Communications in Mathematical Physics - Volume 232

Commun. Math. Phys. 232, 1–18 (2002) Digital Object Identifier (DOI) 10.1007/s00220-002-0737-9

Communications in

Mathematical Physics

Integrable Evolution Equations on the N -Dimensional Sphere A.G. Meshkov1 , V.V. Sokolov2 1 2

Orel State University, Komsomolskaya str., 95, 302026 Orel, Russia Landau Institute for Theoretical Physics, Kosygina str. 2, 117334 Moscow, Russia

Received: 16 July 2001 / Accepted: 25 March 2002 Published online: 14 November 2002 – © Springer-Verlag 2002

Abstract: The problem of a classification of integrable evolution equations on the N-dimensional sphere is considered. We modify the main notions of the symmetry approach such as the formal symmetry and the canonical series of conserved densities to the case of such equations. Using these theoretical results, we solve several special classification problems. The main result is a complete classification of integrable isotropic evolution equations of third order on the sphere. An important class of anisotropic equations is also considered. 1. Introduction In the paper [1] the following equation
 3 3 Ut = Uxx + < Ux , Ux > U + < U, R(U ) > Ux , x 2 2

< U, U >= 1

(1)

was considered. Here U = (U 1 , . . . , U N+1 ) is an unknown vector, R is a constant symmetric matrix. Here and in the sequel < ·, · > stands for the standard scalar product in Euclidean space V . Without loss of generality it can be assumed that R = diag(r1 , . . . , rN+1 ). It was shown that this equation is integrable by the inverse scattering method for any N and R. If N = 2, then (1) is a higher symmetry of the Landau-Lifshitz equation. Besides, Eq. (1) defines an infinitesimal symmetry for the well-known Noemann system [2]
 Uxx = − < Ux , Ux > + < U, R(U ) > U + R(U ), < U, U >= 1, (2) describing the dynamics of a particle on the sphere SN under the influence of field with the quadratic potential U = 21 < U, R(U ) >. More precisely, if we eliminate the derivatives Uxx and Uxxx from (1) using Eq. (2), then the reduced system

2

A.G. Meshkov, V.V. Sokolov

Ut =

 1
< Ux , Ux > + < U, R(U ) > Ux − < Ux , R(U ) > U + R(Ux ) 2

is a symmetry for (2). In this paper we are dealing with a problem of classification of integrable vector evolution equations similar to Eq. (1). Our main tool is the symmetry approach [20, 21, 23, 24] based on the observation that all integrable evolution equations with one spatial variable possess local higher symmetries (or, the same, higher commuting flows). We are developing a specific componentless version of this approach suitable for vector equations. In Sect. 3 the main concepts of the symmetry approach are generalized to the case of equations of the form Ut = fn Un + fn−1 Un−1 + · · · + f1 U1 + f0 U,

Ui =

∂iU , ∂x i

(3)

where U (x, t) is an N -component vector, and fi are scalar functions of variables u[i,j ] =< Ui , Uj >,

i
j,

(4)

0
i, j
n. It is clear that any Eq. (3) is invariant with respect to an arbitrary constant orthogonal transformation of the vector U . Equations of the form (3) are called isotropic. The vector modified Korteweg-de Vries equation Ut = Uxxx + < U, U > Ux ,

(5)

gives us an example of an integrable isotropic equation. It is well known that this equation is integrable by the inverse scattering method for any N. Different examples of integrable vector equations can be found in the papers [4, 5]. Some of them are closely related to such algebraic and geometrical objects as Jordan triple systems and symmetric spaces [3, 8, 9, 11]. In this paper we shall consider Eq. (3) that are integrable for arbitrary dimension N of the vector U . In addition, we assume that the coefficients fi do not depend on N . By virtue of the arbitrariness of N, variables (4) will be regarded as independent. The functional independence of u[i,j ] , i
j is a crucial requirement in all our considerations. If N is fixed, we cannot suppose that. For instance, if N = 3, then the determinant of matrix A with entries aij = u[i,j ] , i, j = 1, 2, 3, 4 identically equals to zero. The signature of the scalar product is inessential for us. Furthermore, the assumptions that the space V is finite-dimensional and the field of constants is R are also unimportant. For instance, U could be a function of t, x, and y and the scalar product be  ∞ < U, V >= U (t, x, y) V (t, x, y) dy. −∞

Thus our formulas and statements are valid also for this particular sort of 1 + 2-dimensional non-local equations. A more restrictive class than Eq. (3) on RN consists of equations Ut = fn Un + fn−1 Un−1 + · · · + f1 U1 + f0 U,

< U, U >= 1,

(6)

U = (U 1 , . . . , U N+1 ), defined on the sphere SN . If R = 0, then (1) belongs to this class.

Integrable Evolution Equations on the N-Dimensional Sphere

3

It is easy to see that the stereographic projection takes any Eq. (6) on SN to some isotropic equation on RN . The converse statement is not true because, in general, the preimage of Eq. (3) on RN under the stereographic projection is non-isotropic on SN . In Sect. 2 we present a complete list of integrable isotropic equations Ut = Uxxx + f2 Uxx + f1 Ux + f0 U,

(7)

on the sphere SN. A sketch of a proof of the corresponding classification theorem is contained in Sect. 4. In order to prove that all equations from the list are really integrable, we find an auto-B¨acklund transformation, involving a “spectral” parameter, for each of the equations (see Sect. 5). Equation (1) with non-trivial R is not isotropic. Nevertheless, equations of such type can be also treated in the framework of our componentless approach. To do this we assume that the coefficients fi of Eq. (3), besides (4), depend on additional variables v[i,j ] =< Ui , R(Uj ) >,

i
j,

(8)

where 0
i, j
n, and R is a constant symmetric matrix. We call such equations anisotropic. In the paper we consider anisotropic equations that are integrable for arbitrary symmetric matrix R. For this reason we regard the union of all scalar products (4) and (8) as a set of independent variables. Section 7 contains new nontrivial examples of integrable anisotropic evolution equations of third order on the N -dimensional sphere. 2. Classification Results for the Isotropic Case In this section we formulate some classification statements concerning integrable evolution equations of third order on the N -dimensional sphere. This classification problem is much simpler than the similar problem on RN. Indeed, the set of independent variables (4) on SN is reduced because of the constraint u[0,0] = 1. Differentiating this identity, we can express all variables of the form u[0,k] , k  1 in terms of the remaining independent scalar products u[i,j ] =< Ui , Uj >,

1
i
j.

(9)

For example, u[0,1] = 0, u[0,2] = −u[1,1] , and so on. Therefore the coefficients of Eq. (7) on SN a priori depend on only three independent variables u[1,1] , u[1,2] , and u[2,2] whereas in the case of RN they are functions of six variables u[0,0] , u[0,1] , u[1,1] , u[0,2] , u[1,2] , and u[2,2] . Theorem 1. Suppose that equation Ut = Uxxx + f2 Uxx + f1 Ux + f0 U,

fi = fi (u[1,1] , u[1,2] , u[2,2] )

(10)

on the sphere < U, U >= 1 possesses an infinite series of commuting flows of the form Uτk = gk Uk + gk−1 Uk−1 + · · · + g1 Ux + g0 U, whose coefficients gi depend on variables (9); following list:

k → ∞,

then this equation belongs to the

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A.G. Meshkov, V.V. Sokolov

Ut = Uxxx

Ut = Uxxx

u[1,2] 3 −3 Uxx + u[1,1] 2

3 + 2





u2[1,2] u[2,2] + 2 +c u[1,1] u[1,1] (1 + a u[1,1] )

 Ux ,

(11)



a 2 u2[1,2] 1 + a u[1,1]

− a (u[2,2] − u2[1,1] ) + u[1,1]

+c

Ux + 3 u[1,2] U, (12)

Ut = Uxxx − 3 Ut = Uxxx − 3 

u[1,2] Uxx + u[1,1]



3 u[2,2] +c 2 u[1,1]

 Ux ,

(q + 1) u[1,2] (q − 1) u[1,2] U Uxx + 3 2 q u[1,1] 2q

 (q + 1) u[2,2] (q + 1) a u[1,2] 2 − + u[1,1] (1 − q) + c Ux , u[1,1] q 2 u[1,1]  where a and c are arbitrary constants, q = ε 1 + a u[1,1] , ε2 = 1. 3 + 2

(13)

(14)

Theorem 2. If Eq. (10) on SN possesses an infinite series of conservation laws (ρk )t = (σk )x , k → ∞, where ρk and σk are functions of variables (9), then this equation belongs to the same list (11)–(14). Remark 1. The constant c can be removed by the Galilean transformation and below we will omit this constant as trivial. The constant a can be reduced to a = 0 or to a = ±1 by an appropriate scaling of x and t. Thus the list contains rather many non-equivalent equations over R. In particular, Eq. (1) with R = 0 coincides with (12), where a = 0. Equation (14) with a = 0 and ε = −1 reads as Ut = Uxxx + (3 u[1,1] + c) Ux + 3 u[1,2] U.

(15)

If a = 0 and ε = 1 then Eq. (14) becomes Ut = Uxxx

  u[1,2] u[2,2] −3 Uxx + 3 + c Ux . u[1,1] u[1,1]

(16)

Remark 2. In the process of the proof of Theorems 1 and 2 (see Sect. 4) it has been checked that all Eqs. (11)–(14) have non-trivial local conservation laws of orders 1,2,3 and 4. Moreover, we have verified that each of these equations possesses a higher symmetry of fifth order. For example, the fifth order symmetry of Eq. (15) has the following form:
 Uτ = U5 + 5 u[1,1] U3 + 15 u[1,2] U2 + 5 3 u[1,1] 2 + 2 u[2,2] + 3 u[1,3] U1

+ 5 6 u[1,2] u[1,1] + 2 u[2,3] + u[1,4] U. We are sure that all our equations have infinite series of symmetries and conserved densities, but of course it should be rigorously proved. From our viewpoint the existence of higher symmetries and / or conservation laws is a very efficient way to list all integrable cases. But there is a little help from symmetries and conservation laws for integrating of a given equation. That is why we find in Sect. 5 auto-B¨acklund transformations for all equations of the list.

Integrable Evolution Equations on the N-Dimensional Sphere

5

Remark 3. The coincidence of the lists from Theorems 1 and 2 shows that the so-called Burgers type equations of the form (10) do not exist on SN . Recall that the Burgers type equations (C-integrable equations in the terminology by F. Calogero [22]) possess higher symmetries but have no higher conservation laws. Remark 4. Equation (13) on RN has been found the papers [4, 6, 11]. This equation is related to vector triple Jordan systems. It is a vector generalization of the well-known Swartz-KdV equation 2 3 vxx vt = vxxx − . 2 vx Remark 5. In the case of one-dimensional sphere Eqs. (11), (12) with a = 0 can be reduced to the potential mKdV equation vt = vxxx + vx3 by the stereographic projection and some point-wise transformations. Equations (11) and (12) with a = −1 are reduced to 1 3 (Q − 4vx2 )2x vt = vxxx − Q vx + , 8 32 vx (Q − 4vx2 )

(17)

where Q(v) = (v 2 + 1)2 . Equation (17) with Q(v) being an arbitrary polynomial of fourth degree is known as the Calogero-Degasperis equation (see [13]). Our particular case corresponds to a trigonometric degeneration of the elliptic curve implicitly involved in (17). Equation (14) is reduced to the following integrable equation: vt = vxxx −

2 6 a vx vxx + 8vx3 . 1 + 4 avx 2

Remark 6. It would be interesting to find a geometrical interpretation of Eqs. (11)–(14) along the lines of [10]. Here we only note that Eqs. (11) and (13) admit the following constraint: < Ux , Ux >= 1. This means that the t-deformation of an initial curve U (x) on the sphere by virtue of these equations preserves the length. 3. Canonical Densities In the papers [12, 20] the concept of formal symmetry for one-component evolution equations of the form ∂iu (18) ∂x i has been introduced. By definition, the formal symmetry (or the formal recursion operator) is a series of the form ut = F (u, u1 , u2 , . . . , un ),

ui =

L = a1 Dx + a0 + a−1 Dx−1 + a−2 Dx−2 + · · · ,

ai = ai (u, u1 , . . . , uni ), (19)

satisfying the following operator relation: Lt = [F∗ , L],

F∗ =

n  ∂iF 0

∂ui

Here Dx is the total derivative operator with respect to x:

Dxi .

(20)

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A.G. Meshkov, V.V. Sokolov

Dx =

∞ 

ui+1

0

∂i , ∂ui

F∗ is the Frech´et derivative of the right-hand side of Eq. (18). It was shown in [12, 19] that the formal symmetry exists for any Eq. (18) possessing an infinite series of local higher symmetries or conservation laws. The residues ρi = res Li are local conserved densities for Eq. (18). They are called the canonical densities. In [19] it was proved that if Eq. (18) has an infinite series of conservation laws, then the canonical densities ρi are trivial for all even i. There is an alternative way [14, 15] to compute the canonical densities. It is based on identities for the logarithmic derivative of a formal eigenfunction for the operator ∂ − F∗ . This algorithm deals with commutative Laurent series in contrast with non∂t commutative series similar to (19). In this section we define an infinite sequence of necessary integrability conditions for Eq. (3). These conditions Dt ρi = Dx θi ,

i = 0, 1, 2, . . .

(21)

have the form of conservation laws, where ρi , θi are some functions of variables (4), which can be recursively found in terms of the coefficients fi of Eq. (3). These conditions are very close to the canonical conservation laws from the papers [12, 20, 21, 7, 14, 15, 23] but do not coincide with them. Our conditions are more convenient for classification problems related to Eq. (3) since they are much simpler than the standard canonical densities for multi-component systems. Theorem 3. (i) If Eq. (3) possesses an infinite series of commuting flows of the form Uτ = gm Um + gm−1 Um−1 + · · · + g1 U1 + g0 U,

(22)

then there exists a formal series L = a1 Dx + a0 + a−1 Dx−1 + a−2 Dx−2 + · · · ,

(23)

satisfying the operator relation Lt = [A, L],

A=

n 

fi Dxi .

(24)

0

Here gi , ai are some functions of variables (4), fi are the coefficients of Eq. (3). (ii) The following functions ρ−1 =

1 , a1

ρ0 =

a0 , a1

ρi = res Li ,

i∈N

(25)

are conserved densities for Eq. (3). (iii) If Eq. (3) possesses an infinite series of conserved densities depending on variables (4), then there exists a series L satisfying (24), and a series S of the form S = s1 Dx + s0 + s−1 Dx−1 + s−2 Dx−2 + · · · such that

(26)

Integrable Evolution Equations on the N-Dimensional Sphere

St + AT S + S A = 0,

7

S T = −S,

(27)

where the superscript T stands for formal conjugation. (iiii) Under the conditions of item (iii) all densities (25) corresponding to even i are trivial i.e., ρ2k = Dx (σk ) for some functions σk of variables (4). Comments. In [21, 7] the notion of the formal symmetry was generalized to the case of systems of evolution equations. In these papers the formal symmetry is a series with matrix coefficients satisfying (20). In our paper both the operators A and L are scalar objects and A does not coincide with the Frech´et derivative F∗ of the right-hand side of the system. In [7–9] Sergey Svinolupov has described integrable cases for several classes of N-component polynomial systems using the existence of the formal symmetry as a necessary condition of integrability. In these papers he imposed very serious restrictions on the structure of the right-hand side and only a collection of unknown constants was to be determined. Any attempts to solve more general classification problems for N-component systems with the help of the standard component-wise approach lead to computational difficulties which cannot be overcome. The use of the scalar series L defined by formula (24) instead the formal symmetry makes possible a complete classification of isotropic integrable systems of the form Ut = U3 + f2 U2 + f1 U1 + f0 U on RN without any assumptions about the structure of the coefficients fi . We are planning to publish a separate paper devoted to this problem. Reduced proof of Theorem 3. In many respects the proof is analogous to one used in [20] for the scalar case. (i) Let us rewrite Eq. (3) and its higher symmetry (22) in the form Ut = A(U ),

Uτ = B(U ),

B=

m 

gi Dxi .

(28)

0

The compatibility of Eqs. (28) implies the following operator identity: Bt − [A, B] = Aτ . For m large enough we can ignore the right-hand side of this relation. In other words, the operator B approximately satisfies (24). But then the first order series Lm = B 1/m also approximately satisfies (24). A rigorous assembling of approximate solutions Lm into one exact solution L can be done in the same way as in [20]. (ii) The statement follows from the known Adler’s formula (see [20]). (iii) Let us represent the variational derivative
∂ρ 
∂ρ   δρ (29) = (−Dx )i Uj + (−Dx )j Ui δU ∂u[i,j ] ∂u[i,j ] i
j

of arbitrary conserved density ρ of order m in the form δρ = Sm (U ), δU where Sm is a scalar differential operator of order 2m, whose coefficients depend on variables (4). As it is well known [28], this variational derivative satisfies the following equation:

8

A.G. Meshkov, V.V. Sokolov



 T  δρ δρ = − A(U ) , δU t δU ∗ where the subscript ∗ means the Frech´et derivative and the superscript T stands for formal conjugation. It follows from this equation that the operator Sm approximately satisfies (27). To conclude the proof of Theorem 3 it suffices to repeat the corresponding reasoning from [20]. Remark 7. In the same way, this theorem can be proved for the isotropic equations on the sphere and for the anisotropic equations. 4. Systems of Third Order. Isotropic Case Finding from (24) coefficients a1 , a0 , a−1 of the series L, it is easy to verify that for Eq. (10) the densities ρ0 and ρ1 are expressed in terms of the coefficients fi by the following formulae: 1 ρ0 = − f2 , (30) 3 1 2 1 1 (31) f − f1 + Dx f2 . 9 2 3 3 The corresponding functions θi can be found from (21). The fact that the left-hand sides of (21) are total x-derivatives imposes rigid restrictions (see below) to the coefficients fi of (7). Expressions for ρi , i > 1 involve the coefficients fk and the functions θj with j
i − 2. Using a technique developed in the papers [14, 15], one can obtain the following recursion formula:

1 ρn+2 = θn − f0 δn,0 − 2 f2 ρn+1 − f2 Dx ρn − f1 ρn 3 ρ1 =

−

 n 1 f2 ρs ρn−s + 3 s=0

−Dx ρn+1 +

n



ρs ρk ρn−s−k + 3

0
s+k
n

n+1 

ρs ρn−s+1

s=0

1 1 ρs ρn−s + Dx ρn , 2 3

n  0,

(32)

s=0

where δi,j is the Kronecker delta and ρ0 , ρ1 are defined by (30), (31). According to this formula,   1 2 2 1 1 2 3 1 1 ρ2 = − f0 + θ0 − f + f1 f2 − Dx f + Dx f2 − f1 3 3 81 2 9 9 2 9 3 and so on. In order to show how to manipulate with conditions (21) we consider Eqs. (10) on SN and present a proof of Theorem 2. To perform the corresponding computations some special Maple routines were written. The equation of the sphere < U, U >= 1 gives rise to the relation < U, Ut >= 0. It follows from this that any Eq. (10) on SN has the following form:

Integrable Evolution Equations on the N-Dimensional Sphere

9

Ut = U3 + f2 U2 + f1 U1 + (f2 u[1,1] + 3 u[1,2] ) U0 . Thus for Eqs. (10) on SN , we have to replace f0 by f2 u[1,1] + 3 u[1,2] in conditions (32). Lemma 1. Suppose Eq. (10) on SN possesses an infinite series of conserved densities depending on variables (4); then the equation has the form Ut = U3 + A u[1,2] U2 + (B u[2,2] + C u2[1,2] + D u[1,2] + E) U1 + (u[1,1] A + 3) u[1,2] U0 ,

(33)

where A, B, C, D, E are some functions of variable u[1,1] . Proof. It follows from (30) and the item (iiii) of Theorem 3 that f2 = Dx (σ0 ) for some function σ0 . Since f2 does not depend on the third order derivative, we see that σ0 may depend on the variable u[1,1] only. Thus f2 = 2σ0 u[1,2] and the function A from (33) coincides with 2σ0 . To specify the form of the coefficient f1 , let us consider the condition Dt (ρ1 ) = Dx (θ1 ), where ρ1 is given by (31). Using the main property of the Euler operator, we δ can eliminate the unknown function θ1 . It is well known [28] that δU Dx (θ ) = 0 for any function θ. Therefore we have  δ
(34) (ρ1 )t = 0. δU It is easily verified that 
 δ
∂f1 2 ∂f1 − Au[1,2] U6 + · · · , (35) (ρ1 )t = 2 Dx δU ∂u[2,2] 3 ∂u[2,2] where the dots denote the terms which do not contain U6 . Equating to zero the coefficient of U6 in (35) we get 2

∂ 2 f1 ∂ 2 f1 u + (u[1,3] + u[2,2] ) [2,3] ∂u[2,2] ∂u[1,2] ∂u2[2,2]

+2

2 ∂f1 ∂ 2 f1 u[1,2] − Au[1,2] = 0. ∂u[2,2] ∂u[1,1] 3 ∂u[2,2]

(36)

This obviously implies ∂ 2 f1 ∂ 2 f1 = = 0. ∂u[2,2] ∂u[1,2] ∂u2[2,2] Taking these relations into account, let us equate the coefficients at u[3,3] u[1,4] U1 and u[2,4] u[1,4] U1 in (35) to zero. It is not hard to check that as the result we obtain  ∂ 3 f1
∂f1 u − 15 =0 [1,1] ∂u[2,2] ∂u3[1,2] and

 ∂ 3 f1
∂f1 2u − 39 = 0. [1,1] ∂u[2,2] ∂u3[1,2]

It follows from these two identities that Lemma 1.

∂ 3 f1 = 0. This concludes the proof of ∂u3[1,2]

10

A.G. Meshkov, V.V. Sokolov

Remark 8. The statement of Lemma 1 is a generalization of the fact (see [23]) that for integrable scalar equations of the form ut = u3 + F (u, u1 , u2 ) the function F is a second degree polynomial in u2 . It follows from (36) that 3B  = A B.

(37)

One more important relation can be derived from δ (ρ2 ) = 0. δU Vanishing of the coefficient of the highest derivative U4 in this condition implies A2 + 2B A u[1,1] + 2A B − 3A = 0.

(38)

At last, the conditions obtained from the coefficients at u[1,6] u[1,2] U1 and u[1,5] U3 in (34) implies that B  B 2 − 2B 3 u[1,1] − 4B 2 B = 0.

(39)

It is easy to find from (37)–(39) that Case 1) or Case 2) or Case 3)

B = µ,

λ , λ = 0, u[1,1] B 2 u[1,1] − 3B = ν, ν = 0,

B=

where µ, λ, ν are constants. 3 Consider, for example, case 2. Relation (37) leads to A = − u[1,1] . Equating to zero the coefficient at u[1,5] U3 in (34), we obtain (2λ − 3)(C u2[1,1] + λ − 3) = 0.

(40)

The coefficient at u1,6] u[1,2] U1 gives us 3C  u[1,1] − 2u2[1,1] C 2 + 2(6 − λ)C = 0.

(41)

Moreover, one can derive from (34) that D = 0 and E is a constant. Thus if λ = 23 , we have the following equation:   u2[1,2] u[1,2] u[2,2] Ut = U3 − 3 U2 + λ + (3 − λ) 2 + c U1 . (42) u[1,1] u[1,1] u[1,1] For any constant λ, this equation satisfies conditions (21) with i = 0, 1, 2, 3, 4. It can be easily checked that the conditions with i = 1 and i = 3 yield non-trivial conserved densities u[1,1] u[2,2] − u2[1,2] ρ1 = u2[1,1]

Integrable Evolution Equations on the N-Dimensional Sphere

11

and ρ3 =

2 2 u2[2,2] u[3,3] (u[1,3] − u[2,2] )2 (9 − λ) (u[1,1] u[2,2] − u[1,2] ) − − + u[1,1] 2 u2[1,1] u2[1,1] u4[1,1]

of second and third orders. It follows from (21) with i = 5 that λ = 3 and we obtain Eq. (16). Note that Eq. (42) with arbitrary λ provides us an example of a non-integrable equation having a local higher conservation law of third order. In the case λ = 23 , the general solution C=

3 2u2[1,1] (1 + a u[1,1] )

of the Bernoulli equation (41) immediately gives rise to Eq. (11). The particular solution C = 0 corresponds to Eq. (13). Case 1 can be subdivided into two subcases: µ = 0 and µ = 0. For both subcases explicit expressions for the coefficients of (33) can be easily obtained. But to specify the values of some constants in these formulae it is needed to use conditions (21) up to i = 6. As the result, we obtain Eqs. (15) and (12). In Case 3 we have B=

3 q +1 , 2 u[1,1]

A=−

 q = ε 1 + a u[1,1] ,

3 q +1 , 2 q u[1,1]

ε2 = 1,

where a = 4ν/9. The remaining coefficients are easily derived from condition (21) with i = 3. As the result, we obtain Eq. (14). 5. B¨acklund Transformations for Equations of the List We have to prove somehow the integrability of all equations from the list (11)–(14) obtained with the help of the necessary integrability conditions (21). The usual way for that is to find Lax representations or Miura transformations between “new” equations from the list and equations known to be integrable. But we choose another possibility. In this section we present first order auto-B¨acklund transformations for all equations from the list. Such a transformation involving an arbitrary parameter allows us to build up both multi-solitonic and finite-gap solutions even if the Lax representation is not known (see [18]). That’s why the existence of an auto-B¨acklund transformation with additional “spectral” parameter λ is a convincing evidence of integrability. For the scalar evolution equations, the auto-B¨acklund transformation of first order is a relation between two solutions u and v of the same equation and their derivatives ux and vx . Writing this constraint as ux = φ(u, v, vx ), we can express all derivatives of u in terms of u, v, vx , . . . , vi , . . . . The last variables are regarded as independent. In the vector case, the independent variables are vectors U, V , V1 , V2 , . . . Vi . . . ,

(43)

and all their scalar products def

v[i,j ] = < Vi , Vj >,

def

wi = < U, Vi >,

i, j  0.

(44)

12

A.G. Meshkov, V.V. Sokolov

In this paper we consider special vector auto-B¨acklund transformations of the form U1 = h V1 + f U + g V ,

(45)

where f, g and h are functions of variables (44) depending on first derivatives at most. Since V lies on the sphere < V , V >= 1, we assume without loss of generality that the arguments of f, g and h are w0 =< U, V >, w1 =< U, V1 >, v[1,1] =< V1 , V1 >. Since < U, U >= 1, and < U, U1 >= 0, it follows from (45) that h=−

f + w0 g . w1

(46)

To find an auto-B¨acklund transformation for Eq. (10), we differentiate (45) with respect to t by virtue of (10) and express all vector and scalar variables in terms of independent variables (43) and (44). By definition of B¨acklund transformation, the expression thus obtained must be identically zero. Splitting this expression with respect to the independent variables different from w0 , w1 , v[1,1] , we derive an overdetermined system of non-linear PDEs for the functions f and g. If the system has a solution depending on an essential parameter λ, this solution gives us the auto-B¨acklund transformation we are looking for. We present below the result of our computations. In the case of Eq. (12) (where we put c = 0) the auto-B¨acklund transformation reads as follows:     1 − λ a w0 w1 (1 + λ a) U1 = −V1 + w1 − w0 G U + + G V, (47) 1 + w0 1 + w0 where

 G=

λ (2 + λ a − λ a w0 )(1 + a v[1,1] ) . (1 + w0 )

In particular, if a = 0, then we have U1 = −V1 +



λ w1 U +V + √ V − w0 U . 1 + w0 1 + w0

(48)

The vector Schwartz-KdV Eq. (13) admits the auto-B¨acklund transformation of the form
 U1 = G w1 (U + V ) − (1 + w0 ) V1 , (49) where  2  1 λ + 1 + w0 G= . 1+ λ 1 + w0 The auto-B¨acklund transformation for Eq. (11) is defined by (49), where  1 2 2 (λ + 1 + w0 )(1 + av[1,1] ) G= + + . 1 + w0 λ λ av[1,1] (1 + w0 )

(50)

(51)

Integrable Evolution Equations on the N-Dimensional Sphere

13

Note that Eq. (13) is a limit of (11) as a → ∞. The corresponding limit value of (51) gives us (50). Formula (51) is not valid if a = 0. To derive from (51) an auto-B¨acklund transformation for Eq. (11) with a = 0, one must put λ = λ /a at first. After that the limit of (51) as a → 0 gives rise to  √ v[1,1] + λ 1 + w0  . G= (1 + w0 ) v[1,1] Finally, the auto-B¨acklund transformation for Eq. (14) is defined by the following expression: U1 = F (V1 − w1 U ) + G (V − w0 U ), where

(52)



 1 + λ2 a(1 − w02 ), p = ε 1 + a v[1,1] ,   λa w12 R w0 w1 F 1 2 G= 2λ + λ(p − 1)R + + w a + , 1 1 − w02 1 − w02 (p − 1)(1 − w02 ) F =

λa w1 + R, p−1

R=

ε and a are the constants from (14) and λ is the B¨acklund parameter. Auto-B¨acklund transformations for Eq. (15) and (16) can be obtained from these formulas by setting a = 0 and ε = −1 or ε = 1 correspondingly. For Eq. (15) the B¨acklund transformation takes the following form:



w1 V − U − 2λ V − w0 U , U1 = V1 + 1 − w0 and for Eq. (16) the B¨acklund transformation is given by   w1 (v[1,1] + 2λ w1 ) w1 V1 . (V − U ) + 1 + 2λ U1 = v[1,1] (1 − w0 ) v[1,1] 6. Soliton Solutions Using the auto-B¨acklund transformations from the previous section, one can find particular solutions of Eqs. (11)–(14). Here we construct soliton type solutions for (12) with c = 0. Let us take for V a constant solution C 1 of (12), where < C 1 , C 1 >= 1. Taking into account that v[i,j ] = 0, wj = 0 for j > 0, we obtain from (47) that Ux = g(C 1 − w0 U ), 

where g=

λ

(53)

2 + λ a − λ a w0 . 1 + w0

It follows from (53) that w0,x = g(1 − w02 ).

(54)

14

A.G. Meshkov, V.V. Sokolov

Using (53), (54), it can easily be checked that Eq. (12) reduces to Ut = λg(C 1 − w0 U ). It means that the solution U depends on x + λt. It is clear that w02
1. Equation (53) has a trivial solution w02 = 1, U = w0 C 1 . The expression U − C 1 w0  1 − w02 is a first integral of (53). Hence, non-trivial solutions of (53) locally can be represented as  U = C 1 w0 + C 2 1 − w02 , where the constant vector C 2 satisfies the conditions < C 2 , C 2 >= 1,

< C 1 , C 2 >= 0.

The simplest way to solve Eq. (54) is to note that this equation is equivalent to the following equation for the function g: gx = λ − g 2 . Analytical properties of the solution U (x + λt) essentially depend on the sign of λ. If λ = −k 2 < 0, then the solution is periodic: √   2(ak 2 − 1) ak 2 − 1 1 2 − 1 + 2C , (55) U =C ak 2 + tan2 ψ cos ψ (ak 2 + tan2 ψ) where a > k −2 , g = −k tan ψ, ψ = k(x − k 2 t). If λ = k 2 > 0, there exists a particular case g 2 = k 2 , where λa = −1 , w0 = tanh ϕ, ϕ = k(x + k 2 t) and U = C 1 tanh ϕ + C 2 cosh−1 ϕ. In the generic case we have two kinds of solutions: (1) if g 2 < k 2 , then a < −k −2 , g = k tanh ϕ, ϕ = k(x + k 2 t) and    |a|k 2 − 1 2(ak 2 + 1) 1 2 − 1 + 2C . U =C 2 2 ak + tanh ϕ cosh ϕ (ak 2 + tanh2 ϕ) (2) if g 2 > k 2 , then a > −k −2 , g = k coth ϕ, ϕ = k(x + k 2 t) and √   ak 2 + 1 2(ak 2 + 1) 1 2 − 1 + 2C . U =C 2 2 ak + coth ϕ sinh ϕ (ak 2 + coth2 ϕ)

(56)

(57)

(58)

We see that the form of solitons and periodic waves described by Eq. (12) essentially depend on their propagation velocities. Really, if a < 0 then the rapid solitons have the form (57) and the slow solitons are of the form (58). If a > 0, then both solitons (58) and periodic waves (55) exist but the latter cannot propagate with small velocities. All solutions (55)–(58) have only two independent components U 1 =< U, C 1 > and U 2 =< U, C 2 >. We present below plots of the initial profiles of U1 and U2 for some solutions:

Integrable Evolution Equations on the N-Dimensional Sphere

15

1 U1

0.5 0 U2

-0.5 -1 -8

-6

-4

-2

0

2

4

6

8

Fig. 1. Soliton solution (57) for a = −1, k 2 = 100/99

1 U1

0.5 0

U2

-0.5 -1 -8

-6

-4

-2

0

2

4

6

8

Fig. 2. Soliton solution (58) for a = −1, k = 2/3

7. Equations on the Sphere. Anisotropic Case In this section we present a list of anisotropic integrable equations similar to Eq. (1). Equation (1) and its symmetries contain both the scalar products uij =< Ui , Uj > and vij =< Ui , R(Uj ) >. Since R is an arbitrary symmetric operator, we regard < U, V > and < U, R(V ) > as two independent scalar products on the same vector space. The theory of canonical densities developed in Sect. 3 can be easily generalized to the case of Eqs. (3), whose coefficients fi depend on variables (4) and (8). The following statement is an extension of Theorems 1 and 2 to the anisotropic case. Theorem 4. Suppose Eq. (10) with fi = fi (u[1,1] , u[1,2] , u[2,2] , v[0,0] , v[0,1] , v[1,1] )

16

A.G. Meshkov, V.V. Sokolov

1 0.5 U2

0 -0.5 U1

-1 -8

-6

-4

-2

0

2

4

6

8

3

4

Fig. 3. Periodic solution (55) for a = 1, k = 1. 05

1

U2 U1

0.5 0 -0.5 -1 -4

-3

-2

-1

0

1

2

Fig. 4. Periodic solution (55) for a = 1, k = 2

on the sphere SN has an infinite series of commuting flows or conserved densities; then this equation is one of (11)–(14) or belongs to the following list: 
3 Ut = U3 + u[1,1] + c v[0,0] U1 + 3 u[1,2] U0 , (59) 2   u2[1,2] u[1,2] 3 u[2,2] c v[1,1] U1 , U2 + + 2 + Ut = U3 − 3 u[1,1] 2 u[1,1] u[1,1] u[1,1]

(60)

  u2[1,2] u[1,2] 3 u[2,2] (v[0,1] + u[1,2] )2 v[1,1] Ut = U3 − 3 U1 . U2 + + 2 − + u[1,1] 2 u[1,1] (u[1,1] + v[0,0] + a) u[1,1] u[1,1] u[1,1] (61)

Integrable Evolution Equations on the N-Dimensional Sphere

17

Equation (59) coincides with (1). Each of the two remaining equations satisfies conditions (21) with i
9 and possesses local conserved densities of orders 1, 2, 3 and 4. We also verified that these equations have fifth order symmetries. Though Eqs. (60) and (61) are definitely integrable, in order to prove this it is necessary to find Lax representations or auto-B¨acklund transformations for them. It will be done in a separate paper. In the case N = 1, after the trigonometric parameterization of the circle u1 =

tan2 (s) − 1 , tan2 (s) + 1

u2 =

2 tan(s) , tan2 (s) + 1

both Eqs. (59) and (60) become st = sxxx + 2 sx3 +

 3
c1 + c2 cos(4s) sx . 4

The latter equation is well known in the theory of integrable PDEs [13, 25]. The rational parameterization u1 =

v2 − 1 , v2 + 1

u2 =

2v v2 + 1

of the circle brings Eq. (61) with N = 1 to the form (17), where Q = α v 4 + β v 2 + α with arbitrary parameters α and β. Thus (61) is an integrable vector generalization of the generic Calogero-Degasperis equation (see Remark 5). Acknowledgements. The authors are grateful to E. V. Ferapontov for useful discussions. This research was partially supported by RFBR grant 02-01-00431, INTAS grant 99-1782, and EPSRC grant GR K99015.

References 1. Golubchik, I.Z., Sokolov, V.V.: Multicomponent generalization of the hierarchy of the LandauLifshitz equation. Theor. and Math. Phys. 124(1), 909–917 (2000) 2. Veselov, A.P.: Finite-gap potentials and integrable systems on the sphere with a quadratic potential. Funct. Anal. and Appl. 14(1), 48–50 (1980) 3. Athorne, C., Fordy, A.: Generalized KdV and MKdV equations associated with symmetric spaces. J. Phys. A. 20, 1377–1386 (1987) 4. Sokolov, V.V., Svinolupov, S.I.: Vector-matrix generalizations of classical integrable equations. Theor. and Math. Phys. 100(2), 959–962 (1994) 5. Sokolov, V.V., Wolf, T.: Classification of integrable polynomial vector evolution equations. J. Phys. A: Math. Gen. 34, 11139–11148 (2001) 6. Svinolupov, S.I., Sokolov, V.V.: Deformations of Jordan triple systems and integrable equations. Theor. and Math. Phys. 108(3), 1160–1163 (1996) 7. Svinolupov, S.I.: On the analogues of the Burgers equation. Phys. Lett. A 135(1), 32–36 (1989) 8. Svinolupov, S.I.: Generalized Schr¨odinger equations and Jordan pairs. Commun. Math. Phys. 143(1), 559–575 (1992) 9. Svinolupov, S.I.: Jordan algebras and generalized Korteweg-de Vries equations. Theor. and Math. Phys. 87(3), 391–403 (1991) 10. Doliwa, A., Santini, P.M.: An elementary characterization of the integrable motions of a curve. Phys. Lett. A 185, 373–384 (1994) 11. Habibullin, I.T., Sokolov, V.V., Yamilov, R.I.: Multi-component integrable systems and non-associative structures. In: Nonlinear Physics: Theory and Experiment, Eds. E. Alfinito, M. Boiti, L. Martina, F. Pempinelli, World Scientific Publisher: 1996, pp 139–168 12. Ibragimov, N.Kh., Shabat, A.B.: Infinite Lie-B¨acklund algebras. Funct. Anal. and Appl. 14(4), 79–80 (1980)

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13. Calogero, F., Degasperis,A.: Spectral transforms and solitons.Amsterdam-NewYork-Oxford: NorthHolland Publ. Co., 1982 14. Chen, H.H., Lee,Y.C., Liu, C.S.: Integrability of nonlinear Hamiltonian systems by inverse scattering method. Phys. Scr. 20(3-4), 490–492, (1979) 15. Meshkov, A.G.: Necessary conditions of the integrability. Inverse Problems 10, 635–653 (1994) 16. Sokolov, V.V.: On the symmetries of evolution equations. Russ. Math. Surv. 43(5), 165–204 (1988) 17. Svinolupov, S.I., Sokolov, V.V., Yamilov, R.I.: Backlund transformations for integrable evolution equations. Dokl. Akad. Nauk SSSR, 271(4), 802–805 (1983) 18. Adler, V.E., Shabat, A.B., Yamilov, R.I.: The symmetry approach to the problem of integrability. Theor. and Math. Phys. 125(3), 355–424 (2000) 19. Svinolupov, S.I., Sokolov, V.V.: Evolution equations with nontrivial conservation laws. Funct. Anal. and Appl. 16(4), 86–87 (1982) 20. Sokolov, V.V., Shabat, A.B.: Classification of integrable evolution equations. Soviet Scientific Reviews, Section C 4, 221–280 (1984) 21. Mikhailov, A.V., Shabat, A.B., Yamilov, R.I.: The symmetry approach to the classification of nonlinear equations. Complete lists of integrable systems. Russ. Math. Surv. 42(4), 1–63 (1987) 22. Calogero, F.: Why Are Certain Nonlinear PDE’s Both Widely Applicable and Integrable?. In: What is integrability? Springer Series in Nonlinear Dynamics, Berlin. Heidelberg New York: SpringerVerlag, 1991 pp 1–62 23. Mikhailov, A.V., Shabat, A.B., Sokolov, V.V.: The symmetry approach to classification of Integrable Equations. In: What is integrability? Springer Series in Nonlinear Dynamics, Berlin Heidelberg, New York: Springer-Verlag, 1991, p. 115–189 24. Fokas, A.S.: Symmetries and integrability. Stud. Appl. Math. 77, 253–299 (1987) 25. Fokas, A.S.: A symmetry approach to exactly solvable evolution equations. J. Math. Phys. 21(6), 1318–1325 (1980) 26. Novikov, S.P., Manakov, S.V., Pitaevskii, L.P., Zakharov, V.E.: Theory of solitons: Inverse Scattering Method. New York: Plenum, 1984 27. Takhtadzhyan, L.A., Faddeev, L.D.: The Hamiltonian Methods in the Theory of Solitons. Berlin: Springer, 1987 28. Olver, P.J.: Applications of Lie Groups to Differential Equations. New York: Springer-Verlag, 1993 Communicated by L. Takhtajan

Commun. Math. Phys. 232, 19–58 (2002) Digital Object Identifier (DOI) 10.1007/s00220-002-0746-8

Communications in

Mathematical Physics

Transfer Operators and Dynamical Zeta Functions for a Class of Lattice Spin Models J. Hilgert1 , D. Mayer2 1

Institut f¨ur Mathematik, Technische Universit¨at Clausthal, 38678 Clausthal-Zellerfeld, Germany. E-mail: [email protected] 2 Institut f¨ ur Theoretische Physik, Technische Universit¨at Clausthal, 38678 Clausthal-Zellerfeld, Germany. E-mail: [email protected] Received: 14 March 2002 / Accepted: 24 June 2002 Published online: 14 November 2002 – © Springer-Verlag 2002

Abstract: We investigate the location of zeros and poles of a dynamical zeta function for a family of subshifts of finite type with an interaction function depending on the parameters λ = (λ1 , . . . , λm ) with 0
λi < 1. The system corresponds to the well known Kac-Baker lattice spin model in statistical mechanics. Its dynamical zeta function can be expressed in terms of the Fredholm determinants of two transfer operators Lβ and Gβ with Lβ the Ruelle operator acting in a Banach space of holomorphic functions, and Gβ an integral operator introduced originally by Kac, which acts in the space L2 (Rm , dx) with a kernel which is symmetric and positive definite for positive β. By relating Lβ via
β closely related to the Kac operator Gβ the Segal-Bargmann transform to an operator G we can prove equality of their spectra and hence reality, respectively positivity, for the eigenvalues of the operator Lβ for real, respectively positive, β. For a restricted range of parameters 0
λi < 21 , 1
i
m we can determine the asymptotic behavior of the eigenvalues of Lβ for large positive and negative values of β and deduce from this the existence of infinitely many non-trivial zeros and poles of the dynamical zeta functions on the real β line at least for generic λ. For the special choice λi = 21 , 1
i
m, we find a family of eigenfunctions and eigenvalues of Lβ leading to an infinite sequence of equally spaced “trivial” zeros and poles of the zeta function on a line parallel to the imaginary β-axis. Hence there seems to hold some generalized Riemann hypothesis also for this kind of dynamical zeta functions. 1. Introduction The transfer matrix method has played an important role in statistical mechanics ever since E. Ising for the first time used this method to solve his 1-dimensional lattice spin model with nearest neighbour interaction. The method was extended later to treat higher dimensional models with arbitrary finite range interactions. The most satisfying theory for this method from the mathematical point of view goes back to D. Ruelle who introduced the so-called transfer operator for 1-dimensional lattice spin systems with

20

J. Hilgert, D. Mayer

arbitrary long range interactions (see [Ru68]). Continued interest in such systems is related to the fact that these systems show up in a rather natural way within the so-called thermodynamic formalism for dynamical systems (see [Ru78]). Thereby the transfer operator can be used for instance to construct invariant measures for such systems and to characterize their ergodic properties (see [Ba00]). Another nice application of this method is to the theory of dynamical zeta functions (see [Ru92, Ru94]). These functions can be interpreted as generating functions for the partition functions of the system constructed in complete analogy to the partition functions of lattice spin systems. It turned out that the transfer operator method had indeed been used already some time ago in the p-adic setup of zeta functions by B. Dwork (see [Dw60] or [Ro86]) who constructed such an operator to show rationality of the Artin-Weil zeta function for projective algebraic varieties over finite fields and proved in this way part of the Weil conjectures (see [We49]). More recently the method also yielded a completely new approach to Selberg’s zeta function, which can also be viewed as a dynamical zeta function for the geodesic flow on surfaces of constant negative curvature (see [Ma91]). Indeed, the aforementioned Artin-Weil zeta function is nothing but the dynamical Artin-Mazur zeta function (see [ArMa65]) for the Frobenius map of the algebraic variety. Typically, such dynamical zeta functions can be expressed in terms of some kind of Fredholm determinant of the transfer operator, which therefore allows a spectral interpretation of the zeros and poles of these functions. The existence of such an interpretation is one of the challenging open problems for all arithmetic zeta functions of number theory and algebraic geometry (see [Be86, Co96, De99]). Obviously such a spectral interpretation is also closely related to another famous open problem for these zeta and more general L-functions, namely the general Riemann hypothesis: one expects that the zeros and poles of such functions are located on critical lines in the complex plane as one does in the special case of the well known Riemann zeta function. Presently it is not known whether such a conjecture makes sense also for general dynamical zeta functions which are, unlike the Selberg- or the Artin-Weil zeta functions, not related to arithmetics. In the present paper we address this problem for the Ruelle zeta function of a certain subshift of finite type which in the physics literature has become known as the Kac-Baker model (see [Ka66]). M. Kac got interested in this model while trying to understand the mathematics behind the phenomenon of phase transitions in systems of statistical mechanics with weak long-range interactions like in the van der Waals gas (see [Ba61]). The model he considered is an Ising spin system on a 1-dimensional lattice with a 2-body interaction given by a finite superposition of terms decaying exponentially fast with the distance between the different spins. His real interest was certainly in a system with a continuous superposition of such exponentially decaying terms to model also interactions decaying only polynomially fast as is the case in the van der Waals gas. However his method did not allow him to treat this limiting case in a rigorous way. We have chosen the Kac model since its dynamical zeta function can be understood rather well by the transfer operator method. On the other hand there seems to be no obvious connection of this zeta function to any arithmetic zeta functions for which a general Riemann hypothesis is known to hold. There exist two rather different transfer operators for this model which allow to express its zeta function as Fredholm determinants of these operators (see [Ma80, ViMa77, Ka66]). Up to now, however, it was not known how these two operators are related to each other. Our investigations show that the Ruelle operator Lβ is equivalent to a modified
β basically through a Segal-Bargmann transformation. This transformaKac operator G

Transfer Operators and Dynamical Zeta Functions

21

tion establishes a unitary map between the Hilbert space of square integrable functions on the real line, where the Kac operator acts, and the Fock space of entire functions on the complex plane square integrable with respect to a certain weight function, to which the Ruelle operator can be restricted. For positive values of the parameter β the operator
β has the same spectrum as Kac’s original operator Gβ . Since this operator Gβ is a G symmetric, positive definite trace class operator for β > 0 its eigenvalues and hence
β and of the Ruelle operator Lβ are positive for also those of the modified operator G
β has real spectrum also for these β-values. Furthermore we can show that the operator G negative β so that the Ruelle operator Lβ indeed has real spectrum for real β. Since Lβ defines a family of trace class operators holomorphic in the variable β we can show that the zeta functions of a whole class of Kac models extend meromorphically to the entire complex β-plane and have infinitely many nontrivial zeros on the real line. For a special case of the parameters we can also show the existence of infinitely many “trivial” zeros of this dynamical zeta function located on a line parallel to the imaginary axis in the complex β-plane. Thus for this function an analogue of the Riemann hypothesis seems plausible. The present paper generalizes analogous results in [HiMa01] for the case of an interaction consisting of a single exponentially decaying term. In detail the present paper is organized as follows: in Sect. 2 we recall the definition of the Kac-Baker models and derive their Ruelle transfer operators. Further, we show how the dynamical zeta function of these models can be expressed through Fredholm determinants of the Ruelle operators. In Sect. 3 we derive, basically following Gutzwiller (see [Gu82]), the Kac operator appropriate to our problem and show that the zeta function can be expressed for positive β also in terms of Fredholm determinants of this Kac operator. In Sect. 4 we show how the kernel of the Kac operator is related to a certain form of Mehler’s formula for the Hermite functions which allows us to diagonalize an integral operator closely related to the Kac operator. In Sect. 5 we introduce the Fock spaces and the Segal-Bargmann transform and show how the Ruelle and a modified Kac operator can be directly related to each other. There we show that for real β the two operators have the same spectrum and give explicit expressions relating the eigenfunctions of the two operators for nonvanishing eigenvalues. In Sect. 6 we derive the asymptotic behavior of the eigenvalues of the Ruelle operator for large positive and negative values of β and apply it to get the results on the location of poles and zeros of the zeta function on the real line. In Sect. 7 we give explicit expressions for the matrix elements of a modified Kac-Gutzwiller operator in the Hilbert space basis given by the Hermite functions which seem best suited for future numerical calculations. 2. The Ruelle Operator for the Kac-Baker Model The generalized Kac model describes a 1-dimensional lattice spin system with a 2-body interaction which is a superposition of finitely many exponentially decaying terms. More precisely, for F := {±1}, ξ = (ξn )n∈Z+ ∈ F Z+ , and i, j ∈ Z+ = {0, 1, 2, . . . } we set φij (ξ ) := −ξi ξj

m  l=1

|i−j |

Jl λ l

,

where m ∈ N and the parameters Jl > 0 and 0 < λl < 1 are fixed and describe the interaction strengths and the different decay rates. The interaction energy of a configuration ξ ∈ F Z+ when restricted to the finite sublattice Z[n−1] = {0, 1, . . . , n − 1} is then given as

22

J. Hilgert, D. Mayer

Un (ξ ) := UZ[n−1] (ξ ) =

∞ n−1  

φi,i+j (ξ ).

i=0 j =1

When inserting the explicit form of φ one gets Un (ξ ) = −

∞ n−1  

m 

ξi ξi+j

i=0 j =1

l=1

j

Jl λ l .

For β ∈ C the partition functions Zn (β) for the finite sublattices Z[n−1] with periodic boundary conditions are defined as 

Zn (β) :=

exp(−βUn (ξ )),

(1)

ξ ∈Per n

where Per n denotes the set of configurations ξ ∈ F Z+ which are periodic with period n. That means ξ ∈ Per n if and only if ξi+n = ξi for all i ∈ Z+ . Defining the shift τ : F Z+ → F Z+ by (τ ξ )i := ξi+1

if ξ = (ξi )i∈Z+ ,

one has Per n = Fix τ n = {ξ ∈ F Z+ : τ n ξ = ξ }. To the dynamical system (F Z+ , τ ) one can associate the Ruelle zeta function ζR (z, β) := exp

∞  zn n=1

n

 Zn (β) .

(2)

 λl Jl |β|c )n . Therefore the seNote that |Un (ξ )| ≤ n m l=1 1−λl =: nc so that |Zn (β)| ≤ (2e 2 ries defining ζR converges in a neighborhood of (0, 0) in C . We will show in Proposition 2.4 that ζR can in fact be extended to a meromorphic function on C2 . To determine the analytic properties of this function one makes use of the transfer operator technique. Note first that the configuration space F Z+ is compact and metrizable with respect to the product topology. For each β ∈ C one can define the Ruelle transfer operator Lβ which will act on the space of observables of the lattice spin system, i.e. on C(F Z+ ), the space of continuous functions on F Z+ . In this framework one sets 

 Lβ f (ξ ) :=



exp(−βU1 (η))f (η).

η∈τ −1 (ξ )

Inserting the explicit expression for U1 (η) one finds 



Lβ f (ξ ) =

 σ =±1

 exp βσ

∞  j =0

ξj

m  l=1

 j +1 

J l λl

f



σ, ξ ,

(3)

Transfer Operators and Dynamical Zeta Functions

23

where (σ, ξ ) := η with η0 = σ and ηj = ξj −1 for all j ∈ N. Generalizing the arguments for the case m = 1 in [Ma80] one introduces the map z = (zl )l=1,..,m : F Z+ → Rm defined by zl (ξ ) :=

∞  i=0

ξi λi+1 l .

Since zl (σ, ξ ) = σ λl + λl zl (ξ ) for all l, the operator Lβ leaves the space of functions z∗ (ϕ) := ϕ ◦ z with ϕ ∈ C(Rm ) invariant. Thus we obtain a factorization C(FO Z+ ) z∗

C(Rm )

Lβ

/ C(F Z+ ) O z∗

/ C(Rm )

of Lβ through C(Rm ), which we will denote again by Lβ and which is of the form   Lβ g (z) = eβJ ·z g(
z + λ) + e−βJ ·z g(
z − λ), (4) where
is the diagonal matrix with diagonal elements λ1 , . . . , λm andJ = (J1 , . . . , Jm). Indeed, by the Ruelle-Perron-Frobenius Theorem (see [Zi00, Chap.4]) the iterates under Lβ of the constant function f (ξ ) = 1 in C(F Z+ ) converge uniformly to the eigenfunction belonging to the leading eigenvalue of Lβ and hence this eigenfunction belongs to the space C(Rm ). Therefore, when restricting the operator to the space C(Rm ) one does not lose the leading eigenvalue which is the most important one from the physical point of view. The physically most satisfying operator is obtained in fact by restricting the domain of the operator still further. From the form of the operator Lβ in the space C(Rm ) we see that, if g is a holomorphic function on the polycylinder   D = z ∈ Cm : |zl | < Rl , l = 1, . . . , m    λl with Rl > 1−λ , then also L g z is such a function. In fact, much more is true: Denote β l by B(D) the Banach space of holomorphic functions on D which extend continuously to the closure D of D. Then, according to [Ma80, Appendix B], we have Proposition 2.1. Lβ : B(D) → B(D) is with respect to the parameter β ∈ C a holomorphic family of nuclear operators of order 0 in the sense of Grothendieck (see [Gr55]). In particular all the Lβ are of trace class.   The trace can be computed using the holomorphic version of the Atiyah-Bott fixed point formula (see [AtBo67] and [Ma80], Appendix B or [Ru94],§1.12): Lemma 2.2. Fix ϕ ∈ B(D) and a continuous map ψ : D → D which is holomorphic on D. Then ψ has a unique fixed point zfix ∈ D and the composition operator A : B(D) → B(D) defined by Ag := ϕ · (g ◦ ψ) is trace class with trace   ϕ zfix   .  trace (A) = det 1 − ψ  zfix  

24

J. Hilgert, D. Mayer

Proposition 2.3. The partition function Zn (β) of the Kac-Baker model can be expressed through the traces of the powers of the Ruelle transfer operator Lβ via  m   1 − λnl trace Lnβ . Zn (β) = l=1

Proof. The defining equation (1) for the partition function Zn (β) can be rewritten as   m n−1  ∞    exp β Jl ξk ξk+i λil . Zn (β) = ξ ∈Per n

l=1

k=0 i=1

Using the fact that ξ ∈ Per n implies ξi+n = ξi for all i ∈ Z+ one gets   m n n−1   Jl   i exp β σk σk+i λl , Zn (β) = 1 − λnl n σ ∈F

l=1

k=0 i=1

where σn+i = σi for all i. On the other hand the nth iterate of the transfer operator Lβ from (4) acting on the Banach space B(D) is given by  n   m n−k n−1 

    Lnβ g (z) = exp β Jl σk λn−k zl + σk σk+i λil l σ ∈F n

×g

 n 

l=1

k=1

k=1 i=1



σi λi +
n z ,

i=1

where

λi

:=

(λi1 , . . .

, λim ). We

apply Lemma 2.2 to the maps ψσ defined by   n  i n ψσ (z) := σi λ +
z i=1

for which the fixed points are given by n −1 zfix σ = (1 −
)

n 

σi λi .

i=1

The result is trace Lnβ

 n n  m  Jl   1 = m exp β σk σi λn−k+i l n 1 − λnl l=1 (1 − λl ) n σ ∈F

l=1

+

k=1 i=1

n−k n−1   k=1 i=1

σk σk+i λil

−

n−k n−1   k=1 i=1

But n n−1   k=1 i=k+1

and hence

σk σi λn−k+i = l

n−k n−1   k=1 i=1

σk σk+i λn+i l

σk σk+i λn+i l

 .

Transfer Operators and Dynamical Zeta Functions

trace Lnβ

25

  m n−1  n n−k  Jl    1 i σ = m exp β σ λ + σk σk+i λil n i l n n) 1 − λ (1 − λ l=1 l l σ ∈F n l=1 i=1 k=1 i=1  n−1  k  . + σk σi λn−k+i l k=1 i=1

Changing the order of summation in the last double sum we finally get   m n  n  Jl   1 trace Lnβ = m exp β σk σk+i λil n 1 − λnl l=1 (1 − λl ) n σ ∈F

which up to the factor

m

In view of the identity 2.3 yields

1 l=1 1−λnl

l=1

k=1 i=1

is just the partition function Zn (β).

 

m 

  nαm l 1 − λnl = α∈{0,1}m (−1)|α | λnα 1 · · · λm Proposition

l=1

Zn (β) =





(−1)|α | trace

α∈{0,1}m

m  l=1

λαl l

n

 Lnβ

,

so that  ∞ n  z  ζR (z, β) = exp n n=1 





(−1)|α | trace

α∈{0,1}m

= exp trace





=



det 1 − z

α∈{0,1}m

l=1

 λαl l

(−1)|α | (−1) log 1 − z

α∈{0,1}m



m 

n  Lβ 

m  l=1

m   l=1

 λαl l Lβ

 λαl l

 Lβ 

(−1)|α|+1 .

Together with Proposition 2.1 this proves the following proposition. Proposition 2.4. The Ruelle zeta function ζR (z, β) defined in (2) for the Kac-Baker model with decay rates λ = (λ1 , . . . , λm ) ∈]0, 1[m can be extended to a meromorphic family (ζR (·, β))β∈C of meromorphic functions on C via the formula ζR (z, β) =



(−1)|α|+1  det 1 − zλα Lβ .

α∈{0,1}m

  Remark 2.5. The analytic properties in the variable z are well known for general lattice spin systems with exponentially fast decaying interactions. For m = 1 Proposition 2.4 was proved in [Ma80].  

26

J. Hilgert, D. Mayer

3. The Kac-Gutzwiller Operator In [Ka59] M. Kac found another operator whose traces are directly related to the partition functions Zn (β) of the Kac-Baker model. Kac did not work with periodic but open boundary conditions and hence his operator has to be modified a bit to give the partition functions we use here. In the case m = 1, M. Gutzwiller already derived this operator (see [Gu82]) and the case of general m  1 can be handled similarly. Just like Gutzwiller and Kac we start with an identity for Gaussian integrals known already to C. Cram´er: For a positive definite (n × n)-matrix A and x ∈ Rn we have (see [Cr46] or the appendix of [Fo89])  1 n 1 1 x· A x − ( ) e2 = (2π) 2 | det B| 2 ex·z e− 2 (z·Bz) dz, (5) Rn

where B = A−1 . Any periodic configuration ξ p ∈ Per n can be extended to a periodic configuration on the entire lattice Z with the same period which we again denote by ξ p . Now consider the (n × n)-matrix A(l) given by (l)

Ai,j = βJl

+∞ 

exp (−γl |i − j + nk|) ,

0 ≤ i, j ≤ n − 1,

k=−∞

where we choose the constants γl so that e−γl = λl with the coupling constants λl of the p p Kac-Baker model. Using ξ pn = ξ0 , . . . , ξn−1 one finds (see [Gu82, §6] for this and the following results on matrix calculations) n−1 +∞

1 p β   p p ξi ξj exp (−γl |i − j |) = Jl ξ n · A(l) ξ pn . 2 2 i=0 j =−∞

But m m n−1 +∞ β    p p |i−j | β  Jl ξ i ξ j λl = −βUn (ξ p ) + n Jl , 2 2 l=1

i=0 j =−∞

(6)

l=1

i.e. the left-hand side is, up to the constant term given by the sum of the Jl , just the interaction energy of the periodic configuration ξ p ∈ Per n for the Kac model.  −1 has for n > 2 the following form: The matrix B(l) = A(l) 

B(l)

cosh γl  −1 2   0  1  .. =  βJl sinh γl  .  0   0 − 21

− 21 cosh γl − 21 .. .

0 − 21 cosh γl .. .

... 0 − 21 .. .

... ... 0 .. .

0 0 ... .. .

... 0 0

0 ... ...

− 21 0 ...

cosh γl − 21 0

− 21 cosh γl − 21

− 21 0 0 .. . 0 − 21 cosh γl

      ,    

Transfer Operators and Dynamical Zeta Functions

27

which for positive β is a positive definite matrix with determinant 4 nγ 2 | det B(l) | = . sinh 2 (2βJl sinh γl )n This formula holds also for n = 1 and n = 2. That B(l) is indeed positive definite one can see as follows: for arbitrary x ∈ Rn one finds  n  n

1  i=1 xi xi−1 (l) 2 , coth(γl )xi − x·B x = βJl sinh γl i=1

where x0 = xn . A simple calculation (see also Proposition 4.1) shows that the following identity holds:

coth(γl )

n  i=1

n  (l)2 xi

 =

(l) (l)

i=1

−

xi xi−1

sinh γl

n γ  l

1 tanh 2 2

(l)2

xi

i=1

(l)2

n



i=1

+ xi−1 +

(l)

(l)

xi − xi−1

sinh γl

2   ,

(7)

  and hence for βJl > 0 and γl > 0 one finds x, B(l) x  0. Inserting (6) into formula (5) one calculates e

m  −βUn ξ p + nβ Jl 2 l=1

= =

m 

e

1 2

l=1 m  



2

n

(2π)− 2

l=1

 ×

2m = nm 2



(2βJl sinh γl ) 1

e− 2



z(l) ·B(l) z(l)

Rn m  l=1

sinh

× Rn



n 2

p

sinh

 ...

e

l

(l)

2 



nγl 2

− 21

nγ

eξ n ·z dz(l)

n

(βπJl sinh γl ) 2



The change z(l)

ξ pn ·A(l) ξ pn

m   l=1

z(l) ·B(l) z(l)

m  

el=1

ξ pn ·z(l)

dz(1) . . . dz(m) .

Rn

√ =: βJl x (l) of integration variables yields p

nβ

m

e−βUn (ξ )+ 2 l=1 Jl n m  nγ  2m  sinh 2 l (βJl ) 2 = nm n 2 2 l=1 (βJl π sinh γl )

   (l) (l) (l)  m √  βJl ξ pn ·x (l) − 21 m βJ · B x x l=1 l l=1 × ... e e dx (1) . . . dx (m) . Rn

Rn

28

J. Hilgert, D. Mayer

Performing the summation over all the periodic configurations ξ ∈ Per n then gives e

nβ 2

m  l=1



Jl

e

−βUn (ξ )

=

ξ ∈Per n

m 

2m



sinh

×

n 

Rn

 m 

cosh

i=1

l=1



Inserting the explicit form of the matrix B(l) = A(l) e

m

l=1 Jl

Zn (β) =

m 

2m 2n(m−1) 



·

n 

Rn

cosh

n

nγl 2

 (l) βJl xi

−1

dx (1) . . . dx (m) .

we find

n

(π sinh γl ) 2  

 m 

i=1 (l)

sinh

 1 exp  − 2

...

Rn

·

l=1



2n(m−1) 2 l=1 (π sinh γl )    (l) (l) (l)  1 m · . . . e− 2 l=1 βJl x ·B x Rn

nβ 2

nγl 2

l=1

m  l=1

 (coth γl ) 

n  i=1

n  (l)

(xi )2 −

i=1

(l) (l)

xi xi−1

sinh γl

   

 (l) βJl xi

dx (1) . . . dx (m) ,

(l)

where x0 = xn . For β  0 we introduce the kernel function 

 cosh

Kβ (ξ , η) :=

m √ 

l=1

 m  1 2 √ βJl ξl cosh βJl ηl

2(m−1)

× exp



− 41

m 

l=1

1

(π sinh γl ) 2

l=1



m l=1

 tanh

γl   2 ξl 2

 + ηl2 +

(ξl −ηl )2 sinh γl



,

(8)

where ξ = (ξ1 , . . . , ξm ) ∈ Rm and η = (η1 , . . . , ηm ) ∈ Rm . We call the associated operator Kβ on L2 (Rm , dξ ) defined by    Kβ (ξ , η)f (η)dη Kβ f (ξ ) = Rm

the Kac-Gutzwiller transfer operator. Note that the kernel  of Kβ decreases fast enough to ensure that Kβ is of trace class with trace Kβ = Rm Kβ (ξ , ξ )dξ . Calculating the trace of the iterates of Kβ and comparing the result to the above formula for the partition function Zn (β) after inserting (7) we find Zn (β) = 2m

m 

sinh

l=1

nγl − nβ m Jl e 2 l=1 trace Kβn . 2

(9)

Transfer Operators and Dynamical Zeta Functions

29

To simplify the expression for Zn (β) we introduce the rescaled Kac-Gutzwiller operator Gβ : L2 (Rm , dξ ) → L2 (Rm , dξ ) defined by Gβ :=

m 

1

(λl eβJl )− 2 Kβ .

(10)

l=1

In view of Zn (β) =



m  l=1

1 − λnl



trace

m 

e

nγl 2

e

−

βJl n 2

l=1

 Kβn ,

which is a simple reformulation of (9) we finally have shown the following proposition. Proposition 3.1. For β  0 the partition function Zn (β) of the Kac-Baker model can be expressed through the traces of the powers of the rescaled Kac-Gutzwiller operator Gβ via Zn (β) =

m  l=1



1 − λnl



trace Gβn .

  An argument similar to the one we used for the Ruelle operator Lβ in the proof of Proposition 2.4 now shows Proposition 3.2. For β  0 the Ruelle zeta function ζR (z, β) for the Kac-Baker model can be written in terms of the modified Kac-Gutzwiller operator via ζR (z, β) =



 (−1)|α|+1 det 1 − zλα Gβ .

α∈{0,1}m

  Since the zeta function ζR (z, β) is meromorphic and its divisor for fixed β uniquely determined, it is not too difficult to see that at least for generic values of the parameters β > 0 and λ ∈]0, 1[m , the spectra of the two operators Lβ and Gβ have to be identical. This indeed has been shown in the case m = 1 already by B. Moritz in (see [Mo89]). We will show however (see Theorem 5.12) that the spectra of the two operators coincide for any real β and all parameters λ. Hence the Fredholm determinants det(1 − zGβ ) and det(1 − zLβ ) of the Kac-Gutzwiller operator and the Ruelle operator coincide on the real axis and extend to a holomorphic function in the entire β plane even if the operator Gβ contrary to the operator Lβ has itself no such analytic continuation to the entire β-plane.

30

J. Hilgert, D. Mayer

4. Hermite Functions and Mehler’s Formula Consider the operators Zj = mxj +

1 ∂ 2π ∂xj

,

Zj∗ = mxj −

1 ∂ 2π ∂xj

,

j = 1, . . . , n,

where mxj denotes the multiplication operator (mg f )(x) = g(x)f (x) in the space L2 (Rm , dx) for g(x) = xj and x = (x1 , . . . , xm ) ∈ Rm . The Hermite functions hα ∈ L2 (Rm , dx) with α = (α1 , . . . , αm ) ∈ N0m are given by (see [Fo89, p.51]) m

h0 (x) = 2 4 e−πx·x ,     m  π |α |  ∗α  24 −1 |α | πx·x ∂ α −2πx·x e hα (x) = e , Z h0 (x) =  √ α! ∂x α! 2 π m whereα = (α1 , . . . , αm ) with αi ∈ N0 for all 1 ≤ i ≤ m, |α| = i=1 αi and ∗α denotes the operator Z ∗α1 · · · Z ∗αm . α! = m α !. Moreover, Z i m i=1 1 The Hermite functions are known to be an orthonormal basis of the Hilbert space L2 (Rm , dx) and the following formula due to Mehler holds (see [Fo89]):

    2 2 2 + 4π w x · y m  −π 1 + w + y x  2 2 , w |α | hα (x)hα (y) = exp  1 − w2 1 − w2 m α∈N0

2 where |w| < 1 and  1−w 2 > 0. From the case m = 1 we then get for λ = (λ1 , . . . , λm ) with 0 < λi < 1 for 1 ≤ i ≤ m the identity m  ∞  l=1 αl =0

λαl l hαl (xl ) hαl (yl )

=

m  l=1



2 1 − λ2l

1



2

· exp

    m  −π 1 + λ2l xl2 + yl2 + 4π λl xl yl l=1

1 − λ2l

.

A simple calculation presented for m = 1 in [HiMa01] shows that the following proposition is true. Proposition 4.1. (i) For λl = e−γl with γl > 0 one has      −π 1 + λ2l x 2 + y 2 + 4πλl xy 1 2 2π 2 − = + y ) cosh γ + xy . (x l sinh γl 2 1 − λ2l (ii)

1 2 sinh γl

       − 21 x 2 + y 2 cosh γl + xy = − 41 x 2 + y 2 tanh γ2l +



(x−y)2 sinh γl

.

Transfer Operators and Dynamical Zeta Functions

31

(iii) For λl = e−γl and xl = ξl 2√1 π , yl = ηl 2√1 π one has the following version of Mehler’s formula: 1 2 1 λ hα (x)hα (y) = λl sinh γl l=1 α∈Nm 0    m γ (ξ − η )2 1 l l l 2 2 (ξl + ηl ) tanh . · exp − + 4 2 sinh γl 

m  

α

l=1

  In Proposition 4.1 (iii) we used only the fact that hα (x) = hα1 (x1 ) · · · hαm (xm ) for α = (α1 , . . . , αm ), x = (x1 , . . . , xm ) (see [Fo89, p.52]) and (λl sinh γl

)−1 .

2 1−λ2l

=

2eγl eγl −e−γl

=

Define next the kernel function    m 1 m 

 2 γl 1 1  2 (ξl − ηl )2 2
K(ξ , η) = 2 ξl + ηl tanh + . exp − 4π sinh γl 4 2 sinh γl l=1

l=1

(11) Then the kernel Kβ (ξ , η) of the Kac-Gutzwiller transfer operator defined in (8) satisfies  Kβ (ξ , η) = cosh

 m 

 βJl ξl cosh

 m 

l=1

 1 2

βJl ηl


, η). K(ξ

(12)

l=1

Lemma 4.2. For an invertible real (m × m)-matrix C and a smooth function a : Rm → ]0, ∞[, consider the map 1

(RC f ) (x) = |det(C)| 2 f (Cx). Then



  RC : L2 Rm , a(C −1 ξ )dξ → L2 Rm , a(x)dx

is an isomorphism of Hilbert spaces. Proof.  Rm

     (RC f ) x 2 a x dx =

 Rm



=

|det(C)| |f (Cx)|2 a(x)dx

|f (ξ )|2 a(C −1 ξ )dξ

Rm

= f 2L2 (Rm ,a(C −1 ξ )dξ ) .  

32

J. Hilgert, D. Mayer

If K : L2 (Rm , dξ ) → L2 (Rm , dξ ) is given by an integral kernel K(ξ , η) then the induced operator KC := RC ◦ K ◦ RC−1 : L2 (Rm , dx) → L2 (Rm , dx) has kernel KC (x, y) = | det(C)|K(Cx, Cy) as one easily verifies by a straightforward calculation using the transformation formula. If c = (c1 , . . . , cm ) ∈ Rm with ci = 0 for 1 ≤ i ≤ m and C is the diagonal matrix with the cl on the diagonal, we simply write Rc for RC and Kc for KC . Note that Cx = (c1 x1 , . . . , cm xm ). Lemma 4.3. Let a : Rm → [1, ∞[ be a smooth function and K be a bounded operator on L2 (Rm , dx) given by the kernel K(x, y) as  (Kf )(x) = K(x, y)f (y)dy. Rm

Then (a) The operator K ◦m√a is an unbounded operator on L2 (Rm , dx) with kernel K(x, y)  a(y). (b) The operator m √1 ◦ K is a bounded operator on L2 (Rm , dx) with kernel √ 1 K(x, y).

a(x)

a

Proof. (a) This follows from (K ◦ m√a f )(x) =



 K(x, y) a(y)f (y)dy

Rm

and

   2     f (x)2 a(x)dx.  a(x)f (x) dx = Rm

Rm

(b) Calculate     1 1  m √1 ◦ Kf (x) =  K(x, y)f (y)dy. K(x, y)f (y)dy = a a(x) a(x) Rm

Rm

  Fix a : Rm → [1, ∞[. Then Lemma 4.3 yields the following commutative diagram: L2 (Rm , dx) o

? _ L2 (Rm , a(x)dx)

K

/ L2 (Rm , dx)



m√a

L2 (Rm , dx) 

K

id

L2 (Rm , dx)

K 

id



/ L2 (Rm , dx) 

m √1

a

/ L2 (Rm , dx)

Transfer Operators and Dynamical Zeta Functions

33

with integral operators K  and K  given by the kernels K  (x, y) = K(x, y) 

1 a(y)

and

K  (x, y) = 

1 a(x)

K(x, y) 

1 a(y)

.

If the kernel |K(x, y)| defines a bounded operator on L2 (Rm , dx), then also the operator K  is bounded. Now consider the Kac-Gutzwiller operator Kβ with kernel Kβ (ξ , η). For s, x ∈ Rm √ we set coshs (x) := cosh(s · x). With J := (J1 , . . . , Jl ) and s = s 0 := 2 βπ J we √ choose the function a := coshs 0 and for c = c0 := 2 π (1, . . . , 1) we obtain √ √ √ Kβ,c0 (x, y) = (2 π)m Kβ (2 πx, 2 π y) √ = (2 π)m Kβ (ξ , η) m

1   γl 2 = 2 coshs 0 (x) coshs 0 (y) (e− 2 ) λα hα (x)hα (y). α∈Nm 0

l=1

 Hence for the kernel Kc0 (x, y) of the operator Kβ,c =: Kc0 , which does not depend on 0 the variable β, one finds

Kc0 (x, y) := 2

m  

γl

(e− 2 )λα hα (x)hα (y).

(13)

l=1 α∈Nm 0

From the fact that the Hermite functions hα determine an orthonormal basis of L2 (Rm , dx) one concludes

1 Kc0 hα (x) = 2λα λ 2 hα (x), (14) i.e., the hα are the complete set of eigenfunctions of the operator Kc0 with eigenvalue ρα := 2λ

α+ 21

=2

m  l=1

αl + 21

λl

.

(15)

In particular, Kc0 is bounded. We will need this result later on. Note that Lemma 4.3 yields the following commutative diagram for the Kac-Gutzwiller operator Kβ :

Kβ L2 Rm , a c−1 0 ◦ ξ dξ Rc0

 Kβ,c0 L2 (Rm , a(x)dx) m√a



 Kβ,c



Kc0

L2 (Rm , dx) id

L2 (Rm , dx)

0

/ L2 (Rm dξ ) Rc0

 / L2 (Rm , dx) id



/ L2 (Rm , dx) 

m √1a

/ L2 (Rm , dx)

34

J. Hilgert, D. Mayer

  2 The functions eπx hα (x) are polynomials of degree α  in x (see [Fo89, p.52]). In view of the estimate    m m  1  |R ||ξ |    i i 2 coshR (ξ ) = cosh Ri ξi
e i=1 , i=1

 one concludes that the functions ξ → hα (ξ ) coshR (ξ ) are in L2 (Rm , dξ ). Hence all the hα are contained in L2 (Rm , coshR (ξ )dξ ). This together with Mehler’s formula in Proposition 4.1 shows immediately that the Kac-Gutzwiller operator Kβ with kernel Kβ (ξ , η) in (12) is symmetric and positive definite for β  0. In fact, 



f, Kβ f = 2

1 m    λl 2 l=1 α∈Nm 0

4π

  λ 



α

Rm

f (ξ ) coshR 0 (ξ )hα



ξ √

2 π



2  dξ   0,

√ where R0,i = βJi , 0
i
m so that the eigenvalues of the operator Kβ are nonnegative for β  0. 5. Fock Space and Segal-Bargmann Transformation For t > 0 consider the Hilbert space HL2 (Cm , µt ) of entire functions F : Cm → C with    F (z)2 µt (z) dz < ∞, F 2t := Cm

where µt denotes the weight function

 2 µt (z) = t m exp −π t z .

