P ROPOSITION 14.5.3 Let β/2 = s/r, s, r ∈ Z≥0 . We have N
eibθl |2 + 2 cos θl |a
l=1
∼ N (2/β)(a
CβEN N →∞
2
−b2 )
(β)
Aa,b ,
(14.104)
where Aa,b := r−(2/β)(a (β)
2
−b2 )
r−1 s−1
G((a + b)/s + ν/r − p/s + 1)G((a − b)/s + ν/r − p/s + 1) . (14.105) G(2a/s + ν/r − p/s + 1)G(ν/r − p/s + 1) ν=0 p=0
It follows that N
eikθl /2
CβEN
l=1
∼
N
|2 − 2 cos θl |ik/2
l=1
CβEN
∼ e−(k
2
/2β) log N
,
thus establishing the Gaussian behavior (14.101) for the respective linear statistics [37]. The result (14.104) can also be used to formulate a β-generalization of the Fisher-Hartwig formula (14.100). Thus for general rational β, in the notation of Propositions 14.5.2 and 14.5.3 one conjectures [201] N
ea(θl )
l=1
CβEN
∼ eg0 N e(2/β)
PR
2 2 r=1 (ar −br ) log N
N →∞
E (β) ,
where E (β) = e(2/β)
P∞
k=1
kgk g−k
R
e−(2/β)(ar +br )g− (θr ) e−(2/β)(ar −br )g+ (θr )
r=1
×
1 − ei(θs −θr )
R −(2/β)(ar +br )(as −bs )
1≤r=s≤R
(β)
Aaj ,bj .
j=1
14.5.4 Value distribution of characteristic polynomials for U (N ) Closely related tothe results of Section 14.5.2 and 14.5.3 are statistical properties of the characteristic polyN nomial Λ(z) := l=1 (λl − z) for the classical group U (N ). Consider first the distribution of the values of log Λ(z) for z on the unit circle. Because of rotational invariance this point can be chosen to be z = −1. We seek the joint distribution of the real and imaginary parts, P (s, t) = δ(s − Re log Λ(−1))δ(t − Im log Λ(−1)) . (14.106) U(N )
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FLUCTUATION FORMULAS AND UNIVERSAL BEHAVIOR OF CORRELATIONS
With the eigenvalues given by {eiθj }j=1,...,N , −π < θj ≤ π, N
Re log Λ(−1) =
Im log Λ(−1) =
j=1
and so
Pˆ (k, l) :=
∞
−∞
ds eiks
∞
−∞
1 θj 2 j=1 N
log |1 + eiθj |,
dt eilt P (s, t) =
N
eilθj /2 |1 + 2 cos θj |ik/2
j=1
U(N )
.
This is precisely the average (14.102) in the case β = 2 with b = l/2, a = ik/2, and so has the same evaluation as specified by (14.103). According to (14.104), for N → ∞ 2 2 Pˆ (k, l) ∼ e− log N (k +l )/4
(for a derivation of this result using the theory of Section 2.8, see [473]), so the following result holds true [354]. P ROPOSITION 14.5.4 Introduce the scaled logarithm of the characteristic polynomial, 2 1/2 ˜ log Λ(z), log Λ(z) = log N ˜ and denote the joint distribution of the real and imaginary parts of log Λ(−1) by P˜ (s, t). We have 1 −(s2 +t2 )/2 e , lim P˜ (s, t) = 2π ˜ and thus the limiting distribution of the values of log Λ(−1) is the complex Gaussian N[0, 1] + iN[0, 1]. N →∞
The Keating-Snaith hypothesis [354] relates this result to the value distribution of the Riemann zeta function on the critical line. The former is an extension of the Montgomery-Odlyzko law (recall Section 7.5.1), and relates the value distribution of ζ( 12 + it) to the value distribution of Λ(z). In particular it is hypothesized that for large t the value distribution of log ζ( 12 + it) is well described by the value distribution of log Λ(z) for N × N unitary random matrices with N = log t/2π (this ensures that the mean spacing between zeros and between eigenvalues is asymptotically equal, but on the other hand is not the identification used in (7.89)). Evidence for this hypothesis is a result of Selberg [516] which gives that for any rectangle B ∈ C " 1 2 2 log ζ(1/2 + it) 1 = t : T ≤ t ≤ 2T, lim e−(x +y )/2 dxdy, (14.107) ∈ B T →∞ T 2 1 B log log T 2 thus exhibiting the complex Gaussian behavior for the logarithm of the characteristic polynomials seen in Proposition 14.5.4 for U (N ), with the identification N = log T.
(14.108)
The value distribution of |Λ(z)| itself is also of interest. Here the corresponding Fourier transform does not lead to a tractable integral. Instead one observes that for p(s) a distribution with support s > 0, knowledge of the Mellin transform (complex moments) ∞ m(λ) := sλ−1 p(s) ds 0
as a function in the complex plane gives 1 p(s) = 2πi
c+i∞ c−i∞
s−x m(x) dx
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CHAPTER 14
via the inverse Mellin transform. Now, for random matrices from U (N ), and with p(s) denoting the distribution of |Λ(−1)|2 , m(a + 1) =
N
|2 + 2 cos θl |a
l=1
U(N )
,
(14.109)
and this in turn has an explicit product of gamma function evaluation which follows from (14.103) (for a discussion of computing the corresponding Mellin transform, see [471]). The asymptotic formula (14.104) with β = 2, r = s = 1, b = 0 shows that for N → ∞ N −a m(a + 1) ∼ 2
G2 (a + 1) . G(2a + 1)
(14.110)
An application to zeta function theory of this result has been given by Keating and Snaith [354], in keeping with their hypothesis relating to (14.107). Thus, after recalling the identification (14.108), one anticipates that the U (N ) result (14.110) should relate to the large T value of 2a 1 1 T 1 + it dt. (14.111) ζ 2 a 2 (log T ) T 0 In zeta function theory this was conjectured to have the structure f (a)A(a), where A(a) is the number theoretic quantity 2 ∞ 1 λ Γ(a + m) 2 −m 1− A(a) = p p m!Γ(a) m=0 primes p
and 42 24024 1 , f (3) = , f (4) = 12 9! 16! (the first two such values are known rigorously), with the other values of f (a) unknown. Indeed, these values are precisely those given by the r.h.s. of (14.110). Thus one is led to interpret this term as due to U (N ) like properties of the Riemann zeros, leading to the conjecture that for general a f (1) = 1,
f (2) =
f (a) = E XERCISES 14.5 N DY
1.
G2 (a + 1) . G(2a + 1)
(i) Note from the results of Exercises 5.1 q.3(i) and (ii) that
eibθl |2 + 2 cos θl |a (1 + xe−iθl )
l=1
E CUEN
=
MN (a + b, a − b, 1) 2 F1 (−N, a + b; b − a − N ; x). MN (0, 0, 1)
(ii) Deduce from this the limiting behavior E DY MN (0, 0, 1) eibθl |2 + 2 cos θl |a (1 + xe−iθl ) = (1 + x)−(a+b) MN (a + b, a − b, 1) l=1 CUEN N
lim
N→∞
and show that this is consistent with the prediction of the Fisher-Hartwig formula (14.100). 2.
(i) Note that with a(θ) of Proposition 14.5.2 specified by g(θ) = 0, R = 2, θ1 = φ1 , θ2 = φ2 , a1 = b1 = 12 , a2 = −b2 = 12 we have DN [ea(θ) ] = eiN(φ2 −φ1 )
N E DY (eiθl − eiφ1 )(e−iθl − e−iφ2 ) l=1
CUEN
.
687
FLUCTUATION FORMULAS AND UNIVERSAL BEHAVIOR OF CORRELATIONS
(ii) Use (5.89) and (5.87) to deduce from this that DN [ea(θ) ] = eiN(φ2 −φ1 )/2
sin((N + 1)(φ2 − φ1 )/2)) . sin((φ2 − φ1 )/2)
As this does not exhibit the N → ∞ behavior (14.100), conclude that the region |Re(ar )| < 12 , |Re(br )| < 1 sufficient for the validity of the latter cannot in general be enlarged in all parameters. 2 3. [332] The objective of this exercise is to use the Szeg¨o asymptotic formula (14.71) to anticipate features of the Fisher-Hartwig asymptotic formula (14.100). (i) In the second expression of Proposition 14.5.2 for a(θ), replace the logarithms by their corresponding power series truncated at the N th term, and read off that the Fourier coefficients of a(θ) are then given by 8 PR −iθr p 1 > χp≤N , p ≥ 1, < gp − p r=1 (ar + br )e p = 0, g0 , ap = > : gp − 1 PR (ar − br )eiθr p χ|p|≤N , p ≤ −1. r=1 |p| (ii) Show from (i) that for N large ∞ X
pap a−p ∼
p=1
∞ X
pgp g−p −
p=1
(ar − br )g+ (θr ) −
r=1 R X
−
R X
R X
(ar + br )g− (θr )
r=1
(ar + br )(ar − br ) log(1 − ei(θr −θr ) ) + log N
r,r =1 r=r
R X
(a2r − b2r ).
r=1
Substitute this in (14.71) to reproduce (14.100) up to the product over the Barnes G-function in E. 4. [368] The aim of this exercise is to determine the distribution of the form factor for the log-gas on a circle. For this purpose, with k = p/N , p ∈ Z consider N DY
ei(u/
E √ √ N ) cos Nkθl +i(v/ N ) sin Nkθl CβEN
l=1
.
(i) Use the cumulant expansion (14.69), (14.70) to deduce that the logarithm of this average has the large N form Z “ 1 ” u2 + v 2 π T 1− . ρ(2) (θ, 0) cos N kθ dθ + O √ 4 N −π (ii) Conclude that the joint distribution of (A, B) =
N N “ 1 X ” 1 X √ cos N kθj , √ sin N kθj N j=1 N j=1
is for large N the Gaussian 2
e−(A
+B 2 )/2σ 2
,
σ2 =
(iii) In the notation of (ii) note that A2 + B 2 =
1 2
Z
π −π
ρT(2) (θ, 0) cos N kθ dθ.
˛ ˛ N 1 ˛ X ikNθj ˛2 e ˛ ˛ N j=1
so that for k = 0
Z ˆ k) = (A2 + B 2 )CβEN = S(N
π −π
ρT(2) (θ, 0) cos N kθ dθ.
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Conclude from the result of (ii) that the distribution of s = A2 + B 2 is, for large N , the exponential ˆ k). distribution with mean S(N 5. [117, 235] With U ∈ U (N ), eigenvalues {eiθl }l=1,...,N , introduce the characteristic polynomial as above (14.106), and in terms of this define PN Z(s) = e−πiN/2 e−i n=1 θn /2 z −N/2 Λ(z). (note that Z(s) is real for |s| = 1). Results from [117] give that for k ∈ Z+ N −k
2
−2k
|Z (1)|2k U ∈U (N)
∼ ˜bk ,
(14.112)
N→∞
where, with [xj ]f (x) denoting the coefficient of xj in the power series expansion of f (x), “ ” ˜bk = (−1)k(k+1)/2 (2k)![x2k ] e−x/2 x−k2 /2 det[Iα+β−1 (2√x)]α,β=1,...,k (cf. (14.110)). (i) Interchange row β with row k − β + 1 in the above determinant, then compare with (8.97) to conclude that “ ” k ˜bk = (−1) (2k)![x2k ] ex/2 E ˜ hard(k) ((0, 4x); ξ = 1; k) , 2 A(k, k)
A(a, μ) := a!
a Y (j + μ − 1)! . j! j=1
(ii) Make use of (8.88) and Proposition 8.3.3 to conclude from (i) that “ Z 4x k ds ” ˜bk = (−1) (2k)![x2k ] exp − (η(s) + k2 ) A(k, k) s 0 where η(s) satisfies the differential equation “ 1” k2 (η − sη ) − 2 = 0 (sη )2 + 4 (η )2 − 64 4 subject to the requirement that it is even in s with η(0) = −k2 .
14.6 ASYMPTOTIC PROPERTIES OF Eβ (n; J) AND Pβ (n; J) 14.6.1 Large s behavior of Eβ (0; s) In (9.90) results of a Coulomb gas argument predicting that for an eigenvalue free region J, log E2 (0; J) is to leading order in |J| proportional to |J|3 , |J| and |J|2 at the soft edge, hard edge and in the bulk, respectively, were required. Here these results will be derived by combining macroscopic electrostatics with a scaling argument [189]. One considers the setting of the one-component log-gas confined to the half-line x > 0 with background charge density given by the power law ρb (x) = −Axμ . The basic hypothesis is that for large s Eβ (0; s) ∼ e−βE(s) ,
(14.113)
where E(s) is the electrostatic energy due to excluding the mobile positive charge from the interval (0, s). Note that e−βE(s) is the Boltzmann factor for the configuration with energy E(s). Now, let ρ(1) (x) denote the charge density of the positive charge (ρ(x) = 0 for x ∈ [0, s]), and let ρˆ(x) denote the total charge, ρˆ(x) = ρ(x) + ρb (x). The latter is constrained by the condition of global charge neutrality ∞ ρˆ(x) dx = 0. (14.114) 0
FLUCTUATION FORMULAS AND UNIVERSAL BEHAVIOR OF CORRELATIONS
With φ(x) = −
∞
−∞
ρˆ(y) log |x − y| dy one then has 1 ∞ ρˆ(x)φ(x) dx. E(s) = 2 0
689
(14.115)
For the region (s, ∞), where there are mobile charges, the potential is a constant which can be taken to be zero. Noting that for x ∈ [0, s], ρˆ(x) = ρb (x) = −Axμ , and making use of (14.114) we see that this latter condition is consistent with the homogeniety property ρˆ(sx) = sμ ρˆ(x).
(14.116)
Substituting in (14.115) gives E(s) = Cs2u+2 ,
C = E(1)
(14.117)
and hence (14.113) gives the prediction Eβ (0; s) ∼ e−βCs
2μ+2
1 2,
.
(14.118)
− 12
and 0 so as to give the known asymptotic form of the density at the soft edge, hard edge With μ = and in the bulk, respectively, found in Chapter 7, this gives the corresponding leading behavior of E2 (0; s) as quoted in (9.90). We remark that the stochastic differential equation characterizations (13.186) and (13.180) of the β-ensembles at the soft edge and the bulk can be used to prove (14.118) in those cases, and furthermore to give the value of C [463], [524], [462]. As already noted, this method can also be used to deduce higher order terms, giving for example the large s expansion in the bulk (9.99). 14.6.2 Large s behavior of Eβbulk (n; s) In the infinite log-gas with a constant background charge density −ρ, the leading large s asymptotics of Eβbulk (n; s)—the probability that an interval of length s contains exactly n eigenvalues—can be determined using an extension of the macroscopic electrostatics argument used above [152], [178]. As in (14.113) the basic hypothesis is that for large s Eβbulk (n; s) ∼ e−βδF ,
(14.119)
where here δF is the change in energy caused by changing the particle density so that the interval of length s contains n particles. Analogous to the hypothesis used in Section 14.2, the change in energy δF is taken to consist of two parts — an electrostatic energy V1 and a free energy V2 . These are calculated from the one-body density ρ(1) (x) according to 1 ∞ V1 = (ρ(1) (x) − ρ)φ(x) dx, (14.120) 2 −∞ ∞ where φ(x) = − −∞ (ρ(1) (y) − ρ) log |x − y| dy, and ∞ ! ρ(1) (x) fβ [ρ(1) (x)] − fβ [ρ] dx. (14.121) V2 = −∞
Since, according to Proposition 4.8.1, the free energy per particle fβ is such that fβ [ρ(1) (x)] − fβ [ρ] = 1 1 β − 2 log(ρ(1) (x)/ρ), we have 1 1 ∞ − V2 = ρ(1) (x) log(ρ(1) (x)/ρ) dx. (14.122) β 2 −∞ The potential φ(x) and density ρ(1) (x) are calculated via two-dimensional macroscopic electrostatics. For
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n < ρs we suppose the n particles are confined to an interval (−b, b) and within that interval the system behaves like a conductor so φ(x) = −c (constant). The quantities b and c are to be calculated. The remaining particles are confined to an open-ended region (−∞, −t) and (t, ∞), where 2t = s. These regions are conductors with φ(x) = 0. The remaining intervals, (−t, −b) and (b, t), contain no mobile particles and so behave as an insulator on which φ will vary continuously between 0 and −c. The density ρ(1) (x) is further constrained by the conditions b ∞ ρ(1) (x) dx = n, (ρ(1) (x) − ρ)dx = 0. (14.123) −b
−∞
Substituting the constraints on φ(x) in (14.120), together with the condition ρ(1) (x) = 0 for b < |x| < t and the first condition in (14.123), then integrating by parts gives t cn dφ(x) V1 = − + ρ dx. (14.124) x 2 dx b To determine dφ/dx we introduce the complex electric field E(z) = −∂φ(z)/∂x + i∂φ(z)/∂y, so that dφ/dx = −ReE(x) (recall Section 1.4.3). Now E(z) is required to be an analytic function of z in the upper half-plane (its value in the lower half-plane is given by symmetry), with real part which vanishes on the conducting regions of the real line. Furthermore, E(z) must have branch points at z = ±t, ±b, it must vanish as |z| → ∞ (since there is no net charge) and as t → ∞ with b = 0 it must equal πiρ (since then the real line is an insulator with uniform charge density ρ and so φ(z) = πy; recall Section 2.7.1). The unique function with these properties is 2 1/2 z − b2 E(z) = πρ i − 2 , t − z2 where the square root is chosen to be positive real on the real axis between (−t, −b) and (b, t). Taking the real part and changing sign gives that for x ∈ (b, t) 2 1/2 dφ x − b2 = πρ 2 , (14.125) dx t − x2 while the density in the interval (−b, b) is given by 1/2 2 1 ! + b − x2 − E (x) − E (x) = ρ 2 ρ(1) (x) = − . 2πi t − x2
(14.126)
Note that b is specified by the first equation in (14.123) with the substitution of (14.126). An integral identity in [270] allows the resulting equation to be rewritten (14.127) n = 2ρt E(k ) − k 2 K(k ) , where K and E are the complete elliptic integrals of the first and second kinds and k 2 = 1 − k with k = b/t. The explicit formula (14.125) allows the evaluation of (14.124), 2
cn π 2 ρ2 2 + (t − b2 ). 2 4 Also, integrating (14.125) from b to t and recalling that φ(t) = 0, φ(b) = −c, gives 1/2 t 2 x − b2 2 c = πρ dx = πρt E (k ) − k K (k ) . t2 − x2 b V1 = −
(14.128)
(14.129)
FLUCTUATION FORMULAS AND UNIVERSAL BEHAVIOR OF CORRELATIONS
691
The integral in (14.121) defining V2 can also be calculated exactly [152]. P ROPOSITION 14.6.1 One has V2 =
1
−
β
1 c, 2
where c = −φ(b) = −φ(x), x ∈ [−b, b]. Proof. Define a function F (z), analytic in the upper half-plane, by F (z) = −
i i 1 (E(z) − E(¯ z )) + (E(z) + E(z)) = ρ + E(z) 2πi 2π π
). Since log(F (z)/ρ) is also analytic in the upper half-plane, and decays of order (on the real axis F (x) = ρ(1) (x) − πi dφ dx 1/|z|2 as |z| → ∞, the residue theorem gives Z ∞ Z ∞ “ ““ i dφ ” i dφ ” ” F (x) log(F (x)/ρ)dx := log ρ(1) (x) − /ρ dx = 0. (14.130) ρ(1) (x) − π dx π dx −∞ −∞ Now for x ∈ / (−t, −b), (b, t) we know that dφ/dx = 0 while for x in these intervals ρ(1) (x) = 0, dφ/dx is positive and ` ´ ` ´ log ρ(1) (x) − πi dφ = log π1 dφ − iπ/2. Taking the real part of (14.130) we therefore have dx dx Z
∞
1 ρ(1) (x) log(ρ(1) (x)/ρ)dx = 2 −∞
„Z
Z t«
−b
+ −t
b
1 dφ dx = − (φ(−b) + φ(b)), dx 2
and the stated result follows upon recalling (14.122).
We see that V1 and V2 are completely specified by (14.128), (14.129) and Proposition 14.6.1. Hence Eβbulk (n; s) is completely specified — all that remains is to calculate the large s behavior of log Eβ (n; s). For this purpose, note that substituting (14.126) in the first equation of (14.123) and expanding for t b shows 2nt 1/2 b∼ . (14.131) πρ Since k = b/t, we see that (k )2 ∼ (2n/πρt). From the known asymptotic expansions for E (k ) and K (k ) as k → 0, (14.129) then gives that 8πρt n log +1 . c ∼ πρt − 2 n Substituting this result in the expressions (14.128) and Proposition 14.6.1 for V1 and V2 , then substituting the sum for δF in (14.119) shows β n β βn 8πρt (πρt)2 + βn + − 1 πρt + 1− − log + 1 , (14.132) log Eβbulk (n; 2t) t,n→∞ ∼ −β 4 2 2 2 2 n tn and this in turn implies Eβbulk (n; 2t) Eβbulk (0; 2t)
∼ c˜β,N t,n→∞ tn
eβnπρt (πρt)βn2 /4+(β/2−1)n/2
(14.133)
for some c˜β,N . We recognize this latter form as precisely the conjectured asymptotic result (9.103) for the same ratio with n fixed, and we recognize (14.132) with n = 0 as agreeing with the first two terms of (9.99). Also of interest is the case 0 ρs − n ρs because of its relationship to charge fluctuations, and in particular the formula (14.89). This case is considered in Exercises 14.6 q.1.
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14.6.3 Eβ,N (0; J) and pβ,N (0; J) for large N, |J| Log-gas arguments can also be used to study the asymptotics of gap probabilities and spacing distributions when N is large but finite. Here one is typically concerned with large deviations, when the size of the gap or spacing (measured in appropriate units) depends on N . Moreover, a rigorous justification of the log-gas heuristics can in some circumstances be given [329]. The simplest situation of this type is the gap probability Eβ,N (0; (−α, α)) in the finite N circular βensemble. Here 0 < α < π is an angle on the circle, and so on average the interval (−α, α) would contain αN/π eigenvalues if not constrained to be a gap. According to the hypothesis (14.113) 2π−α β 2π−α Eβ,N (0; (−α, α)) ∼ exp dθ1 ρˆ(θ1 ) dθ2 ρˆ(θ2 ) log |eiθ1 − eiθ2 | , (14.134) 2 α α where ρˆ(θ) := ρ(1) (θ) − N/2π. An electrostatics evaluation of ρˆ(θ) and the corresponding electrostatic energy is known [147], but we can in fact bypass such a calculation. The reason is that through the Toeplitz determinant asymptotic formula (9.123) we have knowledge of the electrostatic energy in the case β = 2. Since the electrostatic energy is independent of β this gives β α Eβ,N (0; (−α, α)) ∼ exp N 2 log cos . (14.135) 2 2 The next situation to be considered is the right tail large deviation form of the distribution of the largest eigenvalue in the Gaussian β-ensemble, scaled so that the leading order support is the interval (−1, 1). One sees from (8.73) and (8.86) that in the large s asymptotic regime pβ,N (0; (s, ∞)) ∼ ρ(1),N (s).
(14.136)
With s > 1, and thus outside of the interval of leading support, the density is known from (14.82) in the case β = 1, and so we read off that for general β > 0 (14.137) pβ,N (0; (s, ∞)) ∼ exp − βN s s2 − 1 + log(s − s2 − 1) . In the case of the scaled Laguerre ensemble the corresponding asymptotic form of the distribution of the largest eigenvalue outside the leading support is similarly given by (14.83). Finally we consider the left tail large deviation form of the distribution of the largest eigenvalue in the Gaussian β-ensemble, again scaled so that the leading order support is the interval (−1, 1). This is calculated from the log-gas formula pβ,N (0; (s, ∞)) ∼ e−β(U (s)−U(1)) ,
s < 1,
U (s) is the leading order in N portion of the total potential energy for the scaled Gaussian β-ensemble constrained so that the eigenvalues are restricted to be less than s. The working of Exercises 1.4 q.4 gives that such a constraint corresponds to a background density (1.81), and we see from (1.78) that to leading order in N √2N s 1 CN U (s) = , x2 ρb (x) dx − 4 √2N (s−l) 2 where C is given by (1.80). Computing the integral making use of (4.2) then gives that for large N and s < 1 [124] √ 2 s4 5 s + s2 + 3 1 3 1 2 2s 2 2 pβ,N (0; (s, ∞)) ∼ exp − βN − − s 3 + s − s 3 + s − log . 3 27 18 27 2 3 (14.138)
FLUCTUATION FORMULAS AND UNIVERSAL BEHAVIOR OF CORRELATIONS
693
14.6.4 Covariance and variance of spacing distributions in the bulk For a one-dimensional system, the covariance of particular spacing configurations can be related to the variance of related configurations [100]. P ROPOSITION 14.6.2 In the bulk of a one-dimensional system, choose a particular particle at position xi , and denote the positions of the successive particles to the right by xi+1 , xi+2 , . . . . Then, independent of i, pβ (n; s) = δ(si (n) − s),
si (n) := xi+n+1 − xi
(14.139)
(si (n) will be referred to as the nth order spacing), while Cov(si (n), si+r+1 (n )) = Var(si (|r − n| − 1)) + Var(si (r + n + 1)) −Var(si (|r + n − n + 1| − 1)) − Var(si (r))
(14.140)
with the convention that si (−1) = 0. Proof. The first formula is essentially the definition of pβ (n; s). For the second formula, we note from the identity (z1 − z2 )(z3 − z4 ) =
” 1“ (z1 − z4 )2 + (z2 − z3 )2 − (z1 − z3 )2 − (z2 − z4 )2 2
that in general Cov(z1 − z2 , z3 − z4 ) =
” 1“ Var(z1 − z4 ) + Var(z2 − z3 ) − Var(z1 − z3 ) − Var(z2 − z4 ) . 2
The covariance formula (14.140) now follows from the difference formula (14.139) for si (n).
A noteworthy special case of (14.140) is n = n = 0, which gives the covariance of two nearest neighbor spacings (to the right) with r particles in between as 1 Cov(si (0), si+r+1 (0)) = Var(si (r + 1)) − 2Var(si (r)) + Var(si (r − 1)) (14.141) 2 (an alternative derivation of this result is given in Exercises 14.6 q.2). Another special case of interest is r = n = n . This represents the covariance of successive nth order spacings, and (14.140) then reads 1 Var(si (2n + 1)) − 2Var(si (n)) . Cov(si (n), si+n+1 (n)) = (14.142) 2 To make use of the above formulas the value of Var(si (n)) is required. For the random matrix couplings β = 1, 2, 4, and for small values of n, these can be read off from Tables 8.13, 8.14 and the inter-relation (8.161). The corresponding covariances are listed in Table 14.1. In general, since si (n) is not a linear statistic, Var(si (n)) is not easily accessible. However, physically one would expect Var(si (n)) to be asymptotically equal to Var(χ[0,n+1] ) (here we are assuming unit density), where the latter is the variance in the number of particles in an interval of length n + 1. Using the explicit asymptotic formula (14.85), we thus expect for the one-component log-gas that Var(si (n)) ∼
n→∞
2 log n + O(1). π2 β
(14.143)
Substituting in (14.141) then gives 1 1 , π 2 β r2
(14.144)
1 log n. π2 β
(14.145)
Cov(si (0), si+r+1 (0)) ∼ − r→∞
while substituting in (14.142) shows Cov(si (n), si+n+1 (n)) ∼ − n→∞
694
n=0 n=1 n=2 n=3 n=4 n=5 n=6 n=7 n=8
CHAPTER 14
An (1)
An (2)
An (4)
Bn (1)
Bn (2)
−0.077333811 −0.024900730 −0.011437207 −0.006455611 −0.004124830 −0.002857751 −0.002095049 −0.001601116 −0.0012631931
−0.05550500 −0.013895003 −0.005959174 −0.003288239 −0.0020817292 −0.001435997 −0.0010503631 −0.0008017114 −0.0006320362
−0.034643119 −0.007118315 −0.002983845 −0.001640118
−0.077333811 −0.138572479 −0.1784829481 −0.20745438515 −0.23007507345
−0.05550500 −0.089255975 −0.10983253094 −0.12449482474 −0.13586646270
Table 14.1 Tabulation of An (β) := Cov(si (0), si+n+1 (0)), Bn (β) := Cov(si (n), si+n+1 (n)) for small values of n and the random matrix couplings β = 1, 2 and 4. Note that in general A0 (β) = B0 (β).