The Bargmann transform Bt : L2 (Rm , dx) → HL2 (Cm , µt ) defined via   m   2 4 π πt 2 f (x) exp 2π x · z − x 2 − z dx (Bt f ) (z) = t t 2 Rm

determines a unitary operator in the two Hilbert-spaces (see [Fo89, p.47]). In the following we are primarily interested in the case t = 1. In this case we denote the space HL2 (Cm , µt ) simply by Fm and call it the Fock space over Cm . The transform B1 is then denoted by B and hence    m π f (x) exp 2π x · z − π x 2 − z2 dx. (16) (Bf ) z = 2 4 2 Rm

Its inverse B −1 : Fm → L2 (Rm , dx) is given by (see [Fo89, p.45]) 



m π (B −1 F )(x) = 2 4 F (z) exp 2πx · z∗ − π x 2 − z∗ 2 exp −π |z|2 dz, 2 Cm

Transfer Operators and Dynamical Zeta Functions

35

where zl∗ = zl simply is the complex conjugate of zl . An orthonormal basis of Fock space Fm is given by the functions    π |α | α ζα z = (17) z , α!  αi m where zα = m l=1 zi and α = (α1 , . . . , αm ) ∈ N0 . Indeed one has (see [Fo89, p.51]) ζα = Bhα , where as before the hα are the Hermite functions in Rm . This we use to prove the following proposition. Proposition 5.1. For λ = (λ1 , . . . , λm ) ∈ (0, 1)m the bounded operator Mλ : Fm → Fm defined as Mλ := B ◦ Kc0 ◦ B −1 is given by the expression  m  1 Mλ F (z) = 2 λl F (λ1 z1 , . . . , λm zm ) = 2λ 2 F (
z),





l=1

where

1 2

=

1

1 2, 2, . . .

 1

, 2 and
z := (λ1 z1 , . . . , λm zm ).

Proof. Consider first F (z) = ζα (z). In view of (14) and (15) we have   α+ 21 α+ 1  −1  Mλ ζα = B ◦ Kc0 ◦ B ζα = B ◦ Kc0 hα = B 2λ hα = 2λ 2 ζα . But ζα (z) is homogeneous of degree αi in zi and hence λα ζα (z) = ζα (
z). Therefore the claim is true for the basis elements ζα of Fm and hence also for any F ∈ Fm .   For r ∈ Rm define the translation operator τr : L2 (Rm , dx) → L2 (Rm , dx) by   τr f (x) := f (x − r), and for s ∈ R \ {0} define the multiplication operator µs : L2 (Rm , dx) → L2 (Rm , dx) by (µs f ) (x) := sf (x).  α αl ∗ For s ∈ (R \ {0})m we define µs := m l=1 µsl . Since Zl and µs commute, it makes  m ∗ ∗ α α j sense to write (Z + µs ) := j =1 (Zj + µsj ) . Proposition 5.2. For any α ∈ N0m we have

 α Z ∗ α ◦ τr = τr ◦ Z ∗ + µr .

36

J. Hilgert, D. Mayer

Proof. For f ∈ C 1 (Rm ) and 1 ≤ j ≤ m we calculate

1 ∂ )f (x − r) Zj∗ ◦ τr f (x) = (xj − 2π ∂xj   1 ∂ = xj − rj f (x − r) − f (x − r) + rj f (x − r) 2π ∂xj

  = τr ◦ Zj∗ f (x) + τr ◦ µrj f (x)

= τr ◦ Zj∗ + µrj f (x). From this it follows immediately that Z ∗ α ◦ τr = τr ◦ (Z ∗ + µr )α .

 

For the Hermite function h0 (x) one now gets 

Z

∗α





∗

◦ τr h0 (x) = τr ◦ Z + µr

α

h0 (x) = τr

α    α l=0

l

l

µr Z ∗ (α−l) h0 (x),

where we used the notation   α   αm   α1   α l ∗ (α−l)  α1 αm l1 ∗ (αm −lm ) µr Z ··· µr1 · · · µlrmm Z1∗ (α1 −l1 ) · · · Zm = ··· . l l1 lm l=0

l1 =0

lm =0

hα−l we find For Z ∗ (α−l) h0 =:
Z

∗α

◦ τ r h0 =

α    α l=0

l

l µr τr
hα−l .

For s ∈ Rm denote by exps : Rm → R the function defined by exps (x) := es·x . Then one has Proposition 5.3. For α ∈ N0m and s ∈ Rm the following identities hold: s α Z ∗α ◦ mexps = mexps ◦ Z ∗ − , 2π

s α mexps ◦ Z ∗α = Z ∗ + ◦ mexps . 2π Proof. By definition of Zj∗ we obtain for smooth f  1 ∂  s·x e f (x) 2π ∂xj   1 s·x s·x s·x ∂ = xj e f (x) − sj e f (x) + e f (x) 2π ∂xj

sj s·x = es·x Zj∗ f (x) − e f (x) 2π

sj = mexps ◦ Zj∗ − f (x). 2π Iterating this calculation proves the first identity of the proposition. The second identity is proved in the same way.   (Zj∗ ◦ mexps f )(x) = xj es·x f (x) −

From this one derives

Transfer Operators and Dynamical Zeta Functions

37

Proposition 5.4. For s ∈ Rm and
hα = Z ∗ α h0 one has s2

mexps
hα = e 4π

α    α s α−k

k

k=0

τs
hk .

π

2π

h0 = h0 and hence we get Proof. For α = 0 we have


m 2 mexps h0 (x) = es·x 2 4 e−πx m

= 2 4 es·x−πx s 2 s2 m = 2 4 e−π (x− 2π ) + 4π s2

= e 4π τ

2

s 2π

h0 (x).

For general α ∈ N0m we then get using Proposition 5.3 and Proposition 5.2 hα = mexps ◦ Z ∗ α h0 mexps
s α = Z∗ + mexps h0 2π s2 s α = e 4π Z ∗ + ◦ τ s h0 2π 2π α  

2  s α s α−l ∗ l = e 4π Z ◦ τ s h0 2π l 2π l=0

α  

l  α s α−l τ s ◦ Z ∗ + µ s h0 2π 2π l 2π

s2

= e 4π

l=0

=e

l   α    α s α−l  l s l−k τ s ◦ Z ∗ k h0 2π l 2π k 2π

s2 4π

l=0

k=0

  α  l  s α−k τs
hk 2π l k 2π

s2

α

= e 4π

l

l=0 k=0

=e

α α      α l s α−k hk . τs
2π l k 2π

s2 4π

k=0 l=k

But

α     α l l=k

l

k

= 2α−k s2

mexps
hα = e 4π

α 

α  k=0

k

, since

2α−k

α i     αi li l i =k i

li

ki

= 2α i −k i

α  i

ki

, and therefore

  α   s2  α α s α−k s α−k hk = e 4π hk . τs
τs
2π 2π k 2π k π k=0

  Proposition 5.5. For r ∈ Rm and B : L2 (Rm , dx) → Fm the Bargmann transform one has

38

J. Hilgert, D. Mayer π 2

B ◦ τr = e− 2 r mexpπ r ◦ τr ◦ B, where we have denoted the function er·z also by expr . Proof. Using (16) we calculate   m B ◦ τr f (z) = 2 4

 f (x − r)e2πx·z−πx

2 − π z2 2

dx

Rm

=2

m 4



2 − π z2 2

f (y)e2π(y+r)·z−π(y+r)

dy

Rm m



f (y)e2πy·(z−r)−πy

= 24

Rm

=e

πr·z− π2 r 2

= e−

πr2 2

2 − π (z−r)2 2

π 2

eπr·z− 2 r dy





 τr ◦ Bf (z)

mexpπ r ◦ τr ◦ B f (z).

  Consider next the operator in Fm induced from the multiplication operator mexps in   L2 Rm , dx . One finds Proposition 5.6. For s ∈ Rm and F : Cm → C polynomial we have

s 2 s·z s . B ◦ mexps ◦ B −1 F (z) = e 8π e 2 F z + 2π Proof. For the functions
ζα (z) := zα = B h˜ α (z) one finds B ◦ mexps ◦ B −1
ζα = B ◦ mexps
hα   α s2  α s α−k = Be 4π τs
hk 2π k π k=0

=e

s2 4π

 α  s α−k α

k=0

=e

s2 8π

e

s·z 2

k

s2

s·z 2

2

α    α s α−k k=0

= e 8π +

π

s2

e− 8π mexp s ◦ τ



z+

k

π

z−

s 2π

◦ B
hk

s k 2π

s α . 2π

  Since the
ζα form a basis in the space Fm the claim of the proposition is true.

 

Remark 5.7. According to Proposition 5.6 we can view B ◦mexps ◦B −1 as an unbounded operator on Fm which is defined on a dense linear subspace, namely the space of polynomial functions.   For the following we need the densely defined unbounded operator Cs : Fm → Fm defined as

Transfer Operators and Dynamical Zeta Functions

Cs := B ◦ mcoshs ◦ B −1 =

39

1 B ◦ mexps ◦ B −1 + B ◦ mexp−s ◦ B −1 . 2

(18)

From Proposition 5.6 we deduce (Cs F )(z) =

s·z 1 s 2 s·z s s

e 8π e 2 F z + + e− 2 F z − 2 2π 2π

for polynomial F , so indeed Cs is densely defined. Composing this unbounded operator with the bounded operator Mλ of Proposition 5.1 we actually get a bounded operator on Fm as the following proposition shows. Proposition 5.8. For s ∈ Rm , λ = (λ1 , . . . , λm ) ∈ (0, 1)m , and polynomial F ∈ Fm one has      1 s2 s·z s·z  
s
s + e− 2 F
z − , Cs ◦ Mλ F (z) = λ 2 e 8π e 2 F
z + 2π 2π and hence Cs ◦ Mλ extends to a bounded operator Fm → Fm . Proof.

s·z  1 s 2 s·z  s s

C s ◦ Mλ F (z) = e 8π e 2 Mλ F z + + e− 2 (Mλ F ) z − . 2 2π 2π 1

But we have (Mλ F )(z) = 2λ 2 F (
z) and hence the claim follows. α −1

For α ∈ R∗m with R∗ = {r ∈ R : r = 0} define the map να : Fm   να F (z) := F (Az) = F (α1 z1 , . . . , αm zm ) , where α Fm



m



= F :C →C

entire ,

F 2α

:=

Cm

|F (z)|2 e−(π

  → Fm by

m

i=1 |αi zi |

(19)

2)

dz < ∞

and A is the diagonal matrix with diagonal entries α1 , . . . , αm . Then consider the induced operator α −1

να −1 ◦ Cs ◦ Mλ ◦ να : Fm

α −1

→ Fm

.

Inserting the expressions for να and Cs ◦ Mλ one gets  s· A−1 z  

1 s2 ( )
As 2 να −1 ◦ Cs ◦ Mλ ◦ να F (z) = λ e 8π e 2 F
z + 2π   s·(A−1 z)
As + e− 2 F
z − . 2π For 0 = β ∈ C choose next the parameters s = s 0 and α = α 0 with   −1 , 1 ≤ i ≤ m. s0,i = 2 πJi β and α0,i = 2π s0,i We then get

40

J. Hilgert, D. Mayer

m

 1 να −1 ◦ Cs 0 ◦ Mλ ◦ να0 F (z) = (λi exp βJi ) 2



0

i=1

m

m × eβ i=1 Ji zi F (
z+λ)+e−β i=1 Ji zi F (
z−λ) .  1 2 But this operator has up to the multiplicative factor m i=1 (λi exp βJi ) exactly the form of the Ruelle transfer operator of the Kac model for the parameters λ. α −1



−1 The operator να −1 ◦ Cs 0 ◦ Mλ ◦ να 0 is defined in the Hilbert space Fm0 with α0,i = 0

βJi π . All

eigenfunctions of the operator Lβ : B(D) → B(D) besides the ones belonging to the eigenvalue zero belong to this space. This can be seen as follows. From the functional equation ρf (z) = eβJ ·z f (
z + λ) + e−βJ ·z f (
z − λ) one concludes that for ρ = 0 any eigenfunction of Lβ is an entire function in z which m   can grow for z → ∞ at most like eC l=1 |zi | for some positive constant C. Such α functions, however, belong to any of the Fock spaces Fm . Hence the operators Lβ and 1 1 να −1 ◦ Cs 0 ◦ Mλ ◦ να 0 have the same spectra in this space. The eigenm i=1 (λi

exp βJi ) 2

0

functions with eigenvalue zero of the operator Lβ : B(D) → B(D) can be determined explicitly. They are given by the functions fn,α (z) = e

  − β m l=1

Jl 2 z 2λ2l l



m  

exp

l=1

(2nl + 1)π izl 2λl

αl

  with n ∈ N0m and α ∈ Zm such that α  = 1 mod 2 and hence do not belong to the α −1

space Fm0 . Summarizing we have shown Proposition 5.9. For β ∈ C, β = 0 the operators α −1

Fm0 and Cs 0 ◦ Mλ : Fm → Fm are conjugate.

m

l=1 (λl

1

α −1

exp βJl ) 2 Lβ : Fm0 →

 

This leads to α −1

α −1

Proposition 5.10. For β ∈ C, β = 0 the operators Lβ : Fm0 → Fm0 and m

1 1

2 l=1 (λl exp βJl )

mcoshs 0 ◦ Kc0 : L2 (Rm , dx) → L2 (Rm , dx)

are conjugate. Proof. Inserting the definitions of the operator Cs and Mλ we find Cs ◦ Mλ = B ◦ mcoshs ◦ B −1 ◦ B ◦ Kc0 ◦ B −1 = B ◦ mcoshs ◦ Kc0 ◦ B −1 , and hence mcoshs ◦ Kc0 is conjugate to Cs ◦ Mλ . Therefore f ∈ L2 (Rm , dx) is an eigenfunction of mcoshs ◦ Kc0 with eigenvalue 9 iff Bf is an eigenfunction of the operator 1

Cs ◦ Mλ for the same eigenvalue 9. For s = s 0 = 2(πβJ ) 2 , however, the operator

Transfer Operators and Dynamical Zeta Functions

Cs 0 ◦ Mλ : Fm → Fm is conjugate to Proposition 5.9.  

41

m

α −1

1

0 2 i=1 (λi exp βJi ) Lβ : Fm

α −1

→ Fm0 by

Hence one concludes that f ∈ L2 (Rm , dx) is an eigenfunction of the operator 1 m  l=1

(λl exp βJl )

1 2

mcoshs 0 ◦ Kc0



α −1 iff the function να −1 ◦ B f ∈ Fm0 is an eigenfunction of the operator Lβ for the 0 same eigenvalue. If therefore f (x) is an eigenfunction of the operator
β,c := G 0

1 m  l=1

(λl exp βJl )

1 2

mcoshs 0 ◦ Kc0

  in L2 Rm , dx , where the operator Kc0 has been defined in (13), then the corresponding   α −1 α −1 eigenfunction F z of the operator Lβ : Fm0 → Fm0 has the following explicit form:

F (z) = να −1 ◦ Bf (z) 0  2   1 1   2 2   m β π β = 24 f x exp 2πx · ◦ z − π x2 − ◦ z  dx J J π 2 π Rm   !    m m    m βJi β 2 2 4 xi =2 f x exp 2π Ji zi dx. zi − π x − π 2 i=1

Rm

i=1

α −1

α −1

On the other hand given an eigenfunction F = F (z) of the operator Lβ : Fm0 → Fm0 ,
β,c has the form the corresponding eigenfunction f = f (x) of the operator G 0 

  m π F α 0 ◦ z exp 2π x · z∗ − π x 2 − z∗2 f (x) = B −1 ◦ να 0 F (x) = 2 4 2 Cm

× exp(−π|z|2 )dz.

 π Inserting the explicit expression for α 0 = √1J , . . . , √1J β we therefore get f (x) = 2

m 4

!



F Cm

1

π z1 , . . . , βJ1

!

m

π zm βJm





π × exp 2πx · z∗ − πx 2 − (z∗ )2 − π |z|2 dz. 2
β,c depends on β holomorphically and hence deObviously the integral operator G 0 fines a holomorphic family of trace class operators in the Hilbert space L2 (Rm , dx). Its
β,c ) hence is an entire function in the entire β plane Fredholm determinant det(1 − zG 0 coinciding with the Fredholm determinant of the Ruelle operator Lβ . To relate finally

42

J. Hilgert, D. Mayer

the operator mcoshs 0 ◦ Kc0 and its eigenfunctions to those of the Kac-Gutzwiller operator Kβ with kernel Kβ (ξ, η) we use Proposition 5.11. For real β ≥ 0 consider the two operators mcoshs 0 ◦Kc0 and m√coshs ◦ 0 K ◦m√ acting on the space L2 (Rm , dx). Then the following two statements hold: c0

coshs 0

(i) If f ∈ L2 (Rm , dx) is an eigenfunction of the operator mcoshs 0 ◦ Kc0 with eigenvalue ρ = 0, then the function g := √ f is an eigenfunction of the operator coshs 0

m√coshs ◦ Kc0 ◦ m√coshs in L2 (Rm , dx) for the same eigenvalue. 0

0

(ii) Conversely, if g ∈ L2 (Rm , dx) is an eigenfunction of the operator m√coshs ◦ 0  Kc0 ◦ m√coshs with eigenvalue ρ = 0, then the function f = coshs 0 · g is an 0

eigenfunction of the operator mcoshs 0 ◦ Kc0 in L2 (Rm , dx) for the same eigenvalue. f coshs 0

Proof. (i) If f ∈ L2 (Rm , dx) is an eigenfunction of mcoshs 0 ◦Kc0 , then g := √

is

in L2 (Rm , dx) since coshs 0 ≥ 1 and a simple calculation shows that this function is an eigenfunction of the operator m√coshs ◦Kc0 ◦m√coshs for the same eigenvalue. 0

(ii) For φ ∈ L2 (Rm , coshs 0 (x)−1 dx) set  h(x) :=

Rm

0

√ √ ˜ K(2 πx, 2 π y)φ(y) dy.

Then (11) shows that there exist constants c, d > 0 such that |h(x)| ≤ ce−d|x| . In particular, it follows that coshs 0 h ∈ L2 (Rm , dx). If now g ∈ L2 (Rm , dx) is an eigenfunction of the operator m√coshs ◦Kc0 ◦m√coshs 0 0 for the nonzero eigenvalue ρ, then a simple calculation shows that the function f :=

 1 g coshs 0 ∈ L2 Rm , cosh dx is an eigenfunction of the operator mcoshs 0 ◦ Kc0 s 2

0

m

for the eigenvalue ρ = 0 and the above argument applied to φ = ρ(4π )− 2 f shows that f ∈ L2 (Rm , dx).  

This leads us to the main result of this section: α −1

α −1

Theorem 5.12. The Ruelle operator Lβ : Fm0 → Fm0 and the modified Kac-Gutzw
β : L2 (Rm , dξ ) → L2 (Rm , dξ ) with kernel iller operator G
β (ξ , η) := G

m  l=1

− 21

(λl exp βJl )

cosh

 m 


, η) βJl ξl K(ξ

l=1


, η) defined in (11) have the same spectrum. For positive β this spectrum cowith K(ξ incides also with the spectrum of the Kac-Gutzwiller operator Gβ defined in (10) on the Hilbert space L2 (R, dξ ). For nonvanishing eigenvalues the eigenfunctions F (z) and f (ξ ) of Lβ and Gβ can be related to each other as follows:

Transfer Operators and Dynamical Zeta Functions

43

1 1   2 π 2 π  f (ξ ) = F z1 , . . . , zm m √ βJ1 βJm cosh i=1 βJi ξi Cm   √ 1 π · exp πξ · z∗ − ξ 2 − z∗2 − π |z|2 dz, 4 2      m     √ m  4 cosh 2 πβ Jl xl f (2 π x) F (z) = (8π) 

1 2π

m



Rm

 · exp 2 πβ





1

4

l=1

m   l=1

 m β  2 Jl xl zl − πx − · Jl zl dx. 2 2

l=1

Proof. We have to consider the two cases β = 0 and β = 0. For β = 0 the spectra of the two operators are given by {2λα , α ∈ N0m } (see Remark 6.1 and formula (14)) and hence are identical. On the other hand we have seen already in Proposition 5.10 that for  − 21 β = 0 the operators m mcoshs 0 ◦ Kc0 and Lβ are conjugate via the l=1 (λl exp βJl ) map να −1 ◦ B. In fact, we have the commutative diagram 0



Kβ / L2 Rm dξ L2 Rm , dξ Rc0



    L2 Rm , a x dx m√coshs

0

  L2 Rm , dx /



1 m √cosh



  L2 Rm , dx B

Kc0

/

  L2 Rm , dx





Fm να 0

Rc0



/

Mc0

s0

mcoshs

  L2 Rm , dx /

0

B

Fm

/

Cs 0



Fm



α −1

Fm0

B



Lβ

√m

βJl l=1 λl e

/

να 0

α −1

Fm0

 √ c0 = 2 π (1, . . . , 1) and s 0 = 2 βπJ . This proves the first claim. Furthermore Proposition 5.11 shows that for β ≥ 0 the operators mcoshs 0 ◦ Kc0 and m√coshs ◦ Kc0 ◦ 0

44

J. Hilgert, D. Mayer

m√coshs have the same nonvanishing eigenvalues. But the last operator is conjugate to 0

the operator Kβ with kernel Kβ (ξ , η) through the map Rc0 . Hence if f ∈ L2 (Rm , dξ ) 1 2

is an eigenfunction of the integral operator Gβ , then (Rc0 f )(x) = c0 f (C 0 x) is an ei βJl )− 21 m√  √ genfunction of the integral operator m l=1 (λl e coshs 0 ◦ Kc0 ◦ m coshs 0 for the same eigenvalue. But then   1   2 coshs 0 (x) Rc0 f (x) = coshs 0 (x) c0 f (C 0 x) is an eigenfunction of the operator

l=1

eigenvalue. Therefore m





F (z) = 2 4

Rm

· exp 2π

m 

!

i=1

λl eβJl

· mcoshs 0 ◦ Kc0 again for the same

1 2

1   2 c0 f C 0 x

cosh s 0 · x



1

m 

 m βJi β 2 2 x i zi − π x − Ji zi dx π 2 i=1

α −1 0

α −1 0

is an eigenfunctionof the operator Lβ : Fm → Fm for yet again the same eigenvalue. √ √ Inserting c0 = 2 π, . . . , 2 π one therefore finds for F (z)      m      √  m  F (z) = (8π) 4 cosh 2 βπ Jl · x l f 2 π x 

Rm

l=1

 · exp 2 πβ

m   l=1

 m β 2 J l xl zl − π x − Jl zl dx. 2 2

l=1

α −1 0

On the other hand, given an eigenfunction F ∈ Fm of the operator Lβ we know that h(x) = (B −1 ◦να 0 ◦F )(x) is an eigenfunction of the operator m √λ1 exp βJ mcoshs 0 ◦Kc0 . Then by Proposition 5.11 for β ≥ 0 the function √

of the operator m

l=1 (λl

1 exp βJl )

1 2

l

l=1

1 h(x) coshs 0 (x)

l

is an eigenfunction

m√coshs ◦ Kc0 ◦ m√coshs , 0

and hence which is again conjugate to Gβ via Rc−1 0

0



 Rc−1 0

 √

1 coshs 0

·h

(ξ ) is an eigen-

function of Gβ . Inserting all the transforms involved we finally get for the corresponding eigenfunction f = f (ξ ) ∈ L2 (Rm , dξ ),   m ! !  1 π π 1 4 F z , . . . , z · f (ξ ) =  1 m  m 2π βJ1 βJm √ cosh βJl ξl Cm 

l=1

  2 √ 1 2 π ∗2 ∗   · exp dz. πξ · z − ξ − z − π z 4 2

 

Transfer Operators and Dynamical Zeta Functions

45

To discuss the zeros and poles of the Ruelle zeta function for the Kac-Baker models we need the following proposition.
β and Lβ have real spectrum. Proposition 5.13. For β real the operators G Proof. For β = 0 this follows from the explicit form of the eigenvalues. For β > 0 and
β with eigenvalue 9 consider the f ∈ L2 (Rm , dξ ) an eigenfunction of the operator G scalar product     f f
, Gβ f = 9 ,f , coshR 0 coshR 0  where R 0 = βJ . But if 

m 

Rm

then



m 

Rm l=1

1
, η) f (η) dη = 9f (ξ ), (λl exp βJl )− 2 coshR 0 (ξ ) K(ξ

l=1


, η) (λl exp βJl )− 2 coshR 0 (ξ ) K(ξ

f (ξ )

1

coshR 0 (ξ )

dξ = 9

f (η) coshR 0 (η)

,

f (ξ )

2 m
ξ ) and the function
, η) = K(η, since K(ξ coshR 0 (ξ ) belongs to the space L (R , dξ ) if
is real valued we can now calculate
β . Since G f is an eigenfunction of G



f
β f ,G coshR 0





=  =

On the other hand 4.1,   Rm

Rm

=2

f coshR 0 , f

f (ξ ) coshR 0 (ξ )

m  l=1

f (ξ ) Rm



coshR 0 (ξ ) Rm  f (ξ )

coshR 0 (ξ )   f = 9 ,f coshR 0   f =9 ,f . coshR 0 Rm Rm


β (ξ , η)f (η) dη dξ G
β (ξ , η) dξ f (η) dη G

= 0, since according to Mehlers formula in Proposition


β (ξ , η) f (η) dξ dη G − 21

(4π exp βJl )

 α∈Nm 0

  λ 



α

Rm

hα

ξ √

2 π



2  f (ξ ) dξ  > 0


β . But then it is clear that 9 must be identical to 9 and hence for f an eigenfunction of G
β , the function f with real. For β < 0 and f ∈ L2 (Rm , dξ ) an eigenfunction of G cosR 0  R 0 = −βJ is obviously in L2 (Rm , dξ ) and the same reasoning as in the case β > 0 applies.  

46

J. Hilgert, D. Mayer

6. The Ruelle Zeta Function for the Kac-Baker Model

 Recall from Proposition 2.4 that the Ruelle zeta function ζR (z, β) = exp ∞ n=1 with Zn (β) the partition function for the Kac-Baker model can be written as   (−1)|α|+1 det 1 − zλα Lβ ζR (z, β) =

zn n Zn (β)

α∈{0,1}m

with the operator Lβ : B(D) → B(D) given by   Lβ f (z) = eβJ ·z f (λ +
z) + e−βJ ·z f (−λ +
z), where
is the m × m-diagonal matrix with the λj , j = 1, . . . , m, as diagonal elements. Remark 6.1. For β = 0 one then finds for the spectrum σ (L0 ) of L0 ,   σ (L0 ) = 2λα | α = (α1 , . . . , αm ) ∈ N0m . In fact, by induction over |α| one finds a polynomial eigenfunction for each 2λα . In the 2 2 case m = 1 one obtains 1, z, z2 + λ2λ−1 , z3 + λ3λ 2 −1 z and so on. On the other hand, one can show that all eigenfunctions are polynomial, since the derivative of an eigenfunction for the eigenvalue ρ is again an eigenfunction for the eigenvalue ρλ−1 (in the case m = 1) so that infinitely non-vanishing derivatives of an eigenfunction would contradict the compactness of the operator. For m > 1 the argument is similar.   Hence the Ruelle function ζR (z, β) for β = 0 has the form

(−1)|α|+1   . 1 − 2zλα+β ζR (z, 0) = α∈{0,1}m β∈Nm 0

By induction on m one shows ζR (z, 0) =

1 . 1 − 2z

Indeed for m = 1 one has

 ∞  n+1 1 det (1 − zλL0 ) n=0 1 − 2zλ = ζR (z, 0) = . = ∞ n det (1 − zL0 ) 1 − 2z n=0 (1 − 2zλ )

For m = n + 1, on the other hand, one calculates 1+|α |+αn+1

    αn+1 +βn+1 α+β (−1) λ 1 − 2zλn+1 ζR (z, 0) = αn+1 ∈{0,1} α∈{0,1}n βn+1 ∈N0 β∈Nn0





=



α∈{0,1}n βn+1 ∈N0 β∈Nn0

·







α∈{0,1}n βn+1 ∈N0 β∈Nn0

=





βn+1 ∈N0

=

1 . 1 − 2z

1 β

n+1 1 − 2zλn+1

β

n+1 α+β λ 1 − 2zλn+1



(−1)1+|α|





1+|α |−1

βn+1 +1 α+β (−1) 1 − 2zλn+1 λ

 βn+1 ∈N0



β

n+1 1 − 2zλn+1

+1



Transfer Operators and Dynamical Zeta Functions

47

This result does not come as a surprise since for β = 0 the Ruelle zeta function for the Kac-Baker model is just the Artin-Mazur zeta function for the full subshift over two symbols determined by Bowen and Lanford in [BoLa75]. To determine the zeros and poles of the Ruelle function ζR (β) := ζR (1, β)  on the real  β-axis one has to investigate the zeros of the Fredholm determinants det 1 − λα Lβ . Obviously this function takes the special value −1 at the point β = 0. We know already from our discussion of the Kac-Gutzwiller operator Gβ that all its eigenvalues and hence also those of the Ruelle operator Lβ are real for real β and nonnegative for β  0. The poles and zeros of the Ruelle zeta function ζR (1, β) can be located only at those values    −1 , α ∈ {0, 1}m belongs to the spectrum σ Lβ of of β where one of the numbers λα Lβ . For β = 0 we have seen that σ (L0 ) = 2λβ with β ∈ N0m . But 

λα

−1

= 2λβ

⇔

1 = λα+β 2

can be true only for finitely many β ∈ N0m since α ∈ {0, 1}m can take only finitely many values. Furthermore, for infinitely many β ∈ N0m the eigenvalue 2λβ is strictly smaller than (λα )−1 for all α ∈ {0, 1}m . Hence, if we can show that infinitely many eigenvalues 9(β) of Lβ tend to +∞ for |β| → ∞, then the Ruelle zeta function will have infinitely many “nontrivial” zeros and poles on the real β-axis at least for generic values of λ for which possible cancellations of zeros in the quotients of the different Fredholm determinants do not occur. To derive the asymptotic behavior of the eigenvalues of the transfer operator for |β| → ∞ we remark first that the eigenspace of any eigenvalue 9 of Lβ has a basis consisting of eigenfunctions which are also eigenfunctions of the operator P : B(D) → B(D) defined as Pf (z) = f (−z). We call the eigenfunctions with Pf = f even and those with Pf = −f odd. Consider then the two operators L± β : B(D) → B(D) defined via βJ ·z L+ f (λ +
z) + e−βJ ·z f (λ −
z), β f (z) = e

respectively, βJ ·z L− f (λ +
z) − e−βJ ·z f (λ −
z). β f (z) = e − The eigenfunctions of L+ β and Lβ are just the even, respectively odd, eigenfunctions of Lβ . We call the corresponding eigenvalues even, respectively odd. Since Lβ and P commute one therefore finds for the spectrum σ (Lβ ) of Lβ : − σ (Lβ ) = σ (L+ β ) ∪ σ (Lβ ).

Extending a result of B. Moritz (see [Mo89]) for m = 1 to the general case m ∈ N we find for the restricted range 0
λi < 21 for 1
i
m of the parameters λ and arbitrary N  1:

48

J. Hilgert, D. Mayer

Theorem 6.2. (i) For β → ±∞ the N leading even eigenvalues ρα of Lβ behave like

λα (±1)|α | exp βJ · (1 −
)−1 λ . (ii) For β → ±∞ the N leading odd eigenvalues ρα of Lβ behave like

λα (±1)|α |+1 exp βJ · (1 −
)−1 λ . Proof. Consider first the even eigenvalues and their asymptotic behavior for β → +∞. Then L+ β g(z) = exp(βJ · z)g(λ +
z) + exp(−βJ · z)g(λ −
z) = 9 g(z). Writing g(z) = exp(βJ · (1 −
)−1 z)u(z) we get for u the equation

u(λ +
z) + exp −2βJ · (1 −
)−1
z exp(−2βJ · z)u(λ −
z)

= 9 exp −βJ · (1 −
)−1 λ u(z) and hence

where

  u(λ +
z) + exp − 2βJ · (1 −
)−1 z u(λ −
z) = 9u(z), ˜

9˜ = 9 exp −βJ · (1 −
)−1 λ .

Replacing z by λ + z and introducing the function h(z) := u(λ + z) one arrives at the equation

9˜ h(z) = h(
λ +
z) + exp −2βJ · (1 −
)−1 λ

× exp −2βJ · (1 −
)−1 z h(−
λ −
z).     λ2i +
: B D
Define an operator Tβ;λ R → B D
R for D
R the polydisc with Ri > 1−λi for 1
i
m via

+ Tβ;λ h(z) := h(
λ +
z) + exp −2βJ · (1 −
)−1 λ

× exp −2βJ · (1 −
)−1 z h(−
λ −
z).  + Then Tβ,λ is nuclear and its eigenvalues are just the numbers 9˜ = 9 exp − βJ ·      (1 −
)−1 λ , where 9 is an eigenvalue of the operator L+ R → B D
R β . If Tλ : B D
denotes then the composition operator Tλ h(z) = h(
λ +
z), one finds for λ = (λ1 , . . . , λm ) with 0 < λi <

1 2

for all 1 ≤ i ≤ m:

+ lim Tβ,λ − Tλ  = 0,

β→+∞


i such that for all 1
i
m, since for these values of λi one can find R

Transfer Operators and Dynamical Zeta Functions


i > λi > R

49

λ2i . 1 − λi

But the eigenvalues of the operator Tλ can be determined explicitly: they are given by the numbers λα with α ∈ N0m . From this the asymptotic behavior of the leading eigenvalues 9α of L+ β and hence the asymptotic behavior of the leading even eigenvalues of Lβ follows immediately. The proof of the behavior of the odd eigenvalues for β → +∞ follows the same line + of arguments applied to the operator L− β instead of Lβ . For the asymptotic behavior of the even eigenvalues for β → −∞ consider the operator
+ g(z) = exp(−βJ · z)g(λ +
z) + exp(βJ · z)g(λ −
z) L β and the behavior of its eigenvalues for β → +∞. In this case we write an eigenfunction g as

g(z) = exp βJ · (1 +
)−1 z u(z) and get for u the equation

exp(−2βJ · z) exp 2βJ · (1 +
)−1
z u(λ +
z) + u(λ −
z) = 9u(z), ˜   where 9˜ = 9 exp − βJ · (1 +
)−1 λ . Introducing the function h(z) := u(λ +
z) we finally arrive at

9˜ h(z) = h(−
λ −
z) + exp −2βJ · (1 +
)−1 λ

× exp −2βJ · (1 +
)−1 z h(
λ +
z). − Hence 9˜ is an eigenvalue of the operator Tβ;λ : B(D
R ) → B(D
R ) with

− Tβ;λ h(z) = h(−
λ −
z) + exp −2βJ · (1 +
)−1 λ

× exp −2βJ · (1 +
)−1 z h(
λ +
z).


i with λi > R
i > For 0 < λi < 21 , 1
i
m we can find again R " − " − Tλ " = 0, lim "Tβ;λ

λ2i 1−λi

and hence

β→+∞

where Tλ h(z) = h(−
λ −
z), is nuclear of order zero on the Banach space B D
R . The spectrum of Tλ , however, is given by the numbers (−λ)α with α ∈ N0m and hence the leading even eigenvalues 9α   of the operator Lβ behave for β → −∞ like (−1)|α | λα exp − βJ · (1 +
)−1 λ . In exactly the same way one shows  that the leading odd  eigenvalues 9α of Lβ behave for β → −∞ like −(−1)|α| λα exp − βJ · (1 +
)−1 λ .  

50

J. Hilgert, D. Mayer

Theorem 6.2 shows that both for β → ±∞ infinitely many eigenvalues of Lβ tend to +∞. Therefore the determinants det(1 − λα Lβ ) with α ∈ {0, 1}m have infinitely many zeros in the real variable β. Therefore for generic λ with 0 < λi < 21 for 1
i
m the Ruelle zeta function has infinitely many poles and zeros on the real axis. For special values of the parameters λ some eigenfunctions and their eigenvalues for the Ruelle operator Lβ are explicitly known. If λ = λ0 with λ0,i = λ0 = 21 for all 1
i
m, then the identity   

1 1 βJ ·z±βJ ·1 (z ± 1) e = − e−βJ ·z∓βJ ·1 sinh 2βJ · 2 2 and the calculation 1

1

(z + 1) + e−βJ ·z sinh 2βJ · (z − 1) eβJ ·z sinh 2βJ · 2 2

1

1 βJ ·z βJ ·z+βJ ·1 −βJ ·z−βJ ·1 βJ ·z βJ ·z−βJ ·1 = e e + e e −e − e−βJ ·z+βJ ·1 2 2

1

1 2βJ ·z+βJ ·1 −βJ ·1 −βJ ·1 = −e − e−2βJ ·z+βJ ·1 + e e 2 2  1 βJ ·1 2βJ ·z −2βJ ·z e = e −e 2 = eβJ ·1 sinh(2βJ · z) shows that the functions f1,n (z) = Pn (z) sinh(2βJ · z),

(20)

with Pn (z) a polynomial homogeneous of degree n in all the variables zl which is invariant under all translations of the form zl → zl + c for real c, are indeed eigenfunctions of Lβ with eigenvalue 91,n = eβ

m

l=1 Jl

λn0 := 91 λn0 .

The dimension of the space B(D)n of eigenfunctions defined by (20) for fixed n is the dimension of the above space of polynomials and will be calculated in the following proposition. Proposition 6.3. Let K be R or C. The dimension of the space Vm,n,c of homogeneous polynomials f (z1 , . . . , zm ) ∈ K[z1 , . . . , zm ] of degree n in m variables which are invariant under a fixed non-zero translation z → z + c with c ∈ Kn is    m+n−2   for m ≥ 2  n  1 for m = 1 and n = 0    0 otherwise. Proof. Let f ∈ Vm,n,c . For fixed z consider first the polynomial t → f (z − tc) − f (z). It is of degree less than or equal to n and has infinitely many zeros (for t ∈ Z), so it is zero. Thus we have f (z + Kc) = f (z)

∀z ∈ Km .

Transfer Operators and Dynamical Zeta Functions

51

Changing coordinates we may now assume that c = (1, 0, . . . , 0), i.e. f depends only on the variables z2 , . . . , zm . Thus for m = 1 only the constant polynomials satisfy the required invariance, whereas for m ≥ 2 the dimension of Vm,n,c is equal to the dimension  of the space  of homogeneous polynomials of degree n in m − 1 variables and that m+n−2 is . In fact, if d(m, n) denotes the dimension of the space of homogeneous n polynomials of degree n in m variables we obtain the following recursion formula: d(m, n) =

n 

d(m − 1, n − j ).

j =0

Twice induction (first on n, then on m) yields    n   m+j −2 m+n−1 = j n j =0

and d(m, n) =

n  j =0

   n   m+j −2 m+n−1 = . d(m − 1, j ) = j n j =0

  The Ruelle ζR (β) := ζR (1, β) for our special choice of the parameter  zeta function  λ = λ0 = 21 , . . . , 21 has the form ζR (β) =

  m (−1)k+1 det 1 − λk0 Lβ ( k )

m  k=0

as one easily checks. According to Proposition 6.3 the contribution of the eigenspaces B(D)n for the eigenvalues 91,n to the Ruelle function ζR (β) is given by (−1)k+1 ∞  m   m m+r−2  k+r ( k )( r ) 1 − λ 0 91 (21) k=0 r=0

with k + r = n. Remark 6.4. Note that for any β the B(D)n represent the entire eigenspaces for the eigenvalues 91,n . This is true for β = 0 by Remark 6.1 and Proposition 6.3. Indeed the   eigenspace of the eigenvalue 9 = 2λr0 of the operator L0 has dimension m+r−1 m−1 as one can easily check. On the other hand we have     r  m+r −1 m+k−2 = , m−1 k m+k−2

k=0

where is just the dimension of the eigenspace of the eigenvalue 9j,k (β) = k j k 9j (β)λ0 . But for β = 0 the eigenvalues 9j (β) are given by 2λ0 . Since the eigenvalue 91,k (β) is holomorphic in the entire β-plane the dimension of   its eigenspace does not depend on β (see [Kt66], p.68) and is given by m+k−2   m−2 . We will check below that (21) can be reduced to the expression

52

J. Hilgert, D. Mayer

1 − λ 0 91 . 1 − 91

(22)

To see this one needs the following result on binomial coefficients. Proposition 6.5. For all m  2, all r  0 and all l
m − 1 the following identity holds:   2!  m m − 2k + r m

k=0

l

2k

m−1   2 !  m m − 2k + r − 1 = , 2k + 1 l

k=0

where r! is the largest integer less than or equal to r. Proof. The proof is by induction on m, r and l. For m = 2,  r  0rand  l ∈ {0, 1} one finds for the left-respectively right-hand, side: LH S = r+2 + l l , respectively r+1 RH S = 2 l , and hence the two sides of the identity coincide for l ∈ {0, 1}. Next we show that it suffices to show the identity for r ≥ 0. In fact,  assume  nit holds  n for  r. We show that then it holds for r + 1 and hence for all r. Since n+1 = + s s s−1 we find m [m         2! 2]   m m − 2k + r + 1 m m − 2k + r m − 2k + r = + . 2k l 2k l l−1

k=0

k=0

But the right-hand side of this equation is equal to m−1      2 !  m m − 2k + r − 1 m − 2k + r − 1 + 2k + 1 l l−1

k=0

  2 !  m m − 2k + r = , 2k + 1 l m−1

k=0

which proves our claim. Assume next the identity holds up to some m and all l
m − 1. Then we show that the identity of Proposition 6.5 holds for m + 1 and 0
l
m. Indeed for the LHS of the identity one finds m+1   2 !  m + 1 m − 2k + 1

 2 !   m m+1

=



 

m + 2k 2k − 1

k=0

=

l

2k

k=0

m   2!  m m − 2k + 1

k=0

2k

l

 m − 2k + 1 l

m+1   2 !  m m − 2k + 1 + 2k − 1 l

k=1

Transfer Operators and Dynamical Zeta Functions

53

m−1 m−1     2 ! 2 !   m m − 2k m m − 2k − 1 + = 2k + 1 l 2k + 1 l

k=0

k=0

m   2!   m m − 2k   m m − 2k = + . 2k + 1 l 2k l m−1 2 !

k=0

k=0

But this is just   2!  m + 1 m − 2k m

2k + 1

k=0

l

and hence the two sides of the identity coincide. The proposition therefore holds for m + 1, 0
l
m − 1. We have still to show that it holds also for l = m, that means   2 !  m + 1 m + 1 − 2k m+1

k=0

m

2k

  2!  m + 1 m − 2k m

=

2k + 1

k=0

m

.

  But in this case we get LH S = m+1 m respectively RH S = m + 1 for this last equation, and hence the two sides agree. This proves the proposition.   The contribution (21) of the eigenspace B(D)n can be rewritten first as 

1 − λn0 91

m−1 ! m 2 k=0 2k+1



(

m! m 2 k=0 2k



)(m+n−2k−3 n−2k−1 )−

( )(m+n−2k−2 n−2k )

or m−1 !

  2 1 − λn0 91 k=0

m! m 2 k=0 2k



m (2k+1 )(m+n−2k−3 )− m−2

( )(m+n−2k−2 ). m−2

1 For the cases n = 0 and n = 1 one finds the factors 1−9 and 1 − λ0 91 . For n  2, 1 however, Proposition 6.5 shows that all the other eigenvalues 91,n contribute the trivial factor 1 to the zeta function. This proves

 ∞ m  

1 − λk+r 0 91

k=0 r=0

(m)(m+r−2) k

r

(−1)k+1 =

1 − λ 0 91 , 1 − 91

i.e. the reduction of (21) to (22). This factor leads to “trivial” zeros βn of the Ruelle zeta  function with βn = log 2+2πin , where J = m l=1 Jl . This conclusion can be drawn only J if there are no cancellations with other eigenvalues, which at the moment we cannot exclude. For the Kac-Baker model with this special choice of parameters λ = λ0 there exists another Ruelle transfer operator which in a certain sense is simpler than Lβ . These parameters namely describe a Kac-Baker model with an interaction  consisting of one exponentially decreasing term only and whose strength is just J = m l=1 Jl . The Ruelle transfer operator for this model is    
β g(z) = eβJ z g 1 + 1 z + e−βJ z g − 1 + 1 z , L 2 2 2 2

54

J. Hilgert, D. Mayer

where g is now holomorphic in a disc D in the complex plane C. The Ruelle zeta function in terms of the Fredholm determinants of this operator has the form   'β det 1 − 21 L   . ζR (β) = 'β det 1 − L It is easy to see that the function g1 (z) = sinh (2βJ z) is an eigenfunction of the op
β with eigenvalue 91 = exp(βJ ). This eigenvalue leads exactly to the zero of erator L the Ruelle function we discussed above. Indeed any eigenfunction f = f (z) of the
β through Ruelle operator Lβ determines an eigenfunction g = g(z) of the operator L g(z) := f (z, . . . , z) with the same eigenvalue as long as this function does not vanish
β identically. On the other hand given an eigenfunction gk = gk (z) of the operator L with eigenvalue 9k we know that gk is holomorphic in the entire β-plane. Consider then the function fk = fk (z) defined as   m l=1 Jl zl fk (z) := gk J m with J = l=1 Jl . It is entire in Cm and since m     l=1 Jl zl fk ± λ0 +
0 z = gk ±λ0 + λ0 , J it is easy to see that the function fk is an eigenfunction of the operator Lβ with eigenvalue 9k . But with fk also the functions fk,n (z) := Pn (z)fk (z), n = 0, 1, ... with Pn a polynomial homogeneous of degree n in m variables and invariant under translations zl → zl + c for c ∈ R are eigenfunctions of Lβ with eigenvalue 9k,n = 9k λn0 , as we have seen already for the eigenfunction f1,n . Their degree of degeneracy is again given   by m+n−2 . The eigenfunctions g1 (z) and f1,n (z) are only special cases of this quite n general connection between the eigenfunctions and eigenvalues of the two operators Lβ
β . An argument similar to the case k = 1 shows that among all the eigenvalues and L 9k,n = 9k λn0 only 9k,n with n ∈ {0, 1} give nontrivial contributions to the Ruelle zeta function, which coincide exactly with the contributions of the eigenvalues 9k of the
β to the Ruelle zeta function for the model with λ = λ0 . operator L Our discussion shows that there exist infinitely many trivial zeros of the Ruelle zeta function for the Kac model at least for this special parameter λ = λ0 as long as there are not an infinite number of cancellations occurring among the eigenvalues, which one would certainly not expect. The structure of the zeros of the dynamical zeta functions of this family of Kac-Baker models of statistical mechanics hence seems to be very similar to the one well known for arithmetic zeta functions like the Riemann function. It would be interesting to determine their nontrivial zeros and poles and their distribution at least numerically.
β 7. Matrix Elements of the Kac-Gutzwiller Operator G For the numerical determination of the zeros of the Ruelle zeta function ζR (β) =
β seems to be best suited. This was ζR (1, β) the modified Kac-Gutzwiller operator G realized already in the case m = 1 by Gutzwiller in (see [Gu82, §7]). For a numerical study of the eigenvalues and eigenfunctions of a closely related Ruelle operator see

Transfer Operators and Dynamical Zeta Functions

55


α,δ of the operator G
β [Thr94]. It turns out that also for general m the matrix elements G 2 m can be determined explicitly in the basis of L (R , dξ ) given by the Hermite functions   hα . We start with the identity (see [Gu82, (41)])       m m

1 1 2 γi (ξi − ηi )2 2 ξi + ηi tanh + exp − 1 4 2 sinh γi Rm Rm i=1 (4π sinh γi ) 2 i=1   η2 ξ2 ρ2 σ2 · exp ±R · η − − +ρ·ξ − − + σ · η dξ dη 4 2 4 2   m m R2 1 = (2π ) 2 exp + ρ ·
σ ± R · σ ± R ·
ρ − γi , 2 2 i=1

where as before
denotes the diagonal matrix with entries
i,j = λi δi,j . Adding the two identities for ±R one gets       m m

1 2 γi 1 (ξi − ηi )2 2 2 ξi + ηi tanh + exp − 1 4 2 sinh γi Rm Rm i=1 (4π sinh γi ) 2 i=1   η2 ξ2 ρ2 σ2 · cosh(R · η) exp − − +ρ·ξ − − + σ · η dξ dη 4 2 4 2   m m R2 1 + ρ ·
σ − = 2 (2π ) 2 cosh(R · σ + R ·
ρ) exp γi . 2 2 i=1

But (see [Gu82, (37)])  exp −

ξ2 4

−

ρ2 2

 −ρ·ξ

m

= (2π ) 4

 ρα  hα (ξ ), α! α∈Nm 0

and hence  

2

m 

1 1

(4π sinh γi ) 2      m  1  2 γi (ξi − ηi )2 2 ξi + ηi tanh + × exp − 4 2 sinh γi Rm Rm m α∈Nm 0 δ∈N0

i=1

i=1

ρα σ δ · cosh(R · η)  √ hα (ξ ) hδ (η) dξ dη α! δ!   m R2 1 = 2 exp γi cosh(R · (σ +
ρ)). + ρ ·
σ − 2 2 i=1 √ √ √
β (ξ , η) of the Comparing this for R = R0 := β( J1 , . . . , Jm ) with the kernel G modified Kac-Gutzwiller operator in Theorem 5.12 we hence find     ρα σ δ
β (ξ , η) hα (ξ ) hδ (η)  √ dξ dη G m Rm α! δ! m m R α∈N0 δ∈N0

= 2 exp(ρ ·
σ ) cosh(R 0 · (σ +
ρ)).

56

J. Hilgert, D. Mayer


α,δ := (G
β hα , hδ ) this reads In terms of the matrix elements G ρα σ δ
α,δ  √ G = 2 exp(ρ ·
σ ) cosh(R 0 · (σ +
ρ)). α! δ! α∈Nm δ∈Nm   0

0


α,δ = G
α,δ (R , λ) one first solves the To determine from this the matrix elements G 0 problem ρα σ δ Aα,δ  √ = exp(ρ ·
σ ) exp(R 0 · (σ +
ρ)). α! δ! α∈Nm δ∈Nm   0

0

Obviously the right-hand side factorizes into a product of exponentials depending only on the i th coordinates of the different variables. Hence it suffices to solve the equation   αi ∈N0 βi ∈N0

β

  ρ αi σ i Aαi ,βi √ i √i = exp (ρi λi σi ) exp R0,i (σi + λi ρi ) . α i ! βi !

Expanding the exponentials on the right-hand side in σi and ρi leads to M i −µi  αi µi Aαi ,βi (R0,i , λi ) = αi !βi ! λi R0,i ki =1

2ki R0,i

(Mi − µi − ki )! ki ! (µi + ki )!

,

where Mi = max {αi , βi } and µi = |αi − βi |. It is not too difficult to see that M i −µi  αi !βi ! ki =0

=√

2ki R0,i

(Mi − µi − ki )! ki ! (µi + ki )!

 1 Mi !  2 , B µi − Mi , µi + 1; −R0,i αi !βi ! µi !

where B(·, ·; ·) denotes the confluent hypergeometric function. Therefore the matrix  elements Aα,δ = m A l=1 αl ,δl are given by Aα,δ (R 0 , λ) = 

1

µ M!

λα R 0

B µ − M, µ + 1; −R 20 ,

µ! α!δ!

 2 where B µ − M, µ + 1; −R 20 := m B µ − M , µ + 1; −R i i i i=1 0,i .
Finally we then find for the matrix elements Gα,δ of the modified Kac-Gutzwiller
β , operator G

 
α,δ (R , λ) =  1 λα R µ (1 + (−1)|µ| ) M! B µ − M, µ + 1; −R 2 . G 0 0 0 µ! α!δ!
α,δ = 0 only iff |α + δ| = 0 mod 2 which generalizes the result for This shows that G  
β hα , hδ = 0 if |α| mod 2 = |δ| mod 2 m = 1 by Gutzwiller to arbitrary m. Since G also for general m, the Hilbert space L2 (Rm , dξ ) can be decomposed into two subspac
β spanned by the Hermite functions {hα } with |α| = es invariant under the operator G

Transfer Operators and Dynamical Zeta Functions

57


β 0 mod 2, respectively |α| = 1 mod 2. Obviously this property of the operator G corresponds to the fact that the Ruelle operator Lβ leaves invariant the subspaces of the Banach space B(D) spanned by the functions F = F (z) which are even, respectively odd, under the transformation z → −z. This follows from the fact that the SegalBargmann transform B maps the Hermite functions hα to the functions ζα in (17) which under the above transformation z → −z have parity |α| mod 2 as one checks easily.
β to calculate
β in terms of the matrix G One can use the representation of the operator G its traces. For instance one finds  
β =
α,α = 2 trace G G λα B(−α, 1; −βJ ). α∈Nm 0

α∈Nm 0

Because the confluent hypergeometric function B(−n, 1; x) is identical to the Laguerre polynomial Ln (x) = L0n (x) we get by using the generating function for these polynomials the result   m  βJi λi 2
trace Gβ = m , exp 1 − λi i=1 (1 − λi ) i=1

which coincides with trace Lβ . Acknowledgements. This work has been supported by the Deutsche Forschungsgemeinschaft through the DFG Forschergruppe “Zetafunktionen und lokal symmetrische R¨aume”. D.M. thanks the IHES in Buressur-Yvette for partial financial support and the kind hospitality extended to him during the preparation of the paper.

References [ArMa65] Artin, M., Mazur, B.: On periodic points. Ann. Math. 81, 82–99 (1965) [AtBo67] Atiyah, M., Bott, R.: A Lefschetz fixed point formula for elliptic complexes I. Ann. Math. 86, 374–407 (1967) [Ba00] Baladi, V.: Positive Transfer Operators and Decay of Correlations. Singapore: World Scientific, 2000 [Ba61] Baker, G.: One dimensional order-disorder model which approaches a second order phase transition. Phys. Rev. 122, 1477–1484 (1961) [Be86] Berry, M.: Riemann’s zeta function, a model of quantum chaos. Lecture Notes in Physics, Vol. 263, Berlin: Springer Verlag, 1986 [BoLa75] Bowen, R., Lanford, O.E.: Zeta functions of restrictions of the shift transformation. In: Global Analysis, Proc. Symp. Pure Math. Vol. 14, Providence, RI: Am. Math. Soc., 1975 [Co96] Connes, A.: Formule de trace en geometrie non commutative et hypothese de Riemann. C.R. Acad. Sci. Paris Ser I 323, 1231–1236 (1996) [Cr46] Cram´er, H.: Mathematical Methods in Statistics. Princetion, NJ: Princeton University Press, 1946 [De99] Deninger, C.: On dynamical systems and their possible significance for arithmetic geometry. Progr. Math. 171, 29–87 (1999) [Dw60] Dwork, B.: On the rationality of the zeta function of an algebraic variety. Am. J. Math. 82, 631–648 (1960) [Fo89] Folland, G.B.: Harmonic Analysis in Phase Space. Princeton, NJ: Princeton University Press, 1989 ´ [Gr55] Grothendieck, A.: Produits Tensoriels Topologiques et Espaces Nucl´eaires. Mem. Am. Math. Soc. 16, (1955) [Gu82] Gutzwiller, M.: The quantization of a classically ergodic system. Physica 5D, 183–207 (1982) [HiMa01] Hilgert, J., Mayer, D.: The dynamical zeta function and transfer operators for the Kac-Baker model. MPIM-Bonn preprint 2001–86 [Ka59] Kac, M.: On the partition function of a one-dimensional lattice gas. Phys. Fluids 2, 8–12 (1959)

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Kac, M.: The mathematical mechanism of phase transitions. In: Brandeis University Summer Institute in Theoretical Physics, Vol. 1, H. Chretien et al. eds. New York: Gordon & Breach, 1966, pp. 245–305 [Kt66] Kato, T.: Perturbation Theory for Linear Operators. Berlin: Springer Verlag, 1966 [Ma80] Mayer, D.: The Ruelle-Araki transfer operator in classical statistical mechanics. Lecture Notes in Physics 123(1), Berlin: Springer Verlag, 1980 [Ma91] Mayer, D.: The thermodynamic formalism approach to Selberg’s zeta function for P SL(2, Z). Bull. Am. Math. Soc. 25, 55–60 (1991) [Mo89] Moritz, B.: Die Transferoperator–Methode in der Behandlung des Kacschen Spinmodells. Diplomarbeit, RWTH Aachen, 1989 [Ro86] Robba, P.: Une introduction naive aux cohomologies de Dwork. Mem. Soc. Math. France 23, 61–105 (1986) [Ru68] Ruelle, D.: Statistical mechanics of a one dimensional lattice gas. Commun. Math. Phys. 9, 267–278 (1968) [Ru78] Ruelle, D.: Thermodynamic Formalism. London: Addison-Wesley, 1978 [Ru92] Ruelle, D.: Dynamical zeta functions: Where do they come from and what are they good for? In: Proc. Internat. Congress of Math. Phys. X (Leipzig 1991), Berlin: Springer Verlag, 1992, pp. 43–51 [Ru94] Ruelle, D.: Dynamical Zeta Functions for Piecewise Monotone Maps of the Interval. Providence, RI: Am. Math. Soc., 1994 [Thr94] Thron, C.: Taylor series expansions for eigenvalues and eigenfunctions of parametrized composition operators. J. Math. Phys. 35, 2024–2035 (1994) [ViMa77] Vishwanathan, K., Mayer, D.: Statistical mechanics of one-dimensional Ising and Potts models with exponential interactions. Physica 89A, 97–112 (1977) [We49] Weil, A.: Numbers of solutions of equations in finite fields. Bull. Am. Math. Soc. 55, 497–508 (1949) [Zi00] Zinsmeister, M.: Thermodynamic Formalism and Holomorphic Dynamical Systems. Providence, RI: Am. Math. Soc., 2000 Communicated by P. Sarnak

Commun. Math. Phys. 232, 59–81 (2002) Digital Object Identifier (DOI) 10.1007/s00220-002-0740-1

Communications in

Mathematical Physics

Self-Attracting Poisson Clouds in an Expanding Universe Jean Bertoin Laboratoire de Probabilit´es et Mod`eles Al´eatoires, et Institut universitaire de France, Universit´e Pierre et Marie Curie, et C.N.R.S. UMR 7599, 175, rue du Chevaleret, 75013 Paris, France. E-mail: [email protected] Received: 15 February 2002 / Accepted: 8 July 2002 Published online: 7 November 2002 – © Springer-Verlag 2002

Abstract: We consider the following elementary model for clustering by ballistic aggregation in an expanding universe. At the initial time, there is a doubly infinite sequence of particles lying in a one-dimensional universe that is expanding at constant rate. We suppose that each particle p attracts points at a certain rate a(p)/2 depending only on p, and when two particles, say p and q, collide by the effect of attraction, they merge as a single particle p ∗ q. The main purpose of this work is to point at the following remarkable property of Poisson clouds in these dynamics. Under certain technical conditions, if at the initial time the system is distributed according to a spatially stationary Poisson cloud with intensity µ0 , then at any time t > 0, the system will again have a Poissonian distribution, now with intensity µt , where the family (µt , t ≥ 0) solves a generalization of Smoluchowski’s coagulation equation.

1. Introduction There are many models in physics, such as growth by ballistic deposition, diffusion limited aggregation . . . , which involve aggregation of particles upon collision. Perhaps one of the best-known in astrophysics is the so-called free sticky particle system which was used by Zeldovich and others (see e.g. [17, 12, 19] and the references therein) to investigate the formation of large structures in the universe. Roughly, Zeldovich considered particles that move at constant velocity until they collide with some other particles, and collisions are completely inelastic with conservation of mass and momentum. See E et al. [10] and Brenier and Grenier [8] for the analysis of this system. Free sticky particles also bear natural relations with the Burgers equation; see Gurbatov et al. [12], Woyczy´nski [20] and references therein for applications. Of course, such models are mostly interesting in low dimensions as otherwise collisions are unlikely.

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Here, we consider dynamics generalizing systems of free sticky particles, which might serve as an elementary model for clustering by ballistic aggregation1 in an expanding universe. The setting is linear and relativistic, in the sense that each particle has two neighbors, one on its left and one on its right, and at each time we just specify the distances and relative velocities between pairs of neighboring particles. A collision occurs when the distance between two neighboring particles, say p and q, vanishes. Then both particles are removed from the system and replaced by a single particle p ∗ q at the location of the collision, where ∗ : P × P → P is some symmetric map and P stands for some set of particles. Let us now describe the dynamics. Relative velocities result from the combination of two opposite phenomena. On the one hand, we assume that each particle p attracts the others at some rate a(p)/2 depending only on the particle, which induces a contraction of the system. On the other hand, we suppose that the universe is expanding in such a way that if only expansion was taken into account, any two points would go apart from each other with relative velocity proportional to their distance. For the sake of simplicity, we shall assume that the Hubble constant of expansion of the universe is 1, so that ˙ at time t between two combining these two trends, we see that the relative velocity d(t) neighboring particles p and q at distance d(t) from each other, is given by 1 ˙ d(t) = d(t) − (a(p) + a(q)) . 2 This verbal description can easily be made rigorous when one starts with finitely many particles; and we shall first see that under some mild technical conditions, the dynamics can be extended to a large class of infinite systems. Roughly, the main purpose of this work is to point at the following remarkable property for Poisson clouds. Suppose that at the initial time, particles are distributed according to a spatially stationary Poisson cloud on P × R with intensity µ0 , where µ0 is some finite measure on the set of particles P. Informally, this means that we have a doubly infinite sequence of i.i.d. particles with law µ0 /µ0 (P), and the distances between neighboring particles are independent exponential variables with parameter µ0 (P). At the initial time t = 0, let us tag some particle at random and use for every time t ≥ 0 the cluster containing this tagged particle as the reference point for the system. We shall show (again under certain hypotheses which will be specified in the sequel) that at any time t > 0, the system is still distributed according to a Poisson cloud with a certain intensity µt , and the tagged cluster has now a size-biased distribution. The intensity µt solves some variation of Smoluchowski’s coagulation equation:
t f, µt − f, µ0 = ( f, A(µs ) + f, µs ( a, µs − 1)) ds , 0

where f : P → R stands for a generic measurable bounded function, f, µt = f (p)µt (dp) and

1 f, A(µs ) = (f (p ∗ q) − f (p) − f (q)) (a(p) + a(q)) µs (dp)µs (dq) . 2 P P By elementary transformations, the preceding equation can be reduced to a generalization of Smoluchowski’s coagulation equation considered recently by Norris [16]. 1 We stress that such a model does not incorporate Newtownian gravitational force, as the particles in the system generate a velocity field, but no acceleration of gravitational type.

Self-Attracting Poisson Clouds

61

The rest of this paper is organized as follows. Section 2 is devoted to some preliminaries. In particular, we show that the dynamics can be extended to a fairly general class of infinite systems. We also present some basic notions and results about a generalization of Smoluchowski’s coagulation equation studied by Norris [16], and stochastic coalescence. In Sect. 3, we adapt the method of [5] to the study of the evolution of a certain family of random periodic configurations. In particular we establish a connection with some stochastic coalescent process. The main result on the evolution of Poisson clouds is given in Sect. 4. Roughly, the idea is that Poissonian configurations can be obtained as the limit of certain random periodic configurations with large period. This leads us to consider asymptotics for the stochastic coalescent process. In this direction, the intensity µt arises as some hydrodynamic limit of the latter, which yields the connection with Smoluchowski’s coagulation equation (see Norris [15, 16] for similar results). Finally, Sect. 5 is devoted to the so-called additive case when the attraction rate fulfills a(p ∗ q) = a(p) + a(q). We shall see that the variation of Smoluchowski’s coagulation equation can then be solved explicitly, and relate the dynamics with the free sticky particle system. 2. Preliminaries 2.1. Particles, configurations... . Let P denote a set of particles endowed with a sigmaalgebra, such that every singleton is a measurable set. For every measurable function f : P → R+ and every measure µ on P, we shall use the notation
f, µ = f (p)µ(dp) . P

We consider a measurable symmetric map ∗ : P × P → P, where p ∗ q stands for the particle that results from the aggregation of p and q. It is convenient to agree that there exists a particle denoted by ∅ which is neutral for the aggregation in the sense that p ∗ ∅ = ∅ ∗ p = p for every p ∈ P. We suppose we are also given three measurable maps #:P→N

,

m : P → R+

and

a : P → R+ ,

that specify the size, the mass and the attraction rate of particles, respectively. We assume that the size and the mass are additive with regards to aggregation, that is #(p ∗ q) = #(p) + #(q) and m(p ∗ q) = m(p) + m(q)

for all p, q ∈ P .

In particular #(∅) = m(∅) = 0, and for simplicity, we also agree that a(∅) = 0. The notions of size and mass will be used for technical purposes in our analysis. Roughly, the mass will enable us to provide a simple condition for defining rigorously the dynamics. On the other hand, the size will be useful to keep track of the number of collisions that occurred to create some particle in the system at time t. Intuitively, we may think of {p ∈ P : #(p) = 1} as the subset of elementary particles, and if we assume that every particle q = ∅ can be uniquely decomposed into a finite number of elementary particles, then this number is given by the size #(q). Moreover if elementary particles have all unit mass (which can be viewed as a so-called mono-dispersive condition), then the mass and the size coincide; but otherwise these two notions differ.

62

J. Bertoin

Further, it will be convenient to assume that there is some total order on P. Specifically, we shall suppose that the re-ordering of finite sequences of particles is a measurable map P n → P n for every integer n ≥ 2. The choice of a specific order on P is not relevant to this work; its only purpose is to provide a simple mean for labelling finite families of particles. Finally, a typical configuration ω is given by a doubly infinite sequence of particles (pi , i ∈ Z) and strictly positive distances (di , i ∈ Z). More precisely, pi+1 is the right-neighbor of pi and di the distance between the two. We call a configuration ω = ((pi , di ), i ∈ Z) finite if pi = ∅ whenever |i| is sufficiently large. 2.2. Dynamics attraction/expansion. We next describe the evolution of configurations as time passes according to the dynamics attraction/expansion introduced verbally in the Introduction. Let us start at time t = 0 from some finite configuration ω(0) = ω. As long as all the distances between neighboring particles remain strictly positive, the configuration ω(t) at time t > 0 is given by the same sequence of particles as at the initial time, i.e. pi (t) = pi for every i ∈ Z, and the distances between neighboring particles are governed by the differential equation 1 d˙i (t) = di (t) − (a(pi ) + a(pi+1 )) , i ∈ Z. 2 Note that only the nearest neighbors interact with a given particle. Solving this linear differential equation with the initial condition di (0) = di yields 1 di (t) = et di − (et − 1)(a(pi ) + a(pi+1 )) , i ∈ Z. 2 The first collision thus occurs at time t1 which is given by   a(pi ) + a(pi+1 ) et1 = min : a(pi ) + a(pi+1 ) > 2di , i∈Z a(pi ) + a(pi+1 ) − 2di

(1)

and we denote by I1 the set of indices where the minimum is reached:   a(pi ) + a(pi+1 ) = et1 . I1 := i ∈ Z : a(pi ) + a(pi+1 ) − 2di We shall further suppose that I1 does not contain any two consecutive indices, that is every collision involves only two particles 2 . At time t1 , every pair of particles involved into a collision, that is (pi , pi+1 ) for i ∈ I1 , merges as a single particle pi ∗ pi+1 . We obtain a new sequence of particles denoted by (pi (t1 ), i ∈ Z), where the labelling is determined by the following rules. If the particle p0 has not been involved into a collision, i.e. neither −1 nor 0 belongs to I1 , then p0 (t1 ) = p0 . If 0 ∈ I1 (respectively, −1 ∈ I1 ), then p0 (t1 ) = p0 ∗ p1 (respectively, p0 (t1 ) = p−1 ∗ p0 ). So, we may imagine we tagged at the initial time the particle p0 , that the tag is conserved after a collision, and that the label 0 is always assigned to the tagged particle. The labels of the other particles are specified by their initial positions, in the sense that for every i ∈ Z, pi (t1 ) is built from 2 Otherwise one would need either to suppose that the aggregation map ∗ is associative, or to define maps to describe aggregation resulting from a multiple collision. We stress that multiple collisions never occur in the Poissonian setting we will be interested in, i.e. this assumption will be fulfilled with probability one for the random initial conditions that we will consider in the sequel.

Self-Attracting Poisson Clouds

63

particles which were initially at the left of the particles that built pi+1 (t1 ). Finally, the distances at time t1 are defined in an obvious way, specifically di (t1 ) coincides with the distance immediately before time t1 between the particles forming pi (t1 ) and those forming pi (ti+1 ). We may iterate the construction for the next collision times. Although it is easy to define the evolution when we start from a finite configuration, difficulties arise when one wishes to start from an infinite configuration. Indeed, collisions may then occur at arbitrary small times involving particles with large indices, and it is not clear how this affects the evolution. In the rest of this section, we shall show that the dynamics attraction/expansion still make sense for fairly general infinite configurations provided that the attraction rate is bounded from above by the mass, in the sense that a(p) ≤ m(p) for every p ∈ P. (2) More precisely, we shall show that when (2) holds, the evolution can be defined rigorously for a large class of initial configurations by approximation from the finite case. In this direction, we define for an arbitrary configuration ω = ((pi , di ), i ∈ Z) and every (n) integer n ≥ 1, the finite configuration ω(n) = ((pi , di ), i ∈ Z) with (n)

pi

=



pi ∅

for |i| ≤ n, for |i| > n.

For every t ≥ 0, we denote by   (n) (n) ω(n) (t) = (pi (t), di (t)), i ∈ Z the configuration at time t in the dynamics attraction/expansion starting from ω(n) (0) = ω(n) .  ∞ Lemma 1. Let ω = ((pi , di ), i ∈ Z) be a configuration with ∞ i=0 di = i=0 d−i = ∞. Suppose (2) holds and take t ≥ 0 such that for every integer k sufficiently large (1 − e−t )

k i=0

m(pi ) <

k−1 i=0

di and (1 − e−t )

k i=0

m(p−i ) <

k−1

d−i .

i=0

Then there exists a configuration denoted by ω(t) = ((pi (t), di (t)), i ∈ Z), such that for each index i ∈ Z, it holds that (n)

(n)

pi (t) = pi (t) and di (t) = di (t) whenever n is sufficiently large. Proof. Starting from the finite configuration ω(n) , we denote for every i ∈ Z by In (i, t) the set of the indices of the particles at the initial time that have collided by time t to (n) form the particle pi (t). A moment of thought on the dynamics shows that the distance (n) (n) between pi (t) and its right neighbor pi+1 (t) only depends on the pairs (p , d ) for  ∈ In (i, t) ∪ In (i + 1, t), but not on the other pairs in the initial configuration. Therefore, in order to establish the statement, it suffices to check that the cardinal of In (i, t) remains bounded as n increases. So let us suppose that the cardinal of In (i, t) cannot be bounded independently of n. By our assumptions, we may find n sufficiently large such that In (i, t) = {j, . . . , j + k} and

64

J. Bertoin

(1 − e−t )

k

m(pj + ) <

k−1

=0

dj + .

(3)

=0

For every 0 ≤ s ≤ t, denote by D(s) the distance at time s between the particles that contain pj and pj +k , respectively, so in particular D(0) =

k−1

dj +

and

D(t) = 0 .

(4)

=0

We see from the dynamics and the hypothesis (2) for the attraction rates that D(s) satisfies the differential inequality ˙ D(s) ≥ D(s) −

k

m(pj + ) ,

s < tk ,

=0

where tk ≤ t stands for the first instant when all the particles pj , . . . , pj +k have merged (n) to form pi (t). Hence we have D(s) ≥ es D(0) − (es − 1)

k

m(pj + ) ,

s ≤ tk .

=0

We specify this for s = tk using (4) and get D(0) =

k−1 =0

which contradicts (3).

dj + ≤ (1 − e−t )

k

m(pj + ) ,

=0

 

The following variation of Lemma 1 in terms of periodic approximations will be useful in the sequel. Specifically, we now define for an arbitrary configuration ω = ((pi , di ), i ∈ Z) and every integer n ≥ 2, the periodic configuration ωn = ((pn,i , dn,i ), i ∈ Z) such that (pj , dj ) = (pn,i , dn,i ) for j = −n, . . . , −1, 0, 1, . . . , n, i = j + k(2n + 1), and k ∈ Z .

 Again we denote by ωn (t) = (pn,i (t), dn,i (t)), i ∈ Z the configuration at time t in the dynamics attraction/expansion starting from ωn (0) = ωn . Lemma 2. Lemma 1 still holds if we replace the finite configuration ω(n) by the periodic configuration ωn . The proof of Lemma 2 is based on the same argument as for Lemma 1; we omit the details.

Self-Attracting Poisson Clouds

65

2.3. Generalized Smoluchowski’s coagulation equation. In this section, we shall briefly discuss some results about a generalized version of Smoluchowski’s coagulation equation which will arise naturally in our study. We refer to the surveys by Drake [9] and Aldous [1] for general background in this field. Our presentation is based on the paper of Norris [16]; however we stress that only a special type of coagulation is considered here, Norris treats a much more general situation. Let ν be a finite measure on P such that a, ν < ∞. For every bounded measurable function f : P → R, we define

1 f, A(ν) = ν(dp) ν(dq) (f (p ∗ q) − f (p) − f (q)) (a(p) + a(q)) . (5) 2 P P Next, consider a family (νt , t ≥ 0) of finite measures on P such that the map t → νt (Q) is measurable for every measurable set Q ⊆ P and

t
ds νs (dp) νs (dq) (a(p) + a(q)) < ∞ . P

0

P

For every T > 0, one calls (νt , 0 ≤ t < T ) a solution to the (generalized) Smoluchowski’s coagulation equation on [0, T [ if we have
t f, A(νs ) ds (6) f, νt − f, ν0 = 0

for every bounded measurable function f : P → R and 0 ≤ t < T . Next, we turn our attention to a modified version of (6) that will arise in our study. Specifically, we now consider the equation
t f, A(µs )

f, µt f, µ0

− = ds , (7) #, µt #, µ0

#, µs

0 where # stands for the size (cf. Sect. 2.1) and f : P → R denotes a generic bounded and measurable function. The next lemma shows that (7) is closely connected to Smoluchowski’s coagulation equation (6). Lemma 3. Let (µt , t ≥ 0) be a family of finite measures on P with #, µt ∈]0, ∞[ for every t ≥ 0, which solves (7) and such that 1, µt = ce−t ,

(8)

where c > 0 is some fixed real number. Then the following hold: (i) (µt , t ≥ 0) also solves
f, µt − f, µ0 =

t 0

( f, A(µs ) + f, µs ( a, µs − 1)) ds ,

and as a consequence
#, µt = #, µ0 exp

0

t

( a, µs − 1) ds

.

(9)

66

J. Bertoin

(ii) If we set




∞

T = 0

#, µs ds ,

define implicitly the time-substitution τ : [0, T [→ R+ by
τ (t) #, µs ds = t , t
and finally introduce νt = µτ (t) / #, µτ (t) , then (νt , 0 ≤ t < T ) solves Smoluchowski’s coagulation equation (6) on [0, T [, and we can recover (µt , t ≥ 0) from (νt , 0 ≤ t < T ) via the following identities:  
t T = inf t ≥ 0 : 1, νs ds = c , 0

and for t < T , c(1 − e−τ (t) ) =




t 0

1, νs ds ,

µτ (t) =

ce−τ (t) νt . 1, νt

m2 , µ0

< ∞, then (µt , t ≥ 0) is the unique solution to (7) (iii) If (2) holds and which fulfills the requirements above. It is important to note that the solution (µt , t ≥ 0) does not depend on the notion of size, even though it appears in (7). More precisely, neither Eq. (9), nor the way to recover (µt , t ≥ 0) from the solution to Smoluchowski’s equation (6), involve the function #. Proof. (i) On the one hand, we have by (8) and ordinary differential calculus





d 1, µt

d ce−t 1 d #, µt

ce−t = = − 1+ . dt #, µt

dt #, µt

#, µt

#, µt dt On the other hand, (7) gives d dt



1, µt

#, µt

=

1, A(µt )

. #, µt

It follows from (5) that 1, A(µt ) = − 1, µt a, µt = −ce−t a, µt , so by identification, we obtain d #, µt

= ( a, µt − 1) #, µt . dt

(10)

Solving this linear differential equation yields the formula for #, µt stated in Lemma 3(i). Next, we simply write the identity



d f, µt

d f, µt

d f, µt

f, µt d #, µt

= #, µt = #, µt

+ , dt dt #, µt

dt #, µt

#, µt dt

Self-Attracting Poisson Clouds

67

recall from (7) that d dt



f, µt

#, µt

=

f, A(µt )

, #, µt

and use (10) to conclude that d f, µt

= f, A(µt ) + f, µt ( a, µt − 1) . dt (ii) We have on the one hand dτ (t) =

dt , #, µτ (t)

t < T,

and on the other hand, for every real number b > 0 and every measure ρ, f, A(bρ) = b2 f, A(ρ) . Plugging this in (7), we get
f, νt − f, ν0 =

τ (t) 0


f, A(µτ (s) / #, µτ (s) ) #, µτ (s) ds =

t 0

f, A(νs ) ds ,

so (νt , 0 ≤ t < T ) solves Smoluchowski’s coagulation equation (6). To recover (µt , t ≥ 0) from (νt , 0 ≤ t < T ), we simply observe that 1, νt =

dτ (t) 1, µτ (t)

= ce−τ (t) . #, µτ (t)

dt

This readily yields the formulas of (ii). (iii) Finally, the uniqueness follows from Theorem 2.1 in [16] which can be stated in the present setting as follows. If (2) holds, then for any finite measure ν0 on P with m2 , ν0 < ∞ and for any T > 0, there exists a unique solution (νt , 0 ≤ t < T ) to (6) on [0, T [ which starts from ν0 .   2.4. Stochastic coalescent. We introduce here a stochastic coalescent process in the vein of Marcus [14], Lushnikov [13] and Norris [16] which bears close connections with Smoluchowski’s coagulation equation presented in the previous section. We shall work with a finite family of particles, and it will be convenient to assume that any family that can arise from the latter by aggregation consists in distinct particles. This induces no loss of generality as we may always add labels to differentiate between identical particles, i.e. otherwise it suffices to work with some enlarged set of particles. Recall that the set of particles P is endowed with some total order, so we may now compare any two particles that can arise from the initial family. ↓ ↓ For every n ≥ 2 and every sequence p1 , . . . , pn of particles ranked in the decreas↓ ↓ ing order and 1 ≤ i < j ≤ n, denote by (p1 , . . . , pn )i∗j the sequence obtained by ↓ ↓ ↓ ↓ adding a particle pi ∗ pj , removing the particles pi and pj , and re-ordering. We consider the time-homogeneous Markov step process ( (t), t ≥ 0) taking values in the space of ranked finite sequences of particles, with the following transitions. For every

68

J. Bertoin ↓

↓

↓

↓

1 ≤ i < j ≤ n, the jump rate from the state (p1 , . . . , pn ) to the state (p1 , . . . , pn )i∗j is ↓ ↓ a(pi ) + a(pj ) , n−1 and all the other jump rates are zero. The evolution stops at time ζ when all the particles have merged together. ↓ ↓ It is sometimes convenient to identify a ranked finite sequence (p1 , . . . , pn ) as a point measure on P, in the sense that to each particle we associate a Dirac point mass. In particular, for every measurable function f : P → R, we use the notation ↓

f, (p1 , . . . , pn↓ ) :=

n i=1

↓

f (pi ) .