Even with r = 3 the formula (14.144) accurately approximates the exact value given in Table 14.1. More generally we would expect the full distribution of si (n) to be asymptotically equal to that of χ[0,n+1] . It should therefore obey the Gaussian law (14.89). This trend is already seen in the small n data of Tables 8.13, 8.14, in that the skewness and kurtosis is typically decreasing as n increases. Further remarks can be made in relation to both (14.144) and (14.145). Consider first (14.145). Together with (14.143) it implies Cov si−n−1 (n), si (n) + Var(si (n)) + Cov si (n), si+n+1 (n) ∼ O(1). (14.146) This is essentially a charge neutrality result, which says that the charge excess in the interval containing the first n + 1 particles will be compensated by the excess of opposite sign in the two neighboring intervals of n + 1 particles. Regarding (14.144), we note that the asymptotic behavior is identical to the leading non-oscillatory behavior of the charge-charge correlation as given by (14.9). The physical reasoning which led to (14.143) is consistent with this result. An analogous result is that 1 Cov(χ[0,1] , χ[n,n+1] ) = Var(χ[0,n+1] ) − 2Var(χ[0,n] ) − Var(χ[0,n−1] ) , 2 which according to (14.85) also exhibits the asymptotic behavior (14.144). Analytic information is also available on Var(si (n)) for finite n. In particular, French et al. [243] (see also Exercises 14.6 q.3) have refined the physical relationship between Var(si (n)) and Var(χ[0,n+1] ) by compensating for the fact that fixed particles are present at the endpoints in the definition of si (n) but not in χ[0,n+1] . Consequently they have deduced the formula 1 (14.147) Var(si (n)) ≈ Var(χ[0,n+1] ) − . 6 The accuracy of this formula for the log-gas at β = 1, 2 and 4, and small values of n, can be tested by reading off from Tables 8.13, 8.14 and the inter-relation (8.161), and comparing with the corresponding value of Var(χ[0,n+1] ) deduced from (14.84). The results displayed in Table 14.2 demonstrate an accuracy of up to 1 part in 104 . However for larger n the discrepancy increases; for example we have c9 (2) = 0.1585. Related to the topics of this section is the large k form of psoft β (k; s), the latter being the distribution of the
FLUCTUATION FORMULAS AND UNIVERSAL BEHAVIOR OF CORRELATIONS
n
cn (1)
cn (2)
cn (4)
0
0.1606
0.1640
0.1545
1
0.1670
0.1664
0.1603
2
0.1674
0.1665
0.1622
3
0.1672
0.1664
0.1631
4
0.1644
0.1663
0.1635
695
Table 14.2 Tabulation of cn (β) := Var(χ[0,n+1 ) − Var(si (n)) for the random matrix couplings β = 1, 2 and 4. The approximation (14.147) predicts cn (β) ≈ 1/6.
kth largest eigenvalue at the soft edge. To leading order the mean μk of this distribution must satisfy μk 2 |μk |3/2 , ρsoft k∼ (1) (X) dX ∼ 3π −∞ where the second asymptotic form follows from (13.68). It is proved in [281], for β = 2, that with σk2 proportional to (log k)/(βk 2/3 ), as k → ∞ (Xk + |μk |)/σk ∼ N[0, 1]. This is consistent with the data of Tables 9.1 and 9.2. 14.6.5 The Δ3 statistic In the bulk of the spectrum, after appropriate scaling, the mean eigenvalue spacing is a constant 1/ρ. Thus the mean number of eigenvalues n(x) from some (arbitrary) origin to a point x in the spectrum increases linearly with x. The actual number of eigenvalues between 0 and x, which is a staircase function that jumps one unit at the position of each eigenvalue, will deviate from a straight line. On the other hand, there will be a unique line of best fit according to the criterion of least square deviation. For such a straight line within an interval of length , Dyson and Mehta [153] have introduced this deviation as a statistic, denoted Δ3 (the subscript 3 occurs because two similar statistics, denoted Δ1 and Δ2 , were also introduced), characterizing the eigenvalue spectrum. Explicitly 1 /2 Δ3 := minA,B (n(y) − Ay − B)2 dy, (14.148) −/2 where n(y) measures the number of eigenvalues from some arbitrary point a large distance from the interval (−/2, /2). Equating the partial derivatives with respect to A and B in (14.148) to zero to calculate the minimum shows 12 /2 1 /2 A= 3 yn(y) dy, B= n(y) dy. l −/2 l −/2 Minor manipulation then gives /2 1 l/2 12 1 du dv δ(v − u)n(u)n(v) − 3 uvn(u)n(v) − n(u)n(v) Δ3 = −l/2 l −/2 /2 /2 2 1 12uv = 2 n(u) − n(v) . du dv 1 + 2 2 −/2 l −/2
696
CHAPTER 14
Thus if we let N (|u − v|) denote the variance of the number of particles in the interval |u − v|, then by taking the ensemble average we see that /2 /2 1 12uv Δ3 = 2 du dv 1 + 2 N (|u − v|). 2 −/2 l −/2 Finally, changing variables u − v = y, u + v = z and performing the integration over z give [440] 2 y 3 y Δ3 = − 2 + 1 N (y) dy. (14.149) 0 Substituting the asymptotic expansion (14.85) (which assumes ρ = 1) in this expression shows that for large 9 Bβ 1 . Δ3 ∼ 2 log − 2 + π β 4π β 2 E XERCISES 14.6
1. [152], [178] In this exercise the asymptotic form of Eβ (n; s) will be computed in the case 0 s − n s using the theory which led to (14.132). (i) For 0 ρs − n ρs (s = 2t), k = b/t ∼ 1. Thus use known expansions of E(k) and K(k) for k near 0 and 1 in the appropriate formulas given in the text to deduce that ρb ∼
ρs n − ρs + , 2 log(ρs/(n − ρs))
c ∼ π2
(ρs − n) . 2 log(ρs/(n − ρs))
(ii) Substitute the results of (i) in (14.128) and Proposition 14.6.1 to deduce from (14.119) that in the region in question “ ” π2 β (ρs − n)2 , Eβ (n; s) ∼ exp − 4 log ρs and relate this to (14.89). 2.
(i) Use the formula pβ (n; s) =
n d2 X (n − j + 1)Eβ (j; s), 2 ds j=0
which follows from (8.16), to show that Z
∞
s2 pβ (n; s) ds = 2
0
n X
Z (n − j + 1)Iβ (j),
∞
Iβ (p) :=
Eβ (p; s) ds. 0
j=0
(ii) Use the identities (si (n))2 =
n “X
si+j (0)
”2
j=0
to deduce that Z ∞
=
n “ X
si+j (0)
j=0
s2 pβ (n; s) ds = (n + 1)si (0)2 + 2
0
”2
+2
X
si+j (0)si+k (0)
0≤j
n X
(n − j + 1)si (0)si+j (0)
j=1
and compare this with the result of (i) for n = 0, 1, 2, . . . successively to deduce s2i (0) = 2Iβ (0),
si (0)si+j (0) = Iβ (j) (j ≥ 1).
(14.150)
697
FLUCTUATION FORMULAS AND UNIVERSAL BEHAVIOR OF CORRELATIONS
Rewrite (14.150) in the form n “ ” “ ” “ ” X Var si (n) = (n + 1)Var si (0) + 2 jCov si (0), si+n+1−j (0) j=1
and use this inductively to deduce (14.141). 3. [243] Let n(x) represent the microscopic number of particles between 0 and x, where the origin 0 is chosen in the bulk. Then n(x) is a staircase function, increasing by 1 each time x is increased past a particle coordinate. Let n(C) (x) denote some continuous approximation to n(x), and let (x) denote the correction at a particle coordinate xi , so that n(C) (xi ) = n(xi ) + (xi ) with |(xi )| ≤ 21 . Assuming, as is reasonable, that (x) is uniformly distributed and n(x) and (x) are uncorrelated, show that D E D E 1 (n(C) (xi ) − n(C) (xj )) = (n(xi ) − n(xj )) + . 6 (This result forms the basis for the approximate formula (14.147).) 2 3
− −βN u /3 4. [124] Show , where u = 1 − s. √ of (14.138) is to leading order equal to e √ that for s → 1 , the r.h.s. 2N − x from √ the mean position of the largest eigenvalue of the Gaussian Interpret 2N u as the displacement √ β-ensemble without scaling, then write 2N u = X/ 2N 1/6 as is consistent with the soft edge scaling (7.11) to reclaim the leading order term of (9.156). 5. [533], [497] Consider eigenvalues from CUEN . Let ZN be the minimum of all the N spacings between consecutive eigenvalues. It is known that “ “ N 4 ”1/3 ” 3 lim Pr ZN > x = e−x . N→∞ 72π The aim of this exercise is to give an heuristic prediction of this result. (i) Let ZˆN (x) denote the random variable for the number of consecutive spacings less than x. Note that Pr(ZˆN (x) = 0) = Pr(ZN > x). Integrate the leading term of the expansion (8.165) with ρ = N/2π to deduce that as x → 0, the probability that a single spacing is less than x is equal to
cβ (N x)1+β ,
cβ =
(β/2)β ((β/2)!)3 . 2π(β + 1)!(3β/2)!
(ii) From (i), by assuming that to some degree of approximation such spacings are independent, conclude Pr(ZˆN (x) = 0) ≈ (1 − cβ (N x)1+β )N ≈ e−cβ N(Nx)
1+β
.
Now scale x and set β = 2 to obtain the result. 6. [169] Consider a sequence of 2N + 1 particles on a circle of radius (2N + 1)/2π, label them by their scaled angles “ 1 1” N + > xn > · · · x1 > x0 > x−1 > · · · > − N + 2 2 and consider that statistic δn := xn − x0 − n. For p = ±1, ±2, . . . , ±N define the corresponding power spectrum P(p) =
N ˛2 1 ˛˛ X ˛ δn e−2πipn/(2N+1) ˛ . ˛ 2N + 1 n=−N
As a long wavelength (|p| N , N 1), continuum approximation write δn ≈ n(1) (y) is the microscopic density, so that Z ˛2 ˛Z ” 1 ˛ N “ x ˛ (n(1) (y) − y) dy e−2πikx dx˛ ˛ 2N −N 0 Z ˛2 1 “ 1 ”2 ˛˛ N ˛ ≈ n(1) (x)e−2πikx dx˛ , ˛ 2N 2πk −N
P(p) ≈
Rx 0
(n(1) (y) − y) dy, where
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CHAPTER 14
where the second line follows by integration by parts and the assumption that N is large. Deduce from this that “ 1 ”2 ˆ S(2πk), P(p) ∼ 2πk where Sˆ refers to the structure function. In the case of the log-gas, note from (14.8) that this implies P(p) ∼
1 , 2π 2 β|k|
thus exhibiting 1/f noise.
14.7 DYNAMICAL CORRELATIONS 14.7.1 Definition In dynamical many-body systems, the current correlations are fundamental quantities closely related to the density correlations. In particular the macroscopic one-body current J(x; τ ) is related to the one-body dynamical particle density ρ(1) (x; τ ) via the continuity equation ∂ ∂ ρ(1) (x; τ ) = − J(x; τ ). (14.151) ∂τ ∂x The macroscopic one-body current is usually defined as the averaged value of the classical microscopic current jτ (x) =
N dxj (τ ) j=1
dτ
δ(x − xj (τ )).
(14.152)
The latter satisfies ∂ ∂ nτ (x) = − jτ (x), ∂τ ∂x
nτ (x) :=
N
δ(x − xj (τ )),
(14.153)
j=1
and thus the continuity equation is true at a microscopic level. However, in the Fokker-Planck description of Brownian motion the classical microscopic current has no immediate meaning because the velocities do not explicitly occur in (11.15). One way to deduce the analogue of (14.152) is to insist on the applicability of the microscopic continu(1) ity equation. First note that for an observable Aτ1 = A({xj }) measured at parameter value τ1 , and an (2)
observable Bτ2 = B({xj }) measured at parameter value τ2 , the average of the product Aτ1 Bτ2 is given by ∞ ∞ ∞ ∞ 1 (0) (0) (1) (1) (0) dx · · · dxN p0 (x ) dx1 · · · dxN Aτ1 Aτ1 Bτ2 = N ! −∞ 1 −∞ −∞ −∞ ∞ ∞ (2) (2) ×GFP x(0) ; x(1) ) dx1 · · · dxN Bτ2 GFP x(1) ; x(2) ). τ1 ( τ2 −τ1 ( −∞
−∞
(14.154) The delta function initial condition of the Green function and the structure of (11.15) show Gτ1 (x(0) ; x(1) ) = eLτ
N
(1)
δ(xl
(0)
− xl ),
(14.155)
l=1 (1)
where it is understood that L acts on {xl }, and in (11.15) we have set γ = 1. Substituting this formula in
699
FLUCTUATION FORMULAS AND UNIVERSAL BEHAVIOR OF CORRELATIONS
(0)
(0)
(0)
(14.154) allows the integration over {xl } to be carried out. Then substituting (14.155) with {xl }, {xl } (1) (2) (1) replaced by {xl }, {xl } in the resulting expression and integrating over {xl } we obtain ∞ ∞ (2) (2) Aτ1 Bτ2 = dx1 · · · dxN Bτ2 eL(τ2 −τ1 )/γ Aτ1 eLτ1 /γ p0 (x(2) ). (14.156) −∞
−∞
Next we note from Exercises 11.1 q.1 (ii) and (11.15) with γ = 1 that eLτ = e−βW/2 e−τ so we can rewrite (14.156) as ∞ (2) dx1 · · · Aτ1 Bτ2 = −∞
∞
−∞
where A(τ ) := eτ
PN j=1
Π†j Πj /βγ βW/2
e
dxN e−βW/2 B(τ2 )A(τ1 )p0 (x(2) )eβW/2 ,
PN j=1
(2)
Π†j Πj /βγ
Aτ e−τ
PN j=1
Π†j Πj /βγ
(14.157)
(14.158)
and similarly the definition of B(τ ). N The equation (14.158) with Aτ = nτ (x) = j=1 δ(x − xj (τ )) provides a definition of the dynamical microscopic density n(x; τ ) in operator form. Substituting in (14.151) then allows the sought formula for the microscopic current to be obtained. N P ROPOSITION 14.7.1 Let n(x; τ ) be defined by (14.158) with Aτ = nτ (x) = j=1 δ(x − xj (τ )), and similarly define j(x; τ ) with Aτ = jτ (x), jτ (x) to be determined. Then for the continuity equation (14.151) (which now refers to microscopic quantities) to be satisfied we require jτ (x) = −
N i 1 ∂ 1 ∂ . δ(x − xj (τ )) + δ(x − xj (τ )) γβ j=1 i ∂xj i ∂xj
(14.159)
Proof. With n(x; τ ) and nτ as specified it follows from (14.158) that hX i PN † † ∂n(x; τ ) 1 τ PN Π†j Πj , nτ e−τ j=1 Πj Πj /βγ . = e j=1 Πj Πj /βγ ∂τ γβ j=1 N
Comparison with (14.151) and recalling the definition of jτ then shows i ∂ 1 hX † Π Πj , nτ . jτ (x) = − ∂x γβ j=1 N
The stated result now follows from the fact that N X
Π† Πj =
j=1
N X ∂2 + V (x1 , . . . , xN ) ∂x2j j=1
for some V .
14.7.2 Hydrodynamic limit Physical principles involving the continuity equation (14.153) can be used to predict the small k form of the dynamical structure function as given by (13.228) [54]. For a single particle moving in a viscous medium (γ = 1) Newton’s law of motion gives m
dv(τ ) = −v(τ ) + F , dτ
700
CHAPTER 14
where F is the applied force. Hence in an equilibrium situation v(τ ) = F. The many-body analogue of this latter equation is j(x; τ ) = F (x; τ ), where F (x; τ ) now refers to the macroscopic force density. For the log-gas, in the long wavelength regime, the force density will to leading order be of electrostatic origin, implying [145] ∞ ∂ V (x) − j(x; τ ) = −nτ (x) ρ(1) (x ; τ ) log |x − x | dx . (14.160) ∂x −∞ To proceed further one takes the partial derivative with respect to x on both sides, and substitutes for ∂j(x; τ )/∂x on the l.h.s. using the continuity equation (14.153). Next the resulting equation is linearized by writing ρ(1) (x ; τ ) = ρ(1) (x) + δnτ (x) (recall that (14.160) refers to the long wavelength regime so the microscopic quantity nτ (x) is essentially smoothed) and terms of order (δnτ (x))2 are ignored. Using the equilibrium condition ∞ ∂ V (x) − ρ ρ(1) (x) log |x − x | dx = 0, ∂x −∞ the linearized equation then reads
∞ ∂δnτ (x) ∂2 = −ρ 2 δnτ (x ) log |x − x | dx δτ ∂x −∞ ∞ ∂ ∂ = −ρ δn (x ) log |x − x | dx . τ ∂x −∞ ∂x
(14.161)
Introducing Fourier transforms and recalling (14.3) this gives ∂δˆ nτ (k) = −ρπ|k|δˆ nτ (k). ∂τ nτ (0)e−πρ|k|τ , or equivalently Thus δˆ nτ (k) = δˆ ˆ τ (0)e−πρ|k|τ . n ˆ τ (k) = n
(14.162)
ˆ τ ) = 1 ˆ nτ (k). Substituting (14.162) predicts But for a system confined to a region of length L, S(k; L n0 (k)ˆ that for the general β log-gas, in the long wavelength k → 0 limit, ˆ τ ) ∼ S(k; ˆ 0)e−πρ|k|τ , S(k; which when combined with (14.8) implies (13.228).
(14.163)
Chapter Fifteen The two-dimensional one-component plasma The two-dimensional one-component plasma (2dOCP) consists of log-potential charges of the same sign in a two-dimensional domain which contains a smeared out neutralizing background, and so is the twodimensional version of the one-component log-gas. Although only one value of the coupling allows an exact solution, there are a number of different two-dimensional geometries and boundary conditions for which this exact solution is possible. Here the exact solutions for disk, sphere and antisphere geometries are considered, as well as the exact solution for metallic and Neumann boundary conditions. The first three of these allow for interpretations as eigenvalue p.d.f.’s, and as the modulus squared of the many-body wave function formed by free fermions confined to these surfaces in the presence of a particular magnetic field. Also associated with these three geometries are the zeros of three families of random polynomials, although the correlations are given not by determinants, but rather by permanents. For the 2dOCP at general coupling, macroscopic arguments of the type used in Chapter 14 imply a number of sum rules and asymptotic formulas, which can be illustrated on the exact results. The fast decay of the correlations in the bulk is responsible for sum rules which have no one-component log-gas analogues. In the last section, a classification scheme for random matrix ensembles with complex eigenvalues is considered.
15.1 COMPLEX GAUSSIAN RANDOM MATRICES AND POLYNOMIALS 15.1.1 Eigenvalues of complex random matrices The Boltzmann factor for the 2dOCP in a disk has been calculated in Exercises 1.4 q.3 as proportional to N j=1
e−πρΓ|rj |
2
/2
|rk − rj |Γ
(15.1)
1≤j
(recall Γ := q 2 β, where q denotes the magnitude of the charge; in two dimensions it is conventional to keep Γ as the coupling in the Boltzmann factor even if we take q = 1). For the special coupling Γ = 2, this same p.d.f. occurs as the eigenvalue p.d.f. for complex Gaussian random matrices. P ROPOSITION 15.1.1 Consider an N × N random matrix X in which the elements ujk + ivjk are independently distributed with p.d.f. 1 −|ujk |2 −|vjk |2 e . (15.2) π (The set of such matrices is said to define the complex Ginibre ensemble, after [264].) The corresponding eigenvalue p.d.f. for the (complex) eigenvalues zj = xj + iyj is proportional to (15.1) with rj = (xj , yj ), Γ = 2 and ρ = 1/π. Proof. We follow [395, Appendix 35] (see also [266]), as refined by [298], and begin by writing X in terms of its Schur decomposition, X = UTU−1 , (15.3) where U is a unitary matrix which is unique only up to the phase of each column, and T is a triangular matrix with all elements below the diagonal zero and the diagonal elements equal to the eigenvalues. The Schur decomposition (15.3)
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CHAPTER 15
follows by iterating the complex form of (1.176), obtained by Householder transformations (see also Exercises 15.1 q.2). The number of independent real parts of X is 2N 2 , while the number in T is N 2 + N . Consistent with this is that making a specific choice of the phase of each column of U gives the number of independent real variables associated with U as N 2 − N . Now (15.4) U† dXU = dT + U† dUT − TU† dU, and (U† dXU) = (dX) (this latter equation follows from Proposition 3.2.4). With dV := U† dU, the element in position (jk) of the matrix on the r.h.s. of (15.4) is X X dVjl Tlk − Tjl dVlk . (15.5) dTjk + l≤k
j≤l
For j > k, we rewrite (15.5) to read (Tkk − Tjj )dVjk +
“X
dVjl Tlk −
l
X
” Tjl dVlk
(15.6)
j
while for j ≤ k we rewrite it as dTjk + Tjk (dVjj − dVkk ) +
k “X
dVjl Tlk −
l=1 l=j
N X
” Tjl dVlk .
(15.7)
l=j l=k
With the wedge product computed in the order of indices (N 1) ((N − 1) 1) . . . (1 1) (N 2) . . . (1 2), and so on, after recalling that dVjk = −dV¯kj , we see that the differentials are introduced for the first time from the terms outside the bracketed summations in (15.6) and (15.7). Hence the bracketed terms do not contribute to the wedge product, which, after recalling Tjj = zj , is therefore equal to Y r i |zj − zk |2 ∧j dzjr dzji ∧j
where as used previously the superscripts r and i denote real and imaginary parts, respectively. But dVjj must be expressible in terms of {dVjk }j
PN
2
P
2
Finally, the joint distribution of the elements is proportional to e−Tr(XX ) = e− j=1 |zj | − j
j
The dependence on the eigenvalues thus factorizes from the dependence on the other variables, and is indeed proportional to (15.1) with Γ = 2 and ρ = 1/π.
One immediate implication of Proposition 15.1.1 is in relation to the density of eigenvalues of complex Gaussian random matrices. According to Exercises 1.4 q.3 the Boltzmann factor (15.1) results from the 2dOCP in a disk of radius R with a uniform background of charge density −ρ (= −1/π in the eigenvalue analogy). By the principle of local charge neutrality for Coulomb systems, √ this implies that the eigenvalues will uniformly occupy a disk in the complex plane of radius R = N . The validity of this prediction, which will subsequently be established analytically, is supported by displaying graphically the numerical computation of the eigenvalues of a complex Gaussian random matrix (see Figure 15.1). More generally, eigenvalues of random matrices with i.i.d. elements (from a real or complex distribution) of zero mean and
703
THE TWO-DIMENSIONAL ONE-COMPONENT PLASMA
15
10
10
5
5
0
0
–5
–5 –10
–10
–15 –10 –5 –10
–5
0
5
0
5
10 15
10
Figure 15.1 Eigenvalues in the complex plane of a 169 × 169 random matrix with complex Gaussian elements (left most plot) and zeros of a random polynomial with complex coefficients aj chosen from the Gaussian distribution with zero mean and variance 1/j!. Note that the edge of the leading order support |z| < 13 (shaded region) is sharper in the former case.
finite variance are to leading order supported uniformly in a disk about the origin in the complex plane, a result known as the circle law [265], [26], [510]. 15.1.2 Zeros of complex random polynomials The Jacobian j
(15.10)
to the variables {aN , z1 , . . . , zN }, where the zj are the zeros of p(z) [286]. N P ROPOSITION 15.1.2 With (15.10) factorized as p(z) = aN j=1 (z − zj ) we have a0 ∧ · · · ∧ daN −1 ∧ d¯ aN −1 = Jdz1 ∧ d¯ z1 ∧ · · · ∧ dzN ∧ d¯ zN , da0 ∧ d¯ where J = |aN |2N
|zk − zj |2 .
1≤j
Proof. The coefficients are related to the zeros by the formula aj /aN =
X
zi1 zi2 · · · ziN −j =: eN−j
(15.11)
1≤i1 ≤···≤iN −j ≤N
(for the definition, recall (4.132)), while the Jacobian is given by # " ∂ ∂ a a ∂zj k−1 ∂z ¯j k−1 J = det ∂ ∂ a ¯ a ¯ ∂zj k−1 ∂z ¯j k−1
. j,k=1,...,N
Substituting (15.11) in the Jacobian shows that J is a homogeneous polynomial zj } of degree 12 N (N − 1). Q in {zj } and {¯ zk − z¯j ) as a factor. But this It vanishes whenever zj = zk or z¯j = z¯k for any j = k and thus contains j
704
CHAPTER 15
Consider now the case that the coefficients aj in (15.10) are complex Gaussian random variables with mean zero and standard deviation σj , so that the measure associated with the joint distribution of the elements is N j=0
N N 1 2 2 exp − |a | /2σ darj daij . j j 2πσj2 j=0 j=0
(15.12)
Changing variables according to Proposition 15.1.2, this becomes N j=0
N 1 2 exp − |ej |2 /2σN −j 2 2πσj j=0
|zk − zj |
1≤j
2
N
dxj dyj darN daiN
(15.13)
j=1
where e0 := 1 and xj + iyj =: zj . Integrating over arN and aiN shows that the p.d.f. for the roots of p(z) is given by [70] N 2 1 1≤j
where the ej are given in terms of the {zj } by (15.11). Unlike the numerator in (15.14), which represents the Boltzmann factor for particle-particle interaction in the 2dOCP with Γ = 2, the denominator has no simple plasma interpretation as all particle coordinates are coupled. Nonetheless, we will see in Section 15.3.3 that there are some similarities with the plasma system. In particular, choosing σj2 = 1/j! gives a uniform density of zeros in the complex plane with support (to √ leading order) in a disk of radius N . 1. [250] In this exercise we will consider a random matrix J of the form J = H + ivA, where H and A are Gaussian Hermitian random matrices with joint p.d.f.’s for the elements proportional to 1 exp(− 1+τ TrX2 ) (X = H, A), where τ = (1 − v 2 )/(1 + v 2 ).
E XERCISES 15.1
(i) Verify the formulas Tr H2 =
” ” 1“ 1 “ Tr(JJ† ) + Re Tr(J2 ) and Tr A2 = 2 Tr(JJ† ) − Re Tr(J2 ) 2 2v
and thus show that the joint p.d.f. for the elements of J, obtained by changing variables in the joint distributions for the elements of H and vA, is proportional to ” “ 1 Tr(JJ† − τ Re J2 ) . (15.15) exp − 2 1−τ (ii) Decomposing J in terms of its Schur decomposition (15.3), note that the above joint distribution becomes “ exp
−
N N ”” X X 1 “X |zj |2 − τ Re zj2 + |Tjk |2 , 2 1−τ j=1 j=1 j
and hence deduce that the eigenvalue p.d.f. is proportional to “ exp
−
N ”” 1 X“ 2 τ 2 |zj | − (zj + z¯j2 ) 2 1 − τ j=1 2
Y
|zk − zj |2 .
(15.16)
1≤j
x chosen 2. [249] Let XN be an N × N complex matrix, and let z be an eigenvalue with normalized eigenvector x + e1 )/| x + e1 |. The working of Exercises 1.9 q.3 gives with x1 > 0. Define U = 1N − 2vv † , where v = ( – » † z w N−1 U XN = U 0N−1 XN−1
705
THE TWO-DIMENSIONAL ONE-COMPONENT PLASMA
for some N − 1 × 1 vector w and N − 1 × N − 1 matrix XN−1 . (i) Repeat the working which led to (1.179) to show (dXN ) = | det(z1N−1 − XN−1 )|2 dz r dz i (dw N−1 )(dsN−1 )(dXN−1 ), where sN−1 is the first column of UdU with the first entry removed. (ii) Write v = (v1 , q ). The definition of v implies 12 ≤ v12 ≤ 1 and thus 0 ≤ | q |2 ≤ 12 . Note that » – 2| q |2 − 1 −2v1 qT U= , −2v1 q 1N−1 − 2 q q † and from this show
(dsN−1 ) = (a1N−1 + b q q † )(d q) + (2a + b) q q T (d q∗)
where here ∗ denotes complex conjugate and p a = −2 1 − | q |2 ,
1 − 2| q |2 b= p . 1 − | q |2
(iii) Deduce from (ii) that (dsN−1 ) = det(12N−2 + L)(d q ), where » L=
b q q† (2a + b) q ∗ q†
(2a + b) q qT b q ∗ qT
–
» =
b q (2a + b) q∗
(2a + b) q b q∗
–»
q † 0 T
0 T b qT
– .
Now make use of (5.33) to obtain the evaluation det(12N−2 + L) = 22N−2 (1 − | q |2 )N−2 (1 − 2| q |2 ). 3. [156] Use Householder reduction to Hessenberg form, as specified by (2.84), to show that complex Gaussian random matrices with entries independently distributed according to (15.2) are equivalent under unitary conjugation to the Hessenberg matrix [aij ]i,j=1,...,N where all entries are again independently distributed. Explicitly, show that ˜2(N−i) (recall Proposition for i ≤ j, aij is again distributed according to (15.2), while ai+1,i has distribution χ 1.9.1). 4. In this exercise the problem of finding the zeros of a general degree N polynomial p(x) = xN + aN−1 xN−1 + · · · + a1 x + a0 is shown to be equivalent to finding the eigenvalues of the companion matrix 2 3 0 0 ··· ··· 0 −a0 6 1 0 ··· ··· 0 −a1 7 6 7 6 0 1 ··· ··· 0 −a2 7 CN := 6 7. 6 . . 7 . .. 4 .. .. 5 . .. 0 0 · · · · · · 1 −aN−1 (i) Let the zeros of p(x) be {λj }j=1,...,N . Show that T [λpj ]p=0,...,N−1 = λj [λpj ]p=0,...,N−1 . CN
(ii) Conclude from (i) and the Vandermonde determinant formula (1.173) that if the zeros of p(x) are all distinct, T and thus CN is diagonalizable and has eigenvalues λ1 , . . . , λN . CN
706
CHAPTER 15
15.2 QUANTUM PARTICLES IN A MAGNETIC FIELD As well as an interpretation as an eigenvalue p.d.f, the Boltzmann factor (15.1) with Γ = 2 occurs as the modulus squared of the exact ground state wave function for a certain quantum many-body system. This is the quantum many-body system consisting of N non-interacting fermions confined to a plane with a perpendicular magnetic field. All the fermions (electrons) are assumed to have their spin magnetic moment frozen along the direction of the magnetic field, thus implying that the spin of the electrons can be ignored. The demonstration of the analogy requires first revising the quantum mechanics of a single particle in a magnetic field (see, e.g., [112]).
15.2.1 Single particle wave function Consider the setting of a quantum particle confined to the xy-plane. Suppose the particle has mass m and = Bˆ charge −e, and is subject to a perpendicular magnetic field B z, B > 0. The Hamiltonian is then H :=
1 e 2 1 2 1 (−i∇ + A) = Π = wc (a† a + aa† ), 2m c 2m 2
(15.17)
where Πx = −i
∂ e + Ax , ∂x c
l a† = √ (Πx + iΠy ), 2
Πy = −i
∂ e + Ay , ∂y c
l a = √ (Πx − iΠy ), 2
wc := eB/mc (c denotes the speed of light) is called the cyclotron frequency, l := must satisfy magnetic length, and the vector potential A = Bˆ ∇×A z.
(15.18) c/eB is called the (15.19)
Using (15.19) we can check from the definitions (15.18) that the commutation relation [Πx , Πy ] = −i2 /l2 holds, and from this we can check that [a, a† ] = 1.
(15.20)
Use of (15.20) in (15.17) gives that the Hamiltonian can be written in the harmonic oscillator-like form 1 1 = wc aa† − . (15.21) H = wc a† a + 2 2 Thus there are eigenstates |n of H with energy En = (n + 21 )wc , (n = 0, 1, 2, . . . ), referred to as Landau levels, which are given in terms of the ground state |0 by n
(a† ) |n = √ |0 n!
(15.22)
with the state |0 specified by a|0 = 0. Unlike for the harmonic oscillator, these states do not form a complete set (this is not surprising as here the system is two-dimensional). In fact, by studying the classical problem (see Exercises 15.2 q.1) and using the correspondence principle, we can interpret H as proportional√to the square of the radius of the cyclotron orbit, which gives that the eigenstates |n have cyclotron radius 2n + 1l. The classical theory also tells us that once the cyclotron orbit radius is fixed, the only remaining degree of freedom is the center of the orbit. Applying the correspondence principle to the formulas of Exercises 15.2 q.1 for the classical orbit center
707
THE TWO-DIMENSIONAL ONE-COMPONENT PLASMA
gives the quantum center of orbit operators X =x−
l2 Πy ,
Y =y+
l2 Πx .
(15.23)
Using (15.19) and the explicit formulas for Πx and Πy in (15.18) gives the commutation relation [X, Y ] = il2 . This indicates that we should consider the operator X 2 + Y 2 , for then by defining 1 b† = √ (X − iY ), 2l
1 b = √ (X + iY ) 2l
(15.24)
we see that [b, b† ] = 1 and furthermore
1 . X 2 + Y 2 = l2 (bb† + b† b) = 2l2 b† b + 2
2 Thus this operator also has a harmonic oscillator form, and so has eigenstates |m with eigenvalue Rm = 2 (2m + 1)l , m = 0, 1, 2, . . . , which are given in terms of the state |0 (defined as the solution of b|0 = 0) by m
(b† ) |m = √ |0. m!
(15.25)
A direct calculation from (15.18) and (15.24) shows that a, a† commutes with b, b† , and so simultaneous eigenstates of H and X 2 + Y 2 are permitted. Thus for each state (15.22) of specific energy, the operator m (b† ) creates orthogonal states |n, m =
(a† )n (b† )m √ |0, 0 n!m!
with the same energy En = (n + 12 )ωc and eigenvalue of X 2 + Y 2 equal to (2m + 1)l2 . 15.2.2 Many particle ground state The states |m =: ψm (r) given by (15.25) are orthogonal states with the minimal allowed energy 12 wc (i.e., the lowest Landau level) and their interpretation is that they have definite values of the distance from the origin to the center of their cyclotron orbit, which increases with m. The densest N -particle ground state ψ, in which the particles are fermions but otherwise non-interacting, is therefore obtained by constructing a Slater determinant from the states ψ0 (r), . . . , ψN −1 (r), 1 (15.26) ψ(r1 , . . . , rN ) = √ det[ψj−1 (rk )]j,k=1,...,N . N! The immediate task is thus to evaluate the single-particle states (15.25) which from (15.23), (15.24) and (15.18) depend on the particular choice of Ax and Ay . Here we will consider the symmetric gauge = B (−yˆ x + xˆ y). A 2 ∂ ∂ ∂ ∂ ∂ := 12 ( ∂x − i ∂y ) and ∂∂z¯ = 12 ( ∂x + i ∂y ) we can write In the gauge (15.27), with z = x + iy, ∂z ∂ 1 ∂ 1 1 i z¯ + 2l , b= √ z + 2l . a = −√ 2l ∂z 2l ∂ z¯ 2 2
(15.27)
(15.28)
Using in (15.21) the explicit form of a from (15.28) and the corresponding form of a† , we see that H can be written ∂ ωc ∂2 ∂ |z|2 H= − 4l2 + z − z¯ + 2 . (15.29) 2 ∂z∂ z¯ ∂z ∂ z¯ 4l
708
CHAPTER 15
Also, since the state ψ0 (r) := |0 is characterized by a|0 = 0, b|0 = 0 use of (15.28) gives z¯ ∂ ψ0 = − 2 ψ0 ∂z 4l
∂ z ψ0 = − 2 ψ0 . ∂ z¯ 4l
and
(15.30)
z )e−zz¯/4l for any f analytic in z¯, while the second equation has The first equation has solution ψ0 = f (¯ −z z¯/4l2 solution ψ0 = g(z)e for any function g analytic in z. These solutions are only compatible if f and g are constant functions, so after determining the normalization we have 2
ψ0 = √
1
e−zz¯/4l = √
1
2
2πl2
2πl2
e−(x
2
+y 2 )/4l2
.
Substituting this in (15.25) with the explicit form of b† deduced from (15.28) gives (¯ z ) e−(x m
ψm (r) =
2
+y 2 )/4l2 1/2
(2πl2 2m l2m m!)
,
(15.31)
and this substituted in (15.26) together with the Vandermonde formula (1.173) gives 2 2 2 N 1 e−(xj +yj )/4l ψ(r1 , . . . , rN ) = √ N ! j=1 (2πl2 2j−1 l2(j−1) (j − 1)!)1/2
(¯ zk − z¯j ).
(15.32)
1≤j
Comparison with (15.1) shows that |ψ|2 is indeed proportional to the Boltzmann factor for the one-component plasma at Γ = 2 in a disk with ρ = 1/2πl2. As in the case of random matrices, one immediate consequence of the plasma analogy is a prediction for the 1/2 density of the quantum system: to leading order it will be uniform in a disk of radius R = (2N ) l with value 2 1/2πl , and equal to zero outside this radius. This prediction is consistent with the fact√that the maximum distance from the origin to the center of the cyclotron orbit of the single particle states is 2N − 1 l. 15.2.3 Zeros of a random single particle state In Section 15.2.2 an N -body state has been constructed from the first N single-particle states (15.31). Here we will consider a single-particle state φ(r ) constructed as a linear combination of the first N + 1 states (15.31). From the form of these states and their orthogonality, we can write φ(r ) = e−(x
2
+y 2 )/2
p(¯ z ),
p(w) =
N αn √ wn n! n=0
(for convenience we have set 2l2 = 1), where αn = p(z), z n with the inner product defined by ∞ 2 2 1 ∞ dx dy e−(x +y ) f¯g. f, g = π −∞ −∞
(15.33)
(15.34)
Suppose now φ(r) is random in the sense that the coefficients αn in (15.33) are chosen at random from a complex Gaussian distribution with mean zero and variance unity. Then p(w) is equivalent to a polynomial of the form (15.10) in which each aj is chosen from a complex Gaussian distribution with mean zero and variance 1/j!, and so the joint distribution of its zeros is given by (15.14) with σj2 = 1/j!. E XERCISES 15.2
the force is given by − e v × B. 1. For a classical particle of charge −e in a magnetic field B, c
= (i) Suppose the particle is confined to the xy-plane and the magnetic field is perpendicular to this plane, B Bˆ z, B > 0. Show that Newton’s equation of motion implies v˙ x = −ωc vy ,
v˙ y = ωc vx ,
709
THE TWO-DIMENSIONAL ONE-COMPONENT PLASMA
and thus with the initial condition (vx , vy ) = (v0 , 0), vx = v0 cos ωc t,
vy = v0 sin ωc t.
Note that vx2 + vy2 = v02 independent of t. (ii) Use the formulas for vx and vy to show x = x0 +
v0 sin ωc t, ωc
y = y0 −
v0 cos ωc t. ωc
Note that ((x − x0 )2 + (y − y0 )2 )1/2 = v0 /ωc , so the solution corresponds to a circular cyclotron orbit of radius v0 /ωc with center at (x0 , y0 ). (iii) Make the correspondence mvx ↔ Πx , mvy ↔ Πy to interpret Π2x + Π2y as proportional to the square of the cyclotron orbit radius. 2.
(i) [143] Show from (15.29) that the Hamiltonian H in the symmetric gauge can be written „ « ωc “ ∂ 2 Φ ”−1 “ ∂ ∂Φ ”“ ∂ ∂Φ ” H=− −1 − + 2 ∂z∂ z¯ ∂ z¯ ∂ z¯ ∂z ∂z
(15.35)
with Φ = |z|2 /(2l)2 . Also, with Az := Ax + iAy and Az¯ := Ax − iAy , check that Az =
iB 2 ∂Φ (2l ) , 2 ∂ z¯
Az¯ = −
iB 2 ∂Φ (2l ) 2 ∂z
and note too that with dS denoting an element of surface area, dS = (2l)2
∂2Φ dxdy. ∂z∂ z¯
(ii) With H given by (15.35) and Φ general show that „ « ωc “ ∂ 2 Φ ”−1 “ ∂ ∂Φ ” ∂ Φ −Φ =− −2 −1 . e He 2 ∂z∂ z¯ ∂ z¯ ∂ z¯ ∂z Note from this that
z) e−Φ f (¯
(15.36)
is an eigenstate in the lowest Landau level for any f analytic in z¯. ∂ (iii) Verify that J := z ∂z − z¯ ∂∂z¯ commutes with H as defined by (15.35) for general Φ = Φ(|z|2 ), and in these cases the states (15.36) which are simultaneous eigenstates of J are z )n , e−Φ (¯
n ∈ Z≥0 .
(15.37)
3. [205] Consider the wave function “ ” 1 1 1 ψ˜0 = p exp − 2 z z¯ + 2 tanh μ z¯2 , 4l 4l 2l2 π cosh μ 2
z ) for f analytic in z¯ and so belongs to the lowest Landau level. The objective which is of the form e−z z¯/4l f (¯ here is to define operators ˜b and ˜b† such that ˜bψ˜0 = 0 and [˜b, ˜b† ] = 1, and to use the operator ˜b† to construct states orthogonal to ψ˜0 using the analogue of (15.25). (i) Make a Bogoliubov transformation involving the operator b in (15.28) by writing ˜b = αb + βb† with α2 − β 2 = 1 to verify that ˜b† = (cosh μ)b† − (sinh μ)b “1 “1 1 ∂ ” 1 ∂ ” = √ cosh μ z¯ − 2l − √ sinh μ z + 2l 2l ∂z 2l ∂ z¯ 2 2
710
CHAPTER 15
has the sought properties. (ii) By defining ψ˜m =
√1 (˜ b† )m ψ˜0 m!
analogous to (15.25) show that
ψ˜m = Pm (¯ z ) exp
“
−
” 1 1 z z¯ + 2 tanh μ z¯2 , 2 4l 4l
where Pm (z) is a polynomial of degree m defined by the recurrence “ 1 1 dPm (z) ” Pm+1 (z) = √ √ zPm (z) − 2l sinh μ , dz 2 m + 1 l cosh μ
P0 (z) = (2πl2 cosh μ)−1/2 .
Compare this with the Hermite polynomial three-term recurrence 2xHn (x) = Hn+1 (x) + 2nHn−1 (x) and thus conclude that ”−1/2 ” “ ”m/2 “ z 1 “1 Hm p Pm (z) = √ . 2πl2 cosh μ tanh μ m! 2 2l2 sinh 2μ 4. [110], [205] From q.3 (ii) it follows that the Slater determinant ψ constructed from the states ψ˜m , m = 0, 1, . . . , N − 1 is such that |ψ|2 is proportional to “ exp
−
N N ” X 1 X 2 1 2 2 2 (x + y ) + tanh μ (x − y ) j j j j 2l2 j=1 2l2 j=1
Y
|rj − rk |2 .
(15.38)
1≤j
This is proportional to the Boltzmann factor for a 2dOCP with a uniform background of a certain shape. The objective of this exercise is to deduce that the shape is an ellipse. (i) Consider an ellipse Ω with semi-axes A and B such that A = a cosh ξb and B = a sinh ξb . Introduce elliptic coordinates (ξ, η), 0 ≤ ξ ≤ ξb , 0 ≤ η ≤ 2π, related to the Cartesian coordinates by z := x + iy =
a cosh(ξ + iη). 2
Show that the Jacobian J in the change of variables dxdy = Jdξdη is J = (a2 /8)(cosh 2ξ − cos 2η). (ii) Use this formula for J and the expansion [408, Eq. (10.132)] ∞ “ a” X 2 “ − log |z − z | = − ξ + log + cosh nξ cos nη e−nξ cos nη 4 n n=1 ” + sinh nξ sin nη e−nξ sin nη
(15.39)
valid for ξ < ξ (otherwise interchange ξ and ξ on the r.h.s.) to show that Z ” πa2 “ dx dy log |z − z | = − cosh 2ξ + cos 2η − cosh 2ξ cos 2η e−2ξb + const. − 16 Ω = −b1 x2 − b2 y 2 + const., where b1 =
π (1 2
− e−2ξb ), b2 =
π (1 2
+ e−2ξb ).
2 (iii) Use (ii) to interpret the exponent √ in (15.38) as due to a uniformly charged ellipse, charge density −1/2πl , with B/A = e−2μ and a = l 2N .
(iv) Use (ii) to interpret (15.16) as due to a uniformly charged ellipse, charge density −1/(π(1 − √ the exponent in √ τ 2 )), with A = N (1 + τ ), B = N (1 − τ ).
711
THE TWO-DIMENSIONAL ONE-COMPONENT PLASMA
15.3 CORRELATION FUNCTIONS 15.3.1 The 2dOCP in a disk Introducing polar coordinates (r, θ), and using the Vandermonde identity (1.173) we have 2 2 |rk eiθk − rj eiθj | = det[(rj eiθj )k−1 ]j,k=1,...,N .
(15.40)
1≤j
Let ΩR be a disk of radius R centered on the origin, and let f (r) be a function of the polar coordinate r only. Observing the orthogonality R iθ k −iθ j f (r)(re ) (re ) dr = δj,k 2π r2j+1 f (r) dr (15.41) 0
ΩR
the method of proof of Propositions 5.1.1 and 5.1.2 allows the correlations to be computed for a class of Boltzmann factors extending that for the 2dOCP at Γ = 2 in a disk. P ROPOSITION 15.3.1 For the 2dOCP at Γ = 2 subject to a rotationally invariant one-body potential with Boltzmann factor such that e−βU(r1 ,...,rN ) dr1 · · · drN =
N j=1
f (rj )
2
|rk eiθk − rj eiθj |
N
rj drj dθj ,
(15.42)
j=1
1≤j
and confined to a disk of radius R, the n-particle correlation function is given by ⎡ ⎤ N 1/2 j−1 i(θμ −θγ )(j−1) 1 (f (rμ )f (rγ )) (rμ rγ ) e ⎦ ρ(n) (r1 , . . . , rn ) = det ⎣ R 2π j=1 f (r)r2j−1 dr 0
.
(15.43)
μ,γ=1,...,n
It remains to take the thermodynamic limit. Now, for a uniform background in a disk of radius R we have 2 from (1.72) that f (r) = Ae−πρr in (15.42), A constant. In taking the thermodynamic limit there are two distinct cases of interest: the first when r1 , . . . , rn are fixed in the bulk of the system, and the second when r1 , . . . , rn are some finite distance from the surface at |r| = R. The first case is particularly simple [313]. P ROPOSITION 15.3.2 For the n-particle correlation (15.43) with f (r) = Ae−πρr , corresponding to the 2dOCP in a disk with uniform background, 2 2 lim ρ(n) (r1 , . . . , rn ) = ρn det e−πρ(rμ +rγ )/2 eπρzμ z¯γ . (15.44) 2
N →∞ ρ fixed
μ,γ=1,...,n
In particular ρ(1) (r) = ρ,
2 ρ(2) (r1 , r2 ) = ρ2 1 − e−πρ|r1 −r2 | ,
(15.45)
where it is understood ρ(1) and ρ(2) refer to the thermodynamic values. Proof. In (15.43) we have 2 2 N ∞ X X 2 2 1 e−πρ(rμ +rγ )/2 (rμ rγ )j−1 ei(θμ −θγ )(j−1) (πρrμ rγ )j−1 ei(θμ −θγ )(j−1) −πρ(rμ +rγ )/2 lim = e R R 2 N →∞ 2πρ ρ fixed j=1 Γ(j) re−πρr r 2(j−1) dr j=1
0
2
2
= e−πρ(rμ +rγ )/2 eπρzμ z¯γ , where z = reiθ , as required.
712
CHAPTER 15
We will see in Exercises 15.3 q.5 that a general property of g(2) (r) := ρ(2) (0, r)/ρ2 for the 2dOCP is the functional equation [479] g(2) (r) = e±πiΓ/2 e−πρΓr
2
/2
g(2) (ir).
(15.46)
Note that this is indeed satisfied by the exact evaluation. Another property of the exact evaluation is that the truncated two-particle correlation has a Gaussian decay, which is in contrast to the algebraic decay of the same quantity in the one-dimensional log-gas systems. The exact result gives that at Γ = 2 the dimensionless structure function 1/2 1 ˜ S(k) := 1 + ρT(2) (r, 0)eiρ k·r dr (15.47) ρ R2 (cf. (7.3)) k := |k| is given by 2 ˜ S(k) = 1 − e−k /4π .
(15.48)
The evaluation of the surface correlation functions requires a more refined analysis than that of Proposition 15.3.2 [493]. It is convenient to choose as the origin (r, θ) = (R, 3π/2), and to specify points in Cartesian coordinates from this origin, so that 3π xμ + . 2 R Note that in the limit R → ∞ the background then occupies the upper half-plane y > 0. rμ2 = x2μ + (R − yμ )2 ,
θμ ∼ −
(15.49)
P ROPOSITION 15.3.3 With (xμ , yμ ) (μ = 1, . . . , n) as given in (15.49), 1 2 2 lim ρ(n) (r1 , . . . , rn ) = ρn det e−πρ(yμ +yγ ) h (yμ + yγ + i(xμ − xγ )) N →∞ 2 μ,γ=1,...,n ρ fixed where h(z) := with erf x :=
√2 π
x 0
1 1/2 π
∞ 0
e2(2πρ) zt e−t dt 1 2 (1 + erf t) 1/2
2
e−t dt denoting the error function. 2
Proof. Introducing x and y according to (15.49), and replacing j by N + 1 − j in the summation gives 2 2 N 1 X e−πρ(rμ +rγ )/2 (rμ rγ )j−1 ei(θμ −θγ )(j−1) RR 2πρ j=1 re−πρr2 r 2(j−1) dr
0
∼ e
2 −πρ(x2 μ +xγ )/2
e−πρ((R−yμ )
2
+(R−yγ )2 )/2
N X (πρ)N−j e(N−j)(log(R−yμ )+log(R−yγ )) ei(xμ −xγ )(N−j)/R
γ(N − j + 1; N )
j=1
Z
where γ(k; a) :=
a
tk−1 e−t dt
,
(15.50)
0
denotes the incomplete gamma function. Next we make use of the uniform asymptotic expansion „ „ «« j 1 , γ(N − j + 1; N ) ∼ Γ(N − j + 1) 1 + erf √ 2 2N and in this use Stirling’s formula to approximate the gamma function therein by Γ(N − j + 1) ∼ (2πN )1/2 N N−j e−N e−j
2
/2N
.
(15.51)
713
THE TWO-DIMENSIONAL ONE-COMPONENT PLASMA
An expression of the form stated formula follows.
√1 N
PN j=1
√ R∞ f (j/ N ) results. Since this sum tends to 0 f (t)dt in the limit N → ∞, the
In particular, Proposition 15.3.3 gives for the density profile as measured from the wall at y = 0, 1/2 2 ρ −2πρy2 ∞ e2(2πρ) yt e−t dt. (15.52) ρ(1) (y) = 1/2 e 1 π 0 2 (1 + erf t) At contact with the wall (y = 0), the fact that evaluation
d dt
log(1 + erf t) =
2 √2 e−t /(1 + erf π
t) reveals the closed form
ρ(1) (0) = ρ log 2.
(15.53)
We also draw attention to the asymptotic behavior of the truncated two-particle correlation at large distances |x1 − x2 | along the wall. Thus integration by parts of h(z) gives 2ρe−2πρ(y1 +y2 ) . π 2 (x1 − x2 )2 2
ρT(2) (r1 , r2 )
∼
|x1 −x2 |→∞
−
2
(15.54)
15.3.2 Complex random matrices According to Proposition 15.1.1 the only difference between the 2dOCP at Γ = 2, with ρ = 1/π, and complex random matrices from the viewpoint of (15.44) is that in the latter we require R = ∞. By the readily verified formula ∞ N Γ(N ; x) xj−1 −x =e , Γ(j; x) := tj−1 e−t dt = Γ(x) − γ(j; x) (15.55) Γ(N ) (j − 1)! x j=1 the correlations are immediate. P ROPOSITION 15.3.4 For complex random matrices as specified in Proposition 15.1.1, 2 2 Γ(N ; zμ z¯γ ) . ρ(n) (r1 , . . . , rn ) = π −n det e−(rμ +rγ )/2 ezμ z¯γ Γ(N ) μ,γ=1,...,n In the limit N → ∞ the result for the bulk correlations (15.44), with√ ρ = 1/π, is evident. To compute the edge correlation, with r = (x, y) we move the origin to x = 0, y = − N , so that (15.49) applies. Making use then of (15.55) gives the following result. P ROPOSITION 15.3.5 For complex random matrices as specified in Proposition 15.1.1 lim ρ(n) ((x1 , −N + y1 ), . . . , (xn , −N + yn )) 1 2 = π −n det e−(rμ −rγ ) /2 H (yμ + yγ − i(xμ − xγ )) , 2 μ,γ=1,...,n
N →∞
where H(z) :=
√ 1 1 + erf( 2z) . 2π
15.3.3 Correlation functions for the zeros of some complex random polynomials For the p.d.f. (15.14) it is not known how to compute the correlation functions directly. However, by returning to the expression (15.12) for the joint distribution of the coefficients, a closed form expression for ρ(n) can
714
CHAPTER 15
be obtained [288]. Note that the distribution in (15.12) can be regarded as an (N + 1)-dimensional complex −2 Gaussian distribution with covariance matrix L := diag(σ1−2 , . . . , σN +1 ) (see Exercises 15.3 q.3). (0)
(0)
The first step is to introduce complex numbers z1 , . . . , zk (k < (N + 1)/2) and to define 2k linear (0) (0) combinations of the coefficients a0 , a1 , . . . , aN by p(zl ) =: pl and p (zl ) =: pl (l = 1, . . . , k), where the prime denotes differentiation. Then general properties of the Gaussian distribution as revised in Exercises 15.3 q.3 give that in terms of these 2k complex variables (with the other N + 1 − 2k complex variables integrated out), the probability measure (15.12) reduces to k 1 1 1 † −1 i exp − ( p , p ) M ( p , p ) dprl dpil dpr l dpl , (2π)2k det M 2 l=1
where M is the covariance matrix A B , A := [pj p¯l L ]j,l=1,...,k , B := [pj p¯l L ]j,l=1,...,k , C := [pj pl L ]j,l=1,...,k . M= B† C (0)
(0)
(0)
For given points z1 , . . . , zk the above expression gives the p.d.f. for the corresponding values of p(z1 ), (0) (0) (0) (0) (0) . . . , p(zk ) and p (z1 ), . . . , p (zk ). We want to now change variables so that the points z1 , . . . , zk re(0) (0) place the function values p(z1 ), . . . , p(zk ) as the variables. The Jacobian for each such change of variables 2 is simply |pl | so we obtain k 1 1 1 r i † −1 exp − ( p , p ) M ( p , p ) |pl |2 dzlr dzli dpl dpl . (2π)2k det M 2 l=1
Of course the change of variables is only locally one to one, as there will be in general k points giving the i same function value. This shows that if we set p = 0 and integrate over each pr l , pl the k-point correlation function for the zeros will result, giving k 1 1 1 (0) (0) r i 2 0, p )† M−1 (0, p ρ(k) (z1 , . . . , zk ) = ( dp dp |p | exp − ) . (15.56) l l l (2π)2k det M (−∞,∞)2k 2 l=1
P ROPOSITION 15.3.6 The integral formula (15.56) for ρ(k) can be evaluated with the result [288] (0)
(0)
ρ(k) (z1 , . . . , zk ) =
1 per(C − B† A−1 B) πk det A
(15.57)
(recall (4.35) for the definition of a permanent). Proof. Introducing the auxiliary complex vector ν , (15.56) can be rewritten (0)
(0)
ρ(k) (z1 , . . . , zk ) =
Z k Y 1 ∂ 2k 1 i dpr l dpl π 2k det M ∂ν1 · · · ∂νk ∂ ν¯1 · · · ∂ ν¯k (−∞,∞)2k l=1 “ 1 ”˛˛ 1 1 † −1 ∗ × exp − (0, p ) + i − i , ) M (0, p ν ·p ν· p ∗ ˛˛ 2 2 2 ν = 0
)† M−1 (0, p ) = p ∗T N−1 p , where N−1 is where t ∗ denotes the complex conjugate of entries of t. Now write (0, p −1 the k × k matrix which is the lower-right k × k submatrix of M , and so related to the blocks of M by N = C − B† A−1 B. The integral is then of the type evaluated by the formula (15.65) below, so we have (0) (0) ρ(k) (z1 , . . . , zk )
=
“1 ”˛˛ ∂ 2k ∗ exp ν ˛˛ . ν · N det M ∂ν1 · · · ∂νk ∂ ν¯1 · · · ∂ ν¯k 2 ν = 0
“ 2 ”k det N π
(15.58)
715
THE TWO-DIMENSIONAL ONE-COMPONENT PLASMA
Carrying out the differentiation and noting det M = det A det(C−B† A−1 B) = det A det N gives the stated formula.
In general the structure of the ρ(k) for the zeros of random complex polynomials implied by Proposition 15.3.6 is more complicated than the structure exhibited by the ρ(k) for the eigenvalues of complex random matrices as given in Proposition 15.3.2. In particular, in general there is no simple formula for ρT(k) in the random polynomial problem (for an exception see Exercises 15.7 q.3). The simplest correlation is the one body density, when the formula of Proposition 15.3.6 simplifies to read ρ(1) (z) =
p L ¯ pp L pL p p¯ L − p¯ 1 p¯ 1 ∂2 logp¯ pL . = π (p¯ pL )2 π ∂z∂ z¯
(15.59)
In the specific case that σj2 = 1/j!, as occurred in the study of the random single particle state confined to the lowest Landau level (recall Section 15.1.2), we have p¯ pL =
N |z|2l l=0
l!
= e|z|
2
Γ(N + 1; |z|2 ) , Γ(N + 1)
(15.60)
where the second equality follows from (15.55). We know from (15.51) that the asymptotics of Γ(N +1; |z|2 ) switches behavior at |z|2 = N + 1. This indicates that |z|2 = N defines to leading order the boundary of the support of the density √ (which must therefore be equal to 1/π inside the disk, and zero outside). Indeed by writing z = x + i(y − N ) and using (15.51) we see from (15.59) and (15.60) that in the limit N → ∞ 1/π, √ 1 ∂2 1 y → ∞, 1+ ρ(1) (y) = log(1 + erf( 2y)) ∼ 1/4y 2, y → −∞. π 4 ∂y 2 The two-particle correlation function in the case σj2 = 1/j! can also readily be computed in the limit N → ∞. For z1 = x1 + iy1 and z2 = x2 + iy2 fixed one finds [288] 1 f (|z1 − z2 |2 /2), π2 1 d2 2 (sinh2 x + x2 ) cosh x − 2x sinh x x = coth x . (15.61) f (x) := 2 dx2 sinh3 x This result has been proved to be universal, in the sense that it persists for random polynomials with complex coefficients aj drawn from a large class of distributions of mean zero and variance 1/j! [67]. ρ(2) (z1 , z2 ) =
E XERCISES 15.3
1. [320] The aim of this exercise is to compute the asymptotic expansion of the free energy for the 2dOCP in a disk at Γ = 2. (i) Use (4.160) and (1.72) to show that for the 2dOCP in a disk at Γ = 2 βF = − log Z
where
Z
QN :=
dr1 . . . Ω
QN 3 + N 2 log R − N 2 , N! 4
drN e−πρ
PN
j=1
Y
| rj |2
Ω
|rk − rj |2 .
1≤j
(ii) Make use of (15.40) and (15.41) to show that QN = N ! π N (πρ)−N(N+1)/2
N Y j=1
where γ(k; a) is specified by (15.50).
γ(j; N ),
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CHAPTER 15
(iii) Make use of the asymptotic expansions (15.51) and (4.186) (with β = 2) to deduce that for N, R → ∞ with ρ fixed ” “ 1 (15.62) βF ∼ πR2 βfv + 2πRβγ + log (πρ)1/2 R + O(1), 6 where „ « “ ρ ”1/2 Z ∞ ´ ` ρ 1 log βfv = log ρ/2π 2 , βγ = − (1 + erf y) dy (15.63) 2 2π 2 0 (fv is the free energy per volume and γ the surface tension). (iv) [313] Use an appropriate generalization of (4.179) together with (15.45) to show that in the thermodynamic limit the internal energy per particle is given by 1 u = − (log πρ + C), 4 where C is Euler’s constant. 2. [203] Calculate the leading two terms in the X → ∞ asymptotic expansion of erf(Y + iX), and from this and Proposition 15.3.5 conclude that the edge correlations for complex random matrices has the property ρT(2) (y1 , y2 ; x1 − x2 )
∼
|x1 −x2 |→∞
−
f (y1 )f (y2 ) , 4π 2 (x1 − x2 )2
f (y) := π
d ρ(1) (y). dy
(15.64)
3. Consider the integral identity (1.93) with n replaced by 2n and A of the form » – a b A= , −b a uT , v T ) where u and v are n-component vectors, and let * where a and b are n × n matrices, and write bT = ( denote complex conjugation. (i) Show that (1.93) can be written as Z “ 1 “1 ” ” “ ”−1 1 ∗ 1 · z ∗ = (2π)n det C w ∗ ·C−1 w dx1 dy1 · · · dxn dyn exp − z † C z+ w z + w· exp , 2 2 2 2 R2n (15.65) where z= x + i y and w = u + iv . (ii) Let f C denote the normalized average with respect to exp(−(1/2) z † C z ). From (15.65) deduce that X Ci−1 · · · Ci−1 2−p zi1 z¯j1 · · · zip z¯jp C = p P (jp ) 1 P (j1 ) all permutations P of {j1 ,...,jp }
while zi1 · · · zip z¯j1 · · · z¯jq C = 0
for
p = q.
Use the first of these results to deduce that C−1 =
1 [zj z¯k C ]j,k=1,...,N . 2
(iii) Regarding the r.h.s. of (15.65) as a p.d.f. for w—the multivariable complex Gaussian distribution—show that the results of Exercises 1.5 q.1(iii)&(iv) (appropriately modified) hold. 4. [479] The objective of this exercise is to derive the functional formula (15.46). (i) Check that N Y j=3
|r1 eiθ1 − rj eiθj |Γ |r2 eiθ2 − rj eiθj |Γ =
N “ Y j=3
z1 − z¯j )(z2 − zj )(¯ z2 − z¯j ) (z1 − zj )(¯
”Γ/2
717
THE TWO-DIMENSIONAL ONE-COMPONENT PLASMA
is unchanged by the interchange z1 ↔ z2 with z¯1 and z¯2 fixed. z2 , z¯2 )) to show that for a 2dOCP with (ii) Use this property and the definition of ρ(2) (r1 , r2 ) = ρ(2) ((z1 , z¯1 ); (¯ Boltzmann factor N Y
Y
f (rj )
j=1
|rj − rk |Γ =
N Y j=1
1≤j
Y
f (zj , z¯j )
|zj − zk |Γ
1≤j
the functional property ρ(2) ((z2 , z¯1 ); (z1 , z¯2 )) ρ(2) ((z1 , z¯1 ); (z2 , z¯2 )) = f (z1 , z¯1 )f (z2 , z¯2 )((z1 − z2 )(¯ z1 − z¯2 ))Γ f (z2 , z¯1 )f (z1 , z¯2 )((z2 − z1 )(¯ z1 − z¯2 ))Γ holds. Put f (r) = e−πρΓ| r| (15.46).