(11)

The infinitesimal generator G of the stochastic coalescent process pression for functions of the type f, · . Indeed, if we set

has a simple ex-

↓

Gf (p1 , . . . , pn↓ ) =



1 ↓ ↓ 1, (p1 , . . . , pn ) − 1



1≤i<j ≤n



↓

↓

↓

↓

  ↓ ↓ a(p )+a(p )  ,

f (pi ∗ pj )−f (pi )−f (pj )



i

j

(12) then we see from the verbal description of the transitions of

above that

G f, · = Gf . The following statement merely rephrases Kolmogorov’s forwards equation in the present setting. Lemma 4. For every measurable function f : P → R, we have
t E (Gf ( (s))) ds . E ( f, (t) − f, (0) ) = 0

3. Random Periodic Setting 3.1. Some notation for periodic configurations. We call a configuration ω = ((pi , di ), i ∈ Z) periodic if there exists some integer n ≥ 1 such that (pi , di ) = (pi+n , di+n ) for every i ∈ Z. The minimal such n is then called the period of the configuration, and we call ω n-periodic if ω is periodic with period n. It should be plain that the dynamics of attraction/expansion preserves periodicity, in the sense that if the initial configuration ω(0) is n-periodic, and if for k < n, tk denotes the k th instant of collisions, then ω(t) is (n − k)-periodic for every t ∈ [tk , tk+1 [. To take advantage of this elementary observation, we introduce the following notation. If ω = ((pi , di ), i ∈ Z) is periodic, then we say that another configuration ω = ((pi , di ), i ∈ Z) is equivalent to ω if there exists an integer k such that  pi = pi+k

 and di = di+k ,

i ∈ Z.

Self-Attracting Poisson Clouds

69

We denote by ω˜ the class of equivalence of ω; the number of configurations in the class of equivalence ω˜ of course coincides with the period of ω. Similarly, for every permutation σ of {1, . . . , n}, we denote by σ˜ the orbit of σ by the action of cyclic permutations; in other words, σ˜ is the family of permutations of the type (σ (k + 1), . . . , σ (k + n)) for some k = 0, . . . , n − 1, where the addition is taken modulo n. We call σ˜ a cyclic order for n elements, as it will be used to specify the notion of neighbors in Z/nZ. More precisely, if i = σ (j ), the right-neighbor of i for the cyclic order σ˜ is σ (j + 1), and its left-neighbor σ (j − 1). Plainly, this definition does not depend on the choice of the permutation σ in the family σ˜ . Just as in the preceding section, we shall work with a finite family of particles, and assume that any family that can arise from it by aggregation consists in distinct particles. Recall also that P is endowed with some total order. Given an n-periodic configuration ω = ((pi , di ), i ∈ Z), we consider the permutation σ of {1, . . . , n} such that σ (i) is the index of the i th largest particle amongst p1 , . . . , pn for the ordering on P, and set



 ω↓ = (pσ (1) , dσ (1) ), . . . , (pσ (n) , dσ (n) ) , p↓ (ω) = pσ (1) , . . . , pσ (n) . In other words, ω↓ is the rearrangement of the sequence (p1 , d1 ), . . . , (pn , dn ) in the decreasing order of particles, and p ↓ (ω) the projection of ω↓ on P. We call ω↓ a ranked configuration with n particles. Finally, we write  = σ −1 (n) , for the rank of the particle with index 0 in the n-periodic configuration ω when we ↓ re-arrange the sequence of particles in the decreasing order, i.e. p0 = pn = p .  Plainly, if we take another configuration ω in the equivalence class ω, ˜ and denote by σ  the permutation of {1, . . . , n} corresponding to the rearrangement of its elements in the decreasing order of particles, then σ  coincides with σ up to a cyclic permutation, i.e. the cyclic orders σ˜  and σ˜ are the same. Moreover we have also identity of the ranked configurations ω↓ = ω↓ , but of course the rank  of the particle with index 0 for the configuration ω is not the same as that for ω. To summarize, to each n-periodic configuration ω, we associate a ranked configuration ω↓ with n particles, a cyclic order σ˜ for n elements, and the rank  of the particle with index 0. It should be plain that the maps ω˜ → (ω↓ , σ˜ ) and ω → (ω↓ , σ˜ , ) are both bijective. 3.2. Connection with stochastic coalescence. In this section, we introduce randomness for periodic configurations at the initial time t = 0, let the system evolve, and observe the resulting configuration at some later time t > 0. Roughly, our main result (cf. Prop↓ ↓ osition 1 below) can be stated as follows. Fix a finite sequence p1 , . . . , pn of particles ranked in decreasing order. Then pick the particle p0 at random in this sequence according to the size-biased distribution (recall from Sect. 2.1 that there is a map # : P → N which gives the size of each particle and that this map is additive for aggregation), and sample without replacement p1 , . . . , pn−1 according to the equi-probability amongst the remaining particles. Next extend this into an n-periodic sequence indexed by Z, and define a random configuration by choosing i.i.d. exponential distances between neighboring particles. Then as time passes in the dynamics attraction/expansion, the finite family of particles that spans this periodic configuration evolves like the stochastic coalescent process introduced in Sect. 2.4, up to a simple deterministic time change.

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J. Bertoin

Moreover, the conditional distribution of the configuration at time t given its spanning family of particles fits the same description as that for the initial configuration, i.e. we sample the particle p0 (t) according to the size-biased distribution and then the remaining particles without replacement according to the equi-probability, extend this into a periodic sequence and assign i.i.d. exponential distances between neighboring particles. To make a rigorous statement and prepare some notation used in the proof, we now introduce a class of probability distribution on periodic configurations. For every positive ↓ ↓ real number c, every finite sequence p1 , . . . , pn of particles ranked in the decreasing ↓ order with #(pi ) ≥ 1 for some i, and every cyclic order σ˜ on n elements, we denote by ↓

P ((p1 , . . . , pn↓ ); c; σ˜ ) the probability measure on the space of periodic configurations with n particles, which can be described as follows. The ranked configuration ω↓ is given by   ↓ ω↓ = (p1 , ε1 ), . . . , (pn↓ , εn ) , where ε1 , . . . , εn are i.i.d. exponential variables with parameter c > 0. The cyclic order is σ˜ , the rank  of the particle p0 is independent of the εi ’s, and the probability that  = i equals ↓

↓

#(pi )/ #, (p1 , . . . , pn↓ ) ,

i = 1, . . . , n .

In other words, the conditional distribution of the particle p0 given the equivalence class ω, ˜ is a size-biased picked from the sequence p↓ (ω). Finally, we introduce the mixture ↓

P ((p1 , . . . , pn↓ ); c) =

1 ↓ P ((p1 , . . . , pn↓ ); c; σ˜ ) , (n − 1)!

(13)

σ˜

where the sum is taken over the (n − 1)! cyclic orders for n elements. This is the law on n-periodic configurations which has been described informally at the beginning of this section. Recall that ( (t), t ≥ 0) denotes the stochastic coalescent process that was introduced in Sect. 2.4, ζ the instant when this coalescent process is reduced to a single particle, and tk the instant of the k th collision in the dynamics attraction/expansion. We now state our main result. ↓

↓

↓

Proposition 1. Let p1 , . . . , pn be some finite ranked sequence of particles with #(pi ) ↓ ≥ 1 for some i. Let us start from a random periodic configuration ω with law P ((p1 , . . . , ↓ pn ); c). Then the following hold:

 (i) The process p ↓ (ω(t)), t < tn−1 has the same distribution as ( (c(1 − e−t )), ↓ ↓ c(1 − e−t ) < ζ), where the stochastic coalescence starts from (0) = (p1 , . . . , pn ). (ii) For every real number t > 0 and integer k ≥ 2, the conditional distribution of ↓ ↓ ω(t) given p↓ (ω(t)) = (q1 , . . . , qk ) is ↓

↓

P ((q1 , . . . , qk ); ce−t ) . Proposition 1 is closely related to the construction of so-called additive coalescents in [5]. It results from an immediate combination of the following two lemmas.

Self-Attracting Poisson Clouds

71 ↓

↓

Lemma 5. Fix a finite ranked sequence p1 , . . . , pn of particles and some cyclic order σ˜ for n elements, and start from a random periodic configuration ω with law ↓ ↓ P ((p1 , . . . , pn ); c; σ˜ ). For every t > 0, the event that t < t1 (i.e. no collision has occurred before time t) has probability   n 

↓ a(pi ) . exp −c 1 − e−t i=1

↓

↓

Moreover, the conditional distribution of ω(t), given t < t1 , is P ((p1 , . . . , pn ); ce−t ; σ˜ ). ↓

↓

Proof. Fix i = 1, . . . , n and let pj be the right-neighbor of pi for the cyclic order σ˜ , ↓

↓

in the sense that pi = pk and pj = pk+1 for some index k ∈ Z. Then set εi (t) = et εi −

 1 ↓ ↓ a(pi ) + a(pj ) (et − 1) , 2 ↓

so according to (1), εi (t) = dk (t) is the distance at time t between pi = pk and its rightneighbor whenever none of these particles has been involved into a collision before time t. In other words, we have t < t1 if and only if εi (t) > 0 for every i = 1, . . . , n, and in ↓ ↓ that case, dk (t) = εi (t). Since under P ((p1 , . . . , pn ); c; σ˜ ), the variables et ε1 , . . . , et εn are independent and exponentially distributed with parameter ae−t , our claim follows from the absence of memory of the exponential laws.   ↓

↓

Lemma 6. For every 1 ≤ i < j ≤ n, under P ((p1 , . . . , pn ); c), the event that the first ↓ ↓ aggregation occurs at time t1 ∈ [t, t + dt] and involves the particles pi and pj , has probability ↓

ce

−t

↓

a(pi ) + a(pj ) n−1





exp −c 1 − e

−t

n  i=1

 ↓ a(pi )

dt . ↓

↓

Moreover the conditional distribution of ω(t1 ) given this event is P ((p1 , . . . , pn )i∗j ; ce−t ). Proof. First of all, we observe the following. Since at the initial time, the conditional distribution of the particle p0 = p0 (0) given the equivalence class ω(0) ˜ is that of a size-biased sample from p ↓ (ω(0)), our procedure to label particles after a collision and the fact that the size is additive with regards to aggregation entail that at the time t1 of the first collision, the conditional distribution of the particle p0 (t1 ) is still that of a size-biased sample from the ranked sequence p↓ (ω(t1 )) of particles at time t1 . We shall now focus on the case when n ≥ 3 as the easier case n = 2 only requires ↓ straightforward modifications. Denote by &ij the event that the particle pj is the right↓

↓

neighbor of pi at the initial time, and that the first aggregation involves the particles pi ↓ and pj and occurs at time t1 ∈ [t, t + dt]. Since there are at least three particles at the ↓

↓

initial time, on the event &ij , pi cannot be the right-neighbor of pj , so &ij and &j i are disjoint and the event considered in the statement can be expressed as &ij ∪ &j i .

72

J. Bertoin ↓

↓

Then fix some cyclic order σ˜ for which pj is the right-neighbor of pi and work ↓

↓

under P ((p1 , . . . , pn ); c; σ˜ ). Using the same notation as in the proof of Lemma 5 and viewing dt as an infinitesimal increment, the fact that 1 ↓ ↓ ε˙ i (t) = εi (t) − (a(pi ) + a(pj )) 2 shows that the probability of &ij is the same as that of the event 1 ↓ ↓ (a(pi ) + a(pj ))dt} . 2 It now follows from Lemma 5 that the probability of &ij is given by   n  

1 −t  ↓ ↓ ↓ −t a(pi ) + a(pj ) exp −c 1 − e ce a(pi ) dt , 2 {εk (t) > 0 for every k = 1, . . . , n , and εi (t) ≤

i=1

and that the conditional distribution of the class of equivalence ω(t ˜ 1 ) given &ij is that of the class of equivalence ω˜ under ↓

P ((p1 , . . . , pn↓ )i∗j ; ce−t ; σ˜  ) , where σ˜  is the cyclic order on (n − 1) elements induced by σ˜ after the identification of i and j . Plainly, when we pick σ˜ according to the equi-probability on the set of cyclic orders ↓ ↓ on n elements, the event that pj is the right-neighbor of pi has probability 1/(n − 1), and conditionally on that event, σ˜  is distributed according to the equi-probability on the set of cyclic orders on (n − 1) elements. We conclude the probability of &ij under ↓ ↓ P ((p1 , . . . , pn ); c) is   ↓ ↓ n a(pi ) + a(pj ) 

↓ −t −t a(pi ) dt , exp −c 1 − e ce 2(n − 1) i=1

and that the conditional distribution ω(t1 ) given that event is ↓

P ((p1 , . . . , pn↓ )i∗j ; ce−t ) . Since the event considered in the statement can be expressed in the form &ij ∪ &j i with &ij and &j i disjoint, the proof of Lemma 6 is complete.   4. Evolution of Poisson Clouds 4.1. Stationary and size-biased Poisson clouds. We start with defining certain distributions for random configurations. First, given a finite measure µ on P, we say that a random configuration ω = ((pi , di ), i ∈ Z) is distributed as a stationary Poisson cloud with intensity µ if (pi , i ∈ Z) and (di , i ∈ Z) are two independent sequences of i.i.d. variables, where: • each particle pi has the distribution µ/ 1, µ , • each distance di between neighboring particles has the exponential law with parameter 1, µ .

Self-Attracting Poisson Clouds

73

Let us briefly explain the terminology by making the connection with the classical notion of Poisson measure. Let M be a Poisson measure on P × R with intensity µ ⊗ λ, where λ stands for the Lebesgue measure on R. That is, for every measurable set Q ⊆ P and every interval ]x, y] ⊆ R, M(Q×]x, y]) has a Poisson distribution with intensity (y − x)µ(Q); and disjoint sets have independent measures. The atoms of the point measure M are given by pairs (p, x), where p ∈ P is a particle and x ∈ R can be thought of as the location of p. If we condition M to have an atom located at x = 0, denote the corresponding particle by p0 and for every positive integer n by pn (respectively, by p−n ) the nth particle at the right (respectively, at the left) of p0 , and finally by di the distance between pi and pi+1 , then we obtain a random configuration ω = ((pi , di ), i ∈ Z) distributed as a stationary Poisson cloud with intensity µ. Next, recall from Sect. 2.1 that #(p) stands for the size of the particle p ∈ P, and suppose that #(p) ≥ 1 for µ-a.e. p ∈ P and that #, µ < ∞. We say that a random configuration ω = ((pi , di ), i ∈ Z) is distributed as a size-biased Poisson cloud with intensity µ if it fulfills the same requirements as above, except that the particle p0 is now distributed according to the size-biased law of µ. Specifically, (pi , i ∈ Z) and (di , i ∈ Z) are two independent sequences of independent random variables, but now where: the distances di ’s, i ∈ Z, are i.i.d. exponentially distributed with parameter 1, µ , (14) the particles pi ’s, i ∈ Z\{0} are i.i.d. with joint distribution µ/ 1, µ , (15) the particle p0 has the size-biased law of µ, i.e. E(f (p0 )) = #f, µ / #, µ , (16) where f : P → [0, ∞[ stands for a generic measurable function. 4.2. Main result. We are now able to state the main result of this work about the evolution of Poisson clouds in the dynamics attraction/expansion. Let µ0 be some finite measure on P supported by elementary particles 3 , in the sense that #(p) = 1 for µ0 -a.e. p ∈ P. We set 1, µ0 = #, µ0 := c ∈]0, ∞[ , and further assume that m, µ0 ≤ 1 .

(17)

Theorem 1. Suppose that the attraction rate is bounded from above by the mass, i.e. (2) holds, and let us start from a random initial configuration ω = ω(0) which is distributed as a stationary Poisson cloud with intensity µ0 , where µ0 fulfills the requirements above. Then there exists a family (µt , t ≥ 0) of measures on P with 1, µt = ce−t

,

#, µt < ∞ ,

and #(p) ≥ 1 for µt -a.e. p ∈ P ,

3 We stress that this hypothesis induces no loss of generality. Indeed, we may work with an extended set of particles P  := P × N and think of P × {1} as the set of elementary particles of P  . For every p = (p, k) ∈ P  , we set #(p ) = k, and extend the maps m, a and ∗ in an obvious way: m((p, k)) = m(p), a((p, k)) = a(p), and (p, k) ∗ (q, ) = (p ∗ q, k + ). If µ0 is an arbitrary finite measure on P , we may view µ0 as a measure on P  with support in P × {1}, apply Theorem 1 in the P ’-setting, and finally use the projection P  → P since the map # plays no role in the dynamics attraction/expansion.

74

J. Bertoin

and such that for every t ≥ 0, ω(t) has the law of a size-biased Poisson cloud with intensity µt . More precisely, (µs , s ≤ t) solves the modified Smoluchowski’s coagulation equation
t f, µt − f, µ0 = (18) ( f, A(µs ) + f, µs ( a, µs − 1)) ds , 0

for every measurable and bounded function f : P → R, where we used the notation (5). If we assume moreover that m2 , µ0 < ∞, then (18) determines µt . Proof. Recall from Sect. 2.2 that for every integer n ≥ 0, ωn stands for the (2n + 1)-periodic configuration which coincides with ω for the indices −n, . . . , n. Denote by ν2n+1 the distribution of the decreasing rearrangement of p−n , . . . , pn , where the pi ’s are i.i.d. random particles with law c−1 µ0 and c = 1, µ0 . Recall the notation (13) and introduce the following probability measure on the space of (2n + 1)-periodic configurations:
↓ ↓ ↓ ↓ P (µ0 , 2n + 1) := P ((p1 , . . . , p2n+1 ), c)ν2n+1 (dp1 , . . . , dp2n+1 ) . P2n+1

Since µ0 only charges elementary particles, size-biased sampling coincides with uniform sampling, and we see that when ω is a homogeneous Poisson cloud with intensity µ0 , then ωn has the law P (µ0 , 2n + 1). By the law of large numbers, (17) ensures that the sufficient condition of Lemma 1 for the dynamics to be defined up to time t holds a.s. when one starts with the initial configuration ω. Similarly, the event that this sufficient condition holds when one starts with the configuration ωn , has a probability that tends to 1 when n → ∞. So, focussing on this event, we let the configuration ωn evolve according to the dynamics attraction/expansion, and observe the resulting configuration ωn (t) at time t. Denote by pn,i (t) and dn,i (t) the particle with label i and the distance to its rightneighbor in the configuration ωn (t). By Lemma 2, we know that for each index i, pn,i (t) = pi (t) and dn,i (t) = di (t) whenever n is sufficiently large, with probability one. In particular, we deduce from Proposition 1(ii) that the sequence of distances (di (t), i ∈ Z) is independent of the sequence of particles (pi (t), i ∈ Z) and consists in independent exponential variables with parameter ce−t . Hence the condition (14) holds when we replace di by di (t) and 1, µ by ce−t . Next, recall that ωn (t) is periodic and that p ↓ (ωn (t)) stands for the ranked sequence of particles in ωn (t). Introduce the corresponding empirical measure ρn,t which is given in the notation (11) by f, p↓ (ωn (t))

f, ρn,t = . 1, p ↓ (ωn (t))

We condition on the classes of equivalence ω˜ n (t) for n = 1, . . .. According to Proposition 1, the conditional distribution of pn,0 (t) given those classes is that of a size-biased sample from the ranked sequence p ↓ (ωn (t)), that is has the size-biased law of ρn,t , and the particles pn,1 (t), . . . are then sampled without replacement from the remaining particles according to the equi-probability. The fact that with probability one, pn,0 (t) does not depend on n provided that n is chosen large enough, entails that the size-biased law of ρn,t converges a.s. in total variation. According to the forthcoming Lemma 7(i), this implies that ρn,t converges a.s. in total variation towards some probability measure ρt on P. A priori, ρt might depend on the random configuration ω. However, the same argument as in the proof of Lemma 1

Self-Attracting Poisson Clouds

75

shows that the limit of the empirical measure ρn,t should only depend on the (two-sided) tail algebra of the sequence ((pi , di ), i ∈ Z). Since the latter consists in independent variables, Kolmogorov’s 0-1 law implies that ρt must be deterministic. Applying now Lemma 7(ii), we conclude that the distribution of the sequence (pi (t), i ∈ Z) fulfills the conditions (15) and (16) for µ = ce−t ρt . Putting the pieces together, we have shown that the random configuration ω(t) has the distribution of a size-biased Poisson cloud with intensity µt := ce−t ρt . All that is needed now is to relate the latter to the modified Smoluchowski’s coagulation equation, and for this, we rely on elementary properties of the stochastic coalescent introduced in Sect. 2.4. First, combining Proposition 1(i) and Lemma 4, we get in the notation used above that
t     ↓ ↓ E f, p (ωn (t)) − f, p (ωn (0)) = c−1 es E Gf (p ↓ (ωn (s))) ds , (19) 0

where f : P → R is a generic bounded measurable function and Gf has been defined in (12). Next, we observe that, since the size is additive for the aggregation, we have #, p↓ (ωn (s)) = 2n + 1

for every s ≥ 0 .

Writing f, p↓ (ωn (t))

f, p ↓ (ωn (t)) 1, p ↓ (ωn (t))

f, ρn,t

= = , 2n + 1 1, p↓ (ωn (t)) #, p↓ (ωn (t))

#, ρn,t

we deduce from the forthcoming Lemma 7(i) that f, ρt

f, p↓ (ωn (t))

= , n→∞ 2n + 1 #, ρt

lim

where the convergence holds a.s., and also in L1 (P) by dominated convergence. Similarly, we obtain that f, p↓ (ωn (0))

f, ρ0

lim = n→∞ 2n + 1 #, ρ0

and f, A(ρs )

Gf (p ↓ (ωn (s))) = , lim n→∞ 2n + 1 #, ρs

where the notation f, A(ρs ) has been defined by (5). Thus dividing both sides in (19) by 2n + 1, letting n → ∞ and recalling that µs = ce−s ρs , we conclude that (µt , t ≥ 0) solves
t f, µt f, µ0

f, A(µs )

− = ds . #, µt #, µ0

#, µs

0 The proof of Theorem 1 is completed by an appeal to Lemma 3.   Remark . The hypothesis (2) that the mass dominates the attraction rate is only used to ensure that ω(t) can be approached by the periodic configurations ωn (t) in the sense of Lemmas 1 and 2. Thus Theorem 1 would also hold if we drop this hypothesis, provided that we could check directly the validity of such approximations.

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4.3. A lemma on sampling without replacement. In this section, we provide a proof to an elementary lemma about sampling without replacement which was used in the proof of Theorem 1. To simplify the notation, we shall work in the following setting, which is slightly different from that of the preceding section. ↓ ↓ For every integer n ≥ 1, let p1,n , . . . , pk(n),n be a finite (ranked) sequence of parti↓

cles, all with size #(pi,n ) ≥ 1. Denote by k(n)

1 δp↓ ρn := i,n k(n) i=1

the corresponding empirical law. We sample without replacement from this sequence particles p1,n , . . . , pk(n),n as follows. First, we sample pk(n),n according to the size-biased law, viz.   ↓ P pk(n),n = pi,n =

↓

#(pi,n ) k(n) #, ρn

,

i = 1, . . . , k(n) .

Next, we sample p1,n , . . . , pk(n)−1,n from the remaining particles according to the equiprobability, and set p−i,n = pk(n)−i,n for i = 1, . . . , k(n). In particular p0,n = pk(n),n . Lemma 7. Suppose that as n → ∞, k(n) tends to ∞ and the law of p0,n converges in the sense of total variation. Then there is some probability law ρ such that: (i) ρn converges in total variation towards ρ; moreover limn→∞ #, ρn = #, ρ < ∞. (ii) For every integer  ≥ 0 and every bounded measurable functions f− , . . . , f0 , . . . , f : P → R, we have

 lim E f− (p−,n ) · · · f0 (p0,n ) · · · f (p,n ) n→∞
1 = f− (p− ) · · · f0 (p0 ) · · · f (p )#(p0 )ρ(dp− ) #, ρ P 2+1 · · · ρ(dp0 )ρ(dp1 ) · · · ρ(dp ) . Proof. By assumption, for every bounded measurable function f : P → R, E(f (p0,n )) converges as n → ∞. Specifying this for f = 1/# yields that #, ρn converges to some finite limit when n → ∞. Next, we set g = f/#, which is again a measurable bounded function, we have

 f, ρn = #, ρn E g(p0,n ) . This entails that ρn converges in total variation towards ρ. Then, in the identity above, we take f = # ∧ b for some constant b > 0, and let n and then b tend to ∞. We get that #, ρ < ∞, so the statement (i) is proven. Next, we turn our attention to (ii), and for the sake of notational simplicity, we focus on the case when fi ≡ 1 for i = 0, 1. We have 

E f0 (p0,n )f1 (p1,n ) =



↓

↓

↓

f0 (pi,n )f1 (pj,n )#(p1,n ) ,  ↓ #(pi,n )

Self-Attracting Poisson Clouds

77

where both sums in the right-hand side are taken over indices 1 ≤ i, j ≤ k(n) and i = j . We may rewrite the numerator as ↓ ↓ ↓ f0 (pi,n )f1 (pj,n )#(pi,n )    k(n) k(n) k(n) ↓ ↓ ↓ ↓ ↓ ↓ = f0 (pi,n )#(pi,n )  f1 (pj,n ) − f0 (pi,n )f1 (pi,n )#(pi,n ) i=1

j =1

i=1

and the denumerator as

↓

#(pi,n ) = (k(n) − 1)

The stated convergence follows immediately.

k(n) i=1

↓

#(pi,n ) .

 

5. The Additive Case and Free Sticky Particles This section is devoted to the study of the special case when the attraction rate is additive with regards to aggregation, in the sense that a(p ∗ q) = a(p) + a(q) for every p, q ∈ P. In particular we may think of the attraction rate as a mass, i.e. we may take m = a so that condition (2) is automatically fulfilled. We shall first point at a simple connection with systems of free sticky particles provided that in the latter, initial relative velocities between neighboring particles are properly chosen. Next, we shall show that the variation of Smoluchowski’s coagulation equation (18) can be solved explicitly in this setting. Finally, we shall discuss some connection with related results in the literature. 5.1. Connection with the free sticky particle system. We first recall the dynamics for free sticky particle systems. We consider a configuration w(s) that evolves as time s passes as follows. We start at time s = 1 from some finite configuration w(1) = ((pi , di ), i ∈ Z), where as before pi+1 is the right-neighbor of pi and di the distance between the two. Each particle pi receives some initial velocity vi (1), and particles are travelling at constant speed as long as they are not involved into a collision with another particle. Collisions are completely inelastic, with conservation of mass and momentum, that is if the particles p and p collide, they merge as a single particle p ∗ p  at the location of the collision. The mass of this new particle is m(p ∗ p  ) = m(p) + m(p  ), and its velocity m(p)v + m(p  )v  , m(p) + m(p  ) where v and v  are the velocities of the particles p and p  , respectively. We agree to label the particles in the system at time s by (pi (s), i ∈ Z), where as usual pi+1 (s) is the right-neighbor of pi (s), and p0 (s) is the particle built from the cluster of particles at the initial time that contains p0 = p0 (1). The distance between pi+1 (s) and pi (s) is denoted by di (s), and we set w(s) = ((pi (s), di (s)), i ∈ Z). We stress that at any time, the relative velocities in the system only depend on the relative velocities at the initial time, so we may simply specify relative velocities between neighboring particles.

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Proposition 2. Suppose that the attraction rate of a particle coincides with its mass, i.e. a ≡ m, and if the initial relative velocities in the sticky particle system are given in terms of the masses and distances between particles by 1 vi+1 (1) − vi (1) = d˙i (1) = di (1) − (m(pi ) + m(pi+1 )) , 2

i ∈ Z.

(20)

If the initial configurations ω(0) and w(1) coincide, then we have ω(t) = w(et ) for every t ≥ 0. Proof. We work with the system of sticky particles w(·). Whenever s ≥ 1 is smaller than the instant of the first collision, we have 1 d˙i (s) = d˙i (1) = di (1) − (m(pi ) + m(pi+1 )) , 2 and therefore

1 di (s) = di (1) + (s − 1) di (1) − (m(pi ) + m(pi+1 )) 2 s−1 = sdi (1) − (m(pi ) + m(pi+1 )) . 2 This yields the differential equation

1 1 ˙ di (s) = di (s) − (m(pi ) + m(pi+1 )) . s 2 Next, let the first collision occur at time s1 and involve the particles pi and pi+1 . For the sake of simplicity, assume that i ≥ 0, so that our rules for labelling particles yield pi (s1 ) = pi ∗ pi+1 . The relative velocity of this particle as viewed from its left-neighbor is d˙i (s1 −)m(pi+1 ) d˙i−1 (s1 ) = d˙i−1 (s1 −) + m(pi ) + m(pi+1 )

1 (m(pi ) + m(pi+1 )m(pi+1 ) 1 di−1 (s1 ) − (m(pi−1 ) + m(pi )) − = s1 2 2s1 (m(pi ) + m(pi+1 ))

1 1 = di−1 (s1 ) − (m(pi−1 ) + m(pi ) + m(pi+1 )) s1 2

1 1 = di−1 (s1 ) − (m(pi−1 (s1 )) + m(pi (s1 )) . s1 2 Similarly we get 1 d˙i (s1 ) = s1



1 di (s1 ) − (m(pi (s1 )) + m(pi+1 (s1 )) . 2

By iteration on the number of collisions, we see that the dynamics of the system s → w(s) are the same as those governing the evolution of s → ω(log s).  

Self-Attracting Poisson Clouds

79

5.2. Evolution of Poissonian intensities. In this section, it is convenient to assume further that particles are characterized by their masses, so we shall work in the following setting: P = [0, ∞[ and a(p) = m(p) = p for every p ∈ P . (21) We consider some finite measure µ0 on ]0, ∞[ with 1, µ0 = c > 0 and Id, µ0 = 1. In the additive setting specified by (21), let us start from a random initial configuration ω which is distributed as a stationary Poisson cloud with intensity µ0 . We know from Theorem 1 that the configuration at time t, ω(t), is then distributed according to a sizebiased Poisson cloud with a certain intensity denoted by µt . The next statement specifies µt in terms of µ0 . Corollary 1. Under the hypotheses above, we have: (i) (µt , t ≥ 0) solves the Smoluchowski’s coagulation equation for the additive kernel, viz.

1 d f, µt

= (f (p + q) − f (p) − f (q)) (p + q) µt (dp)µt (dq) . dt 2 R+ R+ (ii) µt is specified in terms of µ0 by the identity


 et α(qet , et ) − 1 1 − e−qx µt (dx) = , et (et − 1) ]0,∞[

q ≥ 0,

where for every s > 1, α(·, s) is the inverse bijection of the function


 1 − e−qx µ0 (dx) . q → qs + ]0,∞[

Proof. (i) Recall from Lemma 3(ii) that we can associate to (µt , t ≥ 0) a solution (νt , t ≥ 0) to Smoluchowski’s coagulation equation for the additive kernel, with ν0 = µ0 / 1, µ0 . In particular, there is the differential equation d 1, νt

= − 1, νt Id, νt . dt It is known that the solutions of Smoluchowski’s coagulation equation with the additive kernel are conservative (one also says there is no gelation), i.e. Id, νt = Id, ν0 =

Id, µ0

= 1/c 1, µ0

for all t ≥ 0 ;

see for instance Ball and Carr [2]. We deduce that 1, νt = e−t/c 1, ν0 = e−t/c . In the notation of Lemma 3(ii), it follows that T = ∞ and τ (t) = t/c; and finally that µt = cνct . Since (νt , t ≥ 0) solves Smoluchowski’s coagulation equation for the additive kernel, we deduce by a simple linear time substitution that the same holds for (µt , t ≥ 0). (ii) The formula is obtained by first reducing (i) to the Riemann equation via Laplace transform, and then solving the latter by standard methods; see e.g. Sect. 2 in [6]. Specifically, if we define for every q ≥ 0 and s ≥ 1,


 q 1 − e−qx/s µlog s (dx) , 0(q, s) = − s ]0,∞[

80

J. Bertoin

then 0 is a solution to the Riemann equation, i.e. ∂s 0 + 0∂q 0 = 0. In the notation of the statement, this yields a − α(q, s) 0(q, s) = , s−1 which in turn gives the sought expression for the Laplace transform of µt .

 

5.3. Comments. We conclude our study in the case when the attraction rates are additive with regards to aggregation by connecting our results with related ones in the literature. In short, we have first seen that the model attraction/expansion then coincides up to an exponential time change with a system of free sticky particles in which the initial relative velocities for pairs of neighboring particles are related to theirs masses and distances via the identity (20). We point out that this identity arises naturally, even though it might look rather odd at first sight. Indeed, following e.g. Shandarin and Zeldovich [17] or Gurbatov, Malakov and Saichev [12], consider fluid particles which are uniformly distributed on R at the initial time. Assign initial velocities and let the system evolve according to the dynamics of sticky particles. Suppose that the initial velocities are such that the system observed at some time t > 0 is discrete, in the sense that all particles have merges into clusters whose locations form a discrete set in R. Then, adapting arguments of E et al [10] based on the conservation of mass and momentum, one readily sees that the identity (20) holds for the system at time t. On the other hand, we have shown that the dynamics attraction/expansion are closely connected to Smoluchowski’s coagulation equation and the stochastic coalescence for the additive kernel. It is interesting to recall that the latter have already been used as a model for growth by gravitational clustering in an expanding universe, see Sheth and Pitman [18] and references therein. We also refer to Drake [9] and Golovin [11] where the additive coagulation kernel appears in the study of droplet formation in aerosol. Finally, we pointed out that the dynamics attraction/expansion preserve the Poissonian character of configurations. It was shown in [3] that when a sticky particle system starts from the uniform distribution of masses and random initial velocities of L´evy type (so that roughly, relative velocities are independent and stationary) then at any time, the system can be described in terms of a certain Poisson cloud. In this setting, further connections with the additive coalescent have been described in [4]. References 1. Aldous, D.J.: Deterministic and stochastic models for coalescence (aggregation, coagulation): A review of the mean-field theory for probabilists. Bernoulli 5, 3–48 (1999) 2. Ball, J.M., Carr, J.: The discrete coagulation-fragmentation equation: Existence, uniqueness, and density conservation. J. Statist. Phys. 61, 203–234 (1990) 3. Bertoin, J.: The inviscid Burgers equation with Brownian initial velocity. Commun. Math. Phys. 193, 397–406 (1998) 4. Bertoin, J.: Clustering statistics for sticky particles with Brownian initial velocity. J. Math. Pures Appl. 79, 173–194 (2000) 5. Bertoin, J.: Eternal additive coalescents and certain bridges with exchangeable increments. Ann. Probab. 29, 344–360 (2001) 6. Bertoin, J.: Eternal solutions to Smoluchowski’s coagulation equation with additive kernel and their probabilistic interpretations. Ann. Appl. Probab. 12, 547–564 (2002) 7. Burgers, J.M.: The Nonlinear Diffusion Equation. Dordrecht: Reidel, 1974 8. Brenier, Y., Grenier, E.: Sticky particles and scalar conservation laws. SIAM J. Numer. Anal. 35, 2317–2328 (1998)

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9. Drake, R.L.: A general mathematical survey of the coagulation equation. In: Topics in Current Aerosol Research, Part 2. International Reviews in Aerosol Physics and Chemistry. eds. G.M. Hidy, J.R. Brock, London: Pergameon 1972, pp. 201–376 10. Rykov, E.W., Yu. G., Sinai, Ya.G.: Generalized variational principles, global weak solutions and behavior with random initial data for systems of conservation laws arising in adhesion particle dynamics. Commun. Math. Phys. 177, 349–380 (1996) 11. Golovin, A.M.: The solution of the coagulation equation for cloud droplets in a rising air current. Izv. Geophys. Ser. 5, 482–487 (1963) 12. Gurbatov, S.N., Malakhov, A.N., Saichev, A.I.: Nonlinear Random Waves and Turbulence in Nondispersive Media: Waves, Rays, Particles. Manchester: Manchester University Press, 1991 13. Lushnikov,A.A.: Certain new aspects of the coagulation theory. Izv.Atmos. Ocean Phys. 14, 738–743 (1978) 14. Marcus, A.H.: Stochastic coalescence. Technometrics 10, 133–143 (1968) 15. Norris, J.R.: Smoluchowski’s coagulation equation: uniqueness, non-uniqueness and hydrodynamic limit for the stochastic coalescent. Ann. Appl. Probab. 9, 78–109 (1999) 16. Norris, J.R.: Cluster coagulation. Commun. Math. Phys. 209, 407–435 (2000) 17. Shandarin, S., Zeldovich, Y.: The large scale structures of the universe: turbulence, intermittency, structures in a self-gravitating medium. Rev. Mod. Phys. 61, 185–220 (1989) 18. Sheth, R.K., Pitman, J.: Coagulation and branching processes models of gravitational clustering. Mon. Not. R. Astron. Soc. (1997) 19. Vergassola, M., Dubrulle, B., Frisch, U., Noullez, A.: Burgers’ equation, devil’s staircases and the mass distribution function for large-scale structures. Astron. Astrophys. 289, 325–356 (1994) 20. Woyczy´nski, W.A.: G¨ottingen Lectures on Burgers-KPZ turbulence. Lecture Notes in Maths. 1700, Berlin-Heidelberg New York: Springer, 1998 Communicated by H. Spohn

Commun. Math. Phys. 232, 83–124 (2002) Digital Object Identifier (DOI) 10.1007/s00220-002-0733-0

Communications in

Mathematical Physics

Density of States for Random Band Matrices M. Disertori1∗ , H. Pinson2 , T. Spencer1 1 2

Institute for Advanced Study, Einstein Drive, Princeton, NJ 08540, USA University of Arizona, Tucson, AZ 85721, USA

Received: 6 February 2002 / Accepted: 17 July 2002 Published online: 7 November 2002 – © Springer-Verlag 2002

Abstract: By applying the supersymmetric approach we rigorously prove smoothness of the averaged density of states for a three dimensional random band matrix ensemble, in the limit of infinite volume and fixed band width. We also prove that the resulting expression for the density of states coincides with the Wigner semicircle with a precision 1/W 2 , for W large but fixed. 1. Introduction Random Matrix Theory (RMT) has proved to be relevant in the study of several physical models. It was initially applied to the study of resonance spectra of complex nuclei and later to the study of the quantum properties of weakly disordered conductors, and the spectral properties of quantum systems which are chaotic in their classical limit [1, 2]. RMT also appears in other fields, such as statistics, number theory and random permutations. See for example [3–5] for recent developments. In this article we study the density of states for a class of Hermitian random matrices Hij whose elements are Gaussian with mean zero and covariance
 Hij Hkl = δj k δil Jij , (1.1) where   denotes the average with respect to the probability distribution of H . In the classical case of GUE (Gaussian unitary ensemble) the indices i, j range from 1 to N and Jij = 1/N. For this case, in the limit N ↑ ∞, the density of states (DOS) is given by Wigner’s famous semicircle law   2 1 1 − E4 |E| ≤ 2 (1.2) ρSC (E) = π 0 |E| > 2 . ∗

Supported by NSF grant DMS 9729992

84

M. Disertori, H. Pinson, T. Spencer

The corresponding eigenfunctions are extended. By this we mean that for large N , the 2   2 ψ = O(1/N ). This fact follows from the Unitary participation ratio j ψj4 / j j invariance of the matrix ensenble. The case of diagonal disorder occurs when we set Jij = 0 for any i = j , and Jii = 1. In this case the eigenfunctions are localized (the participation ratio is uniformly bounded away form zero) and the eigenvalue spacings are described by Poisson statistics. Our analysis will focus on band random matrices for which the indices i, j range over a box  ∩ Zd and Jij is small when |i − j | is larger than some fixed band width W (where |i − j | is the distance in Zd ). Note that by varying the band width W these matrices interpolate between classical random matrix ensembles (GUE or GOE) and diagonal disorder. As we let  ↑ Zd the spectral properties of such matrices should be quite similar to that of a random Schr¨odinger operator on a lattice Zd given by H = − + λV (j ),

(1.3)

where V (j ) are independent random variables,  is the discrete Laplacian and λ ≥ 0 is a parameter giving the strength of the disorder. For example in one dimension the spectra of the random Schr¨odinger [6, 7] and the random band matrices [8] are pure point with exponentially decaying eigenfunctions. Therefore we have localization. In two or more dimensions, localization also holds for energies outside some interval depending on d, λ, W . Thus the study of band random matrices may also prove useful for the study of the random Schr¨odinger operators. The goal of this article is to obtain detailed information about the density of states for a special class of random band matrices in 3 dimensions. We shall consider energies at which extended rather than localized states are expected. More precisely let i, j ∈  ∩ Z3 ,  a set of cubes of side W , and define J in (1.1) by  | 1 − |i−j 1 1 W , Jij :=  4πW (1.4) 2 (1+|i−j |) e −W 2  + 1 ij where  is the Laplacian with periodic boundary conditions in the volume  and W is large but fixed. Our estimates are valid uniformly in the size of  ⊆ Z3 . The average density of states is given by

  1 1 ρ (E) = − lim Im . (1.5) π ε↓0 E + iε − H 00 Note that as  ↑ Z3 , the density of states ρ(E) does not depend on the configuration with probability one. The derivative of ρ (E) is d 1 ρ (E) = lim Im R(E + iε; 0, x), π ε↓0 dE

(1.6)

x∈

where

 R(E + iε; 0, x) =

1 E + iε − H



0x

1 E + iε − H

 x0

.

(1.7)

Density of States for Random Band Matrices

85

Note that for x = 0,



1 E + iε − H

 =0

(1.8)

0x

because of the symmetry Hij → −Hij . Our main result is that for large W and E inside the interval [−2, 2], ρ (E) equals the Wigner semicircle distribution (1.2) plus corrections of order W −2 . Moreover R(x) decays exponentially fast and ρ(E) is smooth in E. These results hold for fixed W and are uniform in ε as ε ↓ 0 and in the volume as  ↑ Z3 . See Theorem 2.1 for a precise statement. When |E| > 2 + O(W −1 ) we expect that ρ(E) is smaller than any power of W −1 and that localization holds. For random Schr¨odinger operators given by (1.3) we have the classic bound by Wegner, ρ(E) ≤ const λ−1 for small λ. This estimate is far from optimal since for small λ we expect the density of states to approach that of the Laplacian. Unfortunately there are no uniform bounds on ρ(E) as λ → 0 or estimates on the smoothness of ρ(E) unless either the distribution of V is Cauchy (in which case the density of states can be explicitly computed) or E lies in an interval for which localized states are proved to exist. Note that for both the random Schr¨odinger and the random band matrix ensembles it is conjectured that for d = 3,

2    1   eixp  ρ(E) , π (1.9)  E + iε − H  Dp 2 + ε 0x x where D is the diffusion constant. Here E must be inside [0, 4d] for the case of random Schr¨odinger or inside [−2, 2] for our band matrix ensemble, and both W −1 and λ are small. This paper does not address this important conjecture. Instead we are using the phase oscillations of the Green’s functions to obtain exponential decay for R(x). To establish our results on Green’s functions we use the supersymmetric formalism of K. Efetov [12, 13] which has its roots in earlier work by Wegner [9, 10]. We recommend the survey article of Mirlin [1] and also the paper of Fyodorov and Mirlin [11] which studies random band matrices in 1 dimension. In the mathematics literature, Constantinescu, Felder, Gawedzki and Kupiainen [14] studied a related n orbital model in the case of strong diagonal disorder using similar supersymmetric methods. For the case of one orbital (n = 1) their off diagonal disorder is governed by J = (− + m2 )−1 with m large. This is in the localized regime. In this paper, we study m small. A. Klein studied also the regularity of density of states for random Schr¨odinger by supersymmetric methods [15] but again only at those energies where localization holds. Our approach allows us to take ε ↓ 0 in the density of states independently of the other parameters such as volume and band width. This permits us to see more detailed structure. Earlier work on band matrices fixes ε and then studies the density of states in the limit when the volume and the band width goes to infinity [16–18] (and references therein). The supersymmetric method enables one to explicitly average the Green’s function over the randomness. This technique involves the use of both real and anticommuting variables. However when we perform our estimates all anticommuting variables are integrated out so that the resulting integrals are just over real variables. As a result of this averaging, the problem is converted into a problem in statistical mechanics whose action has approximately the form

86

M. Disertori, H. Pinson, T. Spencer

A(φ) =



W 2 (∇φ)2 (j ) + U (φ)(j ),

(1.10)

j ∈

where the potential U is a function of the field φ(j ) and has two saddle points. In some respects this problem looks like a double well φ 4 interaction. A more careful analysis of the integral over φ shows that one saddle dominates and it yields the Wigner semicircle distribution. The second saddle is suppressed by a determinant as we shall explain later. The large parameter W ensures that the integral is governed by the saddle and its Gaussian fluctuations. There are similar integrals which appear for random Schr¨odinger operators, however the path integral is much more oscillatory and we can not yet control them unless there are long range correlations in the V (j ).  2   The average of (E + iε − H )−1 0x  can also be calculated with the supersymmetric formalism but the statistical mechanics is now more complicated. Instead of two saddle points there is a non-compact saddle manifold and fluctuations have massless modes which are responsible for the power law (1.9). The remainder of this article is organized as follows. In Sect. 2 we give a precise statement of our results. In Sect. 3 we use the supersymmetric formalism to convert averages of the Green’s function to a model in statistical mechanics. The advantage of this representation is that for large W we see that the integral is dominated by two saddle points. These saddle points and their Hessians are discussed in Sect. 4. The following section is devoted to obtaining our results in a box  ⊆ Z3 of side W . In the last section we show that the analysis in the box can be extended to Z3 using a variant of the cluster expansion. Notation. As in the paper we will need to insert many constants in the different bounds we will denote by K any large positive constant, independent from W and , and by c any small positive constant independent from W and . These constants need not be the same in different estimates. Also we will sometimes use the symbol
to indicate that there is a constant factor K on the right side of the inequality (
stands for ≤ K) without writing K explicitly. For complex variables we adopt the following convention: we denote by S ∗ the complex conjugate of a variable S. We will denote the complex conjugate of a matrix A by A∗ , the transpose by At and the adjoint by A† . Finally, for any set  ⊆ Zd , we denote by || the number of elements in .

2. Model As we said in the introduction, we consider the set H of Hermitian matrices H with entries i, j ∈  ⊂ Zd , d > 0. From (1.1) we see that the probability density is P (H ) =

 dHij dHij∗ ij ∈ i<j

2πJij

e

−

|Hij |2 Jij

 dHii − Hii2 e 2Jii , √ 2π J ii i∈

(2.1)

where < is an order relation on  and J is defined in (1.4). With these definitions H is a set of hermitian random band matrices with band width W . Note that in d = 1 for

Density of States for Random Band Matrices

87

 = [1, N ] we have || = N and Jij = N −1 exp[−|i − j |/N ] which is very close to GUE. For any function of H F (H ) we define the average F (H ) as  F (H ) = dH P (H ) F (H ). (2.2) We study the averaged density of states ρ¯ (E):

 
1 Tr 1 1 , = − lim Im G+ ρ¯ (E) =: − lim Im 00 π ε→0+ || E + iε − H π ε→0+

(2.3)

where “Im” indicates the imaginary part and G+ is the retarded Green’s function: G+ :=

1 . E + iε − H

(2.4)

In the following we restrict to d = 3 and we consider energies well inside the spectrum. For technical reasons we also avoid the energy E = 0. Therefore we consider only energies in the interval I =: {E : |E| ≤ 1.8 and |E| > η}

(2.5)

with η > 0. We assume that our region  is a union of cubes in Z3 of side W . The paper is devoted to the proof of the following theorem Theorem 1. For d = 3, there exists a value W0 such that for all W ≥ W0 the averaged density of states ρ¯ (E) is smooth in E, in the interval I, uniformly in W and  (hence also in the limit  ↑ Z3 ):   n ∂ ρ¯ (E)
Cn ∀ n < n0 (W ), (2.6) E where limW ↑∞ n0 (W ) = ∞. Moreover ρ¯ (E) is the semicircle law with a precision 1/W 2 , |ρ¯ (E) − ρSC (E)|


1 , W2

(2.7)

where ρSC (E) is the semicircle law ρSC defined in (1.2). To establish (2.6) we prove that for large |x|  
+  R(x) =  G+ 0x Gx0 ε=0


1 −c |x| e W. W3

(2.8)

Outline of the paper. In Sect. 5 we establish Theorem 1 on a cube  of side W . We use the supersymmetric formalism to write G+ 00  as a functional integral where a saddle point analysis can be performed. Actually there are two saddles. For d = 3 one saddle is suppressed by a factor e−W (note that this is not true for d ≤ 2). The fluctuations around the saddle are controlled using small probability arguments while the integral near the dominant saddle is estimated by a Brascamp-Lieb inequality [19, 20]. Section 6 is devoted to the cluster expansion which enables us to analyze the limit  ↑ Z3 . The cluster expansion expresses G+ 00  as a sum over finite volume contributions Y ⊂ Z3 which are again unions of cubes of side W . We show that large Y terms give small contributions to G+ 00 . This expansion also enables us to prove the bound on R(x).

88

M. Disertori, H. Pinson, T. Spencer

3. Supersymmetric Approach In this section we shall use the algebraic formalism of supersymmetry to express our average Green’s function in terms of a functional integral, which, apart form a determinant, is local. Let J be given by (1.4). Lemma 1. The averaged Green’s function can be written as     
+ 1 t −1 (Eε − ibi ) t −1 G00 = da db exp − (a J a + b J b) 2 (Eε − ai ) i∈

1 × det[J −1 − F (a, b) − F  (a0 , b0 )], (Eε − a0 )  where ai , bi (i ∈ ) are real variables, the measure is da db = j ∈ daj dbj , a t J −1 a = ai Jij−1 aj

(3.1)

(3.2)

i,j ∈

and the same definition holds for bt J −1 b. Finally we defined Eε = E + iε and F (a, b) and F  (a0 , b0 ) are matrices with elements F (a, b)ij := δij

1 , (Eε − ai )(Eε − ibi )

F  (a0 , b0 )ij := δi0 δj 0

1 . (Eε − a0 )(Eε − ib0 )

(3.3)

(3.4)

Note that for each ai there is a pole at ai = E + iε while bi has no singularity (as it appears only in the numerator). This expression is then well defined only for ε > 0. + By the same technique we obtain a similar formula for G+ 0j Gj 0  and in general for + + G+ 0j1 Gj1 j2 ... Gjn 0 . Remarks. Note that if we omit the observable, that is we omit (Eε − a0 )−1 and F  in (3.1), we are actually computing 1 = 1, thus      1 t −1 (Eε − ibi ) t −1 1 = da db exp − (a J a + b J b) det[J −1 − F (a, b)] . 2 (Eε − ai ) i∈

(3.5) Proof. Note that the Green’s function can be written as a functional integral:    −i + ∗ † Gkl = dS dS exp iS (E − H )S det[−i(Eε − H )] Sk Sl∗ , (3.6) ε (2π )||  where we defined dS ∗ dS = j ∈ dSj dSj∗ , for each j ∈  Sj is a complex variable and S † (Eε − H )S = Si∗ (Eε δij − Hij )Sj . (3.7) i,j ∈

Finally the determinant is the normalization factor.

Density of States for Random Band Matrices

89

In order to insert all the H dependence in the argument of the exponential we introduce integrals over Fermionic (anticommuting) fields: det[−i(Eε − H )] = (2π)|| where dχ ∗ dχ = and



∗ j ∈ dχj dχj ,



  dχ ∗ dχ exp iχ † (Eε − H )χ ,

(3.8)

for each j ∈  χj is a complex Fermionic variable

χ † (Eε − H )χ =



χi∗ (Eε δij − Hij )χj .

i,j ∈

(3.9)

Therefore we can write G+ kl

 = −i

  d6∗ d6 exp i6† (Eε − H )6 Sk Sl∗ ,

where, as before, we defined d6∗ d6 = superfield 6j (j ∈ )  6j =

Sj χj





∗ j ∈ d6j d6j

(3.10)

and we have introduced the

  6†j = Sj∗ , χj∗ .

(3.11)

Note that now (Eε − H ) has two additional indices: α, β = 0, 1 corresponding to the αβ Bosonic S or Fermionic χ component of 6: (Eε − H )ij = δαβ (Eε − H )ij . Therefore  6 (Eε − H )6 ≡ 6 †

†

Eε − H 0 0 Eε − H

6 = S † (Eε − H )S + χ † (Eε − H )χ . (3.12)

For supersymmetric formalism and notation we adopted the conventions in the review by Mirlin [1], which are summarized in App. A. Now we can perform the average over H : 





exp −i6† H 6



 1 = exp − Jij (6†i 6j )(6†j 6i ) . 2

(3.13)

ij

To convert this quartic interaction into a quadratic one we perform a Hubbard-Stratonovich transformation: Jij (6†i 6j )(6†j 6i ) = [Ai Jij Aj − Bi Jij Bj + 2Pi∗ Jij Pj ], (3.14) ij

ij

where Ai = Si∗ Si ,

Bi = χi∗ χi ,

Pi = Si∗ χi ,

Pi∗ = Si χi∗ ,

(3.15)

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M. Disertori, H. Pinson, T. Spencer

and there is no sum over i. Note that Ai and Bi are commuting variables while Pi and Pi∗ are anticommuting ones. Now     dai − 1 a t J −1 a−ia t A 1 t − || , exp − A J A = (2π) 2 e 2 √i∈ 2 det J     || dbi − 1 bt J −1 b+bt B 1 exp + B t J B = (2π)− 2 , e 2 √i∈ 2 det J   ∗   i∈ dρi dρi −ρ † J −1 ρ−iρ † P −iP † ρ exp −P † J P = (2π)|| e , (3.16) det J −1 where ai , bi are real Bosonic fields and ρi is a complex Fermionic field for all i ∈ . Therefore     1 t −1 t −1 † −1 † † da db dρ ∗ dρ e− 2 (a J a+b J b+2ρ J ρ)−i6 R6 , (3.17) exp −i6 H 6 = where 6 R6 := †

i∈

 6†i Ri 6i ,

Ri :=

ai ρi∗ ρi ibi

.

(3.18)

Ri is actually a supermatrix, containing both Bosonic and Fermionic variables. For such a matrix we can define the notion of transpose, complex conjugate, determinant and trace and it can be shown that the usual properties of the vector and matrix algebra hold. We summarize the notations in App. A. Using this formalism we have   

 δkl 1 1 ∗ i6† (Eε −H )6 ∗ , (3.19) Sk S l = d6 d6 e −i Eε − Rk 11 Sdet(Eε − Ri ) i

where

  1 (Eε − ai ) ∗ Sdet(Eε − Ri ) = 1 − ρ i ρi , (Eε − ibi ) (Eε − ai )(Eε − ibi ) 

1 Eε − R k

Therefore


 G+ 00 =

11

 −1 1 1 ∗ = . 1 − ρ k ρk (Eε − ak ) (Eε − ak )(Eε − ibk )

(3.20)

(3.21)

  (Eε − ibi ) da db dρ ∗ dρ e (Eε − ai ) i∈    −1 −1  1 ρi∗ ρi ρ0∗ ρ0 1− × 1− . (Eε −a0 )(Eε −ib0 ) (Eε −ai )(Eε − ibi ) (Eε − a0 )





− 21 (a t J −1 a+bt J −1 b+2ρ † J −1 ρ)

i∈

(3.22) The integration over the Fermionic fields can be performed exactly. Using the property: ρi2 = (ρi∗ )2 = 0 ∀i we observe that   −1  ρi∗ ρi ρi∗ ρi 1− = exp , (3.23) (Eε − ai )(Eε − ibi ) (Eε − ai )(Eε − ibi ) therefore the integration over ρ and ρ ∗ reduces to the following expression:

Density of States for Random Band Matrices



dρ ∗ dρ e−ρ

† [J −1 +F (a,b)+F  (a ,b )]ρ 0 0

91

= det [J −1 − F (a, b) − F  (a0 , b0 )],

(3.24)

where F (a, b) and F  (a0 , b0 ) are defined in (3.3–3.4). We obtain then the expression (3.1).

4. Saddle Point Analysis In this section we shall deform the integral (3.1) over aj and bj so that they pass through certain complex saddle points. If we ignore the determinant in (4.9) and the kinetic term, we show that the resulting integrand has a double well structure, with the two wells of the same height. In Sect. 5.1.2 (Lemma 6) we will see that the determinant actually suppresses one of the two saddles by a factor e−W . Saddle points. Observing the integrand in (3.1) we remark that the factor −W 2  in a t J −1 a + bt J −1 b forces the fields a and b to be approximately constant. Therefore if we ignore the determinant, the leading contribution to the integrand (hence also to the saddle structure) is then e

2 2 − || 2 (a +b )



(Eε − ib) (Eε − a)

||

 || = e−[f1 (a)+f2 (b)] ,

(4.1)

where the fields a and b are constant (ai = a, bi = b for all i ∈ ) and we defined a2 + ln(E − a), 2 b2 f2 (b) = − ln(E − ib). 2

f1 (a) =

(4.2)

Note that in this approximation the saddle points for GUE and Random Band Matrix are the same. The critical points of f1 and f2 are given by as = Er ± iEi , bs = −iEr ± Ei ,

(4.3)

where we defined  E E2 E := Er − iEi := −i 1− . 2 4

(4.4)

Note that E satisfies E − E = E ∗,

EE ∗ = 1

∀ |E| < 2 .

(4.5)

Spectrum. Note that, if |E| < 2 the saddles as , bs have non-zero imaginary parts even as ε ↓ 0. For |E| ≥ 2 + O(W −1 ) we expect that the density of states is smaller than any power of W −1 , for W large.

92

M. Disertori, H. Pinson, T. Spencer

Contour deformation. We deform the integration contour in order to pass through a saddle point. To avoid crossing the pole ai = Eε we have to pass through the saddle as = E. On the other hand the choice for bs is arbitrary, as there is no pole in b, but it turns out (see Sect. 5.1) that bs = −iE is the dominant contribution. Note that f1 (E) = f2 (−iE) = (1 − E 2 ) .

(4.6)

Hence the Hessian at this saddle point is B −1 = −W 2  + (1 − E 2 )

(4.7)

Lemma 2. We perform inside (3.1) the translation aj → aj + E, bj → bj − iE ∀ j ∈ 

(4.8)

and take the limit ε ↓ 0. The integral can then be written as 
+   G00 ε=0 = dµB (a, b) det[1 + (D + D0 )B] eV0 + j ∈ Vj ,

(4.9)

where dµB (a, b) (defined in 4.10) is a gaussian measure with covariance B (defined in  4.7), the factor exp[ j Vj ] (4.11) is what remains in the exponential after the Hessian has been extracted and det[1 + DB] (4.12) corresponds to det[J −1 + F ] after the normalization factor det B −1 has been extracted. Finally exp V0 and D0 (4.13–4.14) are the contributions from the observable. More precisely we define the measure as dµB (a, b) = dµB (a) dµB (b), where 1 t −1 B a

dµB (a) = da e− 2 a

√

1 det B

,

1 t

dµB (b) = db e− 2 b B

−1 b

√

1 det B

(4.10)

and B is the Hessian around the saddle, defined in (4.7) . The normalization factor for the measure has been extracted from the determinant. The interactions are given by Vj = Vj (aj ) + Vj (bj ) and Dij = δij Di , where 1 Vj (aj ) =

dt (1 − t) 0

2

aj3

(E ∗ − taj )3 !3

1 Vj (bj ) = −

dt (1 − t)2 0

,

(E ∗

(4.11)

ibj , − tibj )3

Di = [E 2 − F (a + E, b − iE)ii ] 1 = E2 − ∗ (4.12) (E − ai )(E ∗ − ibi )    1 ai ibi . dt + ∗ =− ∗ 2 ∗ (E − tai ) (E − ibi t) (E − tai )(E ∗ − ibi t)2 0 Finally the contributions from the observable are given by V0 and D0 where V0 = − ln(E ∗ − a0 ),

(4.13)

Density of States for Random Band Matrices

93

(D0 )ij = −F  (a0 + E, b0 − iE)ij = −δi0 δj 0

(E ∗

1 . − a0 )(E ∗ − ib0 )

(4.14)

The proof is a straightforward change of variables and a reorganization of the resulting expression. Note that for any |E| < 2 there is no pole in a as the factor E − E − ai is always at a distance at least Ei from zero. For the special value E = 0, a singularity in bi = 1 seems to appear from the factor 1/ i(1−bi ) in the argument of the determinant. This is not a real singularity as there is the same factor in the numerator outside the determinant. Nevertheless to avoid additional technical problems we avoid E = 0 in the following. This is the reason why we chose η > 0 in I in Theorem 1. Properties of the Hessian. The Hessian B −1 (4.7), which is the covariance of the Gaussian measure after the translation, has now a complex mass term:  (1 − E 2 ) = 2 1 −

E2 4

 + iE 1 −

E2 =: m2r + im2i . 4

(4.15)

Note that for |E| < 2 the real part m2r is positive and this ensures the convergence of the integral. In the following, as we will need to treat in a different way the real and imaginary part of B −1 , we introduce the real covariance C, 1 , −W 2  + m2r

(4.16)

B −1 = C −1 + im2i .

(4.17)

C := therefore

Note that C is positive as a quadratic form and pointwise. In momentum space C is written as 1 ˆ k) = 1 , C( d 2 W 2 i=1 (1 − cos ki ) + (mr /W )2 ki = 2π

ni , ||1/d

(4.18)

ni = 0, . . . , ||1/d − 1 .

When  ↑ Zd , ki becomes a continuum variable ki ∈ [0, 2π ]. The spatial decay depends on the dimension. In the particular case of d = 3, 0 < Cij


1

mr

W 2 (1 + |i

− j |)

e−|i−j | W .

(4.19)

The covariance B has the same expression as C, but with an imaginary term in the mass. It is easy to prove that B decays in the same way as C,   Bij 


1 W 2 (1 + |i

mr

− j |)

e−|i−j | W .

(4.20)

94

M. Disertori, H. Pinson, T. Spencer

F1(a) 1

a

0 F2(b) 1

0

εi

2εi

b

Fig. 1. Behavior of F1 (a) and F2 (b)

Properties of the interaction. After the translation the functions f1 , f2 introduced in (4.2) become  1 f1 (a) = − 1 − E 2 aj2 + Vj (aj ), 2 (4.21)  1 2 2 f2 (b) =: − 1 − E bj + Vj (bj ). 2 Note that after the translation there are also constant factors arising from f1 and f2 which cancel. In the following we will insert absolute values in the integral, in order to obtain our estimates. We then have to study the behavior of         F1 (a) =: e−f1 (a)  , F2 (b) =: e−f2 (a)  . (4.22) It is easy to prove that for |E| ≤ 1.8 F1 (a) has only one maximum, in a = 0, of height 1 (see Fig. 1). Note that when 1.8 < |E| ≤ 2 zero is no longer the maximum of F1 (a) and this is why we restrict E to I given by (2.5). Nevertheless, there is still a single saddle point so we expect that by suitable deformation of the contour we should be able to extend our result to the interval |E| ≤ 2 − O(W −1 ). On the other hand, for any value of |E| < 2, F2 (b) has two maxima, which do correspond to the two saddles, one in b = 0 and one in b = 2Ei . Both maxima have height 1 (Fig. 1). We will see in the next section that the second maximum is suppressed by a factor e−W from the determinant appearing in (4.9). 5. Finite Volume Estimate We prove now Theorem 1 in a fixed cube  of side W , with 0 ∈ . We prove the boundness of ρ¯ (E) in Theorem 2, then in Theorem 3 we prove the bounds on the derivatives and on |ρ¯ (E) − ρSC |: these bounds follow by the same technique used for the bounds on ρ¯ (E), with some slight modifications.

Density of States for Random Band Matrices

95

Theorem 2. For  as above, there exists a value W0 such that for all W ≥ W0 and for all E ∈ I, where I is defined in (2.5), the averaged density of states ρ¯ (E) is bounded uniformly in W and , |ρ¯ (E)| ≤ K

(5.1)

Theorem 3. For  as above, there exists a value W0 such that for all W ≥ W0 and for all E ∈ I, where I is defined in (2.5), we have  n  ∂ ρ¯ (E) ≤ Cn ∀ n < n0 (W ), (5.2) E K |ρ¯ (E) − ρSC (E)| ≤ , (5.3) W2 uniformly in  and W . 5.1. Proof of Theorem 2. Inserting the absolute values in the expression (4.9) we have    
+     G  ≤ |dµB (a, b)| det[1 + (D + D  )B] eV0 + j ∈ Vj  (5.4) 0 00 ε=0 The absolute values of dµB and det[1 + (D + D0 )B] are bounded through Lemma 3 and 4 respectively. Lemma 3. The total variation of the complex measure is bounded by |dµB (a, b)|
e

O

|| W3

dµC (a, b).

(5.5)

Proof. The measure dµB (a, b) can be written as    det B −1   dµC (a, b),  |dµB (a, b)| =  det C −1 

(5.6)

where the determinants are the normalization factors for the two measures and can be written as      det B −1    = det 1 + im2 C  . (5.7) i  det C −1  Note that for any matrix A satisfying Tr A† A < ∞ the following inequality is true:   1 †   |det(1 + A)| ≤ eTr A  e 2 Tr A A . (5.8) In our case A = iδC, therefore Tr A is imaginary, and the norm of the first exponential is one. The second exponent gives Tr A† A = m4i Tr C 2 = m4i The bound in (5.5) then follows.

i,j ∈

" !

|i−j | m4i 1 1 − mr W e ≤ . m2r W 4 |i − j |2 W3

i∈

(5.9)

96

M. Disertori, H. Pinson, T. Spencer

Lemma 4. The determinant of 1 + (D + D0 )B is bounded by  ||    det[1 + (D + D  )B] ≤ K W 3 eTr (D+D0 )B  . 0

(5.10)

Proof. The proof is obtained by applying (5.8) and repeating the same arguments as in the lemma above. Note that we applied supab |D(a, b) + D0 (a, b)| = K for some constant K independent from W . ! " Applying the lemmas above we have      || 
+       G  ≤ K W3 dµC (a, b) eTr (D+D0 )B  e j ∈ Vj  , 00 ε=0

(5.11)

where we bounded | exp(V  )| = |E ∗ − a0 |−1 ≤ K, and V  in defined in (4.13). Partitioning the domain of integration. In order to distinguish small field and large field regions we partition the integration domain by inserting 1=

5

χ [I k ]

(5.12)

k=1

as follows: 

5         dµC (a, b) eTr (D+D0 )B  e j ∈ Vj  = Tk , k=1



Tk =:

  (5.13)   Tr (D+D0 )B   j ∈ Vj  dµC (a, b)e  e  χ [I k ],

where χ [I k ] is the characteristic function of the set I k and # " 1 2  I 1 = a, b : |aj |, |bj − bj  | ≤ , ∀j, j ∈  and |b | ≤ 0 1 1 W8 W8 # (5.14) " 1 2 2  I = a, b : |aj |, |bj − bj  | ≤ , ∀j, j ∈  and |b0 − 2Ei | ≤ 1 1 W8 W8 # " 1 I 3 = a, b : bj ∈ R ∀j and ∃ j ∈  s.t. |aj | > , 1 8 W # " 1 1   I 4 = a, b : |aj | ≤ , (5.15) ∀j ∈  and ∃ j, j ∈  s.t. |b − b | > j j 1 1 8 8 W W # " 1 2 5   I = a, b : |aj |, |bj − bj | ≤ . ∀j, j ∈  and |b0 |, |b0 − 2Ei | > 1 1 W8 W8

Density of States for Random Band Matrices

97

Small field region. The first two intervals correspond to the small field region. T1 is the leading contribution and corresponds to the case when all a fields and all b fields are near zero. In this case the interacting terms of the measure do not destroy the log convexity of the Gaussian dµC , therefore we can apply a Brascamp-Lieb inequality [19, 20] which states Brascamp-Lieb Inequality:. Let dµH (x) =: dx1 . . . dxN

1 1 e− 2 H (x) , Z(H )

(5.16)

where x = (x1 , . . . , xN ) ∈ RN , H (x) is a positive function symmetric under x → −x, and the partition function is  1 (5.17) Z(H ) =: dx1 . . . dxN e− 2 H (x) . Then if H  ≥ C −1 > 0 the following inequalities hold:   n > 0, dµH (x) |xi |n ≤ dµC (x) |xi |n   dµH (x) e(f,x) ≤ dµC (x) e(f,x) ,

(5.18) (5.19)

N where dµ C (x) is the free measure with covariance C, f is any vector in R , and (f, x) = i fi xi .

The second term corresponds to the case when all the a fields are near zero and all the b fields are near the second saddle 2Ei (see Fig. 1). In this case we bound the interaction (trace and Vj factors) by sup norm. The large contributions are now suppressed by a small exp[−W ] factor, from the trace bound. Large field region. The last three intervals correspond to the large field region. In all these cases we bound the interaction terms (the trace and Vj ) by sup norm in terms of quadratic and linear expressions in a and b. The large contributions from this bound are then compensated by the small probability factor (as the large field region is very unlikely). Note that the b field bounds are more delicate because of the double well structure (see Fig. 1). Below we analyze the integration restricted to each interval. 5.1.1. Small field region: Leading contribution T1 . We consider the leading contribution T1 . In the region I 1 all the a fields and the b fields are near 0. We apply Re V (aj ) ≤ K |aj |3

if |aj | << 1,

Re V (bj ) ≤ K |bj |3

if |bj | << 1,

(5.20)

and we bounded the Tr(D + D0 )B applying |Dj |
|aj | + |bj |, |D0 |
1.

(5.21)

98

M. Disertori, H. Pinson, T. Spencer

Therefore we can write   dµC (a, b) eK j T1 ≤

|aj |3 +|bj |3

!

eK



j

(|aj |+|bj |)W −2 χ [I 1 ].

Now we insert the cubic and linear contributions in the measure by this definition    | |a j |aj |3 + 2  , (5.22) H (a) =: a t C −1 a − K W j  1 Z(H ) =: da e− 2 H (a) χ [I 1 ], (5.23) dµH (a) =:

1 1 da e− 2 H (a) χ [I 1 ]. Z(H )

(5.24)

The same definitions hold for the b fields. Now we have   !   3 3 −2 Z(H ) 2 dµC (a, b) eK j |aj | +|bj | eK j (|aj |+|bj |)W χ [I 1 ] = , Z0 where the free partition function is Z0 =



1 t −1 C a

da e− 2 a

.

(5.25)

(5.26)

Therefore we actually have to estimate the normalization factor of the interacting measure. This is done through Lemma 5 below. Lemma 5. With Z(H ) and Z0 defined by (5.23) and (5.26) we have Z(H ) ≤ e Proof. Let H (t) defined as

O



|| W3



Z0 .

  | |a j t |aj |3 + 2  H (t)(a) =: a t C −1 a − K W

(5.27)



(5.28)

j

interpolate between H and C −1 . Note that on I1 ,    1 Z(H (1)) d ln dt = ln Z(H (t)) Z(H (0)) dt 0    1  |aj | 3 dt dµH (t) (a) |aj | + 2 = W 0 j     1 |aj | || 3 dt dµCf (a) |aj | + 2 ≤ K 3 , ≤ W W 0

(5.29)

j

where we defined dµH (t) (a) as in (5.24). In the last line we used Brascamp-Lieb (5.18) together with H  ≥ C −1 − f m2r = Cf−1 > 0

(5.30)

Density of States for Random Band Matrices

99

  1 which is valid on I1 and for f = O W − 8 . In general we will use this definition of Cf for f a constant 0 < f < 1 or, when  is a set of cubes, f a diagonal matrix constant on each cube. Now   (5.31) Z(H (0)) = dµC χ [I 1 ] ≤ Z0 . This ends the proof.

" !

Applying Lemma 5 we have T1 ≤ e

O



|| W3



≤ K.

(5.32)

5.1.2. Small field region: Contribution from the second saddle. In this section we show that T2 ≤ e−cW . This means that the fields have actually the same behavior as in a large field region. Note that this property holds only in three dimensions. In the interval I 2 all near the second saddle1 $ the a fields are near 0 and the b fields are 1 2Ei . Recall that Ei = 1 − E 2 /4. Note that for all |aj | ≤ W − 8 and |bj − 2Ei | ≤ W − 8 we have m2r fa aj2 , 2 ! m2 Re V (bj ) ≤ r fb bj2 + (1 − fb )m2r 2Ei bj − Ei 2

Re V (aj ) ≤

(5.33) (5.34)

1

with fa = fb = O(W − 8 ). Note that for b there is a linear contribution coming from the translation to the second saddle. Moreover Tr(D + D0 )B can be bounded applying Lemma 6 below. 1

1

Lemma 6. If |aj | ≤ W − 8 and |bj − 2Ei | ≤ W − 8 , then the real part of [(D + D0 )B]jj is bounded by Re (D + D0 )j Bjj ≤ −c W −2 ,

(5.35)

where c > 0 is some constant independent from W . Proof. Note that

& % Re (DB)jj = Re Dj Re Bjj − Im Dj Im Bjj .

The key point is that, for aj near zero and bj near the second saddle we have  1 , Re Dj = − m2r + O 1 8 W  1 2 Im Dj = − mi + O . 1 W8

(5.36)

(5.37)

Note that these estimates are not true in other regions. If both aj and bj are near zero Dj  0 while for aj or bj far from the saddle we can only say that |Dj | ≤ const. For D0 we only need to know that for any a0 , b0 ∈ R, D0 = O(1).

(5.38)

100

M. Disertori, H. Pinson, T. Spencer

Now, by simple Fourier space analysis we see that 1 , 2 W  1 . =O W3

Re Bjj ≥ c

(5.39)

Im Bjj

(5.40) " !

Inserting these estimates in (5.36) the proof follows. Inserting all these results in T2 we have T2 ≤ e

−c ||2



W

dµC (a, b) e

m2r 2



fa



j

aj2 +fb



j

bj2



e(1−fb )mr 2Ei 2



j

(bj −Ei ) .

(5.41)

We insert the quadratic terms in the measure. The normalization ratio are bounded using Lemma 7 below Lemma 7. For any 0 < f < 1 we have ' det

C −1

(

1 O
e 1−f

Cf−1



f || W2



,

(5.42)

where Cf is defined in (5.30). Proof. Diagonalizing the matrices we can write ' det

C −1

( =

Cf−1

 k

2

3

2 2 i=1 (1 − cos ki )W + mr 2 2 i=1 (1 − cos ki )W + mr (1 − f )

2 3

(5.43)

  1  f m2r Tr Cf0  1 O f || 2 ≤ e W , ≤ e 1−f 1−f

where we defined Cf0 as the covariance Cf where the zero mode has been extracted. This ends the proof. ! " Therefore we can write T2 ≤

O 1 (1−fa )(1−fb ) e

≤ eO(W W

− 81

)



fa || fb || + 2 W2 W



e

−c

|| W2



dµCfb (b) e(1−fb )mr 2Ei 2

e−c W
e−c W ,



j

(bj −Ei ) (5.44)

1

where we inserted || = W 3 and fa = fb = O(W − 8 ) and we applied   2 dµCfb (b) e(1−fb )mr 2Ei j (bj −Ei ) = 1 .