2
/2
, set z1 = 0 and assume a fluid state in the thermodynamic limit to deduce
(iii) Show that (15.46) is equivalent to saying g(2) (r) possesses the functional form g(2) (r) = r Γ e−πρΓr
2
/4
f (r 4 )
for f (x) analytic. 5. [172] In this exercise the eigenvalue density of a matrix with complex eigenvalues will be related to that in a related chiral ensemble. ∂ (i) Note from the Poisson equation (1.40), written in terms of the operators ∂z , ∂∂z¯ defined below (15.27), that for any two-dimensional statistical mechanical system with N particles at (xj , yj ) (j = 1, . . . , N ),
E 1 ∂2 D X 2 ∂2 log(z − zj )(¯ z − z¯j ) = π ∂z∂ z¯ j=1 π ∂z∂ z¯ N
ρ(1) (r) =
Z R2
log |z − w|ρ(1) ((x , y )) dx dy , (15.66)
where z := x + iy, w = x + iy . (ii) In the case of a random matrix X with complex eigenvalues {zj }j=1,...,N , note that N X
“ ” ¯ = Tr log (−1)N H , log(z − zj )(¯ z − z¯j ) = Tr log(z1N − X)(¯ z 1N − X)
(15.67)
j=1
»
where H=
0 ¯T z¯1N − X
z1N − X 0
– .
Let ρH ¯) denote the eigenvalue density of the positive eigenvalues for the class of Hermitian matrices (1) (x; z, z H, and ρX ( r ) the eigenvalue density for the random matrices X. By substituting (15.67) in the first equality (1) of (15.66) deduce the inter-relation Z ∞ 2 ∂2 ρX r) = (log t)ρH ¯) dt. (1) ( (1) (t; z, z π ∂z∂ z¯ 0 6.
(i) [242] Use the result of Exercises 15.2 q.3 to deduce that {Cn (z)}n=0,1,... with Cn (z) := satisfies the orthogonality Z Z ∞ dx −∞
∞ −∞
dy e−x
2
“ τ ”n/2 2
/(1+τ )−y 2 /(1−τ )
“ z ” Hn √ 2τ
p Cm (z)Cn (¯ z ) = πm! 1 − τ 2 δm,n .
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CHAPTER 15
As in going from (15.41) to (15.43), conclude that the correlation kernel for the p.d.f. (15.16) is given by (1 − τ 2 )−n/2 π −n exp
“
−
n ”” N−1 X Cl (zj )Cl (¯ zk ) 1 “X τ 2 2 2 |z | − + z ¯ ) (z . l l l 1 − τ 2 l=1 2 l! l=0
(15.68)
(ii) [250] The √ weakly non-Hermitian limit is specified by writing τ = (1 − α2 /N ), scaling the coordinates z → πz/ N and taking the limit N → ∞. This has the effect of concentrating the eigenvalues in an O(1) neighborhood of the x-axis. Use the asymptotic expansion (7.1) and the result (15.68) to show that the limiting correlation kernel (the scale factor π 2 /N must first be multiplied to account for the scale of dxdy) is equal to ”Z 1 “ π2 2 2 (2π)1/2 e−α u cos πu(z − z ) du. exp − 2 (y 2 + (y )2 ) α α 0
15.4 GENERAL PROPERTIES OF THE CORRELATIONS AND FLUCTUATION FORMULAS 15.4.1 Dipole moment and Stillinger-Lovett sum rules In Chapter 14 macroscopic physical characterizations of log-gases were used to make certain predictions relating to correlations for general β. These considerations carry over to the case of the 2dOCP, or more generally two-dimensional Coulomb systems in their conductive phase [386]. Here, for definiteness, only the case of the 2dOCP will be considered. One-component systems are special because only then is the microscopic particle density n(1) (r) proportional to the microscopic charge density c(1) (r ). We will begin by generalizing (14.12) (generalization is possible because, as distinct from the log-gas on a line, the correlations for the 2dOCP are expected to decay faster than any algebraic power). For this we use the linear response relation (14.1), with the perturbation due to an external charge δq at the origin, l δU = −δq R2 c(1) (r) log |r| dr, and with the observable A = j=1 n(1) (rj − a), which gives Aδq − A0 = βδq
R2
l T n(1) (rj ) dr. log |r + a| c(1) (r)
(15.69)
j=1
With the assumption that the l.h.s. decays faster than any algebraic power for |a| → ∞, and use of the multipole expansion ∞ (−1)p+1 r p cos p(θ − φ) log |r + a| = log |a| + p a p=1 where in polar coordinates r = (r, θ), a = (a, φ) and it is assumed r/a < 1, taking |a| → ∞ in (15.69) gives l T (x − iy)p c(1) (r) n(1) (rj ) dr = 0 (p = 0, 1, . . . ). R2
j=1
Proceeding as in Exercises 14.1 q.1(i) this can be rewritten as l (x − iy)p ρT(l+1) (r1 , . . . , rl , r) dr = − (xj − iyj )p ρT(l) (r1 , . . . , rl ). R2
(15.70)
j=1
In the case p = 0 this is formally the same as (14.12). For p = 1 it is referred to as the dipole moment sum rule. Its validity for Γ = 2 can be checked from Proposition 15.3.2 and (5.9), together with use of polar coordinates. ˆ ˜ For the log-gas on a line the structure function S(k), related to the dimensionless structure function S(k) by (14.7), exhibits the small |k| behavior (14.8). Applying essentially the same linear response argument, but
719
THE TWO-DIMENSIONAL ONE-COMPONENT PLASMA
with a two-dimensional external charge density eik·r , and the two-dimensional generalized function result 2π log |r |e−ik·r dr = 2 , k := |k|, (15.71) − k 2 R one obtains the prediction that k2 . k→0 2πβ
ˆ S(k) ∼
(15.72)
In terms of the notation (14.16), and assuming fast decay of the latter, equating powers of k 2 on both sides shows 2 (15.73) r 2 C(2) (r, 0) dr = − . πβ 2 R This condition on the second moment of C(2) (r, 0) is known as the Stillinger-Lovett sum rule. It can in fact be rigorously derived from the n = 2 Born-Green-Yvon equation, together with the assumptions that (15.70) holds for p = 1, l = 1, 2 and the three-particle correlations have a sufficiently fast decay [387]. 1/2 1/2 ˆ ˜ ˜ Analogous to (14.7), in two dimensions S(k) = ρS(k/ρ ), where S(k/ρ ) is dimensionless. If we 1/2 1/2 ˜ now expand S(k/ρ ) as a power series in k/ρ , and then take ρ → ∞, we see from (15.73) that k2 ˆ . lim S(k) = ρ→∞ 2πβ
(15.74)
This is equivalent to the statement that [316] 1 ∇2 δ(r). lim C(2) (r, 0) = − 2πβ
(15.75)
ρ→∞
For the 2dOCP, C(2) (r, 0) = q 2 ρT(2) (r, 0). In this case, the linear response argument leading to (14.15) can be generalized, implying the fourth moment condition Γ 16 . (15.76) 1 − ρ r4 ρT(2) (r, 0) dr = − (πΓ)3 4 R2 ˆ In fact the expansion of S(k) is known to O(k 6 ) [340]. It has been computed using exact diagrammatic expansion techniques, and implies the sixth moment sum rule 18 8 . (15.77) ρ2 r6 ρT(2) (r, 0) dr = − (Γ − 6) Γ − (πΓ)4 3 R2 The sum rules (15.73), (15.76) and (15.77) can be checked for Γ = 2 on the exact result (15.45).
15.4.2 Contact theorem and related sum rules The sum rule (15.73) is expected to be true of general two-dimensional Coulomb systems in their conductive phase. In the special case of the 2dOCP, a different viewpoint on (15.73) can be obtained by studying the consequence of scaling properties of the partition function. On this theme, we begin by noting from (1.72) that with N 2 2 2 1 ZN = e−(Γ/2)N log R+3ΓN /8 dr1 · · · drN e−πρΓ|rl | /2 |rk − rj |Γ , N ! |r1 |
1≤j
(15.78) ρ = N/πR2 , changing variables rj → Rsj , shows that the dependence on R can be scaled from the integral.
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CHAPTER 15
Hence, analogous to (4.169) one has βP :=
Γ ∂ log ZN = 1 − ρ. 2 ∂(πR ) 4
(15.79)
A simple consequence of this is a formula expressing the density at contact in terms of a moment of the density profile in the finite system [109]. P ROPOSITION 15.4.1 With the density measured inward from the boundary Γρ2 π 2 R 3 Γ ρ=− ρ(1) (0) − 1 − r ρ(1) (R − r) − ρ dr. 4 N 0
(15.80)
Proof. Direct differentiation of (15.78) and use of the general definition of the density (5.1) with n = 1 gives πN Γ “ 1 ”2 ∂ log ZN = ρ(1) (0) + 2 ∂(πR ) 2 πR2
Z
R
” “ r 3 ρ(1) (R − r) − ρ dr.
(15.81)
0
Equating (15.79) and (15.81) gives (15.80).
In the thermodynamic limit (15.80) reduces to a formula for the density at contact with the boundary and the potential drop. P ROPOSITION 15.4.2 For the 2dOCP in the half plane y ≥ 0, the potential drop Δφ is specified by ∞ y(ρ(1) (y) − ρ) dy. (15.82) Δφ = 2π 0
In terms of this quantity, and with βP specified by (15.79), we have ρ(0) = βP + ρΓΔφ.
(15.83)
Proof. Neutrality implies Z
R 0
Z r(ρ(1) (R − r) − ρ) dr =
R
(R − r)(ρ(1) (r) − ρ) dr = 0.
(15.84)
0
Changing variables in (15.81) r → R − r and making use of the above equation implies Z Γρ2 π 2 R 3 r (ρ(1) (R − r) − ρ) dr − N 0 Z Z Z R 3Γ(ρπ)3/2 R 2 Γρ2 π 2 R 3 r(ρ(1) (r) − ρ) dr − r (ρ (r) − ρ) dr + r (ρ(1) (r) − ρ) dr. = 2Γρπ (1) N 1/2 N 0 0 0 For fast decay of ρ(1) (r) − ρ, the second two terms on the r.h.s. tend to zero as N, R → ∞ and so the stated result follows from (15.80).
The so-called contact theorem (15.82) can be verified at Γ = 2 using the exact results (15.52) and (15.53). The idea behind the derivation of Proposition 15.4.1 can be applied to give a formula for the second moment of ρT(2) in the 2dOCP in the finite system [511]. P ROPOSITION 15.4.3 We have 2 ρ(1) (0)/ρ + N ρT(2) (0, R)/ρ2 , r2 ρT(2) (0, r) dr = − πΓ | r|
(15.85)
721
THE TWO-DIMENSIONAL ONE-COMPONENT PLASMA
Proof. Consider Z˜N−1 := e−(Γ/2)N ×
2
log R+3ΓN 2 /8
1 (N − 1)!
Z
Z
|r 1 |
dr1 · · ·
drN−1 |rN −1 |
N−1 Y
|rl |Γ e−πρΓ| rl |
2
/2
l=1
Y
|rk − rj |Γ ,
1≤j
which corresponds to the partition function for the 2dOCP consisting of N particles with one of the particles fixed at the origin. By scaling out the radius we deduce 1 ∂ N −1 Γ Z˜N−1 = ρ(1) (0) − ρρ(1) (0), πR2 4 Z˜N ∂(πR2 )
(15.86)
and direct differentiation gives ∂ Γρ 1 Z˜N−1 = ρ(2) (0, R) + 2R2 Z˜N ∂(πR2 ) Now write
| r |
r 2 ρ(2) (0, r) dr −
Z
Z | r |
Z
r 2 ρ(2) (0, r) dr =
| r|
r 2 ρT(2) (0, r) dr + ρ(1) (0)
Γ ρρ(1) (0). 4
(15.87)
“ 2N ”“ ” ρ(1) (R) − (1 − Γ/4)ρ , 2 Γρ π
where the second equality follows upon use of (15.80), paying attention to the fact that the origin is the boundary in that equation. Substituting in (15.87) shows Z ∂ 1 ˜N−1 = ρ(2) (0, R) + Γρ r 2 ρT(2) (0, r) dr − ρρ(1) (0). (15.88) Z 2R2 | r|
In the thermodynamic limit, with the assumption of fast decay of correlations, it is expected that ρ(1) (0)/ρ → 1 while N ρT(2) (0, R)/ρ2 → 0 at rates faster than any power law, so implying (15.73). 15.4.3 Surface correlations Whereas the particle correlations in the bulk of a two-dimensional Coulomb system in the conductive phase are expected to decay exponentially fast, the same correlations parallel to a boundary will typically exhibit a slow algebraic decay [314]. This is evident from the exact result (15.54) for the asymptotic decay of the truncated two-particle correlation parallel to a hard wall (with a vacuum outside the wall). In fact a universal behavior can be identified if instead of the charge-charge correlation C(2) (r, r ) := c(1) (r)c(1) (r )T one considers the asymptotic surface charge–surface charge correlation ∞ ∞ σ(x)σ(x )T := dy dy Sas (r, r ), 0
0
where Sas (r, r ) is the leading large |x − x | asymptotic form of S(r, r ). (In the case of the soft edge 2dOCP as realized by the eigenvalue PDF for complex Gaussian matrices, the terminals of integration should begin at −∞.) We see from (15.54) and (15.64) that for the 2dOCP at Γ = 2, with a hard wall or soft edge, βσ(x)σ(x )T
∼
|x−x |→∞
−
1 . − x )2
2π 2 (x
(15.89)
This result bears a striking resemblance to the asymptotic behavior of the charge-charge correlation for the log-gas (recall (14.9)). In fact it is identical apart from an extra factor of 21 on the r.h.s. of (15.89).
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CHAPTER 15
Before giving a derivation of this result, let us consider the heuristic reason behind the slow decay. When a charge is isolated near the surface, or any other point within the Coulomb system for that matter, a screening cloud of equal and opposite total charge will form (implying the charge sum rule must be obeyed). However, as distinct from isolating a charge in the bulk, the screening cloud about a charge isolated near the boundary cannot be spherically symmetric. As a result, it will carry a nonzero dipole moment, and it is this which is responsible for the slow decay of the correlation. For the derivation of (15.89) we apply the method of Section 14.3 due to Jancovici [316]. This requires that the macroscopic potential φ(r ) due to the charge distribution in the Coulomb system when an external charge δq, regarded as a perturbation, is fixed at r. There are four cases to consider, depending on the location of the points r and r inside or outside the domain Ω, which for the time being will be taken to be an arbitrary planar region with its boundary forming a simple curve. P ROPOSITION 15.4.4 The macroscopic potential φ(r ) defined above is given by ⎧ δq log |r − r | + δq/C, r, r ∈ Ω, ⎪ ⎪ ⎨ δq log |r − r | + δqF (r ), r ∈ Ω, , r ∈ / Ω, φ(r ) = δq log | r − r |, r ∈ / Ω, , r ∈ Ω, ⎪ ⎪ ⎩ δq log |r − r | + δqG(r, r ), r, r ∈ /Ω where C is the capacitance, δqG(r, r ) is the potential at the point r due to the charge at r and the induced surface charge, and δqF (r ) is the potential at r due to the surface charge. Derivation. For both r and r in the system, a screening cloud of charge −δq will surround the charge δq at r. From r |. Also, due to charge large distances this appears as a point charge, and so creates a potential at r equal to δq log |r − conservation, a charge δq will spread itself around the boundaries creating a constant potential δq/C, where C is the capacitance. Thus Φ(r ) − Φ(r )0 = δq log |r − r | + δq/C. (15.90) For an infinite system in two dimensions 1/C will diverge (e.g., for a disk of radius R, 1/C = − log R). However, we will see below that only the derivatives ∂ 2 /∂rμ ∂rν are relevant to the derivation of (15.89). Now suppose r is in the system, but r is outside. Reasoning as above gives Φ(r ) − Φ(r )0 = δq log |r − r | + δqF (r ),
(15.91)
where δqF (r ) is the potential at r due to the surface charge. In the case r is outside and the observation point r is in, there is no screening cloud but there will be an induced surface charge of zero total charge with a potential cancelling that due to the external charge, so that (15.92) Φ(r ) − Φ(r )0 = δq log |r − r |. The final case, when both r and r are outside the conductor, follows by definition of G.
Substituting the results of Proposition 15.4.4, via the linear response relation (14.1), in the formula (14.43) for the electric field–electric field correlation and proceeding as in the derivation of (14.45) we see that 1 ∂ 2 G(r, r ) βσ(r )σ(r )T = − , (15.93) (2π)2 ∂rn ∂rn r,r ∈surface where rn is the direction normal to the surface. It remains to specify G(r, r ). In the case of the half-plane boundary, the method of images gives G(z, z ) = − log |z − z | + log |z − z¯ |. Substituting in (15.93) gives the formula (15.89). As another explicit example, consider the case of a disk of radius R. Then the method of images gives for G the expression (15.174) below, which when substituted in
723
THE TWO-DIMENSIONAL ONE-COMPONENT PLASMA
(15.93) implies βσ(r )σ(r )T = −
1 , 2π 2 (2R sin θ2 )2
(15.94)
where θ is the angle between r and r . 15.4.4 Distribution of a linear statistic In general the Fourier transform of the p.d.f. for a linear statistic A = Pˆ (k) =
N
eika(rl )
N j=1
a(rj ) is given by
(15.95)
l=1
(recall Section 14.4). In the case of the OCP in a disk with a soft wall at Γ = 2, or equivalently the problem of Gaussian random matrices with complex entries, the quantity (15.95) can be evaluated exactly for rotationally N symmetric linear statistics A = j=1 a(|rj |). Furthermore, by scaling the variables so that the support of the system is the unit disk, the limiting behavior of the distribution can be explicitly determined [194]. √ P ROPOSITION 15.4.5 Let e−βW be given by (15.1) with rj → rj / N , and defined Pˆ (k) by (15.95) N (generalized to a two-dimensional domain) with a linear statistic A = j=1 a(|rj |). Then for N → ∞ 2 2 Pˆ (k) ∼ eikμ e−k σ /2
with
1
ra(r) dr,
μ = 2N 0
σ2 =
1 2
1
r(a (r))2 dr.
0
Proof. From (15.95) R Pˆ (k) =
R2
R Q 2 dr1 · · · R2 drN e−N| rl | +ika( rl ) 1≤j
(15.96)
Introducing the Vandermonde determinant as in the proof of Proposition 15.3.1 allows the integrals in this expression to be evaluated, with the result √ N R ∞ −s l−1 ika( s/N) Y e s e ds 0 R∞ . Pˆ (k) = −s sl−1 ds e 0 l=1 To analyze the N → ∞ limit, we change variables s → ls and expand the integrand about its large l maximum at s = 1. This gives N √ √ Y 2 2 Pˆ (k) ∼ eika( l/N) e−k (a ( l/N )) . l=1
Writing the product of exponentials as the exponential of a sum shows that in fact Riemann sums occur, and their leading behaviors written as integrals gives the stated formula.
The result of Proposition 15.4.5 exhibits two key features in common with the fluctuation formula (14.68) for the log-gas. These are that the distribution is a Gaussian, and that the variance is O(1). Indeed the argument used to deduce that the distribution will be a Gaussian in the infinite density (macroscopic) limit applies immediately to the two-dimensional OCP, with the key formula (14.61) from macroscopic electrostatics giving an exact linear relationship between an external potential and the corresponding induced charge density. No assumption of a rotationally invariant statistic is required. Let us now turn our attention to the computation of the variance in the case of general Γ and no assumed symmetry for the linear statistic. With q = 1 for convenience, according to the formulas (14.38) and (14.16)
724
CHAPTER 15
we have
dr1
Var(A) = Ω
dr2 a(r1 )a(r2 )C(2) (r1 , r2 ),
(15.97)
Ω
where Ω denotes the unit disk. The significance of this is that in the infinite density limit we know that C(2) exhibits universal behavior in the bulk and at the boundary. In the bulk, from (15.75), we have 1 ∂ ∂ ∂ ∂ C(2) (r1 − r2 , 0) = δ(r1 − r2 ). + i − i 2πβ ∂x(1) ∂y (1) ∂x(2) ∂y (2) Substituting in (15.97), and integrating by parts (ignoring possible boundary terms, which are treated separately below) gives ∂a(x, y) 2 ∂a(x, y) 2 1 2 dxdy. (15.98) + σbulk = 2πβ Ω ∂x ∂y In the special case a(r) = a(|r|), β = 2, this reproduces the exact result of Proposition 15.4.5 for σ 2 . At the surface, in terms of polar coordinates, we have from (15.94) that ∂2 1 θ1 − θ2 C(2) (r1 − r2 , 0) = − 2 log sin δ(r1 − 1)δ(r2 − 1). 2π β ∂θ1 ∂θ2 2 Substituting in (15.97), integrating by parts and introducing the Fourier series (14.68) gives 2 = σsurface
∞ 2 n|an |2 , β n=1
a(1, θ) =
∞
an einθ .
(15.99)
n=−∞
2 vanishes for a(r) = a(|r|), but in general Var(A) is equal to the sum of (15.98) and (15.99), Note that σsurface which are both O(1). A simple check on these predictions is possible. Consider the definition (15.96) of Pˆ (k), suitably generalized to account for general β. We notice that with the special choice of linear statistic a(x, y) = c10 x + c01 y, the dependence on k can be factored out of the integral by completing the square, giving the exact formula 2 2 2 Pˆ (k) = e−k (c10 +c01 )/2β
independent of N . Thus the distribution is indeed Gaussian and the variance is given by σ 2 = (c210 + c201 )/β. 2 This agrees with the prediction from the general formulas (15.98) and (15.99), since they give σbulk = (c210 + 2 2 2 2 2 c01 )/2β and σsurface = (c10 +c01 )/2β and thus sum to σ . For the case β = 2 with a soft wall (corresponding to the eigenvalues of complex Gaussian matrices considered in Section 15.1) the above predictions have been rigorously proved [468], [15]. N The variance of a linear statistic A = j=1 a(rj /α) in the infinite system, with α → ∞, is considered in Exercises 15.4 q.1. 1. Consider the formula (15.97) for the variance of the linear statistic and suppose Ω = R2 . In the case that a(r) → a(r/α), use (15.75) to show that Z ““ ∂a(x, y) ”2 “ ∂a(x, y) ”2 ” 1 dxdy. lim Var(A) = + α→∞ 2πβ R2 ∂x ∂y
E XERCISES 15.4
By using the convolution theorem in (15.97) and using (15.71), obtain the alternative formula Z 1 1 lim Var(A) = |ˆ a(k)|2 |k|2 dk α→∞ 2πβ (2π)2 R2 (cf. (14.54)).
THE TWO-DIMENSIONAL ONE-COMPONENT PLASMA
725
2. Use (15.64) to verify (15.89) in the case of the 2dOCP with a soft edge. 3. Applying the expansion (7.5) to the exact result of Proposition 15.3.2 gives the expansion [313] “ 2 1 g(2) (r, 0) = −e−πρr + (Γ − 2) Ei(−πρr 2 ) − Ei(−πρr 2 /2) 2 “ ” −πρr 2 1 2 +e Ei(πρr /2) − (log πρr 2 + C) + O((Γ − 2)2 ), 2 R∞ where C denotes Euler’s constant and Ei(x) := x (e−t /t) dt. Deduce from this that [340] 1 “ πΓρ ”j ρ 2
Z
j “X 2k − 1 j” r 2j ρT(2) (r, 0) dr = −j! + (Γ − 2)j! − + O((Γ − 2)2 ), k + 1 2 R2 k=0
and so verify that the sum rules (15.70) (in the case l = 1, p = 0), (15.73), (15.76) and (15.77) are obeyed to the same order.
15.5 SPACING DISTRIBUTIONS 15.5.1 The bulk The orthogonality (15.41) allows some two-dimensional analogues of spacing distributions to be computed explicitly in terms of the incomplete gamma functions. This contrasts with the one-dimensional case, where we saw in Chapters 8 and 9 that the calculation of spacing distributions requires Painlev´e transcendents. Now, for one-dimensional systems we have defined Eβ (n; s) as the probability that an interval of length s contains exactly n particles, and Pβ (n; s) as the p.d.f. for the event that our randomly chosen particle is a distance s from another particle, and there are exactly n particles in between. For a two-dimensional system in a fluid state in the bulk, similar quantities can be defined. D EFINITION 15.5.1 In a two-dimensional statistical mechanical system at inverse temperature β, introduce the notation Eβ (n; α) (Fβ (n; α)) denote for the probability that a randomly chosen point (particle) has exactly n particles within a radius α. Furthermore, let Pβ (n; α) (Qβ (n; α)) denote the probability density that a randomly chosen point (particle) is a distance α from another particle, and there are exactly n particles within this distance. In fact Pβ can be deduced from Eβ , and Qβ from Fβ . P ROPOSITION 15.5.2 We have d d Qβ (n; α) = − Fβ (n; α) + Qβ (n − 1; α) Pβ (n; α) = − Eβ (n; α) + Pβ (n − 1; α), dα dα valid for n = 0, 1, 2, . . . with Pβ (−1; α) = Qβ (−1; α) = 0. Proof. Let δα denote a small change in α. We see from the definitions that δα Pβ (n − 1; α) + Eβ (n; α) − Eβ (n; α + δα) = δαPβ (n; α), which implies the first formula. The second result follows similarly.
For a finite system in the canonical ensemble we have 1 N dr1 · · · drn drn+1 · · · drN e−βW (r1 ,...,rN ) , Eβ (n; α) = ˆ n ZN Ωα Ωα Ωα Ωα
726
CHAPTER 15
where Ωα denotes the disk of radius α centered at the origin and Ωα := Ω − Ωα . Note that each Eβ (n; α) can be calculated via the generating function N N 1 n dr1 . . . drN e−βW (r1 ,...,rN ) . u Eβ (n; α) = +u +u (15.100) ZˆN l=1 Ωα Ωα Ωα Ωα n=0 2
For the 2dOCP in a disk at Γ = 2 we have e−βW given by (15.42) with f (r) proportional to e−πρr . Using the same integration technique which led to (15.43), and recalling the definitions (15.50) and (15.62), then allows (15.100) to be evaluated with the result N N ! Γ(j; πρα2 ) − Γ(j; N ) + uγ(j; πρα2 ) n u E2 (n; α) = . (15.101) γ(j; πρR2 ) n=0 j=1 Taking the N → ∞ limit, we obtain a closed form expression for the generating function [186]. P ROPOSITION 15.5.3 For the 2dOCP at Γ = 2 in the thermodynamic limit we have ∞
un E2 (n; α) =
n=0
∞ Γ(j; πρα2 ) + uγ(j; πρα2 ) . Γ(j) j=1
Thus E2 (0; α) =
∞ Γ(j; πρα2 ) , Γ(j) j=1
E2 (n; α) = E2 (0; α)
n γ(jl ; πρα2 ) . Γ(jl ; πρα2 )
1≤j1 <···<jn <∞ l=1
Similar formulas can be derived for F2 (0; α) and F2 (n; α)/F2 (0; α) (see Exercises 15.5 q.2). In particular 2
F2 (0; α) = eπρα E2 (0; α).
(15.102)
According to Proposition 15.5.2 we can use this formula to compute Q2 (0; α) which is the p.d.f. for the nearest neighbour spacing between the particles in the bulk. The large α asymptotics of E2 (0; α) and E2 (n; α)/E2 (0; α) (for fixed n) can be determined from the above explicit formulas. In the former case this requires taking the logarithm and breaking the resulting summation over j into three regions, (i) 1 ≤ j ≤ J,
(ii) J + 1 ≤ j ≤ [α2 ],
(iii) [α2 ] + 1 ≤ j,
where J := α2 − α1+ , 0 < 1. In each region we approximate Γ(j + 1; x) by a different asymptotic expansion [512], j2 e−x xj+1 j 2j (i) Γ(1 + j; x) ∼ 1− + + O x−j (x − j)2 (x − j)3 (x − j)4 2 e−τ x 1 j 1 − erf τ +√ − (ii) Γ(1 + j; x) ∼ Γ(1 + j) 2 τ 2 2πj x − j 1/2
where τ := (x − j − j log(x/j)) , while in region (iii) the expansion (15.51) is used. After some further manipulation, the following result is obtained [186]. P ROPOSITION 15.5.4 With μ = (πρ)
1/2
α and erfc x := 1 − erf x denoting the complementary error
727
THE TWO-DIMENSIONAL ONE-COMPONENT PLASMA
function, we have
1 1 log E2 (0; α) ∼ − μ4 − μ2 log μ + 1 − log 2π μ2 4 2 ∞ 1/2 1/2 t2 +2 μ log π te erfc t dt + 0
∞
log
0
1 (1 + erf t) dt + o(μ) 2
1 = − μ4 − μ2 log μ + 0.081μ2 − 2.294μ + o(μ), 4 where the decimal digits are accurate to three places, as μ → ∞. A macroscopic argument of the type used in Section 14.6 can be used to predict the leading term in this expansion (see Exercises 15.5 q.3). The asymptotic expansion of E2 (n; α)/E2 (0; α) for fixed n is obtained from the term j1 = 1, j2 = 2, . . . , jn = n in the formula of Proposition 15.5.3. Using the formula Γ(j + 1; x) ∼ xj e−j (1 + j/x + O(1/x2 )) valid for j fixed and x large, which follows from (i) above, we deduce that 2 1 Γ(j + 1) Γ(j + 1)eα γ(j + 1; α2 ) j = − 1 ∼ 1 − + O Γ(j + 1; α2 ) Γ(j + 1; α2 ) α2j α2 α4 and thus obtain the sought asymptotic expression [186]. P ROPOSITION 15.5.5 We have
2 n−1 E2 (n; α) n(n − 1) enα 1 ∼ , Γ(j + 1) n(n−1) 1 − + O 2 4 E2 (0; α) 2α α α j=0
valid for n fixed and α large.
15.5.2 The edge We next turn our attention to the distribution of the largest eigenvalue (in modulus) for complex Gaussian matrices. Now, the probability that all eigenvalues are inside the disk Ωα of radius α about the origin is given by N 2 1 E2 (0; (α, ∞)) = dr1 · · · drN e−|rl | |rk − rj |2 , ZˆN Ωα Ωα 1≤j
l=1
where
ZˆN :=
R2
dr1 · · ·
R2
drN
N
e−|rl |
2
l=1
|rk − rj |2 .
1≤j
The working required for Exercises 15.3 q.1(ii) shows that the integrals can be evaluated with the result N γ(j; α2 ) E2 (0; (α, ∞)) = . Γ(j) j=1
(15.103)
From this the following limit formula can be deduced [466]. P ROPOSITION 15.5.6 Let γN = log(N/2π) − 2 log log N , and αN (x) =
√ √ √ N + γN /2 + x/2 γN . We
728
CHAPTER 15
have lim log E2 (0; (αN (x), ∞)) = exp(− exp(−x)).
(15.104)
N →∞
interpretation tells us that to leading order the support of the eigenvalues is a disk of radius Proof. The 2dOCP √
√
N. Writing α = N + fN (x)/2 with fN (x) linear in x, setting N → N − j + 1, then making use of the uniform expansion (15.51) shows N ”” X “ “√ “ j fN (x) ”” 1 1“ + √ N + fN (x), ∞ ∼ log 1 + erf √ log E2 0; 2 2 2 2N j=1 Z ∞ √ 1 log (1 + erf t) dt. ∼ 2N 2 fN (x)
For this to be of order unity, we require fN (x) → ∞ as N → ∞. Estimating the integrand in this limit gives r 2 ”” “ “√ N e−(fN (x)) /2 1 log E2 0; N + fN (x), ∞ ∼− . 2 2π (fN (x))2 √ With 21 fN (x) = αN (x) − N the result (15.104) follows.
Although not the setting of the present problem, the distribution in (15.104) is well known as the extremal distribution exhibited by the largest of N independent random variables, for N → ∞ and after suitable scaling. In fact this coincidence persists for the distribution of the kth largest eigenvalue in magnitude [467] (k = 1, 2, . . . ) telling us that after appropriate scaling these magnitudes are in fact statistically independent. Pk l 1. [275] Let e(k; x) := l=0 x /l!. Use the formula (15.55) to show that the formula for E2 (0; α) in Proposition 15.5.3 can be written as
E XERCISES 15.5
E2 (0; α) =
∞ “ Y
” 2 e−πρα e(j; πρα2 ) .
j=1
2
2. Follow the method used to derive Proposition 15.5.3 to show that F2 (0; α) = eπρα E2 (0; α), and F2 (n; α) = F2 (0; α)
n Y γ(jl + 1; πρα2 ) . Γ(jl + 1; πρα2 )
X
1≤j1 <···<jn <∞ l=1
3. [319] In this exercise the leading large α form of the probability Eβ (0; α) of there being no eigenvalues in a disk of radius α—the hole probability—for the 2dOCP at general Γ will be deduced. (i) Hypothesize that the creation of a hole of radius α, with α 1, causes at r = α a neutralizing uniform excess surface charge density Q = πρα2 . (ii) Calculate the electrostatic energy of the surface charge/surface charge, and surface charge/disk background interaction (for the latter use can be made of (1.71)), and substitute in (14.113) to deduce 2 2
Eβ (0; α) ∼ e−Γ(πρα α→∞
) /8
.
729
THE TWO-DIMENSIONAL ONE-COMPONENT PLASMA
15.6 THE SPHERE 15.6.1 Poisson equation The pair potential Φ for two-dimensional unit charges on a sphere of radius R must obey the charge neutral Poisson equation on the sphere, 1 ∂ ∂Φ 1 ∂2Φ 1 sin θ + 2 2 = −2πδS ((θ, φ), (θ , φ )) + , (15.105) 2 R sin θ ∂θ ∂θ 2R2 R sin θ ∂φ2 where θ refers to the usual azimuthal angle, φ to the polar angle and δS ((θ, φ), (θ , φ )) is the delta function on the sphere, defined by the property π 2π 2 R dθ sin θ dφf (θ, φ)δS ((θ, φ), (θ , φ )) = f (θ , φ ). (15.106) 0
0
Note from (15.106) that the integral over the sphere of the r.h.s. of (15.105) gives 0. The interpretation is that as well as there being a charge at (θ , φ ) there is a uniform neutralizing background contributing to the potential at (θ, φ); without charge neutrality, the Poisson equation on the sphere—a compact surface—does not have a solution. With θ = 0 (source at the north pole) we see from the rotational invariance of the sphere about the azimuthal axis that Φ must be independent of φ (of course the solution must also be independent of φ ). Furthermore, for small θ we must reclaim the potential in a plane with free boundary conditions, and thus Φ ∼ − log |Rθ|. It is easy to check by substitution in (15.105) that the required solution in this circumstance is Φ ((θ, φ), (0, φ )) = − log (2R sin(θ/2)) . Geometrically 2R sin(θ/2) is the chord length between the points (θ, φ) and (0, φ ) on the sphere. Hence, by rotational invariance of the sphere, for a general source point (θ , φ ), Φ must be given by minus the logarithm of the chord length. Writing r and r for the vector from the center of the sphere to the points with spherical coordinates (θ, φ) and (θ , φ ) we thus have Φ ((θ, φ), (θ , φ )) = − log |r − r |. Explicitly, if α denotes the relative angle between the vectors r and r , then Φ ((θ, φ), (θ , φ )) = − log (2R sin(α/2)) . 2
By noting sin (α/2) =
1 2 (1
(15.107)
− cos α), cos α = r · r /R and that in Cartesian coordinates 2
r = (R sin θ cos φ, R sin θ sin φ, R cos θ), a short calculation verifies that (15.107) can be rewritten as Φ ((θ, φ), (θ , φ )) = − log (2R|u v − uv |) ,
(15.108)
where u, v (and similarly u , v ) are the Cayley-Klein parameters u := cos(θ/2)eiφ/2 , v := −i sin(θ/2)e−iφ/2 .
(15.109)
The total potential of an N -particle system is calculated from the pair potential according to the method of Section 1.4.1. Due to the rotational invariance of the sphere the potential energy for the particle-background interaction only contributes a constant. From the total potential we find that the Boltzmann factor for the
730
CHAPTER 15
system is
1 2R
N Γ/2 eΓN
2
/4
|uk vj − uj vk |Γ .
(15.110)
1≤j
15.6.2 Correlation functions At Γ = 2 use of the Vandermonde determinant (1.173) shows that the Boltzmann factor is proportional to N
(cos(θj /2))2(N −1)
j=1
1≤j
2 v k−1 j det . uj j,k=1,...,N
With ΩR denoting the sphere of radius R, noting the orthogonality π m n 2 g(θ)(v/u) (¯ v /¯ u) dΩ = 2πR δm,n g(θ)(tan θ/2)m+n sin θ dθ, ΩR
(15.111)
0
the method of proof of Proposition 15.3.1 gives that at Γ = 2 the correlation function is given in terms of the Cayley-Klein parameters by [104] N −1 ρ(n) ((θ1 , φ1 ), . . . , (θn , φn )) = ρn det ((uk , vk ), (uj , vj )) , (15.112) j,k=1,...,n
¯k + vj v¯k . A feature of (15.112) is that it is invariant with respect to a where (uk , vk ), (uj , vj ) := uj u rotation of the sphere. This follows from the fact that a rotation of the sphere corresponds to the multiplication of the vectors (u, v) by a unitary matrix, and that the inner product ·, · is invariant with respect to such a transformation. For the particles in the neighborhood of the north pole, we can substitute (15.109) for the Cayley-Klein parameters in (15.112), set θj = rj /R and take the limit N, R → ∞ (N/4πR2 = ρ). In this limit the neighborhood of the north pole becomes a plane, and (rj , φj ) specifies the position of the particles in polar coordinates. A straightforward calculation shows that the n-particle correlation (15.112) tends to the planar bulk n-particle correlation specified in Proposition 15.3.2, as expected. 15.6.3 Magnetic analogy = Bˆ In the theory of a charged particle confined to a sphere of radius R with a radial magnetic field B r at the surface (due to a magnetic monopole at the center of the sphere), the operator formalism of Section 15.2.1 is no longer applicable since there it is assumed xy-coordinates are used. The situation is further complicated such that by the fact (see, e.g., [549]) that it is not possible to define a singularity free vector potential A ∇ × A = Bˆ r over the entire sphere. Instead, the sphere is divided into overlapping regions (a) 0 ≤ θ < π/2 + δ, 0 ≤ φ < 2π, (b) π/2 − δ < θ ≤ π, 0 ≤ φ < 2π, where 0 ≤ δ ≤ π/2. Using the general formula 1 ∂Ar ˆ ∂Aθ ∂Ar ˆ ∂ = ∇×A (Aφ sin θ) − rˆ + θ − sin θ φ , R sin θ ∂θ ∂φ ∂φ ∂θ = Bˆ valid for r = R, we see that the equation ∇ × A r is satisfied in the region (a) and (b) by choosing BR BR (1 − cos θ) , Ab = 0, 0, − (1 + cos θ) (15.113) Aa = 0, 0, sin θ sin θ respectively, where the coordinates in (15.113) refer to the spherical coordinates (r, θ, φ).
731
THE TWO-DIMENSIONAL ONE-COMPONENT PLASMA
In the overlap region the wave functions implied by different vector potentials are related, and this implies a restriction on the allowed value of B. Now, using the formula ∇f =
1 ∂f ˆ 1 ∂f ˆ θ+ φ R ∂θ R sin θ ∂φ
a and A b are so related and the solution of the a = Ab + ∇f , where f = 2BR2 φ. But if A we see that A Schr¨odinger equation 1 e 2 − i∇ + A ψ = Eψ (15.114) 2m c = A b is given by ψb , then the solution of (15.114) with the same value of E but with vector potential A =A a is given by ψb e−ief /c . In the present situation this means vector potential A ψa (θ, φ) = ψb (θ, φ)e−i2BeR
2
φ/c
.
Furthermore ψa and ψb must both be periodic of period 2π in φ. This is only possible if c , n = 1, 2, . . . , (15.115) 2eR2 which is the quantization condition for the strength of the magnetic monopole. =A a . An algebraic formalism is Our task is to determine the minimal energy states of (15.114) with A := r × (−i∇ + e A), which is a type of angular momentum operator. possible by the introduction of Λ c satisfy the commutation relation Direct calculation gives that the components of Λ n z , [Λx , Λy ] = i Λz − 2 R B=n
together with the relations obtained by cyclic permutation of the indices. This tells us that if we define =Λ + n rˆ L 2 then we obtain the canonical commutation relations for angular momentum, [Lx , Ly ] = iLz , [Ly , Lz ] = iLx , [Lz , Lx ] = iLy . by noting The Hamiltonian operator in (15.114) can be written in terms of L 2 2 2 2 2 2 = Λ2 + n = R2 −i∇ + e A +n , L 4 c 4 2C · C) 2 . Thus we have × C) 2=B 2 − (B where we have used the vector identity (B 2 2 1 2 − n ψ(θ, φ) = Eψ(θ, φ). L Hψ(θ, φ) = 2mR2 4
(15.116)
(15.117)
(15.118)
(15.119)
2 are l(l + 1)2 , The angular momentum algebra (see Exercises 15.6) gives that the allowed values of L 1 3 l = 0, 2 , 1, 2 , . . . . However, the original Hamiltonian (15.114) must have non-negative eigenvalues, so the smallest allowed value of l is n/2, which from (15.119) and (15.115) is seen to correspond to E = 12 wc . As is well known, and revised in Exercises 15.6, for a given value of l the allowed values of Lz are m with m = −l, −l + 1, . . . , l. Now, from (15.116) we can deduce that n ∂ − , Lz = − i ∂φ 2
732
CHAPTER 15
which has corresponding eigenfunctions ψm (θ, φ) = Am (θ)ei(m−n/2)φ .
(15.120)
2, The function Am (θ) can be determined by writing out the explicit form of L 2 ∂ 1 n 1 ∂ ∂ n2 2 2 sin θ + + (cos θ − 1) i L = − + sin θ ∂θ ∂θ 4 sin2 θ ∂φ 2
(15.121)
(cf. (15.105)). Substituting (15.120) and (15.121) in the eigenvalue equation (15.119) with E = 12 wc gives that Am (θ) must satisfy the equation 5 2 6 n n 2 cos θ − m Am (θ) = 0. Am (θ) + cot θAm (θ) + (15.122) − 2 sin θ With m = −n/2, direct substitution verifies that A−n/2 (θ) = C−n/2 sinn (θ/2) is a solution of (15.122), where the constant C−n/2 is a normalization. The solution for m = −n/2 + 1, −n/2 + 2, . . . , n/2 can be constructed from A−n/2 by applying the raising operator ∂ n ∂ + i cot θ + (1 − cos θ) . L+ = Lx + iLy = eiφ ∂θ ∂φ 2 sin θ This gives Am (θ) = Cm cosn/2+m (θ/2) sinn/2−m (θ/2), which when substituted in (15.120) shows that ψm (θ, φ) = Cm u
n/2+m n/2−m −inφ/2
v
e
,
Cm =
1/2 n n/2 + 1 , 4πR2 n/2 + m
(15.123)
where u and v are the Cayley-Klein parameters (15.109) (the value of the normalization constant can be determined by making use of the trigonometric Euler integral of Exercises 4.1 q.1(i)). Setting N = n + 1, an N -particle state can be formed by constructing a Slater determinant from the states (15.120), ψ ((θ1 , φ1 ), . . . , (θN , φN )) = Ce−i(N −1)
PN j=1
φj /2
N
2 1 vjN −1 det (uj /vj )k−1 j,k=1,...,N .
(15.124)
j=1 2
Use of the Vandermonde formula and comparison with (15.110) shows that |ψ| is proportional to the Boltzmann factor for the 2dOCP on a sphere at Γ = 2.
15.6.4 Stereographic projection The 2dOCP on a sphere can be mapped onto a 2dOCP in the plane with a particular one-body potential. The mapping is achieved by making a stereographic projection from the south pole of the sphere to a plane tangent to the north pole (see Figure 15.2 and cf. Section 2.5). This is specified by the equation θ z = 2Reiφ tan , z = x + iy. (15.125) 2 Using this to rewrite the argument of the logarithm in (15.108) and comparing with (15.107) we see that 2R sin
θ θ α = cos |z − z | cos , 2 2 2
(15.126)
733
THE TWO-DIMENSIONAL ONE-COMPONENT PLASMA
N=(0,0)
(x,y)
(θ,ϕ) θ
θ/2 S
Figure 15.2 Cross-section of the stereographic projection from the south pole of the sphere to the plane tangent to the north pole
while calculation of the relevant Jacobian gives dS := R2 sin θdθdφ =
1 dxdy. (1 + |z|2 /4R2 )2
(15.127)
The Boltzmann factor (15.110) therefore transforms according to Γ |uk vj − uj vk | dS1 · · · dSN 1≤j
=
N j=1
1 (1 + |zj
|2 /(4R2 ))2+Γ(N −1)/2
|zj − zk |Γ dr1 · · · drN .
(15.128)
1≤j
N Writing the one-body terms on the r.h.s. of (15.128) in the form exp(−β j=1 V (rj )), we see that the effective one body potential is V (r ) = ( 12 (N − 1) + Γ2 ) log(1 + |z|2 /(4R2 )). Substituting this into the (flat space) Poisson equation (1.40) shows that the corresponding background density is 2 1 1 1 (N − 1) + . 2 2 2πR 2 Γ (1 + |z| /(4R2 ))2 The Hamiltonian for the quantum particle on a sphere in the presence of a perpendicular magnetic field can also be transformed by the stereographic projection [143]. First, we note from (15.119) and (15.121) that the a (15.113) (recall this has a singularity at the south pole, which will Hamiltonian with the vector potential A be projected to infinity) is given by R 2 2 2 ∂ 1 ∂ 1 ∂ H= sin θ + + − . i (cos θ − 1) 2mR2 sin θ ∂θ ∂θ l sin2 θ ∂φ The change of variables (15.125) gives the projected Hamiltonian as |z|2 ∂ ∂ |z|2 |z|2 2 ∂ 2 2 2 − 4l 1 + + 1+ − z¯ + 2 . z H= 2m 4R2 ∂z∂ z¯ 4R2 ∂z ∂ z¯ 4l
(15.129)
734
CHAPTER 15
This Hamiltonian is of the form (15.35) with R 2 |z|2 . log 1 + Φ= l (2R)2 It therefore follows from Exercises 15.2 q.2 that the states −(R/l)2 |z|2 1+ z¯p , (2R)2
p ∈ Z≥0
(15.130)
(15.131)
are orthogonal states in the lowest Landau level. Choosing n = N − 1 in the quantization condition (15.115), and forming a Slater determinant out of the states (15.131) with p = 0, 1, . . . , N − 1, shows that the absolute value squared of the corresponding N -particle free fermion state, when multiplied by the product of measures (15.127), is proportional to (15.128).
15.6.5 Random matrix analogy The p.d.f. corresponding to (15.128) with Γ = 2, R =
1 2
can be realized as an eigenvalue p.d.f. [366].
P ROPOSITION 15.6.1 Let A, B be complex Gaussian matrices as specified by Proposition 15.1.1. The eigenvalue p.d.f. of A−1 B is proportional to N l=1
1 (1 + |zl |2 )N +1
|zk − zj |2 .
(15.132)
1≤j
Proof. We follow [298]. From (3.82) we have that the distribution of Y := A−1 B is proportional to 1 . det(1 + YY † )2N Introduce now the Schur decomposition (15.3) for Y to conclude that the distribution of the upper triangular matrix T =: TN therein is proportional to det(1N
1 + TN T†N )2N
Y
|zk − zj |2 ,
(15.133)
1≤j
where zj = Tjj are the eigenvalues of Y. The task then is to integrate out the strictly upper triangular elements of TN , ˜ N , say, and thus compute T Z Y 1 ˜ N ). |zk − zj |2 IN (z1 , . . . , zN ), IN (z1 , . . . , zN ) := (dT (15.134) det(1 + TN T†N )2N N 1≤j
(15.135)
where p ≥ n (this ensures convergence). For these integrals a recurrence can be obtained by proceeding in a way similar to the derivation of the result (b) in Proposition 2.5.1. Let u denote the last column of T˜n . We can then write – » 1n−1 + Tn−1 T†n−1 + u u† zn u , (15.136) 1n + Tn T†n = † 2 z¯n u 1 + |zn |
735
THE TWO-DIMENSIONAL ONE-COMPONENT PLASMA
showing “ det(1n + Tn T†n ) = (1 + |zn |2 ) det 1n−1 + Tn−1 T†n−1 + “ = (1 + |zn |2 ) det(1n−1 + Tn−1 T†n−1 ) 1 +
uu† ” 1 + |zn |2
” 1 u† (1n−1 + Tn−1 T†n−1 )−1 u , 2 1 + |zn |
where the last equality follows by noting that the final term in the determinant of the previous line is of rank 1. Substituting this last equality in (15.135) shows Z 1 1 ˜ n−1 ) (dT In,p (z1 , . . . , zn ) = (1 + |zn |2 )p det(1n−1 + Tn−1 T†n−1 )p Z “ ”−p 1 × (d u) 1 + u† (1n−1 + Tn−1 T†n−1 )−1 u . (15.137) 2 1 + |zn | The change of variables
v = (1 + |zn |2 )−1/2 (1n−1 + Tn−1 T†n−1 )−1/2 u
then gives the recurrence In,p (z1 , . . . , zn ) =
Cn,p , In−1,p−1 (z1 , . . . , zn−1 ), (1 + |zn |2 )p−n+1 Z
where Cn,p =
(dv) . (1 + v †v )p
(15.138)
(15.139)
Iterating (15.138) with n = 2N , p = N shows IN (z1 , . . . , zN ) ∝
N Y l=1
1 , (1 + |zl |2 )N+1
which when substituted in (15.134) implies (15.132).
15.6.6 Zeros of a random spin state 2 are l(l + 1)2 , l = 0, 1 , 1, 3 , . . . . In the quantum theory of We have noted above that the eigenvalues of L 2 2 2 equal to l(l + 1)2 are said to have spin l. angular momentum applied to spin, eigenstates with a value of L According to (15.123) a general state Φ(θ, φ) of spin N/2 (and thus n = N ) is then of the form 1 N/2 + 1 1/2 Φ(θ, φ) := (−i sin θ/2)N e−iN φ p(eiφ cot θ/2), N 4πR2 N N 1/2 αj z j , (15.140) p(z) := j j=0 N where N := ( j=0 |αj |2 )1/2 . Consider the function p(eiφ cot θ/2). Now the mapping z = eiφ cot θ/2 can be interpreted geometrically as a stereographic projection from the north pole of a unit diameter sphere to a plane passing through the equator of the sphere. This means that if p(z) is factorized with zeros at z1 , . . . , zN , then p(eiφ cot θ/2) vanishes at the value of (θj , φj ) on the sphere corresponding to the stereographic projection of the zj . Such a parametrization of a spin state is due to Majorana [382]. With all the coefficients of (15.140) chosen as independent Gaussian random variables with mean zero and unit variances, the density of zeros can be computed from (15.59) with σj2 = Nj , while the two-point function follows by simplifying the general formula of Proposition 15.3.6. To compute the density we note
736
CHAPTER 15
that p¯ pL =
N N j=0
j
|z|2j = (1 + |z|2 )N
and thus ρ(1) (z) =
1 N . π (1 + |z|2 )2
But with z = eiφ cot θ/2, dS =
1 dx dy (1 + |z|2 )2
(recall (15.127)), so in terms of the spherical coordinates (θ, φ) N , π and thus the zeros are uniformly distributed on the unit diameter sphere [288], [71]. ρ(1) (θ, φ) =
15.6.7 Ground state The ground state configuration for the 2dOCP in the infinite plane is expected to be the triangular lattice [12]. The question of the configuration with minimum energy can also be addressed in the finite system; indeed in the case of a disk with a uniform background of variable density it will be seen in Section 15.8.2 that this problem arises in studying the configurations of vortices in a rotating superfluid. On the unit sphere, the problem of minimizing the energy of a general repulsive pair potential of the form v(ˆ r1 , rˆ2 ) = −sgn(α)|ˆ r1 − rˆ2 |−α = −sgn(α)|2(1 − rˆ1 · rˆ2 )|−α/2
(15.141)
has attracted much attention (see, e.g., [474]). In general the configuration of minimum energy can only be determined numerically. Associated with the configuration is a tiling of Dirichlet (Voronoi) cells Dj , (j = 1, . . . , N ), which are defined as the boundary of the region of the sphere closest to the jth lattice point, Dj := {x ∈ S 2 : |x − xj | = min |x − xk |}. 1≤k≤N
For general potentials (15.141) it is observed numerically that all Dirichlet cells are hexagons or pentagons, and furthermore that for N large enough there are precisely 12 pentagons. In fact this latter fact can be understood [474] as a consequence of Euler’s relation (1.107). P ROPOSITION 15.6.2 A tiling of the sphere consisting only of hexagons and pentagons, in which there are 3 edges from each vertex, must have exactly 12 pentagons. Proof. Let p denote the number of hexagons and q denote the number of pentagons so that F = p+q. Since each edge is shared by 2 cells, we have E = 12 (6p +5q), while the fact that each vertex is shared by three cells gives V = 13 (6p+5q). Furthermore, the Euler number for a sphere is 2. Substituting the above formulas for F , E and V in (1.107) with g = 0 gives q = 12. Note that the faces of a conventional soccer ball conform to Proposition 15.6.2. The logarithmic potential 1 v(ˆ r1 , rˆ2 ) = − log |ˆ r1 − rˆ2 | = − log |2(1 − rˆ1 · rˆ2 )| (15.142) 2 has the special property that the dipole moment of the configuration of minimum energy must vanish [57].
737
THE TWO-DIMENSIONAL ONE-COMPONENT PLASMA
P ROPOSITION 15.6.3 Let rˆ1 , . . . , rˆN denote a configuration of minimum energy for the pair potential N (15.142) on the unit sphere. Then j=1 rˆj = 0. Proof. The force experienced by particle i due to particle j is given by −∇i log |ri − rj | =
ri − rj . |ri − rj |2
For an equilibrium configuration the total force must be directed radially in the direction of rˆi . Taking the dot product ri − rˆj |2 = 2(1 − rˆi · rˆj ) shows that the magnitude is 12 (N − 1) independent of i. Summing the with rˆi and noting |ˆ resulting expression over i gives N N X X 1 rˆi − rˆj = rˆi . (N − 1) |ˆ ri − rˆj |2 2 i,j=1 i=1 i=j
But the left-hand side is antisymmetric with respect to interchanging labels i and j and so vanishes, implying the result.
E XERCISES 15.6
1. (i) [104] Consider the Boltzmann factor (15.110) with Γ = 2. Use the orthogonality (15.111) to show „ ZˆN = N !R2N
1 2R
«N e
N 2 /2
N−1 Y „ p=0
Γ(N − p)Γ(p + 1) 4π Γ(N + 1)
« .
(15.143)
(ii) [320] Use Stirling’s formula and the asymptotic expansion (4.186) with β = 2 to deduce that for N, R → ∞ with ρ fixed βF := −
” “ ZˆN 1 1 1 log = 4πR2 βfv + log (4πρ)1/2 R + − 2ζ (−1) + o(1), N N! 3 12
(15.144)
where βfv is given by (15.63). 2.
(i) For Hermitian operators satisfying the canonical commutation relations for angular momentum (15.117), show that [L2 , Lz ] = 0,
[L2 , L± ] = 0,
[Lz , L± ] = ±L± ,
L∓ L± = L2 − L2z ∓ Lz ,
(15.145)
where L± = Lx ± iLy . (ii) Suppose Lz |ψ = μ|ψ, where μ is a simple (i.e., nondegenerate) eigenvalue. From the first commutation relation in (i) note that L2 |ψ = λ|ψ for some λ ≥ μ2 , and from the second and third commutation relations show that L2 (L± |ψ) = λL± |ψ,
Lz (L± |ψ) = (μ ± )(L± |ψ),
with λ ≥ (μ±)2 . From this last inequality deduce that there exist highest |ψ+ and lowest |ψ− eigenstates of L2 and Lz for which L+ |ψ+ = L− |ψ− = 0. (iii) Use the final relation in (i) to deduce that (L2 − L2z ∓ Lz )|ψ± = 0 and thus (λ − μ2± ∓ μ± ) = 0, where Lz |ψ+ = μ± |ψ+ . From this and the second displayed equation in (ii), deduce that μ+ − μ− = 2l for some l = 0, 12 , 1, 32 , . . . , and so μ+ = −μ− = l. (iv) From the results of (iii) conclude that the allowed values of the eigenvalues of L2 are λ = l(l + 1)2 , l = 0, 12 , 1, 32 , . . . and that for each value of l there exist simultaneous eigenfunctions of L2 and Lz which have eigenvalue of Lz given by μ = m, where m is an integer −l ≤ m ≤ l.
738
CHAPTER 15
15.7 THE PSEUDOSPHERE 15.7.1 Geometry The sphere of radius R can be characterized by the fact that its Gaussian curvature is equal to 1/R2 . D EFINITION 15.7.1 For a surface parametrized by the coordinates (τ, φ), and possessing a line element ds of the form (ds)2 = (dτ )2 + (f (τ )dφ)2 , the Gaussian curvature κ is defined as κ=−
1 ∂2f . f ∂τ 2
(15.146)
Parametrizing the sphere by (θ/R, φ) gives (ds)2 = (dθ)2 + (R sin(θ/R)dφ)2
(15.147)
and formula (15.146) gives κ = 1/R2 as stated. A closely related, but very different, two-dimensional surface is the pseudosphere (see, e.g., [42]). The pseudosphere is characterized by having constant negative Gaussian curvature κ = −1/a2 , and is of infinite extent. Furthermore, unlike the sphere, it cannot in its entirety be embedded in three-dimensional Euclidean space. Instead, the pseudosphere is naturally embedded in the three-dimensional Minkowski space with coordinates (y0 , y1 , y2 ) in which the distance from the origin is defined as −y02 + y12 + y22 . The equation −y02 + y12 + y22 = −a2
(15.148)
(set of points at “distance” −a2 from the origin in Minkowski space) defines a surface with two branches, intersecting the y0 -axis at y0 = ±a, respectively. Note that in Euclidean space (15.148) defines a hyperboloid. In Minkowski space the upper hyperboloid is called the pseudosphere. The equation (15.148) can be parametrized by y0 = a cosh τ,
y1 = a sinh τ cos φ,
y2 = a sinh τ sin φ.
(15.149)
The Minkowski space line element (ds)2 = −(dy0 )2 + (dy1 )2 + (dy2 )2 then becomes (ds)2 = a2 (dτ )2 + a2 sinh2 τ (dφ)2 .