(5.45)

Density of States for Random Band Matrices

101

5.1.3. Large field region. This is the region selected by the intervals I 3 (one a field large) I 4 (one pair of b fields with |bj − bj  | large) and I 5 (all b fields far from both saddles). We apply the following inequalities:  1   , (5.46) Re (D + D0 )i Bii ≤ sup |(D + D0 )i | |Bii |
O W2 a,b m2r fa aj2 , 2 m2 Re V (bj ) ≤ r fb bj2 + O(1 − fb ), 2 Re V (aj ) ≤

(5.47) (5.48)

with 1/2 < fa < 1, fb = 1 − W −3 . These estimates are true for any value of aj and bj ∈ R. On the other hand, when we are in the interval I 5 , all b fields must be far from both saddles and the interaction is exponentially small, therefore we gain an additional small factor:  m2r 1 2 2 Re V (bj )|I 5 ≤ . (5.49) fb bj + O(1 − fb ) − c 1 2 W8 Note that the factor O(1 − fb ) comes from the contribution of the second saddle (see Fig. 1). Therefore we can write O



||



+(1−f )||

b T3 + T4 + T5 ≤ e W 2   #   2 " 1 m2r 2 3 −4 j aj +fb j bj 2 fa χ [I 3 ] + χ [I 4 ] + e−c W W . · dµC (a, b) e

(5.50) We insert the quadratic terms in the measure: 

T3 + T4 + T5 ≤







(f +f )|| 1 O a 2b O || +(1−fb )|| W e e W2 (1 − fa )(1 − fb ) # "  1 3 −4 , · dµCfa (a) dµCfb (b) χ [I 3 ] + χ [I 4 ] + e−c W W

(5.51) where we defined Cfa and Cfb as in (5.30) and we applied Lemma 7. To bound the contributions from I 3 and I 4 we apply the following lemma. 1

1

Lemma 8. The probability of having one |aj | > W − 8 or one pair |bj − bj  | > W − 8 is exponentially small  1 2 −4 (5.52) dµCa (a) χ [I 3 ]
W 3 e−c W W ,  1 2 −4 (5.53) dµCb (b) χ [I 4 ]
W 6 e−c W W .

102

M. Disertori, H. Pinson, T. Spencer

Proof. We consider first the integral for a     1 dµCfa (a) χ [I 3 ] ≤ dµCfa (a) χ |aj | > W − 8 j

≤

 j

≤



2e

dµCfa (a) 1 2 2 x (Cfa )jj

e−xaj + exaj 1+e

e−x W

!

1 −2 x W − 8

− 18

e−x W


W 3 e−c W

− 81

1 2 W−4

,

(5.54)

j 1

where we applied (Cfa )jj = O(1/W 2 ) and we set x = O(W − 8 W 2 ). The same proof holds for the b field. In this case the presence of a difference bj − bj  is crucial to ensure that the factor [(Cfb )jj + (Cfb )j  j  − 2(Cfb )jj  ] is of order W −2 and does not depend on the mass (which could be very tiny for Cfb ). The factor W 6 comes from the sum over " j and j  . ! Putting together all the factors we have 

T 3 + T4 + T5 ≤







(f +f )|| 1 O a 2b O || +(1−fb )|| W e W2 e (1 − fa )(1 − fb ) # " 1 1 2 −4 3 −4 · (W 3 + W 6 )e−c W W + e−c W W


e−c W

1 3W − 4

,

(5.55)

where we have inserted || = O(W 3 ), fa = 3/4 and fb = 1 − (1/W 3 ). Note that there is an additional factor W 3 from the zero mode (1 − fb )−1 of the determinant. 5.1.4. Sum over the different regions. Summing the bounds on different intervals we have finally    3− 1 
1 + e−c W + e−c W 4 ≤ K. ε=0


+  G 00

(5.56)

This completes the proof of (5.1). 5.1.5. Large volume. It is straightforward to extend the above estimates to the case when  is a union of cubes. Corollary 1. The density of states in a union of cubes  is bounded by |ρ¯ (E)|
e

O



|| W3



.

(5.57)

Proof. In each cube we apply the bounds above. The result is written as a quadratic form exp[v T Cf v], where v is a vector which depends on the bounds on each particular cube and f is now a diagonal matrix which is constant on each cube. The key point is that Cf ≤

1 . −W 2 N + (1 − f )m2r

(5.58)

Density of States for Random Band Matrices

103

where N is a Laplacian with Neumann boundary conditions on the cubes, and decouples the cubes automatically. Now we can perform the estimates in each cube separately. This completes the proof of (5.57). ! " 5.2. Proof of Theorem 3. To prove this result we integrate by parts to generate perturbative terms. To control the remainder we apply the bounds of Theorem 2. 5.2.1. Semicircle law. We prove that ρ(E) ¯ = ρSC with a precision of order W −2 :  1 ρ¯ (E) = ρSC + O . (5.59) W2 Note that ρ¯ (E) = − π1 ImG+ 00 , therefore we have to study  1 G+ dµB (a, b) ∗ eV det[1 + (D + D0 )B]. 00  = (E − a0 )

(5.60)

We have to perform a few steps of perturbative expansion on the observable (E ∗ − a0 )−1 and D0 . These are more clear if we write the determinant as a Fermionic integral,  ! ∗  det[1 + (D + D0 )B] = dµB (ρ ∗ , ρ) e−ρ ρD 1 − D0 ρ0∗ ρ0 , (5.61) where we defined dµB (ρ ∗ , ρ) = det B e−ρ ρ ∗ ρD =

j

and Dj and as

D0

† B −1 ρ

,

(5.62)

ρj∗ ρj Dj ,

(5.63)

are introduced in (4.12) and (4.14). The density of states is then written G+ 00  =



dµB (a, b, ρ ∗ , ρ) eV −ρ

∗ ρD

O0 ,

(5.64)

where we defined dµB (a, b, ρ ∗ , ρ) = dµB (a, b)dµB (ρ ∗ , ρ) and the observable O0 is ! 1 O0 = ∗ 1 − D0 ρ0∗ ρ0 (E − a0 )  1 1 1 D0 ρ0∗ ρ0 . dt ∗ − ∗ (5.65) = E + a0 2 (E − ta ) (E − a ) 0 0 0 The first term is a constant and gives the semicircle law − π1 ImE = ρSC . Note that we apply  ∗ dµB (a, b, ρ ∗ , ρ) eV −ρ ρD = 1. (5.66) The remaining two terms give the corrections   1   1 ∗ , dt ∗ δρ1 = dµB (a, b, ρ ∗ , ρ) eV −ρ ρD a0 (E − ta0 )2 0  % & ∗ δρ2 = dµB (a, b, ρ ∗ , ρ) eV −ρ ρD −D0 ρ0∗ ρ0 .

(5.67) (5.68)

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M. Disertori, H. Pinson, T. Spencer

Estimate of δρ2 . We first consider the estimate on the second integral, as it is the easiest one. We partition the integral domain inserting (5.12) as in Sect. 5.1 and we perform the Fermionic integral in a different way depending on the region. Near the first saddle (interval I 1 ) we apply   1 ∗ − dµB (ρ ∗ , ρ) e−ρ ρD ρ0∗ ρ0 = det[1 + DB]. (5.69) B −1 + D 00 −2 ). It is easy to see that for aj and bj near zero Dj  0 and |(B −1 + D)−1 00 | = O(W Therefore we have       −D0 1 1 V  
1 , |δρ2 (I )|
2 dµB (a, b) e det[1 + DB] (5.70) W (E ∗ − a0 )  W2

where we applied the same bounds as in Sect. 5.1.1. In the other regions I k , k = 0 we cannot apply (5.69) as (B −1 + D)−1 is not well defined (D is big and may cancel B −1 ). Therefore we apply  ∗ − dµB (ρ ∗ , ρ) e−ρ ρD ρ0∗ ρ0 = det M, (5.71) where M is the matrix 1 + DB with the row 0 substituted with B: Mij = (1 + DB)ij Mi0 = Di Bi0 M0j = B0j

i, j = 0, i = 0, ∀ j.

(5.72)

If we apply (5.8) we obtain the same bounds as in Sect. 5.1. Therefore performing the same bounds as in Sects. 5.1.2–5.1.3, we have |δρ2 (I k )|
e−c W . (5.73) k =1

Hence |δρ2 |


1 1 + e−c W
. 2 W W2

(5.74)

Estimate of δρ1 . Now we consider the first error term. Before inserting the partition (5.12) and integrating over the Fermionic integrals we have to perform one step of integration by parts  ∗ δρ1 = B0k dµB (a, b, ρ ∗ , ρ) eV −ρ ρD k



! δ δ × Vk − ρk∗ ρk Dk + δ0k δak δa0



1 0

 1 . dt ∗ (E − ta0 )2

(5.75)

Note that | δaδ k Vk |
|ak |2 + |ak |3 . In the region around the first saddle (I 1 ), applying the Brascamp-Lieb inequality (5.18), these fields give a factor W −2 . In the other regions they

Density of States for Random Band Matrices

105

are bounded by the exponential mass decay. The contribution from ρk∗ ρk is estimated as in δρ2 above. Therefore ' ( 1 1 |δρ1 |
|B0k | + |B00 |
, (5.76) W2 W2 k

where we applied (5.59)



k

|B0k | ≤ const and |B00 | = O(W −2 ). This ends the proof of

5.2.2. Smoothness. Now we consider the derivatives. Note that it is easier to compute the derivatives on the starting expression G+ 00  than directly on the functional integral (5.60). The derivative at order n is given by
 
∂En ρ¯ (E) = −n!(−1)n π1 Im (Gn )00 ∝ Im G0j1 ...Gjn 0 . (5.77) j1 ,...,jn

Applying the supersymmetric approach and the saddle point analysis as in Sects. 3–4, we can write for instance R(x) as R(x) = G0x Gx0  = O0 Ox SU SY − O0 SU SY Ox SU SY , where we defined


 F (a, b, ρ ∗ , ρ) SU SY =:



dµB (a, b, ρ ∗ , ρ) eV −ρ

∗ ρD

F (a, b, ρ ∗ , ρ)

(5.78)

(5.79)

and the observables are (5.65) and Ox =

(E ∗

! 1 1 − Dx ρx∗ ρx . − ax )

(5.80)

A similar formula holds for the general case. We perform now integration by parts starting from O0 until we have a path of connected vertices that connects 0 to j or we have enough vertices to extract a factor W −3 for each observable Oj . This factor ensures that we can sum over the position of j inside the cube . Note that, as in general we will have to estimate products of fields, both Fermionic and Bosonic, we will need the two lemmas below. Lemma 9. Let us consider the average of the product of p Fermionic fields  p    ∗ ρik ρj∗k . dµB (ρ ∗ , ρ) e−ρ ρD

(5.81)

k=1

Note that ik and jk are not necessarily equal. This integral gives different estimates depending on the region we are considering. If a and b are near zero we have    p    p!2 ∗   ρi∗k ρjk  ≤ | det(1 + DB)| χ [I 1 ]. (5.82)  dµB (ρ ∗ , ρ) e−ρ ρD χ [I 1 ]   W 2p k=1

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On the other hand, in the other regions I s , s = 1 we have   p   ∗ ρj∗k ρik = σ [detJ I M] χ [I s ], dµB (ρ ∗ , ρ) e−ρ ρD χ [I s ]

(5.83)

k=1

where M is the matrix 1 + DB, where the rows i1 ,...ip are substituted by the corresponding rows of B, and the columns j1 ,...jp are substituted by the corresponding columns of DB: Mij = (1 + DB)ij Mijk = Di Bijk Mik j = Bik j

i = i1 , ...ip , j = j1 , ...jp , i = i1 , ...ip , ∀ j.

(5.84)

Finally I are the set of indices I = {i1 , ..., ip } J = {j1 , ..., jp } and det J I M is the determinant of the matrix M without the rows j1 , , ...jp and the columns i1 , ..., ip , and σ is a sign. This new determinant can be bounded as usual |det M J I |
e

O



|| W3



     ep e i ∈I ∪J (DB)ii  .

(5.85)

Proof. To obtain the first bound (5.82) we apply (5.69) for a product of Fermionic fields. The result is the determinant of a p×p matrix whose elements are (B −1 +D)−1 ij with i = i1 , ..., ip and j = j1 , ...jp . This determinant is easily bounded by p!2 sup |(B −1 +D)−1 ij |.

−2 ) we obtain the result. Applying |(B −1 + D)−1 ij | = O(W The second expression (5.83) is easily obtained using the anticommuting properties of the Fermionic fields. Finally (5.85) holds because the only error terms come from the absence of a term 1 in p diagonal elements. ! "

Lemma 10. We consider the integral  Ia (n1 , ..., np ) =:

p     V dµC (a, b) e  |ajk |nk eRe Tr DB ,

(5.86)

k=1

where p > 0, nk > 0 for all k and n =



k

 Ia (n1 , ..., np )[I 1 ]
n!

nk . Then K W

n

,

(5.87)

Ia (n1 , ..., np )[I 2 ]
K n e−cW , 7 $ 4 nk ! e−cW Ia (n1 , ..., np )[I q ]
K n

(5.88) q > 2.

(5.89)

k

If instead of a fields we have b fields the result is the same, but in the large field region we pay a larger factor, because we have a very small mass remaining in the covariance 3

Ib (n1 , ..., np )[I q ]
K n W 2 n

$ k

nk ! e−cW

7 4

q > 2.

(5.90)

Density of States for Random Band Matrices

107

Proof. As in Sect. 5 we partition the integration domain 1 = When we are near the first saddle ( I 1 ) we write |aj1 |n1 ... |ajp |np ≤

6

q=1 χ [I

p 1 nk |ajk |n . n

q ].

(5.91)

k=1

Now we can apply Brascamp-Lieb inequality as stated in (5.18). In the region near the second saddle ( I 2 ) we can bound the field a by a constant. In the large field region we bound the fields a using a fraction of the exponential decay of the mass term  1 2 2 K n √ n n! e 2 δmr aj , (5.92) |aj | ≤ √ δ where δ > 0 is a small constant δ < 1 which must be smaller than the mass of Cfa . Note that for the b fields in the region I 3 or I 4 δ must be of order δ = O(W −3 ) as this is the mass of Cfb . This completes the proof. ! "

6. Infinite Volume Limit In this section we shall establish bounds on G+ 00  and the exponential decay of R(x) 3 uniformly as  ↑ Z . This is done by a standard method (see [21] or [22], ch.III.1) in statistical mechanics called the cluster expansion. These expansions are possible when there is a single dominant saddle point (in our case a = b = 0) whose fluctuations are close to that of a massive Gaussian i.e. a Gaussian whose covariance B has exponential decay. We are going to use a standard expansion with a few modifications. By supersymmetry some terms of the expansion are one (see Lemma 11) thus simplifying the expression. On the other hand for technical reasons the treatment of the covariance in the measure is slightly different from the usual one. We prove the following theorem Theorem 4. There exists W0 such that for all W > W0 lim↑Z3 ρ¯ (E) is bounded in I uniformly in W      lim ρ¯ (E) ≤ K (6.1) ↑Z3  for some constant independent from W . Moreover      lim ∂ n ρ¯ (E) ≤ Cn ∀ n < n0 (W ), ↑Z3 E    lim ρ¯ (E) = ρSC (E) + O W −2 , ↑Z3

uniformly in  and W . In particular, for x = 0,     |x|  lim R(x)| ≤ K e−c W . ↑Z3  W3

(6.2) (6.3)

(6.4)

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M. Disertori, H. Pinson, T. Spencer

Outline of the proof. Note that the exponential decay (4.20) of B means that regions at a distance higher than W are approximately decoupled. As the observable depends only on the fields at zero or at zero and x we expect that all interactions take place in a volume of order W 3 around i = 0. To exploit this fact we partition  in cubes of side W forming the lattice L. For that we introduce the function  1 if i ∈  χ (i) = (6.5) 0 otherwise  satisfying ∈L χ (i) = χ (i). In the following we call root cube the cube containing i = 0 and we denote it by 0 . The cluster expansion expresses G+ 00  and R(x) as a sum of finite volume contributions. Let Y be a union of cubes  containing zero (zero and x when we estimate R(x)). Then the cluster expansion gives
+ cY ZY , (6.6) G00 ε=0 = Y '0

ZY

where is a functional integral over fields aj , bj , ρj and ρj∗ , j ∈ Y , and cY is a coefficient which is exponentially small in the length of the shortest tree spanning the cubes of Y . The main purpose of this section is to give a precise description of (6.6) and provide estimates to establish the convergence of the sum. Note that in conventional statistical mechanics there is usually an additional factor ZY c with Y c = \Y . However we show in Lemma 11 below that ZY c = 1. The factors ZY are similar to a partition function except that there are derived vertices aj3 , bj3 or ρj∗ ρj (aj + bj ) present. We shall show, using the ideas of Sect.5.1.5, that 1

ZY ≤ (KW − 6 )|Y | .

Lemma 11. If we restrict to the set of cubes Y c = \Y and there is no observable contribution we have   ∗ (6.7) ZY c = dµBY c (a, b, ρ ∗ , ρ) e j ∈ (Vj −ρj ρj Dj ) = 1, where dµBY c (a, b, ρ ∗ , ρ) is defined after (5.64) and BY c is the covariance B restricted to the volume Y c . Proof. We perform the translation aj → aj − E, bj → bj + iE for all j ∈  . Note −1 2 2 that, for a general  ⊂ , B  = −W  + (1 − E ). Therefore the translation gives some linear and constant terms. The constant terms are cancelled when we add the contributions from the a and b fields. By performing the inverse Hubbard-Stratonovich transformation we obtain 

 † R = d6∗ d6 ei6 (Eε +A−H )6 = 1, (6.8) 1

where the average  1 is computed with the probability distribution (2.1) with covariance J˜ instead of J , with J˜ij = B −1 + E 2 . The matrix A is a diagonal matrix † 6† A6 = 6 i 6 i Ai , Ai = E (J˜ik − δik ). (6.9) i

This completes the proof.

" !

k

Density of States for Random Band Matrices

109

6.1. Cluster expansion. We derive the cluster expansion formula. The result is stated in Lemma 12 below. We construct the expansion by an inductive argument. First we want to test if there is any connection between the root cube 0 and some other  ∈ . For that purpose we introduce an interpolated covariance B(s1 ) with 0 ≤ s1 ≤ 1, which satisfies B(1) = B while B(0) decouples the root cube 0 from the rest of the volume. The easiest choice for B(s) is B(s)ij = sBij for i ∈ 0 and j ∈ \0 , or vice versa, and B(s)ij = Bij otherwise. For technical reasons we choose the following (less natural) interpolation rule, B(s1 )−1 = C(s1 )−1 + im2i , where

 C(s1 )ij =

s1 Cij Cij

if i ∈ 0 , j ∈  = 0 , otherwise .

(6.10)

or vice versa

(6.11)

The reason we use this definition of B(s) is that we do not want to mix the real and imaginary part in B −1 in order to apply later the same estimates of Sect. 5. Note that (6.11) is equivalent to the definition   (6.12) C(s1 ) = s1 C + (1 − s1 ) C0 0 + Cc0 c0 , (C )ij =

& 1% χ (i)Cij χ (j ) + χ (i)Cij χ (j ) , 2

(6.13)

where c0 = L\0 . Therefore C(s) is still a positive operator, as it is a convex combination of the positive operators C and C . This fact is essential to ensure the convergence of the integrals. With the interpolated covariance we define   ∗ (6.14) F [s1 ] = dµB(s1 ) (a, b, ρ ∗ , ρ) e j ∈Y c (Vj −ρj ρj Dj ) O0 .
 Note that for s1 = 1, F [s1 ]s1 =1 = G+ 00 ε=0 . Now we apply a first order Taylor formula to F [s1 ],  1 F [s1 ]s1 =1 = F [s1 ]s1 =0 + ds1 ∂s1 F [s1 ] . (6.15) 0

The first term F [s1 ]s1 =0 = F0 corresponds to decoupling 0 from the rest of the volume. The derivative in the second term of (6.15) gives   ∗ (6.16) ∂s1 F [s1 ] = ∂s1 dµB(s1 ) (a, b, ρ ∗ , ρ) e j (Vj −ρj ρj Dj ) O0 . Using integration by parts we have   ∗ ∂s1 B(s1 )ij ∂s1 dµB(s1 ) (a, b, ρ , ρ) = dµB(s1 ) (a, b, ρ ∗ , ρ) 

ij

δ δ δ δ δ δ × + + ∗ δai δaj δbi δbj δρj δρi

(6.17)

 .

(6.18)

110

M. Disertori, H. Pinson, T. Spencer

The derivative δ/δai may apply to exp[Vi − ρi∗ ρi Di ] or to the observable O0 (this only for i = 0):    δ j Vj δ e = e j Vj Vi (ai ) , (6.19) δai δai    ∗ δ − j ρj∗ ρj Dj δ e = Di (ai bi ) (−ρi∗ ρi )e− j =i ρj ρj Dj , (6.20) δai δai δ δ  O0 = (−ρ0∗ ρ0 ) D . (6.21) δa0 δa0 0 The same definitions hold for δ/δbi . The Fermionic derivative δ/δρi may apply to exp[−ρi∗ ρi Di ] or to the observable O0 (this only for i = 0):  ∗ δ − j ρj∗ ρj Dj e = Di ρi∗ e− j =i ρj ρj Dj , δρi δ O0 = ρ0∗ D0 . δρ0

Similar formulas hold for δ/δρi∗ . Therefore % & ∂s1 F [s1 ] = ∂s1 B(s1 )i1 j1 F [s1 ]((i1 j1 )),

(6.22) (6.23)

(6.24)

i1 ,j1

where  F [s1 ]((i1 j1 )) =

 δ δ δ δ δ δ dµB(s1 ) (a, b, ρ , ρ) + + ∗ δai1 δaj1 δbi1 δbj1 δρj1 δρi1    ∗ × e j (Vj −ρj ρj Dj ) O0 . (6.25) 

∗

Let us consider the propagator ∂s1 B(s1 )i1 j1 extracted by the Taylor formula. If we choose the easiest interpolating rule, that is B(s)ij = sBij when i and j are in different cubes and i or j ∈ 0 , the derivative is not zero only for i1 ∈ 0 and j1 ∈ 0 , or vice versa. Hence ∂s1 B(s1 )i1 j1 connects explicitly 0 to a different cube. With the definition (6.10) the derivation is different and instead of one line we extract three ' ( 1 ∂s1 B(s1 )i1 j1 = ∂s1 C(s1 )−1 + im2i i j ' ' (1 1 ( & % 1 1 = ∂s1 C(s1 ) k k  1 1 1 + im2i C(s1 ) 1 + im2i C(s1 )   i1 k1 k1 j1 k1 ,k1 = G(s1 )i1 k1 Ck1 k1 G(s1 )k1 j1 , (6.26) 1 =0 k1 ∈0 k1 ∈1

where we used (6.11) and G(s) is G(s1 ) =:

1 . 1 + im2i C(s1 )

(6.27)

Density of States for Random Band Matrices

111

For each term (k1 , k1 ) with k1 ∈ 0 and k1 ∈ 1 we say there is a strong connection between 0 and 1 . We denote this by drawing a line from 0 to 1 . Note that these points do not correspond to any derivative inside the functional integral, as the only derivatives occur on i1 and j1 . If i1 and j1 belong to some cube  ∈ 0 ∪ 1 they give some additional strong connections. Therefore the first step of the induction extracts a link l1 , associated to four points i1 , j1 , k1 , k1 , connecting 0 to a set of one, two or three new cubes depending on the ˜ 1 . The different possible positions of i1 and j1 . We call this set the generalized cube  ˜ 1 are shown in Fig. 3. Now links inside  ∂s1 F [s1 ] =

(i1 ,j1 ) (k1 ,k1 )

G(s1 )i1 k1 Ck1 k1 G(s1 )k1 j1 F [s1 ]((i1 j1 )) .

(6.28)

Note that the functional integral after ∂s1 Bs1 has been extracted is a function only of (i1 , j1 ) and not of (k1 , k1 ), as only the first two indices correspond to a functional derivative inside the integral. Now we fix the points (i1 , j1 ), (k1 , k1 ) corresponding to a strong connection between ˜ 1 . We want to test if there is any connection between the set  ˜ 0,1 = 0 ∪  ˜1 0 and  ˜ 0,1 ). We introduce then a new parameter s2 in C(s1 ) in and any other cube  ∈ \( the functional integral:  C(s1 , s2 )ij =

s2 C(s1 )ij C(s1 )ij

˜ 0,1 , j ∈  ˜ 0,1 , if i ∈  otherwise .

or vice versa

(6.29)

As for C(s1 ) we can write C(s1 , s2 ) as a convex combination of positive operators  C(s1 , s2 ) = s2 C(s1 ) + (1 − s2 ) C˜ 0,1 ˜ 0,1 (s1 ) + C˜ c



˜ c (s1 ) 0,1 0,1

,

(6.30)

where C(s1 ) and C (s1 ) are positive. Therefore C(s1 , s2 ) is still positive. Then F [s1 ]((i1 j1 )) = F [s1 , s2 ]((i1 j1 ))s2 =1 . We apply again the first order Taylor formula  F [s1 , s2 ]((i1 j1 ))s2 =1 = F [s1 , s2 ]((i1 j1 ))s2 =0 +

1 0

ds2 ∂s2 F [s1 , s2 ]((i1 j1 )) . (6.31)

˜ 0,1 : As before F [s1 , s2 ]((i1 j1 ))s2 =0 corresponds to a functional integral restricted to  F [s1 , s2 ]((i1 j1 ))s2 =0 = F˜ 0,1 [s1 ]((i1 j1 )). The other term gives ∂s2 F [s1 , s2 ]((i1 j1 )) =



(i2 ,j2 )(k2 ,k2 ) ˜ 0,1 ,k  ∈ ˜c k2 ∈ 2 0,1



G(s1 , s2 )i2 k2 C(s1 )k2 k2 G(s1 , s2 )k2 j2

× F [s1 , s2 ]((i1 j1 ),(i2 j2 )) .



(6.32)

We repeat this argument until we construct all the possible connected components containing the root cube. This is a finite sum, for  fixed.

112

M. Disertori, H. Pinson, T. Spencer

Definitions. We give now some more precise statements. We define a generalized cube ˜ as a set of one, two or three disjoint cubes in . A generalized polymer Y˜ is then a  ˜ A tree T on Y˜ is a set of links l1 ,.. ln connecting the disjoint set of generalized cubes . generalized cubes in Y˜ and forming no loops (See Fig. 2). We call the set of all these cubes the polymer Y contained in Y˜ . Each link lr corresponds to four connected vertices ir , jr , kr , kr . The corresponding propagators are Gir ,kr , Ckr ,kr and Gkr ,jr defined in ˜ and tree T (6.34) and the corresponding links are shown in Fig. 3. Note that the same  may correspond to several different polymers Y (see Fig. 2). We apply then the following formula.
 Lemma 12. G+ 00 ε=0 can be written as


 G+ 00 ε=0

=

Y˜ '0 T onY˜

 ×

˜ |−1 |Y r=1

˜ |−1  |Y 1 r=1

0

dsr

(ir ,jr ) (kr ,kr )

MT (s) 

G(s)ir kr Ckr kr G(s)kr jr  FY [s]({ir , jr }),

(6.33)

where we sum over the generalized polymers Y˜ and over the ordered trees T on Y˜ with root 0 . Note that the ordering on a tree is the ordering on its links lr , r = 1, ...|Y | − 1. The product over r is then the product over the links in the tree. The points (ir , jr ) (kr , kr ) ˜ ∈ Y˜ and the links connecting them. Each tree link fix the number of cubes inside each  lr is associated to the parameter 0 ≤ sr ≤ 1. The product Gir kr Ckr kr Gkr jr corresponds to ∂sr B(sr ) for each link lr ∈ T and the factor MT (s) is the product of s factors extracted

~ ∆1

~ ∆3

∆0 ~ ∆2 ∆1

∆0 ∆’2

∆1

∆3 ∆’1 ∆2

a.

∆3

∆’3 ∆’’2 b.

∆’’3

∆’3

∆0 ∆2

c.

Fig. 2. The two polymers and tree structures in (b) and (c) both correspond to the same generalized polymer Y˜ and tree T in (a)

Density of States for Random Band Matrices

i1

k’1 k1

∆0

113

k1

k’1

∆0

∆1

j1

j1

∆1 i1

a. k1

∆’1

b.

k’1

∆0

∆1

i1

j1

∆ 1’

∆"1

c. Fig. 3. Some examples of links of type 1,2 and 3; the two point vertex is a filled dot, the three point one is an empty dot

by the derivatives ∂r B(sr ). Finally FY [s]({ir , jr }) is the functional integral remaining after the propagators ∂sr B(sr ) have been extracted. More precisely, for each r the points (ir , jr ) (kr , kr ) must satisfy the constraint ˜ r  for some r  < r, kr ∈  ˜ r , and ir , jr may belong to any  ˜ r  with r  ≤ r. The kr ∈  propagators C(s) G(s) are defined as G(s) =:

1 , 1 + im2i C(s)

C(s)ij =: sij Cij ,

(6.34)

where sii sij sij sij

= 1, =1  = rk=r sk =0

˜ r, if ∃ r s.t. i, j ∈   ˜ r, j ∈  ˜ r , if ∃r < r s.t. i ∈  if i ∈ Y, j ∈ Y .

(6.35)

The remaining functional integral is  FY [s]({ir , jr }) = dµB(s) (a, b, ρ ∗ , ρ)   n  δ δ δ δ δ δ  j ∈Y (Vj −ρj∗ ρj Dj )  e × + + ∗ O0 , δair δajr δbir δbjr δρjr δρir r=1

(6.36) where B(s) is B(s)−1 := C(s)−1 + im2i .

(6.37)

Note that, as we constructed the tree, the order on the tree lines ensures that, for each ˜ r to some  ˜ r  with r  < r. 1 ≤ r ≤ n, the tree line lr connects 

114

M. Disertori, H. Pinson, T. Spencer

Proof. The proof follows directly from the inductive procedure we explained above.

" !

Now we can bound |G00 | inserting the absolute value inside (6.33). Therefore we have to bound |




 G+ 00 ε=0 |

≤

Y˜

×

˜ |−1   |Y 1 i=1

T onY˜ ˜ |−1  |Y r=1

0

dsi



|MT (s)|

(ir ,jr ) (kr ,kr )

 |G(s)ir kr | |Ckr kr | |G(s)kr jr | |FY [s]({ir , jr })| .

(6.38)

6.2. Bound on the functional integral. We bound the remaining functional integral |FY [s]({ir , jr })| by a generalization of the arguments for the finite volume case (Sec.5). Definitions. We call Vd the set of vertices derived by the cluster expansion appearing in (6.36): Vd = {j ∈ Y : ∃ 1 ≤ r ≤ |Y˜ | − 1 s.t. j = jr or j = ir }.

(6.39)

For each j ∈ Vd we call dj (a), dj (b), dj (ρ), dj (ρ ∗ ) the number of derivatives δ/δaj , δ/δbj , δ/δρj or δ/δρj∗ respectively. As we see from (6.19–6.22) these derivatives apply to V (aj ), V (bj ) or Dj ρj∗ ρj . We also have a contribution from the observable O0 , but only for j = 0. Note that, by the properties of anticommuting variables, dj (ρ), dj (ρ ∗ ) can be only 0 or 1. On the other hand there is no limit to dj (a), dj (b). We call dj = dj (a) + dj (b) + dj (ρ) + dj (ρ ∗ ) the total number of derivatives in j . This number is actually fixed by the choice of {(ir , jr )}r . For each j ∈ Vd we have to study   d (a) d (b) d (ρ ∗ ) dj (ρ) ∂ρj (6.40) eVj (a)+Vj (b) (1 − ρj∗ ρj Dj ) . ∂ajj ∂bjj ∂ρ ∗j j

In the small field region we need to extract some structure. We can actually write the derivative as ∗ r (a) r (b) r (ρ) ajj bjj ρjj ρj∗ rj (ρ ) C(aj )C(bj ), (6.41) rj (a),rj (b) rj (ρ),rj (ρ ∗ )

where rj (a), rj (b), rj (ρ), rj (ρ ∗ ) are the number of fields remaining after the derivatives have been performed. Note that rj (ρ) and rj (ρ ∗ ) can take only the values 0 and 1. On the other hand it is easy to see that rj (a) ≤ 3dj (b), and the same holds for b. Moreover the parameter nj = dj + rj ≥ 3, except for j = 0 (and j = x if we are considering R(x)). The factors C(a) and C(b) no longer depend on the Fermionic fields. By analytic tools we can show that in the small field region I 1 , |C(a)| ≤ K dj dj (a)!

|C(b)| ≤ K dj dj (b)!.

(6.42)

Density of States for Random Band Matrices

115

In the large field region (that means in I q for q > 1), we do not need to extract the whole structure, as the fields are large and the small factors come from the probability. We only need to extract explicitly the Fermionic fields; therefore we write the derivatives as ∗ r (ρ) ρjj ρj∗ rj (ρ ) C(aj )C(bj ). (6.43) rj (ρ),rj (ρ ∗ )

Again by analytic tools we can show that in the

    |C(a)C(b)| ≤ K dj dj (a)!dj (b)! eVj (a)+Vj (b)  .

(6.44)

In this region we have no factors rj . Nevertheless we define rj = 0 if dj ≥ 3 and rj = 3 − dj otherwise. In this way we ensure nj ≥ 3 for all j ∈ Vd . With these definitions and results we prove the following theorem. Theorem 5. |FY [s]({ir , jr })| ≤

|Y | K1







K n r ! 2

{rj }j ∈Vd ∈Y

where K1 , K2 are constants and r =



 j ∈Vd ∩



j ∈Vd ∩ rj ,

n =

1 dj ! W

rj

 ,

(6.45)



j ∈Vd ∩ nj .

Proof. The proof is a generalization of the arguments for Theorems 2 and 3 in Sect. 5. Note that Lemmas 3 and 4 hold also after substituting B and C with B(s) and C(s). Then we can introduce the partition (5.13) in each cube separately     1= χ [Ik ] . (6.46) 

k

We perform the bounds in each cube as in Sect. 5. Note that, as the derivations bring several Bosonic and Fermionic fields out of the exponential we have to use some of the ideas of Theorem 3. First we have to perform the integral over the Fermionic variables extracting the correct factors W −1 . Now we have many different cubes, each in a different interval, therefore we cannot apply (5.82) as in Lemma 9, as (B −1 + D)−1 is not be well defined. We apply then a generalization of (5.83) which allows to extract also the small factors. We have to compute the integral     − j ∈Y \V ρj∗ ρj (Dj +D0 ) d dµB(s) (ρ ∗ , ρ) e       ×  



  ρj   

j ∈Vd : rj (ρ)=1 rj (ρ ∗ )=0



  ρj∗   

(ρ ∗ )=1

j ∈Vd : rj rj (ρ)=0



  ρj ρj∗  . 

(6.47)

j ∈Vd : rj (ρ)=rj (ρ ∗ )=1

We partition now the set of j ∈ Y as U = {j ∈ Y | j ∈

Vd }, D = {j ∈ Y | j ∈ Vd , rj (ρ ∗ ) = 1 or rj (ρ) = 1}.

(6.48)

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M. Disertori, H. Pinson, T. Spencer

Note that the points j ∈ Vd with rj (ρ ∗ ) = rj (ρ) =  1 do not give any contribution to the integral. For each cube we introduce r (ρ) = i∈Vd ∩ rj (ρ) and r (ρ ∗ ) =  ∗ i∈Vd ∩ rj (ρ ). Note that the fields ρj are columns in the resulting matrix and the ∗ fields ρj are rows. With these definitions we have the following lemma. Lemma 13. By multi-linearity of the determinant, the Fermionic integral (6.47) above can be written as    r (ρ)r (ρ) r (ρ ∗ )r (ρ ∗ ) σ det M, (6.49) ∗ W r (ρ)+r (ρ ) 

where σ is a sign depending on how we order the rows and column in M, and M is the block matrix  Muu Mud M= . (6.50) Mdu Mdd Muu is a matrix corresponding to the elements (M)ij with both i, j ∈ U, Mdd to the elements (M)ij with both i, j ∈ D and Mud and Mdu are the mixed terms. Note that if j ∈ Vd and rj (ρ) = 1 but rj (ρ ∗ ) = 0, j appears only in a column of Mdd and the element Mjj is not present in the matrix. Therefore we order the lines and columns of Mdd : (Mdd )ij = (M)li cj , where li and cj ∈ Vd . With these definitions the matrix elements are % & (Muu )ij = 1 + (D + D0 )B(s) ij , (Mud )ij = Di B(s)icj r(cW) (ρ) , j (6.51) (Mdd )ij = r W(ρ ∗ ) B(s)li cj r(cW) (ρ) , (Mdu )ij = r W(ρ ∗ ) B(s)li j , (li )

j

(li )

j

j

where (i) is the cube containing the vertex i. Proof. The proof follows from the properties of the anticommuting variables and determinants. ! " Now we can insert absolute values inside the Bosonic integral. If we bound det M as in (5.8) we obtain the same bound as in Lemma 4, with an additional error term. The precise statement is given in Lemma (14) below. Lemma 14. The determinant of the matrix M defined as in (6.50–6.51) satisfies the bound | det M|
eRe Tr(Muu −1) eO(|Y |+|Vd |) .

(6.52)

Proof. Using (5.8) we have 1

† (M−1)

| det M| ≤ eRe Tr(M−1) e 2 Tr(M−1)

.

(6.53)

Now applying the definitions (6.50) we can write Re Tr (M − 1) = Re Tr (Muu − 1) + Re Tr (Mdd − 1), Tr (M − 1)† (M − 1) = Tr (Muu − 1)† (Muu − 1) + Tr (Mdd − 1)† (Mdd − 1)   + Tr (Mdu − 1)† (Mud − 1) + (Mud − 1)† (Mdu − 1) . (6.54)

Density of States for Random Band Matrices

117

By inserting the definitions (6.51) and using the decay of Bij , it is easy to see that all terms are bounded by a constant per cube except for Tr(Mdd − 1)† (Mdd − 1)
r (ρ) + r (ρ ∗ ). (6.55) 

This completes the proof.

" !

Now we perform the estimates as in Sec. 5. As in Sec.5.1.5, in order to decouple the estimates on different cubes we write all the bounds in terms of quadratic forms like v T Cf (s)v, where v is some vector and Cf (s) = C(s)−1 − f m2r > 0 as in (5.30), but f is now a diagonal matrix which is constant on each cube. Then we can apply C(s) <

1 , −W 2 N + m2r

(6.56)

where N is the discrete Laplacian with Neumann boundary conditions on the cubes. This operator decouples automatically different cubes. Note that to estimate the contribution from the small field region we have to apply (5.87), Lemma 10. To extract the factors W −rj in the large field cubes we use the exponential factors e−W . 6.3. Sum over the clusters. We perform now the sum over the clusters Y . We split the sum in several pieces. First fixing the cubes  we sum over the points ir , jr , kr , kr in the cubes. Note that after this operation is done, there is still a small factor associated to each cube. The factorials arising from the combinatorics are beaten by a piece from the decay of GCG. The remaining piece of the decay is used to sum over the cube positions, following the tree structure. Finally we sum over the tree choice T using the fact that we have a small factor per cube. To perform all these bounds we need now to study the spatial decay of the propagators C, B and G. We know already the spatial decay of C(s) (see (4.19)). The decay of G is given by the following lemma. Lemma 15. The propagator G = (1 + im2i C(s))−1 decays as |Gij | ≤ δij +

mr |m2i | 1 e− W |i−j | W 2 1+|i−j |

+

4 2 |mi | −f e W 3 mr

mr W

|i−j |

,

(6.57)

where f = inf[1/2, g] and g is some constant independent from W . Proof. By a Combes-Thomas argument we prove that + + + + 1 + −1 + R +R +≤2 + 1 + im2i C(s) +

(6.58)

for R a multiplication operator defined as R|x >= exp[µx]|x > and µ any vector with |µ| < gmr /W . Now G can be written as ' ' ( ( 1 1 2 4 = δij − imi C(s)ij − mi C(s) . (6.59) C(s) 1 + im2i C(s) 1 + im2i C(s) ij

ij

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M. Disertori, H. Pinson, T. Spencer

We need to study the decay of the last term. Actually we see that     1   µ|x−y| C(s) C(s)  e   1 + im2i C(s) ij     1  −1  = R C(s) C(s)R    1 + im2i C(s) ij   ' (     1  −1  −1 −1 =  R C(s)R R R R C(s)R    1 + im2i C(s)

ij

2 = |(V , AW )| ≤ *V * *W * *A* ≤ , W3  where we defined (V , AW ) = k,k  Vk Akk  Wk  and Vk = (R −1 C(s)R)ik ,

Wk  = (R −1 C(s)R)k  j ,

(6.60)

A = R −1 (1 + im2i C(s))R . (6.61)

We chose µ such that µ|x − y| = µ(x − y), and µ = |µ| ≤ gmr /W , and µ ≤ mr /2W . The last condition must ensure that the exponential decay of C(s) controls the exponentials from R −1 and R. This completes the proof. ! " Note that this lemma gives also the decay of B(s) |B(s)ij | ≤

mr 1 1 e− W |i−j | W 2 1+|i−j |

+

2 2 |mi | −f e W 3 mr

mr W

|i−j |

(6.62)

as B(s) = (G − 1)i/m2i . 6.3.1. Extracting small factors. Before performing the estimates we extract some factors from the propagators GCG for each tree line, to offset the factorials eventually arising in the estimates and to ensure we have a small factor per vertex. p

Factorials. Constant powers of factorials such as d , for p fixed, can be beaten using a piece of the decay of CGC. Note that each tree line lr connects different cubes, therefore we have d disjoint cubes hooked to the cube  by the tree T . When d is large, since we are in a finite dimensional space, many of these cubes must be very far from . It 1

is easy to see that half of the d cubes must be at a distance from  of order W d3 . 4

Therefore we gain a factor exp[−c d3 ] which can beat any constant power of factorials. 

˜ |−1 |Y



 e

−kr | −ε |ir W

e

|k −k  | −ε r W r

r=1

Note that the constant K depends on p.

e

|k  −j | −ε r W r



  K n ≤ . n  !p 

(6.63)

Density of States for Random Band Matrices

119

W factors. We need a factor W −1 for each field hooked to a derived vertex j ∈ Vd . We extract then a factor W −1 from each G propagator:    ˜ |−1 |Y   1 r  |G(s)ir kr | |G(s)kr jr | W  r=1    ˜ |−1 |Y   K nj . = (6.64) |W G(s)ir kr | |W G(s)kr jr |  W r=1

j ∈Vd

Note that nj ≥ 3 for all j ∈ Vd except for j = 0. As the each cube has volume W 3 , this ensure that we can choose the position of each vertex without paying any factor W . Finally, after extracting a fraction ε of the exponential decay and the factors W −1 we separate in the remaining factors the polynomial and exponential decay    ˜ |−1 |Y n  f f f − d( , ) − d( , )   (kr ) (jr ) (kr ) e W (kr ) ˜ ir kr C˜ kr ,k  G ˜ k  jr   , e− W d((ir ) ,(kr ) ) e W G r r r=1

r=1

(6.65) where  ˜ ij =: δir kr W 2 + G

1 1+|i−j |

+

1 W



C˜ ij =:

W −4 1+|kr −kr |

,

(6.66)

where f  = f mr − ε is the remaining mass and d(,  ) is the distance between the center of the cube  and  . ˜ and C˜ we sum 6.3.2. Sum over the vertex positions. Now using the decay 1/|i − j | in G over the positions of all vertices inside their cube (the cube is fixed). Each line of the cluster expansion corresponds to four vertices ir , jr , kr , kr , where kr and kr correspond to two point vertices and must belong to different cubes while ir , jr may belong to the same cube. For each j = ir or j = jr we distinguish two cases: – j contracts to j  and j  has never been extracted before in the cluster expansion. Then we say j  is new with respect to j . – j contracts to j  and j  has already been extracted before by the cluster expansion. Then we say j  is old with respect to j . We consider first the case kr = ir and kr = jr so that the factors δir kr and δjr kr disappear. Note that we sum over the position of ir (jr ) only when it is new. We consider the different cases. ir = jr and both ir and jr new. Then we sum over ir and jr ,



kr ∈(kr ) ir ∈(ir )

˜ ir kr G

kr ∈(k  ) r

C˜ kr ,kr

jr ∈(jr )

˜ k  jr ≤ W 5 = G r



5

W2

Therefore we pay a factor W 5/2 for ir and the same factor for jr .

 

 5 W2 .

(6.67)

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M. Disertori, H. Pinson, T. Spencer

ir = jr and ir new, jr old. Then we sum only over ir , kr ∈(k  )



˜ k  jr G r

kr ∈(kr )

r



C˜ kr ,kr

˜ ir kr ≤ W 2 . G

(6.68)

ir ∈(ir )

Therefore we pay a factor W 2 for ir and no factor for jr . ir = jr and ir and jr old. Then ir and jr are both fixed,



˜ ir kr W −4 G

˜ k  jr ≤ O(1), G r

kr ∈(k  ) r

kr ∈(kr )

(6.69)

where we bounded C˜ kr ,kr ≤ W −4 . ir = jr and ir new. Then we sum over ir ,       2 ˜ ir k  + ˜ G Ckr ,kr r   kr ∈(kr ) kr ∈(k  )   ir ∈(ir ) r





˜ ir kr G

ir ∈(ir ) |ir −kr |<|ir −kr |

|ir −kr |≥|ir −kr |

    2     

≤ W 2. (6.70)

Therefore we pay a factor W 2 for ir . ir = jr and ir old. Then ir is fixed,

˜ ir kr W −4 G

kr ∈(k  )

kr ∈(kr )

˜ k  jr ≤ O(1), G r

(6.71)

r

where we bounded C˜ kr ,kr ≤ W −4 . When ir = kr or jr = kr it is easy to see that the same estimations hold. Note that for each j = ir (j = jr ), with j = 0, we pay a factor W 5/2 when it is new and some constant K in any other case. Therefore, applying nj ≥ 3, we have 5 2

W K

nj −1

1 = W nj

'

(nj

K W

1− 2n5

j

 ≤

K 1

W6

nj

.

(6.72)

1

Therefore we have a factor W − 6 for each tree line hooked to j . This means we have 1 ˜ in Y˜ . The case j = 0 is special as nj ≥ 0. a factor W − 6 for each generalized cube  Nevertheless the position of 0 is fixed so that we do not pay the factor W 5/2 . Therefore for j = 0 we have K n0

1 . W n0

(6.73)

Density of States for Random Band Matrices

121

6.3.3. Combinatorial bounds. The combinatoric inside each cube costs a factor ˜ K d (d !)(r !) ≤ K d (d !)4 = K |Y | (d !)4 . (6.74) r ≤3d

The factorials are beaten by a piece of the exponential decay of the tree lines (6.63), while the constant factor will be bounded later when we will sum over the tree structure. 6.3.4. Sum over the cube positions and the tree structure. We sum over the cube positions using the exponential decay and following the tree from the leaves towards the ˜ root. The result is K |Y | . The remaining sum is now 
+  G

00 ε=0

 
|Y˜ |

|−1  |Y ˜



orders i=1 T unordered

1 0

dsi |MT (s)|

 K  1

˜ Y˜ ∈

W6

,

(6.75)

where we have split the sum over ordered rooted trees as the sum over unordered rooted trees, and the sum over orders. This last sum is performed by the integral over the interpolating factors (see [21] or [22], Lemma III.1.1) We give here a sketch of the proof. Lemma 16. The sum over all the orders on the tree T is bounded using the interpolating factors si as follows: |−1  |Y ˜

1 0

orders i=1

dsi |MT (s)| = 1.

(6.76)

˜ ˜ ˜ Proof. We introduce the variables εij for all i j ∈ Y . Then we introduce the function F (ε) =: (ij )∈T (1 + εij ), where T is unordered. Now we perform the tree expansion as we did in Lemma 12, Sect. 6.1. We define εij (s1 ) = s1 εij if i or j = 0 and εij (s1 ) = εij otherwise. We apply the first order Taylor expansion and we go on until we extract all the εij . The term proportional to all εij is then 

|−1  |Y ˜

εij

orders i=1

(ij )∈T

1 0

dsi |MT (s)|.

(6.77)

 If we expand F (ε) in powers of ε we see that the term (ij )∈T εij has coefficient 1. Therefore by comparing powers of ε we obtain (6.76). ! " Finally the sum over the structure can be written as   d˜ d˜ i 0   ˜   g |Y | = g g . . . , |Y˜ |

T unordered

d˜ ≥0 i=1 0

1 6

(6.78)

d˜ ≥0 i  =1 i

where we defined g = K/W < 1 and for each generalized cube we sum over the coordination number. We start summing from the leaves going towards the root. The leaves give 1 1 1 1 gd = g if g 2 ≤ g2, < 1. (6.79) g 1−g 1−g d≥0

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M. Disertori, H. Pinson, T. Spencer

The following step gives g

d≥0

1

d

g2 = g

1−g

1

1 2

Then we repeat inductively. Finally

1

1

≤ g2,

if g 2

1

1 − g2

< 1.

˜

g |Y | ≤ K + 1,

(6.80)

(6.81)

T |Y˜ | unordered

where the constant comes from the bound for n = 0. This completes the proof of the first part of Theorem 4, namely the boundness of the density of states (6.1). 6.4. Smoothness and exponential decay. To bound the derivatives of the density of states (6.2), and in particular the decay of R(x) (6.4), we perform the cluster expansion as in Sect. 6.1. For R(x), note that contributions where Y does not contain both 0 and x are cancelled. When 0 and x both belong to Y we can extract the exponential decay directly from the tree lines GCG. For derivatives ∂En ρ¯ (E) the idea is the same. Only contributions from Y containing all the observables are not cancelled. The fine structure (the factor W −3 in R(x)) are then extracted by a few steps of perturbative expansion as in Sect. 5.2.2. In the same way we can prove the semicircle law behavior (6.3). This completes the proof of Theorem 4.

Appendix A Supersymmetric formalism We summarize the conventions and notations we adopted in this work (they are based on the review by Mirlin [1]). Anticommuting variables. Let us introduce a set of 2N variables χ1 , . . . , χN , χ1∗ , . . . , χN∗ with the following properties: χi χj = −χj χi ,

χi∗ χj = −χj∗ χi , (χi∗ )∗ = −χi ,

χi∗ χj∗ = −χj∗ χi∗ ,

(χi χj )∗ = χi∗ χj∗ .

(A.1) (A.2)

With these properties χ1 , . . . , χN is a set of complex anticommuting variables and χ1∗ , . . . , χN∗ are their complex conjugates. The integration is defined by     1 ∗ dχi 1 = dχi 1 = 0, dχi χi = dχi∗ χi∗ = √ (A.3) 2π With these definitions we introduce a vector and its adjoint as usual   χ1 !   χ =  ...  χ † = χ1∗ , · · · , χN∗ χN

(A.4)

Density of States for Random Band Matrices

123

Now χ † χ is a real commuting variable and   M † dχi∗ dχi e−χ Mχ = det 2π

(A.5)

i

for any matrix M. Supervectors and supermatrices. A supervector is defined as   S1  ..   .    ! S  ∗ 6= N 6† = S1∗ , · · · , SN , χ1∗ , · · · , χN∗ , χ  1  .   ..  χN

(A.6)

where Si are the commuting and χi are the anticommuting components. Similarly a supermatrix is a matrix with both commuting and anticommuting entries  aσ M= , (A.7) ρ b where a and b are ordinary matrices while σ and ρ have anticommuting elements. αβ We identify the element of a supermatrix by four indices Mij , where α, β specify in which sector we are: (0,0) corresponds to a (Boson-Boson); (1,1) corresponds to b (Fermion-Fermion); (0,1) corresponds to σ (Boson-Fermion); (1,0) corresponds to ρ (Fermion-Boson). (i.j ) identify the matrix element inside each sector. For example Mij00 = aij . The notions equivalent to trace and determinant are supertrace and superdeterminant SdetM = det(a − σ b−1 ρ) det b−1 .

StrM = Tra − Trb,

(A.8)

With these definitions we have Str ln M = ln SdetM,  

d6∗ d6 e−6

† M6

d6∗ d6 6α,k 6∗β,l e−6

† M6

(A.9)

= SdetM −1 , αβ

= (M −1 )kl SdetM −1 .

(A.10)

(A.11)

Note that some properties are different from that of the usual matrices, in particular: Sdet zM = SdetM

(A.12)

for any complex number z. Finally from these formulas one can derive the inverse of the supermatrix M (A.7): ' ( !−1 !−1 −1 a − σ b−1 ρ − a − σ b−1 ρ σb −1   !−1 −1 !−1 −1 M = , (A.13) −b−1 ρ a − σ b−1 ρ b σb 1 + ρ a − σ b−1 ρ which is the usual block matrix inversion formula.

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M. Disertori, H. Pinson, T. Spencer

Acknowledgements. We thank Rowan Killip for many discussions and suggestions related to this paper. These discussions lead us to improve the proof, in particular by introducing Brascamp-Lieb inequalities.

References 1. Mirlin, A.: Statistics of energy levels and eigenfunctions in disordered and chaotic systems: Supersymmetry approach. Phys. Rep. 326, 259 (2000) cond-mat/0006421 2. Atland, A., Offer, C.R., Simons, B.D.: In: Supersymmetry and Trace Formulae. Chaos and Disorder. I.V. Lerner, J.P. Keating, D.E. Khmelnitskii (eds), 1999, p. 17 3. Bleher, P.M., Its, A.R. eds.: Random Matrix Models and their Applications. MSRI Publications, Vol. 40 4. Katz, N.M., Sarnak, P.: Zeroes of Zeta functions and symmetry. Bull. Am. Math. Soc. 36(1), 1 (1999) 5. Johansson, N.K.: Shape fluctuations and Random matrices. Commun. Math. Phys. 209, 437 (2000) 6. Pastur, L., Figotin, A.: Spectra of Random and Almost-Periodic Operators. Berlin-Heidelberg-New York: Springer-Verlag, 1992 7. Aizenman, M., Schenker, J., Friedrich, R.M., Hundertmark, D.: Finite-volume fractional-moment criteria for anderson localization. To appear in Commun. Math. Phys. 8. Schenker, J.: Private communication, to be published 9. Wegner, F.J.: The mobility edge problem: Continuous symmetry and a conjecture. Z. Phys. B 35, 207 (1979) 10. Sch¨afer, L., Wegner, F.J.: Disordered system with n orbitals per site: Lagrange formulation, hyperbolic symmetry, and goldstone modes. Z. Phys. B 38, 113 (1980) 11. Fyodorov, Y.V., Mirlin, A.: Scaling properties of localization in random band matrices: A σ -model approach. Phys. Rev. Lett. 67 (18), 2405 (1991) 12. Efetov, K.B.: Supersymmetry and theory of disordered metals. Adv. Phys. 32(1), 53 (1983) 13. Efetov, K.B.: Supersymmetry in Disorder and Chaos. Cambridge: Cambridge University Press, 1997 14. Constantinescu, F., Felder, G., Gawedzki, K., Kupiainen, A.: Analyticity of density of states in a gauge-invariant model for disordered electronic systems. J. Stat. Phys. 48(3/4), 365 (1987) 15. Klein, A.: The supersymmetric replica trick and smoothness of the density of states for random Schr¨odinger operators. In: Operator Theory: Operator Algebras and Applications, Part 1, Proceedings of symposia in pure mathematics Vol. 51, 1990, p. 315 16. Molchanov, S.A., Pastur, L.A., Khorunzhii, A.M.: Limiting eigenvalue distribution for band random matrices. Theor. Math. Phys. 90(2), 108 (1992) 17. Khorunzhi, A.M., Pastur, L.A.: Limits of infinite interaction radius, dimensionality and the number of components for random operators with off-diagonal randomness. Commun. Math. Phys. 153, 605 (1993) 18. Khorunzhi, A.M., Kirsch, W.: On asymptptic expansions and scales of spectral universality in band random matrix ensembles. To appear in Commun. Math. Phys. 19. Brascamp, H.J., Lieb, E.H.: On extension of the Brunn-Minkowski and Pr´ekopa-Leindler theorems, including inequalities for log concave functions and with an application to the diffusion equation. J. Funct. Anal. 22, 366 (1976) 20. Spencer, T.: Scaling, the free field and statistical mechanics. Proc. Symp. Pure Math. 60, 373 (1997) 21. Brydges, D.: A Short Course on Cluster Expansions. Les Houches session XLIII, 1984, Elsevier Science Publishers, 1986 22. Rivasseau, V.: From Perturbative to Contructive Renormalization. Princeton Series in Physics, 1991 Communicated by A. Kupiainen

Commun. Math. Phys. 232, 125–155 (2002) Digital Object Identifier (DOI) 10.1007/s00220-002-0727-y

Communications in

Mathematical Physics

Weak Disorder Localization and Lifshitz Tails Fr´ed´eric Klopp∗ D´epartement de Math´ematique, Institut Galil´ee, U.M.R. 7539 C.N.R.S, Universit´e de Paris-Nord, 99 venue J.-B. Cl´ement, 93430 Villetaneuse, France. E-mail: [email protected] Received: 7 December 2001 / Accepted: 26 July 2002 Published online: 29 October 2002 – © Springer-Verlag 2002

Abstract: This paper is devoted to the study of localization of discrete random Schr¨odinger Hamiltonians in the weak disorder regime. Consider an i.i.d. Anderson model and assume that its left spectral edge is 0. Let γ be the coupling constant measuring the strength of the disorder. For γ small, we prove a Lifshitz tail type estimate and use it to derive localization in a band starting at 0 going up to a distance γ 1+η (0 < η < η0 ) of the average of the potential. In this energy region, we show that the localization length at energy E is bounded from above by a constant times the square root of the distance between E and the average of the potential. R´esum´e. Dans cet article, nous e´ tudions la localisation a` faible d´esordre pour des op´erateurs de Schr¨odinger al´eatoires discrets. Consid´erons un mod`ele d’Anderson i.i.d. dont le bord spectral gauche vaut 0. Soit γ la constante de couplage mesurant le d´esordre. Pour γ petit, nous d´emontrons une estim´ee de type estim´ee de Lifshitz pour la densit´e d’´etats, et nous utilisons cette estim´ee pour prouver que le spectre de cet op´erateur est localis´e dans un intervalle allant de l’´energie 0 jusqu’`a une distance de l’ordre de γ 1+η (0 < η < η0 ) de la moyenne du potentiel. Dans cette r´egion d’´energie, la longueur de localisation a` une e´ nergie E est major´ee par une constante fois la racine de la distance s´eparant E de la moyenne du potentiel. 0. Introduction The main purpose of the present paper is to study the spectral behavior of discrete random Schr¨odinger operators in the weak disorder regime. We prove exponential decay for the Green’s function of such operators in a neighborhood of finite spectral edges and give a lower bound on the rate of decay of the eigenfunctions associated to eigenvalues in this neighborhood. As a consequence of such a bound, we derive exponential localization ∗ It is a pleasure to thank M. Aizenman for his explanations on the paper [4]. The author also gratefully acknowledges support of the FNS 2000 “Programme Jeunes Chercheurs”

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and dynamical localization at finite band edges as well as Poissonian statistics for the eigenvalue spacing. Let us start with a prototypical example. Consider the discrete Anderson model acting on 2 (Zd ) defined by Hω,γ = − d + γ Vω ,

(0.1)

where − d ≥ 0 is the free discrete Laplace operator
(− d u)n = (um − un ) for u = (un )n∈Zd ∈ 2 (Zd ), (m,n) nearest neighbors in Zd

γ is a positive coupling constant and Vω is the random potential (Vω u)n = ωn un

for u = (un )n∈Zd ∈ 2 (Zd ).

(0.2)

We assume that the random variables (ωn )n∈Zd are independent identically distributed, bounded from below and non-trivial. It is well known (see e.g. [6, 21]) that, under these assumptions, there exists γ ⊂ R such that, for almost every ω = (ωn )n∈Zd , the spectrum of Hω,γ is equal to γ . Moreover, γ is given by γ = σ (− d ) + γ · supp(ω0 ) = [0, 4d] + γ · supp(ω0 ). Let − be the infimum of γ . To fix ideas, let us assume that inf(ω0 ) = 0. This is no restriction on generality as it can be achieved by shifting energy by a constant. In this case, one has − = 0. Our purpose is to study the spectral properties of Hω,γ near 0. Therefore, we assume that the random variables (ωn )n∈Zd are square integrable, i.e. E(|ω0 |2 ) < +∞ and that their common distribution admits a bounded density, say g. In the sequel, P(·) and E(·) respectively denote the probability and expectation with respect to the random variables (ωn )n∈Zd . Define ω = E(ω0 ). We prove Theorem 0.1. Fix η ∈ (0, d/(6d + 4)) and s ∈ (0, 1). There exists γη,s > 0 and a > 0 such that, for γ ∈ (0, γη,s ), the Green’s function of Hω satisfies, for (m, n) ∈ Zd × Zd and E ∈ Iγ ,η := [0, γ (ω − γ η )],  s  1 √   (0.3) sup E  δn , (Hω,γ − E − iε)−1 δm  ≤ e−a |E−γ ω||m−n| . a ε∈R   Here, δn is the vector in 2 Zd with all coordinates equal to 0 except the nth that is equal to 1. It is well known (see e.g. [24, 2, 4]) that the bound (0.3) has strong implications on the spectral properties of Hω,γ in the energy interval Iγ ,η ; in particular, it is known that it implies • Exponential localization ([24, 3]). ω-almost surely, the spectrum of Hω,γ in the interval Iγ ,η is purely punctual, and the corresponding eigenfunctions are exponentially localized. Moreover, the decay rate of the off-diagonal behavior of the Green’s function gives an estimate on the rate of decay of the eigenfunctions, i.e. an eigenfunction ψ corresponding to an eigenvalue E in Iγ ,η satisfies lim inf −

|n|→+∞

log |ψ(n)| ≥ a |E − γ ω|. |n|

(0.4)

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Estimate (0.4) gives a lower bound on the Lyapunov exponent, hence, an upper bound on the localization length; we see that this upper bound increases as E approaches γ ω. This is expected as it is expected that the nature of the spectrum changes at an energy between 0 and γ ω called the mobility edge (see e.g. [1]). In this respect, it would be interesting to show a lower bound on the localization length having the same behavior as the upper bound obtained here. In the recent paper [23], the localization length in the bulk of the spectrum of Hω,γ was analyzed for γ small in dimensions 1 and 2; in dimension 1, see also [21]. It was proved that it is of size γ −2 . It would be interesting to understand the transition between that regime and the one described in the present paper. • Dynamical localization ([1, 2]). If !η (Hω,γ ) denotes the spectral projector of Hω,γ on the interval Iγ ,η , then, for some a˜ > 0, one has

  √    1  itHω,γ E sup  δn , e !η Hω,γ δm  ≤ e−a˜ |E−γ ω||m−n| . (0.5) a˜ t∈R That is, wave packets with energy in Iγ ,η do not spread. • Absence of level repulsion ([20]). In the energy range Iγ ,η , the local level spacings for Hω,γ have Poisson statistics. More precisely, let "L be the cube of center 0 and side length 2L + 1 in Zd , and let Hω,γ |"L be the Hamiltonian Hω,γ restricted to the cube "L with Dirichlet boundary conditions. Hω,γ |"L is a (2L + 1)d -dimensional Hermitian matrix. Let E1 (L) ≤ · · · ≤ E(2L+1)d (L) be its eigenvalues. Define the integrated density of states of Hω by

# eigenvalues of Hω,γ |" ≤ E Nγ (E) = lim . (0.6) #"→+∞ #" The limit in (0.6) exists ω-a.e.; it is non-random and non-decreasing ([6, 21]). Its derivative with respect to E is a positive measure; it is the density of states of Hω and we denote it by dNγ . As the common distribution of the random variables (ωn )n∈Zd admits a bounded density, the measure dNγ admits a bounded density, i.e. dNγ = nγ (E)dE; this is a consequence of the celebrated Wegner estimate (see [26, 6])). To define the local level spacings, we fix an energy E and assume nγ (E) > 0. We rescale the eigenvalues   d Ej (L) − E ˜ , Ej (L, E) = (2L + 1) nγ (E) and define the point process ξ(L, E) by ξ(L, E) =

d (2L+1)


j =1

δE˜ j (L,E) ,

where δx is the Dirac measure on R. Theorem 0.1 and the main result of [20] imply that, for γ sufficiently small, for Lebesgue almost every energy in γ ∩ Iγ ,η , the point process ξ(L, E) converges in distribution to the standard Poisson point process.

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In the recent paper [25] that triggered our interest in this question, W.M. Wang proved that the result of Theorem 0.1 and its consequences hold in a smaller energy interval. Using a weak disorder expansion of the resolvent, she proved that, for η > 0 fixed, for γ sufficiently small, (0.3) holds true for energies in the range [0, γ 1+η ]. Our method of proof is completely different from the one used in [25]. It relies on an estimate of the integrated density of states near the spectral edges that is related to Lifshitz tails; namely, under the assumptions made above, we prove Theorem 0.2. Fix η ∈ (0, d/(6d + 4)). Then, there exists ε > 0 and γη > 0 such that, for γ ∈ (0, γη ), one has −ε

Nγ (γ (ω − γ η )) ≤ e−γ . In the sequel, we prove a analogue of Theorems 0.2 and 0.1 for more general operators than Hω . Theorem 0.2 can also be extended to continuous random Schr¨odinger operators [13] (and, possibly, to other random models) and, thus, can be used to derive localization results and bounds of the type (0.4) or (0.5) for these classes of random operators. To complete this section, let us now briefly sketch the proof of Theorem 0.2. To simplify the argument, we restrict ourselves to the case when the random variables are bounded. One first reduces the question of estimating the integrated density of states to computing the probability of finding an eigenvalue below the energy γ (ω − γ η ) for Hω restricted to a cube "N of side-length N (N is not too large, e.g. it is of size an inverse power of γ ). So, we want to estimate the probability that there exists a normalized vector ϕ supported in "N such that − ϕ, ϕ + γ Vω ϕ, ϕ ≤ γ (ω − γ η ). As both − and Vω are non-negative, this implies − ϕ, ϕ ≤ Cγ , Vω ϕ, ϕ ≤ ω − γ η .

(0.7) (0.8)

Let ϕˆ be the “Fourier transform” of ϕ. By (0.7), the part of ϕˆ outside of a ball of radius L−1 is of L2 -norm
(γ L2 )1/2 (here and in the rest of the paper, a
b means a ≤ Cb for some positive constant C). We pick L so that γ 1/2 L  γ η (here and in the rest of the paper, a  b means a/b → 0). Cutting ϕ off in momentum space outside of a ball of radius L−1 , we make an error of size  γ η . So, in (0.8), up to this error, we can assume that the “Fourier transform” of ϕ has support in a ball of radius L−1 . The uncertainty principle then states that, modulo a small error, ϕ can be replaced with a function that is constant over cubes of size L  L; the error is estimated by means of Lemma 6.2 and is of size L /L. We pick L so that the error term L /L  γ η . This now enables us to transform condition (0.8) into  


1  (ωk − ω) aj
−γ η and aj = 1 + O(γ η ). (0.9)  d (L )  j

|k−kj |≤L

j

Here, the sites kj are the centers of the balls where ϕ is “constant”; and aj is the square of the L2 -norm of ϕ on that ball.

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The probability of the event (0.9), say P , is then estimated using large deviations 2η  d techniques; the exponential estimate one gets is P
e−γ (L ) . Saturating the two error −1/2+η  estimates given above, we get L  γ and L  γ −1/2+2η (here and in the rest −1 of the paper, a  b means C b ≤ a ≤ Cb for some positive constant C). Hence, P is small if γ 2η+2dη−d/2 → +∞ when γ → 0. This is the case if η < d/(4d + 4). In Theorem 0.2, the limit value for η is d/(6d + 4); the difference comes from the fact that we only assumed that the random variables are square integrable; hence, one has to take their possible unboundedness into account in the error estimates and in the large deviation estimation. We now describe our results in their full generality.

1. The Results Let H be a translational invariant Jacobi matrix with exponential off-diagonal decay that is H = ((hk−k  ))k,k  ∈Zd such that, H0: • h−k = hk for k ∈ Zd and for some k = 0, hk = 0. • there exists c > 0 such that, for k ∈ Zd , |hk | ≤

1 −c|k| e . c

(1.1)

The infinite matrix H defines a bounded self-adjoint operator on 2 (Zd ). Using the Fourier transform, it is easily seen that H is unitarily equivalent to the multiplication by the function θ → h(θ ) defined by h(θ ) =




hk eikθ where θ = (θ1 , . . . , θd ),

k∈Zd

acting as an operator on L2 (Td ), where Td = Rd /(2π Zd ) (the Lebesgue measure on Td is normalized so that the constant function 1 has norm 1). The function h is real analytic on Td . We assume H1: the minima of h : Td → R are quadratic non-degenerate.
(1 + cos(θi )). In this case, assumpIf H is the free Laplace operator, then h(θ ) = 1≤j ≤d

tion (H1) is satisfied. Let Vω be the diagonal matrix defined by (0.2) with entries the independent identically distributed real valued random variables (ωk )k∈Zd that satisfy H2: the random variables (ωk )k∈Zd are non-trivial, are lower semi-bounded and for some r ∈ (1, +∞], they satisfy ∀r  < r,





E(|ω0 |r )1/r < +∞.

(1.2)

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At the cost of adding a constant to H , we may assume that 0 is the essential infimum of the random variables (ωk )k∈Zd . To state our results, we define the constant η(r) 0 < η(r) =

 d d d 2−r ≤ where c(r) = + sup ,0 . 4(d + 1 + c(r))) 4(d + 1) 2(r − 1) r −1 (1.3)

Pick γ > 0 and define the self-adjoint operator Hω,γ = H + γ Vω .

(1.4)

Let Nγ (E) be the integrated density of states of Hω,γ ; it is defined by (0.6). Our main result is Theorem 1.1. Assume (H1) and (H2). Fix η ∈ (0, η(r)). Then, there exists γη > 0 and ε > 0 such that, for γ ∈ (0, γη ), one has −ε

Nγ (γ (ω − γ η )) ≤ e−γ .

(1.5)

Let us now comment on our results. Since the work of I.M. Lifshitz (see [16, 17], and also [18]), it is well known that, near the bottom of the spectrum of random Schr¨odinger operators, the eigenvalue distribution, i.e. the density of states has a large deviation regime, the so-called “Lifshitz tails”. In Theorem 1.1, we show that, in the weak disorder limit, the zone where the large deviation regime holds goes quite far, almost up to the average of the potential. It would be very interesting to understand what happens if one goes further up in energy, i.e. when one gets closer to γ ω. One can expect to enter some kind of “central limit theorem” zone where the behavior of the density of states (and certainly of other spectral quantities) changes dramatically. This is the energy region where the mobility edge is expected. Our second result concerns the decay of the Green’s function. To deduce estimates on the Green’s function of Hω from Theorem 1.1, we use the results of [4]; hence, we need a regularity condition on the distribution of the random variables (ωk )k∈Zd , namely we assume H3: the common distribution of the random variables (ωk )k∈Zd is H¨older continuous, i.e. that there exists τ ∈ (0, 1) and C > 0 such that, for a < b, one has P({ω0 ∈ [a, b]}) ≤ C|b − a|τ . Theorem 1.1 implies Theorem 1.2. Assume (H1), (H2) and (H3). Fix η ∈ (0, η(r)) and s ∈ (0, τ/4). There exists γη,s > 0 and a > 0 such that, for γ ∈ (0, γη,s ), the Green’s function satisfies, for (m, n) ∈ Zd × Zd and for E ∈ [0, γ (ω − γ η )],   −1 s  1 −a √|E−γ ω||m−n|  δm  ≤ e . sup E  δn , Hω,γ − E − iε a ε∈R

(1.6)

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As a consequence of estimate (1.6), we obtain exponential and dynamical localization, as well as an upper bound on the localization length (see the paragraph following Theorem 0.1 in the introduction, as well as [24, 3, 1, 2]). As for the Poisson behavior of the level spacings, though in [20], the argument is presented in the case when H = − d , it extends as is to the case we are dealing with here. Hence, when (H3) holds with τ = 1, the Wegner estimate (see e.g. [6]) ensures the existence of the density of states nγ (E); if nγ (E) > 0, the level spacings follow a Poisson distribution asymptotically. We note here that Theorem 0.1 (resp. Theorem 0.2) is a special case of Theorem 1.1 (resp. Theorem 1.2). Let us shortly comment on our assumptions. First, one can weaken the decay assumption on the random variables (ωk )k∈Zd and only assume that they are integrable; i.e. we assume that H2’: the random variables (ωk )k∈Zd are non-trivial, lower semi-bounded and satisfy E(|ω0 |) < +∞. Then, one proves Theorem 1.3. Assume (H1) and (H2’). Fix η ∈ (0, 1). There exists γη > 0 and ε > 0 such that, for γ ∈ (0, γη ), one has −ε

Nγ (γ (ω − η))) ≤ e−γ . This implies Theorem 1.4. Assume (H1), (H2’) and (H3). Fix η ∈ (0, 1) and s > 0. There exists γη,s > 0 and a > 0 such that, for γ ∈ (0, γη,s ), the Green’s function satisfies, for (m, n) ∈ Zd × Zd and for E ∈ [0, γ (ω − η)],   −1 s  1 −a √|E−γ ω||m−n|  δm  ≤ e . sup E  δn , Hω,γ − E − iε a ε∈R Let us notice here that one can also relax assumption (H1) on the background operator h. From the proof given in the present paper, it is clear that, at the expense of changing η(r), Theorems 1.1 and 1.3 still hold if one only assumes that h reaches its minimum at finitely many isolated points and that the minimum is attained in a polynomial way. In dimension 2, using the techniques developed in [14], one can prove that Theorems 1.1 and 1.3 hold for any non-constant, analytic function h. 1.1. The layout of the paper. To end this section, we describe the layout of the paper. In Sect. 2, we show how to deduce Theorem 1.2 from Theorem 1.1. In Sect. 3, we reduce Theorem 1.1 to an equivalent statement for bounded random variables depending on γ . Section 4 is devoted to the proof of that statement. The section begins with a description of the periodic approximations, Sect. 4.1; we recall some facts of Floquet theory in Sect. 4.2. Then, we estimate the density of states of the periodic approximations in two steps; first, we reduce this question to estimating the probability of certain events (expressed only in terms of the random variables) in Sects. 4.3 and 4.4. In Sect. 4.5, we carry out this estimation following the ideas outlined at the end of the introduction. The paper is completed by an appendix, Sect. 6, where we gather some useful technical results.

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2. The Proof of Theorem 1.2 To deduce Theorem 1.2 from Theorem 1.1, we use the finite volume localization criterion developed in Theorem 3.1 of [4]. To be able to use this criterion, we first need to check that, for some 0 < s < τ , our random variables are s-regular in the sense of Definition 3.1 in [4], i.e. the s-decoupling property (1.7) in [4]; this is done in Sect. 6.3. Let us note that, if h is a trigonometric polynomial, the s-regularity is not necessary as we can use Theorem 1.1 of [4]; this applies in particular when H is the free Laplace operator. Let us now describe the finite volume criterion. Therefore, we consider the resolvent of Hω,γ restricted to a cube with Dirichlet boundary conditions. More precisely, pick D "L ⊂ Zd , the cube centered at 0 of side-length 2L + 1. Let Hω,γ |"L be the random Hamiltonian Hω,γ restricted to the box "L with Dirichlet boundary conditions, i.e., D Hω,γ |"L = !"L Hω,γ !"L , where !"L is the projector onto the sites in "L . Rephrased with our notations, to check the localization criterion (3.8) developed in [4], it suffices to check that, for γ sufficiently small, for some cube "L , one has s  
  C0 L2d γ −2s e−c|m−n| E  δ0 , (E + iε − Hω,γ |"L )−1 δm  eδ(E)|n|/C0 < 1, m∈"L n∈Zd \"L

(2.1) where C0 is a constant depending only on h and on the random variables (ωγ )γ ∈Zd . Let us prove the bound (2.1); therefore, we use Theorem 2.1 ([14]). There exists C > 0 such that, for L ≥ 1, γ ∈ [0, 1] and E ∈ R, one has   D P Hω,γ ≤ CLd Nγ (E). |"L admits an eigenvalue below energy E In [14], Theorem 2.1 was proved in dimension 2; the proof carries over to arbitrary dimension without a change. We do not repeat it here. Fix η as in Theorem 1.2  and fix η  < η < η(r). Define 4γ ,η ,L = { there exists an 

eigenvalue of Hω,γ |"L ≤ γ ω − γ η . Using Theorem 1.1 in conjunction with Theorem 2.1 we obtain that there exists γη > 0 and ε > 0 such that, for 0 < γ < γη , −ε/2 , one has 1 ≤ L ≤ eγ P(4γ ,η ,L ) ≤ Cedγ

−ε/2

e−γ

−ε

1

−ε

≤ e− 2 γ .