(15.150)
Replacing τ by τ /a puts (15.150) into the form required in Definition 15.7.1, and (15.146) gives κ = −1/a2 as remarked. Note also that making the replacements a → iR,
τ → iθ
(15.151)
in the modified form of (15.150) reclaims the formula (15.147) for the sphere. Like the sphere, the pseudosphere is homogeneous, in the sense that any two points can be connected by generalized translations (isometries). These can be constructed from a rotation Rφ0 and boost Tτ0 specified by the orthogonal matrices (in Minkowski space) ⎞ ⎛ ⎛ ⎞ 1 0 0 cosh τ0 sinh τ0 0 Rφ0 = ⎝ 0 cos φ0 − sin φ0 ⎠ , (15.152) Tτ0 = ⎝ sinh τ0 cosh τ0 0 ⎠ . 0 0 1 0 sin φ0 cos φ0 15.7.2 The plasma system After these preliminary remarks on the geometry of the pseudosphere, let us turn our attention to the 2dOCP confined to this surface [321]. First, we require the solution of the corresponding Poisson equation. Now, in terms of the coordinates (τ, φ), the Laplacian on the pseudosphere can be written down from the Laplacian on the boundary of the sphere (l.h.s. of (15.105)) by making the replacements (15.151). Thus the Poisson
THE TWO-DIMENSIONAL ONE-COMPONENT PLASMA
equation on the pseudosphere reads 1 ∂ ∂Φ 1 ∂2Φ sinh τ + = −2πδ((τ, φ), (τ , φ )), a2 sinh τ ∂τ ∂τ a2 sinh2 τ ∂φ2 where the delta function on the r.h.s. is defined by the property ∞ 2π 2 dτ sinh τ dφ f (τ, φ)δ((τ, φ), (τ , φ )) = f (τ , φ ) a 0
739
(15.153)
0
2
(a sinh τ is the Jacobian associated with the change of variables (15.149)). Note that, with the pseudosphere being an infinite surface, there is no need to have the r.h.s. of the Poisson equation (15.153) charge neutral, as distinct from (15.105). For τ = 0, the solution of (15.153) must be independent of φ and φ . With the boundary condition that Φ → 0 as τ → ∞, the solution is verified to be τ . (15.154) Φ((τ, φ), (0, φ )) = − log tanh 2 Noting that aτ is the geodesic distance from the point (τ, φ) to the point (0, φ ), homogeneity of the pseudosphere implies that s , (15.155) Φ((τ, φ), (τ , φ )) = − log tanh 2a where s is the geodesic distance between (τ, φ) and (τ , φ ). To obtain an explicit formula for s we note s (15.156) cosh = cosh τ cosh τ − sinh τ sinh τ cos(φ − φ ), a which can be deduced from the analogous formula for the sphere cos
1 s = 2 r · r = cos θ cos θ + sin θ sin θ cos(φ − φ ) a R
(recall the formula below (15.107)) by making the replacements s → is and θ → iτ , while sinh2 (s/2a) = 1 2 (1 − cosh(s/a)). This implies tanh τ2j eiφj − tanh τ2k eiφk Uj Vk − Uk Vj (zj − zk )/2a sjk = = (15.157) = tanh τ ¯j Uk − V¯j Vk 1 − zj z¯k /4a2 , 2a 1 − tanh 2j tanh τ2k eiφj −iφk U where U and V are the analogues of the Cayley-Klein parameters (15.109), τ τ U = cosh eiφ/2 , V = sinh e−iφ/2 (15.158) 2 2 while τ (15.159) z := 2a tanh eiφ 2 (cf. (15.125)). In Euclidean space the transformation (15.159) represents a stereographic projection from a hyperboloid into a disk (the Poincar´e disk) of radius R = 2a. Substituting (15.157) in (15.155) shows that the pair potential is formally the same as the pair potential (15.174) below for a charge inside a disk of radius R = 2a with metallic boundary conditions. However the systems are different because the pseudosphere is not a flat space. The explicit formula (15.157) allows the total potential energy for a system of N equal charges confined to the portion of the pseudosphere 0 ≤ τ ≤ a0 , 0 ≤ φ ≤ 2π, in the presence of a uniform background of charge density −η, (the particle density is given by ρ = N/|Ω|, |Ω| = 2πa2 (cosh a0 − 1), but because of the analogy with the metal wall problem, it is not required that ρ = η). However it is not possible to then
740
CHAPTER 15
calculate the free energy from the formula βF = − alim →∞ 0 ρ fixed
1 log ZN . N
The problem is that the length of the boundary at τ = a0 is 2πa sinh a0 . For a0 → ∞ this is of the same order ea0 as the volume, so, contrary to the disk and semiperiodic boundary conditions, the boundary contribution to βF cannot be distinguished from the bulk contribution. Alternative ways of accessing the bulk free energy and bulk energy per particle are discussed in [321] (see also [317] for computation of the pressure using the Maxwell stress tensor); they essentially rely on the observation that the pair potential (15.155) decays exponentially fast as a function of τ , so virial expansion techniques from the theory of fluids with short range potentials can be used. We will not address this problem, but focus attention on the calculation of the correlation functions.
15.7.3 The correlation functions For the one-component system confined to the portion of the pseudosphere 0 ≤ τ ≤ a0 , 0 ≤ φ ≤ 2π, the N general formulas of Section 1.1 give that the background-particle potential is given by j=1 V (τj ) with 2π a0 tanh τ2 − tanh τ2 eiφ 2 1 dτ sinh τ dφ log V (τ ) = a2 η 2 1 − tanh τ2 tanh τ2 e−iφ 0 0 τ a0 − log sinh a0 + log 2 cosh2 . = 2πa2 η cosh a0 log tanh 2 2 The corresponding Boltzmann factor is thus 2
PN
2
e−βU = e−ΓC1 e−ΓN C2 e−Γ2πa η j=1 log cosh τj /2 tanh τj eiφj − tanh τk eiφk Γ 2 2 × 1 − tanh τj tanh τk eiφj −iφk , 2
1≤j
(15.160)
2
where C1 and C2 are independent of N . The explicit form of C1 does not affect the correlations, and the fact that C2 → −∞ as a0 → ∞ is all that is relevant regarding C2 . Note also that unlike the metal wall problem below, the self-energy term is not included. In terms of the variable z defined in (15.159) the pseudosphere surface element dS is given by dS =
dr (1 − |z|2 /4a2 )2
(15.161)
(cf. (15.127)). Hence the generalized grand partition function associated with the Boltzmann factor (15.160) can be written ∞ N Γ2πa2 η−1 (e−ΓC2 ζ)N 1 − |zj |2 /4a2 dr1 · · · drN a(rj ) WN Γ , ΞΓ [a; ζ] = e−ΓC1 N! (Ω2a tanh a /2 )N j=1 N =0
0
where WN Γ :=
N
1 2 /(2a)2 1 − |z | j j=1
1≤j
(z − z )/2a Γ k j 1 − zj z¯k /4a2
and Ωα denotes the disk of radius α about the origin. At the special coupling Γ = 2, making use of the identity (4.34) and the Fredholm formula (5.32) shows that we have Ξ2 [a; ζ] = e−ΓC1 det(1 + ζK),
(15.162)
741
THE TWO-DIMENSIONAL ONE-COMPONENT PLASMA
where K is the integral operator defined by the mapping rule 4πa2 η−1 a(r )f (z ) −2C2 K[f (z )](z) := e 1 − |r |2 /4a2 dr . 1 − z z¯ /4a2 Ω2a tanh a0 /2 Analogous to the situation in Section 15.8 below for the OCP in a disk with metallic boundary conditions, the form (15.162) implies , ρ(n) (r1 , . . . , rn ) = det G(rj , rk ) j,k=1,...,n
where G(r, r ) is given by (15.179). Thus the quantity ˜ r , r ) := (1 − r2 /4a2 )2πa2 η−1/2 G(r, r )(1 − r2 /4a2 )−2πa2 η+1/2 G( satisfies the integral equation ˜ r1 , r3 )(1 − r32 /4a2 )−2πa2 η+1/2 + ζe−2C2 G(
2
Ω2a tanh a0 /2
(1 − r22 /4a2 )2πa
η−1/2
˜ r1 , r2 ) G( dr2 1 − z2 z¯3 /4a2
2 1 = ζe−2C2 (1 − r12 /4a2 )−2πa η+1/2 . 2 1 − z1 z¯3 /4a
Note that this implies (1 − r12 /4a2 )−2πa
2
η+1/2
˜ r1 , r3 )(1 − r32 /4a2 )2πa2 η−1/2 G(
is analytic in z1 and z¯3 . Furthermore G itself has the symmetry (15.181). Taking the limit a0 → ∞, using the fact that then C2 → −∞, and setting z3 = 0 shows 2 ˜ r1 , r2 ) dr2 = (1 − r12 /4a2 )2πa2 η−1/2 . (1 − r22 /4a2 )2πa η−1/2 G(
(15.163)
Ω2a
This has a solution of the required type, ˜ r1 , r2 ) = (1 − r12 /4a2 )2πa2 η−1/2 (1 − r22 /4a2 )2πa2 η−1/2 f (z1 z¯2 ), G( where f must be such that
? f (0) = 1 Ω2a
2
(1 − r22 /4a2 )4πa
η−1
dr2 = η.
The fact that (15.163) is independent of ζ indicates that the particle density must equal the background ˜ r , r) = η = ρ and f is uniquely specified with the result density. Hence G( 2
2
2 2 2πa ρ−1/2 (1 − r22 /4a2 )2πa ˜ r1 , r2 ) = ρ (1 − r1 /4a ) G( (1 − z1 z¯2 /4a2 )4πa2 ρ−1
ρ−1/2
.
Projecting back onto the pseudosphere and recalling (15.161), (15.158) and (15.159), we see that 2 , ρ(n) ((τ1 , φ1 ), . . . , (τn , φn )) = ρn det ((Uk , Vk ), (Uj , Vj ))−4πa ρ−1 j,k=1,...,n
(15.164)
¯k −Vj V¯k . This expression is essentially identical to (15.112) for the sphere where (Uk , Vk ), (Uj , Vj ) := Uj U 2 (write N = 4πR ρ and R → ia).
742
CHAPTER 15
15.7.4 Random matrix analogy It is to be shown in Exercises 15.7 q.2 (ii) that for the one-component plasma with background density η on the pseudosphere, and with pair potential s , (15.165) Φ((τ, φ), (τ , φ )) = − log sinh 2a the Boltzmann factor is proportional to N
1 − |zj |2 /(2a)2
Γ(4πηa2 +1)/2
j=1
|zj − zk |Γ ,
(15.166)
1≤j
|zj | < 2a. For the coupling Γ = 2 this same Boltzmann factor, with 4πηa2 + 1 suitably chosen, is also proportional to the eigenvalue p.d.f. for a certain class of random matrices [559]. P ROPOSITION 15.7.2 Let S ∈ U (N + n) be chosen to be Haar distributed. The eigenvalue p.d.f. of the top N × N sub-block of S, SN , say, is proportional to (15.166) with Γ = 2, (2a)2 = 1 and 4πηa2 + 1 = n − 1. Proof. Following [209], which requires n ≥ N , we begin with the result of Proposition 3.8.2 with N → N + n and n1 = n2 = N . This tells us that the distribution of SN is proportional to det(1N − SN S†N )n−N . Introducing the Schur decomposition SN = UTN U−1 , with U and TN as specified in (15.3), and recalling the change of variables formula (15.9), we see the eigenvalue p.d.f. for SN is proportional to Z Y 2 ˜ N ). |zk − zj | JN (z1 , . . . , zN ), JN (z1 , . . . , zN ) := det(1N − TN T†N )n−N (dT (15.167) 1≤j
The fact that SN is a sub-block of a unitary matrix tells us that the p.d.f. is supported on |zj | < 1 (j = 1, . . . , N ). We will now proceed as in the proof of Proposition 15.6.1 and compute JN by recurrence. For this purpose, for m ≥ p introduce Z 1 ˜ m) (dT (15.168) Im,p (z1 , . . . , zm ) := det(1m − Tm T†m )p ˜ m , we can write (cf. (15.135)). Analogous to (15.136), with u denoting the last column of T 1m −
Tm T†m
» =
1m−1 + Tm−1 T†m−1 + uu† † z¯m u
zm u 1 − |zm |2
– ,
(15.169)
which gives det(1m − Tm T†m )
“ = (1 − |zm |2 ) det(1m−1 − Tm−1 T†m−1 ) 1 +
” 1 u† (1m−1 − Tm−1 T†m−1 )−1 u . 2 1 + |zm |
Substituting in (15.168) and proceeding as in the derivation of (15.138) shows Jm,p (z1 , . . . , zm ) =
˜m,p C Jm−1,p−1 (z1 , . . . , zm−1 ), (1 − |zm |2 )p−m+1
˜m,p . Iterating (15.170) with m = N , p = N − n shows for a suitable C JN (z1 , . . . , zN ) ∝
N Y (1 − |zl |2 )n−1 l=1
and this substituted in (15.167) gives the stated result.
(15.170)
743
THE TWO-DIMENSIONAL ONE-COMPONENT PLASMA
E XERCISES 15.7
1. [321] In this exercise it will be argued that for a general Coulomb system on the pseudosphere, replacing the pair potential (15.155) by (15.165) does not affect the correlation functions in the thermodynamic limit. This helps explain the relationship between (15.164) and (15.112), as making the replacement a → iR gives that the potential (15.165) maps to − log sin(s/2R), which according to (15.107) is the Coulomb potential on a sphere. (i) Use P (15.157) to show that for the pair potential (15.165), the total potential energy of a system of charges qi ( i qi = 0) can be written „ « ˛ (z − z )/2a ˛ 1 X 1X 1X 2 ˛ i ˛ j 2 2 + − qi qj log ˛ q log(1 − (r /2a) ) − q log(1 − (ri /2a)2 ), ˛ i i 2 1 − zi z¯j /4a2 2 i 2 i i i =j
which can be interpreted as a Coulomb system in a disk of radius 2a with metallic boundary conditions (the second term corresponds to the self energy), also coupled to an external potential − 21 qi2 log(1 − (ri /2a)2 ). (ii) Use the formula
sjk (|zj − zk |/2a) = 2a (1 − (rj /2a)2 )1/2 (1 − (rk /2a)2 )1/2 to show that for the pair potential (15.165) the total potential energy of the system in (i) is sinh
−
1X 2 1X qi qj log(|zi − zj |/2a) − q log(1 − (ri /2a)2 ), 2 2 i i i =j
which is the same as in (i) apart from the image terms. (iii) Recalling (15.161), argue that for βqi2 /2−2 ≤ 0 at least, the density will be infinite at the boundary |z| = 2a in the thermodynamic limit. Conclude that the image particles will be perfectly screened from the interior of the disk and so can be ignored. 2.
(i) Show that for the one-component plasma with background density η and pair potential (15.165), the Boltzmann factor is proportional to (15.166). (ii) Consider now a quantum particle on the pseudosphere in the presence of a perpendicular magnetic field. = (0, 0, Ba (1 − cosh τ )), repeat the working which led to (15.129) to show With the vector potential A sinh τ that the projected Hamiltonian is given by (15.35) with Φ=−
“ a ”2 l
“ |z|2 ” log 1 − . (2a)2
(iii) Use the result of Exercises 15.2 q.2 to conclude that the states “
|z|2 ”(a/l) p z¯ , (2a)2 2
1−
p ∈ Z≥0 ,
(15.171)
are orthogonal states in the lowest Landau level. Form an N -particle state ψ as a Slater determinant from the single-particle states with p = 0, 1 . . . , N − 1, and note that |ψ|2 is proportional to the Boltzmann factor in (i) with Γ = 2 provided 4πηa2 + 1 = 2(a/l)2 . (iv) Set a = 1/2, (a/l)2 = L/2 − 1 in (15.171). Note that the normalized state is then equal to „“ «1/2 L − 1 ”“ L + p − 1 ” π
p
(1 − |z|2 )L/2−1 z¯p .
Analogous to (15.140), a general linear combination of these states with p = 0, . . . , N + 1 is specified by φ(z) :=
1 “ L − 1 ”1/2 (1 − |z|2 )L/2−1 p(z), N π
p(z) :=
N “ X L + j − 1 ”1/2 j=0
j
αj z j .
(15.172)
744
CHAPTER 15
With N → ∞ and L = 1, the correlation function for the zeros of p(z), which are given in general by the permanent formula (15.57), can be written as a determinant [443] h i 1 (15.173) ρ(n) (z1 , . . . , zn ) = π −n det 2 (1 − zj z¯k ) j,k=1,...,n (this relies crucially on the Borchardt identity (4.36)). Use the general formula (15.43) to show that this coincides with the N → ∞ correlation function for the 2dOCP in a unit disk without a background.
15.8 METALLIC BOUNDARY CONDITIONS Metallic boundary conditions along some boundary ∂Ω specify that the electrostatic potential must vanish along ∂Ω. Physically, such boundary conditions arise when a perfect conductor occupies the region outside ∂Ω. Prescribing the value of the potential along a boundary is also referred to as Dirichlet boundary conditions. With Ω a disk of radius R, and a perfect conductor occupying the region r > R, the electrostatic potential which satisfies the 2d Poisson equation (1.40) inside the disk, vanishes on the boundary of the disk and is symmetric in r and r is (z − z )/R (15.174) Φ(r, r ) = − log 1 − z z¯ /R2 (for R = 1 cf. (14.48)). The metal wall thus has the well-known property of creating an effective charge of opposite sign at the image point R2 /¯ z . Consider now the total energy of a one-component system, with a uniform background of density η in the region r < R. For hard wall boundary conditions, as considered in Exercises 1.4.1 q.3, it has been assumed that η is equal to the particle density ρ so that the system is overall charge neutral. However, as noted in Section 5.6.2, for metal boundary conditions, the image charges of opposite sign allow for a grand canonical formalism in which the background density is fixed independent of the number of particles, with the latter being summed over. For such a system with a fixed number N of particles of unit charge, the total potential energy consists of the particle-particle energy (zj − zk )/R ; U1 = − log 1 − zj zk /R2 1≤j
the self-energy
U1 ,
which is defined by U1 :=
N 1 Φ( r lim , r ) − log | r − r | , j j 2 j=1 r →rj
(15.175)
(recall Section 2.7.2) and so is equal to U1 =
1 N log(1 − rj2 /R2 ) + log R; 2 j=1 2 N
the particle-background energy U2 =
N j=1
V (rj ),
π V (r) = − η(R2 − r2 ), 2
(15.176)
which can be calculated from the fact that V (r) must satisfy the Poisson equation ∇2 V (r) = 2πη, together with the boundary condition V (R) = 0; and the background-background potential, which is calculated from
745
THE TWO-DIMENSIONAL ONE-COMPONENT PLASMA
the formula for V (r) using (1.43) to be given by U3 =
(πηR2 )2 . 8
Observe that the Boltzmann factor e−βU1 (r) ∝ 1/|1 − r2 /R2 |Γ/2 for a single particle and its image is not integrable at |r| = R for Γ ≥ 2. To prevent short distance collapse in these cases the particles can be confined to the region |r| < R − , > 0. The corresponding generalized grand partition function is then given by ΞΓ [a; ζ] = e−Γ(πηR ) /8 ∞ 2 2 ζ N 1 Γπη(R2 −r12 )/2 × d r a( r )e · · · drN a(rN )eΓπη(R −rN )/2 1 1 Γ/2 N ! ΩR− R ΩR− N =0 Γ N (zj − zk )/R 1 × (15.177) 1 − zj z¯k /R2 , 2 2 Γ/2 (1 − |z | /R ) j j=1 2 2
1≤j
where ΩR− denotes the disk of radius R − centered at the origin. At the special coupling Γ = 2, it is possible to write the second line of (15.177) as a determinant, and to proceed to express Ξ2 [a; ζ] as a determinant. P ROPOSITION 15.8.1 For Γ = 2 we have Ξ2 [a; ζ] = e−(πηR
2 2
) /4
det[1 + ζK],
where K is the integral operator defined by the mapping rule 2 2 f (z ) 1 a(r )eπη(R −r ) dr . K[f (z )](z) := R ΩR− 1 − z z¯ /R2 Proof. Setting xj = zj /R and yj = z¯j /R in the Cauchy double alternant (4.34) shows that N Y j=1
1 (1 − |zj |2 /R2 )
Y 1≤j
h i 1 |(zj − zk )/R|2 = det . (1 − zj z¯k /R2 )2 1 − zj z¯k /R2 j,k=1,...,N
(15.178)
Substituting this in (15.177) and using the Fredholm formula (5.32) gives the stated result.
15.8.1 Correlation functions Substituting the formula of Proposition 15.8.1 in (5.20) and using (5.31) gives ρ(n) (r1 , . . . , rn ) = det[G(rj , rk )]j,k=1,...,n , where G(r, r ) := r |ζK(1 + ζK)−1 |a=1 |r .
(15.179)
Thus G satisfies the integral equation 2 2 ζ G(r1 , r3 ) + eπη(R −r3 ) R
ΩR−
G(r1 , r2 ) ζ eπη(R −r3 ) d r = . 2 1 − z2 z¯3 /R2 R 1 − z1 z¯3 /R2 2
2
(15.180)
A useful symmetry of G is that ¯ r , r). G(r, r ) = G(
(15.181)
This follows by noting ζK(1+ζK)−1 = (1+ζK)−1 ζK and writing out the corresponding integral equation for G. With r1 and r3 fixed, the limit R → ∞ gives the correlation functions in the bulk. The integral equation
746
then reduces to
CHAPTER 15
R2
G(r1 , r2 ) dr2 = 1,
which has solution satisfying (15.181) G(r1 , r2 ) = ηe−πη(r1 +r2 )/2 eπηz1 z¯2 . Notice that there is no longer any dependence on the fugacity ζ. The corresponding distribution functions are identical to the distribution functions in the bulk with hard wall boundary conditions and η = ρ, as expected. 2
2
15.8.2 Vortex analogy The theory of two-dimensional incompressible fluid flow (here two-dimensional means that the flow is the same through any plane in the fluid parallel to the reference plane), has close analogies with two-dimensional electrostatics, and hence two-dimensional Coulomb systems. This can be seen from the continuity equation (x, y), which expresses in a mathematical form the assumption that ∇ · V (x, y) = 0 for the velocity field V the fluid is incompressible. Because V (x, y) is divergence-free, it is possible to choose a function Ψ(x, y) = ∇×(Ψˆ such that V z) (Ψ is known as a stream function; we will see below that motion of the fluid particles occurs along constant values of the stream function). This means that ∇2 Ψ(x, y) = −ξ(x, y),
ξ(x, y)ˆz = ∇ × V (x, y).
The function ξ(x, y) is called the vorticity — only when ξ(x, y) = 0 is there a net circulation about the point (x, y). This equation is the same as the two-dimensional Poisson equation satisfied by an electrostatic potential Ψ(x, y) due to a continuous charge distribution ξ(x, y)/2π. Thus we can write 1 Ψ(x, y) = G(r, r )ξ(r ) dr , (15.182) 2π Ω where G(r, r ) satisfies the 2d Poisson equation with delta function source term (1.40). In free boundary conditions we would have G(r, r ) = − log |r − r |. But for a fluid confined to a region (x, y) · n = 0, where n is the unit normal at the point (x, y) on the boundary. Ω the boundary condition is V Now, in terms of ξ(x, y), ∂ ∂ 1 ,− V (x, y) = G(r, r )ξ(r ) dr ∂y ∂x 2π Ω which shows that the boundary condition can be satisfied if ∂ ∂ G(r, r ), − G(r, r ) = 0 n · ∂y ∂x for r ∈ ∂Ω. But (−ny , nx ) is tangent to the boundary at (x, y), so this is equivalent to saying the directional derivative of G(r, r ) vanishes along the boundary, which in turn implies G(r, r ) must be constant for r ∈ ∂Ω. Choosing the constant to be zero shows that this is equivalent to metallic boundary conditions in the corresponding electrostatic problem. In particular, for a disk of radius R, G(r, r ) is therefore given by (15.174). To further exploit the analogy (see, e.g., [111]), consider the total kinetic energy of the fluid 1 E= ρ V 2 dr, 2 Ω where ρ denotes the mass density. Use of the vector identity × B) =B · (∇ × A) −A · (∇ × B), ∇ · (A
747
THE TWO-DIMENSIONAL ONE-COMPONENT PLASMA
= Ψˆ = ∇ × A, shows we can write with A z, B 1 × (Ψˆz)) dr. E= ρ Ψ(r)ξ(r) − ∇ · (V 2 Ω But from the divergence theorem the second integral can be rewritten as @ × (Ψˆz)) · n ds, (V ∂Ω
which vanishes since Ψ vanishes on the boundary. Hence, after making use of (15.182), we have ρ dr dr ξ(r)ξ(r )G(r, r ), E= 4π Ω Ω
(15.183)
which is formally proportional to the expression for the potential energy of a continuous charge distribution ξ(r) in Ω with metallic boundary conditions. In the case that there are N vortices with a small support centered at rj (j = 1, . . . , N ), one writes ξ(r ) =
N
ξj (r − rj ) ≈
j=1
N
Γj δ(r − rj ).
(15.184)
j=1
The quantities Γj represent the circulation about the point rj , since, with C denoting a small circle about rj and D the corresponding disk, @ (∇ × V ) · zˆ dr = ξ(r ) dr = Γj . V · dr = C
D
D
Substituting (15.184) for ξ in the formula (15.183) for E gives E=
N ρ 2 ρ Γj Γk G(rj , rk ) + Γ Gfinite (rj , rj ), 2π 4π j=1 j
(15.185)
j
where Gfinite represents the finite part of the self-energy (the infinite part which results from the point approximation (15.184) is disregarded, as in electrostatics). This expression is proportional to that for the potential energy of a system of charges of strength Γj in Ω with metallic boundary conditions. For Ω a disk of radius R, another case of interest arises when the whole fluid is rotated with angular velocity ω. Taking the rotating system as the frame of reference, we see that we should subtract ω from ξ(r) in (15.183). This implies that in the formula (15.185) there is an additional one-body term −
N ρω Γj (R2 − rj2 ), 4 j=1
(15.186)
as well as a constant. Comparing with (15.176), we see that this additional term corresponds to a uniform background in the analogy with charges in a disk with metallic boundary conditions.
15.9 ANTIMETALLIC BOUNDARY CONDITIONS The metallic boundary condition is the → ∞ boundary condition at the interface between a vacuum and a dielectric material with dielectric constant . To see this, with the dielectric material on the outside of some boundary ∂Ω, and n ˆ denoting the (outward) normal, we recall from macroscopic electrostatics that the
748
CHAPTER 15
potential Φ must satisfy the boundary condition ∂Φ ∂n ˆ
z→∂Ω−
∂Φ = . ∂n ˆ z→∂Ω+
(15.187)
In the limit → ∞ these equations can only be satisfied if Φ = C (C constant) for z ∈ ∂Ω+ , and thus by continuity of Φ, Φ = C for z ∈ ∂Ω. The other extreme limit is = 0. Although this is unphysical since ≥ 1, it is mimicked by situations in which the dielectric constant of the material inside the system is much greater than that outside. With = 0, (15.187) gives that the normal derivative of the potential must vanish at the boundary, a situation referred to as Neumann boundary conditions. Suppose now that the dielectric material occupies the outside of a disk of radius R. Then for the source point r inside the disk one can verify that the solution of the 2d Poisson equation (1.40) inside the disk, subject to the boundary condition (15.187), is |z − z | Φ(r, r ) = − log . (15.188) |R − zz /R|( −1)/( +1) Here the arbitrary additive constant has been chosen so that (15.188) reduces to (1.41) for = 1 and to (15.174) for → ∞. The formula (15.188) exhibits the well-known feature that the dielectric material creates an effective particle of charge −( − 1)/( + 1) at the image point R2 /¯ z . In the case that = 0, this image particle has charge of the same sign and strength as the test particle—such a boundary condition will therefore be referred to as anti-metallic. We will consider a one-component system formed by placing N particles of unit charge in a disk of radius R with the anti-metallic boundary condition. Inside the disk we assume that there is a uniform neutralizing background of density ρ. A short calculation using the formulas of Section 1.4.1, and taking into consideration the self energy as defined by (15.175), shows that the total energy of such a system is U=
N N2 1 3 πρrj2 −log(1−rj2 /R2 ) − log R− N 2 + 2 8 2 j=1
log |zk −zj ||1−zj z¯k /R2 | (15.189)
1≤j
(note that the constant terms are the same as for hard wall boundary conditions calculated in Exercises 1.4 q.3). The corresponding Boltzmann factor is thus Ae−πΓρ
PN j=1
rj2 /2
|zk − zj |Γ |1 − zj z¯k /R2 |Γ
2
((1/2) log R−3/8)
(1 − rj2 /R2 )Γ/2 ,
(15.190)
j=1
1≤j
where A = e−ΓN
N
.
15.9.1 Random matrix analogy In the case of a dielectric material with = 0 in the region y < 0, the pair potential is Φ(r, r ) = − log |z − z ||z − z¯ | . For a one-component system confined to the semi-disk |z| < R, y > 0, with a neutralizing background of density ρ = N/( 12 πR2 ), we see that the corresponding Boltzmann factor is proportional to N j=1
e−πΓρrj /2 |zj − z¯j |Γ/2 2
|zk − zj |Γ |zk − z¯j |Γ .
(15.191)
1≤j
With Γ = 2, this same expression (up to a factor of certain ensemble of random matrices [264], [395].
N j=1
|zj − z¯j |) occurs as the eigenvalue p.d.f. for a
749
THE TWO-DIMENSIONAL ONE-COMPONENT PLASMA
P ROPOSITION 15.9.1 Let X be an N × N random matrix with real quaternion elements, and suppose the (l) four real numbers ujk (l = 1, . . . , 4) in each such element are chosen independently with p.d.f. 2 1 − P4l=1 (u(l) jk ) . e 2 π For X a 2N × 2N matrix with complex elements, the eigenvalues λj of X come in complex conjugate pairs ¯ j ), and the p.d.f. of the N eigenvalues with Im(λj ) > 0 is proportional to (15.191) with Γ = 2 (λj+N = λ N and ρ = 2/π, multiplied by an additional self-energy type factor j=1 |zj − z¯j |.
Proof. The mechanism for the eigenvalues occurring in complex conjugate pairs is the same as that for the Kramer’s degeneracy discussed in Exercises 1.3 q.1. Formally, the result for the p.d.f. follows from the p.d.f. for complex random matrices Y 2 1 − PN j=1 |λj | |λk − λj |2 , e C 1≤j
¯ j for j = 1, . . . , N . This prescription can be verified by starting with by simply replacing N by 2N and putting λj+N = λ the Schur decomposition (15.3) where now U has real quaternion elements, and proceeding as in the proof of Proposition 15.1.1 (see [395]).