(2.2)

By a Combes-Thomas estimate, Lemma 6.1, we get that, for ω ∈ 4γ ,η,L and E ∈ [0, γ (ω − γ η )], one has √   η   (2.3)  δn , (E + iε − Hω,γ |"L )−1 δm  ≤ Cγ −1−η e− |E−γ (ω−γ )||m−n|/C , for C ≥ C˜ 0 , where C˜ 0 > 0 is a constant independent of γ and η. Define δ(E) :=  |E − γ (ω − γ η )|. We notice that, as E − γ ω ≥ γ 1+η and η < η , for γ sufficiently small, one has 1 δ(E) ≥ |E − γ ω|. (2.4) 2

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Pick 0 < s < s  < 1. Using H¨older’s inequality and the bound (2.10) in [4], namely

 s      −1 E  δn , (E + iε − Hω,γ |"L ) δm  ≤ Cγ −s , Eq. (2.3) gives, for some ε > 0,  s    E  δn , (E + iε − Hω,γ |"L )−1 δm   s    = E  δn , (E + iε − Hω,γ |"L )−1 δm  1{ω∈4γ ,η,L }  s    +E  δn , (E + iε − Hω,γ |"L )−1 δm  1{ω∈4γ ,η,L } ≤ Cγ −1−η e−δ(E)|m−n|/C

 s  s/s      −1 +E  δn , (E + iε − Hω,γ |"L ) δm  P(4γ ,η,L )(s −s)/s 1

−ε

≤ Cγ −1−η e−δ(E)|m−n|/C + Cγ −s e− 2 γ .

(2.5)

Summing (2.5) over m ∈ "L for n ∈ Zd \ "L , and taking "L of side-length 1 ≤ L ≤ −ε/2 , for γ sufficiently small, we obtain eγ s  
  CL2d γ −2s e−c|m−n| E  δ0 , (E + iε − Hω,γ |"L )−1 δm  eδ(E)|n|/(8C) m∈"L n∈Zd \"L

≤ CL3d γ −3s eδ(E)L/(8C) e−γ

−ε

+ CL2d γ −2s−1−η · S,

(2.6)

where S :=




e−c|m−n| e−δ(E)|m|/C eδ(E)|n|/(8C)

m∈"L n∈Zd \"L

=




|m|≤L |n|≥2L

+




+

|m|≤L/2 |n|≥L

≤ Ce−L/C Ld−1 +




e−c|m−n| e−δ(E)|m|/C eδ(E)|n|/(8C)

L/2≤|m|≤L L≤|n|≤2L




e−c|m−n| e−δ(E)|m|/(8C)

L/2≤|m|≤L L≤|n|≤2L

≤ Ce−L/C Ld−1 + CLd e−δ(E)L/(8C) . −ε/2

(2.7)

for 2ρ > 1 + η(r), for E ∈ [0, γ (ω − γ η )], one has If we take γ −ρ ≤ L ≤ eγ −ε δ(E)L ≥ γ for some ε > 0 and γ sufficiently small. Hence, using this in (2.7) and (2.6), for γ sufficiently small, we obtain  s 
  CL2d γ −2s e−c|m−n| E  δ0 , (E + iε − Hω,γ |"L )−1 δm  eδ(E)|n|/(8C) < 1/16. m∈"L n∈Zd \"L

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Taking (2.4) into account, we obtain for C ≥ C˜ 0 , that there exists γC,η > 0 such that, for 0 < γ < γC,η , (8C)L2d γ −2s




e−c|m−n|

m∈"L n∈Zd \"L

 s  √   E  δ0 , (E + iε − Hω,γ |"L )−1 δm  e |E−γ ω||n|/(8C) < 1. So, the finite volume criterion (2.1) is satisfied if we take C so that 8C > C0 . Then, Theorem 3.1 of [4] implies Theorem 1.2. 3. The Proof of Theorem 1.1 To prove Theorem 1.1, we first bring ourselves back to the case when the random variables are bounded. Of course, except if the (ωn )n∈Zd are bounded, the new random variables and, a fortiori, their supremum depend on γ . Let η ∈ (0, η(r)) be fixed. As there exists r  < r such that η < η(r  ) < η(r) and   E(|ω0 |r )1/r < +∞, we may assume that E(|ω0 |r )1/r < +∞. Pick λ > 0 and, for n ∈ Zd , consider the random variables ωn (λ) = inf(ωn , λ). They are independent, identically distributed and bounded. Let us estimate the difference E (ω0 − ω0 (λ)). Therefore, we write  
0 ≤ E(ω0 − ω0 (λ)) = E (ω0 − λ)1{ω0 ≥λ} ≤ λN P(N λ < ω0 ≤ (N + 1)λ) N≥1

= λ lim

M→+∞

M


N [P(ω0 > N λ) − P(ω0 > (N + 1)λ)]

N=1

= λP(ω0 > λ)  + λ lim

M→+∞

MP(ω0 > (M + 1)λ) −

= λP(ω0 > λ) − λ




M


 P(ω0 > N λ)

N=2

P(ω0 > N λ).

(3.1)

N≥2

The convergence in (3.1) is guaranteed by the Markov bound P(ω0 > N λ) ≤

E(ω0r ) . (N λ)r

This and (3.1) imply that, if we set ω(λ) := E(ω0 (λ)), then 0 ≤ ω − ω(λ) ≤

E(ω0r ) . λr−1

(3.2)

Weak Disorder Localization and Lifshitz Tails

135

On 2 (Zd ), consider the random operator Hω,γ (λ) = H + γ Vω (λ), where Vω (λ) is the diagonal matrix having entries (ωn (λ))n∈Zd . One has Hω,γ (λ) ≤ Hω,γ . Hence, if Nγ (E; λ) denotes the integrated density of states of Hω,γ (λ), we have   Nγ (γ (ω − γ η )) ≤ Nγ (γ (ω − γ η ); λ) ≤ Nγ γ (ω(λ) − γ η + C/λr−1 ); λ . 

In the last step, we have used (3.2). In particular, if we set λ = λ(γ ) = γ −η /(r−1) , where η < η < η(r), for γ sufficiently small, we get    Nγ (γ (ω − γ η )) ≤ Nγ γ (ω(λ) − γ η ); λ(γ ) . (3.3) Hence, Theorem 1.1 is an immediate corollary of 

Theorem 3.1. Fix η < η < η(r) and set λ = λ(γ ) = γ −η /(r−1) as above. Then, there exists γη > 0 and ε > 0 such that, for γ ∈ (0, γη ), one has   −ε Nγ γ (ω(λ) − γ η ); λ(γ ) ≤ e−γ . 4. The Proof of Theorem 3.1 From now on, to simplify the notations, except in the case of possible confusion, we do not explicitly write the dependence of the random variables (ωn (λ))n∈Zd in λ; we call them (ωn )n∈Zd ; we just keep in mind the fact that they are bounded by λ = λ(γ ). We also do so for all the objects involving these random variables; e.g., the operator Hω,γ (λ) is called Hω,γ , etc. Let us now briefly explain the proof of Theorem 3.1. First, we introduce a periodic approximation scheme (that has proved useful for related issues in [11, 15]). This scheme delivers a sequence of very quickly converging finite volume approximations to the density of states. To analyze the finite volume density of states, we recall the Floquet theory of periodic operators. We use it to reduce the problem of estimating the density of states to the probability of estimating a well chosen event. 4.1. Periodic approximations. Let (ωn )n∈Zd be a realization of the random variables N , a periodic operator acting on 2 (Zd ) by defined above. Fix N ∈ N∗ . We define Hω,γ

N N |δl+n δl+n | . Hω,γ = H + Vω,γ =H+ ωn n∈Zd2N +1

l∈(2N+1)Zd

Here, Zd2N +1 = Zd /(2N + 1)Zd , δl = (δj l )j ∈Zd (δj l is the Kronecker symbol) and |uu| is the orthogonal projection on a unit vector u. N , we define the integrated density of states denoted by For the periodic operator Hω,γ N by (0.6) (see e.g. [22]); it is a non-decreasing function. Its derivative, dN N , is a Nω,γ ω,γ positive measure and, it satisfies, for ϕ ∈ C0∞ (R),    
 1 N N N δ . (4.1) ϕ(x)dNω,γ (x) = , ϕ(H )δ ϕ, dNω,γ = k k ω,γ (2N + 1)d R d k∈Z2N +1

One has

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Lemma 4.1. Pick η > 0. There exists ν0 ∈ (0, 1) and ρ > 0 such that, for γ ∈ [0, 1], E ∈ R, ν ∈ (0, ν0 ) and N ≥ ν −ρ , one has     −η −η N N E Nω,γ (E − ν) − λ(γ )e−ν ≤ Nγ (E) ≤ E Nω,γ (E + ν) + λ(γ )e−ν . (4.2) Proof of Lemma 4.1. In [11], we have proved a result similar to Lemma 4.1 in dimension 1. To prove Lemma 4.1, we first prove a “smoothed out” version of the same result, namely Lemma 4.2. There exists C > 1 such that, for ϕ ∈ C0∞ (R), for K ∈ N, K ≥ 1, and N ∈ N∗ , we have  

   J   CK K N 2 d ϕ  |E ϕ, dNω,γ − (ϕ, dN ) | ≤ λ(γ ) (1 + x ) J (x) . sup  N d x x∈R 0≤J ≤K+d+2

(4.3) Proof of Lemma 4.2. The proof follows the lines of the proof of Lemma 1.1 in [12] (and corrects some misprints). For the reader’s convenience, we reproduce it here. We know (see [21, 6]) that    (ϕ, dN ) = ϕ(λ)dN = E δ0 , ϕ(Hω,γ )δ0 . R

Averaging (4.1) in ω and using the fact that the random variables (ωn )n∈Zd are i.i.d. and the translation invariance of H , we get     N N E ϕ, dNω,γ = E δ0 , ϕ(Hω,γ (4.4) )δ0 .   N )δ − δ , ϕ(H So that we want to estimate δ0 , ϕ(Hω,γ 0 0 ω,γ )δ0 . Therefore we use Helffer-Sj¨ostrand’s formula ([8]) that reads  ∂ ϕ˜ i ϕ(Hω ) = (4.5) (z) · (z − Hω )−1 dz ∧ dz, 2π C ∂z where ϕ˜ is an almost analytic extension of ϕ (see [19]), i.e. a function satisfying 1. for z ∈ R, ϕ(z) ˜ = ϕ(z); 2. supp(ϕ) ˜ ⊂ {z ∈ C; |Im(z)| < 1}; 3. ϕ˜ ∈ S({z ∈ C; |Im(z)| < 1}); ∂ ϕ˜ (x + iy) · |y|−n (for 0 < |y| < 1) is bounded in 4. the family of functions x → ∂z S(R) for any n ∈ N; more precisely, there exists C > 1 such that, for all p, q, r ∈ N, there exists Cp,q > 0 such that    

  ∂ q ϕ    p ∂q ∂ ϕ ˜   sup sup x (x + iy)  ≤ C r Cp,q sup sup x p |y|−r ·  (x) . q q   ∂x ∂z ∂x 0<|y|≤1 x∈R q  ≤r+q+2 x∈R p ≤p

(4.6)

Weak Disorder Localization and Lifshitz Tails

137

Equations (4.4) and (4.5) then yield 

        ∂ ϕ˜  1   N  (z) |δ0 , (z − H N )−1 ) − (ϕ, dN ) ≤ E E (ϕ, dNω,γ ω,γ   4π C ∂z   −(z − Hω,γ )−1 δ0 |dxdy     1   ∂ ϕ˜ 
≤ |ωk − ω[k]N | E (z)  ∂z  4π C

|k|≥N

N −1 |δ0 , (z − Hω,γ ) δk | |δk , (z − Hω,γ )−1 δ0 |dxdy



(4.7) where z = x + iy and [k]N ≡ k mod (2N + 1) (the point being chosen in [−N, N ]d ). By a Combes-Thomas argument (see Sect. 6.1), we know that there exists C > 1 such that, uniformly in (ωn )n∈Zd and N ≥ 1, we have, for Im(z) = 0, N −1 |δ0 , (z − Hω,γ ) δk | + |δk , (z − Hω,γ )−1 δ0 | ≤

C e−|Im(z)|·|k|/C . |Im(z)|

Hence, as the random variables (ωn )n∈Zd are bounded by λ(γ ), (4.7) gives, for some C > 1,         ∂ ϕ˜  1   N  e−|Im(z)|·N/C dxdy. E ϕ, dNω,γ − (ϕ, dN ) ≤ Cλ(γ )  (z) d+2 ∂z |Im(z)| C Taking into account the properties of almost analytic extensions (4.6), for some C > 1, for K ≥ 1 and N ≥ 1, we get      1   N K+1 |y|K e−|y|·N/C dxdy E ϕ, dNω,γ − (ϕ, dN ) ≤ λ(γ )C 1 + x2 R×[−1,1]   J   2 d ϕ   · sup (1 + x ) d J x (x) x∈R 0≤J ≤K+d+2



CK ≤ λ(γ ) N

This completes the proof of the Lemma 4.2.

K sup

x∈R 0≤J ≤K+d+2

  J   (1 + x 2 ) d ϕ (x) .   dJ x

" !

N and dN are Let us now complete the proof of Lemma 4.1. First note that, as dNω,γ γ supported in I := [0, C +λ(γ )] (for some C > 0), we only need to prove Lemma 4.1 for energies E ∈ I . Pick ϕ, a Gevrey class function of Gevrey exponent α > 1 (see [10]); assume, moreover, that ϕ has support in (−1, 1), that 0 ≤ ϕ ≤ 1 and that ϕ ≡ 1 on (−1/2, 1/2]. Let E ∈ I and ν ∈ (0, 1), and set · ϕE,ν (·) = 1[0,E] ∗ ϕ . ν

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F. Klopp

Then, by Lemma 4.2 and the Gevrey estimates on the derivatives of ϕ, we get that there exist C > 1 such that, for N ≥ 1, K ≥ 1 and 0 < ν < 1, we have K+d+2

    C(K + d + 2)1+α   N . E ϕE,ν , dNω,γ − (ϕE,ν , dNγ ) ≤ λ(γ ) Nν

(4.8)

We optimize the right-hand side of (4.8) in K. Therefore, we take N ≥ 1 and 0 < ν < 1 so that Nν satisfy log(N ν/C) ≥ (1 + α)[1 + log(d + 3)];

(4.9)

then, we choose K ∈ N∗ so that (1 + α) log(K + d + 2) ≤ log(N ν/C) − log 2 ≤ (1 + α) log(K + d + 3). Plugging this into (4.8), for N ≥ 1 and 0 < ν < 1 so that N ν satisfy (4.9), we obtain N |E((ϕE,ν , dNω,γ )) − (ϕE,ν , dNγ )| ≤ λ(γ )e−(Nν/2C)

1/(1+α) /2

.

Pick η > 0. Now, if we choose N ≥ ν −1−η , we get that there exist ν0 > 0 such that for 0 < ν < ν0 , we have N |E((ϕE,ν , dNω,γ )) − (ϕE,ν , dNγ )| ≤ λ(γ )e−ν

−η/(2α)

.

(4.10)

By definition, ϕE,ν ≡ 1 on [0, E], and ϕE,ν has support in [−ν, E + ν] and is bounded N and dN are positive measures, we have by 1. As dNω,γ γ       N N N ≤ E Nω,γ E Nω,γ (E) ≤ E ϕE,ν , dNω,γ (E + ν) ,   E Nγ (E) ≤ E((ϕE,ν , dNγ )) ≤ E(Nγ (E + ν)).

(4.11)

Hence, by (4.10) and (4.11), we obtain   N Nγ (E) ≤ (ϕE,ν , dNγ ) = E ϕE,ν , dNω,γ    N + (ϕE,ν , dNγ ) − E ϕE,ν , dNω,γ   −η/(2α) N ≤ E Nω,γ (E + ν) + λ(γ )e−ν and Nγ (E) ≥ (ϕE−ν,ν , dNγ )      N N + (ϕE−ν,ν , dNγ ) − E ϕE−ν,ν , dNω,γ = E ϕE−ν,ν , dNω,γ   −η/(2α) N ≥ E Nω,γ (E − ν) − λ(γ )e−ν . This completes the proof of Lemma 4.1 as η > 0 was chosen independent of α. ! "

Weak Disorder Localization and Lifshitz Tails

139

N , we need some Floquet theory that we de4.2. Some Floquet theory. To analyze Hω,γ velop now. We denote by F : L2 ([−π, π ]d ) → 2 (Zd ) the standard Fourier series transform. For u ∈ L2 ([−π, π ]d ), we have
(Hˆ ω,γ u)(θ ) = (F ∗ Hω,γ Fu)(θ ) = h(θ )u(θ ) + γ ωj (!j u)(θ ), j ∈Zd

where (!j u)(θ ) =

1 eij θ (2π)d

 [−π,π]d

e−ij θ u(θ )dθ.

 d    π π ⊗ , 2N+1 Define the unitary equivalence U : L2 [−π, π ]d → L2 − 2N+1   2 Zd2N +1 by (U u)(θ ) = (uk (θ ))k∈Zd , where the (uk (θ ))k∈Zd are defined by 2N +1 2N +1
u(θ ) = eikθ uk (θ ), (4.12) k∈Zd2N +1

2π Zd periodic. 2N + 1 The functions (uk )k∈Zd are computed easily; if the Fourier coefficients of u are 2N +1 denoted by (uˆ l )l∈Zd , then one gets
uˆ k+(2N+1)l ei(2N+1)lθ . (4.13) uk (θ ) = and the functions (θ → uk (θ ))k∈Zd

are

2N +1

l∈Zd

The operator

N F ∗U ∗ U FHω,γ

acts on

L2

 d    π π − 2N+1 , 2N+1 ⊗ 2 Zd2N+1 ; it is the

multiplication by the matrix N Mω,γ (θ ) = H N (θ ) + γ VωN ,

where H N (θ ) =

  hj −j  (θ ) (j,j  )∈Zd

2

2N +1

Here, the functions (hk )k∈Zd

2N +1

+ 1)d

and VωN =



ωj δjj 

(4.14) 

 2 (j,j  )∈ Zd2N +1

.

(4.15)

are the components of h decomposed according to (4.12).

+ 1)d -matrices

The (2N × (2N H N (θ ) and VωN are non-negative matrices. N with This immediately tells us that the Floquet eigenvalues and eigenvectors of Hω,γ

  Floquet quasi-momentum θ i.e. the vectors, u = uj j ∈Zd , solution to the problem 

N u = λu, Hω,γ uj +k = e−ikθ uj for j ∈ Zd , k ∈ (2N + 1)Zd



N (θ ) are the eigenvalues and eigenvectors of the (2N + 1)d × (2N + 1)d matrix Mω,γ continued to Zd quasi-periodically. For E ∈ R, one has    1 N N Nω,γ (E) = D eigenvalues of Mω,γ (θ ) in [0, E] dθ. (4.16) d (2π) [− 2Nπ+1 , 2Nπ+1 ]d

140

F. Klopp

d Considering H as (2N + 1)-periodic on  that the Floquet eigenvalues of  Z , we see 2πk H (for the quasi-momentum θ ) are h θ + 2N+1 ; the Floquet eigenvalue k∈Zd2N +1   2π k d h θ + 2N +1 is associated to the Floquet eigenvector uk (θ ), k ∈ Z2N+1 defined by

 2π k   1 −i θ+ 2N +1 j uk (θ ) = e . (2N + 1)d/2 j ∈Zd 2N +1

In the sequel, the vectors in 2 (Zd2n+1 ) are given by their components in the orthonormal basis (uk (θ ))k∈Zd . The vectors of the canonical basis denoted by (vl (θ ))l∈Zd have 2N +1 2N +1 the following components in this basis vl (θ ) =

1 (2N + 1)d/2

We define the vectors (vl )l∈Zd

e

  2π k i θ+ 2N +1 l k∈Zd2N +1

.

by

2N +1

 2π kl  1 ei 2N +1 . d/2 k∈Zd2N +1 (2N + 1)

vl = e−ilθ vl (θ ) =

(4.17)

4.3. Estimating the density of states of the periodic approximations. Define the event N 4(E, γ , N ) = {ω; ∃θ ∈ Rd such that Mω,γ (θ ) has an eigenvalue in [0, E]}.

By Lemma 4.1 and (4.16), Theorem 3.1 is a consequence of Proposition 4.1. Fix 0 < η < η < η(r) and ρ > d. Then, there exists γη,ρ > 0 and ε > 0 such that, for γ ∈ (0, γη,ρ ), one has −ε

P[4(γ (ω − γ η ), γ , N )] ≤ e−γ ,

(4.18)

where 







2N + 1 = [γ −1/2+2η +η /(r−1) ]o · [γ −η −η /(r−1) ]o · [γ −ρ ]o . Here, [·]o denotes the largest odd integer smaller than Notice that by the definition of η(r), as η < η < η(r), one has −η − η /(r − 1) < 0 and −1/2 + 2η + η /(r − 1) < 0. Let us now derive Theorem 1.1 from Proposition 4.1. Pick η as in Theorem 1.1 and η < η < η(r). Let ρ > d be larger than the exponent ρ obtained in Lemma 4.1 for α = 1. By (4.16) and (4.18), one has      N E Nω,γ γ ω − γη  1  N = E[D{eigenvalues of Mω,γ (θ ) in [0, γ (ω − γ η )]}]dθ (2π )d [− 2Nπ+1 , 2Nπ+1 ]d  1  ≤ P[4(γ (ω − γ η ), γ , N )](2N + 1)d dθ (2π )d [− 2Nπ+1 , 2Nπ+1 ]d 

= P[4(γ (ω − γ η ), γ , N )].

Weak Disorder Localization and Lifshitz Tails

141

N (θ ) has rank (2N + 1)d . Applying Proposition 4.1, for Here, we used the fact that Mω,γ γ sufficiently small, we get 

−ε

N (γ (ω − γ η ))] ≤ Ce−γ . E[Nω,γ 

Plugging this into (4.2) for E = γ (ω − γ η ) and ν = γ 1+η , as Nγ is increasing and as η < η , for γ sufficiently small, we obtain         −ε −ε  N γ ω − γη + e−γ ≤ e−γ Nγ γ ω − γ η ≤ E Nω,γ with ε  < ε. This completes the proof of Theorem 3.1.

" !

4.4. The proof of Proposition 4.1. The proof of Proposition 4.1 follows roughly the lines of the proof of the upper bound in [11], Sect. 2.3. The main difference comes from the fact that, in the present case, the supremum norm of the random variables depends on the energy interval on which we want to control the density of states or the probability of the presence of eigenvalues. Pick η, ρ as in Proposition 4.1. Define 



2L + 1 = [γ −1/2+2η +η /(r−1) ]o

and





2K  + 1 = [γ −η −η /(r−1) ]o · [γ −ρ ]o .

Notice that 2N + 1 = (2K  + 1)(2L + 1). By assumption (H0), h is real analytic on the compact torus Td ; it admits a compact set of minima. As the minima are quadratic non-degenerate, they are isolated points. Let Z be the finite set of these minima i.e. Z = h−1 ({0}) = {θ1 , . . . , θM }. By assumption (H1), we know that there exists C > 0 such that, for θ ∈ Td , h(θ ) ≥ C

inf

1≤J ≤M

|θ − θJ |2 .

(4.19)

Then, we prove Lemma 4.3. Pick 0 < η < η < η(r) and N as in Proposition 4.1. Let L and K  be defined as above. There exists C > 0 and γ0 > 0 such that, for 0 < γ < γ0 , we have        (4.20) 4J,J ,k  , 4(γ (ω − γ η ), γ , N ) ⊂ |k  |≤K 

1≤J ≤J  ≤M

where, for 1 ≤ J ≤ J  ≤ M and |k  | ≤ K  , we define the events     
   η   γ 1    i(θJ −θJ  )l    (2L +1)+l  − ω)e ≥ 4J,J ,k = ω; (ω k  16(M + 1)2  .  (2L + 1)d     |l |≤L

Before proving this lemma, let us use it to complete the proof of Proposition 4.1.   Therefore, we need to estimate the probability of the events 4J,J ,k . This is done in

142

F. Klopp

Proposition 4.2. There exists ε > 0 such that, for γ sufficiently small, for 1 ≤ J, J  ≤ M and for |k  | ≤ K  , we have P(4J,J

 ,k 

−ε

) ≤ e−γ .

(4.21)

As K  ≤ γ −ρ−1 and 1 ≤ J, J  ≤ M, using Lemma 4.3, summing the estimates (4.21) over (J, J  , k  ), we obtain (4.18). Proof of Proposition 4.2. Proposition 4.2 is a large deviation estimate. The only “difficulty” comes from the fact that, in the present case, the random variables (ωn )n∈Zd depend on γ , hence, on the size of the sample of random variables we are considering. Nevertheless, the standard ideas to obtain large deviation estimates do apply (see, e.g. [7]). 

Recall that ω0 = ω0 (λ) and λ = γ −η /(r−1) . Let us first estimate the variance of (ωn )n∈Zd . One needs to consider two cases. If the exponent r in assumption (H2) satisfies r ≥ 2, one has sup E((ω0 (λ) − ω(λ))2 ) < +∞. λ>1

(4.22)

If 1 < r < 2, we compute E(ω0 (λ)2 ) = E(ω02 1{ω0 ≤λ} ) + λ2 P(ω0 > λ) ≤ λ2−r E(ω0r ) + λ2 E(ω0r )/λ−r ≤ 2λ2−r E(ω0r ).

(4.23)

J,J  ,k 

Let us now estimate P(4 ). We only do the explicit computations for J = J  ;  the case J = J is treated in the same way. As the random variables (ωn )n∈Zd are real, we only have to estimate the probability that    
  1 γη        (ω − ω) cos((θ − θ )l ) ≥ J k (2K +1)+l J   2 (2L + 1)d     32(M + 1) |l |≤L

and the probability that    
  1 γη        (ω − ω) sin((θ − θ )l ) ≥ . J k (2K +1)+l J    d 2 (2L + 1)     32(M + 1) |l |≤L

We estimate the first probability; the second estimate is done in the same way. Therefore, we have to estimate the probability that (2L


1 γη (ωk  (2K  +1)+l  − ω) cos((θJ − θJ  )l  ) ≥ d + 1)   32(M + 1)2

(4.24)

|l |≤L

and the probability that (2L


1 γη (ωk  (2K  +1)+l  − ω) cos((θJ − θJ  )l  ) ≤ − . d 32(M + 1)2 + 1)   |l |≤L

Both estimates being done in the same way, we only do the computations for (4.24).

Weak Disorder Localization and Lifshitz Tails

143

To simplify the notations, let R = (2L + 1)d and, reindex the points {l  ; |l  | ≤ L } as {l  (U ); 1 ≤ U ≤ R}. Define ω˜ U = (ωk  (2K  +1)+l  (U ) − ω) cos((θJ − θJ  )l  (U )). The random variables (ω˜ U )1≤U ≤R are independent, centered (i.e. E(ω˜ U ) = 0) and bounded by λ. Estimating the probability of (4.24) is estimating the probability of R 1
ω˜ U ≥ γ η /C0 , R U =1

where C0 = 32(M + 1)2 . For t > 0, as the random variables (ω˜ U )1≤U ≤R are independent, we compute % & %  % && R R 1
C0
η η P ω˜ U ≥ γ /C0 ≤ E exp t −γ + ω˜ U R R U =1

=e

−tγ η

R ' U =1



U =1



tC0 E exp ω˜ U R

 .

Now, pick t such that C0 tλ ≤ R; then, for 1 ≤ U ≤ R, one has %



 2 &

tC0 tC0 tC0 E exp ≤ E 1+ ω˜ U ω˜ U + ω˜ U R R R %

&

  tC0 2 tC0 2 ≤ 1+ C(λ) ≤ exp C(λ) , R R

(4.25)

(4.26)

where C(λ) satisfies C(λ) ≥ sup E((ω˜ U )2 ).

(4.27)

1≤U ≤R

Plugging (4.26) into (4.25), we obtain & %



 R tC0 1
η η ω˜ U ≥ γ /C0 ≤ exp −t γ − C(λ) . P R R U =1

Let us now distinguish the two cases: 1. If 1 < r < 2, one can take C(λ) = Cλ2−r . Recalling that 

R ∼ γ −d/2+dη (2+1/(r−1)) ,



λ ∼ γ −η /(r−1) ,

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F. Klopp 



and taking 2C0 tλ = R, i.e. t ∼ γ −d/2+2dη +(d+1)η /(r−1) , for γ sufficiently small, we obtain 

  1 tC0    C(λ) ≥ tγ η 1 − γ −η+η ≥ γ −d/2+2dη +(d+1)η /(r−1)+η . t γη − R 2 By definition, −d/2 + 2dη(r) + (d + 1)η(r)/(r − 1) + η(r) = 0; hence, as 0 < η < η < η(r), one has − d/2 + 2dη + (d + 1)η /(r − 1) + η < 0. This completes the proof of Proposition 4.2 in the case 1 < r < 2. 2. If r ≥ 2, the variance of ω˜ U is bounded by C uniformly in 1 ≤ U ≤ R. Taking   2CC0 tλ = Rγ η , i.e. t ∼ γ −d/2+2dη +(d+1)η /(r−1)+η , for γ sufficiently small, we obtain 

1 tC0   η C ≥ tγ η (1 − 1/2) ≥ γ −d/2+2dη +dη /(r−1)+2η . t γ − R 2 By definition, −d/2 + 2dη(r) + dη(r)/(r − 1) + 2η(r) = 0; hence, as 0 < η < η < η(r), one has − d/2 + 2dη + dη /(r − 1) + 2η < 0. This completes the proof of Proposition 4.2 in the case 2 ≤ r.

" !

4.5. The proof of Lemma 4.3. To prove Lemma 4.3, we follow the argument sketched at the end of the introduction. We define 







2L + 1 = [γ −1/2+2η +η /(r−1) ]o · [γ −η −η /(r−1) ]o

and

2K + 1 = [γ −ρ ]o .

Notice that 2N + 1 = (2K + 1)(2L + 1). Pick ω ∈ 4(γ (ω − γ η ), γ , N ). Hence, there exists θ ∈ Rd and a =




ak uk (θ )

k∈Zd2n+1

such that

() 2 • &a&2 (Zd ) = k∈Zd2N +1 |ak | = 1,  N 2N +1  • Mω (θ )a, a 2 (Zd ) ≤ γ (ω − γ η ), 2N +1 • uk (θ ) is defined in section 4.2. Using (4.14) and the positivity of H N (θ ) and VωN , we obtain that     VωN a, a 2  d  ≤ ω − γ η . H N (θ )a, a 2  d  ≤ γ (ω − γ η ) and  Z2N +1

 Z2N +1

(4.28)

Weak Disorder Localization and Lifshitz Tails

145

As a is normalized, the second inequality in (4.28) yields    VωN − ωI d N a, a 2 d ≤ −γ η ,

(4.29)

 (Z2N +1 )

where I d N is the identity matrix. We now use the first inequality in (4.28) to get more information on a. For 1 ≤ J ≤ M  d π π and θ ∈ − 2N+1 , 2N+1 , by (4.19), we know that, for some C > 0 and for γ small enough  



   2π k  1 2π k 1−2η   /C . (4.30)  2N + 1 − θJ  ≥ 2L + 1 '⇒ h θ + 2N + 1 ≥ γ For 1 ≤ J ≤ M, let kJ ∈ Zd be the unique vector satisfying 2π kJ − (2N + 1)θJ ∈ [−π, π )d . Define a J by  ak if |k − kJ | ≤ K, J (a )k = 0 if not. For γ sufficiently small, the vectors (a J )1≤J ≤M are pairwise orthogonal. Indeed, they have disjoint support as, for J = J  , one has |kJ − kJ  | ≥ N/C (for some C > 0) and K = o(N ). By the first estimate in (4.28) and by (4.30), we get * *2 M * *
  * J* a * ≤ Cγ · γ −1+2η = Cγ 2η . (4.31) *a − * * d J =1

2 (Z2N +1 )

Plugging this into (4.29), we get that, for some C > 0 and γ small enough & %M &, %M + 
  
 N N J J a , a ≤ −γ η 1 − Cγ η −η . Vω − ωI d J =1

J =1

2 (Zd2N +1 )

Expanding the vectors (aJ )1≤J ≤M in the basis (vl )l defined in (4.17), we get   M



   (ωl − ω)|a J , vl |2 + 2Re  (ωl − ω)a J , vl a J , vl  1≤J <J  ≤M l∈Zd

J =1 l∈Zd

2N +1

2N +1

η −η

η

≤ −γ (1 − Cγ

).

(4.32) aJ

We now translate each of the by kJ so as to center its support at 0. The vector thus obtained we again call a J ; then, (4.32) becomes M



(ωl − ω)|a J , vl |2

J =1 l∈Zd 2N +1



 + 2Re 







1≤J <J  ≤M l∈Zd

2N +1

η

≤ −γ (1 − Cγ

η −η

).

e

2iπ(kJ −kJ  )l 2N +1

  (ωl − ω)a J , vl a J , vl 

146

F. Klopp 

We apply Lemma 6.2 to each a J ; as K/K  ∼ γ η+η /(r−1) , we thus obtain a˜ J ∈ 2 (Zd2N+1 ) such that 

• &a J − a˜ J &2 (Zd • for

l

∈

2N +1 )

Zd2L +1

and



≤ Cγ η +η /(r−1) &a J &2 (Zd

k

∈

Zd2K  +1 ,

2N +1 )

;

we have

a˜ J , vl  +k  (2L +1)  = a˜ J , vk  (2L +1) ; • &a J &2 Zd



2N +1

= &a˜ J &2 Zd

2N +1

(4.33)

. 

Recalling that the random variables (ωk )k∈Zd are bounded by Cγ −η /(r−1) and that M is fixed, we get that, for some C > 0 and γ small enough M



(ωl − ω)|a˜ J , vl |2

J =1 l∈Zd

2N +1









 + 2Re 

e

2iπ(kJ −kJ  )l 2N +1

  (ωl − ω)a˜ J , vl a˜ J , vl 

1≤J <J  ≤M l∈Zd

2N +1

η

≤ −γ (1 − Cγ Hence, if ε :=

(η

η −η

).

− η)/2, for γ sufficiently small, one has

M



(ωl − ω)|a˜ J , vl |2

J =1 l∈Zd

2N +1



 + 2Re 







e

2iπ(kJ −kJ  )l 2N +1

  (ωl − ω)a˜ J , vl a˜ J , vl 

1≤J <J  ≤M l∈Zd η

2N +1

ε

≤ −γ (1 − γ ). So that, by (4.33), we get  M

 1   (2L + 1)d d  J =1 k ∈Z  2K +1




+

% 




 (ωl  +k  (2L +1) − ω) (2L + 1)d |a˜ J , vk  (2L +1) |2

l  ∈Zd2L +1 

2Re S(J, J , k )e

2iπ(kJ −kJ  )k  2K  +1

a˜

J

&



 , vk  (2L +1)  a˜ J , vk  (2L +1)

1≤J <J  ≤m k  ∈Zd2K  +1

≤ −γ η (1 − γ ε ),

where S(J, J  , k  ) =

1 (2L + 1)d


l  ∈Zd2L +1

(ωl  +k  (2L +1) − ω)e

2iπ(kJ −kJ  )l  2N +1

.



Weak Disorder Localization and Lifshitz Tails

Hence, M







 

J =1 k  ∈Zd

2K  +1

≤2




147

 (2L

1 + 1)d




 (ωl  +k  (2L +1) − ω) (2L + 1)d |a˜ J , vk  (2L +1) |2

l  ∈Zd2L +1

  S(J, J  k  ) (2L + 1)d |a˜ J , vk  (2L +1) a˜ J , vk  (2L +1) | − γ η (1 − Cγ ε ).

1≤J <J  ≤m k  ∈Zd2K  +1

(4.34) If we define (J, J  , k  ) = 2iπ kJ as | 2N +1 − θJ | ≤

(2L

1 2N+1 ,

1 + 1)d






(ωl  +k  (2L +1) − ω)ei(θJ −θJ  )l ,

l  ∈Zd2L +1

we get that

|(J, J  , k  ) − S(J, J  , k  )| = O(γ ρ ). So, as &a˜ J & = &a J & ≤ 2, by (4.34), for γ sufficiently small, we obtain   M


1  (ωl  +k  (2L +1) − ω) (2L + 1)d |a˜ J , vk  (2L +1) |2   + 1)d (2L d d   J =1 k ∈Z  2K +1




≤2

l ∈Z2L +1

  (J, J  , k  ) (2L + 1)d |a˜ J , vk  (2L +1) a˜ J  , vk  (2L +1) | − γ η /2.

1≤J <J  ≤M k  ∈Zd2K  +1

(4.35) This implies that, for some 1 ≤ J ≤ J  ≤ M and k  ∈ Zd2K  +1 , we have      
1 γη   i(θ −θ )l  J J    (ω − ω)e  ≥ l +k (2L +1)  d  (2L + 1)  16(M + 1)2   l  ∈Zd  2L +1

that is ω ∈

  4J,J ,k .

This completes the proof of Lemma 4.3.

" !

5. The Proof of Theorems 1.3 and 1.4 To obtain Theorem 1.3, one uses Theorem 1.1 when r = +∞, i.e. when the random variables (ωn )n∈Zd are bounded. Do as in Sect. 2; pick λ large enough so that the random variables (ωn (λ))n∈Zd defined in Sect. 2 satisfy ω(1 − η/2) ≤ E(ω0 (λ)) = ω(λ). Then, with the same notations as in Sect. 2, we get   Nγ (γ (ω − η)) ≤ Nγ (γ (ω − η); λ) ≤ Nγ γ ω(λ)(1 − η/2); λ . Theorem 1.3 then immediately follows from Theorem 1.1 with r = +∞. Theorem 1.4 is deduced from Theorem 1.3 in the same way as Theorem 1.2 is deduced from Theorem 1.1; we do not repeat the proof.

148

F. Klopp

6. Appendix This section is devoted to results used in the course of this paper.

6.1. Combes-Thomas estimates. Combes-Thomas estimates are estimates on the decay of the Green’s kernel of an operator H , see [5, 1, 9]. The ones we need in the present paper are stated in Lemma 6.1. Let Hω,γ be as in Sect. 1. Then, there exists C > 0 (depending only on h) such that, for any ω, γ > 0 and any " ⊂ Zd , one has • if E ∈ C \ R, |Im(E)| ≤ 1, for m, n ∈ ",   −1   δm  ≤  δn , E − Hω,γ |"

C e−|Im(E)||m−n|/C . |Im(E)|

• if E < inf(σ (Hω,γ |" )) or E > sup(σ (Hω,γ |" )), for m, n ∈ ", one has   C 2   −1 e−ρ(E)|m−n|/C , δn , (E − Hω,γ |" ) δm  ≤ ρ(E)

(6.1)

where ρ(E) = inf

 ( dist(E, σ (Hω,γ |" )), c/2 .

Here, c is defined in assumption (H0). Proof. The proof of the first part of this lemma is well known (see e.g. [1]). The proof of the second part may be less well known; we give it for the reader’s convenience. To fix ideas assume E < inf(σ (Hω,γ |" )). Pick a ∈ Rd , |a| < c, where c is defined in (H0). On the domain c (Zd ) = {u = (un )n∈Zd ; ∃N > 0, un = 0 if |n| > N }, consider the operator Ta defined by (Ta u)n = ea·n un (u = (un )n∈Zd ). Recall that !" is the projector on the subset " ⊂ Zd . We note that Ta !" = !" Ta ,

Ta−1 = T−a

and

Ta∗ = Ta .

(6.2)

It is convenient to consider the operator Ta on the Fourier side, i.e. to conjugate them by the discrete Fourier transformation. We keep the same notations. For u, a trigonometric polynomial, one has (Ta u)(θ ) = u(θ − ia). Ta can be extended to the space of functions analytic in the strip |Imθ | < c. By (6.2), one has T−a (h + γ Vω )Ta = h(· + ia) + γ Vω . As h is real analytic, one has h(θ + ia) = h(θ ) + a 2 gr (θ, a) + i(a · ∇h(θ ) + a 2 gi (θ, a)), where gi and gr are bounded uniformly for |a| < c/2.

(6.3)

Weak Disorder Localization and Lifshitz Tails

149

Equations (6.2) and (6.3) imply that, for u = !" u ∈ 2 (Zd ), |T−a !" (h + Vω − E)!" Ta u, u|2 = |T−a (h + Vω − E)Ta u, u|2 = |T−a (h + Vω − E + a 2 gr (·, a))Ta u, u|2 +|T−a (a · ∇h(θ ) + a 2 gi (·, a))Ta |u, u|2 ≥ |T−a !" (h + Vω − E +a 2 gr (·, a))!" Ta u, u|2 .

(6.4)

Pick a such that |a| = ρ(E)/C, where ρ(E) is defined in Lemma 6.1 and C > 0 is such that ρ(E) sup |gr (θ, a)| ≤ C 2 /2, |a|≤c/2 θ∈Rd

then !" (h + Vω − E + a 2 gr (·, a))!" ≥

1 !" ρ(E)2 !" . 2

Hence,

|T−a !" (h + Vω − E)!" Ta u, u|2 ≥

ρ(E) 2

4 &u&2 .

This implies that T−a !" (h + Vω − E)!" Ta is invertible and its inverse is bounded by 4/ρ 2 (E). Let us now compute |δn , (!" (h + Vω − E)!" )−1 δm | = |Ta δn , (T−a (!" (h + Vω − E)!" )Ta )−1 T−a δm |



2 2 2 2 ≤ &T−a δm &&Ta δn & = e−a(m−n) . ρ(E) ρ(E) Taking a = " !

ρ(E) (m − n), we obtain (6.1). This completes the proof of Lemma 6.1. C|m − n|

6.2. The key lemma. The following lemma is a very simple but quite convenient quantitative version of the uncertainty principle for the Fourier transformation on finite Abelian discrete groups. Lemma 6.2 ([11]). Assume N, L, K, K  L are positive integers such that • 2N + 1 = (2K + 1)(2L + 1) = (2K  + 1)(2L + 1) • K < K  and L < L. Pick a = (an )n∈Zd

2N +1

∈ 2 (Zd2N+1 ) such that, for |n| > K, an = 0. Then, there exists

a˜ ∈ 2 (Zd2N +1 ) such that 1. &a − a& ˜ 2 (Zd

2N +1 )

≤ CK,K  &a&2 (Zd

2N +1 )

where CK,K 



K/K  →0

K/K  .

150

F. Klopp

2. For l  ∈ Zd2L +1 and k  ∈ Zd2K  +1 , we have a, ˜ vl  +k  (2L +1) 2 (Zd ) = 2N +1   a, ˜ vk (2L +1) 2 (Zd ) . 2N +1 3. &a&2 (Zd ) = &a& ˜ 2 (Zd ) . 2N +1

2N +1

A one-dimensional version of this result was proved in [11]. The proof in arbitrary dimension is very similar to that in dimension 1; for the reader’s convenience, as this proof is quite short, we reproduce it here. Proof. By definition of (vm ) (see (4.17)), for m ∈ Zd2N+1 , one has a, vm 2 (Zd

2N +1 )

=

1 (2N + 1)d/2




2iπ nm

an e− 2N +1 .

n∈Zd2K+1

We decompose m = l  + k  (2L + 1), where k  ∈ Zd2K  +1 and l  ∈ Zd2L +1 so that a, vm 2 (Zd

2N +1 )

=




1  d/2 (2K + 1) (2L + 1)d/2

(2L

+ =

1 1   d/2  + 1) (2K + 1)d/2

(2K 

1 + 1)d/2 










2iπ nk 2iπ nl − 2K  +1 − 2N +1

e

n∈Zd2K+1

 =

an e




an e



2iπ nk − 2K  +1

n∈Zd2K  +1 2iπ nl 





e− 2N +1 − 1 an e

2iπ nk  − 2K  +1

 

n∈Zd2K  +1

 1 K   a, v  k 2 Zd (2L + 1)d/2 2K  +1    1  , + D l a, vkK 2  d  d/2  Z2K  +1 (2L + 1)

where • a is seen as an element of 2 (Zd2K  +1 ),   • vkK  d is the orthonormal basis of 2 (Zd2K  +1 ) defined by k ∈Z2K  +1



vkK =

1 (2K  + 1)d/2



2iπ kk 

e 2K  +1

 k∈Zd2K  +1



,

• For l  ∈ Zd2L +1 , D l is the (2K  + 1)d × (2K  + 1)d diagonal matrix (acting on   2 Zd2K  +1 ) with the diagonal entries  l dnn

=

2iπ nl 

e− 2N +1 − 1 if |n| ≤ K 0 if not.

Weak Disorder Localization and Lifshitz Tails

151

Obviously, 

sup l  ∈Zd2L +1

&D l &





≤

L 2 (Zd2K  +1 )

sup k∈Zd2K+1 l  ∈Zd2L +1

   − 2iπ kl   e 2N +1 − 1 = CK,K   

and CK,K  We define


a˜ =

n∈Zd2N +1



K/K  →0

K . K

  1 K   vn , a, v [n]  L 2 Zd (2L + 1)d/2 2K  +1

where [n]L is the unique point k  ∈ Zd such that n − k  (2L + 1) ∈ (−L , L ]d . Then, a˜ satisfies Lemma 6.2. Indeed, as the vectors (vn )n∈Zd form an orthonormal basis of 2N +1

2 (Zd2N +1 ), point 2 is obvious by the definition of a. ˜ Point 3 follows from  2   
1    2  K   ˜ 2 Zd  = &a& a, v  [n]L 2 d  + 1)d   Z2K  +1  (2L 2N +1 d n∈Z2N +1

1 = · (2L + 1)d  (2L + 1)d


n∈Zd2K  +1

 2         = &a&2  d .  a, vkK 2  d 2 Z2N +1   Z2K  +1 

Let us check point 1; we compute  2   
 1   2  K     − a, vn  2 d a, v &a − a& ˜ 2 Zd  =  [n]L 2 d  Z2N +1   + 1)d/2   Z2K  +1 (2L 2N +1 d N∈Z2N +1

=




l  ∈Zd2L +1

=




l  ∈Zd2L +1

(2L

1 + 1)d


k  ∈Zd2K  +1





|D l a, vkK 

  |2 2 Zd2K  +1

1  &D l a&22 (Zd ) (2L + 1)d 2K  +1

2 2 ≤ CK,K  &a& 2  d  Z

2N +1

.

This completes the proof of Lemma 6.2.

" !

6.3. The s-decoupling property. Let us recall the s-decoupling property (1.7) of [4]. A probability distribution dP satisfies the s-decoupling property with some 0 < s < 1, if there exists C > 0 such that

152

F. Klopp

 ∀a, b, c ∈ C,

1 s R |x − a|

       x − b s  x − b s 1  dP (x) ≤ C  dP (x).   dP (x)  x − c  s R |x − a| R x−c (6.5)

We prove Lemma 6.3. If dP , the common probability distribution of the random variables (ωn )n∈Zd satisfies assumption (H2’) and (H3) and the essential infimum of dP is 0, it is s-regular for 0 < s < τ/4 (where τ is the exponent defined in assumption (H3)). The purpose of this result is only to be used in Sect. 2; we did not try to extend it to its maximal generality. Proof. Define     x − b s 1   dP (x). dP (x) and ψ (b, c) = s   s R |x − a| R x−c

 φs (a) =

By the Cauchy-Schwartz C.3 of [4], it suffices to show that the √ inequality, as in Lemma √ continuous functions φ2s (a)/φs (a) and ψ2s (b, c)/ψs (b, c) are bounded. To prove this, we only need to study the behavior of the φs and ψs at infinity. Let us start with φs . We may assume |a| large. One has  0

+∞

1 dP (x) ≥ |x − a|s



1 0

1 1 dP (x) ≥ |a|−s , s |x − a| C

(6.6)

where C > 0 is independent of a, |a| > 1. On the other hand, 

+∞

0

1 dP (x) = I1 + I2 + I3 , |x − a|s

where, by assumption (H2’), 

|a|/2

I1 = 0

1 dP (x) ≤ C|a|−s , |x − a|s  2|a| I2 = |a|/2

 I3 =

+∞ 2|a|

1 dP (x) ≤ C|a|−s−1 , |x − a|s

1 dP (x). |x − a|s

Let us estimate I2 . By (H2’) and (H3), we know that, for 0 ≤ u ≤ v, P ([u, +∞)) ≤ C|u|−1

and

P ([u, v]) ≤ C|u − v|τ .

Interpolating between these two estimates, for µ ∈ [0, 1], we get P ([u, v]) ≤ C|u|−µ |u − v|τ (1−µ) .

(6.7)

Weak Disorder Localization and Lifshitz Tails

We rewrite  |a| I2 = = ≤

1 dP (x) + |x − a|s

|a|/2 +∞
 (1−2−k−1 )|a|

−k n=1 (1−2 )|a| +∞  (1−2−k−1 )|a|
−k n=1 (1−2 )|a|



1 |x − a|

153

2|a|

1 dP (x) |x − a|s |a| +∞  (1+2−k+1 )|a|
dP (x) + s

1 dP (x) |x − a|s

−k n=1 (1+2 )|a| +∞  (1+2−k+1 )|a|


1 dP (x) + |x − |a||s

−k n=1 (1+2 )|a|

1 dP (x). |x − |a||s

Hence, using (6.7), if −s + τ (1 − µ) < 0, we get I2 ≤ C

+∞


(2−k−1 |a|)−s ((1 − 2−k )|a|)−µ (2−k−1 |a|)τ (1−µ)

n=1 +∞ 


−s 

τ (1−µ)  −µ  1 + 2−k |a| 2−k |a|

2−k |a|

+C

n=1

≤ C|a|−s−µ+τ (1−µ)

1 . 1 − 2−s+τ (1−µ)

We pick s < τ/2 and µ > 1/2 such that −s + τ (1 − µ) < 0 and −µ + τ (1 − µ) < 0; hence, we get I2 ≤ C|a|−s . Hence, 

+∞ 0

1 dP (x) = I1 + I2 + I3 ≤ C|a|−s . |x − a|s

(6.8)

This and (6.6) proves that, if s < τ/2, for some Cs > 1 independent of a, |a| ≥ 1, one has 1 |a|−s ≤ φs (a) ≤ Cs |a|−s . Cs So, for s < τ/4, the continuous function

√ φ2s (a)/φs (a) is bounded.

Let us now study ψs . The study is quite similar to the one done for φs . Let us first assume that |c| ≥ 1. For the upper bound, we assume s < τ/2 < 1/2 and use the Cauchy-Schwartz inequality to get &1/2  

 1/2 %     x − b s  1 2s 2s     ψs (b, c) = |x − b| dP (x)    dP (x) ≤  dP (x) R x−c R R x−c

 1/2 1 ≤C s |x|2s + |b|2s dP (x) |c| R 1 ≤ C s (1 + |b|s ). (6.9) |c|

154

F. Klopp

In the second step, we have used estimate (6.8). To get the lower bound, we write  

   1/2   x − b s 1  b s 1   ψs (b, c) ≥ (6.10)  x − c  dP (x) ≥ C  c  + |c|s . 0 This and (6.9) proves that, if s < τ/2, for some Cs > 1 independent of b, c, |c| ≥ 1, one has

  

 s  b 1  b s 1 1   + s ≤ ψs (b, c) ≤ Cs   + s . Cs  c  |c| c |c| For |c| small, say, |c| ≤ 1, using the same computations as in (6.9) and (6.10), we get     x − b s 1 s s   C(|b| + 1) ≥  x − c  dP (x) ≥ C (|b| + 1). R √ This proves that, for s < τ/4, the continuous function ψ2s (b, c)/ψs (b, c) is bounded. We have thus completed the proof of Lemma 6.3. ! " References 1. Aizenman, M.: Localization at weak disorder: Some elementary bounds. Rev. Math. Phys. 6, 1163– 1182 (1994) 2. Aizenman, M., Graf, G.M.: Localization bounds for an electron gas. J. Phys. A 31(32), 6783–6806 (1998) 3. Aizenman, M., Molchanov, S.: Localization at large disorder and at extreme energies: An elementary derivation. Commun. Math. Phys. 157, 245–278 (1993) 4. Aizenman, M., Schenker, J.H., Friedrich, R.M., Hundertmark, D.: Finite-volume fractional-moment criteria for Anderson localization. Commun. Math. Phys. 224(1), 219–253 (2001) Dedicated to Joel L. Lebowitz 5. Combes, J.M., Thomas, L.: Asymptotic behavior of eigenfunctions for multi-particle Schr¨odinger operators. Commun. Math. Phys. 34, 251–270 (1973) 6. Cycon, H.L., Froese, R.G., Kirsch, W., Simon, B.: Schr¨odinger Operators. Berlin: Springer Verlag, 1987 7. Dembo, A., Zeitouni, O.: Large Deviation Techniques and Applications. Boston: Jones and Bartlett Publishers, 1992 8. Helffer, B., Sj¨ostrand, J.: On diamagnetism and the De Haas-Van Alphen effect. Ann. de l’Institut Henri Poincar´e, s´erie Physique Th´eorique 52, 303–375 (1990) 9. Hislop, P.D.: Exponential decay of two-body eigenfunctions: A review. In Proceedings of the Symposium on Mathematical Physics and Quantum Field Theory (Berkeley, CA, 1999), San Marcos, TX: Southwest Texas State Univ 2000, pp. 265–288 (electronic) 10. H¨ormander, L.: The Analysis of Linear Partial Differential Operators. Heidelberg: Springer Verlag, 1983 11. Klopp, F.: Band edge behaviour for the integrated density of states of random Jacobi matrices in dimension 1.J. Stat. Phys. 90(3–4), 927–947 (1998) 12. Klopp, F.: Internal Lifshits tails for random perturbations of periodic Schr¨odinger operators. Duke Math. J. 98(2), 335–396 (1999) 13. Klopp, F.: Weak disorder localization and Lifshitz tails: Continuous Hamiltonians. Technical report, Universit´e Paris-Nord, 2001. To appear in Ann. IHP 14. Klopp, F., Wolff, T.: Lifshitz tails for 2-dimensional random Schr¨odinger operators. To appear in J. d’Analyse Math. 15. Klopp, F.: Precise high energy asymptotics for the integrated density of states of an unbounded random Jacobi matrix. Rev. Math. Phys. 12(4), 575–620 (2000) 16. Lifshitz, I.M.: Structure of the energy spectrum of impurity bands in disordered solid solutions. Sov. Phys. JETP 17, 1159–1170 (1963) 17. Lifshitz, I.M.: Energy spectrum structure and quantum states of disordered condensed systems. Sov. Phys. Uspekhi 7, 549–573 (1965)

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18. Lifshitz, I.M., Gredeskul, S.A., Pastur, L.A.: Introduction to the theory of disordered systems. NewYork: Wiley, 1988 19. Mather, J.N.: On Nirenberg’s proof of Malgrange’s preparation theorem. In: Proceedings of Liverpool Singularities-Symposium I, Number 192 in Lecture Notes in Mathematics, Berlin: Springer Verlag, 1971 20. Minami, N.: Local fluctuation of the spectrum of a multidimensional Anderson tight binding model. Commun. Math. Phys., 177(3), 709–725 (1996) 21. Pastur, L., Figotin, A.: Spectra of Random and Almost-Periodic Operators. Berlin: Springer Verlag, 1992 22. Reed, M., Simon, B.: Methods of Modern Mathematical Physics, Vol IV: Analysis of Operators. New-York: Academic Press, 1978 23. Schlag, W., Shubin, C., Wolff, T.; Frequency concentration and localization lengths for the Anderson model at small disorders. To appear in J. d’Analyse Math., 2001 24. Simon, B., Wolff, T.: Singular continuous spectrum under rank one perturbations and localization for random hamiltonians. Commun. Pure and Appl. Math. 39, 75–90 (1986) 25. Wang, W. M.: Localization and universality of Poisson statistics for the multidimensional Anderson model at weak disorder. Invent. Math. 146(2), 365–398 (2001) 26. Wegner, F.: Bounds on the density of states in disordered systems. Zeit. f¨ur Physik B 44, 9–15 (1981) Communicated by M. Aizenman

Commun. Math. Phys. 232, 157–188 (2002) Digital Object Identifier (DOI) 10.1007/s00220-002-0732-1

Communications in

Mathematical Physics

Quantum Spheres and Projective Spaces as Graph Algebras Jeong Hee Hong1,∗ , Wojciech Szymanski ´ 2,∗∗ 1

Department of Applied Mathematics, Korea Maritime University, Busan 606–791, South Korea. E-mail: [email protected] 2 Mathematics, The University of Newcastle, NSW 2308, Australia. E-mail: [email protected] Received: 31 January 2001 / Accepted: 29 July 2002 Published online: 7 November 2002 – © Springer-Verlag 2002

Abstract: The C ∗ -algebras of continuous functions on quantum spheres, quantum real projective spaces, and quantum complex projective spaces are realized as Cuntz-Krieger algebras corresponding to suitable directed graphs. Structural results about these quantum spaces, especially about their ideals and K-theory, are then derived from the general theory of graph algebras. It is shown that the quantum even and odd dimensional spheres are produced by repeated application of a quantum double suspension to two points and the circle, respectively. Contents 0. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 1. Preliminaries on Graph Algebras . . . . . . . . . . . . . . 1.1 Definition . . . . . . . . . . . . . . . . . . . . . . . 1.2 Gauge action . . . . . . . . . . . . . . . . . . . . . 1.3 K-Theory . . . . . . . . . . . . . . . . . . . . . . . 1.4 Ideals . . . . . . . . . . . . . . . . . . . . . . . . . 2. Quantum 3-Sphere and Projective Spaces . . . . . . . . . 2.1 Quantum SU (2) . . . . . . . . . . . . . . . . . . . . 2.2 Quantum SO(3) . . . . . . . . . . . . . . . . . . . . 2.3 Quantum CP 1 . . . . . . . . . . . . . . . . . . . . . 3. Quantum 2-Sphere and Projective Space . . . . . . . . . . 3.1 Quantum S 2 . . . . . . . . . . . . . . . . . . . . . . 3.2 Quantum RP 2 . . . . . . . . . . . . . . . . . . . . . 4. Quantum Odd Dimensional Spheres and Projective Spaces 4.1 Quantum S 2n−1 . . . . . . . . . . . . . . . . . . . .

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∗ Supported by grant No. R04–2001–000–00117–0 from the Korea Science & Engineering Foundation. ∗∗ Partially supported by the Research Management Committee of the University of Newcastle.

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4.2 Quantum RP 2n−1 . . . . . . . . . . . . . . . . . . . 4.3 Quantum CP n−1 . . . . . . . . . . . . . . . . . . . 5. Quantum Even Dimensional Spheres and Projective Spaces 5.1 Quantum S 2n . . . . . . . . . . . . . . . . . . . . . 5.2 Quantum RP 2n . . . . . . . . . . . . . . . . . . . . 6. Quantum Double Suspension . . . . . . . . . . . . . . . . Appendix A. Another Graph for Quantum S 2n−1 . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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0. Introduction Topological structure of a compact manifold M is completely determined by the isomorphism class of the commutative C ∗ -algebra C(M) of continuous complex valued functions on M. This is, roughly speaking, the context of the Gelfand Theorem. Thus, it is a commonly accepted viewpoint that the topological aspects of a compact quantum manifold should be captured by the corresponding unital noncommutative C ∗ -algebra [44]. Probably the simplest, as well as the most important, compact manifolds are spheres and projective spaces. The first one of them to be quantized was the 3-sphere [45]. To this end Woronowicz perturbed the commutation relations satisfied by the coordinate functions on S 3 (identified with the SU (2) group) and analyzed the resulting universal C ∗ -algebra. Building on this fundamental work, soon afterwards Podle´s constructed quantum versions of the 2-sphere and the projective space RP 3 (identified with the SO(3) group) as quantum homogeneous spaces [31]. Again, the C ∗ -algebras of continuous functions on these spaces were defined as universal with respect to certain relations (see [32, 26]). A few years later Vaksman and Soibelman introduced quantum odd dimensional spheres S 2n−1 as quantum homogeneous spaces SU (n)/SU (n − 1), giving
 ∗ 2n−1 generators and relations for the C -algebras C Sq [42] (see also [29] and [46, 22]).
 In the same article the C ∗ -algebras C CPqn−1 of continuous functions on quantum complex projective spaces
were defined as the fixed point algebras for suitable actions  2n−1 of the circle group T on C Sq . Quite recently a quantum analogue of the classical construction of the real projective space RP 2 as the quotient of S 2 by the antipodal Z2 -action was found by Hajac in [14]. In December 2000 Connes and Landi gave a general procedure which yields quantizations of even dimensional quantum spheres [7]. However, their quantization parameter θ is not real but a complex number of modulus one. The C ∗ -algebra C(Sθ4 ) of continuous functions on the quantum 4-sphere of Connes and Landi is further analyzed in [10]. In a remarkable paper [43], V´arilly shows that quantum even dimensional spheres of Connes and Landi are homogeneous spaces of quantum special orthogonal groups1 . Multiparameter generalizations of the quantum 4-sphere Sθ4 appear in the works of Sitarz [40], and Brzezi´nski and Gonera [5]. A quite different construction of yet another quantum 4-sphere is given by Bonechi, Ciccoli and Tarlini in [4]. Namely, they define C(Sq4 ), with real parameter q, to be the fixed point algebra of a certain action of the quantum SU (2) group on the quantum 7-sphere of Vaksman and Soibelman. Reading existing descriptions of the C ∗ -algebras of continuous functions on the above mentioned quantum manifolds one discovers that all of them are essentially defined as 1

We are grateful to Joseph V´arilly for useful comments on recent developments in quantization.

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universal algebras for finite sets of generators and relations. Analysis of the structure of these C ∗ -algebras is typically carried through ingenious ad hoc manipulations with the generators and relations. However, all of these C ∗ -algebras exhibit certain similarities. A significant first step towards finding common properties of these algebras was taken by Sheu, who has recently studied odd dimensional quantum spheres, quantum complex projective spaces and quantum special unitary groups in the framework of groupoids [37–39]. In the present article, we show that the C ∗ -algebras of continuous functions on quantum odd dimensional spheres, quantum complex projective spaces, quantum 2-sphere and quantum RP 3 and RP 2 spaces are all isomorphic to Cuntz-Krieger algebras corresponding to suitable directed graphs. We then define C ∗ -algebras of continuous functions on quantum even dimensional spheres and quantum real projective spaces, and show that all of them are isomorphic to graph algebras as well. In the forthcoming article [17], the C ∗ -algebras of continuous functions on quantum lens spaces of all odd dimensions are realized as C ∗ -algebras of directed graph. Our results allow one to study these important quantum spaces with the help of the very powerful and by now well established theory of graph algebras. One of the key advantages of the theory is the great ease with which it is possible to understand structure and compute invariants of the algebras just by looking at the underlying graphs, or the corresponding matrices [9, 8, 19, 24, 23, 2, 13, 41, 14, 35, 30, 11, 12, 1, 18]. In particular, it is straightforward to calculate the K-groups, the Ext-groups, and find the primitive spectrum of any graph algebra. Consequently, we are able to show that the K-theories of the quantum spheres and projective spaces are identical with those of their classical counterparts. Our article is organized as follows. In Sect. 1, we recall for the reader’s convenience some basic facts about C ∗ -algebras of directed graphs, essential to what follows. In Sect. 2, we show that the C ∗ -algebras of continuous functions on the quantum 3-sphere (identified with the quantum SU (2) group), quantum RP 3 (identified with the quantum SO(3) group) and quantum CP 1 are isomorphic to the C ∗ -algebras of suitable directed graphs. In the special case of the quantum 3-sphere this description was essentially already obtained by Woronowicz in his original article [45, Theorem A2.2], where he showed that all C ∗ -algebras C(SUq (2)), q ∈ [0, 1), are isomorphic. The point here is that it is easy to recognize C(SU0 (2)) as a graph algebra. In Sect. 3, we prove analogous results about the quantum 2-spheres of Podle´s and quantum real projective plane RP 2 . In Sect. 4, we show that the C ∗ -algebra C(Sq2n−1 ) of continuous functions on the quantum odd dimensional sphere, constructed by Vaksman and Soibelman [42], can be realized as the graph algebra C ∗ (L2n−1 ) corresponding to a finite directed graph which we denote L2n−1 . (In Appendix A, we give a different, smaller graph L˜ 2n−1 such that the C ∗ -algebras C ∗ (L˜ 2n−1 ) and C ∗ (L2n−1 ) are isomorphic.) Then we show that the C ∗ -algebra C(CPqn−1 ) of continuous functions on the quantum complex projective space is isomorphic with the fixed point algebra of the canonical gauge action of the circle group on C ∗ (L2n−1 ), and itself it is a graph algebra C ∗ (Fn−1
). The graph Fn−1 ∗ is infinite, though with finitely many vertices. The C -algebra C RPq2n−1 of continuous functions on the quantum real projective space is constructed as the fixed point algebra corresponding to the restriction of the gauge action to {±1}. It turns out that in the absence of sinks (and this is the case with odd
dimensional quantum spheres) this  (2) (2) ∗ fixed point algebra is naturally isomorphic with C L2n−1 , where L2n−1 is the graph

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of paths of length two in L2n−1 . That is, L2n−1 has the same set of vertices as L2n−1 (2)

and the edges in L2n−1 are in a natural one-to-one correspondence with paths of length two in L2n−1 . In Sect. 5, we define and discuss even dimensional quantum spheres and real projective spaces. The C ∗ -algebras resulting from our construction are closely related to the quantum spheres of Podle´s [31, 28, 36] and the corresponding projective spaces [14, 15]. Podle´s defined the quantum 2-sphere as a suitable homogeneous space of the quantum SU (2) group, and it turned out that it is also a homogeneous space of the quantum SO(3) group. It seems that a higher dimensional quantum sphere S 2n should appear as a homogeneous space of the quantum SO(n + 1) group, as shown by V´arilly in the case of the quantum spheres Sθ2n of Connes and Landi. Unfortunately, we are not able to present such a definition at the moment. However, the classical 2n-sphere has a natural imbedding into the (2n + 1)-sphere, and such an imbedding also exists of the quantum 2-sphere into the
quantum 3-sphere. That is, there exists 
a very natural ∗ 3 2 . Taking this C -algebra homomorphism of C Sq with image isomorphic to C Sqc
 as our starting point, we define the C ∗ -algebra C Sq2n of continuous functions on the
 quantum 2n-sphere as a suitable quotient of C Sq2n+1 , and show that this C ∗ -algebra is isomorphic to the graph algebra C ∗ (L2n ), where graph with
finite 
L2n is a suitable 2 2 two sinks. Then, imitating the construction of C RPq from C Sqc , we define the 
C ∗ -algebra C RPq2n of continuous functions on the quantum real projective space
 RP 2n as the fixed point algebra for the antipodal action of Z2 on C Sq2n . It turns out
 that C RPq2n is isomorphic with C ∗ (L 2n ), where L 2n is a suitable graph naturally related to L2n . In Sect. 6, we describe a quantum double suspension using the language of extensions of C ∗ -algebras. We then show that inside the class of unital graph algebras the quantum double suspension has a very simple realization. Namely, if E is a graph with finitely many vertices then the quantum double suspension of C ∗ (E) may be identified with another graph algebra C ∗ (S 2 E). The graph S 2 E is obtained by adding one additional vertex to E and adding one edge from this extra vertex to any other one (including itself). Applying this process to one point we get the Toeplitz algebra, i.e. the C ∗ -algebra of continuous functions on the quantum disc. Applying it to two points we get the C ∗ -algebra of continuous functions on the quantum 2-sphere of Podle´s, and continuing in like manner we produce even dimensional quantum spheres. Repeated application of the quantum double suspension to the circle yields odd dimensional quantum spheres of Vaksman and Soibelman. Incidentally, it is a corollary to our results that the C ∗ -algebras of continuous functions on quantum odd dimensional spheres and real projective spaces are groupoid C ∗ -algebras. Indeed, the C ∗ -algebra of a graph without sinks comes from a groupoid [24, 23, 30]. However, it is our opinion that the graph theoretic description of a C ∗ -algebra is far more convenient than the groupoid one. In fact, the directed graph is essentially a generator for the groupoid. It is a much smaller, easier to understand and handle object, which nevertheless gives all the information. In closing, let us mention a limitation of our approach. Namely, the C ∗ -algebras of continuous functions on quantum special unitary groups SU (n) with n greater than 2

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cannot be realized as Cuntz-Krieger algebras of directed graphs. This follows easily from the composition series for C(SUq (n)) given by Sheu in [37, 38] and the description of the primitive spectrum of a graph algebra in [1]. Indeed, the primitive spectrum of a graph algebra cannot topologically contain higher dimensional tori Tk , k ≥ 2. 1. Preliminaries on Graph Algebras 1.1. Definition. We recall the definition of the C ∗ -algebra corresponding to a directed graph [13]. Let E = (E 0 , E 1 , r, s) be a directed graph with (at most) countably many vertices E 0 and edges E 1 , and range and source functions r, s : E 1 → E 0 , respectively. ∗ ∗ C (E) is, by0 definition, the universalC -algebra1 generated by families of projections Pv | v ∈ E and partial isometries Se | e ∈ E , subject to the following relations: (G1) (G2) (G3) (G4) (G5)

Pv Pw = 0 for v, w ∈ E 0 , v = w. Se∗ Sf = 0 for e, f ∈ E 1 , e = f . Se∗ Se = Pr(e) for e ∈ E 1 . 1 Se Se∗ ≤ Ps(e) for e ∈ E . ∗ Se Se for v ∈ E 0 , provided {e ∈ E 1 | s(e) = v} is finite and Pv = e∈E 1 : s(e)=v

non-empty.   Universality in this definition means that if {Qv | v ∈ E 0 } and Te | e ∈ E 1 are families of projections and partial isometries, respectively, satisfying conditions (G1–G5), then there exists aC ∗ -algebra homomorphism from C ∗ (E) to the C ∗ -algebra generated   0 1 by Qv | v ∈ E and Te | e ∈ E such that Pv → Qv and Se → Te for v ∈ E 0 , e ∈ E1. 1.2. Gauge action. It is an immediate consequence of the universal property that there exists a gauge action γ : T → Aut(C ∗ (E)), defined on the generators by γt (Pv ) = Pv ,

γt (Se ) = tSe .

Existence of the gauge action is equivalent to the universality. Indeed, the following gauge invariant uniqueness theorem holds. If φ : C ∗ (E) → B is a surjective C ∗ -algebra homomorphism and there exists an action β : T → Aut(B) such that φ ◦ γt = βt ◦ φ for all t ∈ T, then φ is an isomorphism. For finite graphs without sinks this follows from the analysis in [8, Remark 3], but the explicit statement of the theorem (and proof) appears first in [19, Th. 2.3]. Generalizations to infinite graphs with sinks are given in [2, Th. 2.1] and [35, Th. 2.7]. The fixedpoint algebra C ∗ (E)γ for the gauge action is equal to the closed span of the projections Pv | v ∈ E 0 and all elements of the form Sα Sβ∗ , where α = (e1 , . . . , ek ) and β = (f1 , . . . , fk ) are two paths of the same length k ∈ N in E. (For α = (e1 , . . . , ek ) we denote Sα = Se1 Se2 · · · Sek .) It can be shown that it is an AF -algebra. The restriction of γ to {±1} gives rise to an action of Z2 on C ∗ (E). We denote the corresponding fixed point algebra by C ∗ (E)γ−1 . In order to describe this algebra we define a new graph E (2) , corresponding to E, as follows. The set of vertices of the graph E (2) coincides with E 0 , while the set of edges equals the collection of all paths of length 2 in E, that is {(e, f ) | e, f ∈ E 1 , r(e) = s(f )}. The source of (e, f ) is s(e) and the range is r(f ). We recall that a graph is row-finite if every vertex emits at most finitely many edges, and a vertex is called sink if it emits no edges at all.

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Proposition 1.1. If E is a row-finite graph without sinks then there exists a unique C ∗ algebra isomorphism
 C ∗ E (2) → C ∗ (E)γ−1 such that Pv → Pv

and

S(e,f ) → Se Sf

for all v ∈ E 0 and e, f ∈ E 1 with r(e) = s(f ).   Proof. It follows from the universal property of C ∗ E (2) that the map φ : S(e,f ) →   Se Sf , φ : Pv → Pv extends to a C ∗ -algebra homomorphism φ : C ∗ E (2) → C ∗ (E) since target Se Sf, Pv satisfy relations (G1–G5) for the graph E (2). Clearly,  ∗the  (2)  elements ∗ γ −1 φ C E ⊆ C (E) . The algebra C ∗ (E)γ−1 is generated by all elements of the form Se Sf , Se Sf∗ , with e, f ∈ E 1 . Indeed, C ∗ (E) is the closed span of {Se1 · · · Sek Sf∗1 · · · Sf∗m | ei , fj ∈ E 1 , k, m ∈ N}. The conditional expectation C ∗ (E) → C ∗ (E)γ−1 , x → 21 (x + γ−1 (x)), kills to 0 all Se1 · · · Sek Sf∗1 · · · Sf∗m with k + m odd, and does not move those with k + m even. Any Se1 · · · Sek Sf∗1 · · · Sf∗m with k + m even is a product of elements of the form Se Sf , (Se Sf )∗ and Se Sf∗ . Moreover, since E is row-finite and has no sinks, we may write Se Sf∗ =



Se Sg Sg∗ Sf∗

g∈E 1 , s(g)=r(e)

and, hence, C ∗ (E)γ−1 is generated by elements of the form Se Sf . Thus, φ is surjective. Since C ∗ (E)γ−1 is generated by Se Sf , e, f ∈ E 1 , there exists a well defined action β : T → Aut(C ∗ (E)γ−1 ) such that βt (x) = γ√t (x), x ∈ C ∗ (E)γ−1 , t ∈ T. We have φ ◦ γt = βt ◦ φ for all t ∈ T and thus φ is injective by the gauge invariant uniqueness theorem [2, Th. 2.1]. Uniqueness of such a homomorphism is obvious.   1.3. K-Theory. Computation of the K-theory of a graph algebra is ultimately based on the classical result of Cuntz [8, Prop. 3.1]. As originally formulated, it covers the case of a finite graph (finite matrix) without sinks (no zero rows in the matrix) in which every vertex-simple loop has an exit (condition (I) of [9]). Incidentally, none of the graphs appearing in the present article satisfies all of these requirements. Cuntz’s fundamental theorem was later extended to row–finite graphs without sinks in [24, Cor. 6.12], to arbitrary row-finite graphs in [35, Th. 3.2], to arbitrary graphs with finitely many vertices in [41, Prop. 2], and the general case was finally presented in [12, Th. 3.1]. All graphs appearing in the present article are covered by [41, Prop. 2]. The formula for the K-groups looks as follows. Let VE be the collection of all those vertices which are not sinks and emit finitely many edges, and let ZVE and ZE 0 be free abelian groups on free generators VE and E 0 , respectively. We have K0 (C ∗ (E)) K1 (C ∗ (E))

∼ = coker(KE ), ∼ = ker(KE ),

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where KE : ZVE → ZE 0 is the map defined on generators as    KE (v) =  r(e) − v. e∈E 1 : s(e)=v

We would like to stress that this formula applies to arbitrary countable graphs. 1.4. Ideals. Cuntz’s analysis of the ideal structure of C ∗ -algebras of finite graphs (finite matrices) in [8] is of fundamental importance. A complete description of ideals for finite graphs without sinks satisfying condition (II) is given in [8, Th. 2.5]. Condition (II) of [8] guarantees that all closed 2-sided ideals are invariant under the gauge action of the circle group. The analysis in [8, Remark 3] is a step towards understanding of the ideal structure in the case of arbitrary finite graphs without sinks. This task was completed in [19, Th. 3.5 and Th. 4.7]. These results were extended to locally finite graphs without sinks satisfying condition (K) (a graph analogue of condition (II)) in [24, Th. 6.6], and to arbitrary graphs satisfying condition (K) in [11, Th. 3.4 and Th. 4.6]. Gauge invariant ideals of row-finite graphs are described in [2, Th. 4.1]. A complete description of all ideals and quotients, and the hull-kernel topology on the primitive spectrum, for an arbitrary graph algebra is now available [1, 18] (see also [16] for a concise summary of the results). Taking into account the great variety of possible graphs and their C ∗ -algebras it is not surprising that this description is rather complicated. Below, we give a brief summary of the ideal structure of C ∗ -algebras corresponding to the two classes of graphs arising in the present article. Namely, we will only be concerned with graphs which are either row-finite or do not have any loops. Let E be such a directed graph. At first we describe closed 2-sided ideals of C ∗ (E) invariant under the gauge action, as well as the corresponding quotients. To this end we consider hereditary and saturated subsets of E 0 . A subset K ⊆ E 0 is hereditary and saturated if the following two conditions are satisfied: (HS1) If v ∈ K, w ∈ E 0 and there exists a path from v to w then w ∈ K. (HS2) If v ∈ E 0 emits finitely many edges and is not a sink and for each e ∈ E 1 with s(e) = v we have r(e) ∈ K, then v ∈ K. We denote by *E the collection of all hereditary and saturated subsets of E 0 . Any hereditary and saturated set K gives rise to a gauge invariant ideal generated by {Pv | v ∈ K} and denoted JK . If E is row-finite then the quotient C ∗ (E)/JK is naturally isomorphic to C ∗ (E/K), where E/K denotes the restriction of the graph E to E 0 \ K. (In general, the graph E/K corresponding to C ∗ (E)/JK might be somewhat more complicated [1].) It is shown in [2, 1] that if E is either row-finite or does not have any loops then there exists a bijection between *E and the collection of all gauge invariant ideals of C ∗ (E), given by the following two maps: K → JK ,

J → {v ∈ E 0 | Pv ∈ J }.