15.9.2 Correlation functions At the special coupling Γ = 2, the products in (15.190) can be written in terms of the Vandermonde determinant (1.173) of dimension 2N with xj = zj /R, xj+N = R/¯ zj (j = 1, . . . , N ), and the free energy and correlation functions calculated exactly [493]. First, for the products in the Boltzmann factor we have N N (zj /R)k−1 2 2 −N 2 2N −1 |zk − zj | (1 − zj z¯k /R ) = R (¯ zj ) det . (15.192) (R/¯ zj )k−1 j=1,...,N 1≤j
j=1
j,k=1
k=1,...,2N
The determinant has a similar structure to the one appearing in (6.14), which suggests we proceed using skew orthogonal polynomials. The appropriate skew symmetric inner product for this purpose is 2π 2 1 R f |gs = z) − f (R/¯ z )g(z/R) , (15.193) dr r2N e−πρr dθ e−iθ(2N −1) f (z/R)g(R/¯ 2 0 0 z := reiθ . With respect to this inner product we can check that Q2n+1 (z) = z 2N −1−n ,
Q2n (z) = z n
(n = 0, . . . , N − 1)
(15.194)
are skew orthogonal with normalization π (15.195) qn = R2N +1 N −n−1 γ(n + 1; N ) − N n−2N γ(2N − n; N ) . 2 Here we have relaxed the condition that Qj (z) be of degree j, and replaced it by the requirement that Qk−1 (zj /R) (zj /R)k−1 det ∝ det . Qk−1 (R/¯ zj ) j=1,...,N (R/¯ zj )k−1 j=1,...,N k=1,...,2N
k=1,...,2N
The analogue of the formulas of Propositions 6.1.6 and 6.1.7 apply for the evaluation of the partition function and correlation functions (see Exercises 15.9 q.1). In particular, for the density ρ(1) (z) in the finite system these formulas imply 2 N −1 z )2N −1−n − (z/R)2N −1−n (R/¯ z )n (¯ z )2N −1 e−πρ|z| (z/R)n (R/¯ . ρ(1) (z) = πR2N +1 N −n−1 γ(n + 1; N ) − N n−2N γ(2N − n; N ) n=0
(15.196)
750
CHAPTER 15
Taking the limit N, R → ∞ with z and ρ fixed gives for the bulk density ρ(z) = ρ as expected. If instead the coordinates are chosen as in (15.49) and then the limit N → ∞ taken (which requires using (15.51)), we obtain for the density profile as measured from the boundary at y = 0, ∞ 1/2 2 2 sinh(2πρ) yt 2 ρ(1) (y) = ρe−2πρy g(2y), g(y) := √ dt. (15.197) e−t π −∞ erf t With the particle coordinates fixed in the bulk of the system, we find that the general n-particle correlation deduced from the quaternion determinant formalism reduces to the bulk expression of Proposition 15.3.2 in the thermodynamic limit. For the n-particle correlation in the neighborhood of the boundary, using the coordinates (15.49) and the asymptotic expansion (15.51), we find ρ(n) (r1 , . . . , rn ) = ρn 2 2 e−πρ(yj +yk ) g(yj + yk + i(xj − xk )) ×qdet 2 2 e−πρ(yj +yk ) g(yj − yk − i(xj − xk ))
2
2
−e−πρ(yj +yk ) g(yj − yk + i(xj − xk )) 2 2 e−πρ(yj +yk ) g(yj + yk − i(xj − xk ))
, j,k=1,...,n
(15.198) where g(u) is defined in (15.197). 1. (i) Use the method of the proof of Proposition 6.1.6 (with C(x) replaced by the identity) to derive the formula – » N N−1 N Y Y Y Y 2 S(zj , zk ) I(zj , zk ) e−πρ|zl | |zk − zj |2 (1 − zj z¯k /R2 ) = 2ql qdet , D(zj , zk ) S(zk , zj ) N×N
E XERCISES 15.9
l=1
1≤j
where with w(z) ˜ := e−πρ|z|
j,k=1
2
S(z, u) := w(z) ˜ w(u) ˜
/2 (N−1/2)
z¯
N−1 X m=0
I(z, u) := −w(z) ˜ w(u) ˜
” 1 “ u) − Q2m+1 (z/R)Q2m (R/¯ u) , Q2m (z/R)Q2m+1 (R/¯ 2qm
N−1 X m=0
D(z, u) := w(z) ˜ w(u) ˜
l=0
N−1 X m=0
” 1 “ Q2m (z/R)Q2m+1 (u/R) − Q2m+1 (z/R)Q2m (u/R) , 2qm
” 1 “ z )Q2m+1 (R/¯ u) − Q2m+1 (R/¯ z )Q2m (R/¯ u) . Q2m (R/¯ 2qm
(ii) With {Qj (x)}j=0,1,...,2N−1 skew orthogonal with respect to the skew inner product (15.193), note that the analogue of the formulas (6.17) hold and thus so does the analogue of Proposition 6.1.7. 2. From Proposition 15.9.1 √ we know that the eigenvalue p.d.f. for random matrices with real quaternion elements is, after scaling zj → zj / 2, N 1 Y −|zj |2 e |zj − z¯j |2 QN j=1
Y
|zk − zj |2 |zk − z¯j |2 ,
1≤j
where Im(zj ) ≥ 0 and −N
Z
QN = 2
Z dx1 dy1 . . .
R2
dxN dyN R2
N Y l=1
2
2
e−(xl +yl ) |zl − z¯l |2
Y 1≤j
|zk − zj |2 |zk − z¯j |2 .
751
THE TWO-DIMENSIONAL ONE-COMPONENT PLASMA
(i) Show from the Vandermonde determinant (1.173) N Y (¯ zl − z l ) l=1
»
Y
2
zjk−1 (¯ zj )k−1
2
|zk − zj | |zk − z¯j | = det
1≤j
– . j=1,...,N k=1,...,2N
(ii) Define the skew symmetric inner product Z 2π Z “ ” 2 1 ∞ f |gs = dr re−r dθ (z − z¯) f (z)g(¯ z ) − f (¯ z )g(z) , 2 0 0 and verify that with respect to this inner product Q2n (z) = 2n n!
n X z 2l , 2l l! l=0
Q2n+1 (z) = z 2n+1
(n = 0, . . . , N − 1)
are skew orthogonal with normalization qn = πΓ(2n + 2). (iii) Use the results of (i) and (ii) to show QN =
N−1 Y
N−1 Y
qn = π N
n=0
Γ(2n + 2).
n=0
2
z ), 2qn replaced by Qn (z), (iv) Use the formulas of q.1 with w(z) ˜ = e−|z| /2 (z − z¯)1/2 and Qn (z/R), Qn (R/¯ Qn (¯ z ), qn , respectively, as specified above to show that in the limit N → ∞ ρ(n) (r1 , . . . , rn ) =
n Y
2
e−| rj | 2yj qdet
j=1
where f (w, z) =
»
f (zj , z¯k ) f (¯ zj , z¯k )
−f (zj , zk ) f (zk , z¯j )
– , n×n
∞ ““ z ”n+1/2 “ w ”n+1/2 ” X i , I (wz) − n+1/2 (2π)1/2 n=0 w z
with Ia (z) denoting the Bessel function of pure imaginary argument, specified by the integral (8.67) for a ∈ Z, and more generally by the series In+1/2 (x) :=
∞ X (x/2)2k+n+1/2 . k!Γ(k + n + 3/2) k=0
(v) [344] Verify that ∂f i = zf + ezw , ∂z π Add these two equations together to deduce f (w, z) = e(w
2
+z 2 )/2
∂f i = wf − ezw . ∂w π A(z − w),
A(0) = 0,
for some A(x), and substitute this back in either of the equations to deduce 2 ∂A(s) i = e−s /2 . ∂s π
Hence conclude
“z − w” 2 2 i . f (w, z) = √ e(w +z )/2 erf √ 2π 2
(15.199)
752
CHAPTER 15
15.10 EIGENVALUES OF REAL RANDOM MATRICES For N × N random matrices with real entries there is a nonzero probability that there will be exactly k real eigenvalues for each k = 0, . . . , N , provided N and k have the same parity (this latter constraint is required because the complex eigenvalues must occur in complex conjugate pairs). In the case that the entries of such a matrix, A say, are independent standard Gaussians (the matrix A is then said to belong to the real Ginibre ensemble) the joint p.d.f. for the k real and N −k complex eigenvalues can be calculated exactly [371], [155]. For this purpose one uses the real Schur decomposition A = QRQT ,
(15.200)
where Q is an N × N orthogonal matrix with elements of the first row positive while ⎤ ⎡ λ1 · · · R1,k R1,k+1 · · · R1,m ⎥ ⎢ .. .. .. .. ⎥ ⎢ . . . ··· . ⎥ ⎢ ⎢ λk Rk,k+1 · · · Rk,m ⎥ ⎥. R=⎢ ⎢ Zk+1 · · · Rk+1,m ⎥ ⎥ ⎢ ⎥ ⎢ .. .. ⎦ ⎣ . .
(15.201)
Zm In (15.201) all elements not explicitly shown are zero, m = (N + k)/2, while Rij is of size p × q with ⎧ 1 × 1 if i ≤ k, j ≤ k, ⎪ ⎪ ⎨ 1 × 2 if i ≤ k, j > k, p×q = 2 × 1 if i > k, j ≤ k, ⎪ ⎪ ⎩ 2 × 2 if i > k, j > k. Furthermore, the λj (j ≤ k) are the real eigenvalues of A, while the 2 × 2 matrices Zj have the structure xj bj , bj , cj > 0, Zj = (15.202) −cj xj such that the complex eigenvalues of A are xj ± iyj , yj = bj cj (see Exercises 15.10 q.2). ˜ denote the strictly upper triangular part of R, involving only entries labeled Rj,k in (15.201). We Let R ˜ has N (N − 1)/2 − (N − k)/2 independent variables, while {λi } has k independent variables, see that R {Zi } has 3(N − k)/2 independent variables and Q has N (N − 1)/2 independent variables. This adds up to the N 2 independent parameters of A. The change of variables (15.200) can be carried out according to the following result [155].
P ROPOSITION 15.10.1 We have (dA) = 2(N −k)/2
T ˜ |λ(Rpp ) − λ(Rjj )|(dR)(Q dQ)
j
(N +k)/2
×
|bl − cl |dλ1 · · · dλk (dZ),
l=k+1
where λ(Rll ) = λl for l ≤ k, while λ(Rll ) = xl ± iyl for l > k, and
(N +k)/2
(dZ) =
j=k+1
dxj dbj dcj .
(15.203)
753
THE TWO-DIMENSIONAL ONE-COMPONENT PLASMA
Proof. Analogous to (15.4) one has QT dA Q = dR + QT dQ R − RQT dQ. With dO := QT dQ the ij (block) element on the r.h.s. is dRij + dOij Rjj − Rii dOij +
X
dOil Rlj −
l<j
X
Ril dOlj
(15.204)
l>i
(cf. (15.5)). Arguing now as in the derivation of (15.8) shows that the wedge product of the off-diagonal elements in (15.204) is Y ˜ ˜ |λ(Rkk ) − λ(Rjj )|(dO)(d R), (15.205) j
˜ is as defined below (15.202), while (dO) ˜ refers to the product of (block) off-diagonal differentials in dO. where R Noting that – » 0 doi dOii = −doi 0 for some doi shows that with i = j > k (15.204) has the explicit form » – dbi dxi + (bi − ci )doi . −dci dxi + (ci − bi )doi The wedge product of these entries is therefore equal to Y
(N+k)/2
2(N−k)/2
Y
(N+k)/2
|bl − cl |(dZ)
l=k+1
doi .
(15.206)
i=k+1
Multiplying together (15.206) and (15.205), and noting too that for i = j < k (15.204) reduces to dλi , gives the result.
Suppose now that 1 N 2 /2 T e−Tr(AA )/2 (15.207) 2π so that the elements of A are independent real Gaussians with distribution N[0, 1]. Changing variables according to (15.200) the corresponding eigenvalue p.d.f. can be computed. Pr(A) =
P ROPOSITION 15.10.2 For N × N real Gaussian matrices distributed according to (15.207), with k real eigenvalues λ1 , . . . , λk and (N − k)/2 complex conjugate pairs of eigenvalues xj ± iyj (j = 1, . . . , (N − k)/2) the eigenvalue p.d.f. is equal to 1 2(N −k)/2 } ∪ {x ± iy } ) Δ({λ l l=1,...,k j j j=1,...,(N −k)/2 k!((N − k)/2)! 2N (N +1)/4 N Γ(l/2) l=1 ×e−
Pk j=1
λ2j /2
e
P(N −k)/2 j=1
(yj2 −x2j )
(N −k)/2
√ erfc( 2yj ),
(15.208)
j=1
m where Δ({zp }p=1,...,m ) := j 0} gives the probability that a matrix from the ensemble (15.207) has exactly k real eigenvalues. Proof. Substituting for A using (15.200) and making use of (15.201), (15.202) shows T
e−Tr(AA
)/2
= e−
P i<j
2 rij /2 −
e
Pk
j=1
λ2 j /2 −
e
P(N −k)/2 j=1
2 2 (x2 j +yj +δj /2)
,
(15.209)
754
CHAPTER 15
where δj := bj − cj
(15.210)
˜ Furthermore, a direct calculation gives and [rij ] = R. dbj dcj = q
2yj δj2 + 4yj2
dyj dδj .
(15.211)
Multiplying (15.209) and (15.203), and further changing variables according to (15.210), making use of (15.211) we see that the eigenvalues separate from the other variables apart from the integration over δj . For this one uses e−y
2
Z
√ √ 2 |δ|e−δ /2 p dδ = 2πey erfc( 2y), δ 2 + 4y 2 2
∞ −∞
which can be derived by appropriate changes of variables. To calculate the precise form of the constants, note from (1.38) in the case β = 1 that Z N Y 1 (QT dQ) = π N(N+1)/4 , Γ(j/2) j=1 while the integration over {rij } are simple Gaussians.
Setting k = N in (15.208), and integrating over the λj using (1.163) in the case β = 1 shows that the probability of all eigenvalues being real is equal to 2−N (N −1)/4 .
Generalized partition function Let the p.d.f. (15.208) be denoted p({λl }l=1,...,k ; {xj ± iyj }j=1,...,(N −k)/2 ), and specify the corresponding generalized partition function by ∞ ∞ k Zk,(N −k)/2 [u, v] = dλ1 · · · dλk u(λl ) dx1 dy1 · · · dx(N −k)/2 dy(N −k)/2 −∞
−∞
l=1
R2+
R2+
(N −k)/2
×
v(xl , yl )p({λl }l=1,...,k ; {xj ± iyj }j=1,...,(N −k)/2 ).
l=1
This can be evaluated in terms of the series expansion of a particular Pfaffian, analogous to the expression of Proposition 6.7.1. P ROPOSITION 15.10.3 Let {pl−1 (x)}l=1,...,N be a set of monic polynomials, with pl−1 (x) of degree l − 1. Let ∞ ∞ 2 2 αj,k = dx u(x) dy u(y) e−(x +y )/2 pj−1 (x)pk−1 (y)sgn (y − x), −∞ −∞ √ 2 2 βj,k = 2i dxdy v(x, y)ey −x erfc( 2y) pj−1 (x + iy)pk−1 (x − iy) − pk−1 (x + iy)pj−1 (x − iy) . R2+
(15.212) For k, N even we have Zk,(N −k)/2 [u, v] =
2N (N +1)/4
1 N
l=1 Γ(l/2)
[ζ k/2 ]Pf[ζαj,l + βj,l ]j,l=1,...,N .
(15.213)
755
THE TWO-DIMENSIONAL ONE-COMPONENT PLASMA
Proof. We have 2 [pl−1 (λj–)]j=1,...,k “ ” » Δ {λl }l=1,...,k ∪ {xj ± iyj }j=1,...,(N−k)/2 = det 4 pl−1 (xj +iyj ) pl−1 (xj −iyj )
3 5
j=1,...,(N−k)/2
.
l=1,...,N
With the λs ordered λk > · · · > λ1 , the absolute value signs about Δ in (15.208) can be removed, provided one multiplies by i(N−k)/2 . Substituting for Δ according to the above formula, we can apply the method of integration over alternate variables. With k assumed even, this gives Zk,(N−k)/2 [u, v] = i(N−k)/2 Ak,N
√ ×erfc( 2yl )
k! (k/2)!
Z
Z
Z R2 +
Z
∞
dx1 dy1 · · · Z
∞
dλ2 −∞
−∞
dλ4 · · ·
Y
(N−k)/2 R2 +
dx(N−k)/2 dy(N−k)/2
l=1
2 " R λ2j
6 6 dλk det 6 4 −∞ ∞
2
2 u(λ)e−λ /2 pl−1 (λ) dλ −∞
u(λ2j
»
−λ2 /2 2j )e p
#
l−1 (λ2j )
–
pl−1 (xj +iyj ) pl−1 (xj −iyj )
2
v(xl , yl )eyl −xl
3 j=1,...,k
j=1,...,(N−k)/2
7 7 7 5 l=1,...,N
where Ak,N denotes the numerical prefactor in (15.208). Expanding the determinant according to its definition, then performing the remaining integrations shows Zk,(N−k)/2 [u, v] = Ak,N
k/2 X Y k! ε(P ) aP (2l−1),P (2l) (k/2)! P ∈S l=1 N
where
Z aj,k = Z
∞
dx u(x)e−x
2
Z /2
−∞
dxdy v(x, y)ey
bj,k =
pk−1 (x)
2 −x2
R2 +
x
dy u(y)e−y
2
Y
N/2
bP (2l−1),P (2l) ,
l=k/2+1
/2
pj−1 (y),
−∞
√ erfc( 2y)pj−1 (x + iy)pk−1 (x − iy).
Imposing the restriction P (2l) > P (2l − 1) and using the notation of the statement of (15.213) this reads Zk,(N−k)/2 [u, v] = 2(k−N)/2 Ak,N
k! (k/2)!
X
ε(P )
P ∈SN P (2l)>P (2l−1)
k/2 Y l=1
Y
N/2
αP (2l−1),P (2l)
βP (2l−1),P (2l)
l=k/2+1
= 2(k−N)/2 Ak,N k!((N − k)/2)![ζ k/2 ]Pf[ζαj,l + βj,l ]j,l=1,...,N (cf. (6.159)), where the final equality, which immediately implies (15.208), can be verified directly from the definition of the Pfaffian.
With N even and ZN [u, v] :=
N/2
ZN [u, v] =
k=0
Z2k,(N −2k)/2 [u, v], we read off from (15.213) that [491]
2N (N +1)/4
1 N l=1
Γ(l/2)
Pf[αj,k + βj,k ]j,k=1,...,N .
(15.214)
N/2 Similarly we see that ZN (ζ) := k=0 ζ k Z2k,(N −2k)/2 [1, 1], which is the generating function for the probability p2k,N = Z2k,(N −2k)/2 [1, 1] of there being exactly 2k real eigenvalues, has the Pfaffian form 1 + β ] . (15.215) ZN (ζ) = Pf[ζα j,k j,k j,k=1,...,N N u=v=1 2N (N +1)/4 l=1 Γ(l/2) If we choose, say, pj (x) = xj in (15.212) all the integrals defining αj,k , βj,k can be computed explicitly
,
756
CHAPTER 15
[155] (see also the proof of Proposition 15.10.4 below). Furthermore, we can check that by the symmetry of the integrand α2j,2k |u=1 = α2j−1,2k−1 |u=1 = 0,
β2j,2k |v=1 = β2j−1,2k−1 |v=1 = 0.
Thus the entries of the Pfaffian in (15.215) vanish in a checkerboard fashion. As in deducing (8.132) from (8.134) we can therefore write the Pfaffian as a determinant of half the size [10] 1 ZN (ζ) = | + β | , (15.216) det ζα N 2j−1,2k u=1 2j−1,2k v=1 j,k=1,...,N/2 2N (N +1)/4 l=1 Γ(l/2) giving a computable formula for the p2k,N . From this viewpoint the general N exact result pN,N = 2−N (N −1)/4 follows by noting from the coefficient of ζ N/2 in (15.215) that pN,N =
2N (N +1)/4
1 N l=1
Γ(l/2)
det[α2j−1,2k |u=1 ]j,k=1,...,N/2 .
Choosing pj (x) = Rj (x), where {Rj (x)} are the skew orthogonal polynomials for the β = 1 Gaussian weight, as discussed in Section 6.3 the determinant becomes diagonal, implying the sought result.
Correlation functions Since the number of real eigenvalues is a variable, the summed up generalized partition function (15.214) is the appropriate quantity to use in the functional derivative formula for the correlation functions. The latter may involve both real and complex eigenvalues. In the case that it involves only real eigenvalues (to be denoted ρr(n) ) the functional differentiation formula reads δn 1 ZN [u, 1] . (15.217) ρr(n) (x1 , . . . , xn ) = ZN [1, 1] δu(x1 ) · · · δu(xn ) u=1 To compute (15.217), suppose furthermore that the polynomials {pl−1 (x)}l=1,2,... in Proposition 15.10.3 have been chosen so that the matrix [(αj,k + βj,k )|u=v=1 ]j,k=1,...,2N evaluates to the block diagonal structure of (6.2). This is equivalent to supposing that {pl−1 (x)}l=1,2,... are skew orthogonal with respect to the skew product implied by the matrix elements. Now, the only term in the matrix elements dependent on u is αj,k , 2 and this is proportional to the β = 1 inner product (6.61) with e−V (x) = e−x /2 (Gaussian case). As a consequence the functional differentiation formula (15.217) must give a formula for ρr(n) which is structurally identical to that implied by Propositions 6.3.3 and 6.3.2 for ρGOE (n) . Hence [214], [89], [497] r S (xj , xk ) I˜r (xj , xk ) r (15.218) ρ(n) (x1 , . . . , xn ) = qdet Dr (xj , xk ) S r (xk , xj ) with e−y2 /2 Φ2k (x)p2k+1 (y) − Φ2k+1 (x)p2k (y) , uk k=0 y ∂ r 1 S (x, y), I˜r (x, y) = sgn(y − x) − S r (x, z) dz. Dr (x, y) = ∂x 2 x N/2−1
S r (x, y) =
In (15.219) uk−1 := (α2k−1,2k + β2k−1,2k )|u=v=1 ,
Φk (x) =
∞
−∞
sgn(x − y)pk (y)e−y
2
/2
(15.219) (15.220)
dy.
The crucial difference between ρr(n) and ρGOE (n) is the explicit form of the skew orthogonal polynomials,
THE TWO-DIMENSIONAL ONE-COMPONENT PLASMA
757
which in the former case are given by the following result. P ROPOSITION 15.10.4 Consider the matrix X := [(αj,k + βj,k )|u=v=1 ]j,k=1,...,2N . The polynomials {pl (x)}l=0,... with the property that X evaluates to the block diagonal structure (6.2) are specified by p2j+1 (x) = x2j+1 − 2jx2j−1 ,
(15.221)
√ (α2k−1,2k + β2k−1,2k )|u=v=1 = 2 2πΓ(2k − 1).
(15.222)
p2j (x) = x2j , and furthermore
Proof. For general {pj }j=0,1,... set
pj , pk := αj,k |u=1 + βj,k |v=1 .
For any even p2j and odd p2j+1 , by changing variables x → −x, y → −y in the definition of αj,k , and changing variables θ → π − θ in the definition of βj,k it is easy to see that p2j , p2k = p2j+1 , p2k+1 = 0. It remains to verify that with pj as stated in (15.221), p2j−1 , p2k = 0 for j = k, and that for j = k the normalization (15.222) results. This is an immediate corollary of the explicit formula j −2j+k+3/2 j!Γ(k + 1/2), j ≥ k, (15.223) x2j+1 , x2k = 0, j < k. To derive (15.223), use can be made of recurrences satisfied by the corresponding integrals [215]. However the details are lengthy, and so will not be given in full. Briefly, using integration by parts one finds the recurrences “ 3” , α2j+4,2k+1 |u=1 = (2j + 2)α2j+2,2k+1 |u=1 − 2Γ j + k + 2 “ ” 3 α2j+2,2k+3 |u=1 = (2k + 1)α2j+2,2k+1 |u=1 + 2Γ j + k + , 2 ” “ √ 3 − 2 2π(j + k + 1)!δj+1,k , β2j+4,2k+1 |v=1 = (2j + 2)β2j+2,2k+1 |v=1 + 2Γ j + k + 2 “ √ 3” β2j+2,2k+3 |v=1 = (2k + 1)β2j+2,2k+1 |v=1 − 2Γ j + k + + 2 2π(j + k + 1)!δj+1,k . 2 With γj,k = xj , xk := αj,k |u=1 + βj,k |v=1 it follows that √ γ2j+4,2k+1 = (2j + 2)γ2j+2,2k+1 − 2 2π(j + k + 1)!δj+1,k (j ≥ 0, k ≥ 1), √ γ2j+2,2k+3 = (2k + 1)γ2j+2,2k+1 + 2 2π(j + k + 1)!δj+1,k (j ≥ 0, k ≥ 0). √ With initial condition γ2,1 = −2 2π, these recurrences can be verified to have the solution (15.223).
Substituting the result of Proposition 15.10.4 in (15.219) shows 2 2 Γ(N − 2; xy) e−(x +y )/2 1 √ N −1 x2 /2 γ(N/2 − 1/2; x2 /2) √ ( 2y) + exy , (15.224) e sgn(x) S r (x, y) = 2 Γ(N − 1) Γ(N − 1) 2π where use has been made of (15.50) and (15.55). According to (15.218), setting x = y in this gives the density √ of the real eigenvalues. Plots show that to leading order the support of the density is in the region |x| < N and indicate the boundary layer to be O(1). This suggests that for N → ∞ the correlations are well defined in the neighborhood of this edge. Indeed use of (15.51) shows 2 √ √ X + Y e−Y 1 1 −(X−Y )2 /2 r lim S ( N + X, N + Y ) = √ e 1 − erf √ + √ (1 + erf X) . (15.225) N →∞ 2π 2 2 2 2
758
CHAPTER 15
For X, Y → −∞ (15.225) reveals the bulk limiting form 2 1 lim S r (X, Y ) = √ e−(X−Y ) /2 , 2π
(15.226)
N →∞
√ which with X = Y implies that the density of real eigenvalues is the constant 1/ 2π. One consequence ∞ of this is that the expected number of real eigenvalues, EN := −∞ ρr (x) dx, has the large N behavior EN ∼ 2N/π. In fact direct integration of (15.224) in the case x = y leads to the evaluation [156] 2 Γ(N + 1/2) 1 (15.227) EN = + 2 F1 (1, −1/2; N ; 1/2), 2 π Γ(N ) and from this the asymptotic expansion of EN can be systematically generated. Application of Proposition 15.10.4 to the calculation of the correlation between complex eigenvalues is given in Exercises 15.10 q.3. Although not considered here, the mixed correlation involving both real and complex eigenvalues can be computed [89] by working similar to that in the proof of Proposition 6.7.2. Consider the probability Pr(Rs ), say, that either there is no real eigenvalue, or all real eigenvalues are less than s. This is given in terms of the generalized partition function (15.214) according to Pr(Rs ) = ZN [χx∈(−∞,s) , 1].
(15.228)
With f r denoting the 2 × 2 matrix integral operator on (−∞, ∞) with kernel equal to the block matrix exhibited by (15.218), and χ(−∞,s) = diag[χ(−∞,s) , χ(−∞,s) ], the reasoning which led to (9.18) implies 2 Pr(Rs ) = det[12 − f r χ(−∞,s) ]. However, from the perspective √ of numerical computation it is easier to work directly with (15.228). A case of particular interest is s = N , which corresponds to the probability that either no eigenvalue is real, or all real eigenvalues are inside the leading order support. The task of computing such probabilities first arose in the context of stability analysis of biological webs [374], [391] (see also Exercises 15.10 q.4). E XERCISES 15.10
1. [170] A polynomial of the form f (z) = z N + a1 z N−1 + · · · + aN ,
aN = eiφ
aN−j = aN a ¯j ,
is said to be self-reciprocal. Such polynomials have a nonzero probability of having zeros on the unit circle, while zeros off the unit circle occur in pairs ρj eiθj , (1/ρj )eiθj . Suppose N is odd, and let there be L zeros {αj = eiδj }j=1,...,L on the unit circle (L odd) and 2M zeros {βj = ρj eiθj , 1/β¯j = (1/ρj )eiθj }j=1,...,M off the unit circle. The Jacobian J for the change of variables from Re a1 , Im a1 , . . . Re a(N−1)/2 , Im a(N−1)/2 , φ specifying the coefficients, to {δj }j=1,...,L , {ρj , θj }j=1,...,M specifying the zeros has been calculated to be equal to M ˛ ˛Y 1 ˛ ˛ Δ(β1 , 1/β¯1 , . . . , βM , 1/β¯M , α1 , . . . , αL )˛, 2M −(N−1)/2 ˛ ρ m m=1 where Δ is as in (15.208). Relate this to the Boltzmann factor for a certain one-component log-potential system is a unit disk with anti-metallic boundary conditions. 2. [155] Let XN be a real N × N matrix. Let α ± iβ, β > 0 be a pair of complex eigenvalues with corresponding y |2 = 1 and c > 0. eigenvectors x ± ic y , where | x|2 = | (i) Show that x· y = 0 and XN [x y ] =
[ x
y]
»
α −βc
β/c α
– .
(15.229)
Show too that the effect of interchanging x and y is to replace c → −1/c in (15.229). Conclude that the decomposition (15.229) for | x|2 = | y |2 = 1, the first component of x, y positive and x· y = 0 is unique provided c ≥ 1.
759
THE TWO-DIMENSIONAL ONE-COMPONENT PLASMA
(ii) Show from the result of Exercises 1.9 q.3 that there is a Householder matrix U1 such that U1 x = e1 , and y= y , where y is a unit vector with first component 0. Now use the fact that x· y = 0 to show that U1 construct a Householder transformation U2 of the form – » T 0N−1 1 , 0N−1 V where V is an N − 1 × N − 1 Householder transformation with the property U2 y = e2 , to deduce that U2 U1 [ x y] = [e1
e2 ].
(iii) Conclude from the results of (i) and (ii) that for Q = U2 U1 an orthogonal matrix we have – – » » Z Y2×N−2 α b1 , QT , Z := XN = Q 0N−2×2 XN−2 −b2 α √ where b1 , b2 > 0, b1 b2 = β. 3. [214], [155] In this exercise the explicit form of ρc(n) is given. (i) Let f c denote the 2 × 2 matrix integral operator on (−∞, ∞) with kernel equal to f c (w, z) and let g = diag [g(z), g(z)]. With the polynomials {pl−1 (x)}l=1,2,... chosen as in Proposition 15.10.4, and q2j−2 (z) := −p2j−1 (z), q2j−1 (z) := p2j−2 (z), use the method of Proposition 6.1.9 to show Zn [1, 1 + g] = det(12 + f c g), where, with z = x + iy, w = u + iv, and uk as in (15.219), f c (w, z) := 2iev
2
−u2
√ erfc( 2v)
»
¯ z) S c (w, S c (w, z)
−S c (w, ¯ z¯) −S c (w, z¯)
– ,
S c (w, z) :=
N X pj−1 (w)qj−1 (z) j=1
u[(j−1)/2]
.
(ii) Use the theory of the paragraph including (6.31) to deduce from (i) that ρc(n) ((x1 , y1 ), . . . , (xn , yn )) = qdet[f c ((xj , yj ), (xk , yk ))]j,k=1,...,n and thus in particular r ρc(1) ((x, y))
=
√ 2 Γ(N − 1, x2 + y 2 ) 2y 2 ye erfc( 2y). π Γ(N − 1)
(15.230)
4. Let y(t) be an N × 1 column vector, and let B be an N × N random real Gaussian matrix in which the entries are chosen from N[0, σ]. Consider the matrix linear differential equation d y(t) y (t). = (−1N + B) dt (i) Show that for y(t) → 0 as t → ∞, the real parts of all eigenvalues of B must be less than 1.