We now turn to the description of primitive ideals. Key objects used in the classification of primitive ideals of graph algebras are maximal tails, defined as follows. A non-empty subset M ⊆ E 0 is called a maximal tail if the following three conditions are satisfied: (MT1) If v ∈ E 0 , w ∈ M and there is a path in E from v to w then v ∈ M.

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(MT2) If v ∈ M emits finitely many edges and is not a sink in E then there exists an edge e ∈ E 1 such that s(e) = v and r(e) ∈ M. (MT3) For any v, w ∈ M there is a y ∈ M such that there exist paths in E from v to y and from w to y. The collection M(E) of all maximal tails is a disjoint union of its two subcollections Mγ (E) and Mτ (E), defined as follows. A maximal tail M belongs to Mγ (E) if and only if every vertex simple loop (e1 , e2 , . . . , ek ) (where ei ∈ E 1 , r(ei ) = s(ei+1 ), r(ek ) = s(e1 ) and r(ei ) = r(ej ) for i = j ) whose vertices s(ei ) all belong to M has an exit e ∈ E 1 (that is, s(e) ∈ {s(e1 ), . . . s(ek )} but e ∈ {e1 , . . . , ek }) with r(e) ∈ M. Otherwise M belongs to Mτ (E). It can be shown that each maximal tail from Mγ (E) gives rise to a primitive ideal of C ∗ (E) invariant under the gauge action, and each maximal tail from Mτ (E) gives rise to a circle of primitive ideals none of which is invariant under the gauge action. Let Prim(C ∗ (E)) denote the set of all primitive ideals of C ∗ (E). If E is either row-finite or does not have any loops then there exists a bijection Mγ (E) ∪ (Mτ (E) × T) ↔ Prim(C ∗ (E)). See [19, 24, 2, 1, 18] for the details. 2. Quantum 3-Sphere and Projective Spaces 2.1. Quantum SU (2). The classical SU (2) group and the 3-sphere are homeomorphic as topological spaces. Assuming the same holds on
thequantum level, we may think of the C ∗ -algebra C(SUq (2)) as of the C ∗ -algebra C Sq3 of continuous functions on the quantum 3-sphere. We begin by recalling the definition of the quantum SU (2) group [45]. For q ∈ [0, 1) the C ∗ -algebra C(SUq (2)) of continuous functions on the quantum SU (2) group is defined as the universal C ∗ -algebra generated by two elements a, b, subject to the following relations: a ∗ a + b∗ b = I, ∗

ab = qba,

2 ∗

ab∗ = qb∗ a, b∗ b = bb∗ .

aa + q b b = I,

We use symbols a and b rather than Woronowicz’s original α and γ since we prefer to reserve the symbol γ for the gauge action on a graph algebra. It is well known that all C(SUq (2)), q ∈ [0, 1), are isomorphic as C ∗ -algebras [45, Th. A2.2]. In this section, we show that this C ∗ -algebra can be described as a CuntzKrieger algebra C ∗ (L3 ) corresponding to the following directed graph L3 . We denote the edges of this graph e1,1 , e1,2 and e2,2 in the manner consistent with the general case discussed in Sect. 4.1. e1,1

L3

e2,2

. . ....................................... ....................................... ...... .. ...... .. ..... ..... ..... ..... ... ... ... ... ... ... . . ... ... ... ... ... ... ... ..... . ... ... .... ... ... . .. ... . ... . .. ... . . . . ... . . .... . . . . . . . . . . ..... ..... 1,2 .... .... ....... . .............................................................................................................................................................. ..

• v1

e

• v2

Quantum Spheres and Projective Spaces as Graph Algebras

165

Recall that C ∗ (L3 ) is the universal C ∗ -algebra generated by projections Pv1 , Pv2 and partial isometries Se1,2 , Se1,1 , Se2,2 , subject to the following relations: Pv1 Pv2 = 0, Pv1 = Se∗1,1 Se1,1 = Se1,1 Se∗1,1 + Se1,2 Se∗1,2 ,

Pv2 = Se∗1,2 Se1,2 = Se∗2,2 Se2,2 = Se2,2 Se∗2,2 .

The following Proposition 2.1 is essentially a reformulation of [45, Th. A2.2]. Proposition 2.1. For q ∈ [0, 1) there exists a C ∗ -algebra isomorphism C(SUq (2)) → C ∗ (L3 ) such that: (1) If q ∈ (0, 1) then a → b →

∞  n=0 ∞  n=0

1 − q 2(n+1) −

 1 − q 2n (Se1,1 + Se1,2 )n (Se∗1,1 + Se∗1,2 )n+1 ,

q n (Se1,1 + Se1,2 )n Se2,2 (Se∗1,1 + Se∗1,2 )n .

(2) If q = 0 then a → Se∗1,1 + Se∗1,2 , b → Se2,2 . Proof. By virtue of [45, Th. A2.2] and formulae [45, (A2.2) and (A2.3)] it suffices to prove the proposition in the case q = 0. Indeed, there exists a C ∗ -algebra homomorphism φ : C(SU0 (2)) → C ∗ (L3 ) such that φ(a) = Se∗1,1 + Se∗1,2 and φ(b) = Se2,2 , since the target elements satisfy the relations for SU0 (2) generators. On the other hand, there exists a C ∗ -algebra homomorphism φ −1 : C ∗ (L3 ) → C(SU0 (2)) such that φ −1 (Pv1 ) = a ∗ a, φ −1 (Pv2 ) = bb∗ , φ −1 (Se1,1 ) = a ∗ (I − bb∗ ), φ −1 (Se2,2 ) = b and φ −1 (Se1,2 ) = a ∗ bb∗ , since the target elements satisfy relations (G1–G5) for the graph L3 . One may verify that φ ◦ φ −1 = id and φ −1 ◦ φ = id.   In view of Proposition 2.1 one may study the C ∗ -algebra of continuous functions on the quantum SU (2) group with the help of the very extensive machinery which has been developed for Cuntz-Krieger algebras of directed graphs. The ideal structure of the C ∗ -algebra C(SUq (2)) was determined by Woronowicz in [45]. Nevertheless, we would like to point out that this structure is immediately clear from our identification of C(SUq (2)) with C ∗ (L3 ) and the known results on ideals of graph algebras. For example, {v2 } is a hereditary and saturated set in L3 . The corresponding ideal is isomorphic to C(T) ⊗ K (with K the compacts on a separable Hilbert space) and the quotient is C(T). Thus, we get the exact sequence 0 −→ C(T) ⊗ K −→ C(SUq (2)) −→ C(T) −→ 0.

(2.1)

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J. H. Hong, W. Szyma´nski

We have M(L3 ) = Mτ (L3 ) = {{v1 }, {v1 , v2 }} and hence the primitive spectrum of C(SUq (2)) consists of two disjoint copies of the circle T. One of them is closed and the closure of a point on the other one contains the entire opposite circle. The K-theory of the C ∗ -algebra C(SUq (2)) was calculated by Masuda, Nakagami and Watanabe in [27, Th. 2.4]. However, this is also automatic from our Proposition 2.1. Indeed, the K0 and K1 groups of C ∗ (L3 ) are given by the cokernel and kernel, respectively, of the map KL3 : Z2 → Z2 corresponding to the matrix   00 . 10 Hence K0 (C(SUq (2))) ∼ = Z, K1 (C(SUq (2))) ∼ = Z. It is also possible to understand the group structure of the quantum SU (2) group in terms of the generators of C ∗ (L3 ). Recall that the comultiplication 1q : C(SUq (2)) → C(SUq (2)) ⊗ C(SUq (2)) is a

C ∗ -algebra

homomorphism such that 1q (a) = a ⊗ a − qb∗ ⊗ b, 1q (b) = b ⊗ a + a ∗ ⊗ b.

The counit

ε : C(SUq (2)) → C

is a character such that ε(a) = 1 and ε(b) = 0. The counit is given on the generators of C ∗ (L3 ) by ε(Pv1 ) = ε(Se1,1 ) = 1,

ε(Pv2 ) = ε(Se1,2 ) = ε(Se2,2 ) = 0.

In particular, ε does not depend on q. It is more difficult to determine the values of comultiplication 1q . However, Lance gives in [25, (4.3) and Th. 4.1] (see also [47]) an explicit description of a unitary map Uq : 32 (N × Z × Z × Z) → 32 (N × Z × N × Z), where N = {0, 1, . . . }, such that Uq (ρ(x) ⊗ I )Uq∗ = 1q (ρ(x)) for all x ∈ C(SUq (2)), where ρ : C(SUq (2)) → B(32 (N × Z)) is the faithful representation constructed by Woronowicz in [45]. If {ξ(k, m) | k ∈ N, m ∈ Z} is an orthonormal basis of 32 (N × Z) then we have ρ(Pv1 ) : ξ(k, m) → ρ(Pv2 ) : ξ(k, m) → ρ(Se1,1 ) : ξ(k, m) → ρ(Se1,2 ) : ξ(k, m) → ρ(Se2,2 ) : ξ(k, m) →

(1 − δk,0 )ξ(k, m), δk,0 ξ(k, m), (1 − δk,0 )ξ(k + 1, m), δk,0 ξ(k + 1, m), δk,0 ξ(k, m + 1),

with δ being the Kronecker symbol. With the help of this unitary Uq it is possible to find an explicit formula describing the values of the comultiplication 1q on the generators Se1,1 , Se1,2 , Se2,2 .

Quantum Spheres and Projective Spaces as Graph Algebras

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2.2. Quantum SO(3). The classical SO(3) group and the real projective space RP 3 are homeomorphic as topological spaces. Assuming the same holds on the quantum level, we may think of the C ∗ -algebra C(SOq (3)) as of the C ∗ -algebra C(RPq3 ) of continuous functions on the quantum real projective space RP 3 . The C ∗ -algebra C(SOq (3)) of continuous functions on the quantum SO(3) group may be defined as the C ∗ -subalgebra of C(SUq (2)) generated by all products xy with x, y ∈ {a, a ∗ , b, b∗ } [31, 32, 26]. Let γ : T → Aut(C ∗ (L3 )) be the gauge action. Identifying C(SUq (2)) with C ∗ (L3 ), as in Proposition 2.1, we have γt (a) = ta and γt (b) = tb. It is not difficult to verify that C(SOq (3)) is precisely the fixed point algebra for the Z2 action on C(SUq (2)) ∼ = C ∗ (L3 ) given by γ−1 (the antipodal action of Z2 on the quantum
 (2)

3-sphere). Thus, by virtue of Proposition 1.1, C(SOq (3)) is isomorphic to C ∗ L3

.

(2)

The graph L3 of paths of length two in L3 looks as follows: ... ... ................................ ................... ................. ............ ..... ...... . ..... ..... .... ... ... ... ... .. . ... . . ... . ... . ..... . .. .... ... .. .. ... .. ... ... ... .. ... . . . . ... ... .. .. . . . . . ..... 1,1 1,2 ...... ..... ..... ...... .................................................................................................................................................................... ........ .... ...... .......... ......... . . . ............. . . . . . . . . . ....................................................... .

(e1,1 , e1,1 ) .......... (2)

L3

(e2,2 , e2,2 )

• v1

(e

,e

)

(e1,2 , e2,2 )

• v2

(2)

The ideal structure of C(SOq (3)) can be readily understood from the graph L3 . Indeed, the same argument as in the case of the graph L3 applies. The primitive spectrum of C(SOq (3)) is homeomorphic with the primitive spectrum of C(SUq (2)), and the exact sequence 0 −→ C(T) ⊗ K −→ C(SOq (3)) −→ C(T) −→ 0

(2.2)

is the same as for C(SUq (2)). However, the C ∗ -algebras C(SUq (2)) and
C(SO  q (3)) are (2)

not isomorphic (cf. [26]). In our setting, the K0 and K1 groups of C ∗ L3

by the cokernel and kernel, respectively, of the map KL(2) : 3 the matrix   00 . 20 Hence K0 (C(SOq (3)) ∼ = Z ⊕ Z2 , K1 (C(SOq (3)) ∼ = Z, exactly as in the classical case of RP 3 [20].

Z2

→

Z2

are given

corresponding to

168

J. H. Hong, W. Szyma´nski

2.3. Quantum CP 1 . The C ∗ -algebra of continuous functions on the quantum complex 1 projective space CP  3  is defined in [42] as the fixed point algebra for an action of the circle group on C Sq , which under the isomorphism from Proposition 2.1 is identified   with the gauge action γ : T → Aut(C ∗ (L3 )). Thus, the C ∗ -algebra C CPq1 is isomorphic with the AF core subalgebra C ∗ (L3 )γ of C ∗ (L3 ). As this algebra is the closed span of elements of the form Sα Sβ∗ , with α, β paths of the same length in E, it is not difficult to observe that C ∗ (L3 )γ is nothing but the minimal unitization of the compacts  (the one-point compactification of the quantum plane). It is then easy to show that C CPq1 is isomorphic with C ∗ (F1 ), where F1 is the following graph: F1

v1 •...................................................................................................................•.. v2 (∞)

Here and in what follows we use the label (∞) to indicate infinitely many edges from one vertex to the other. Thus, in F1 there are infinitely many edges from v1 to v2 and hence F1 is not row-finite. 3. Quantum 2-Sphere and Projective Space 3.1. Quantum S 2 . Quantum  2-spheres were defined and investigated by Podle´s in [31].
2 of continuous functions on the quantum equatorial sphere The C ∗ -algebra C Sq∞ (case c = ∞) is defined as the universal C ∗ -algebra generated by two elements A and B, subject to the following relations. B ∗ B = I − A2 , BB ∗ = I − q 4 A2 ,

A = A∗ , BA = q 2 AB.

2 ) is defined as the universal Here we allow q ∈ [0, 1). If c ∈ (0, ∞) then C(Sqc generated by two elements A and B, subject to the following relations:

C ∗ -algebra

B ∗ B = A − A2 + cI, BB ∗ = q 2 A − q 4 A2 + cI,

A = A∗ , BA = q 2 AB.

It is known that all these C ∗ -algebras are isomorphic. It turns out that they can be described as the Cuntz-Krieger algebra of the following graph, which we denote L2 . We use the edge and vertex labelling consistent with the general case discussed in Sect. 5.1. e L2

.. ........................ ...... ... ............ ..... .... ... ... .. . ... .. ... .... .. . 1,1 .... .. .. ... . . ... .. . . ..... ..... ...... ..... ............ .1 .......... ........... .... ........ . . . . ..... . . .... ............. 1,1 ................... ..... 1,2 ..... .. ..... ..... . . ..... . ....

g

w1•

v •

g

•w 2

The proof of Proposition  2  3.1, below, relies implicitly on the description of irreducible representations of C Sqc [31, Prop. 4].

Quantum Spheres and Projective Spaces as Graph Algebras

169

Proposition 3.1. For q ∈ [0, 1) and c ∈ (0, ∞] there exists a C ∗ -algebra isomorphism
 2 C Sqc → C ∗ (L2 ) such that: (1) If c = ∞ and q ∈ (0, 1) then A →

∞  n=0

B →

∞ 


n q 2n (Se1,1 + Sg1,1 + Sg1,2 )n (Pw1 − Pw2 ) Se∗1,1 + Sg∗1,1 + Sg∗1,2 , 

n=0


n+1 (1 − q 4 )q 4n  (Se1,1 + Sg1,1 + Sg1,2 )n Se∗1,1 + Sg∗1,1 + Sg∗1,2 . 1 − q 4n+4 + 1 − q 4n

(2) If c = ∞ and q = 0 then A → Pw1 − Pw2 , B → Se∗1,1 + Sg∗1,1 + Sg∗1,2 . (3) If c ∈ (0, ∞) and q ∈ (0, 1) then A →

B →

∞ √ 1  2n q (Se1,1 + Sg1,1 + Sg1,2 )n ((1 + 1 + 4c)Pw1 2 n=0
n √ + (1 − 1 + 4c)Pw2 ) Se∗1,1 + Sg∗1,1 + Sg∗1,2 , ∞  n=0

+

n+1  
∗ − ∗ ∗ S (Se1,1 + Sg1,1 + Sg1,2 )n λ+ (c)P + λ (c)P + S + S w1 w2 n n e1,1 g1,1 g1,2

 √
∗ c Se1,1 + Sg∗1,1 + Sg∗1,2 ,

where we denoted  λ± n (c) =

 √ √ q 2(n+1) q 2(n+1) (1 ± 1 + 4c) 1 − (1 ± 1 + 4c) 2 2   √ √ q 2n q 2n − c+ (1 ± 1 + 4c) 1 − (1 ± 1 + 4c) . 2 2 c+

(4) If c ∈ (0, ∞) and q = 0 then √ √ 1 1 (1 + 1 + 4c)Pw1 + (1 − 1 + 4c)Pw2 , 2 2 √
∗ ∗ ∗ B→  c Se1,1 + Sg1,1 + Sg1,2 . A →

Proof. Since the cases (2) and (4), corresponding to q = 0, are trivial we only consider the remaining two cases (1) and (3).  2 In both cases (1) and (3) there exists a C ∗ -algebra homomorphism φ : C Sqc →   2 . To construct C ∗ (L2 ), as above, since φ(A), φ(B) satisfy the relations defining C Sqc the inverse of φ we use the universal property of C ∗ (L2 ).

170

J. H. Hong, W. Szyma´nski

At first we consider the case c = ∞. As shown in [31], 1 and −1 are isolated points of the spectrum of A. Thus, if χ±1 are the functions of {±1}, respec 2 characteristic  tively, then χ±1 (A) are projections in C Sq∞ , and we have φ(χ1 (A)) = Pw1 and ∗ ∗ φ(χ−1 (A)) = Pw decomposition (in some faithful rep 2 .2 Let  B = U |B ∗| be the polar resentation of C Sq∞ ). Since BB = I − q 4 A2 ≥ (1 − q 4 )I > 0, U is an isometry  2  belonging to C Sq∞ , and one may verify that φ(U ) = Se1,1 + Sg1,1 + Sg1,2 . We now observe that the assignment Pv1 → I − χ1 (A) − χ−1 (A), Pw1 → χ1 (A), Pw2 → χ−1 (A), Se1,1 → U (I − χ1 (A) − χ−1 (A)), Sg1,1 → U χ1 (A) and Sg1,2 → χ−1 (A) extends to 2 ), since φ −1 (P ), φ −1 (P ), a C ∗ -algebra homomorphism φ −1 : C ∗ (L2 ) → C(Sq∞ v1 w1 −1 −1 −1 −1 ∗ φ (Pw2 ), φ (Se1,1 ), φ (Sg1,1 ) and φ (Sg1,2 ) satisfy the relations defining C (L2 ). One may check that φ and φ −1 are indeed inverses of one another. Now √ we consider the case c ∈ (0, ∞). We have Sp(A) ⊆ [θ− , θ+ ], with θ± = 1 points of the spectrum and hence χθ+ (A) 2 (1 ± 1 + 4c). Both θ + and θ− are isolated  2 and χθ− (A) are projections in C Sqc . One may verify that φ(χθ+ (A)) = Pw1 and φ(χθ− (A)) = Pw2 . Since the solutions to q 2 x − q 4 x 2 + c = 0 are outside of the interval [θ− , θ+ ], 0 is not in the spectrum of BB ∗ = q 2 A − q 4A2 + cI . Thus, with 2 . One may check that B ∗ = U |B ∗ | the polar decomposition, U is an isometry in C Sqc √ φ(U ) = c(Se1,1 + Sg1,1 + Sg1,2 ). Then the assignment Pv1 → I − χθ+ (A) − χθ− (A), Pw1 → χθ+ (A), Pw2 → χθ− (A), Se1,1 → U (I − χθ+ (A) − χθ− (A)), Sg1,1 → U χθ+ (A)  2 and Sg1,2 → χθ− (A) extends to a C ∗ -algebra homomorphism C ∗ (L2 ) → C Sqc , inverse to φ.   2 ) follows from the exact sequence The ideal structure of C(Sqc


 2 0 −→ K ⊕ K −→ C Sqc −→ C(T) −→ 0,

(3.1)

explicitly given by Sheu in [36] and implicitly contained in [31]. This exact sequence is also immediate from Proposition 3.1. Indeed, {w1 , w2 } is a hereditary and saturated set in L2 . The corresponding ideal is isomorphic with the direct sum of two copies of the compacts, and the quotient is C(T). 2 ) were found in [28]. However, they are much more easily The K-groups of C(Sqc calculated with the help of the general machinery of graph algebras. Indeed, the K0 and K1 groups of C ∗ (L2 ) are isomorphic with the cokernel and kernel, respectively, of the map KL2 : Z → Z3 corresponding to the matrix   0 1. 1 Hence

 2 ∼ K0 C Sqc = Z2 ,

 2 = 0. K1 C Sqc So far we have concentrated on the case c > 0. Podle´s considered also quantum 2 with c = 0 [31]. He showed, in particular, that C(S 2 ) is isomorphic to the spheres Sq0 q0

Quantum Spheres and Projective Spaces as Graph Algebras

171

minimal unitization of the compacts. As we have already seen this is also the C ∗ -algebra of continuous functions on the quantum complex projective space CP 1 . That is, the 2 and the quantum complex projective space CP 1 are homeomorphic. quantum sphere Sq0 q We close this section by describing a very natural imbedding of the quantum 2-sphere into the quantum 3-sphere, as follows. In the classical case q = 1 the real and imaginary parts of the generators a and b of C(SU (2)) correspond to the cartesian coordinate functions on the 3-sphere. Thus, the imbedding S 2 :→ S 3 with the last coordinate 0 corresponds on the level of algebras to the surjective homomorphism C(S 3 ) → C(S 2 ) whose kernel is the ideal generated by b − b∗ . We apply this approach to the quantum case q ∈ [0, 1) as well. Therefore, let J˜3 be the 2-sided closed ideal of C(SUq (2)) generated by b − b∗ . Identifying C(SUq (2)) with C ∗ (L3 ), as in Proposition 2.1, we see that J˜3 is transformed onto the ideal J3 of C ∗ (L3 ) generated by Se2,2 − Se∗2,2 . We denote by ψ3 : C ∗ (L3 ) → C ∗ (L3 )/J3 the natural surjective homomorphism. We omit an easy proof of the following proposition. A more general result is proved in Proposition 5.1. Proposition 3.2. There exists a C ∗ -algebra isomorphism C ∗ (L2 ) → C ∗ (L3 )/J3 such that Pv1 → ψ3 (Pv1 ), 1 Pw1 → ψ3 (Pv2 + Se2,2 ), 2 1 Pw2 → ψ3 (Pv2 − Se2,2 ), 2

Se1,1 → ψ3 (Se1,1 ),
 1
Sg1,1 → ψ3 Se1,2 Se22,2 + Se2,2 , 2
 1
Sg1,2 → ψ3 Se1,2 Se22,2 − Se2,2 . 2

3.2. Quantum RP 2 . Quantum real projective plane was defined algebraically by Hajac in [14]. The C ∗ -algebra C(RPq2 ) of continuous functions on the quantum real projective plane was analyzed in detail in [15]. It was shown there that C(RPq2 ) may be realized 2 ) given by A → −A, as the fixed point algebra for the antipodal Z2 action on C(Sq∞ B → −B. Below, we reexamine this construction from the point of view of the graphical approach we have taken in this article. Let γ−1 be the antipodal automorphism of the quantum 3-sphere, as defined in Sect. 2.2, and let ψ3 : C ∗ (L3 ) → C ∗ (L3 )/J3 be the homomorphism defined above Proposition 3.2. Since ker ψ3 is γ−1 -invariant there exists a unique order 2 automorphism η of C ∗ (L2 ) such that η ◦ ψ3 = ψ3 ◦ γ−1 . We have η(Se1,1 ) = −Se1,1 and η(Sg1,i ) = −Sg1,3−i  2 from Proposition 3.1 this η is the for i = 1, 2. With the identification C ∗ (L2 ) ∼ = C Sqc antipodal action of Z2 on the quantum 2-sphere, determined by A → −A, B → −B in 2 ) of continuous functions on the quantum the case c = ∞. Thus, the C ∗ -algebra C(RPqc real projective plane may be defined as the fixed point algebra C ∗ (L2 )η . It turns out that this C ∗ -algebra is isomorphic to the Cuntz-Krieger algebra of the following graph, which we denote L 2 . We use the labels for the vertices and edges of this graph which are consistent with the general case of L 2n , defined in Sect. 5.2.

172

J. H. Hong, W. Szyma´nski .... .................................. ...... . ..... ..... ... 1,1 ... ... . ... ... ... .... . ... .. ... .. .. ... . . ... . . . ..... . .... ...... ...... ..............1 ............ ....... .... .... . . .. .. ......... ......... .... .... 1,1 ...... ..... 1,2 ... .. ... ... ... ... ..... .

(e

L 2

, e1,1 )

v •

f

f

• w

The proof of the following Proposition 3.3 is very similar to that of Proposition 1.1 and thus it is omitted. A more general result is proved in Proposition 5.2. Proposition 3.3. There exists a C ∗ -algebra isomorphism
 2 C ∗ (L 2 ) → C ∗ (L2 )η ∼ = C RPqc such that  Pv1 , Pv1 → Pw → Pw1 + Pw2 ,

S(e1,1 ,e1,1 ) →  Se1,1 Se1,1 , Sf1,1 →  Se1,1 (Sf1,1 + Sf1,2 ), Sf1,2 →  Sf1,1 − Sf1,2 .

  2 has been analyzed in [15]. However, in view of the isoThe C ∗ -algebra C RPq∞   2 ∗ ∼ morphisms C RPqc = C (L2 )η ∼ = C ∗ (L 2 ) its properties follow much more easily from the general theory of graph algebras. In particular, the ideal structure of C ∗ (L 2 ) is immediately clear. Indeed, {w} is a hereditary and saturated set in L 2 . The corresponding ideal, generated by Pw , is isomorphic to the compacts and the quotient is the C ∗ -algebra of the graph with one vertex and one edge, which is isomorphic to C(T). Thus we have the exact sequence
 2 0 −→ K −→ C RPqc −→ C(T) −→ 0,

(3.2)

found in [15]. However, as shown in [15], this is not the usual Toeplitz algebra. Indeed, the K0 and K1 groups of C ∗ (L 2 ) are isomorphic to the cokernel and kernel, respectively, of the map KL 2 : Z → Z2 corresponding to the matrix   0 . 2 Hence

 2 ∼ K0 C RPqc = Z ⊕ Z2 ,

 2 = 0, K1 C RPqc just as in the classical case of RP 2 [20].

Quantum Spheres and Projective Spaces as Graph Algebras

173

4. Quantum Odd Dimensional Spheres and Projective Spaces   4.1. Quantum S 2n−1 . The C ∗ -algebra C Sq2n−1 of continuous functions on the quantum sphere S 2n−1 is given in [42] as the universal C ∗ -algebra generated by elements z1 , z2 , . . . , zn , subject to the following relations. zj zi = qzi zj zj∗ zi = qzi zj∗

for i < j, for i = j,  zi∗ zi = zi zi∗ + (1 − q 2 ) zj zj∗

(4.1) (4.2) for i = 1, . . . , n,

(4.3)

j >i

n  i=1

zi zi∗ = I.

(4.4)

If n = 2 then the above relations are the same as those defining C(SUq (2)), if one identifies z1 ↔ a ∗ and z2 ↔ b. Vaksman and Soibelman assumed that q ∈ (0, 1), but the above relations make sensefor q = 0 as well. We will show below that for any n = 1, 2, . . . the C ∗ -algebra C Sq2n−1 is isomorphic with C ∗ (L2n−1 ). The graph  L2n−1 has n vertices {v1 , . . . , vn } and n(n + 1)/2 edges ni=1 {ei,j | j = i, . . . , n} with s(ei,j ) = vi and r(ei,j ) = vj . For example, if n = 4 then the corresponding graph L7 looks as follows: e1,1

.... ... ... ... .............................. ............................. ............................. ............................. ..... . ..... ..... . ..... . ... ... ... ... ... ... ... ... ... ... ... ... ... .. .. .. .... .... .... .... . ... ... ... ... .. .. .. .. ... ... ... ... .. .. . .. . . . . . . .. .... . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1,2 ....... ....... . . . . . .................................................................................................................................................................................................................................................................................................................................................................................................. .......... ............. .......... ............... .................... 1 ............................................................................ 1,3 . 2 .............................................................................................. . 3 4 ........................ ..... ..................................................................................................................... . ........... .... . ............ ........................................................................ .. . . . . . . . . . . . .............. ........ . . . . . . . . . ................ . . . . ............ ...................... ................................................................................................................ ....

v•

L7

e

e

v•

v•

v•

e1,4

Lemma 4.1. Let q ∈ (0, 1), and let 32 (Nn−1 ×Z) be a Hilbert space with an orthonormal basis {ξ(k1 , . . . , kn−1 , m) | ki ∈ N, m ∈ Z}. The following hold: (1) There exists a ∗-representation
 π : C Sq2n−1 → B(32 (Nn−1 × Z)) such that

1 − q 2(k1 +1) ξ(k1 + 1, k2 , . . . , kn−1 , m),

k1 +···+kj −1 1 − q 2(kj +1) π(zj ) : ξ(k1 , . . . , kn−1 , m) → q π(z1 ) : ξ(k1 , . . . , kn−1 , m) →

ξ(k1 , . . . , kj −1 , kj + 1, kj +1 , . . . , kn−1 , m), π(zn ) : ξ(k1 , . . . , kn−1 , m) → q k1 +···+kn−1 ξ(k1 , . . . , kn−1 , m + 1), for j = 2, . . . , n − 1 and k1 , . . . , kn−1 ∈ N, m ∈ Z. (2) There exists a ∗-representation ρ : C ∗ (L2n−1 ) → B(32 (Nn−1 × Z))

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J. H. Hong, W. Szyma´nski

such that ρ(Pvn ) : ξ(k1 , . . . ρ(Pvj ) : ξ(k1 , . . . ρ(Sen,n ) : ξ(k1 , . . . ρ(Sej,n ) : ξ(k1 , . . .

, kn−1 , m) → δk1 ,0 · · · δkn−1 ,0 ξ(k1 , . . . , kn−1 , m), , kn−1 , m) → δk1 ,0 · · · δkj −1 ,0 (1 − δkj ,0 )ξ(k1 , . . . , kn−1 , m), , kn−1 , m) → δk1 ,0 · · · δkn−1 ,0 ξ(k1 , . . . , kn−1 , m + 1), , kn−1 , m) → δk1 ,0 · · · δkn−1 ,0 ξ(k1 , . . . , kj −1 , kj + 1, kj +1 , . . . , kn−1 , m), ρ(Sei,j ) : ξ(k1 , . . . , kn−1 , m) → δk1 ,0 · · · δkj −1 ,0 (1 − δkj ,0 ) ξ(k1 , . . . , ki−1 , ki + 1, ki+1 , . . . , kn−1 , m),

for j = 1, . . . , n − 1, i = 1, . . . , j , and k1 , . . . , kn−1 ∈ N, m ∈ Z. Here δ is the Kronecker symbol.   Proof. This follows from the universal properties of the algebras C Sq2n−1 and C ∗ (L2n−1 ), since the target operators satisfy the defining relations (4.1–4.4) and (G1– G5) for the graph L2n−1 , respectively.     Lemma 4.2. Let q ∈ (0, 1) and let D2n−1 (q) be the C ∗ -subalgebra of C Sq2n−1 generated by {zj zj∗ | j = 1, . . . , n}. Then D2n−1 (q) is abelian and unital, and the map zj zj∗ → xj extends to a C ∗ -algebra isomorphism between D2n−1 (q) and C(A2n−1 (q)), where A2n−1 (q) consists of (1, 0, . . . , 0) and all those (x1 , . . . , xn ) ∈ [0, 1]n for which there exist m ∈ {2, . . . , n} and k1 , . . . , km−1 ∈ N such that x1 = 1 − q 2k1 , xi = (1 − q 2ki )q 2(k1 +...+ki−1 ) ,

xm = q 2(k1 +...+km−1 ) , xr = 0, for i = 2, . . . , m − 1 and r = m + 1, . . . , n.

Proof. Relations (4.1) and (4.2) imply that D2n−1 (q) is abelian, and (4.4) implies it is unital. We consider a character of D2n−1 (q) such that zi zi∗ → xi for i = 1, . . . , n. Proceeding by induction on j we show that x1 , . . . , xj , . . . , xn have the desired form. Let j = 1. From (4.3) and (4.4) we get z1∗ z1 = I − q 2 + q 2 z1 z1∗ . ..

z1∗ z1 ........ 1

1 − q2

.. ... .. ... ................................................................................................................... .... .. . ... ........ ..... .. ... ........ ......... .... ........ ... ... ............... .......... . . ... . . . . . . . ... . .. ... ..... ........ ... ..... ........ .... ... . . . . . . . . . ... . . . ...... ..... . . . . . . ... ..... . . ... . ........ ... ... ............... . . ... ............... ....... ....... ....... .................. . . ... .... . ... . . . ... ... . ... . . ... . ... . ... . ... . . ... . .... ... . . ... . ... ... . . . ... . ... . . . ... ..... . ... . . ... . ... . ... . . ... . ... . .. ... ......... .... .. ... ...... ......................................................................................................................................................

1 Since Sp(z1∗ z1 ) ∪ {0} = Sp(z1 z1∗ ) ∪ {0} this implies

z1 z1∗

Quantum Spheres and Projective Spaces as Graph Algebras

175

Sp(z1 z1∗ ) ⊆ {1 − q 2k1 | k1 ∈ N} ∪ {1}. Thus, either x1 = 1 − q 2k1 for some k1 ∈ N or x1 = 1. In the latter case we have x2 = . . . = xn = 0 by (4.4). The inductive step is an immediate consequence of the following observations. Firstly, if there exist natural numbers k1 , . . . , kj −1 such that x1 = 1 − q 2k1 and j −1  2k 2(k +···+k ) 1 i i−1 xi = (1 − q )q for i = 2, . . . , j − 1, then 1 − xi = q 2(k1 +···+kj −1 ) . Secondly, (4.3) and (4.4) imply that

i=1



zj∗ zj = (1 − q 2 ) 1 −

j −1  i=1

 zi zi∗  + q 2 zj zj∗ .

(4.5)

Let ϑj −1 be the C ∗ -algebra homomorphism of C(A2n−1 (q)) into itself such that ϑj −1 (zi zi∗ ) = xi for i = 1, . . . , j − 1 and ϑj −1 (zi zi∗ ) = zi zi∗ for i = j, . . . , n. Since Sp(ϑj −1 (zj∗ zj )) ∪ {0} = Sp(ϑj −1 (zj zj∗ )) ∪ {0}, (4.5) implies that       j −1 j −1       Sp(ϑj −1 (zj zj∗ )) ⊆ (1 − q 2kj ) 1 − xi  | kj ∈ N ∪ 1 − xi .     i=1

i=1

Consequently, either xj = (1 − q 2kj )q 2(k1 +...+kj −1 ) for some kj ∈ N or xj = q 2(k1 +···+kj −1 ) . In the former case we can continue our induction, while in the latter we have xj +1 = . . . = xn = 0 by (4.4) and there is nothing more to prove. In order to complete the proof of the lemma it suffices to show that for any element (x1 , . . . , xn ) of A2n−1 (q) there exists a character of D2n−1 (q) such that zj zj∗ → xj for j = 1, . . . , n. This however follows from Lemma 4.1 since one may verify that the joint spectrum of the operators π(z1 z1∗ ), . . . , π(zn zn∗ ) is exactly A2n−1 (q).   Remark 4.3. Pusz and Woronowicz studied twisted canonical commutation relations (TCCR) in [34] (see also [33]). Their relations are satisfied by the (possibly unbounded) creation and annihilation operators covariant with respect to the action of the quantum SU (n) group on the Fock space. The TCCR are very similar to (4.1–4.4), and perhaps more general. Pusz and Woronowicz classified irreducible representations of the TCCR and also proved in their setting a result similar to our Lemma 4.2, by examining purepoint common spectra of suitable operators. However, C ∗ -algebras were not introduced in [34]. The precise relationship between the quantum spheres of Vaksman and Soibelman and representations of the TCCR of Pusz and Woronowicz is not yet fully clear to us, but it certainly deserves clarification2 . Before stating and proving the main result of this section we introduce the following notation. For j = 1, . . . , n − 1 and k1 , . . . , kj ∈ N we denote kj  k1  n k2  n n    T (k1 , . . . , kj ) = Se1,i Se2,i ··· Sej,i  , (4.6) i=1

an element of C ∗ (L

i=2

2n−1 ). As before, we denote by χ1

i=j

the characteristic function of {1}.

2 We are grateful to Professors Pusz and Woronowicz for bringing their papers [34, 33] to our attention, after the main results of the present article had been reported in Warsaw in November/December, 2000.

176

J. H. Hong, W. Szyma´nski

Theorem 4.4. For q ∈ [0, 1) there exists a C ∗ -algebra isomorphism 
φ : C Sq2n−1 → C ∗ (L2n−1 ). (1) If q ∈ (0, 1) then  φ : zn →

q k1 +···+kn−1 T (k1 , . . . , kn−1 )Sen,n T (k1 , . . . , kn−1 )∗ ,

k1 ,... ,kn−1 ∈N

φ : zi →



q

k1 +···+ki−1



1 − q 2(ki +1)

−

 1 − q 2ki

k1 ,... ,ki ∈N

  n  T (k1 , . . . , ki )  Sei,j  T (k1 , . . . , ki )∗ , j =i

for i = 1, . . . , n − 1, and the inverse φ −1 : C ∗ (L2n−1 ) → C(Sq2n−1 ) is given by φ −1 : Pvn → χ1 (zn zn∗ ),     n n   φ −1 : Pvj → χ1  zr zr∗  − χ1  zr zr∗  , r=j

φ

−1

: Sen,n →

φ

−1

: Sej,n → (1 − q ) 

φ −1 : Sei,j

r=j +1

zn χ1 (zn zn∗ ), 2 −1/2

zj χ1 (zn zn∗ ),     n n   → zi |zi |−1 χ1  zr zr∗  − χ1  zr zr∗  , r=j

for j = 1, . . . , n − 1 and i = 1, . . . , j . (2) If q = 0 then φ : zi →

n 

r=j +1

Sei,j ,

j =i

for j = 1, . . . , n − 1, and the inverse φ −1 : C ∗ (L2n−1 ) → C(S02n−1 ) is given by φ −1 : Pvj → zj zj∗ , φ −1 : Sei,j → zi zj zj∗ , for j = 1, . . . , n and i = 1, . . . , j . Proof. The case q = 0 being trivial we concentrate on the case q ∈ (0, 1). At first we observe that φ −1 is well defined on the generators. Indeed, it follows from n  Lemma 4.2 that for each j = 1, . . . , n the spectrum of zi zi∗ contains 1 as an isoi=j

lated from Lemma 4.2 that if i ≤ j then the
restriction of |zi| to
point. Also,  it follows

n n n ∗ ∗ −1 ∗ χ1 χ1 r=j zr zr −χ1 r=j +1 zr zr is invertible and, hence, |zi | r=j zr zr −
 n ∗ χ1 makes sense. r=j +1 zr zr

Quantum Spheres and Projective Spaces as Graph Algebras

177


 The rest of the proof is routine. Namely, the universal properties of C Sq2n−1 
and C ∗ (L2n−1 ) imply that there exist C ∗ -algebra homomorphisms φ : C Sq2n−1 → 
C ∗ (L2n−1 ) and φ −1 : C ∗ (L2n−1 ) → C Sq2n−1 with the required values on the generators. Indeed, the target elements satisfy relations (4.1–4.4) and (G1–G5), respectively. It is then not very difficult to verify that φ and φ −1 , as defined in the theorem, are inverses of one another.   Remark 4.5. Although we do not need it in this paper, it can be shown that both representations π and ρ from Lemma 4.1 are faithful. Furthermore, the isomorphisms from Theorem 4.4 are given by φ = ρ −1 ◦ π and φ −1 = π −1 ◦ ρ, respectively.   The ideal structure and the K-theory of the C ∗ -algebras C Sq2n−1 were determined by Vaksman and Soibelman in [42]. However, these also follow easily from the general theory of graph algebras. For example, {vn } is a hereditary and saturated set in L2n−1 . The corresponding ideal is C(T) ⊗ K, and the quotient is isomorphic to C ∗ (L2n−3 ) ∼ = C(Sq2n−3 ). Consequently, there exists an exact sequence

 0 −→ C(T) ⊗ K −→ C Sq2n−1 −→ C Sq2n−3 −→ 0. (4.7) The primitive spectrum consists of n copies of the circle (with a suitable topology), and according to [35] the K0 and K1 groups of C ∗ (L2n−1 ) are given by the cokernel and kernel, respectively, of the map KL2n−1 : Zn → Zn corresponding to the n-by-n matrix 

 0 0  0 0 . ..   . 1 1 1 ··· 0 0 1 1 1 1 ··· 1 0

0 1  1 1  . . . 1

0 0 1 1 .. .

0 0 0 1 .. .

0 0 0 0 .. .

··· ··· ··· ···

0 0 0 0 .. .

Consequently, we have K0 (C(Sq2n−1 )) ∼ = Z, 2n−1 ∼ K1 (C(S )) = Z. q

4.2. Quantum RP 2n−1 . Taking a cue from the construction of the quantum real projective space RP 3 , described in  Sect. 2.2, we now define for n = 1, 2, . . . and q ∈ [0, 1)  the C ∗ -algebra C RPq2n−1 of continuous functions on the odd dimensional quantum   real projective space as the fixed point algebra of the antipodal action of Z2 on C Sq2n−1 determined by zj → −z  j for j = 1, . . . , n. Taking into account Theorem 4.4 it is not difficult to see that C RPq2n−1 coincides with the fixed point algebra of the restriction to Z2 = {±1} of the gauge action γ : T → Aut(C ∗ (L2n−1 )). Then, by virtue of Proposition 1.1,

Z2
 (2) ∼ C RPq2n−1 = C Sq2n−1 (4.8) = C ∗ (L2n−1 )γ−1 ∼ = C ∗ L2n−1 .

178

J. H. Hong, W. Szyma´nski (2)

For example, if n = 4 then L7 is the following graph: .. .. .. .. ............................ ............................ ............................ ............................ ..... .. ..... .. ..... .. ..... .. ... ... ... ... ... ... ... ... ... ... ... ... .. .. .. .. ..... ..... ..... ..... ... ... ... ... .. .. .. .. . . . . . . . . . . . .. . . .. . . ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..................... ... ..................... ... ..................... ... ................................... .................................. .................................. .. .... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. .. .. .. .. . ....... .......... ................................................................................................................................................................................................................................................................................................................................................................................................................................ ..................... .................. ......... .................. ..................................... ............................. ........................................... 1 2 ................................................................................................................................................................................. . 3 4 .................................... ............................. ................................ .................. ..................... ....... ... ....................... ................ .................................. ............ ................... ........................ ....... ..... .......................... .... ......................... ........................ ............................... .................................................................................................. .................................. ................................................................................... . .......................................... . . . . . . . . . . . . . . . . . . . .............................. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ................. ...... ................ ........................ ...... ............................................................................................................ ..................................................... .................. ... ..... .... ........ ........... ......... ............ ........... ......... .......... ................ ............. ......... ............ ........................ ..................................... . . .. ... ............... ........................ . ...................... ...................................................... .................................. ........................................................ ..

(2)

v •

L7

v•

•v

•v

  It is then easy to understand the ideal structure of C RPq2n−1 . For example, the set (2)

{vn } is hereditary and saturated in L2n−1 . The corresponding ideal, generated by Pvn , is
   (2) ∼ isomorphic to C(T) ⊗ K, and the quotient is C ∗ L = C RP 2n−3 . Thus, there q

2n−3

exists an exact sequence 

0 −→ C(T) ⊗ K −→ C RPq2n−1 −→ C RPq2n−3 −→ 0.

(4.9)



(2) (2) We have M L2n−1 = Mτ L2n−1 = {{v1 , . . . , vj } | j = 1, . . . , n}. Thus, as a   set, the primitive spectrum of C RPq2n−1 consists of n disjoint copies of the circle. As  
topological spaces, equipped with the hull-kernel topology, Prim C RPq2n−1 and  
Prim C Sq2n−1 are homeomorphic. The K0 and K1 groups are given by the cokernel and kernel, respectively, of the map KL(2) : Zn → Zn corresponding to the n-by-n matrix 2n−1



0 2 3 2 .. .

0 0 2 3 .. .

0 0 0 2 .. .

          2 2 2  2 2 2   2 2 2 2 2 2

0 0 0 0 .. . 2 2 2 2

··· ··· ··· ··· ··· ··· ··· ···

0 0 0 0 .. . 0 2 3 2

0 0 0 0 .. .

0 0 0 0 .. .

0 0 0 0 .. .



      .   0 0 0  0 0 0   2 0 0  3 2 0

Consequently,

 K0 C RPq2n−1 ∼ = Z ⊕ Z2n−1 ,

 K1 C RPq2n−1 ∼ = Z. The cokernel has generators , 1)t and (0, 1, 0, . . . , 0)t of orders ∞ and 2n−1 ,

(1, . . .  2n−1 has generators [I ] of infinite order and [Pv2 ] of respectively. Thus, K0 C RPq order 2n−1 .

Quantum Spheres and Projective Spaces as Graph Algebras

179

  4.3. Quantum CP n−1 . The C ∗ -algebra C CPqn−1 of continuous functions on the quantum complex projective space was defined by Vaksman
and Soibelman in [42] as the fixed  point algebra for the action of the circle group on C Sq2n−1 determined on the gener
 ators by (t, zj ) → tzj for t ∈ T, j = 1, . . . , n. Under the isomorphism C Sq2n−1 ∼ = ∗ C (L2n−1 ) from Theorem 4.4 this action becomes nothing but the gauge action γ : T → Aut (C ∗ (L2n−1 )). Consequently, it is the core AF -subalgebra of C ∗ (L2n−1 ). It is not difficult to show that it is isomorphic to the graph algebra
 C CPqn−1 ∼ (4.10) = C ∗ (Fn−1 ) = C ∗ (L2n−1 )γ ∼ corresponding to the graph Fn−1 , defined as follows. The vertices of Fn−1 are {v1 , . . . , vn }, and for any pair i, j ∈ {1, . . . , n} with i ≤ j there are infinitely many edges from vi to vj . For example, if n = 4 then F3 looks as follows: F3

v2 v3 v4 v1 •.................................................................................................................................................•...........................................................................................................................................................•................................................................................................................................................•.. ................... ............ (∞) ............ ............... (∞) .............. .............. ........ ................ (∞) ............. ....... ........... ..... ................. ........ ..................... ............................. ... ......... .. ..................................................................... ................. ......... ............................................................................................. . .......... .......... ........... ........... ............. ................ ............. ...................... ................ ................................................................................................... ...

(∞)

(∞)

(∞)

Thus, applying the general theory of graph algebras we may easily reproduce
Vaksman  and Soibelman’s results pertaining to the ideal structure and the K-theory of C CPqn−1 . For example, {vn } is a hereditary and saturated set in Fn−1 . The corresponding ideal, generated by Pv
n , is isomorphic with the compacts, and the quotient is isomorphic to  ∗ n−2 ∼ C (Fn−2 ) = C CP . Consequently, we have the exact sequence q



 0 −→ K −→ C CPqn−1 −→ C CPqn−2 −→ 0.

(4.11)



 It follows that the primitive spectrum Prim C CPqn−1 consists of n points (with the suitable topology). The K-groups of C ∗ (Fn−1 ) are isomorphic to the cokernel and kernel, respectively, of the map KFn−1 : 0 → Zn and, hence,

 K0 C CPqn−1 ∼ = Zn ,

 K1 C CPqn−1 = 0. 5. Quantum Even Dimensional Spheres and Projective Spaces 5.1. Quantum S 2n . In this section, we use an analogy with the classical imbedding S 2n :→ S 2n+1 to define a quantum sphere S 2n . The case n = 1 was discussed at the ˜ end
closed  3.1. Now let q ∈ [0, 1). We consider the 2-sided  ideal J2n+1 of
of Sect. 2n+1 ∗ 2n+1 ∗ generated by zn − zn . With the identification of C Sq with C (L2n+1 ), C Sq ∗ as in Theorem 4.4, J˜2n+1 is transformed onto the ideal J2n+1 of C (L2n+1 ) generated

180

J. H. Hong, W. Szyma´nski


 by Sen+1,n+1 − Se∗n+1,n+1 . We define the C ∗ -algebra C Sq2n of continuous functions on 
the quantum sphere S 2n to be the quotient C Sq2n+1 /J˜2n+1 . Thus, C(Sq2n ) is isomorphic to C ∗ (L2n+1 )/J2n+1 . The of this algebra follows easily from Proposition 
structure 5.1, where we show that C Sq2n is isomorphic to C ∗ (L2n ). The graph L2n has vertices  {v1 , . . . , vn }∪{w1 , w2 } and edges ni=1 {ei,j | j = i, . . . , n}∪{gi,k | i = 1, . . . , n, k = 1, 2} with s(ei,j ) = s(gi,k ) = vi , r(ei,j ) = vj and r(gi,k ) = wk . For example, if n = 3 then the graph L6 looks as follows: e1,1......................................................... L6

e

e

.............. ............. 2,2.................. .......................... 3,3.................. ........................... . ... ... ... .. ... ... . ... . ... . . . ... ... ... .. .. .. ... .. .. .. ... ... ... ..... ..... ..... . . . . . ... ... ... ... . . . . .. . . ... ... ... . . .. . . . ... . . . . . ... ... . . .. . . ..... . . . . . . . . . . . . . . ...... 1,2 ......... 2 ........ 2,3 ......... 3 ........ ...... ............ 1 ..................................................................................................................................................................................................................................... ... .................... .... ........ ............. .. .................... ... ..... ..... ................... .... . . . . . . . . . . . . . . . ....... .............................. ....... ...................................................... ....... . ..... ... . .... ..... ....... ....... .... ................. ....... ........ . . . . . .. . ... . . . . . . ....... ..... ... .......... .... 1,3 ............................. . ........ . .. . . . . . . . . ... . . . ....... ..... .... ... . ........ . . . . . . . . . . . . . . . . . . ............ ........... ..... ...... ... . 2,1 . . . . . . . . . . . . . ....... 2,2 . ... ..... ............ . . ............. . . . . . . . . .... .......... ....... . . ...... ........... ..... .... ...... .......... ............ ..... ..... 1,1 ..... ... 3,2 ............ ....... ..... ..... ....... . . . . . ... . .. . . . . . . . . . ....... ..... ... ........... . ... . . . . . . . . . ....... .... .... ... ..................... . . . . . . . 3,1 1,2 ....... .... ... ... ................ . . . . . . . . ........... .......

e

v •

• w1

e

v •

e

g

g

v •

g

g

g

g

•

w2

We denote by ψ2n+1 : C ∗ (L2n+1 ) → C ∗ (L2n+1 )/J2n+1 the natural surjective homomorphism. Proposition 5.1. There exists a C ∗ -algebra isomorphism


 C ∗ (L2n ) → C ∗ (L2n+1 )/J2n+1 ∼ = C Sq2n

such that Pvi → ψ2n+1 (Pvi ), 1 Pw1 → ψ2n+1 (Pvn+1 + Sen+1,n+1 ), 2 1 Pw2 → ψ2n+1 (Pvn+1 − Sen+1,n+1 ), 2 Sei,j → ψ2n+1 (Sei,j ),

 1 Sgi,1 → ψ2n+1 Sei,n+1 Se2n+1,n+1 + Sen+1,n+1 , 2

 1 Sgi,2 → ψ2n+1 Sei,n+1 Se2n+1,n+1 − Sen+1,n+1 , 2 for i = 1, . . . , n and j = i, . . . , n. Proof. There exists a C ∗ -algebra homomorphism φ : C ∗ (L2n ) → C ∗ (L2n+1 )/J2n+1 with the prescribed values on the generators because the target elements satisfy the defining relations (G1–G5) for the graph L2n . In order to construct its inverse we observe that there exists a C ∗ -algebra homomorphism µ˜ : C ∗ (L2n+1 ) → C ∗ (L2n ) such

Quantum Spheres and Projective Spaces as Graph Algebras

µ˜ : Pvi µ˜ : Pvn+1 µ˜ : Sei,j µ˜ : Sei,n+1 µ˜ : Sen+1,n+1

→  →  →  →  →

181

Pvi , Pw1 + Pw2 , Sei,j , Sgi,1 + Sgi,2 , Pw1 − Pw2 ,

that for i = 1, . . . , n and j = i, . . . , n, because the target elements satisfy the defining ˜ it induces the homomorrelations (G1–G5) for the graph L2n+1 . Since J2n+1 ⊆ ker(µ) phism µ : C ∗ (L2n+1 )/J2n+1 → C ∗ (L2n ). One can verify that φ and µ are inverses of one another.  
 It is now easy to understand the ideal structure of C Sq2n ∼ = C ∗ (L2n ). For example, the set {w1 , w2 } is hereditary and saturated in L2n . The corresponding 2-sided closed ideal, generated by
 Pw1 + Pw2 , is K ⊕ K, and the quotient is isomorphic with ∗ 2n−1 ∼ C (L2n−1 ) = C S . Thus, there exists an exact sequence q



 0 −→ K ⊕ K −→ C Sq2n −→ C Sq2n−1 −→ 0,

(5.1)

describing an imbedding of the quantum S 2n−1 into the quantum S 2n . The primitive
 spectrum of C Sq2n consists of n copies of the circle and two points.



 In view of Proposition 5.1 the groups K0 C Sq2n and K1 C Sq2n can be

calculated as the cokernel and kernel, respectively, of the map KL2n : Zn → Zn+2 , corresponding to the following (n + 2)-by-n matrix:   0 0 0 0 ··· 0 0 1 0 0 0 ··· 0 0   1 1 0 0 ··· 0 0 . . . . .. ..  . . . .  . . . . . . .   1 1 1 1 ··· 0 0 1 1 1 1 ··· 1 0   1 1 1 1 ··· 1 1 1 1 1 1 ··· 1 1 Consequently, we have

 K0 C Sq2n ∼ = Z2 ,

 K1 C Sq2n = 0. Furthermore, [I ] and [Pw1 ] (or [Pw2 ]) are free generators of the K0 group.

5.2. Quantum RP 2n . Motivated by the construction of C(RPq2 ) from Sect. 3.2 and the 
imbedding from Proposition 5.1, we define the C ∗ -algebra C RPq2n of continuous functions on the quantum real projective space RP 2n as the fixed point algebra for

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 the antipodal action η˜ of Z2 on C Sq2n+1 /J˜2n+1 determined by η˜ : zi + J˜2n+1 → −zi + J˜2n+1 , for i = 1, . . . , n + 1. The isomorphisms of Theorem 4.4 and Proposition 5.1 transform η˜ into the action η of Z2 on C ∗ (L2n ), determined by η : Pvi → Pvi , η : Pwk → Pw3−k , η : Sei,j → −Sei,j , η : Sgi,k → −Sgi,3−k , for i = 1, . . . , n, j = 1, . . . , n and k = 1,
2. This definition makes sense for all q ∈ [0, 1).   It turns out that C RPq2n is isomorphic with C ∗ L 2n , with the graph L 2n

defined as follows. The vertices of L 2n are {v1 , . . . , vn } ∪ {w}. The edges of L 2n  are {(h1 , h2 ) | h1 , h2 ∈ L12n , r(h1 ) = s(h2 ), r(h2 ) = w1 , w2 } ∪ ni=1 {fi,k | k = 1, . . . , n + 2 − i} with s(h1 , h2 ) = s(h1 ), r(h1 , h2 ) = r(h2 ), s(fi,k ) = vi and r(fi,k ) = w. For example, if n = 3 then L 6 looks as follows: (e1,1 , e1,1 )

L 6

(e2,2 , e2,2 )

(e3,3 , e3,3 )

....... ......... ......... ............... ............. ............................ ............... ............. ... ... ..... ..... ..... ... ... ... ... ... ... ... ... ... ... .... .... .... ... ... ... . . .. .. .. . . ... . . . . . . . . . . . . . . . . ... . ....................................................................... ... . ....................................................................... ... ... . . . . . . . . . . . . . . . . . . . . . . . . . . . .. ............ . ..... ....... .. . . . . . . . . ......................................................................................................................................................................2 . . . . . . . . . . . . . . . . . .................................................................................................................................................... .......................... ....... ........ .... ........... .......................... ........ .................................... 1 ................................................................................................................................ 3 .. ...... ................................. ...... ... ......... ...... ......................... ..................................................................................................................................................................... ................ . ...... ................ ......... .......... ................. .. .... . ... .. .......... ...... . . . . . . . . . . . . . . . . . . . . . . .................................................................................. ............ ...... .... ...... .. ... ...... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .................. ...... ... ...... .... .. .................................................................................... ....... .... ..... .... ..... ... ... ..... ...... ..... .... ... .. ... .. ... .. ..... ...... ... ... ..... ...... . .... .. .... .... . ... .... ...... .......... .. . . . . . . .. ... ... ...... ........ ... .. .. ... .... .. .... ... ... ..... .. .. .. 3,1 .... .... .. . .. ... .... ........ ........ 1,4 ... .... .. .... ... . . ... ... ........... ...... . . . . . ... ... ...... ....... ..... ........ ... ........ ... ....... .. ... . ... ..... ..... ... ..... 3,2 ........... .... ... ........ ......... ......... .... ........ .. ... ... .. ... ........... . ... ..... ... .... .... .... ...... ......... ........ . . . . . . . . . . . . . ...... ..... ...... ..... ... .. .. ..... ...... . . . . . . . . . . . . . . . . . 1,1 ................ ........... .......... ......... .... ... ... ...... ....... ....... ....... ...... ..... ... .. ..... ..... ....... ...... ..... .. .. ..... ...... ........ .............. .. ..... .......... ....... ........ . ........... ................................................................ ..................................

v

v •

•

• v

f

f

f

f

• w

Proposition 5.2. If n = 1, 2, . . . and q ∈ [0, 1) then there exists a C ∗ -algebra isomorphism 
  C ∗ L 2n → C ∗ (L2n )η ∼ = C RPq2n such that Pvi →  Pw →  S(h1 ,h2 ) → Sfi,k → Sfi,n+2−i →

Pvi , Pw1 + Pw2 , Sh1 Sh2 , Sei,i+k−1 (Sgi+k−1,1 + Sgi+k−1,2 ), Sgi,1 − Sgi,2 ,

for i = 1, . . . , n, k = 1, . . . , n + 1 − i and h1 , h2 ∈ L12n with r(h1 ) = s(h2 ), r(h2 ) = w1 , w2 .   Proof. By virtue of the universal property of C ∗ L 2n , the map defined on the generators above extends to a C ∗ -algebra homomorphism φ : C ∗ L 2n → C ∗ (L2n )η . Indeed, the target elements belong to the fixed point algebra and satisfy the defining relations (G1–G5) for the graph L 2n .

Quantum Spheres and Projective Spaces as Graph Algebras

183

Surjectivity of φ follows from the fact that the C ∗ -algebra C ∗ (L2n )η is generated by elements of the set $ # W = Sh1 Sh2 | h1 , h2 ∈ L12n , r(h1 ) = s(h2 ), r(h2 ) = w1 , w2 ∪{Sgi,1 − Sgi,2 | i = 1, . . . , n} ∪{Sei,i+k−1 (Sgi+k−1,1 + Sgi+k−1,2 ) | i = 1, . . . , n, k = 1, . . . , n + 1 − i}. Indeed, let G(x) = 21 (x + η(x)) be the conditional expectation from C ∗ (L2n ) onto C ∗ (L2n )η . The C ∗ -algebra C ∗ (L2n )η is generated by all images G(Sx Sy∗ ), with x = (x1 , . . . , xk ) and y = (y1 , . . . , ym ) finite paths in L2n . But a non-zero element G(Sx Sy∗ ) must be equal to a scalar multiple of one of the following; 1. Sx Sy∗ (if k + m is even and r(x) = r(y) is different from w1 , w2 ),   2. Sx1 · · · Sxk−1 Sgi,r + (−1)k Sgi,3−r ,
 3. Sg∗i,r + (−1)m Sg∗i,3−r Sy∗m−1 · · · Sy∗1 ,
 4. Sx1 · · · Sxk−1 Sgi,r Sg∗j,r + (−1)k+m Sgi,3−r Sg∗j,3−r Sy∗m−1 · · · Sy∗1 , with i, j = 1, . . . , n and r = 1, 2. It is a somewhat tedious but not difficult matter to verify that each of 1–4 may be written as a product of elements from W ∪ W ∗ , and this gives surjectivity of φ. The homomorphism φ is injective by virtue of the Cuntz-Krieger uniqueness theorem [2, Th. 3.1], since all vertex simple loops in the graph L 2n have exits and φ(Pu ) = 0 for all vertices u ∈ {v1 , . . . , vn , w} of L 2n .   
With help of Proposition 5.2 we can now determine the ideal structure of C RPq2n .

For example, {w} is a hereditary and saturated set in L 2n . The corresponding 2-sided closed ideal, generated by Pw
, is isomorphic with the compacts and the corresponding    (2) 2n−1 . Thus, there is an exact sequence C RP quotient is isomorphic to C ∗ L2n−1 ∼ = q 

0 −→ K −→ C RPq2n −→ C RPq2n−1 −→ 0.

(5.2)

    Since Mγ L 2n consists of one element {v1 , . . . , vn , w} and Mτ L 2n
consists  of n elements {{v1 , . . . , vk } | k = 1, . . . , n}, the primitive spectrum of C RPq2n is composed of n copies of the circle and one additional point. The K0 and K1 groups of C ∗ (L 2n ) are given by the cokernel and kernel, respectively, of the map KL 2n : Zn → Zn+1 corresponding to the (n + 1)-by-n matrix   0 0 0 0 ··· 0 0 0 0  2 0 0 0 ··· 0 0 0 0    3 2 0 0 ··· 0 0 0 0   3 2 0 ··· 0 0 0 0  2  . .. .. .. .. .. .. ..   .  . . . . . . . .  .   2 2 2 ··· 2 0 0 0  2  2 2 2 2 ··· 3 2 0 0    2 2 2 2 ··· 2 3 2 0 n + 1 n n − 1 n − 2 ··· 5 4 3 2

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J. H. Hong, W. Szyma´nski

Consequently,

 K0 C RPq2n ∼ = Z ⊕ Z2n ,

 K1 C RPq2n = 0. The cokernel has generators . , 1)t and (0, 1, 0, . . . , 0)t of orders ∞ and 2n , re

(1, . . 2n spectively. Hence, K0 C RPq has generators [I ] of infinite order and [Pv2 ] of n order 2 .

6. Quantum Double Suspension In this section, we define a quantum analogue of the classical double suspension of a compact space. We then show that our construction has a very simple graphical realization. In examples, we show that repeated application of the quantum double suspension to the circle, the two-point space and the one-point space, respectively, yields quantum analogues of the odd dimensional spheres, the even dimensional spheres and the even dimensional balls, respectively. If A is a compact topological space then its suspension SA is the quotient of A×[0, 1] with both A × {0} and A × {1} collapsed to single points. Thus the double suspension S 2 A is the quotient of A×[0, 1]2 with each of the sets: A×[0, 1]×{0}, A×[0, 1]×{1}, A × {0} × {x} and A × {1} × {x} for x ∈ (0, 1), collapsed to single points. The C ∗ -algebra C(S 2 A) of continuous functions on the double suspension is an essential extension 0 −→ C(A) ⊗ C∞ (R2 ) −→ C(S 2 A) −→ C(T) −→ 0,

(6.1)

where C∞ (R2 ) denotes the C ∗ -algebra of continuous functions on R2 which vanish at infinity. In order to define a quantum analogue of the classical double suspension we replace the C ∗ -algebra C∞ (R2 ) with its quantum analogue K, the compact operators on a separable Hilbert space. Moreover, it is necessary to make some additional assumptions about the nature of the extension, and we formulate them in terms of the Busby invariant [6], [3, Sect. 15.2]. We denote; by M(A ⊗ K) the multiplier C ∗ -algebra of A ⊗ K, by µ the canonical surjection from M(A ⊗ K) onto M(A ⊗ K)/A ⊗ K, by {ui,j | i, j ∈ N} a self-adjoint system of matrix units in K, by T an isometry with range of codimension one, and by z the identity function on the circle T. Definition 6.1. Let A be a unital C ∗ -algebra. Its quantum double suspension is defined as the unital C ∗ -algebra S 2 A for which there exists an essential extension 0 −→ A ⊗ K −→ S 2 A −→ C(T) −→ 0

(6.2)

such that the corresponding Busby invariant β : C(T) → M(A ⊗ K)/A ⊗ K satisfies β : z → µ(I ⊗ T ).

(6.3)

Quantum Spheres and Projective Spaces as Graph Algebras

185

It follows from Definition 6.1 that the index map ∂ind : K1 (C(T)) → K0 (A ⊗ K) and the exponential map ∂exp : K0 (C(T)) → K1 (A ⊗ K) satisfy ∂ind : [z] → −[I ⊗ u0,0 ], ∂exp : [I ] → 0.

(6.4) (6.5)

It turns out that within the class of graph algebras the quantum double suspension has a very simple realization. Indeed, let E be a directed graph with finitely many vertices {v1 , . . . , vn }. We define a new graph S 2 E, called the quantum double suspension of E, as follows. The vertices of S 2 E are {v1 , . . . , vn } ∪ {v0 }. The edges of S 2 E are E 1 ∪ {f0 , . . . , fn } with s(fi ) = v0 and r(fi ) = vi . f S2E

. ......

....... ....... ....... ...... . . ....... ......

..... .. ............ ... ................ ............ .. .. ..... ... .... 1 . . . . ... .. . ... . . . . . . . ... ....... . ... . . . . . . . . . . . . 0...... .. ... ............ . . . . . . . . . . . . . . . . . . . ... ... ............... ..... ... ... 1 2 ............ .......................................................... . .. . . . . . .. . ... . . . . . . . ........................................................ ..... . . . . ... ....... ............................................... . ................... . ... .. ............. 2 . ............. . ............. 0 .. ............. ... . .................. . . ....................... .............. ... n ... .............. ............. . .. ... ... n .. .. ..... .... .. . ...... ...... . . .... . . ...... . ....... ....... ....... .. ....

f

• v

f

f

•v •v .. .

E

•v

Proposition 6.2. If E is a graph with finitely many edges and S 2 E is its quantum double suspension graph, then C ∗ (S 2 E) is the quantum double suspension C ∗ -algebra of C ∗ (E). Proof. The set {v1 , . . . , vn } is hereditary and saturated in S 2 E. Let J denote the corresponding ideal of C ∗ (S 2 E), generated by nk=1 Pvk . We claim that J is isomorphic n to C ∗ (E) ⊗ K. Indeed, let u0 = nk=1 Pvk , u1 = nk=1 Sfk , ui = Sfi−1 k=1 Sfk for 0 ∗ i = 2, 3, . . . , and let ui,j = ui uj for i, j ∈ N. Then {ui,j | i, j ∈ N} is a self adjoint system of matrix units in K and the assignment x ⊗ ui,j → ui xu∗j extends to a C ∗ -algebra isomorphism C ∗ (E) ⊗ K → J . Since the quotient C ∗ (S 2 E)/J is isomorphic to C(T) we have an exact sequence 0 −→ C ∗ (E) ⊗ K −→ C ∗ (S 2 E) −→ C(T) −→ 0. For this extension we have β : z → µ(I ⊗ Sf0 ), as required.

 

Example 6.3. Let A = T be the circle and let E be the graph consisting of one vertex and one edge, so that C ∗ (E) ∼ = C(A). On the classical level, we have S 2 A = S 3 and repeated application of the double suspension yields all odd dimensional spheres S 2 S 2n−1 = S 2n+1 . On the other hand, on the quantum level, we have S 2 E = L3 so that C ∗ (S 2 E) ∼ = C(Sq3 ). Since S 2 L2n−1 = L2n+1 we see that repeated application of the quantum double suspension to the classical circle yields quantum odd dimensional spheres. Example 6.4. Let A = S 0 be the space consisting of two points and let E be the graph consisting of two vertices and no edges, so that C ∗ (E) ∼ = C(A). On the classical level we have S 2 A = S 2 and repeated application of the double suspension yields all even dimen2 sional spheres S 2 S 2n = S 2(n+1) . On the other hand,  2on the quantum level, S E = L2 so ∗ 2 ∗ that C (S E) is isomorphic to the C -algebra C Sqc , c = 0, of continuous functions

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J. H. Hong, W. Szyma´nski

on the Podle´s sphere. Since S 2 L2n = L2(n+1) we see that repeated application of the quantum double suspension to the space of two points yields quantum even dimensional spheres. Example 6.5. Let A be the space consisting of a single point and let E be the graph consisting of one vertex and no edges, so that C ∗ (E) ∼ = C(A). On the classical level S 2 A = D is the closed disc, and repeated application of the double suspension yields all even dimensional closed balls S 2 D = B 4 , S 2 B 2n = B 2(n+1) . On the other hand, on the quantum level, C ∗ (S 2 E) is isomorphic to the Toeplitz algebra, i.e. the C ∗ -algebra of continuous functions on the quantum (closed) disc [21]. Therefore, it appears rea sonable to us to define the C ∗ -algebra C Bq2n of continuous functions on the quantum 2n-ball as the C ∗ -algebra of the graph obtained by the quantum double suspension applied from this definition that
repeatedly
 n times to the one-point graph E. It
follows
 2n 2n ∼ K0 C Bq = 0. = Z (with the generator [I ]) and K1 C Bq

Appendix A. Another Graph for Quantum S 2n−1   We showed in Theorem 4.4 that the C ∗ -algebras C Sq2n−1 , q ∈ [0, 1), and C ∗ (L2n−1 ) are isomorphic. However, the same Cuntz-Krieger algebra may usually be represented  by many distinct graphs. In this Appendix we show that C Sq2n−1 is also isomorphic to C ∗ (L˜ 2n−1 ), where L˜ 2n−1 is the following graph. The vertices of L˜ 2n−1 are {v1 , . . . , vn } and the edges are {f1 , . . . , fn } ∪ {e1 , . . . , en−1 } with r(fi ) = s(fi ) = s(ei ) = vi and r(ei ) = vi+1 . f1

L˜ 2n−1

• v1

fn−1

f2

............................. ............................. .... .... ....... ....... ... ... ... ... ... ... ..... ..... ... ... .. .. .. .. . ... . ... .. . .... . . . . . . . . ...... ....... .. ... . . ..... 2 . . . . . . ........................................................1 . . . . ............................ .......... .................................

e

• v2

e

... ...

e

fn

............................. ............................. .... .... ....... ....... ... ... ... ... ... ... ..... ..... ... ... .. .. .. .. . ... n−1 . ... .. . n−2 ....... . . . . . . . . ...... ... ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .................................. .......... ............................................................... ..........

•

vn−1

e

• vn

  By virtue of Theorem 4.4 it suffices to prove that C ∗ (L˜ 2n−1 ) is isomorphic to C S02n−1 .
 Proposition A.1. There is a C ∗ -algebra isomorphism σ : C ∗ (L˜ 2n−1 ) → C S02n−1 such that  zi zi∗ , σ : Pvi → σ : Sfi →  zi zi zi∗ , σ : Sej →  zj zj∗+1 ,


for i = 1, . . . , n and j = 1, . . . , n − 1. The inverse σ −1 : C S02n−1 → C ∗ (L˜ 2n−1 ) is given by σ −1 : zn → Sfn , σ −1 : zk → Sfk + Sek σ −1 (zk+1 ), for k = n − 1, . . . , 1.

Quantum Spheres and Projective Spaces as Graph Algebras

187

Proof. Homomorphism σ exists by virtue of the universal property of C ∗ (L˜ 2n−1 ), since ˜ the target elements satisfy conditions (G1–G5) for
the graph  L2n−1 . Homomorphism 2n−1 −1 , since the target elements σ exists by virtue of the universal property of C S0 satisfy relations (4.1–4.4) with q = 0. One can easily check that σ ◦ σ −1 = id and  σ −1 ◦ σ = id.  Acknowledgements. We are grateful to Professors Nicola Ciccoli, Alain Connes, Joachim Cuntz, Piotr Hajac, Giovanni Landi, Wiesław Pusz, Joseph V´arilly and Stanisław Woronowicz for useful comments.

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25. Lance, E.Ch.: An explicit description of the fundamental unitary for SU (2)q . Commun. Math. Phys. 164, 1–15 (1994) 26. Lance, E.Ch.: The compact quantum group SO(3)q . J. Operator Theory 40, 295–307 (1998) 27. Masuda, T., Nakagami, Y., Watanabe, J.: Noncommutative differential geometry on the quantum SU (2), I: An algebraic viewpoint. K-Theory 4, 157–180 (1990) 28. Masuda, T., Nakagami, Y., Watanabe, J.: Noncommutative differential geometry on the quantum two sphere of Podle´s. I: An algebraic viewpoint. K-Theory 5, 151–175 (1991) 29. Nagy, G.: On the Haar measure of the quantum SU (N) group. Commun. Math. Phys. 153, 217–228 (1993) 30. Paterson, A.L.T.: Graph inverse semigroups, groupoids and their C ∗ -algebras. J. Operator Theory, to appear 31. Podle´s, P.: Quantum spheres. Lett. Math. Phys. 14, 193–202 (1987) 32. Podle´s, P.: Symmetries of quantum spaces. Subgroups and quotient spaces of quantum SU (2) and SO(3) groups. Commun. Math. Phys. 170, 1–20 (1995) 33. Pusz, W.: Twisted canonical anticommutation relations. Rep. Math. Phys. 27, 349–360 (1989) 34. Pusz, W., Woronowicz, S.L.: Twisted second quantization. Rep. Math. Phys. 27, 231–257 (1989) 35. Raeburn, I., Szyma´nski, W.: Cuntz-Krieger algebras of infinite graphs and matrices. Preprint, 1999 36. Sheu, A.J.-L.: Quantization of the Poisson SU (2) and its Poisson homogeneous space — the 2-sphere. Commun. Math. Phys. 135, 217–232 (1991) 37. Sheu, A.J.-L.: Compact quantum groups and groupoid C ∗ -algebras. J. Funct. Anal. 144, 371–393 (1997) 38. Sheu, A.J.-L.: Groupoids and compact quantum groups. In: Quantum Groups and Quantum Spaces (Warsaw 1995), Banach Center Publ. Vol. 40, Warsaw: Inst. Math. Polish Acad. Sci., 1997 39. Sheu, A.J.-L.: Groupoid approach to quantum projective spaces. In: Operator Algebras and Operator Theory (Shanghai, 1997), Contemp. Math. Vol. 228, Providence, RI: Am. Math. Soc., 1998, pp. 341–350 40. Sitarz, A.: More noncommutative 4-spheres. Lett. Math. Phys. 55, 127–131 (2001) 41. Szyma´nski, W.: On semiprojectivity of C ∗ -algebras of directed graphs. Proc. Amer. Math. Soc. 130, 1391–1399 (2002) 42. Vaksman, L.L., Soibelman, Y.S.: Algebra of functions on quantum SU (n + 1) group and odd dimensional quantum spheres. Algebra-i-Analiz 2, 101–120 (1990) 43. V´arilly, J.C.: Quantum symmetry groups of noncommutative spheres. Commun. Math. Phys. 221, 511–523 (2001) 44. Woronowicz, S.L.: Pseudospaces, pseudogroups and Pontryagin duality. In: Proceedings of the International Conference on Mathematical Physics (Lausanne 1979). Lecture Notes in Physics Vol. 116, Berlin: Springer-Verlag, 1980, pp. 407–412 45. Woronowicz, S.L.: Twisted SU (2) group. An example of a non-commutative differential calculus. Publ. Res. Inst. Math. Sci. 23, 117–181 (1987) 46. Woronowicz, S.L.: Tannaka-Krein duality for compact matrix pseudogroups. Twisted SU (N) groups. Invent. Math. 93, 35–76 (1988) 47. Woronowicz, S.L.: Quantum SU (2) and E(2) groups. Contraction procedure. Commun. Math. Phys. 149, 637–652 (1992) Communicated by A. Connes

Commun. Math. Phys. 232, 189–221 (2003) Digital Object Identifier (DOI) 10.1007/s00220-002-0702-7

Communications in

Mathematical Physics

Comments on the Stress-Energy Tensor Operator in Curved Spacetime Valter Moretti Department of Mathematices, Trento University, 38050 Povo (TN), Italy. E-mail: [email protected] Received: 24 September 2001/Accepted: 14 May 2002 Published online: 22 November 2002 – © Springer-Verlag 2002

Abstract: The technique based on a ∗-algebra of Wick products of field operators in curved spacetime, in the local covariant version proposed by Hollands and Wald, is strightforwardly generalized in order to define the stress-energy tensor operator in curved globally hyperbolic spacetimes. In particular, the locality and covariance requirement is generalized to Wick products of differentiated quantum fields. Within the proposed formalism, there is room to accomplish all of the physical requirements provided that known problems concerning the conservation of the stress-energy tensor are assumed to be related to the interface between the quantum and classical formalism. The proposed stress-energy tensor operator turns out to be conserved and reduces to the classical form if field operators are replaced by classical fields satisfying the equation of motion. The definition is based on the existence of convenient counterterms given by certain local Wick products of differentiated fields. These terms are independent from the arbitrary length scale (and any quantum state) and they classically vanish on solutions of the Klein-Gordon equation. Considering the averaged stress-energy tensor with respect to Hadamard quantum states, the presented definition turns out to be equivalent to an improved point-splitting renormalization procedure which makes use of the nonambiguous part of the Hadamard parametrix only that is determined by the local geometry and the parameters which appear in the Klein-Gordon operator. In particular, no extra added-byhand term gαβ Q and no arbitrary smooth part of the Hadamard parametrix (generated by some arbitrary smooth term “w0 ”) are involved. The averaged stress-energy tensor obtained by the point-splitting procedure also coincides with that found by employing the local ζ -function approach whenever that technique can be implemented. 1. Introduction In [1–3] the issue is addressed concerning the definition of Wick products of field operators (and time-ordered products of field operators) in curved spacetime and remarkable results are found (see Sect. 3). The general goal is the definition of the perturbative

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S-matrix formalism and corresponding renormalization techniques for self-interacting quantum fields in curved spacetime. The definition proposed by Hollands and Wald in [3] also assumes some locality and covariance requirements which (together with other properties) almost completely determine local Wick products. Some of the results on the Wick polynomials algebra presented in [3] are straightforward generalizations of Minkowski-spacetime results obtained by D¨utsch and Fredenhagen in [4]. A more general approach based on locality and covariance is presented in [5]. Using the machinery introduced in [3], a stress-energy tensor operator could be defined, not only its formal averaged value (also see [1] and comments in [2], where another definition of stress-energy operator was proposed in terms of a different definition of Wick products). However, the authors of [3] remark that such a stress-energy operator would not satisfy the conservation requirement. This paper is devoted to show that, actually, a natural (in the sense that it coincides with the classical definitions whenever operators are replaced by classical fields) definition of a well-behaved stress-energy tensor operator may be given using nothing but local Wick products defined by Hollands and Wald, provided one considers the pointed-out problem as due to the interface between the classical and quantum formalism. The way we follow is related to the attempt to overcome some known drawbacks which arise when one tries to define a natural point-splitting renormalization procedure for the stress-energy tensor averaged with respect to some quantum state. Let us illustrate these well-known drawbacks [6, 7]. Consider a scalar real field ϕ propagating in a globally hyperbolic spacetime (M, g). Assume that the field equations for ϕ are linear and induced by some Klein-Gordon operator P = − + ξ R + m2 and let ω be a quantum state of the quantized field ϕ. ˆ A widely studied issue is the definition of techniques which compute averaged (with respect to ω) products of pairs of field operators ϕˆ evaluated at the same event z. In practice, one is interested in formal objects like ϕ(z) ˆ ϕ(z) ˆ ω . The point-splitting procedure consists of replacing classical terms ϕ(z)ϕ(z) by some argument-coincidence limit of an integral kernel representing a suitable quantum two-point function of ω. (1) A natural choice involves the Hadamard two-point function1 , Gω (x, y), which is regular away from light-related arguments for Hadamard states (see 2.2). The cure for ultraviolet divergences which arise performing the argument-coincidence limit consists (1) of subtracting the “singular part” of Gω (x, y), (see 2.2), before taking the coincidence limit (x, y) → (z, z). This is quite a well-posed procedure if the state is Hadamard since, in that case, the singular part of the two-point functions is known by definition and is almost completely determined by the geometry and the K-G operator. The use of such an approach for objects involving derivatives of the fields, as the stress-energy tensor, turns out to be more problematic. The na¨ıve point-splitting procedure consists of the following limit: Tˆµν (z)ω =

lim

(x,y)→(z,z)

Dµν (x, y)[G(1) ω (x, y) − (Zn (x, y) + W (x, y))], (1)

where Zn is the expansion of the singular part of Gω in powers of the squared geodesic distance s(x, y) of x and y truncated at some sufficiently large order n, and Dµν (x, y) is a non-local differential operator obtained by point-splitting the the form of the stressenergy tensor [6, 7] (see (3) in 2.1). W is an added smooth function of x, y. 1 The use of the Hadamard function rather than the (Wightman) two-point functions is a matter of taste, since the final result does not depend on such a choice as a consequence of the bosonic commutation relations.

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That procedure turns out to be plagued by several drawbacks whenever D = dim(M) is even (D = 4 in particular). Essentially, (a) the produced averaged stress-energy tensor turns out not to be conserved (in contrast with Wald’s axioms on stress-energy tensor renormalization [7]) and (b) it does not take the conformal anomaly into account [7] which also arises employing different renormalization approaches [6]. (c) The choice of the term W turns out to be quite messy. Indeed, a formal expansion of W is known in terms of powers of the squared geodesic distance [8], but it is completely determined only if the first term W0 (x, y) of the expansion is given. However, it seems that there is no completely determined natural choice for W0 (see discussion and references in [8, 7, 9]). It is not possible to drop the term W if D is even. Indeed, an arbitrary length scale λ is necessary in the definition of Zn and changes of λ give rise to an added term W . As a minor difficulty we notice that (d) important results concerning the issue of the conservation of the obtained stress-energy tensor [8] required both the analyticity of the manifold and the metric in order to get convergent expansions for the singular part of (1) Gω . The traditional cure for (a) and (b) consists of by hand improving the prescription as Tˆµν (z)ω =

lim

(x,y)→(z,z)

Dµν (x, y)[G(1) ω (x, y) − (Zn (x, y) + W (x, y))] + gµν (z)Q(z), (1)

where Q is a suitable scalar function of z determined by imposing the conservation of the final tensor field. A posteriori, Q seems to be determined by the geometry and P only. Coming back to the stress-energy operator, one expects that any conceivable definition should produce results in agreement with the point-splitting renormalization procedure, whenever one takes the averaged value of that operator with respect to any Hadamard state ω. However, the appearance of the term Q above could not allow a definition in terms of local Wick products of field operators only. In Sect. 2 we prove that it is possible to “clean up” the point-splitting procedure. In fact, we suggest an improved procedure which, preserving all of the relevant physical results, is not affected by the drawbacks pointed out above. In particular, it does not need added-by-hand terms as Q, employing only mathematical objects completely determined by the local geometry and the operator P . The ambiguously determined term W (not only the first term W0 of its expansion) is dropped, barring the part depending on λ as stressed above. Finally, no analyticity assumptions are made. Our prescription can be said to be “minimal” in the sense that it uses the local geometry and P only. The only remaining ambiguity is a length scale λ. We also show that the presented prescription produces the same renormalized stress-energy tensor obtained by other definitions based on the Euclidean functional integral approach. In Sect. 3 we show that the improved procedure straightforwardly suggests a natural form of the stress-energy tensor operator written in terms of local Wick products of operators which generalize those found in [2, 3]. This operator is conserved, reduces to the usual classical form, whenever field operators are replaced by classical fields satisfying the field equation, and agrees with the point-splitting result if one takes the averaged value with respect to any Hadamard state. To define the stress-energy tensor operator as an element of a suitable ∗-algebra of formal operators smeared by functions of D(M), we need to further develop the formalism introduced in [3]. This is done in the third section, where we generalize the notion of local Wick products given in [3] to differentiated local Wick products proving some technical propositions.

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Concerning notations and conventions, throughout the paper a spacetime, (M, g), is a connected D-dimensional smooth (Hausdorff, second-countable) manifold with D ≥ 2 and equipped with a smooth Lorentzian metric g (we adopt the signature −, + · · · , +).  denotes the Laplace-Beltrami-D’Alembert operator on M, locally given by ∇µ ∇ µ , ∇ being the Levi-Civita covariant derivative associated with the metric g. A spacetime is supposed to be oriented, time oriented and in particular globally hyperbolic [10] also if those requirements are not explicitly stated. Throughout µg denotes the natural positive √ measure induced by the metric on M and given by −g(x)dx 1 ∧ · · · ∧ dx n in each coordinate patch. The divergence, ∇ · T , of a tensor field T is defined by (∇ · T )α... β... = ∇µ T µ α... β... = ∇ µ Tµ α... β... in each coordinate patch. Finally, throughout the paper, “smooth” means C ∞ .

2. Cleaning up the Point-Splitting Procedure 2.1. Classical framework. Consider a smooth real scalar classical field ϕ propagating in a smooth D- dimensional globally hyperbolic spacetime (M, g). P ϕ = 0 is the equation of motion of the field, the Klein-Gordon operator P being P = − + ξ R(x) + V (x) = − + m2 + ξ R(x) + V  (x), def

def

(2)

where ξ ∈ R is a constant, R is the scalar curvature, m2 ≥ 0 is the mass of the field and V  : M → R is any smooth function. The symmetric stress-energy tensor, obtained by variational derivative with respect to the metric of the action2 [10], reads
 Tαβ (x) = ∇α ϕ(x)∇β ϕ(x) − 21 gαβ (x) ∇γ ϕ(x)∇ γ ϕ(x) + ϕ 2 (x)V (x) 
  + ξ Rαβ (x) − 21 gαβ (x)R(x) + gαβ (x) − ∇α ∇β ϕ 2 (x).

(3)

Concerning the “conservation relation” of Tαβ (x), if P ϕ = 0, a direct computation leads to 1 ∇ α Tαβ (x) = − ϕ 2 (x)∇β V  (x). 2

(4)

It is clear that the right-hand side vanishes provided V  ≡ 0 and (4) reduces to the proper conservation relation. The trace of the stress-energy tensor can easily be computed in terms of ϕ 2 (x) only. In fact, for P ϕ = 0, one finds gαβ (x)T

αβ

 ξD − ξ  − V (x) ϕ 2 (x), (x) = 4ξD − 1 

(5)

where ξD = (D − 2)/[4(D − 1)] defines the conformal coupling: For ξ = ξD , if V ≡ 0 and m = 0, the action of the field ϕ turns out to be invariant under local conformal transformations (g(x) → λ(x)g(x), ϕ(x) → λ(x)1/2−D/4 ϕ(x)) and the trace of Tαβ (x) vanishes on field solutions by (5). 2 For ξ = 1/6, in (four dimensional) Minkowski spacetime this tensor coincides with the so-called “new improved” stress-energy tensor [11].

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2.2. Hadamard quantum states and Hadamard parametrix. From now on A(M, g) denotes the abstract ∗-algebra with unit 1 generated by 1 and the abstract field operators ϕ(f ) smeared by the functions of D(M) := C0∞ (M, C). The abstract field operators enjoy the following properties where f, h ∈ D(M) and E is the advanced-minus-retarded fundamental solution of P which exists in globally hyperbolic spacetimes [12]. (a) Linearity: f → ϕ(f ) is linear, (b) Field equation: ϕ(Pf ) = 0, (c) CCR: [ϕ(f ), ϕ(h)] = E(f ⊗h)1, E being the advanced-minus-retarded bi-solution [12], (d) Hermiticity: ϕ(f ) = ϕ(f )∗ . An algebraic quantum state ω : A(M, g) → C on A(M, g) is a linear functional which is normalized (ω(1) = 1) and positive (ω(a ∗ a) ≥ 0 for every a ∈ A(M, g)). The GNS theorem [13] states that there is a triple (Hω , +ω , ,ω ) associated with ω. Hω is a Hilbert space with scalar product , ω . +ω is a ∗-algebra representation of A(M, g) which takes values in a ∗-algebra of unbounded operators defined on the dense invariant linear subspace Dω ⊂ Hω 3 . The distinguished vector ,ω ∈ Hω satisfies both +ω (A(M, g)),ω = Dω and ω(a) = ,ω , +ω (a),ω ω for every a ∈ A(M, g). Different GNS triples associated to the same state are unitary equivalent. From now on, ϕˆω (f ) denotes the closeable field operator +ω (ϕ) and Aω (M, g) denotes the ∗-algebra +ω (A(M, g)). Wherever it does not produce misunderstandings we write ϕˆ instead of ϕˆω and ,  instead of , ω . The Hadamard two-point function of ω is defined by (+) G(1) ω = Re Gω , def

(6)

(+)

Gω being the two-point function of ω, i.e., the bilinear map on D(M) × D(M) G(+) ˆ )ϕ(g), ˆ ω . ω : f ⊗ g → ω(ϕ(f )ϕ(g)) = ,ω , ϕ(f We also assume that ω is globally Hadamard [12, 7, 14], i.e., it satisfies the (+) Hadamard requirement. Gω ∈ D (M × M) and takes the singularity structure of the (global) Hadamard form in a causal normal neighborhood N of a Cauchy surface / of M. (+) (+) In other words, for n = 0, 1, 2, . . . , the distributions Gω − χ Zn ∈ D (M × M), (+) n can be represented by functions of C (N × N ). Zn is the Hadamard parametrix truncated at the order n and defined on test functions supported in Cz × Cz for every z ∈ M, Cz being a convex normal neighborhood of z (see [12, 16] for the definition of χ and N ). Since we are interested in the local behavior of the distributions we ignore the smoothing fuction χ in the following because χ (x, y) = 1 if x is sufficiently close to y. The propagation of the global Hadamard structure in the whole spacetime [15] (see [12, 7, 16]) entails the independence of the definition of Hadamard state from /, N and χ . It (1) (+) (1) also implies that Gω (as well as Gω ) is a smooth function, (x, y) → Gω (x, y) away from the subset of M × M made of the pairs of points x, y such that either x = y or they are light-like related. If Cz is a convex normal neighborhood of z, using the Hadamard condition and the content of Appendix A, one proves that Re(Z (+) ) is represented by a 3

The involution being the adjoint conjugation on Hω followed by the restriction to Dω .

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smooth kernel Z(x, y) if s(x, y) = 0 and the map (x, y) → Gω (x, y) − Zn (x, y) can be continuously extended into a function of C n (Cz × Cz ). For s(x, y) = 0, U (x, y) |s(x, y)| (1) (2) Zn (x, y) = βD D/2−1 if D is even, (7) + βD V (n) (x, y) ln s (x, y) λ2 T (n) (x, y) (1) Zn (x, y) = βD θ (s(x, y)) D/2−1 if D is odd. (8) s (x, y) θ (x) = 1 if x > 0 and θ (x) = 0 otherwise. The smooth real-valued functions U, V (n) , T (n) are defined by recursive (generally divergent) expansions in powers of the (signed) squared geodesical distance s(x, y) and are completely determined by the (i) metric and the operator P . βD are numerical coefficients. λ > 0 is an arbitrarily fixed length scale. Details are supplied in Appendix A. 2.3. Classical ambiguities and their relevance on quantum ground. Let us consider the point-splitting procedure introduced in Sect. 1.1 by (1). The differential operator Dµν (x, y) (written in (10) below putting η = 0 therein) is obtained by point-splitting the classical expression for the stress-energy tensor (3) [6, 7]. The crucial point is that the classical stress-energy tensor may be replaced by a classically equivalent object which, at the quantum level, breaks such an equivalence. In particular, classically, we may re-define (η) Tµν (z) = Tµν (z) + η gµν (z) ϕ(z)P ϕ(z), def

(9)

where η ∈ R is an arbitrarily fixed pure number and Tµν (z) is given by (3). It is obvious (η) that Tµν (z) = Tµν (z) whenever ϕ satisfies the field equation P ϕ = 0. Therefore, there is no difference between the two tensors classically speaking and no ambiguity actually takes place through that way. On quantum ground things dramatically change since ϕ(x)P ˆ ϕ(x) ˆ ω = 0, provided the left-hand side is defined via the point-splitting procedure (see also [3] where the same remark appears in terms of local Wick polynomials). Therefore the harmless classical ambiguity becomes a true quantum ambiguity. Actually, we argue that, without affecting the classical stress-energy tensor, the ambiguity we found can be used to clean up the point-splitting procedure. By this way, the general principle “relevant quantum objects must reduce to corresponding well-known classical objects in the formal classical limit, i.e., when quantum observables are replaced by classical observables”, is preserved. (η) The operator used in the point-splitting procedure corresponding to Tµν is obtained by means of a point-separation and symmetrization of the right-hand side of (3) and (9). It reads     def 1 β β (η) D(z)αβ (x, y) = δαα (z, x)δβ (z, y)∇(x)α  ∇(y)β  + δαα (z, y)δβ (z, x)∇(y)α  ∇(x)β  2   1  µ − gαβ (z) g γ γ (z)δ(z, x)γ  δ(z, y)νγ ∇(x)µ ∇(y)ν + V (z) 2 

 gαβ (z)
1 + ξ Rαβ (z) − gαβ (z)R(z) + x + y 2 2   1  α  β β − δα (z, x)δβ (z, x)∇(x)α  ∇(x)β  + δαα (z, y)δβ (z, y)∇(y)α  ∇(y)β  2   gαβ (z)
+η (10) Px + Py , 2

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δ(v, u) is the operator of the geodesic transport from Tu M to Tv M. We aim to show that there is a choice for η, ηD , depending on the dimension of the spacetime manifold D only, such that (ηD ) Tˆµν (z)ω =

def

(η )

lim

(x,y)→(z,z)

D D(z)µν (x, y)[G(1) ω (x, y) − Zn (x, y)],

(11)

is physically well behaved. To this end a preliminary lemma is necessary. 2.4. A crucial lemma. The following lemma plays a central rˆole in the proof of Theorem 2.1 concerning the properties of the new point-splitting prescription. The coefficients of the expansion of U in (7), Uk (z, z), which appear below are defined as in Appendix A. Lemma 2.1. In a smooth D-dimensional (D ≥ 2) spacetime (M, g) equipped with the differential operator P in (2), the associated Hadamard parametrix (7), (8) satisfies the following identities, where the limits hold uniformly. (a) If n ≥ 1, lim

(x,y)→(z,z)

Px Zn (x, y) =

lim

(x,y)→(z,z)

Py Zn (x, y) = δD cD UD/2 (z, z).

(12)

Above δD = 0 if D is odd and δD = 1 if D is even and def

cD = (−2)D/2−1

D+2 , (4π )D/2

(13)

(b) If D is even and n ≥ 1 or D is odd and n > 1, lim

(x,y)→(z,z)

µ

Px ∇(y) Zn (x, y) =

lim

(x,y)→(z,z)

µ

µ

∇(x) Py Zn (x, y) = δD kD ∇(z) UD/2 (z, z) (14)

with def

kD = (−2)D/2−1

D . 2(4π )D/2

(15)

(c) Using the point-splitting prescription to compute ϕ(z)P ˆ ϕ(z) ˆ ˆ ϕ(z) ˆ ω and P (ϕ(z)) ω,  def lim P(y) G(1) ϕ(z)P ˆ ϕ(z) ˆ ω = ω (x, y) − Zn (x, y) = −δD cD UD/2 (z, z), (x,y)→(z,z)



def

(P ϕ(z)) ˆ ϕ(z) ˆ ω =

lim

(x,y)→(z,z)



P(x) G(1) ω (x, y) − Zn (x, y)

(16) = −δD cD UD/2 (z, z). (17)

In particular ϕ(z)P ˆ ϕ(z) ˆ ˆ ϕ(z) ˆ ω = P (ϕ(z)) ω. Proof. See Appendix B.

 

Remark . With our conventions, when D is even, the anomalous quantum correction to the trace of the stress-energy tensor is −2cD UD/2 (z, z)/(D +2) [20] (and coincides with the conformal anomaly if V ≡ 0, ξ = ξD in (2)). Notice that the coefficients Uk (z, z) do not depend on either ω and the scale λ used in the definition of Zn . We conclude that ϕ(z)P ˆ ϕ(z) ˆ ω (i) does not depend on the scale λ, (ii) does not depend on ω and (iii) is proportional to the anomalous quantum correction to the trace of the stress-energy tensor.

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2.5. The improved point-splitting procedure. Let us show that the point-splitting procedure (11) produces a renormalized stress-energy tensor which is well behaved and in agreement with Wald’s four axioms (straightforwardly generalized to the case V  ≡ 0 when necessary) for a particular value of η uniquely determined. Theorem 2.1. Let ω be a Hadamard quantum state of a field ϕ on a smooth globally-hyperbolic D-dimensional (D ≥ 2) spacetime (M, g) with field operator (2). If (η) D(z)µν (x, y) is given by (10), consider the symmetric tensor field and the scalar field locally defined by def

(η) z → Tˆµν (z)ω =

def

z → ϕˆ 2 (z)ω,λ =

(η)

lim

D(z)µν (x, y)[G(1) ω (x, y) − Zn (x, y)],

(18)

lim

[G(1) ω (x, y) − Zn (x, y)],

(19)

(x,y)→(z,z) (x,y)→(z,z)

where, respectively, n ≥ 3 and n > 0. The following statements hold. (η)

(a) Both z → Tˆµν (z)ω and z → ϕˆ 2 (z)ω are smooth and do not depend on n. Moredef

over, if (and only if ) η = ηD = D[2(D + 2)]−1 , they satisfy the analogue of (4) for all spacetimes 1 (ηD ) ∇ µ Tˆµν (z)ω = − ϕˆ 2 (z)ω ∇ν V  (z). 2

(20)

(η )

(b) Concerning the trace of TˆµνD (z)ω , it holds   2cD ξD − ξ (ηD ) g µν(z) Tˆµν  − V (x) ϕˆ 2 (z)ω − δD UD/2 (z, z), (21) (z)ω = 4ξD − 1 D+2 The term on the last line does not depend on the scale λ > 0 used to define Zn and coincides with the conformal anomaly for ξ = ξD , V ≡ 0. def

(c) If D is even, η ∈ R, Qη,ηD (z) = δD (η − ηD )cD UD/2 (z, z), (ηD ) (η) Tˆµν (z)ω = Tˆµν (z)ω + gµν (z)Qη,ηD (z)

holds. (d) Changing the scale λ → λ > 0 one has, with obvious notation, 

λ (ηD ) (ηD ) ˆ ˆ Tµν (z)ω,λ − Tµν (z)ω,λ = δD ln tµν (z), λ

(22)

(23)

where the smooth symmetric tensor field t is independent from either the quantum state, λ and λ , is conserved for V  ≡ 0 and it is built up, via standard tensor calculus, by employing the metric and the curvature tensors at z, m, ξ , V  (z) and their covariant derivatives at z. (e) If (M, g) is the (D = 4) Minkowski spacetime, V  ≡ 0 and ω is the Minkowski (η ) vacuum, there is λ > 0 such that TˆµνD (z)ω,λ = 0 for all z ∈ M. If m = 0 this holds for every λ > 0. Proof. See Appendix B.

 

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Def. 2.1 (Quantum averaged stress-energy tensor and field fluctuation). Let ω be a Hadamard quantum state of a field ϕ in a smooth globally-hyperbolic D-dimensional (D ≥ 2) spacetime (M, g) with field operator (2). Referring to Theorem 2.1, the tensor def (η ) field defined in local coordinates by z → Tˆµν (z)ω = TˆµνD (z)ω and the scalar field z → ϕˆ 2 (z)ω , are respectively called the quantum averaged stress-energy tensor in the state ω and the quantum field fluctuation of the state ω. Remarks. (1) The point-splitting renormalization defined above turns out to be in agreement with Wald’s axioms. This can be realized by following the same discussion, developed in [7] concerning the standard point-splitting prescription and using the theorem above. (2) The need of adding a term to the classical stress-energy tensor to fulfill the conservation requirement can be heuristically explained as follows. As in [6], let us assume that there is some functional of the metric corresponding to the one-loop effective action:

def Sω [g] = i ln Dg ϕ e−iS[ϕ,g] , where S is the classical action associated with P , and ω enters the assignment of the integration domain. In this context, the averaged stress-energy tensor is defined as 2 δSω [g] Tˆµν (z)ω = − √ , −g(z) δg µν (z) where the functional derivative is evaluated at the actual metric of the spacetime. The conservation of the left-hand side is equivalent to the (first order) invariance under diffeomorphisms of Sω [g]. The relevant point is that the measure Dg ϕ in general must be supposed to depend on the metric [17, 6, 18]. Changing g into g by a diffeomorphism, one gets, assuming the invariance of Sω [g] and making explicit the dependence of the measure on the metric  

2 δJ [ϕ, g, g ] µ  0 = Dg ϕ ∇ µ √ − i∇ T (z) e−iS[ϕ,g] , | µν g =g −g(z) δg  µν (z) where J [ϕ, g, g ]Dg ϕ = Dg ϕ, J [ϕ, g, g] = 1. The conserved quantity is a term corresponding to the classical stress-energy tensor added to a further term depending on the functional measure  

2eiSω [g] δJ [ϕ, g, g ] (η=0) −iS[ϕ,g]  ∇ µ Tˆµν = 0. (z)µ + i √ e | Dg ϕ g =g δg  µν (z) −g(z) Therefore the term we found, ηD gµν (z)ϕ(z)P ϕ(z)ω , added to the classical stressenergy tensor should be related to the second term in the brackets above. (3) The functional approach can be implemented via Wick rotation in the case of a static spacetime with compact Cauchy surfaces for finite temperature (1/β) states and provided V does not depend on the global Killing time. Within that context, the (1) Euclidean section turns out to be compact without boundary and Gω has to be replaced with the unique Green function Gβ , with Euclidean Killing temporal period β, of the operator obtained by Wick rotation of P . One expects that the following identity holds: −√

  δSE,β [g] 2 (ηD ) D(z)ab (x, y) Gβ (x, y) − Zn (x, y) , = lim ab (x,y)→(z,z) −g(z) δg (z)

(24)

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where we have replaced Lorentzian objects by corresponding Euclidean ones and a, b denote tensor indices in a Euclidean manifold. In a sense, (24) can actually be rigorously proven as stated in the theorem below. Indeed, the point-splitting procedure in the right-hand side can be implemented also in the Euclidean case because the parametrices Zλ,n (λ being the length scale used in the definition of the parametrices) can be defined also for Euclidean metrics using the same definition given above, omitting θ (s(x, y)) in (8) and dropping | | in the logarithm in (7). On the other hand, the left-hand side of (24) may be interpreted, not depending on the right-hand side, as an Euclidean ζ -function (ζ ) regularized stress-energy tensor Tab (z)β,µ2 which naturally introduces an arbitrary mass scale µ (see [20] where σ (x, y) indicates s(x, y)/2). We remind the reader that, in the same hypotheses, it is possible to define a ζ -function regularization of the field (ζ ) fluctuation, ϕˆ 2 (z)β,µ2 (see [20] and references therein). Theorem 2.2. Let (M, g) be a smooth spacetime endowed with a global Killing time-like vector field normal to a compact Cauchy surface and a Klein-Gordon operator P in (2), where V  does not depend on the Killing time. Consider a compact Euclidean section of the spacetime (Mβ , gE ) obtained by (a) a Wick analytic continuation with respect to the Killing time and (b) an identification of the Euclidean time into Killing orbits of period β > 0. Let Gβ be the unique Green function of the Euclidean Klein-Gordon operator defined on C ∞ (Mβ ) obtained by analytic continuation of P and assumed to be strictly positive. The equations (ζ )

Tab (z)β,µ2 = Tab (z)β,λ , (ζ ) ϕˆ 2 (z)β,µ2

= ϕ 2 (z)ω,λ

(25) (26)

hold, where λ = cµ−1 , c > 0 being some constant on the right-hand sides, and the (1) right-hand sides of (25) and (26) are defined as in Def. 2.1, using Gβ in place of Gω and the Euclidean parametrix. Sketch of proof. The left-hand side of (25) coincides with (ν )

Tˆab D (z)ω,λ + gab (z)QνD ,ηD (z), where νD = (D−2)/(2D), as shown in Theorem 4.1 of [20] provided (using  = c = 1) λ coincides with µ−1 with a suitable positive constant factor. (The smooth term W added to the parametrix which appears in the cited theorem can be completely re-absorbed in the logarithmic part of the parametrix as one can directly show). Equation (22) holds true also in the Euclidean case as one can trivially show and thus the thesis is proven. The proof of (26) is similar.   3. The Stress-Energy Operator in Terms of Local Wick Products [1–3] contain very significant progress in the definition of perturbative quantum field theory in curved spacetime. Those works take advantage of the methods of microlocal analysis [21] and the wave front set characterization of the Hadamard requirement found out by Radzikowski [14]. In [1] it is proven that, in the Fock space generated by a quasifree Hadamard state, a definition of Wick polynomials (products of field operators evaluated at the same event) can be given with a well-defined meaning of operator-valued distributions. That is obtained by the introduction of a normal ordering prescription

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with respect to a chosen Hadamard state. In the subsequent paper [2], it is shown that quantum field theory in curved spacetime gives rise to “ultraviolet divergences” which are of the same nature as in Minkowski spacetime. This result is achieved by a suitable generalization of the Epstein-Glaser method of renormalization in Minkowski spacetime used to analyze time ordered products of the Wick polynomial, involved in the perturbative construction of interacting quantum field theory. However the performed analysis shows that quantities which appear at each perturbation order in Minkowski spacetime as a renormalized coupling constant are replaced, in curved spacetime, by functions whose dependence upon the spacetime points can be arbitrary. In [3] generalizing the content of [4] and using ideas of [1, 2], it is found that such ambiguity can be reduced to finitely many degrees of freedom by imposing a suitable requirement of covariance and locality (which is an appropriate replacement of the condition of Poincar´e invariance in Minkowski spacetime). The key step is a precise notion of a local, covariant quantum field. In fact, a definition of local Wick products of field operators in agreement with the given definition of local covariant quantum field is stated. Imposing further constraints concerning scaling behavior, appropriate continuity properties and commutation relations, two uniqueness theorems are presented about local Wick polynomials and their time-ordered products. The only remaining ambiguity consists of a finite number of parameters. Hollands and Wald also sketch a proof of existence of local Wick products of field operators in [3]. The local Wick products they found make use of the Hadamard parametrix only and turn out to be independent from any preferred Hadamard vacuum state. In principle, by means of a straightforward definition to local Wick products of a differentiated field, these local Wick products may be used to define a well-behaved notion of the stress-energy tensor operator. However, as remarked in [3] such a definition would produce a non-conserved stress-energy tensor. In this section, after a short review of the relevant machinery developed in [3], we prove how such a problem can be overcome generalizing ideas of Sect. 2. 3.1. Normal products and the algebra W(M, g). From now on, referring to a globally hyperbolic spacetime (M, g) equipped with a Klein-Gordon operator (2), we assume dim(M) = 4 and V  ≡ 0 in (2). In the following, for n = 1, 2, . . . , D(M n ) denotes the space of smooth compactly-supported complex functions on M n and Dn (M) ⊂ D(M n ) indicates the subspace containing the functions which are symmetric under interchange of every pair of arguments. In the remaining part of the work we make use of some mathematical tools defined in microlocal analysis. (See Chap. VIII of [21] concerning the notion of wave front set and [14] concerning the microlocal analysis characterization of the Hadamard requirement.) Preserving the usual seminorm-induced topology on D(M), all definitions and theorems about distributions u ∈ D (M) (Chap. VI of [21]) can straightforwardly be re-stated for vector-valued distributions and in turn, partially, for operator-valued distributions on D(M). That is, respectively, continuous linear maps v : D(M) → H, H being a Hilbert space, and continuous linear maps A : D(M) → A, A being a space of operators on H (with common domain) endowed with the strong Hilbert-space topology. The content of Chap. VIII of [21] may straightforwardly be generalized to vector-valued distributions. In this part we consider quasifree [12, 7] states ω. Referring to 2.2, this means that the n-point functions are obtained by functionally differentiating with respect to f the formal identity 1

ω(eiϕ(f ) ) = e− 2 ω(ϕ(f )ϕ(f )) .

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In that case there is a GNS Hilbert space Hω which is a bosonic Fock space, ,ω ∈ Hω is the vacuum vector therein, operators ϕ(f ˆ ) are essentially self-adjoint on Dω if f ∈ D(M) is real and Weyl’s relations are fulfilled by the one-parameter groups generated by operators ϕ(f ˆ ). Let us introduce normal Wick products defined with respect to a reference quasifree Hadamard state ω [2, 3]. Fix a GNS triple for ω, (Hω , +ω , ,ω ) and consider the algebra of operators with domain Dω , Aω (M, g) (see 2.2). From now on, we write ϕˆ instead of ϕˆω whenever it does not give rise to misunderstandings. For n ≥ 1, define the symmetric operator-valued linear map, Wˆ ω,n : Dn (M) → Aω (M, g), given by the formal symmetric kernel Wˆ ω,n (x1 , . . . , xn ) def

=: ϕ(x ˆ 1 ) · · · ϕ(x ˆ n ) :ω       1 ω(x,y)f (x)f (y)dµg (x)dµg (y)+i ϕ(z)f ˆ (z)dµg (z)  n n 2   1 def   δ e =  i(−g(xj ))1/2 δf (x1 ) · · · δf (xn )  j =1

, f ≡0

(27) where the result of the formal functional derivative is supposed to be symmetrized, and (1) thus only the symmetric part of ω, i.e., Gω , takes place in (27). ϕ(x) ˆ is the formal def kernel of ϕ(= ˆ ϕˆω ), ω(x, y) is the formal kernel of ω. Finally define Wˆ ω,0 = I the unit of Aω (M, g). The operators Wˆ ω,n (h) can be extended (or directly defined) [2, 3] to a dense invariant subspace of Hω , the “microlocal domain of smoothness” [3], Dω ⊃ Dω , which is contained in the self-adjoint extension of every operator ϕ(f ˆ ) smeared by real f ∈ D(M).4 From now on we assume that every considered operator is defined on Dω Dω enjoys two relevant properties. (a) Every map h → Wˆ ω,n (h), h ∈ Dn (M), defines a symmetric operator-valued distribution. (b) Those operator-valued distributions may give rise to operators which can be interpreted as products of field operators evaluated at the same event. This is because every Wˆ ω,n can be smeared by a suitable class of distributions and, in particular, Wˆ ω,n (f δn ) can be interpreted as :ϕˆ n (f ) :ω if f ∈ D(M) and δn is the distribution:

def h(x1 , . . . , xn )δn (x1 , . . . , xn )dµg (x1 ) · · · dµg (xn ) = h(x, x, . . . , x)dµg (x). M

M

Let us summarize the proof of this remarkable result following [3]. By Lemma 2.2 in [2], if B ∈ Dω the wave front set of the vector-valued distributions t → Wˆ ω,n (t)B, W F Wˆ ω,n (·)B [21], is contained in the set Fn (M, g) = {(x1 , k1 , . . . , xn , kn ) ∈ (T ∗ M)n \ {0}|ki ∈ Vx−i , i = 1, . . . , n}, (28) def

4 Therefore, Weyl’s commutation relations, and thus bosonic commutation relations on D , are ω preserved.

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+/−

Vx denoting the set of all nonzero time-like and null co-vectors at x which are future/past directed. Theorem 8.2.10 in [21] states that if the wave front sets of two distributions u, v ∈ D (N ), N being any manifold, satisfy W F (u) + W F (v)  {0}, then a pointwise product between u and v, u  v can be unambiguously defined giving rise to a distribution of D (N ). The theorem can be straightforwardly generalized to vector-valued distributions. In our case we are allowed to define the product between a distribution t and a vector-valued distribution Wˆ ω,n (·)B provided W F (t) + Fn (M, g)  {0}. To this end define  def  En (M, g) = t ∈ Dn (M) | supp t is compact, W F (t) ⊂ Gn (M, g) , where def

∗

Gn (M, g) = T M

n

 x∈M

(Vx+ )n

∪

 x∈M

 (Vx− )n

.

W F (t) + Fn (M, g)  {0} for t ∈ En (M, g) holds. By consequence the product, t  ˆ (·)B can be defined for every B ∈ Dω and it Wˆ ω,n B, of the distributions  t and Wω,n  ˆ is possible to show that t  Wω,n B (f ) ∈ Dω for every f ∈ Dn (M). In turn, varying B ∈ Dω , one straightforwardly gets a well-defined operator-valued distribution t  Wˆ ω,n . Summarizing: if t ∈ En (M, g), n ∈ N, thereis a well-defined operator-valued symmetric distribution Dn (M)  f → t  Wˆ ω,n (f ), with values defined in the dense invariant domain Dω . To conclude we notice that if t ∈ En (M, g), Wˆ ω,n can be smeared by t making use of the following definition. Since, for all B ∈ Dω , supp (t  Wˆ ω,n B) ⊂ supp t 5 , take f ∈ Dn (M) such that f (x1 , . . . , xn ) = 1 for (x1 , . . . , xn ) ∈ supp t and define the operator, with domain Dω ,   def Wˆ ω,n (t) = t  Wˆ ω,n (f ). It is simply proven that the definition does not depend on the f used and the new smearing operation becomes the usual one for t ∈ Dn (M) ⊂ En (M, g). Finally, since f δn ∈ En (M, g) if f ∈ D(M), the following operator-valued distribution is welldefined on Dω , def f → :ϕˆ n (f ) :ω = Wˆ ω,n (f δn ).

: ϕˆ n (f ) :ω is called the normal ordered product of n field operators with respect to ω. Generalized normal ordered Wick products of k fields, :ϕˆ n1 (f1 ) · · · ϕˆ nk (fk ) :ω are similarly defined [3]. Given a quasifree Hadamard state ω and a GNS representation, Wω (M, g) is the ∗-algebra generated by I and the operators Wˆ ω,n (t) for all n ∈ N and t ∈ En (M, g) with 5 It can be shown using the continuity of the product with respect to the H¨ ormander pseudo-topology and Theorem 6.2.3 of [21] which assures that each distribution is the limit in that pseudo-topology of a sequence of smooth functions and the fact that the convergence in the pseudo-topology implies the usual convergence in D .

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def involution given by Wˆ ω,n (t)∗ = Wˆ ω,n (t)†
Dω (= Wˆ ω,n (t)). Aω (M, g) turns out to be a sub ∗-algebra of Wω (M, g) since one finds that ϕˆω (f ) = :ϕ(f ˆ ) :ω for f ∈ D(M). Different GNS triples for the same ω give rise to unitary equivalent algebras Wω (M, g) by GNS’s theorem. However, if ω, ω are two quasifree Hadamard states, Wω (M, g), Wω (M, g) are isomorphic (not unitary in general) under a canonical ∗-isomorphism αω ω : Wω (M, g) → Wω (M, g), as shown in Lemma 2.1 in [3]. These ∗-isomorphisms also satisfy, αω ω ◦ αω ω = αω ω and αω ω (ϕˆω (t)) = ϕˆω (t), but in general, for n > 1, αω ω (:ϕˆ n (t) :ω ) = :ϕˆ n (t) :ω . One can define an abstract ∗-algebra W(M, g), isomorphic to each ∗-algebra Wω (M, g) by ∗-isomorphisms αω : W(M, g) → Wω (M, g) such that, if ω, ω are quasifree Hadamard states, αω ◦ αω−1 = αω ω . As above A(M, g) is ∗-isomorphic to a sub ∗-algebra of W(M, g) and αω (ϕ(t)) = :ϕ(t) ˆ :ω . Elements Wω,n (t) and :ϕ n (f ) :ω are defined in W(M, g) via (27).

3.2. Local Wick products. Following [3], a quantum field in one variable E is an assignment which associates with every globally hyperbolic spacetime (M, g) a distribution E[g] taking values in the algebra W(M, g). E, is called local and covariant [3] if it satisfies the following Locality and Covariance requirement. For any embedding χ from a spacetime (N, g ) into another spacetime (M, g) which is isometric (thus g = χ ∗ g) and causally preserving6 , iχ (E[g ](f )) = E[g](f ◦ χ −1 ) for all f ∈ D(N ) holds. Above iχ : W(N, g ) → W(M, g) is the injective ∗-algebra homomorphism such that if ω is a quasifree Hadamard state on (M, g) and ω (x, y) = ω(χ (x), χ (y)), we have, iχ (Wω ,n (t)) = Wω,n (t ◦ χn−1 )

for all n ∈ N, t ∈ En (N, g ), def

(29)

where χ −1 is defined on χ (N ) and (t ◦ χn−1 )(x1 , . . . , xn ) = t (χ −1 (x1 ), . . . , χ −1 (xn )). The generalization to a (locally and covariant) quantum field in n-variables is straightforward. It is worth stressing that the notion of local covariant field is not trivial. For instance, any assignment of the form (M, g) → ω(M, g), where ω(M, g) are quasifree Hadamard states, does not define a local covariant quantum field by the map (M, g) → : ϕ 2 : ω(M,g) [3]. In [3], Hollands and Wald sketched a proof of existence of local and covariant quantum fields in terms of local Wick products of field operators. Let us review the construction of these Wick products also making some technical improvements. As M is strongly causal [24, 10], there is a topological base of open sets N such that each N is contained in a convex normal neighborhood, each inclusion map i : N → M is causally preserving and each N is globally hyperbolic with respect to the induced metric. We call causal domains these open neighborhoods N . Let N ⊂ M be a causal domain. The main idea to build up local Wick products [3] consists of a suitable use of the Hadamard parametrix which is locally and covariantly defined in the globally hyperbolic spacetime (N, g
N ) in terms of the metric [3]. In fact, it is possible to define a suitable distribution H ∈ D (N × N ) such that 6

That is χ preserves the time orientation and J + (p) ∩ J − (q) ⊂ χ(N) if p, q ∈ χ (N ).

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every distribution H − Zn
N×N is a function of (x, y) ∈ N × N which is smooth for x = y and with vanishing derivatives for x = y up to the order n. Then define the def

elements of W(N, g
N ), WH,0 = 1 and WH,n given by (27) with ω replaced by H and ϕˆ replaced by ϕ ∈ W(N, g
N ). These distributions enjoy the same smoothness (1) properties of Wˆ ω,n for every quasifree Hadamard state ω because Gω
N×N − H = (1) (Gω − Zn )
N×N − (H − Zn
N×N ) is smooth on N × N and all of its derivative (of any order) must vanish at x = y since n is arbitrary. In particular every WH,n can be smeared by distributions of En (N, g
N ). The local Wick products (on N ) found by Hollands and Wald in [3] are the elements of W(N, g
N ) of the form, with f ∈ D(N ), : ϕ n (f ) :H = WH,n (f δn ). def

A few words on the construction of H are necessary. H is given as follows: H = Re(H (+) ). def

(30)

Using definitions and notation as in Appendix A, the distribution H (+) ∈ D (N × N ) (+) is defined, in the sense of the H-prescription, by a re-arrangement of the kernel of Zn (55) with D = 4, i.e., (1)

β4

U (x, y) sH,T (x, y) (2) . + β4 V (∞) (x, y) ln sH,T (x, y) λ2

(31)

(+)

(Similarly to Zn , H (+) does not depend on the choice of the temporal coordinate T .) Above

+∞ 1 s(x, y) def  s k (x, y). (x, y) ψ U V (∞) (x, y) = k+1 2k−1 k! αk k=0

ψ : R → R is some smooth map with ψ(x) = 1 for |x| < 1/2 and ψ(x) = 0 for |x| > 1 and αk > 0 for all k ∈ N. The series above converges to a smooth function which vanishes with all of its derivatives at x = y, provided the sequence of αk tends to zero sufficiently fast (see [22]). From now on we omit the restriction symbol
N and
N×N whenever these are implicit in the context. Our aim is to extend the given definitions to the whole manifold M (and not only N) and generalize to a differentiated field the notion of local Wick products. We have a preliminary proposition. Proposition 3.1. Referring to the given definitions, the sub ∗-algebra of W(N, g), WH (N, g), generated by WH,n (t), t ∈ En (N, g), n = 0, 1, . . . . (a) WH (N, g) coincides with W(N, g) itself and (b) is naturally ∗-isomorphic to the sub ∗-algebra of W(M, g) whose elements are smeared by distributions with support in N n . In this sense WH,n (t) ∈ W(M, g), n ∈ N. Proof. (a) Fix a Hadamard state ω in (N, g) and generate W(N, g) by elements Wω,n (t). Define the ∗-isomorphism α : WH (N, g) → W(N, g) as in the proof of Lemma 2.1 in def

(1)

[3] with d = Gω − H . (The reality of d is assured by the fact that H is real.) α, in fact, is the identity map in WH (N, g). (b) It is a direct consequence of the existence of the natural injective ∗-homomorphism defined in Lemma 3.1 in [3].  

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In order to define local Wick products of field operators, consider n linear differential operators Ki , acting on functions of D(M), with the form Ki = a(i0) + ∇a(i1) + ∇a2(i2) + · · · + ∇aL(iL ) , def

i

(32)

where a(i0) ∈ C ∞ (M; C) and, for k > 0, a(ik) is a smooth complex contravariant tensor field of order k defined on M. ∇ak : D(M) → D(M) is defined, in each local chart, by k ∇a(x) = a µ1 ...µk (x)∇µ1 (x) · · · ∇µk (x) .

tn [K1 , . . . , Kn , f ] ∈ D (M n ) is the compactly supported in N n distribution with formal kernel tn [K1 , . . . , Kn , f ](x1 , . . . , xn ) = Kn(xn ) t Kn−1 (xn−1 ) . . . t K1(x1 ) f (x1 )δn (x1 , . . . xn ),

def t

(33)

where the right-hand side is supposed to be symmetrized in x1 , . . . , xn . Above f ∈ D(N ), D(N ) being identified with the subspace of D(M) containing the functions with support in N. The transposed operator t Ki is defined as usual by “covariant” integration by parts with respect to Ki [23]. As a general result, W F (∂u) ⊂ W F (u) and W F (hu) ⊂ W F (u) if h is smooth. By consequence, for every f ∈ D(N ) and operators Ki , tn [K1 , . . . , Kn , f ] ∈ En (M, g) because W F (tn [K1 , . . . , Kn , f ]) ⊂ W F (f δn ) ⊂  {(x1 , k1 ; . . . ; xn , kn ) ∈ T ∗ M n \ {0} | i ki = 0} which is a subset of Gn (M, g). This result enables us to state the following definition. Def. 3.1 (Local wick products of (differentiated) fields I). Let N be a causal domain in a globally hyperbolic spacetime M with H defined in (30). The local Wick product of n (differentiated fields) generated by n operators Ki and f ∈ D(M) with supp f ⊂ N is def

:K1 ϕ · · · Kn ϕ(f ) :H = WH,n (tn [K1 , . . . , Kn , f ]) ∈ W(M, g).

(34)

The definition can be improved dropping the restriction supp f ⊂ N as follows. A preliminary lemma is necessary. Lemma 3.1. Referring to Def. 3.1, the following statements hold: (a) The local Wick products on a causal domain N ⊂ M, do not depend on the arbitrary terms ψ and {αk } used in the definition of H (but may depend on the length scale λ). (b) If N  ⊂ M is another causal domain with N ∩ N  = ∅ and :K1 ϕ · · · Kn ϕ(f ) :H  denote a local Wick product of differentiated fields operators defined on N  using the same length scale λ as in N , then :K1 ϕ · · · Kn ϕ(f ) :H = :K1 ϕ · · · Kn ϕ(f ) :H  and for any choice of operators Ki and f ∈ D(N ∩ N  ). Proof. See Appendix B.

 

(35)

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Def. 3.2. (Local Wick products of (differentiated) fields II). Referring to Def. 3.1, consider an open cover {Ni } of M made of causal domains with distributions Hi defined with the same scale length λ. Take a smooth partition of the unity {χij }, with supp χij ⊂ Oij ⊂ Ni , {Oij } being a locally finite refinement of {Ni }. The local Wick product of n (differentiated) fields generated by n operators Ki and f ∈ D(M) is the element of W(M, g), def  :K1 ϕ · · · Kn ϕ(χij f ) :Hi . (36) :K1 ϕ · · · Kn ϕ(f ): = i,j

Remark . Only a finite number of non-vanishing terms are summed in the right-hand side of (36) as a consequence of the locally finiteness of the cover {Oij } and the compactness of supp f . Moreover, by (a) of Lemma 3.1 and the linearity on En (M, g) of the involved distributions, the given definition does not depend on the functions ψ and constants {αk } used in the definition of H . By (b) of Lemma 3.1 the definition is independent from the choice of the cover and on the partition of the unity. The (differentiated) local Wick products enjoy the following properties. Proposition 3.2. Let (M, g) be a four-dimensional globally hyperbolic spacetime with a Klein-Gordon operator (2) with V  ≡ 0. Given n > 0 operators Ki , the following statements hold: (a) Given a, b ∈ C, f, h ∈ D(M), :ϕ(f ) : = ϕ(f ), :K1 ϕ · · · Kn ϕ(f ) :∗ = K1 ϕ · · · Kn ϕ(f ) :, :K1 ϕ · · · Kn ϕ(af + bh) : = a :K1 ϕ · · · Kn ϕ(f ): +b :K1 ϕ · · · Kn ϕ(h): .

(37) (38) (39)

(b) If ω is a quasifree Hadamard state on M, define :K1 ϕˆω · · · Kn ϕˆω (f ) ∈ : Wω (M, g) with f ∈ D(M), by Def 3.2 using the operators ϕˆω (h) of a GNS representation of ω; :K1 ϕˆω · · · Kn ϕˆω (f ):= αω (:K1 ϕ · · · Kn ϕ(f ):)

(40)

holds. Moreover, varying f ∈ D(M), the left-hand side gives rise to an operator-valued distribution f →:K1 ϕˆω · · · Kn ϕˆω (f ) : defined on the dense invariant subspace Dω . (c) For f ∈ D(M), :K1 ϕˆω · · · Kn ϕˆω (f ) :⊂:K1 ϕˆω · · · Kn ϕˆω (f ):† .

(41)

(d) If ω, ω are Hadamard states on M and f ∈ D(M), αω,ω (:K1 ϕˆω · · · Kn ϕˆω (f ):) = :K1 ϕˆω · · · Kn ϕˆω (f ) : .

(42)

Remark . (d) does not hold for normal products defined w.r.t. any quasifree Hadamard state ω. Sketch of proof. (a) is direct consequences of the given definitions, the reality of H and the linearity of all the involved distributions on En (M, g). (b) The continuity with respect to the strong Hilbert-space topology is the only non-trivial point. It can be shown as follows. Take a sequence of functions {fj } ⊂ D(M) with fj → f in D(M). In particular, this implies that there is a compact K with supp fj , supp f ⊂ K for j > j0 .

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By Def. 3.2, it is sufficient to prove the continuity when the supports of test functions belong to a common causal domain N ⊂ M, i.e., αω (WH,n (tn [K1 , . . . , Kn , fj ]))B → αω (WH,n (tn [K1 , . . . , Kn, f ]))B if fk → f in D(N ) and B ∈ Dω . By the definition of W (M, g) and the isomorphism αω (see 3.1), it is sufficient to show that tn [K1 , . . . , Kn , fj ] ∗ n → tn [K 1 , . . . , Kn , f ] in the closed conic set On = {(x1 , k1 ; . . . ; xn , kn ) ∈ T M \ {0} | i ki = 0} which contains the wave front set of all involved distributions, if fj → f in D(N ). The proof of the required convergence property is quite technical and it is proven in Appendix B. (c) is a trivial consequence of the fact that αω is a ∗- isomorphism and the definition of the involution on W(M, g). (d) is a trivial consequence of (40) and the identity αω,ω = αω ◦ αω−1 .   We can state a generalized locality and covariance requirement. A differentiated quantum field in one variable E is an assignment which associates with every globally hyperbolic spacetime (M, g) and every smooth contravariant tensor field on M, A (with fixed order) a distribution E[g, A] taking values in the algebra W(M, g). E, is said to be local and covariant if it satisfies the following: Locality and Covariance requirement for differentiated fields. For any embedding χ from a globally hyperbolic spacetime (N, g ) into another globally hyperbolic spacetime (M, g) which is isometric and causally preserving, iχ (E[g , A ](f )) = E[g, A](f ◦ χ −1 )

(43)

holds for all f ∈ D(N ) and all smooth vector fields A on M, A denoting (χ −1 )∗ Aχ(N) . The generalization to a (locally and covariant) quantum field in n-variables and depending on several smooth contravariant vector fields is straightforward. We conclude this part by showing that the introduced differentiated local Wick polynomials are local and covariant. Theorem 3.1. Take n ∈ {1, 2, . . . } and, for every i ∈ {1, . . . , n}, take integers Li = 0, 1, . . . . Let E be the map which associates with every globally hyperbolic spacetime (M, g) and every of class smooth contravariant vector fields on M, {a(ij ) }i=1,...,n, j =0,...Li , the (abstract) distribution f →:K1 ϕ · · · Kn ϕ(f ) :, where f ∈ D(M) and each Ki being defined in (32) using the fields a(ij ) . E is a locally and covariant differentiated quantum field in one variable. Sketch of proof. By Def. 3.2 the proof reduces to check (43) making use of spacetimes (N, g ) and (M, g) which are causal domains. In that case, if H  and H are the distributions (30) on N and M respectively, one finds H  (x, y) = H (χ (x), χ (y)) (provided the length scale λ is the same in both cases). Representing generators Wω,n in terms of generators WH,n as indicated in the proof of Proposition 3.1, one also gets that the injective ∗-algebra homomorphism iχ : W(N, g ) → W(M, g) (29) satisfies iχ (WH  ,n (t)) = WH,n (t ◦ χn−1 ). Referring to (33) and (32), we adopt the notation, def

tn [g, a(ij ) , f ] = tn [K1 , . . . , Kn , f ]. With the obtained results and using (34), (43) turns out to be equivalent to WH,n (tn [g , (χ −1 )∗ a(ij ) , f ]◦χn−1 ) = WH,n (tn [g, a(ij ) , f ◦χ −1 ]) for all n ∈ N, f ∈ D(N ) and all smooth tensor fields a(ij ) on M. That identity holds because tn [g, a(ij ) , f ◦ χ −1 ] = tn [g , (χ −1 )∗ a(ij ) , f ] ◦ χn−1 by the definition of distributions tn [K1 , . . . , Kn , f ] (33) and g = χ ∗ g.  

Stress-Energy Tensor Operator in Curved Spacetime

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3.3. The stress-energy tensor operator. From now on, :K1 ϕ(x) · · · Kn ϕ(x): indicates the formal kernel of the one-variable distribution f →:K1 ϕ · · · Kn ϕ(f ):. Using that notation and interpreting h ∈ C ∞ (M) as a multiplicative operator, we also define :h(x)ϕ n (x): = :K1 ϕ(x) · · · Kn ϕ(x): where K1 = h and Ki = I if i = 2, . . . n, def def

:h(x)∇X ∇Y ϕ 2 (x) : = 2 :h(x)ϕ(x) ∇X ∇Y ϕ(x) : +2 :h(x)∇X ϕ(x) ∇Y ϕ(x) : . Let {Z(a) }a=0,1,2,3 be a set of tetrad fields, i.e., four smooth contravariant vector fields def

def

defined on M such that g(Z(a) , Z(b) )(x) = ηab , where ηab = ηab = ca δab everywhere (there is no summation with respect to a) with c0 = −1 and ca = 1 otherwise. Making use of fields Z(a) , we define def



def



:h(x)g(∇ϕ, ∇ϕ)(x): =

ηab :h(x)∇Z(a) ϕ(x) ∇Z(b) ϕ(x):,

a,b

:h(x)ϕ(x)ϕ(x): =

ηab :h(x)ϕ(x) ∇Z(a)∇Z(b) ϕ(x):

a,b

−



ηab :h(x)ϕ(x) ∇

a,b

∇Z(a) Z(b)

 ϕ(x): .

These definitions do not depend on the choice of the tetrad fields and reduce to the usual ones if the field operators are replaced by classical fields. Finally, def

:h(x)ϕ 2 (x): = 2 :h(x)g(∇ϕ, ∇ϕ)(x): +2 :h(x)ϕ(x)ϕ(x): . Theorem 2.1 strongly suggests the following definition. Def. 3.3. (The stress-energy tensor operator). Let (M, g) be a four-dimensional smooth globally-hyperbolic spacetime equipped with a Klein-Gordon operator (2) with V  ≡ 0. Let X, Y be a pair of smooth vector fields on M. The stress-energy tensor operator with respect to X, Y and f ∈ D(M), :TX,Y (f ):, is defined by the formal kernel 1 m2 def :TX,Y (x): = :∇X ϕ(x)∇Y ϕ(x) : − :gX,Y (x)g(∇ϕ, ∇ϕ)(x) : − :gX,Y (x)ϕ 2 (x): 2 2

1 1 2 − :gX,Y (x)R(x)ϕ (x) :λ + ξ : RX,Y (x) − gX,Y (x)R(x) ϕ 2 (x) : 2 2 1 + ξ :gX,Y (x)ϕ 2 (x): −ξ :gX,Y (x)∇X ∇Y ϕ 2 (x): + :gX,Y (x)ϕ(x)P ϕ(x):, (44) 3 def

def

where gX,Y (x) = g(X, Y )(x), RX,Y (x) = R(X, Y )(x), R being the Ricci tensor and def

:gX,Y (x)ϕ(x)P ϕ(x) : = − :gX,Y (x)ϕ(x)ϕ(x) : + :gX,Y (x)(R(x) + m2 )ϕ 2 (x): . (45) Remarks. (1) We have introduced the, classically vanishing, term :gX,Y (x)ϕ(x)P ϕ(x) :. Its presence is crucial to obtain the conservation of the stress-energy tensor operator using

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the analogous property of the point-splitting renormalized stress-energy tensor as done in the proof of the theorem below. (2) The given definition depends on the choice of a length scale λ present in the distribution H used to define the local Wick products of field operators. To conclude our analysis we analyze the interplay between the above-introduced stress-energy tensor operator and the point-splitting procedure discussed in Sect. 2. Concerning the issue of the conservation of the stress-energy tensor, we notice in advance that, if T is a second-order covariant symmetric tensor field,



 
ab − f (∇ · T )X dµg = f T∇⊗X dµg + TZ(a) ,X ∇ · f Z(b) η M

M

a,b

M

 +f TX,∇Z(a) Z(b) dµg

(46)

for all f ∈ D(M) and all smooth contravariant vector fields X on M. Above (∇ · T )X = (∇ µ Tµν )X ν and T∇⊗X = Tµν ∇ µ X ν in the abstract index notation. Therefore the conservation requirement ∇ · T ≡ 0 is equivalent to the requirement that the right-hand side of (46) vanishes for all f ∈ D(M) and smooth contravariant vector fields X on M. We have a following conclusive theorem where, if ν is a quasifree Hadamard state, :Tˆν X,Y (f ) :, :ϕˆν2 (f ) : respectively represent :TX,Y (f ) : and :ϕ 2 (f ) : in Wν (M, g) in the sense of (b) in Proposition 3.2. Theorem 3.2. Let (M, g) be a four-dimensional smooth globally- hyperbolic spacetime equipped with a Klein-Gordon operator (2) with V  ≡ 0. Let λ > 0 be the scale length used to define local Wick products of field operators and {Z(a) }a=1,...,4 a set of tetrad fields. Considering the given definitions, the statements below hold for every f ∈ D(M). (a) For every h ∈ C ∞ (M), :h(x)ϕ(x)P ϕ(x) : does not depend on λ and turns out to be a smooth function. In particular if U2 (x, x) is defined as in Appendix A, 3 :hϕP ϕ(f ) := 4π 2



M

h(x)U2 (x, x)f (x)dµg (x) 1.

(47)

(b) Take a quasifree Hadamard state ν and let ω be any (not necessarily quasifree) Hadamard state represented by Bω ∈ Dν ⊂ Hν in a GNS representation of ν. For every pair of contravariant vector fields X, Y ,

 ˆ (48) Bω , :TνX,Y (f ) : Bω = TˆXY (z)ω f (z)dµg (z), ν

M  Bω , :ϕˆν2 (f ) : Bω = ϕˆ 2 (z)ω f (z)dµg (z), (49) ν

M

holds, TˆXY (z)ω = Tˆµν (z)ω X µ (x)Y ν (z) and ϕˆ 2 (z)ω denotes the fields obtained by the point-splitting procedure Def. 2.1. (c) The stress-energy tensor operator is conserved, i.e., for every contravariant vector field X on M, :(∇ · T )X (f ) := 0

(50)

Stress-Energy Tensor Operator in Curved Spacetime

209

holds, where, following (46), def

:(∇ · T )X (f ) : = − :T∇⊗X (f ):   

 − ηab :TZ(a) ,X ∇ · f Z(b) : + :TX,∇Z(a) Z(b) (f ): .

(51)

a,b

(d) The trace of the stress-energy tensor operator satisfies 

ηab :TZ(a) ,Z(b) (f ): =

a,b

6ξ − 1 :ϕ 2 (f ) : − :(m2 + ξ R)ϕ 2 (f ): 2

1 + U2 (x, x)f (x)dµg (x) 1. 4π 2 M

(52)

(e) If 0 < λ = λ, with obvious notation, :TX,Y (f ) :(λ) − :TX,Y (f ) :(λ ) = ln

λ λ

M

tX,Y (x)f (x)dµg (x) 1,

(53)

where the smooth, symmetric, conserved tensor field t is that introduced in (23). Proof. We start by proving (49) which is the simplest item. It is obvious by Def. 3.2 that we may reduce to consider f ∈ D(N ), where N ⊂ M is a causal domain. We have   Bω , :ϕˆν2 (f ): Bω = lim Bω , :ϕˆν2 (sj ): Bω , j →∞

ν

ν

where {sj } ⊂ D(N 2 ) is a sequence of smooth functions which converge to t2 (I, I, f ) = f δ2 in the H¨ormander pseudo-topology in a closed conic set in N × (R4 \ {0}) containing W F (f δ2 ). Such a sequence does exist by Theorem 8.2.3 of [21]. Above we have used the continuity of the scalar product as well as the continuity of the map 2 t2 (I, I, f ) → :ϕˆν (f ) : Bω since Bω ∈ Dν . On the other hand we may choose each sj of the form j cj hj ⊗ hj , where the sum is finite, cj ∈ C and hj , hj ∈ D(N ). This is because, using Weierstrass’ theorem on uniform approximation by means of polynomials in Rm , it turns out that the space of finite linear combinations h ⊗ h as above is dense in D(N × N ) in its proper seminorm-induced topology (viewing N as a subset of R4 because of the presence of global coordinates). We leave the trivial details to the reader. With that choice one straightforwardly finds  (1) Bω , :ϕˆν2 (sj ) : Bω = (G(1) ω − H )(sj ) = sj (Gω − H ), ν

(1)

where we have used the fact that both Gω) −H and sj are smooth. Since the convergence in the H¨ormander pseudotopology imply the convergence in D (N ), we finally get  Bω , :ϕˆν2 (f ) : Bω = lim sj (G(1) ω − H) j →∞ ν

= (G(1) ω − H )(x, y)f (x)δ2 (x, y)dµg (x)dµg (y). N×N

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The achieved result can be re-written in final form taking (19) into account and noticing that Zn − H is C n (N × N ), it vanishes with all of the derivatives up to the order n for x = y and n may be fixed arbitrarily large. In this way we get

 2 Bω , :ϕˆν (f ): Bω = ϕˆ 2 (x)ω f (x)dµg (x) ν

M

which is nothing but our thesis. Using the same approach one may prove (48) as well as

" ! h(x)ϕ(x)P ˆ ϕ(x) ˆ Bω , :hϕˆν P ϕˆν (f ): Bω ν = ω f (x)dµg (x) , M

where f ∈ D(N ) and h ∈

C ∞ (M).

In other words, by Lemma 2.1,

" ! h(x)c4 U2 (x, x)f (x)dµg (x) . Bω , :hϕˆν P ϕˆν (f ): Bω ν = − M

The right-hand side does not depend on Bω which, it being Hadamard as ν (but not necessarily quasifree), may range in the dense subspace of the Fock space Hω containing n-particle states with smooth modes [3]. Finally, using the fact that the Hilbert space is complex one trivially gets the operator identity on Dω ,

h(x)c4 U2 (x, x)f (x)dµg (x)I. :hϕˆν P ϕˆν (f ): = − M

By Def. 3.2. such an identity holds true also for f ∈ D(M), then


 −1 h(x)c4 U2 (x, x)f (x)dµg (x), :hϕP ϕ(f ): = αν :hϕˆν P ϕˆν (f ): = M

because α is an algebra isomorphism. We have proven item (a). Items (e) and (d) may be proven analogously starting from (48), in particular (d) is a direct consequence of (b) in Theorem 2.1. Let us prove the conservation of the stress-energy tensor operator (50). To this end, we notice that (48) together with item (a) of Theorem 2.1 for V  ≡ 0 by means of the procedure used to prove (47) implies the operator identity on Hν ,    
:Tˆν ∇⊗X (f ) : + ηab :Tˆν Z(a) ,X ∇ · (f Z(b) ) : + :Tˆν X,∇Z(a)Z(b) (f ) : = 0 , (54) a,b

for any Hadamard state ω, X, Y smooth vector fields and Z(a) tetrad fields. This identity  entails (50) by applying αν−1 on both sides.  4. Summary and Final Comments We have shown that a definition of the stress-energy tensor operator in curved spacetime is possible in terms of local Wick products of field operators only by adding suitable terms to the classical form of the stress energy tensor. Such a definition seems to be quite reasonable and produces results in agreement with well-known regularization procedures of averaged quantum observables. The added terms in the form of the stress-energy tensor operator enjoy three remarkable properties. (1) They classically vanish, (2) they are written as local Wick products of field operators, (3) they are in a certain sense universal, i.e., they do not depend on the scale λ and belong to the center of the algebra W(M,g).

Stress-Energy Tensor Operator in Curved Spacetime

211

The issue whether or not it could be possible to define a stress-energy tensor operator free from these terms is related to the issue of the existence of Hadamard singular bidistributions, defined locally and somehow “determined by the local geometry only”. The positiveness seems not to be a requirement strictly necessary at this level. The appearance of the so-called conformal anomaly shared by the various regularization techniques and related to the presence of the terms we found could be in contrast to the existence of such local bisolutions. However, no proof, in any sense, exists in literature to the knowledge of the author. As a final comment we suggest that the universal terms : hϕP ϕ(f ) :, or similar terms, may be useful in studying other conservation laws within the approach based on local Wick products, e.g., conserved currents in (non-)Abelian gauge theories and related anomalies.

Appendix A If (M, g) is a smooth Riemannian or Lorentzian manifold, an open set C ⊂ M is said to be a normal convex neighborhood if there is an open set W ⊂ T M, W = {(q, v) | q ∈ C, v ∈ Sq }, Sq ⊂ Tq M being a starshaped open neighborhood of the origin, such that expW : (q, v) → expq v is a diffeomorphism onto C × C. It is clear that C is connected and there is only one geodesic segment joining any pair q, q  ∈ C, completely contained in C, i.e., t → expq (t ((expq )−1 q  )) t ∈ [0, 1]. Moreover if q ∈ C, {eα |q } ⊂ Tq M is a basis, t = t α eα |q → expq (t α eα |q ), t ∈ Sq defines a set of coordinates on C centered in q which is called a normal Riemannian coordinate system centered in q. In (M, g) as above, s(x, y) indicates the squared geodesic distance of x def

from y: s(x, y) = gx (expx−1 y, expx−1 y). By definition s(x, y) = s(y, x) and s turns out to be smoothly defined on C × C if C is a convex normal neighborhood. The class of the convex normal neighborhood of a point p ∈ M defines a fundamental system of neighborhoods of p [23, 24]. With the signature (−, +, . . . , +), we have s(x, y) > 0 if the points are space-like separated, s(x, y) < 0 if the points are time-like related and s(x, y) = 0 if the points are light related. In Euclidean manifolds s defined as above is everywhere nonnegative. (+) The distribution Zn is defined by the following integral kernel in the sense of the + usual H → 0 prescription: (1)

βD

U (x, y) sH,T (x, y) (2) + βD V (n) (x, y) ln sH,T D/2−1 (x, y) λ2

(1)

βD

T (n) (x, y) D/2−1

sH,T

(x, y)

if D is odd,

if D is even,

(55)

(56)

def

s k (x, y) = (s(x, y))k . The cut branch in the logarithm is fixed along the negative real def

axis, moreover sH,T = s(x, y) + 2iH(T (x) − T (y)) + H 2 , T being any global tempo(+) ral function defined on M increasing toward the future. The distribution Zn does not depend on the choice of T (see [12] and Appendix A3 of [16]). λ ∈ R is a length scale

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arbitrarily fixed, (1) def

βD = −

O( D2 − 1) 4π

(57)

D 2

D

β

(−1) 2

(2) def

=

D

(4π) 2 O( D2 )

.

(58)

U, V (n) , T (n) admit the following expansions in powers of s(x, y). If D is even, (D−4)/2 

1 Uk (x, y) s k (x, y), (4 − D|k) k=0 

 n D 1 def (n)  V (x, y) = 2  − 1 (x, y)s k (x, y), UD k 2 2 k! 2 +k−1 def

U (x, y) = RD

(59)

(60)

k=0

where R2 = 0 and RD = 1 if D > 2, and T

(n)

def

(x, y) =

n+(D−3)/2  k=0

def

1 Uk (x, y) s k (x, y), (4 − D|k)

(61)

def

if D ≥ 3 is odd. (α|0) = 1 and (α|k) = α(α + 2) · · · (α + 2k − 2). For any open convex normal neighborhood C in M there is exactly one sequence of C ∞ (C × C) real valued functions Uk , used in the expansions above, satisfying the differential equations on C × C: −Px Uk−1 (x, y) + gx (∇(x) s(x, y), ∇(x) Uk (x, y)) + (M(y, x) + 2k)Uk (x, y) = 0, (62) with the initial conditions U−1 (x, y) = 0 and U0 (x, x) = 1. The function M is defined def

as M(x, y) = 21 x s(x, y) − D , with D the dimension of the manifold. The proof of existence and uniqueness is trivial using normal coordinates centered in x. The coefficients Uk (x, y) can be defined, in the same way, also if the metric is Euclidean. They coincide, barring numerical factors, with the so-called Hadamard-MinakshisundaramDeWitt-Seeley coefficients. If C  , C are convex normal neighborhoods and C  ⊂ C, the restriction to C  of each Uk defined in C coincides with the corresponding coefficient directly defined on C  . There is significant literature on coefficients Uk , in relation with heat-kernel theory and ζ -function regularization technique [20]. Since Hadamard, heat-kernel Seeley-deWitt coefficients appear in the literature with different definitions, we give below the relations among these coefficients. Our coefficients Uk are those defined in Chapter III of [22] and indicated by U(k) therein, moreover: (G)

(F )

Uk (x, y) = (4 − D|k)Uk (x, y) = (−1)k ak (x, y)/2k = 2k Uk (x, y). (G)

The coefficients Uk are those defined in (5.47)–(5.48) p. 154 of [25] with L = −P (+P if one uses the signature (+, −, . . . , −) in Lorentzian manifolds); ak (x, y) are the heatkernel Seeley-deWitt coefficients defined in (3) and (4) of [19] with A0 = P ; U (F ) (x, y) are the Hadamard coefficients defined in (6.2.2) and (6.2.5) of [23] where P corresponds to our −P due to the use of the signature (+, −, . . . , −). Finally it is useful to know

Stress-Energy Tensor Operator in Curved Spacetime

213

that a0 (x, y) = (V V M (x, y))1/2 where V V M is the so-called VanVleck-Morette determinant (e.g. see [19]). As in 2.2 Z(x, y) indicates the kernel of Re(Z (+) ) which is smooth for s(x, y) = 0 in every convex normal neighborhood Cz  x, y. It is possible to show that the coefficients are symmetric, i.e., if x, y ∈ C, Un (x, y) = Un (y, x) [19] and thus, since s(x, y) = s(y, x), Zn (x, y) = Zn (y, x) holds for any n ≥ 0 and s(x, y) = 0. The recurrence relations (62) have been obtained by requiring that the sequence Zn defines a local y-parametrized “approximated solution” of Px S(x, y) = 0 [25]. That solution is exact if one takes the limit Z = limn→∞ Zn of the sequence provided the limit exists. This happens in the analytic case, but in the smooth general case the sequence may diverge. Actually, in order to produce an approximated/exact solution for D even, a smooth part W has to be added to Z, S = Z + W , and also W can be expanded in powers of s [25]. Different from the expansion of Z which is completely determined by the geometry and the operator P , the expansion of W depends on its first term W0 (corresponding to s 0 ) and there is no natural choice of W0 suggested by P and the local geometry. Finally if D is even and for whatever choice of W0 there is no guarantee for producing a function Z + W (provided the limits of corresponding sequences exist) which is solution of field equations in both arguments: This is because in general W (x, y) = W (y, x) also if W0 (x, y) is symmetric [8]. Appendix B Referring to 2.2, the properties (b) and (c) respectively imply the relations in M × M, (1) G(1) ω (x, y) = Gω (y, x),

Px G(1) ω (x, y)

=

Py G(1) ω (x, y)

(63) = 0,

(64)

which hold when x = y are not light-like related. These relations are useful in the following. Proof of Lemma 2.1. (a) In the following we take advantage of the identity, where f and g are C 2 functions, P (fg) = f P g − (f )g − 2g(∇f, ∇g) . Suppose D > 2 even, using the identity above and the definition of Zn , one finds, for either x, y time-like related or space-like separated, (1) Px U (x, y) Px Zn (x, y) = βD − U (x, y)(x) s 1−D/2 s D/2−1

1−D/2 − 2gx (∇(x) s , ∇(x) U (x, y)) 

|s| |s| − V (n) (x, y)(x) ln 2 λ2 λ  (n) gx (∇(x) s, ∇(x) V (x, y)) −2 . s (2))

+ βD

(Px V (n) (x, y)) ln

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Using (62) for n ≥ 1 we have  |s| |s| (n) (2) Px Zn (x, y) = −βD − (ln 2 )Px V (n) (x, y) + ((x) ln 2 )V  (x, y) λ λ  gx (∇(x) s, ∇(x) V  (n) (x, y)) , +2 s where V

(n)

def

(x, y) =

n 

Vk (x, y)s k (x, y)

k=1

with def

Vk (x, y) =

 D 1 UD 2  − 1 (x, y) . k 2 2 k! 2 +k−1 (2)

Expanding the derivatives and using (62) once again, if n ≥ 1, one gets that , −(βD )−1 Px Zn (x, y) equals

gx (∇(x) s(x, y), ∇(x) s(x, y)) (x) s(x, y) 4 − V1 (x, y) + 2 V1 (x, y) s s s s |s| + 2gx (∇(x) s, ∇(x) V1 (x, y)) + |s|n O1,n (x, y) ln 2 + |s|n−1 O2,n (x, y) λ + |s|n−1/2 O3,n (x, y) , (65) where Ok,n are smooth in a neighborhood of (z, z) and the last two terms appear for n > 1 only. Using gx (∇(x) s(x, y), ∇(x) s(x, y)) = 4s(x, y), one finds (2)

−(βD )−1 Px Zn (x, y) = ((x) s(x, y) − 4)V1 (x, y) + 8V1 (x, y) + 2gx (∇(x) s, ∇(x) V1 (x, y)) + |s|n O1,n (x, y) ln + |s|n−1 O2,n (x, y) + |s|n−1/2 O3,n (x, y),

|s| λ2 (66)

and thus, since (x) s(x, y) → 2D and ∇(x) s(x, y) → 0 as (x, y) → (z, z), lim

(x,y)→(z,z)

Px Zn (x, y) = cD UD/2 (z, z) ,

which is part of (12) for D even. Zn (x, y) = Zn (y, x) implies the remaining identity in (12). If D = 2, x, y are either time-like related or space-like separated and n ≥ 1, the proof is essentially the same. One directly finds  |s| |s| (2) Px Zn (x, y) = −β2 (−Px V (n) (x, y)) ln 2 + V (n) (x, y)(x) ln 2 λ λ  (n) gx (∇(x) s, ∇x V (x, y)) +2 , s with V (n) (x, y) =

def

n  k=0

Vk (x, y)s k (x, y)

Stress-Energy Tensor Operator in Curved Spacetime

and def

Vk (x, y) =

215



D 1 2  − 1 (x, y). UD 2 2k k! 2 +k−1

Using V0 = U0 (D = 2) and (62) for k = 0, one gets (66) once again. If D is odd, n ≥ 1 and x, y are either space-like separated or time-like related, (62) entails Px Zn (x, y) = θ (s(x, y))|s(x, y)|n−1/2 On (x, y), where On is smooth in a neighborhood of (z, z). Therefore, lim

(x,y)→(z,z)

Px Zn (x, y) = 0,

(67)

which is part of (12) for D odd; the other part is a trivial consequence of the symmetry as above. Notice that the proof shows also that the limit is uniform in the three cases treated because |s(x, y)| uniformly tends to 0 as (x, y) → (z, z). (b) The proof follows a very similar procedure as that used in the proof of (a). One µ obtains that lim(x,y)→(z,z) Px ∇(y) Zn (x, y) equals (2)

−βD



µ

µ

µ



(2D − 4)∇(y) V1 (x, y) + 8 ∇(y) V1 (x, y) − 4 ∇(x) V1 (x, y) µ

x=y=z

.