√ (ii) Deduce from (15.225) and (15.230) √ that for σ = 1 and N large the spectral radius is equal to N + O(1). Use this to show that for σ = / N , and with N → ∞, the condition of (i) holds provided 0 < < 1. 5. [371], [215] Let S be an element of the GOE. Let A be an antisymmetric real Gaussian matrix with joint distribution of its elements proportional to exp(−Tr A2 /2). With 0 < τ < 1 and c := (1 − τ )/(1 + τ ) define random matrices X according to √ 1 X = √ (S + cA). (15.231) b (i) When τ = 0, b = 1 observe that X is a real Gaussian matrix with distribution (15.207), while when τ = 1, b = 1, X is a member of the GOE.
760
CHAPTER 15
(ii) Write S and A in terms of X and XT , and also show from (15.231) that √ √ 2 (dX) = 2N(N−1)/2 ( c)N(N−1)/2 ( b)−N (dS)(dA), to deduce from the joint distribution of the elements of S and A that the joint distribution of the elements of X is equal to “ “ ”” b Aτ,b exp − Tr XXT − τ Tr X2 (15.232) 2(1 − τ ) (cf. (15.15)) where √ √ 2 2 Aτ,b = ( c)−N(N−1)/2 ( b)N (2π)−N /2 . (iii) Denote (15.208) by Pk,(N−k)/2 ({λj }j=1,...,k ; {xj ± iyj }j=1,...,(N−k)/2 ), which corresponds to the eigenvalue p.d.f. in the case √ τ = 0, b = 1, conditioned so that there are exactly k real eigenvalues. For these matrices scale X → bX/(1 − τ )1/2 to obtain that for matrices with p.d.f. A0,1 bN
2
/2
(1 − τ )−N
2
/2 −bTr XXT /2(1−τ )
e
,
(15.233)
the eigenvalue p.d.f. is equal to bN/2 (1 − τ )−N/2 “ √ ” √ √ ×Pk,(N−k)/2 { bλj /(1 − τ )1/2 }j=1,...,k ; { bxj /(1 − τ )1/2 ± i byj /(1 − τ )1/2 }j=1,...,(N−k)/2 . By comparing (15.233) to (15.232) deduce that for matrices with p.d.f. (15.232) the eigenvalue p.d.f, conditioned so that there are exactly k eigenvalues, is equal to (N−k)/2 k “ τb “ X ”” X Aτ,b (1 − τ )N(N−1)/2 exp λ2j + 2 (x2j − yj2 ) A0,1 2(1 − τ ) j=1 j=1 √ √ √ 1/2 ×Pk,(N−k)/2 ({ bλj /(1 − τ ) }j=1,...,k ; { bxj /(1 − τ )1/2 ± i byj /(1 − τ )1/2 }j=1,...,(N−k)/2 ) √ √ ˛ ˛ 2(N−k)/2 ( b)N(N+1)/2 ( 1 + τ )N(N−1)/2 ˛ ˛ = ˛Δ({λl }l=1,...,k ∪ {xj ± iyj }j=1,...,(N−k)/2 )˛ QN N(N+1)/4 k!((N − k)/2)! 2 Γ(l/2) l=1 (N−k)/2 “r 2b ” P(N −k)/2 2 Pk Y 2 (yj −x2 j) ×e−b j=1 λj /2 eb j=1 erfc yj . 1−τ j=1
15.11 CLASSIFICATION OF NON-HERMITIAN RANDOM MATRICES In our studies of Hermitian random matrices in Chapters 1 and 3, a total of ten distinct ensembles were identified on the basis of symmetry constraints. Furthermore, these were shown to be identical to the Hermitian part of the ten infinite families of symmetric spaces. The task of classifying non-Hermitian random matrices according to symmetries has been undertaken in [59], [379]. D EFINITION 15.11.1 Let A be a square matrix and let p, c, q and k be unitary matrices of the same size as A and with p2 = 1,
cT c† = ±1,
q2 = 1,
¯ = ±1. kk
The matrix A is said to have a symmetry of P -type, C-type, Q-type and K-type respectively if A = −pAp† ,
A = ±cAT c† ,
A = −qA† q† ,
¯ †, A = kAk
respectively. For A to have two or more of these symmetries, the symmetries are required to commute. In addition to the ten classes of Hermitian random matrices already catalogued, in [379] consideration
761
THE TWO-DIMENSIONAL ONE-COMPONENT PLASMA
of these symmetries gave rise to twenty types of non-Hermitian random matrices. Three of these are the real, complex and real quaternion Ginibre ensembles, corresponding to k = 1, no symmetry and k = Z2N , respectively. We will single out just three others, obtained by requiring that the Ginibre matrices anticommute with the P -type symmetry 1p Op×q Oq×p −1q (with the sizes doubled in the quaternion case). This gives the non-Hermitian matrices Op×p Ap×q Bq×p Oq×q
(15.234)
with the elements of A := Ap×q and B := Bq×p real, complex and real quaternion. With the matrices A and B chosen according to a Gaussian measure, this class of random matrices have been applied to studies of QCD [439], [9]. Following these references, we’ll take up the problem of computing the eigenvalue p.d.f. in the case the elements of (15.234) are complex. In numerical linear algebra, for square matrices A, B the decomposition A = QTA Z† ,
B = QTB Z†
is called the QZ decomposition. Here Q, Z are unitary matrices, while TA , TB are upper triangular matrices with diagonal elements (TA )jj , (TB )jj , such that (TA )jj /(TB )jj are the eigenvalues of B−1 A. The matrices Q and Z are unique provided the entries of the first row are chosen to be positive. A variation of this is the joint decomposition of p × q, q × p (p ≥ q) matrices A, B given by [439] B = V† TB U† ,
A = UTA V,
(15.235)
where TA , TB are p × q, q × p upper triangular matrices and U, V are unitary. With the diagonal entries of TA , TB denoted {xk }, {yk }, respectively, and the nonzero eigenvalues of the product AB denoted {−zk2} so that the eigenvalues of (15.234) are {±izk }, we have −zk2 = xk yk . The matrix V is unique provided the entries of the first row are chosen to be positive, while U requires this condition, but then is only unique up to multiplication on the right by 1q Oq×(p−q) O(p−q)×q S for S a (p − q) × (p − q) unitary matrix. For A and B having all entries independent standard complex Gaussians, we want to compute the distribution of the independent eigenvalues {izk } of (15.234). The first step is to compute the Jacobian for the change of variables implied by (15.235). ˜ A, T ˜ B denote the strictly upper triangular part of TA , TB . We have P ROPOSITION 15.11.2 Let T (dA)(dB) =
q l=1
|xl |2(p−q)
|zk2 − zj2 |2
1≤j
q
˜ A )(dT ˜ B )(U† dU)(V† dV). (15.236) dxrl dxil dylr dyli (dT
l=1
Proof. From (15.235) we compute (U† AV† )jk = (dTA )jk +
X † X (U dU)jl (TA )lk − (TA )jl (VdV† )lk , l≤k
(VBU)jk = (dTB )jk +
X l≤k
j≤l
†
(VdV )jl (TB )lk −
X
(TB )jl (U† dU)lk
j≤l
(cf. (15.5)). We want to take the wedge product of all the independent differentials in these equations. By choosing the
762
CHAPTER 15
order of doing this so that a triangular structure results, as done below (15.7), we see that the only contributing terms are (TA )kk (U† dU)jk − (TA )jj (VdV† )jk ,
(TB )kk (VdV† )jk − (TB )jj (U† dU)jk
(15.237)
and (dTA )jk ,
(dTB )jk
for j > k and j ≤ k, respectively. Furthermore, for j > q, k ≤ q only the term (TA )kk (U† dU)jk of (15.237) contributes. Recalling that (TA )jj = xj , (TB )jj = yj , xj yj = −zj2 , we see by taking the wedge product of the contributing terms that (15.236) results.
In terms of the decomposition (15.235) the joint p.d.f. for A and B is proportional to e−
PN
j=1 (|xj |
2
+|yj |2 )−
P
˜
j
2
+|(T˜B )jk |2 )
.
˜ B ) in (15.236) therefore only contributes a constant, as does the integration ˜ A ) and (dT The integration of (dT † † over (U dU)(V dV). The variable xj is independent of yj and so to deduce the distribution of the zj we must substitute dy 2 z 2 j yj = −zj2 /xj , dyjr dyji = dzjr dzji = 4 dzjr dzji dz xj and integrate over xj . Our remaining task then is to compute 2 2 2 |x|−2+2(p−q) e−|x| −|z| /|x| dxr dxi .
(15.238)
R2
Changing to polar coordinates gives an integral representation of the modified Bessel function of the second kind Kν (w) results, showing that (15.238) is proportional to |z|2(p−q) Kp−q (2|z|2 ), and giving the sought eigenvalue p.d.f. [439]. P ROPOSITION 15.11.3 Consider matrices (15.234) in the case that the elements of Ap×q and Bp×q are independent standard complex Gaussians. The eigenvalues come in ± pairs, which we write as {±izk }k=1,...,q . The p.d.f. for {zk }, defined on zkr ≥ 0 (k = 1, . . . , q) is proportional to q
|zl |2(p−q+1) Kp−q (2|zl |2 )
|zk2 − zj2 |2 .
1≤j
l=1
We remark that the method of derivation of Proposition 15.3.1 can be used to write down the corresponding n-point correlation functions as an n × n determinant. Consider in particular the case p = q, and change variables wj = −zj2 to the eigenvalues of the matrix product AB. According to Proposition 15.11.3 the latter have p.d.f. proportional to p l=1
Making use of the definite integral ∞ 0
K0 (2|wl |)
|wk − wj |2 .
1≤j
rν+2j−1 Kν (2r) dr =
1 Γ(j)Γ(j + ν) 4
(15.239)
THE TWO-DIMENSIONAL ONE-COMPONENT PLASMA
and recalling (15.199) we see that the limiting bulk correlation kernel is given by 1/2 2 K0 (2|u|)K0 (2|v|) I0 (2(u¯ v )1/2 ). π It follows from (15.240) and the large |z| expansions [541] π 1/2 ez , K (z) ∼ e−z I0 (z) ∼ 0 2z (2πz)1/2
763
(15.240)
(15.241)
that the eigenvalue density ρ(1) (z) exhibits the large |z| form 1/(2π|z|). This could have been anticipated, as it is generally true that for a product of M independent random complex N × N Gaussian matrices (N large), the decay of the bulk eigenvalue density will be proportional to |z|−2+2/M [103]. Furthermore, with |r0 | large, writing u → r0 + (2πr0 )1/2 u and v → (2πr0 )1/2 v in (15.240), forming the product with its complex conjugate, and making use of (15.241) we reclaim the bulk form of the two-point correlation function as given in (15.45). We would expect the scaled n-point correlations, n ≥ 3, to similarly reduce α to (15.43). Related to this is the fact that for the general one body term f (r) = e−cr , α ≥ 2, in (15.42) (asymptotically (15.239) corresponds to the case α = 1) , scaling of the general n-point correlation to (15.43) has been explicitly demonstrated [528].
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Index
L-functions, 307 Δ3 statistic, 695 β-ensemble circular, 371 Gaussian, 43, 147 Jacobi, 129–130, 145, 665 circular, 130–132, 152 Laguerre, 127–129, 150, 664 fixed trace, 105 τ -function, 342, 374 q-Morris identity, 172 q-integral, 169 affine Weyl group, 337 Airy function asymptotic expansion, 299, 394 incomplete, 287 integral representation, 286 alternate variables, 269 integration method, 254 inter-relations, 140 B¨acklund transformations, 338–342 Baik-Deift-Johansson theorem, 498 Barnes G-function, 184, 683, 687 bead process, 460 Bessel stochastic operator, 639 Bessel function asymptotic expansion, 295 differential equation, 348 identities, 290, 294, 348 incomplete, 292 integral representation, 292 pure imaginary argument, 751 three term recurrence, 292 Binet-Cauchy formula, 62, 261 binomial coefficients classical, 582 generalized, 582, 601 binomial theorem classical, 534, 555 generalized, 534, 555, 593, 602 biorthogonal ensemble, 277 functions, 223 polynomials, 375
Blanske product, 634 Bogoliubov transformation, 709 Bogoliubov-deGennes equation, 100 Boltzmann factor, 20–23, 28, 68, 69, 89, 112, 126, 187, 223, 224, 443, 701, 710, 711, 717, 733, 748 self energy, 84 Borchardt identity, 744 Borodin-Okounkov identity, 468–470 boundary conditions Dirichlet, 744 doubly periodic, 215, 298 metal wall, 213 Neumann, 748 semiperiodic, 212, 297 Brownian carousel, 636 motion, 452, 475, 510, 635, 638, 639 Dyson model, 505, 630, 641 harmonic potential, 506 overdamped, 509 Calogero-Sutherland model exchange terms, 521 Green function, 518 root structure, 528 Schr¨odinger operator, 516–518, 564 canonical average, 197, 673 capacitance, 722 Carlson’s theorem, 136, 361, 362, 677 Cartan matrix, 337, 338 Catalan numbers, 36, 40 Cauchy ensemble, 68–71, 126, 314, 321, 363 normalization, 174 identity, 455, 465 dual, 455, 478 weight, 68, 201 Cauchy-Binet formula, 230 Cayley transformation, 69, 71, 73, 117 Cayley-Klein parameters, 729, 732, 739 characteristic polynomial, 46, 48, 129, 146, 149, 152, 154, 195, 197, 685 Chebyshev inequality, 41 polynomial, 208 chemical potential, 185 Cherednik operator, 523, 546, 583, 620
786
type BC, 528 chiral ensemble eigenvalue p.d.f., 86 Christoffel-Darboux integral form, 203 kernel, 381, 384, 387 summation formula, 189, 192, 207, 283, 376 circle law, 703 circular ensemble, 56–294 normalization, 173 parameter-dependent, 512 classical groups, 208–209, 212, 352, 481, 485, 674 correlations, 294 eigenvalue p.d.f., 71 classical weight, 202, 249 COE, 57 eigenvalue p.d.f., 59 gap probability, 366 composition, 543, 550, 563 diagram, 553 dominance ordering, 544 modulus, 544 parts, 543 condition number, 87, 363 conductive phase, 658 convergence, 381 almost sure, 41 corner growth model, 475 correlation coefficients, 111 correlation functions, 186 β = 1, 251 β = 4, 236 β = 2, 186 asymptotic behavior, 298–301, 309, 313, 317, 320, 603, 651, 654 asymptotic expansion, 713, 716 charge-charge, 661, 721 circular β-ensemble, 601 complex random matrices, 713 complex random polynomials, 713 density-density, 640–644 dynamical β = 2, 540 β = 2, 530 Gaussian β-ensemble, 608 Jacobi β-ensemble, 605 Laguerre β-ensemble, 606 one-component plasma disk, 711–713 metallic boundary conditions, 745 pseudosphere, 740 sphere, 730 perturbation, 284–286, 309, 725 real Ginibre matrices, 756 surface, 712, 721–723 two-particle, 182 correlation kernel, 195, 227 Coxeter relations, 337 CSE, 57 eigenvalue p.d.f., 59 CUE, 57
INDEX
eigenvalue p.d.f., 59, 60 cumulant expansion, 673 determinant Cauchy double alternant, 138, 142, 143, 199, 570, 745 confluent Vandermonde, 51 cycle expansion, 189 evaluation, 413 expanded form, 193 Fredholm, 380, 470, 745 Hankel, 674 identity, 195 Jacobi’s theorem, 230, 471 Laplace expansion, 63, 188, 198, 347, 383 quaternion, 237 Slater, 707, 710, 732, 743 Toeplitz, 375, 414, 468, 470, 496, 673 Vandermonde, 50, 60, 187, 205, 229, 255, 268, 411, 443, 445, 460, 467, 569, 587, 708, 711, 723, 730, 749, 751 type B, C, D, 210, 454, 456 determinantal process, 223, 681 L-ensemble, 229 k-component system, 229 dilogarithm, 309 dimers brickwork lattice, 456 Dirac operator, 85 Dirichlet integral, 48 distribution beta, 153, 154 Cauchy, 153 Dirichlet, 138, 140, 144, 152, 154 gamma, 140, 150 Dixon-Anderson density, 138, 150, 152, 589 integral, 138, 633 generalization, 156 DMPK equation, 515 Dotsenko-Fateev integral, 163, 652 Dunkl kernel type A, 616, 617 operator type A, 522, 552, 620 type B, 527 transform, 626 Dynkin diagram, 337, 338 Dyson constant term identity, 177 integral, 173 electric field complex, 24, 107, 690 elementary transposition, 550 entanglement, 103 error function, 712 complementary, 727 Euler angles, 61 integral, 28, 133, 165, 206 product formula, 301 relation, 36, 37, 736
787
INDEX
fatgraph, 37 Fisher-Hartwig asymptotics, 683 formula, 686 Floquet operator, 55 fluctuations charge, 680 Gaussian, 672–677 potential, 681 Fokker-Planck equation, 506, 509 operator Green function, 628 Fuchsian differential equation, 168 fugacity, 213 functional differentiation, 193–194, 254, 280 β = 4, 241–244 Gambier identity, 398 gamma function duplication formula, 36, 184 generalized, 556 incomplete, 712 Stirling’s formula, 32, 382, 712 gap probability asymptotics, 407, 420, 424, 608, 688 bulk, 357 eigenvalue expansion, 403 evenness symmetry, 365, 409 hard edge, 354 inter-relations, 363, 370 orthogonal symmetry, 365, 370 soft edge, 353 spectrum singularity, 357 symplectic symmetry, 370 Gaussian ensemble, 1, see GOE,GUE,GSE bulk correlations, 283–284, 308, 316 global density, 289, 310, 318, 663 normalization, 173 parameter-dependent, 505 shifted mean, 42–534 soft edge, 286, 310, 318 weight, 200, 224, 234 Gaussian curvature, 738 Gaussian distribution multivariable, 716 generating function gap probability, 329, 384, 401, 412, 419 two-species, 385 lattice path, 442, 445, 449 spacing distribution, 329 Toeplitz matrix, 469 Geronimus relations, 130 Gessel-Viennot theorem, 440 GOE, 1–3, 44 eigenvalue p.d.f., 5–10 energy landscapes, 30–32 shifted mean, 43 Gram-Schmidt orthogonalization, 92, 96, 97, 207, 236, 546, 566
GSE, 11 eigenvalue p.d.f., 13 GUE, 11, 41–42, 206, 228 eigenvalue p.d.f., 13 minor process, 234–235, 288, 461 Haar form, see volume form Hadamard inequality, 383 Hamilton equations, 333, 342, 392, 403 Hammersley process, 487, 497 Hardy-Littlewood conjecture, 306 harmonic approximation, 181 Hecke algebra, 529 type A degenerate, 562 isomorphism, 561 Hellman-Feynman formula, 406 Hermite ensemble generalized, 371 perturbed kernel, 287–288 Plancheral-Rotach formula, 289 polynomial, 200, 710 asymptotic expansion, 286 generalized, 613–619 generalized generating function, 615 generalized integration formulas, 617 generalized orthogonality, 614 generalized recurrences, 614 integral representation, 225, 228 nonsymmetric, 609 summation formula, 532 symmetric, 617, 627 weight, 226 Hessenberg form, 76–80 Hirota symbol, 220 Householder transformation, 43 hyperbolic plane, 636 hypergeometric equation confluent, 343, 349 Gauss, 343 hypergeometric function 2 F1 , 496 2 F1 , 207 Gauss summation, 594 generalized, 592, 602, 606 holonomic, 595 integral representation, 597 Kummer relations, 595, 605 two-variable sets, 596 inclusion-exclusion principle, 230 increasing subsequences, 472, 489 inner product skew symmetric, 236, 251, 749 integral equation, 22–24, 28, 29, 89, 107, 194, 670 Fredholm expansion, 196, 244 integral operator, 195, 404, 415, 426, 741 Fredholm, 369, 384, 409 matrix, 242, 256 universal equations, 396 interpolation nonsymmetric Jack polynomial, 559
788
isomonodromic deformation, 334, 430 isomorphism type A, 527 type B, 527 Ito’s formula, 636 Jack polynomials Pκ , Pκ∗ isomorphism, 581 antisymmetric, 567–573 Cauchy product, 569, 593 determinant eigenoperator, 580 dual Cauchy product, 585 Hessenberg determinant, 577 integral eigenoperator, 590 interpolation vanishing property, 563 interpolation nonsymmetric, 558 interpolation symmetric, 579 Macdonald automorphism, 583 negative α, 578 nonsymmetric, 543–549 normalization, 554 specialization formula, 553 Pieri formula, 585, 589 renormalized, 592 Rodrigues formula, 563 shift operators, 571 symmetric, 564–575 normalization, 566, 573 Jacobi circular ensemble, 125–126 spectrum singularity, 293, 313, 321 ensemble, 111, 208, 461, 467, 470 gap probability, 350, 365 global density, 113 hard edge, 293, 312, 319, 612 normalization, 174 parameter-dependent, 514 matrix, 192 polynomial, 200, 565 nonsymmetric, 624 weight, 200, 234 Jacobian, 5, 6, 9, 45, 67, 79, 82, 92, 94, 102, 117, 121, 128, 132, 138, 152, 514, 714 Janossy density, 384 Kac-Rice formula, 31 Kadell’s lemma, 171, 179 Kardar-Parisi-Zhang, 501 Karlin-MacGregor formula, 452 Keating-Snaith hypothesis, 685 kernel Airy, 393–398, 432, 468 Bessel, 399–403 integrable, 426 resolvent, 387 sine, 386–391 KP equation, 438 hierarchy, 220 Kramer’s degeneracy, 749
INDEX
Lagrange interpolation, 177 inversion, 41 Laguerre ensemble, 91, 470, 495, 497 eigenvalue p.d.f., 103 global density, 291, 313, 321, 663 hard edge, 290, 312, 319 log-gas analogy, 91 normalization, 174 soft edge, 291, 313, 321 multiple function, 226 perturbed kernel, 228 Plancheral-Rotach formula, 291 polynomial, 200 asymptotic expansion, 290 generalized, 619 generalized generating function, 621 generalized integration formulas, 622 generalized orthogonality, 619 generalized recurrences, 620 identities, 316 nonsymmetric, 611 summation formula, 532 symmetric, 623, 628 weight, 200, 224, 226, 234 Laguerre-Freud method, 377 Landauer formula, 119 Langevin equation, 509 Laplace equation, 667 Laplacian, 507 generalized, 525, 618 last passage percolation, 472 lattice paths conjugate, 445, 448 continuous, 451–453, 535 nonintersecting, 440–451 single move, 449 triangular lattice, 450 wall, 446 continuous, 456 Lax pair, 430 level curvature, 512 Lie algebras, 14, 88, 103 linear response, 658, 666, 670, 672, 718, 722 linear statistic, 665, 723–724 Liouville’s theorem, 137, 160, 163, 164, 215, 428 lock step model, 440 log-gas, see Boltzmann factor doubly periodic boundary conditions, 75 free energy, 180 multicomponent, 661 one-component, 21, 665 partition function, 139 semi-periodic boundary conditions, 73 two-component, 278 Macdonald conjecture, 176 magnetic field angular momentum, 731 cyclotron orbit, 706, 709
789
INDEX
Landau level, 706, 709, 715, 734, 743 magnetic monopole, 731 plane, 706 sphere, 730–732 symmetric gauge, 707 Majorana parametrization, 735 map graphical, 35, 108 planar, 108 Mar˘cenko-Pastur law, 106–108 Maslov phase, 304 matrix 0,1, 477, 491 antisymmetric, 202, 238, 244, 506 antisymmetric Hermitian, 18, 73, 102 antisymmetric quaternion real Hermitian, 19 antisymmetric unitary, 59 bidiagonal, 127, 130 chiral, 85–89 companion, 705 complex weakly non-Hermitian, 718 complex Ginibre, 701 ensembles inter-relations, 140, 156 GUE, 452 Hermitian, 139 Hessenberg, 81, 130 integral, 33–39, 64, 94, 123, 629 biorthogonal ensembles, 534 character expansion, 533 duality, 630 Harish-Chandra, 532 Itzykson and Zuber, 532 inter-relation, 271 minor, 155, 232 non-negative symmetric, 383 permutation, 492 quaternion real, 12, 749 random external source, 505 real Ginibre, 752 real orthogonal, 8, 56, 58, 71, 81, 371 real symmetric, 139, 506 reflection, 119 scattering, 53–55, 66 self-dual quaternion, 55, 237 superimposed ensembles, 274 symmetric, 55 transmission, 119 tridiagonal, 43, 47, 129, 130, 147 unitary, 17, 54, 58, 143, 371, 685 Hurwitz construction, 61–64 unitary symmetry, 330 unitary symplectic equivalent, 13, 58, 59, 71, 81 Wigner-Smith, 98 Wishart, 90, 99, 105, 127, 468 correlated, 110, 534 parameter-dependent, 513 Maxwell stress tensor, 740
Mehta integral, 173, 177, 178 Mellin transform, 685 metal wall Boltzmann factor, 74 metric form, 9, 60, 86, 121, 507 Minkowski space, 738 monodromy, 434 matrices, 430 preserving, 430 Montgomery conjecture, 301–302 Montgomery-Odlyzko law, 302, 685 Morris constant term identity, 556 integral, 126, 134, 210, 684 moveable singularities, 330 multiple orthogonal polynomials, 225 Narayana number, 109 octonions, 14 one-component plasma correlation functions Neumann boundary conditions, 749 dipole moment, 736 disk, 702 ellipse, 710 ground state, 736 metallic boundary condition, 743 metallic boundary conditions, 744 Neumann boundary conditions, 747 pseudosphere, 738–740 quantum analogy, 706–708, 730 random matrix analogy, 701, 734, 742, 748 sphere, 729 vortex analogy, 746 operator antisymmetrization, 567 identity, 510 lowering-type, 551 raising-type, 550 symmetrization, 565 ordering Bruhat, 544 partial, 544, 563 reverse lexicographical, 577 oriented digital boiling, 479 orthogonal polynomial Heine formula, 190 three term recurrence, 191 orthogonal polynomials, 187–191 monic, 187 Rodrigues formula, 235 skew, 236, 749, 756 skew β = 1, 263 skew β = 4, 246 Painlev´e classical solution, 342 equation, 330 σ-form, 331, 403 master equation, 333 discrete, 372
790
II, 394, 416, 502 V, 399 XXXIV, 433 property, 331 Hamiltonian formulation, 333 partition, 443 frequency, 566 skew diagram, 585 partition function, 20 generalized, 193, 242, 280, 457, 754 grand, 213 generalized, 745 permanent, 142, 714 Pfaffian, 238, 482, 485, 754 Laplace expansion, 244 Pfaffian process, 241 Pieri formula, 487 plane partitions, 458 Poincar´e disk, 739 Poisson equation, 20, 74, 75, 733, 746 pseudosphere, 739 sphere, 729 kernel, 66 process, 489 Poisson kernel, 636 polynuclear growth model discrete, 464 droplet, 488, 500 flat, 493 semicontinuous, 475 symmetrized, 480–487 Pr¨ufer phases, 49 prime number theorem, 305 principal component analysis, 90 prolate spheroidal functions, 405 pseudosphere, 738 quantum transport, 670 queue, 474 quotient space, 58, 63 random matrix, see matrix permutation, 488 symmetrized, 492–494 polynomial, 153 complex, 703 quantum analogy, 708, 735 projection, 139, 155 rational function, 138, 153, 154 words, 491 random turns model, 449 real quaternion, 12 reflection coefficients, 375 replica trick, 30 Ricatti equation, 331, 342 Riemann hypothesis, 302 generalized, 308 zeros, 301, 686 zeta function, 301, 685
INDEX
critical strip, 303 trivial zeros, 303 Riemann-Hilbert problem, 25, 378, 426 root system, 175 RSK correspondence, 466, 480, 483, 484, 497 continuous, 466, 470 dual, 477 stochastic recurrences, 471 Sch¨utzenberger dual, 480 evacuation, 487 Schlesinger equations, 428 Schmidt decomposition, 104 Schr¨odinger operator integrable, 519, 524, 527 Schur decomposition, 701, 742, 752 Schur polynomial, 444, 464, 478, 485, 569, 580, 585, 629 evaluation formula, 445 specialization formula, 445 screening cloud, 659, 722 Selberg correlation integrals, 592–601 density, 146 integral, 125, 133, 351 Anderson’s derivation, 137, 178 Aomoto’s derivation, 165, 219 complex, 165 Dotsenko and Fateev’s derivation, 160 generalized, 558, 573, 592 Selberg’s derivation, 133 self energy, 89 Shannon entropy, 104 shooting eigenvectors, 49 shot noise power, 666 sine kernel, 458 singular value decomposition, 86, 89, 116, 120, 124 soft edge, 498 spacing distribution asymptotics, 688 covariance and variance, 693 gap probability, 328 hard edge, 607 inter-relations, 633 numerical computation, 356–357, 360, 370, 394 two-dimensional, 725–727 asymptotics, 728 edge, 727 specific heat, 180 spectrum singularity, 126 stereographic projection, 68, 152, 732 Stieltjes minimum, 30 Jacobi polynomial, 115, 127 Laguerre polynomial, 110 Stieltjes-Wigert polynomials, 213 stochastic Loewner evolution, 516 stochastic sine equation, 409, 636 structure function, 284, 309, 317, 654, 659, 712, 718 functional equation, 655 Sturm sequences, 48 sum rule
791
INDEX
charge, 659 contact theorem, 719 dipole moment, 718 fourth moment, 719 sixth moment, 719 Stillinger-Lovett, 718 sum rules, 332, 372 summation formula classical β = 1, 265 classicalβ = 4, 250 minor, 261 surface tension, 716 Sylvester’s identity, 345 symmetric function complete, 445 elementary, 168, 442, 469, 582 monomial, 564 power sum, 576 symmetric space, 58, 629 Szeg¨o asymptotic formula, 470, 673, 683 tableau oscillating, 448 semi-standard, 443, 463, 465, 585 standard, 449, 490 theta function, 215 three-term recurrence, 145, 149, 151, 710 tiling Aztec diamond, 462 hexagon, 458 time reversal, 2–3, 54, 55, 86, 100, 104, 119
Toda lattice equation, 218, 344 totally asymmetric simple exclusion process, 477 trace formula, 304 transfer matrix, 50, 124 triangular action, 544 structure, 547, 565 unfolding, 3 universality, 6 value distribution, 684 vicious walkers, 440, 449 Virasoro algebra, 219 constraints, 220, 435 volume form, 7, 57, 64, 69, 92 weak coupling, 183 wedge product, 6–7, 128, 753 Weierstrass P-function, 331 Weingarten function, 65 Wick’s theorem, 33 Wiener-Hopf operator, 469 Wigner semicircle law, 24, 36, 41, 289, 663 Wigner surmise, 332, 367, 372 wireless communication, 91 WKB asymptotics, 406 zonal polynomials, 629