(68)

µ

(One has to differentiate (66) with respect to ∇(y) and use ∇(y) x s(x, y))|x=y=z = 0 and ∇(x)µ ∇(y)ν s(x, y)|x=y=z = −2gµν (z).) Finally one notices that V1 is proportionα V (z, z) = 2∇ α V (x, y)| al to UD/2 and, since Un (x, y) = Un (y, x), ∇(z) 1 x=y=z also (y) 1 holds. Using that in (68) the thesis (b) arises. For D = 2 and D odd the proof is the same with trivial modifications. (c) The proof directly follows from (a) and (64).   (1)

Proof of Theorem 2.1. (a) (x, y) → Gω (x, y) − Zn (x, y) is C n in a whole neighborhood of z (also for light-like related arguments). Since we want to apply the second (η) order operator D(z)µν to it, we need to fix n ≥ 2, however we also want to derive the obtained stress-energy tensor and thus we need n ≥ 3. With n ≥ 3 the map above is (η) C n at least and z → Tˆµν (z)ω is C n−1 . Finally, since the latter map does not depend ∞ on n it must be C also if n ≥ 3 is finite. The independence from n is a consequence def

of lim(x,y)→(z,z) n,n (x, y) = 0, where n,n (x, y) = Zn (x, y) − Zn (x, y), which holds true for any pair n, n ≥ 3 as the reader may straightforwardly check and prove by induction. The remaining part of (a) may be proven as follows. For any C 3 function (x, y) → O(x, y) symmetric under interchange of x and y we have the identity  
 (η) µ ∇(z) D(z)µν O(x, y)|x=y=z = −Px ∇(y)ν O(x, y)|x=y=z + η∇(z)ν Px O(x, y)|x=y=z 1 − O(z, z)∇ν V  (x). 2

(69)

Indeed, if ϕ does not satisfy the field equation, (4) reads, 1 ∇ α Tαβ (x) = −(P ϕ)(x)∇β ϕ(x) − ϕ 2 (x)∇β V  (x). 2

(70)

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Such an identity can be obtained by using the form of the stress-energy tensor and the symmetry of O(x, y) = ϕ(x)ϕ(y) only. So it holds true for each symmetric sufficiently smooth map (x, y) → O(x, y). The proof of (69) is nothing but the proof of (70) taking (1) the added term proportional to η into account. Then put O(x, y) = Gω (x, y)−Zn (x, y) into (69) with n ≥ 3; this is allowed because Zn is symmetric by construction (see (1) Appendix A) and Gω satisfies (63). The first line of the right-hand side of (69) reduces to lim

(x,y)→(z,z)

Px ∇(y)ν Zn (x, y) − η∇(z)ν

lim

(x,y)→(z,z)

Px Zn (x, y)

because of (64). Both terms above can be computed by Lemma 2.1 finding 1 (η) ∇ µ Tˆµν (z)ω = δD (kD − ηcD )∇ν UD/2 (z, z) − ϕˆ 2 (z)ω ∇ν V  (z). 2 The former term in the right-hand side vanishes if and only if η = kD /cD , i.e., η = ηD . This concludes the proof of (a). (b) Directly from the form of the stress-energy tensor, one finds that, if P ϕ = 0, (5) reads  

ξD − ξ D αβ 2 gαβ (x)T (x) =  − V (x) ϕ (x) + 1 − (P ϕ)(x)ϕ(x). (71) 4ξD − 1 2 The same results holds if ϕ(x)ϕ(y) is replaced by any sufficiently smooth symmetric function O. Therefore, if O is as above, similarly to (71) we get   ξD − ξ (η) µν g (z)D(z)µν O(x, y)|x=y=z = (z) − V (z) O(z, z) 4ξD − 1

D + 1− (72) + ηD (Px O(x, y))|x=y=z . 2 (1)

If η = ηD and O(x, y) = Gω (x, y) − Zn (x, y), using (64) and Lemma 2.1, we get the identity in (b). −(2cD /(D + 2))UD/2 (z, z) is the conformal anomaly for V ≡ 0 and m = 0, ξ = ξD because it coincides with the heat-kernel coefficient aD/2 (z, z)/(4π )D/2 [20]. This can be seen by direct comparison of recursive equations defining both classes of coefficients (see references in [20]). This concludes the proof of (b). (c) The proof is direct by employing the given definitions. (d) For D odd the proof of the thesis is trivial. Hence assume D even. In that case, with obvious notation,   n λ  ck s k (x, y)Uk−1+D/2 (x, y), Zλ,n (x, y) − Zλ ,n (x, y) = 2 ln λ k=0

(η)

where ck are numerical coefficients defined above. lim(x,y)→(z,z) D(z)(x,y) (Zλ − Zλ ) is a polynomial of coefficients Uk (z, z) and their derivatives. These coefficients do not depend on the state, are proportional to heat-kernel ones, and thus are built up as indicated in the thesis [9]. Notice that the obtained tensor field t must be conserved because of the difference of two conserved tensor fields if V  ≡ 0. (e) For m = 0 the proof of the thesis is trivial because Zn in Minkowski spacetime does not contain the logarithmic term and does not depend on both λ and n; moreover , if (1) ω is Minkowski vacuum Gω (x, y) = Zn (x, y) and thus the renormalized stress-energy

Stress-Energy Tensor Operator in Curved Spacetime

217

tensor vanishes. Let us consider the case m > 0. In that case, the smooth kernels of the two-point function are given by, in the sense of the analytic continuation if s(x, y) < 0,  #  4m # G(+) K1 m sH,T (x, y) ω (x, y) = lim+ H→0 (4π)2 sH,T (x, y) def

with sH,T (x, y) = s(x, y) + 2i(T (x) − T (y)) + H 2 , where H → 0+ indicates the path to approach the branch cut of the squared root along the negative real axis if s(x, y) < 0. T indicates any global time coordinate increasing toward the future. K1 is a modified Bessel function. The corresponding Hadamard function can be expandend as $ 2γ 2

% 2γ 2

m2 s 4 e m s m2 e m s (1) 2 Gω (x, y) = 1+ + ln + s f (s) ln (4π)2 s (4π)2 8 4 4   2 2 m 5m s − 1+ + s 2 g(s), 2 (4π) 16 where f and g are smooth functions and γ is Euler-Mascheroni’s constant. Similarly $ % m2 s s 4 m2 2 3 Zλ,3 (x, y) = 1 + ln 2 , s + C s + + C 2 3 2 2 (4π) s (4π) 8 λ where C2 and C3 are constants. Therefore $  % 2 2γ 2

 m2 s 5m2 s m2 m2 λ e m (1) Gω − Zλ,3 (x, y) = 1+ 1+ ln − (4π)2 8 4 (4π )2 16 +s 2 [h(s) + k(s) ln s] , where h and k are smooth functions. Trivial computations lead to  2 2γ 2

 λ e m m4 7 (η4 ) Tˆµν ln (z)ω,λ = − − gµν (z) . 3(4π)2 4 4 7

Posing λ2 = 4e 4 −2γ m−2 , the right-hand side vanishes.

 

Proof of Lemma 3.1. (a) The thesis can be proved working in a concrete algebra Wω (N, g), using operators :K1 ϕˆ · · · Kn ϕ(f ˆ ) :H ∈ Wω (N, g), where ω is a fixed quasifree Hadamard state and ϕˆ = ϕˆω . Notice that, for every B ∈ Dω , Wˆ H,n+1 (x1 , . . . , xn+1 )B equals      Wˆ H,n (x1 , . . . , xn )Wˆ H,1 (xn+1 )B − Wˆ H,n (x1 , . . . , xˆl , xn+1 )BH (xl , xn+1 ) , S

l

S

where S indicates the symmetrization with respect to all arguments and xˆl indicates that the argument is omitted. We prove the thesis, which is true for n = 1, by induction. If H  is defined as H but with a different choice of ψ and
{αk }, for each B ∈ Dω , the formula above and our inductive hypothesis imply that :K1 ϕˆ · · · Kn+1 ϕ(f ˆ ) :H −  ˆ ) :H B reduces to :K1 ϕˆ · · · Kn+1 ϕ(f  

 Wˆ H,n (x1 , . . . , xˆl , xn+1 )BS(xl , xn+1 ) Mn

l

S

×t Kn+1 (xn+1 ) . . . t K1(x1 ) f (x1 )δn (x1 , . . . xn+1 ),

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where S(x, y) is a smooth function which vanishes for x = y together with all of its derivatives. As Wˆ H,n (x1 , . . . , xˆl , xn+1 )B is singular on the diagonal we cannot directly conclude that the integral vanishes. However, there is a sequence of smooth vector-valued functions {Uj } which tends to the distribution in (· · · )S in the sense of H¨ormander pseudo-topology in a closed conic set containing the wave front set of the distribution. It is simply proven that the smearing procedure of distributions by means of distributions defined in 3.1 is continuous in the sense of H¨ormander pseudo-topology. Therefore
:K1 ϕˆ · · · Kn+1 ϕ(f ˆ ) :H − :K1 ϕˆ · · · Kn+1 ϕ(f ˆ ) :H  B can be computed as the limit 



Mn

 Uj (x1 , . . . , xˆl , xn+1 )S(xl , xn+1 )

l

S

× t Kn+1 (xn+1 ) . . . t K1(x1 ) f (x1 )δn (x1 , . . . xn+1 ), for j → ∞. However, as each Un is regular, we can conclude that each term of the sequence above vanishes and this proves the thesis because B ∈ Dω is arbitrary. (b) By (a) we may assume that the distributions H and H  are constructed using the same function ψ and the same sequence of numbers {αk }. Define L = N ∩ N  and take f ∈ D(L). Since the convex normal neighborhoods define a topological base, L is the union of convex normal neighborhoods. In turn, it implies that the compact set suppf ⊂ L admits a finite covering {Ui } made of convex normal neighborhoods contained in L and thus both in N and N . Therefore, if x, y ∈ Ui , the squared geodesic distance s(x, y) computed by viewing x, y as elements of N agrees with the analogue by viewing x, y as elements of N  (also if N ∩ N  may not be convex normal). By consequence H and H  induce the same distribution on each Ui . {Ui } is locally finite and thus there is a smooth partition of the unity {χj }j subordinate  to {Ui }. By linearity we have :K1 ϕ · · · Kn ϕ(f ) :H = j :K1 ϕ · · · Kn ϕ(χj f ) :H = j :K1 ϕ · · · Kn ϕ(χj f ) :H  =:K1 ϕ · · · Kn ϕ(f ) :H  which concludes the proof.   Proof of part of (b) in Proposition 3.2. We want to show that, if N ⊂ M is a causal domain then, fj → f in D(N ) entails tn [K1 , . . . , Kn , fj ] → tn [K1 , . . . , Kn , f ] in the sense of H¨o rmander pseudo-topology in the conic set On = {(x1 , k1 ; . . . ; xn , kn ) ∈ T ∗ M n \ {0} | i ki = 0} which contains the wave front set of all involved distributions. Since there is a coordinate patch covering N , ξ : N → O (normal Riemannian coordinates centered on some p ∈ N ), we can make use of the Rn -distribution definition of convergence (see Definition 8.2.2. in [21]). Therefore, in the following, f, fj and tn [K1 , . . . , Kn , f ], tn [K1 , . . . , Kn, fj ] have to be understood as distributions in D (O), and D (O n ) respectively, O being an open subset of R4 . Posing uj = tn [K1 , . . . , Kn , fj ] and u = tn [K1 , . . . , Kn , f ], we have to show that (1) uj → u in D (O n ), and this is trivially true by the given definitions since fn → f in D(O), and, (2), & 'j (k)| → 0, − ψu sup |k|N |ψu(k) V

as j → +∞

(73)

for all N = 0, 1, 2, . . . , ψ ∈ D(O) and V , closed conic set7 in R4n such that On ∩ (supp ψ × V ) = ∅. 7

A conic set V ⊂ Rm is a set such that if v ∈ V , λv ∈ V for every λ > 0.

(74)

Stress-Energy Tensor Operator in Curved Spacetime

219

( v denotes the Fourier transform of v. From now on K denotes a generic vector in R4n of the form (k1 , . . . , kn ), ki ∈ R4 and, in components ki = (ki1 , ki2 , ki3 , ki4 ). We leave it to & j and ψu & are polynomials the reader to show that, with the definitions given above, ψu in the components of K, whose coefficients smoothly depend on k1 + · · · + kn , i.e., 

& j (k) = ψu

aj r11 ,... ,rn4 (k1 + · · · + kn )

4 n   i=1 m=1

r11 ,... ,rn4 ∈N

(kim )rim .

(75)

An analogous identity concerning u and coefficients ar11 ,... ,rn4 (k1 + · · · + kn ) hold by omitting j in both sides. (Above 0 ∈ N and only a finite number of functions aj r11 ,... ,rn4 and ar11 ,... ,rn4 differ from the null function.) Moreover fj → f in D(O) implies that, for every N ∈ N and rim ∈ N,   (76) sup |x|N aj r11 ,... ,rn4 (x) − ar11 ,... ,rn4 (x) → 0. x∈R4

With the given notations, our thesis (73) reduces to )  4   4 *      N sup |K|  aj r11 ,... ,rn4 ki − ar11 ,... ,rn4 ki K∈V

i=1

r11 ,... ,rn4 ∈N

×

n  4  i=1 m=1

i=1

  (kim )rim  → 0

(77)

as j → +∞ for all N ∈ N and V which is closed in R4n conic and such that V ∩ {K ∈ R4n \ {0} |

n 

ki = 0, } = ∅ .

(78)

i=1

We want to prove (77) starting from (76). Consider a linear bijective map A : K → Q ∈ R4n , where Q = (q1 , . . . , qn ) and q1 = p1 +· · ·+pn . The functions x → bs11 ,... ,sn4 (x) and x → bj s11 ,... ,sn4 (x) which arise when translating (75) (and the analog for u) in the variable Q, i.e., & j (k) = ψu

 s11 ,... ,sn4 ∈N

bj s11 ,... ,sn4 (q1 )

4 n   i=1 m=1

(qim )sim ,

are linear combinations of the functions x → aj r11 ,... ,rn4 (x) (with coefficients which do not depend on j ) and vice versa, therefore (76) entails   sup |x|N bj s11 ,... ,sn4 (x) − bs11 ,... ,sn4 (x) → 0 (79) x∈R4

for every N ∈ N and sim ∈ N. Since linear bijective maps transform closed conic sets into closed conic sets and |Q| ≤ ||A|||K|, |K| ≤ ||A−1 |||Q|, our thesis (77) is equivalent to      4 n       sup |Q|N  bj s11 ,... ,sn4 (q1 ) − bs11 ,... ,sn4 (q1 ) (qim )sim  → 0 (80) Q∈U s11 ,... ,sn4 ∈N  i=1 m=1

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as j → +∞ for all N ∈ N and U closed in R4n conic and such that U ∩ {Q ∈ R4n \ {0} | q1 = 0} = ∅ .

(81)

It is possible to show that if U ∈ R4n is a set which fulfills (81) and U is conic and closed in R4n , then there is p ∈ N \ {0} such that U ⊂ Up , where the closed set Up is defined by  % $ +  1 |q2 |2 + · · · + |qn |2 , Up = Q ∈ R4n  |q1 | ≥ p   and thus Up ∩ Q ∈ R4n \ {0} | q1 = 0 = ∅ for every p ∈ N \ {0}. The proof is left to the reader (hint: U is conic and satisfies (81) then, reducing to a compact neighborhood of the origin of R4n , one finds a sequence of points of U which converges to some point x ∈ {Q ∈ R4n \ {0} | q1 = 0} this is not possible because U = U and thus x ∈ U ∩ {Q ∈ R4n \ {0} | q1 = 0}). If (80) holds on each Up , it must hold true on each conic closed set U which fulfills (81). The validity of (80) on each Up is a direct consequence of (79) and the inequalities which hold on Up , + |Q| ≤ 1 + p 2 |q1 | and |qrs | ≤ |q1 |/p for r=2,3, . . . ,n, s=1,2,3,4.   Acknowledgements. comments.

I am grateful to Romeo Brunetti and Klaus Fredehagen for helpful discussions and

References 1. Brunetti, R., Fredenhagen, K., K¨ohler, M.: The microlocal spectrum condition and Wick polynomials of free fields on curved spacetimes. Commun. Math. Phys. 180, 633 (1996) [gr-qc/9510056] 2. Brunetti, R., Fredenhagen, K.: Microlocal analysis and interacting quantum field theories: Renormalization on physical backgrounds. Commun. Math. Phys. 208, 623 (2000) [math-ph/9903028] 3. Hollands, S., Wald, R.M.: Local Wick polynomials and time ordered products of quantum fields in curved spacetime. Commun. Math. Phys. 223, 289 (2001) [gr-qc/0103074] 4. D¨utsch, M., Fredenhagen, K.: Algebraic quantum field theory, perturbation theory, and the loop expansion. Commun. Math. Phys. 219, 5 (2001) [arXiv:hep-th/0001129]; D¨utsch, M., Fredenhagen, K.: Perturbative algebraic field theory, and deformation quantization. Fields Inst. Commun. 30, 151 (2001) [arXiv:hep-th/0101079] 5. Brunetti, R., Fredenhagen, K., Verch, R.: The generally covariant locality principle: A new paradigm in local quantum field theory. [math-ph/0112041] 6. Birrell, N.D., Davies, P.C.: Quantum Fields in Curved Space. Cambridge, UK: Univ. Pr., 1982, 340p 7. Wald, R.M.: Quantum Field Theory in Curved Space–Time and Black Hole Thermodynamics. Chicago, USA: Univ. Pr., 1994, 205p 8. Wald, R.M.: Trace anomaly of a conformally invariant quantum field in curved space–time. Phys. Rev. D 17, 1477 (1978) 9. Fulling, S.A.: Aspects of quantum field theory in curved space–time. Cambridge, UK: Univ. Pr., 1989, 315p 10. Wald, R.M.: General Relativity. Chicago, USA: Univ. Pr., 1984, 491p 11. Callan, C.G., Coleman, S., Jackiw, R.: A new improved energy-momentum tensor. Ann. Phys. (NY) 59, 42 (1970) 12. Kay, B.S., Wald, R.M.: Theorem on the uniqueness and thermal properties of stationary, nonsingular, quasifree states on space–times with a bifurcate Killing horizon. Phys. Rept. 207, 49 (1991) 13. Haag, R.: Local Quantum Physics. Berlin-Heidelberg, Germany: Springer-Verlag, 1996, 2nd edition, 390p 14. Radzikowski, M.J.: Micro-local approach to the Hadamard condition in quantum field theory on curved space–time. Commun. Math. Phys. 179, 529 (1996); Radzikowski, M.J.: A Local to global singularity theorem for quantum field theory on curved space–time. Commun. Math. Phys. 180, 1 (1996)

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15. Fulling, S.A., Sweeny, M., Wald, R.M.: Singularity structure of the two point function in quantum field theory in curved space–time. Commun. Math. Phys. 63, 257 (1978) 16. Sahlmann, H., Verch, R.: Microlocal spectrum condition and Hadamard form for vector-valued quantum fields in curved spacetime. Rev. Math. Phys. 13, 1203 (2001) [math-ph/0008029] 17. Hawking, S.W.: Zeta function regularization of path integrals in curved space–time. Commun. Math. Phys. 55, 133 (1977) 18. Fujikawa, K.: Comment on chiral and conformal anomalies. Phys. Rev. Lett. 44, 1733 (1980) 19. Moretti, V.: Proof of the symmetry of the off-diagonal Hadamard/Seeley-deWitt’s coefficients in C ∞ Lorentzian manifolds by a ‘local Wick rotation’. Commun. Math. Phys. 212, 165 (2000) [gr-qc/9908068] 20. Moretti, V.: One-loop stress–tensor renormalization in curved background: The relation between zeta-function and point-splitting approaches, and an improved point-splitting procedure. J. Math. Phys. 40, 3843 (1999) [gr-qc/9809006] 21. H¨ormander, L.: The Analysis of Linear Partial Differential Operators I. Berlin Heidelberg, Germany: Springer-Verlag, 1990, 2nd edition, 442p 22. G¨unther, P.: Huygens’ Principle and Hyperbolic Equations. Boston, USA: Academic Press, Inc., 1988, 848p 23. Friedlander, F.G.: The Wave Equation on a Curved Space–Time. Cambridge, UK: Univ. Pr., 1975, 262p 24. Beem, J.K., Eherlich, P.E., Easley, K.L.: Global Lorentzian Geometry (second edition). New York, USA: Marcel Dekker, 1996 25. Garabedian. G.: Partial Differential Equations. New York, USA: Wiley, 1964, 672p Communicated by H. Nicolai

Commun. Math. Phys. 232, 223–277 (2003) Digital Object Identifier (DOI) 10.1007/s00220-002-0713-4

Communications in

Mathematical Physics

Partial Dynamical Systems and the KMS Condition Ruy Exel1,∗ , Marcelo Laca2,∗∗,∗∗∗ 1 2

Departamento de Matem´atica, Universidade Federal de Santa Catarina, 88040-900 Florian´opolis SC, Brazil Department of Mathematics, University of Newcastle, NSW 2308, Australia.

Received: 21 November 2000 / Accepted: 31 May 2002 Published online: 22 November 2002 – © Springer-Verlag 2002

Abstract: Given a countably infinite 0–1 matrix A without identically zero rows, let OA be the Cuntz–Krieger algebra recently introduced by the authors and TA be the Toeplitz extension of OA , once the latter is seen as a Cuntz–Pimsner algebra, as recently shown by Szyma´nski. We study the KMS equilibrium states of C ∗ -dynamical systems based on OA and TA , with dynamics satisfying σt (sx ) = Nxit sx for the canonical generating partial isometries sx and arbitrary real numbers Nx > 1. The KMSβ states on both OA and TA are completely characterized for certain values of the inverse temperature β, according to the position of β relative to three critical values, defined to be the abscissa of convergence of certain Dirichlet series associated to A and the N (x). Our results for OA are derived from those for TA by virtue of the former being a covariant quotient of the latter. When the matrix A is finite, these results give theorems of Olesen and Pedersen for On and of Enomoto, Fujii and Watatani for OA as particular cases.

Contents 1. 2. 3. 4. 5. 6. 7. 8.

KMS States for Graded C ∗ -Algebras . . . . . . . . . . Graded Algebras Given by Partial Actions . . . . . . . Algebras Graded over the Free Group . . . . . . . . . Partial Actions of the Free Group . . . . . . . . . . . . Toeplitz–Cuntz–Krieger Algebras for Infinite Matrices Partial Representations . . . . . . . . . . . . . . . . . Unital and Non-Unital Algebras . . . . . . . . . . . . Scaling States and the Partition Function Z(β) . . . . .

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227 230 232 235 236 240 243 245

∗ Partially supported by CNPq ∗∗ Supported by the Australian Research Council ∗∗∗ Current address: Mathematics and Statistics, University ofVictoria, P.O. Box 3045 STN CSC,Victoria,

B.C., Canada V8W 3P4

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9. Existence of Finite Type Scaling States . . . . . . . . . . . . . . . 10. Irreducible Matrices and the Fixed-Target Partition Function Zy (β) 11. The Structure of T
A . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . 12. Invariant and Subinvariant States on Q 13. States on Q . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14. The Fixed-Source-and-Target Partition Function Zxy (β) . . . . . . 15. Energy Bounded Below . . . . . . . . . . . . . . . . . . . . . . . 16. An Example of Behavior at the Critical Point . . . . . . . . . . . 17. KMS States on OA . . . . . . . . . . . . . . . . . . . . . . . . . 18. The Finite Dimensional Case . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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250 252 254 258 263 266 268 270 271 274 276

Our motivation for the present work stems from the perception that the Cuntz–Krieger algebras for infinite matrices studied in [EL] naturally provide C ∗ -dynamical systems with interesting KMS state structure. The main phenomena in which we are interested are those intrinsically associated to the infinite dimensionality of the matrices but our approach also gives fresh insight into some salient features that have not been observed or emphasized enough even in the finite dimensional case, particularly with respect to the consideration of nonperiodic dynamics and the symmetries of the equilibrium states. The earliest ancestor of our results is an intriguing theorem of Olesen and Pedersen’s [OP], which appeared in the late seventies amidst a flurry of examples and counterexamples triggered by the advent of the Cuntz algebras On [C], stating that the periodic gauge action on On admits a unique KMS (equilibrium) state, whose inverse temperature is β = log n. This was followed by a result of Evans [Ev, 2.2] relating the existence of KMS states for a non-necessarily periodic gauge action of On with the fact that certain parameters defining the action have the same sign. Shortly afterwards Cuntz and Krieger [CK] came up with their C ∗ -algebras OA and, in the ensuing flurry, the theorem of Olesen and Pedersen was duly generalized by Enomoto, Fujii, and Watatani [EFW], who proved, among other things, that when the matrix A is irreducible (and not a permutation) the gauge action on OA admits a unique KMS state, at inverse temperature equal to the logarithm of the spectral radius of A. Despite the explicitness of the computations involved those early results have up to until recently been largely regarded as curious counterexamples rather than as sources of interesting new phenomena to be explored. Part of the reason for this derives from [OP, Theorem 1], according to which the dynamics involved are “nonphysical” because they have no (weak, approximate) Hamiltonian. Recently, however, the interest in KMS states of Cuntz-Krieger algebras has been renewed, mainly in [PWY] where results along the lines of [EFW] have been obtained for periodic full dynamics on unital C ∗ -algebras, and where the KMS condition is linked to a variational principle for the entropy. The main purpose and the methods of the present work are of a different nature: we aim to study the KMS equilibrium states of C ∗ -dynamical systems that are inspired on the periodic gauge action of R on OA , but which are more general in three important aspects: • we allow the matrix A to be countably infinite as in [EL]; • we focus on the Toeplitz extension TA rather than on OA itself; and, • while still dealing with dynamics having the generating partial isometries as eigenvectors, we allow the possibility of different eigenvalues and thus of nonperiodicity, as in [Ev].

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In order to deal effectively with the new situation we must first spend some effort in developing the necessary approach and technical tools, and this involves realizing our C ∗ -algebras as crossed products by partial actions of a countably generated free group, and characterizing KMS states of such crossed products in terms of a certain invariance property of probability measures under the partial action, using techniques analogous to those of [L] for semigroup actions. There is no significant extra cost in carrying out this first task in the slightly more general context of C ∗ -algebras that are topologically graded over free groups, and we do so in the first few sections. We then specialize to our main setting, which we would like to describe briefly next. Given a matrix A of zeros and ones over a countable set G, we consider certain one-parameter groups of gauge automorphisms of three closely related C ∗ -algebras, namely T OA , TA , and OA . OA is the generalized Cuntz–Krieger algebra introduced by the authors in [EL] for an arbitrary infinite 0–1 matrix A. T OA was also introduced in [EL] as an auxiliary tool to study OA , and TA is the Toeplitz extension of OA , once the latter is seen as a Cuntz–Pimsner algebra as shown by Szyma´nski [Sz]. All of the above three algebras have canonical generating sets consisting of partial isometries, say {sx }x∈G and, given a choice of positive real numbers {N (x)}x∈G , there are one-parameter groups of gauge automorphisms satisfying σt (sx ) = N (x)it sx ,

t ∈ R.

Clearly these are subgroups of the canonical gauge action of the torus TG . Since KMS states are known to be σ -invariant, one expects them to be invariant under the (compact) closure of {σt } inside this torus, and thus to factor through the conditional expectation onto the fixed point algebra. One of the biggest surprises we find here is that the KMS states under analysis are shown (Theorem 8.2) to factor through the conditional expectation onto a much smaller subalgebra which can be identified as the fixed point algebra for a coaction of the infinitely generated free group. We do not explore this coaction here except for the fact that it leads to a highly useful conditional expectation. Nevertheless it is remarkable that the KMS condition seems to impose the preservation of symmetries way beyond what is expected at first. The small subalgebra mentioned above is actually a commutative algebra and hence the search for KMS states boils down to a study of measures on its spectrum. Especially when dealing with the case of TA , to which we dedicate the biggest share of our attention, there is a natural dichotomy breaking the spectrum into a “finite” part, denoted f , and an “infinite” part ∞ (see 8.6). The measures considered are therefore classified in finite type or infinite type according to whether they assign full mass to the finite or to the infinite part of the spectrum. The behavior of KMS states at inverse temperature β, and hence also of the measures which determine them, strongly depends on the relative position of β with respect to three critical inverse temperatures. In order to describe these let us denote by PA the set of all admissible words (with respect to the given matrix A) in the alphabet G, by y xy PA the admissible words ending in y, and by PA the admissible words beginning in x and ending in y. We then introduce three (families of) Dirichlet series of one variable β, namely Z(β) =

 µ∈PA

N (µ)−β ,

Zy (β) =

 y µ∈PA

N (µ)−β

Zxy (β) =

 xy µ∈PA

N (µ)−β

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where N (µ) is defined by N (µ1 ) · · · N (µk ) when µ is the admissible word µ = µ1 · · · µk . Every Dirichlet series has an abscissa of convergence which marks the lower end of its interval of convergence. Accordingly we denote by βc the abscissa of convergence of Z(β) and this turns out to be the first important critical inverse temperature. We prove that all KMSβ states correspond to finite type measures for β above this critical point. In addition we are able to describe these measures in very concrete terms and hence all KMSβ states are concretely exhibited (see 9.7). In the case of an irreducible matrix A the Dirichlet series Zy , for y ∈ G, satisfy a “solidarity” property in the sense that, for a given β, either they all converge or they all diverge. Therefore there is a single abscissa of convergence, denoted β˙c , which does not depend on y. Another solidarity property holds among the Zxy , in turn defining a third critical value β¨c . Since each of Z(β), Zy (β), and Zxy (β) is a subseries of the previous one it is clear that their abscissa of convergence satisfy β¨c ≤ β˙c ≤ βc . Still speaking of the irreducible case we prove (Theorem 10.6) that all KMSβ states are of infinite type for β below β˙c and that there are no KMSβ states at all for β below β¨c (Theorem 14.5). The following diagram illustrates these results:

β¨c

0 None at all (Theorem 14.5)

β˙c

βc

Only infinite type (Theorem 10.6)

∞ Only finite type (Corollary 9.7)

KMS states and critical inverse temperatures

The results sketched in this diagram are the strongest results we can offer under the sole assumption that A is irreducible but there are several strengthenings we can provide under extra hypotheses. For instance, we show (Theorem 15.2) that there are no KMSβ -scaling states at all for β < β˙c under the hypothesis that inf x∈G N (x) > 1. Unfortunately there is nothing we can say about the interval between β˙c and βc , which remains conspicuously absent from our conclusions. Not being able to deal with it we may at least identify a rather common situation in which it collapses (Proposition 15.4), and this is when there is a finite target set, i.e. a finite set {y1 , . . ., yn } ⊆ G such that for every x ∈ G one has A(x, yi ) = 1 for at least one i (see (10.1, FTS)). A related problem which we could not resolve is whether or not finite type KMS states can coexist with infinite type ones in the case of an irreducible matrix. We therefore leave these as open problems. Nevertheless under all of the hypotheses mentioned so far our theory gives a complete description of KMS states for all inverse temperatures, except for the critical inverse temperature βc . At βc quite different things can happen. There are examples in which there is a single KMSβc state (Theorem 18.4) but there are also examples in which infinitely many such states exist (Sect. 16). Even though our main focus is on TA we can provide some useful information about the KMS states on OA as well. Since OA is a covariant quotient of TA , the set of KMS states on OA correspond to the set of KMS states on TA which factor through OA . We therefore take up the problem of characterizing the latter set (Theorem 17.2) giving several equivalent ways to describe it.

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For a finite matrix A one may obviously say a lot more than in the general case. As it turns out we give a complete characterization of all KMS states on TA for a finite irreducible matrix (Theorem 18.4). With respect to the KMS states on OA for a finite A, we completely characterize its KMS states even if A is not irreducible (Theorem 18.5). In particular when considering the periodic gauge action, i.e. when N (x) = e for all x, our methods can be easily applied to recover the results of Olesen, Pedersen [OP], and Evans [Ev] on the KMS states of On as well as the result obtained by Enomoto, Fujii, and Watatani [EFW] for the gauge action on OA . After the present paper circulated as a preprint we have found out about a recent result by J. Zacharias [Z, 4.3] in which KMS states on OA are characterized for non-periodic gauge actions in the case of a finite irreducibe matrix A. In the course of the research reported here we were deeply influenced by Vere–Jones paper [V] which generalizes the classical Perron–Frobenius theorem to the case of infinite matrices. But, because of the difference between our emphasis on Dirichlet series and Vere-Jones’s emphasis on power series, among other reasons, we were often impeded to use these results in a straightforward way. The present work culminates a project started in December 1996 and continued through several short visits of R.E. to Newcastle and of M.L. to Florian´opolis, and we would both like to thank the members of both departments for the hospitality provided to the visitor of turn. We also gratefully acknowledge funding from CNPq and from the Australian Research Council. 1. KMS States for Graded C ∗ -Algebras Let B be a C ∗ -algebra and let G be a discrete group. We shall say B is graded over G, or simply G-graded, if we are given a linearly independent family {Bg }g∈G of closed linear subspaces of B such that, for all g, h ∈ G, (i) Bg Bh ⊆ Bgh , (ii) Bg∗ = Bg −1 , and  (iii) g∈G Bg is dense in B. The main examples are given by crossed products algebras, including the case of partial actions which will be discussed in some detail below. We shall fix throughout a G-graded C ∗ -algebra B. Moreover we will fix a strongly continuous one-parameter group σ = {σt }t∈R of automorphisms of B such that each σt restricted to each Bg is a multiple of the identity. One necessarily has that for each g in G there is a positive real number N (g) such that σt (b) = N (g)it b,

t ∈ R, b ∈ Bg .

(1.1)

We shall also assume that N is a group homomorphism from G to the multiplicative ∗ of positive reals (this is in fact necessarily the case if B B = {0} for all g group R+ g h and h). Obviously σ is determined by N . However under the present hypothesis it is not clear whether there exists a one-parameter group σ satisfying (1.1) for each group homomor∗ . This existence question will be dealt with when we discuss the phism N : G → R+ case of crossed-products by partial actions below. Nevertheless, even though partial actions are among our main examples, we stress that we are only assuming, for the time being, that B is G-graded and that σ satisfies ∗. (1.1) for some group homomorphism N : G → R+

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Recall that an element b ∈ B is said to be entire analytic (with respect to σ ) [BR, 2.5.20] if the map t ∈ R → σt (b) extends to an entire analytic function on the complex plane. For simplicity we shall refer to entire analytic elements simply as analytic elements. Also recall from [BR, 5.3.1] that a state ψ on B is said to be a σ -KMS state at value β ∈ (0, ∞) – the inverse temperature in Mathematical Physics terminology – or simply a KMSβ state if for any pair of elements a and b in a given norm-dense σ -invariant *-subalgebra of analytic elements of B one has ψ(aσiβ (b)) = ψ(ba).

(1.2)

By the last sentence of [BR, 5.3.7] one has that, for a KMS state, the identity above in fact holds for every a in B and every analytic element b in B. In the special case β = ∞, KMSβ states are called ground states and are defined in a slightly different fashion but by [BR, 5.3.19] they are the states on B such that for every pair of analytic elements a and b in B, sup |ψ(aσz (b))| < ∞.

(1.3)

Imz≥0

In order to verify a state to be a ground state the most economical way is to verify (1.3) only for a and b in a norm-dense subspace of analytic elements of B (cf. [Pe, 8.12.3] and [L, Remark 11]). Given g ∈ G it is clear from (1.1) that each b in Bg is analytic and moreover σz (b) = N (g)iz b,

(1.4) 

for z ∈ C. It follows by linearity that the algebraic direct sum g∈G Bg consists of analytic elements. In addition the latter is clearly a norm-dense σ -invariant *-subalgebra of B. We will therefore use this algebra whenever we need to verify the KMS condition, both for finite and infinite values of β. The following is a characterization of σ -KMS states on B. 1.5. Proposition. Suppose that B is G-graded and that σ satisfies (1.1) for a group ∗ . Let ψ be a state on B and let β ∈ (0, ∞). Then homomorphism N : G → R+ (i) ψ is a KMSβ state if and only if ψ(ab) = N (g)β ψ(ba) whenever a ∈ B, g ∈ G, and b ∈ Bg . (ii) ψ is a ground state if and only if ψ(BBg ) = {0} whenever N (g) < 1. Proof. It will be convenient to keep in mind that plugging z = iβ in (1.4) gives σiβ (b) = N (g)−β b. Suppose that ψ is a KMSβ state and let a ∈ B and b ∈ Bg . We then have ψ(ba) = ψ(aσiβ (b)) = N (g)−β ψ(ab), proving the forward implication in (i). Conversely, suppose that ψ satisfies the equality in (i). Given a ∈ B and b ∈ Bg we then have ψ(ba) = N (g)−β ψ(ab) = ψ(aσiβ (b)), proving that (1.2)  holds for our choice of a and b. By linearity we see that the same is true for any b ∈ g∈G Bg . Since

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the latter is a norm-dense σ -invariant *-subalgebra of analytic elements we see that ψ is a KMSβ state. Let us now deal with (ii). Given a ∈ B and b ∈ Bg we have |ψ(aσz (b))| = |N (g)iz ψ(ab)| = N (g)−Imz |ψ(ab)|.

(†)

Observe that this is bounded on the upper half plane as a function of z if and only if either N(g) ≥ 1 or ψ(ab) = 0. Thus if ψ is a ground state and N (g) < 1 we must have ψ(ab) = 0, proving the forward implication in (ii). Conversely, if ψ(BBg ) = 0 whenever N (g) < 1 then (†)  is always bounded on Im z ≥ 0. It follows by linearity that (1.3) holds for any b ∈ g∈G Bg . Since the latter is a norm-dense set of analytic elements we see that ψ is a ground state.   It should be noted that (1.5.i) implies that, for β < ∞, a KMSβ state restricted to Be must be a trace. In contrast, ground states need not restrict to traces on Be . In fact, if σ is the trivial action of R on B, corresponding to N (g) ≡ 1, then any state on B is a ground state and one can clearly manufacture examples in which ψ|Be is not a trace. Recall from [E2, 3.4] that B is said to be topologically G-graded if there exists a positive contractive conditional expectation E : B → Be

 which vanishes on every Bg for g = e. From [E2, 3.5] it follows that g∈G Bg is a topological direct sum in the sense that the canonical projections onto the Bg ’s extend to bounded linear maps Eg : B → Bg . If B is topologically G-graded and we are given a state φ on Be the composition ψ := φ ◦ E is a state on B. Our next result is intended to discuss the conditions under which ψ is a σ -KMS state on B. 1.6. Proposition. Assume that B is topologically G-graded with conditional expectation ∗ . Let φ be a state E and that σ satisfies (1.1) for a group homomorphism N : G → R+ on Be and set ψ = φ ◦ E. Also let β ∈ (0, ∞). Then (i) ψ is a KMSβ state if and only if φ(ab) = N (g)β φ(ba) for every g ∈ G, a ∈ Bg −1 , and b ∈ Bg . (ii) ψ is a ground state if and only if φ(Bg −1 Bg ) = {0} whenever N (g) < 1. Proof. The forward implication in (i) follows immediately from  (1.5.i). Conversely let a ∈ B and b ∈ Bg . By considering first the case in which a ∈ g∈G Bg it is easy to see that E(ab) = Eg −1 (a)b, and E(ba) = bEg −1 (a). Therefore ψ(ab) = φ(E(ab)) = φ(Eg −1 (a)b) = N (g)β φ(bEg −1 (a)) = N (g)β φ(E(ba)) = N (g)β ψ(ba). That ψ is a KMSβ state now follows from (1.5.i). The forward implication in (ii) follows immediately from (1.5.ii). Conversely let a ∈ B and b ∈ Bg with N (g) < 1. We then have ψ(ab) = φ(E(ab)) = φ(Eg −1 (a)b) = 0. Thus ψ is a ground state by (1.5.ii).

 

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2. Graded Algebras Given by Partial Actions One of the main sources of examples of topologically G-graded algebras is the theory of partial actions. Accordingly this section is devoted to reviewing the results of this theory which are relevant to us. The reader should consult [E1, M, E3, E4] for more details. Recall that a partial action of a discrete group G on a C ∗ -algebra A is a pair   $ = {Dg }g∈G , {θg }g∈G such that, for each g in G, Dg is a closed two sided ideal of A, θg : Dg −1 → Dg is a *-isomorphism, and for all g and h in G one has (i) De = A and θe is the identity automorphism of A, (ii) θg (Dg −1 ∩ Dh ) = Dg ∩ Dgh , and (iii) θg (θh (a)) = θgh (a) for all a ∈ Dh−1 ∩ Dh−1 g −1 . Throughout this section we shall fix a partial action as above. Our goal is to construct a G-graded algebra from this data. Let L denote the set of all functions f : G → A such that f (g) ∈ Dg for all g ∈ G and moreover g∈G f (g) < ∞. Clearly L is a Banach space under the norm f  =



f (g) ,

f ∈ L.

g∈G

 It is often convenient to denote by g∈G ag δg the function f such that f (g) = ag . Employing this notation we define a Banach *-algebra structure on L by means of the operations   (aδg ) ∗ (bδh ) = θg θg1 (a)b δgh , (aδg )∗ = θg1 (a ∗ )δg

and (2.1)

1

for a ∈ Dg and b ∈ Dh . See the references given above for the proof that L is indeed a Banach *-algebra under these operations. 2.2. Definition. The crossed-product of the C ∗ -algebra A by the group G under the partial action $ is the enveloping C ∗ -algebra of the Banach *-algebra L described above. We denote it as A $ G, or simply A  G if $ is understood. 2.3. Proposition. The collection of subspaces {Bg }g∈G of A  G given by Bg = Dg δg makes A  G into a topologically G-graded algebra. Proof. The construction of A  G is precisely that of the cross-sectional C ∗ -algebra of the Fell bundle formed by the Bg ’s under the operations defined by (2.1). The result is then a consequence of [E2, 3.2 and 2.9].  

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Observe that A is canonically isomorphic to Be via the map a → aδe . We will therefore identify A and Be and hence think of the conditional expectation as the unique bounded map E : A  G → A such that E



= ae .

ag δg

g∈G

Let us now deal with the question of defining the dynamics on A  G in terms of a given group homomorphism N . ∗ be a group homomorphism. Then there exists a 2.4. Proposition. Let N : G → R+ strongly continuous one-parameter group σ of *-automorphisms of A  G such that

σt (b) = N (g)it b for all t ∈ R, g ∈ G, and b ∈ Bg . Proof. For each t in R consider the linear operator σt on L given by σt



ag δg

g∈G

=



N (g)it ag δg .

g∈G

It is easy to see that each σt is an isometric *-isomorphism of L. It is also clear that σt σs = σt+s for all t and s in R, so that σ gives a group homomorphism σ : R → Aut(L), which is obviously strongly continuous. The result now follows by extending each σt to a *-isomorphism of the enveloping C ∗ -algebra A  G.   This puts us in the context of Sect. 1 and so we may apply (1.6) to characterize the KMS states on A  G that factor through the conditional expectation E described above. The following result facilitates checking conditions (i) and (ii) of (1.6): 2.5. Proposition. Let $ be a partial action of the discrete group G on a C ∗ -algebra A ∗ be a and consider the standard grading {Bg }g = {Dg δg }g of A  G. Let N : G → R+ group homomorphism. (i) If β ∈ (0, ∞) and g ∈ G then φ(ab) = N (g)β φ(ba), ∀a ∈ Bg 1 , b ∈ Bg

(†)

if and only if φ is a trace and φ(θg (a)) = N (g) β φ(a), ∀a ∈ Dg 1 . (ii) Bg 1 Bg = Dg 1 .

(‡)

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Proof. We begin with the forward implication in (i). That φ is a trace follows from (†) applied to the case g = e. Given a ∈ Dg 1 choose an approximate identity {ui }i for Dg and observe that φ(θg (a)) = lim φ(ui θg (a)) = lim φ((ui δg ) (aδg 1 )) i

i

= N (g) β lim φ((aδg 1 ) (ui δg )) = N (g) β lim φ(aθg 1 (ui )) = N (g) β φ(a). i

i

Conversely, suppose that φ is a trace satisfying (‡). Given a ∈ Dg 1 and b ∈ Dg we have that aδg 1 ∈ Bg 1 and bδg ∈ Bg and       φ (aδg 1 )(bδg ) = φ θg 1 (θg (a)b) = N (g)β φ θg (a)b     = N (g)β φ b θg (a) = N (g)β φ (bδg )(aδg 1 ) . Since aδg 1 and bδg are generic elements of Bg 1 and Bg , respectively, we have proven (†). We leave the proof of (ii) to the reader.   Observe that (2.5.i.(‡)) says that, on the suitable domain, φ is scaled when composed with θg . If one considers a global action (i.e. a partial action for which each Dg = A, as in the classical situation) then this scaling property cannot hold in non-trivial situations because the norm of φ is necessarily invariant. Nevertheless if one deals with partial actions this obstruction disappears allowing for many interesting examples as we shall see below. See also [L] for examples arising as semigroup crossed products. 3. Algebras Graded over the Free Group We will be mostly interested in the case where the group G is the free group F on a (possibly infinite) set G of generators. When speaking of F we will often employ the following standard notations: • If g ∈ F we will denote by |g| the length of g, that is, the number of generators appearing in the reduced decomposition of g. • F+ will refer to the positive cone of F, that is, the subsemigroup of F generated by G, including the unit group element. • We will usually denote the elements of F by g, h, k; the elements of G by x, y, z; and the elements of F+ by µ, ν. Gradings over F occasionally satisfy two additional properties which we would now like to recall from [E2]. 3.1. Definition. A grading {Bg }g∈F of a C ∗ -algebra B is said to be: (i) semi-saturated if Bgh = Bg Bh (closed linear span) whenever g, h ∈ F are such that |gh| = |g| + |h|. (ii) orthogonal if Bx∗ By = {0} whenever x, y ∈ G and x = y. The following lemma is the main result of this section and is the key to our characterization of KMS states on Cuntz–Krieger algebras. 3.2. Lemma. Let B be a C ∗ -algebra which is F-graded by means of a semi-saturated orthogonal grading {Bg }g∈F . Also let σ be a strongly continuous one-parameter group ∗. of automorphisms of B satisfying (1.1) for some group homomorphism N : F → R+ Suppose that N (x) > 1 for all x ∈ G and let ψ be a KMSβ state on B for β in the interval (0, ∞]. Then ψ(Bg ) = {0} for all g ∈ F with g = e.

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Proof. Initially observe that the hypothesis on N implies that N (µ) > 1 for all µ ∈ F+ \ {e}. Since there exists a slight asymmetry between the cases of finite and infinite β let us first assume that β ∈ (0, ∞). In this case we claim that ψ(Bµ ) = {0} for every µ ∈ F+ \ {e}. To see this note that Bµ = Be Bµ (closed linear span) by [BMS, 1.7], so it suffices to show that ψ(ab) = 0 whenever a ∈ Be and b ∈ Bµ . We then have, using (1.5.i) twice, that ψ(ab) = N (µ)β ψ(ba) = N (µ)β N (e)β ψ(ab) = N (µ)β ψ(ab). But, since N (µ) = 1 and β = 0, we have that ψ(ab) = 0 as claimed. Clearly we also have that ψ(Bµ 1 ) = ψ(Bµ∗ ) = ψ(Bµ ) = {0}, so that we have proven the statement for all g ∈ F+ ∪ F+1 \ {e}. Since our grading is semi-saturated and orthogonal we have that Bg = {0} for every g ∈ F which is not of the form g = µν 1 , where µ, ν ∈ F+ and |g| = |µ| + |ν| as a moment’s reflection will easily show. We therefore assume that g is of the above form. We will proceed by induction on m = min{|µ|, |ν|}. If m = 0 then either g = µ or g = ν 1 and the conclusion follows as above. So assume that m ≥ 1 and write µ = xµ and ν = yν  , where x, y ∈ G and µ , ν  ∈ F+ . Since the grading is semi-saturated we have that Bg = Bµ Bν∗ = Bx Bµ Bν∗ By∗ and hence we will be done once we prove that ψ(bx bµ bν∗ by∗ ) = 0 whenever bi ∈ Bi for i = x, y, µ , ν  . We have by (1.5.i) that ψ(bx bµ bν∗ by∗ ) = N (y 1 )ψ(by∗ bx bµ bν∗ ).

(‡)

If on the one hand x = y then by∗ bx = 0 by orthogonality and (‡) vanishes. If on the other hand x = y then by∗ bx bµ bν∗ ∈ By∗ Bx Bµ Bν∗ ⊆ Bµ Bν∗ . By the induction hypothesis ψ vanishes on Bµ Bν∗ and so again (‡) vanishes. This concludes the proof of the case β < ∞. Assume now that β = ∞ and hence that ψ is a ground state. As before we only need to prove that ψ(Bg ) = 0 for a nontrivial g of the form µν 1 with µ, ν ∈ F+ and |g| = |µ| + |ν|. If ν = e then N (ν) < 1 and Bg = Bµ Bν 1 by semi-saturatedness. So we have that ψ(Bg ) = ψ(Bµ Bν 1 ) = {0} by (1.5.ii). If ν = e then µ = e and ψ(Bg ) = ψ(Bµ∗ ) = ψ(Bµ 1 ) = {0} as seen above.   Observe that in the above proof, when considering finite values of β, we only needed that N (µ) = 1 for µ ∈ F+ \ {e} and hence the result does hold under this weakened hypothesis. However we have not seen how to reach the same conclusion for ground states. In the topologically graded case we may use (1.6) and (3.2) to get the following very precise characterization of KMS states: 3.3. Theorem. Let B be a C ∗ -algebra which is topologically F-graded by means of a semi-saturated orthogonal grading {Bg }g∈F . Also let σ be a strongly continuous one-parameter group of automorphisms of B satisfying (1.1) for a given group homomorphism

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∗ . Suppose that N (x) > 1 for all x ∈ G and let β ∈ (0, ∞]. Then the N : F → R+ formula φ → φ ◦ E

defines an affine homeomorphism from the set S defined below onto the set of KMSβ states on B. (i) If β ∈ (0, ∞) then S is the set of traces φ on Be such that φ(ab) = N (x)β φ(ba),

∀x ∈ G, a ∈ Bx 1 , b ∈ Bx .

(ii) If β = ∞ then S is the set of states φ of Be such that φ(Bx Bx 1 ) = {0},

∀x ∈ G.

Proof. Under the hypothesis that φ lies in the set S described in (i) we claim (see (1.6.i)) that φ(ab) = N (g)β φ(ba), ∀g ∈ F, a ∈ Bg 1 , b ∈ Bg . Clearly this holds for g = e because φ is a trace. Next consider the case g = x 1 ∈ G 1 . So let a ∈ Bg 1 and b ∈ Bg . Then b ∈ Bx 1 and a ∈ Bx so that φ(ba) = N (x)β φ(ab) from where we obtain φ(ab) = N (x 1 )β φ(ba) = N (g)β φ(ba). Proceeding by induction on |g| let |g| > 1 and write g = hx with x ∈ G ∪ G 1 and |g| = |h| + 1. Given a ∈ Bg 1 and b ∈ Bg suppose that b = bh bx , where bh ∈ Bh and bx ∈ Bx . Then abh ∈ Bx 1 and φ(ab) = φ(abh bx ) = N (x)β φ(bx abh ) = . . . Moreover bx a ∈ Bh 1 and hence, by the induction hypothesis, the above equals . . . = N (x)β N (h)β φ(bh bx a) = N (g)β φ(ba). Since the grading is semi-saturated the linear combinations of the b’s considered is dense in Bg and hence the claim is proven. By (1.6.i) φ is a KMSβ state. On the other hand, under the hypothesis that φ satisfies the property described in (ii) we claim (see (1.6.ii)) that φ(Bg 1 Bg ) = {0},

∀g ∈ G such that N (g) < 1.

We have already mentioned that, as a consequence of semi-saturatedness, Bg = {0} unless g = µν 1 , where µ, ν ∈ F+ and |g| = |µ| + |ν|. We therefore suppose that g has this form. Since N (g) = N (µ)N (ν) 1 < 1 and N (µ) ≥ 1 we must have ν = e. So write ν = xν  with x ∈ G and ν  ∈ F+ . By semi-saturatedness we have Bg 1 Bg = Bν Bµ 1 Bµ Bν

1

= Bx Bν  Bµ 1 Bµ Bν  1 Bx

1

⊆ Bx Be Bx

1

⊆ Bx Bx 1 ,

whence φ(Bg 1 Bg ) = {0}. By (1.6.ii) φ is a ground state. Since φ = (φ ◦ E) |Be the correspondence is 1–1. In order to show surjectiveness let ψ be a KMSβ state on B. Then,  letting φ = ψ|Be , we have that ψ = φ ◦ E, which is easily proven by checking on g∈G Bg with the help of (3.2). Finally we have that φ is in S by the forward implications in (1.6). Clearly φ → φ ◦ E is an affine map. Moreover this is obviously a continuous map (with respect to the weak topologies as usual). Since S is compact we actually have a homeomorphism.  

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4. Partial Actions of the Free Group In this section we will show how to put together the results of the previous sections in order to determine the KMS states on the crossed product C ∗ -algebra resulting from a partial action of the free group with suitable properties. We therefore fix throughout this section a C ∗ -algebra A and a partial action   $ = {Dg }g∈F , {θg }g∈F of the free group F on a possibly infinite set of generators G. The crossed-product algebra A  F is therefore topologically graded via the subspaces Bg = Dg δg by (2.3). We begin by determining conditions for this grading to be semi-saturated and orthogonal. 4.1. Proposition. The above grading of A  F is: (i) semi-saturated if and only if Dgh ⊆ Dg whenever |gh| = |g| + |h|. (ii) orthogonal if and only if Dx ∩ Dy = {0} for all x, y ∈ G with x = y. Assume, in addition, that A is abelian with spectrum a locally compact space X and that $ is obtained by means of a partial action   α = {Ug }g∈F , {αg }g∈F of F on X (cf. [EL, Sect. 2], [E4]). Then the grading of A  F is: (a) semi-saturated if and only if Ugh ⊆ Ug whenever |gh| = |g| + |h|. (b) orthogonal if and only if Ux ∩ Uy = ∅ for all x, y ∈ G with x = y. Proof. Given g, h ∈ F we have

    Bg Bh = (Dg δg )(Dh δh ) = θg θg1 (Dg )Dh δgh = θg Dg 1 Dh δgh ,

  so that our grading is semi-saturated if and only if θg Dg 1 ∩ Dh = Dgh when |gh| = |g| + |h|. However it is an axiom for partial actions that θg (Dg 1 ∩ Dh ) = Dg ∩ Dgh . So semi-saturatedness is equivalent to Dg ∩ Dgh = Dgh which is the same as saying that Dgh ⊆ Dg . If we plug g = x 1 and h = y in the equation displayed above, where x, y ∈ G and x = y, then we see that our grading is orthogonal if and only if Dx ∩ Dy = {0}. As for the part in which A is assumed abelian note that Dg = C0 (Ug ) and also that Dg ∩ Dh = C0 (Ug ∩ Uh ) for all g and h. Therefore Dgh ⊆ Dg ⇐⇒ Ugh ⊆ Ug and Dx ∩ Dy = {0} ⇐⇒ Ux ∩ Uy = {0}.   It is perhaps worth mentioning that the condition |gh| = |g| + |h| in the free group means that g ≤ gh in the sense that g is an initial segment in the reduced decomposition of gh. Therefore condition (4.1.i) above can be rephrased as saying that Dg is decreasing as a function of g in the sense that g ≤ k ⇒ Dk ⊆ Dg .

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4.2. Definition. We shall say that $ is a semi-saturated partial action if condition (4.1.i) above holds. We shall say that $ is an orthogonal partial action when (4.1.ii) is satisfied. Having understood the relationship between graded algebras and partial actions regarding semi-saturatedness and orthogonality we may now present our main abstract result. 4.3. Theorem. Let $ be a semi-saturated orthogonal partial action of the free group F on a C ∗ -algebra A. On A  F consider the standard grading {Bg }g∈F and conditional expectation E : A  F → A. Also let {N (x)}x∈G be a collection of real numbers in the interval (1, ∞). Then there exists a unique strongly continuous one-parameter group σ of automorphisms of A  F such that σt (b) = N (x)it b,

∀x ∈ G, b ∈ Bx .

The σ -KMS states at inverse temperature β on A  F are precisely those of the form ψ = φ ◦ E, where φ is a state on A satisfying: (i) if β < ∞ : φ is a trace and φ(θx (a)) = N (x) β φ(a) for all x ∈ G and all a ∈ Dx 1 . (ii) if β = ∞ : φ(Dx ) = {0} for all x ∈ G. ∗ and hence we may apply Proof. Extend N to a group homomorphism N : F → R+ (2.4) to conclude that σ exists as required. Given that the action is semi-saturated, so is the grading by (4.1.i). With this it is easy to see that σ is uniquely determined by the identity displayed in the statement. We may now apply (3.3) and thus all we need to do is show that the set S described in (3.3.i–ii) can be alternatively characterized by conditions (i–ii) here, but this is precisely the purpose of (2.5).  

5. Toeplitz–Cuntz–Krieger Algebras for Infinite Matrices Throughout this and the remaining sections of this work we shall fix a countable (meaning finite or countably infinite) set G and a matrix A = {A(x, y)}x,y∈G with entries in the set {0, 1}, having no identically zero rows. It is in terms of A that we will introduce the algebras which will be the object of our main applications. A is the universal Recall from [EL] that the Toeplitz–Cuntz–Krieger algebra1 TO ∗ unital C -algebra generated by a family of partial isometries {sx }x∈G subject to the requirement that their initial projections qx = sx∗ sx and final projections py = sy sy∗ satisfy the following conditions for all x and y in G: CK1 ) qx qy = qy qx , CK2 ) sx∗ sy = 0, if x = y, CK3 ) qx sy = A(x, y)sy . A is the crossed-product C ∗ -algebra for a partial group action According to [EL, 4.6] TO which we would now like to briefly describe. See [EL] for full details. 1 This algebra was denoted TO in [EL] but will be denoted TO A here for reasons which will become A apparent soon.

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TOA be the closed subset of the compact topological space 2F 5.1. Definition. Let2  given by    ξ ∈ 2F : e ∈ ξ, ξ is convex, 
TOA = ifg ∈ ξ there is at most one x ∈ G such that gx ∈ ξ, and .    ifg ∈ ξ, y ∈ G, and gy ∈ ξ thengx 1 ∈ ξ ⇔ A(x, y) = 1
TOA given by For each g ∈ F let 5g be the clopen subset of 
TOA : g ∈ ξ } 5g = {ξ ∈  and consider the homeomorphism αg : ξ ∈ 5g Then

−→ gξ ∈ 5g .

1

  α := {5g }g∈F , {αg }g∈F


TOA in the sense of [EL, Sect. 2] (see also [M, E3], is a partial group action of F on 
TOA ), namely and [E4]). This induces a partial action of F on C(   $ = {Dg }g∈G , {θg }g∈G , given by Dg = C0 (5g ) and θg : f ∈ Dg

1

→ f ◦ αg

1

∈ Dg .

A is isomorphic to C(
TOA ) The already mentioned Theorem 4.6 of [EL] asserts that TO  F under an isomorphism that maps each sx to 15x δx , where 15x is the characteristic function of 5x . In order to define the next two algebras which are relevant to our study it is convenient to introduce the following notation: given finite subsets X, Y ⊆ G we let   A(X, Y, z) = A(x, z) (1 − A(y, z)), z ∈ G, x∈X

and q(X, Y ) =

y∈Y

 x∈X

qx



(1 − qy ).

y∈Y

  Observe that the 0–1 vector A(X, Y, z) z∈G is simply the coordinatewise product of the row-vectors of A indexed by X, and the boolean negation of the row-vectors indexed by Y .
A is the quotient of TO A Recall from [EL] that the (unital) Cuntz–Krieger algebra O obtained by imposing the following extra relation in addition to CK1−3 :  CK4 ) q(X, Y ) = z∈G A(X, Y, z)pz whenever X, Y ⊆ G are finite sets such that A(X, Y, z) is finitely supported as a function of z. 2


TO here. This space was denoted τA in [EL] but will be denoted  A

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As explained in the first section of [EL], condition CK4 is formally derived from multiplying together sufficiently many occurrences of the Cuntz–Krieger relations [CK] so that the infinite sums involved become finite.
A " C(
OA )  F, where 
OA is the α-invariTheorem 7.10 of [EL] asserts that O
ant subset of TOA obtained by taking the closure of the set of unbounded elements (cf. Definition 5.5 in [EL]). As before there is an isomorphism which maps each sx to 15x ∩
O δx . A We shall be concerned here with yet another C ∗ -algebra, denoted T
A (see also [FLR, A and O
A in the sense that the quotient map TO A → O
A Sz]), which sits in between TO alluded to above factors through T
A . 5.2. Definition. Given a countable set G and a 0–1 matrix A = {A(x, y)}x,y∈G with no identically zero rows we denote by T
A the universal unital C ∗ -algebra generated by a family of partial isometries {sx }x∈G subject to conditions CK1−3 and CK04 ) q(X, Y ) = 0 whenever X and Y are finite subsets of G such that A(X, Y, z) is identically zero as a function of z.
A as the Cuntz–Pimsner algebra [Pi] of In [Sz, Theorem 5] Szyma´nski has realized O a bimodule, and has shown that T
A is the corresponding Toeplitz extension [Sz, Theorem 10] (the case in which A is the edge matrix of a graph had been dealt with in [FLR]). Because of this and other reasons related to the interesting features of KMS states on T
A , we will gradually concentrate our attention on T
A . Observe that CK04 , seen as a set of relations, is a subset of CK4 since the identity in the latter reduces to q(X, Y ) = 0 when A(X, Y, · ) ≡ 0. Given that CK04 is less restrictive A , T
A , and O
A are “decreasing” in than CK4 we therefore have that the algebras TO the sense that each is a quotient of the preceding one. We would now like to describe T
A as the crossed-product algebra for a partial group action in order to be able to study its KMS states using the tools developed in the previous sections. In preparation for this we need to recall some terminology from [EL].
TOA and g ∈ ξ recall from [EL, 5.5 and 5.6] that Given ξ ∈  • the root of g relative to ξ is the subset of G defined by Rξ (g) = {x ∈ G : gx 1 ∈ ξ }.
TOA is the unique maximal (finite or infinite) word in • the stem of an element ξ ∈  the alphabet G such that all of its finite initial subwords (interpreted as elements of F+ ) belong to ξ . • ξ is said to be bounded if its stem is finite. Otherwise ξ is said to be unbounded. We shall also make use of the topological space 9A obtained by taking the closure, within the compact space 2G , of the set of columns of A. Observe that a column of A is a 0–1 vector and hence it may indeed be seen as an element of 2G . Moreover, any subset of G such as Rξ (g) can, and often will, also be interpreted as belonging to 2G in the usual way.
TA be the set of all ξ ∈ 
TOA such that either ξ is unbounded, or it 5.3. Theorem. Let 
TA is a compact subspace is bounded and Rξ (ω) ∈ 9A , where ω is the stem of ξ . Then 
TOA . Moreover, letting for each g ∈ F, of 
TA = {ξ ∈ 
TA : g ∈ ξ } 5τg := 5g ∩ 

Partial Dynamical Systems and the KMS Condition

239

and

αg : ξ ∈ 5τg 1 → gξ ∈ 5τg  
TA such that the crossedwe have that {5τg }g∈F , {αg }g∈F is a partial action of F on 
TA )  F is isomorphic to T
A under an isomorphism which maps each 15τ δx product C( x to sx . Proof. Given finite subsets X and Y of G let fX,Y : 2F → {0, 1} be defined by   1   1 fX,Y (ξ ) = x ∈ξ y ∈ /ξ , x∈X

y∈Y

where the brackets correspond to the obvious boolean valued function. Following [EL,
TOA
TA consist of the set of all ξ ∈  Sect. 7] and [ELQ, 4.4] it suffices to show that  1 such that fX,Y (g ξ ) = 0 for all g ∈ ξ and all pairs X, Y of finite subsets of G such that A(X, Y, · ) ≡ 0.
TA and suppose by contradiction We begin by proving the inclusion “⊆”. So let ξ ∈  1 that fX,Y (g ξ ) = 1 for some g ∈ ξ , where X and Y are as above. We have   1   1 1 = fX,Y (g 1 ξ ) = /ξ , gx ∈ ξ gy ∈ x∈X

y∈Y

and so for all x ∈ X and y ∈ Y one has that gx 1 ∈ ξ and gy 1 ∈ / ξ . These translate to x ∈ Rξ (g) and y ∈ / Rξ (g). Consider the neighborhood of Rξ (g) within 9A given by / c, ∀x ∈ X, ∀y ∈ Y }. V (X, Y ) = {c ∈ 2G : x ∈ c, y ∈ We claim that V (X, Y ) contains at least one column of A. The argument here breaks into two cases: if, on the one hand, g is the (finite) stem of ξ then the claim follows
TA and hence that Rξ (g) is in the closure 9A of the set from the hypothesis that ξ ∈  of columns of A. If on the other hand g is not the stem of ξ then there exists j ∈ G such
TOA that gx 1 ∈ ξ ⇔ A(x, j ) = 1 and that gj ∈ ξ . It follows from the fact that ξ ∈  th hence that Rξ (g) coincides with the j column of A, which therefore lies in V (X, Y ). The claim is therefore proven and we may then pick j such that the j th column of A belongs to V (X, Y ). It follows that A(x, j ) = 1 for x ∈ X and A(y, j ) = 0 for y ∈ Y so that A(X, Y, j ) = 1. This contradicts the fact that A(X, Y, · ) ≡ 0.
TOA be such that fX,Y (g 1 ξ ) = 0 In order to prove the reverse inclusion let ξ ∈ 
TA . If ξ is unbounded whenever g ∈ ξ and A(X, Y, · ) ≡ 0. We want to prove that ξ ∈  then this follows by definition. So we assume that ξ is bounded and we must show that Rξ (ω) ∈ 9A , where ω is the stem of ξ . Suppose by contradiction that this is not so and hence there exists a “basic” neighborhood of Rξ (ω) of the form V (X, Y ) containing no column of A. Since Rξ (ω) is obviously in V (X, Y ), one has that ωx 1 ∈ ξ and ωy 1 ∈ / ξ for all x in X and y in Y , which says that   1   ωx 1 ∈ ξ ωy ∈ fX,Y (ω 1 ξ ) = / ξ = 1. (†) x∈X

y∈Y

Consider the equation  x∈X

A(x, j )

 y∈Y

(1 − A(y, j )) = 1

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in the unknown j . The solutions consist, of course, of those j ’s such that for all x ∈ X and y ∈ Y one has that A(x, j ) = 1 and A(y, j ) = 0. Thus j is a solution if and only if the j th column of A belongs to V (X, Y ). By assumption there is no such column and hence neither are there solutions. In other words A(X, Y, · ) ≡ 0. By hypothesis we therefore have that fX,Y (w 1 ξ ) = 0 which contradicts (†).  
TA which will be relevant to us later and this is perhaps the There is a subspace of  right place to introduce it.
e be the subset of 
TA consisting of all ξ ∈ 
TA whose stem 5.4. Proposition. Let 
TA in the sense that there exists a continuous
e is a retract of  is equal to e. Then 
e such that r(ξ ) = ξ for all ξ ∈ 
e . Moreover such a function
TA →  function r :  can be chosen so that Rr(ξ ) (e) = Rξ (e).
TOA have trivial stem and be such that Rη (e) =
TA let η ∈  Proof. Given any ξ in 
TA . To see this we Rξ (e). Such an element exists by [EL, 5.14]. We claim that η ∈  have to consider two cases: on the one hand if ξ has trivial stem then η = ξ by the
TA . If, on the other hand, the uniqueness part of [EL, 5.14] and hence obviously η ∈  stem of ξ is not trivial then there exists some j ∈ G ∩ ξ . It follows from x

1

∈ ξ ⇔ A(x, j ) = 1

that Rξ (e) coincides with the j th column of A and hence Rξ (e) ∈ 9A . Thus Rη (e) ∈ 9A
TA . implying that η ∈ 
e which clearly restricts
TA to  Define r(ξ ) = η thus obtaining a function from 
e has a
e . It now remains to prove that r is continuous. Given that  to the identity on  product topology it is enough to verify that the map  
TA → g ∈ r(ξ ) ∈ {0, 1} ξ ∈ is continuous for all g ∈ F. Write g = xg  with x ∈ G ∩ G 1 and |g| = 1 + |g  |. Suppose
TOA and since x ∈ first that x ∈ G. Since r(ξ ) is convex by being in  / r(ξ ) because the 1 stem of r(ξ ) is trivial we must have g ∈ r(ξ ) = 0 for all ξ . Suppose  now  that x ∈ G . Using convexity as well as [EL, 5.11] we may prove that g ∈ r(ξ ) = g ∈ ξ and we again have continuity of our map.   6. Partial Representations We will now consider the map

S : F → T
A

defined (cf. [EL, Sect. 3]) as follows: if x is in G put S(x) = sx and S(x 1 ) = sx∗ . For a general g ∈ F write g = x1 x2 . . . xn in reduced form, that is, each xk ∈ G ∪ G 1 and xk+1 = xk1 , and set S(g) = S(x1 ) · · · S(xn ). The key feature of S (cf. [EL, 3.2]) is that it is a partial representation of F in the sense that • S(e) = I , • S(g 1 ) = S(g)∗ , and

Partial Dynamical Systems and the KMS Condition

241

• S(g)S(h)S(h 1 ) = S(gh)S(h 1 ), for all g, h ∈ F (see also [E4]). We will use, for each g ∈ F, the notation • pg = S(g)S(g)∗ , and • qg = S(g)∗ S(g). One may prove [E4, 2.4.iii] that the pg and qg form a commutative set. Also each S(g) is a partial isometry with initial and final projections qg and pg , respectively. It is easy to see (cf. also [EL, 3.2]) from the definition of S that it is a semi-saturated partial representation in the sense that • S(g)S(h) = S(gh) whenever |gh| = |g| + |h|. Moreover it is clearly also an orthogonal partial representation in the sense that • S(x)∗ S(y) = 0 whenever x, y ∈ G are such that x = y. Partial isometries in general tend to behave very badly, rarely satisfying any algebraic properties at all. For instance the square of a partial isometry may not be a partial isometry. However, as we shall see, the fact that our partial isometries sx are assembled into a partial representation will make it much easier for us to deal with them. Actually this was the main technical tool in bringing the Cuntz–Krieger algebras for infinite matrices to life [EL].
us now relate  Let  S to the crossed-product structure of TA . Recall from (5.3) that
TA whose crossed-product is isomor{5τg }g∈F , {αg }g∈F is a partial action of F on  phic to T
A in such a way that each sx corresponds to 15τx δx . At the algebra level let us agree to denote by θg : C0 (5τg 1 ) → C0 (5τg ) the *-isomorphism given by θg (f ) = f ◦ αg 1 for all f ∈ C0 (5τg 1 ). 6.1. Proposition. Let g ∈ F. Then (i) S(g) = 15τg δg , (ii) pg = 15τg , (iii) qg = 15τ 1 , g

(iv) S(g)aS(g)∗ = θg (a) for all a ∈ C0 (5τg 1 ),


TA ), and (v) S(g)aS(g)∗ = θg (qg a) for all a ∈ C( (vi) θg (qg ) = pg . Proof. In order to prove (i) we will use induction on |g|. If |g| = 0 the result is obvious. If g = x ∈ G we have S(x) = sx = 15τx δx , as already mentioned. It is an easy exercise to show that this implies the result also for g ∈ G 1 . So suppose that |g| > 1 and write g = rs with |g| = |r| + |s| and |r|, |s| < |g|. Using that S is semi-saturated and the induction hypothesis we have   S(g) = S(r)S(s) = (15τr δr )(15τs δs ) = θr θr 1 (15τr )15τs δrs      = θr 15τ 1 15τs δrs = θr 15τ 1 ∩5τs δrs = 1  τ τ δrs . r

r

αr 5

On the other hand observe that   αr 5τr 1 ∩ 5τs = 5τr ∩ 5τrs = 5τrs ,

∩5s r 1

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where the first equality follows from the fact that α is a partial action and the second by semi-saturatedness (see (4.1.a)). Therefore S(g) = 15τrs δrs = 15τg δg , concluding the proof of (i). In order to prove (ii) we have   pg = S(g)S(g)∗ = (15τg δg )(15τ 1 δg 1 ) = θg θg 1 (15τg )15τ 1 δe g g   = θg 15τ 1 δe = 15τg δe = 15τg , g


TA ). Point (iii) follows from (ii) where we identify, as usual, a and aδe for all a in C( simply by replacing g by g 1 . Regarding (iv) let a ∈ C0 (5τg 1 ). We have     S(g)a = (15τg δg )(aδe ) = θg θg 1 (15τg )a δg = θg 15τ 1 a δg = θg (a)δg . g

Therefore S(g)aS(g)∗ = (θg (a)δg )(15τ 1 δg 1 ) = θg (a15τ 1 )δe = θg (a)δe = θg (a). g

g


TA ) that As for (v) we have for all a ∈ C( S(g)aS(g)∗ = S(g)S(g)∗ S(g)aS(g)∗ = S(g)qg aS(g)∗ = θg (qg a), by (iv) because qg a = 15τ 1 a ∈ C0 (5τg 1 ). Finally (vi) follows by plugging a = 1 in g (v).   We now wish to name a subalgebra of T
A which will play an important role alongside the partial representation S above.
be the subalgebra of T
A generated by the set {qx : x ∈ 6.2. Definition. We will let Q G} ∪ {1}.
TA ). We will now discuss certain properties Note that T
A is in fact a subalgebra of C(
of S in relation to Q. 6.3. Proposition. For µ, ν in F+ one has (i) If |µ| = |ν| but µ = ν then S(µ)∗ S(ν) = 0. (ii) If |µ| ≥ 1 and z is the last generator in the reduced decomposition of µ then qµ = εqz , where ε is either 1 or 0 according to whether µ is admissible (i.e. A(µi , µi+1 ) = 1 for all i = 1, . . ., |µ| − 1) or not. ∗

(iii) If |µ| and z are as  in (ii) then S(µ)   QS(µ) ⊆∗ Cqz ⊆ Q. ∗ ∗.


(iv) If |µ| ≤ |ν| then S(µ)QS(µ) S(ν)QS(ν) ⊆ S(ν)QS(ν) Proof. Statements (i–ii) follow from claims 2 and 1, respectively, in the proof of [EL, Proposition 3.2]. To prove (iii) let x ∈ G and observe that S(µ)∗ qx S(µ) = S(xµ)∗ S(xµ) = qxµ = εqz , by (ii). Now let x1 , . . ., xn ∈ G and observe that, since S(µ) is a partial isometry, we have that S(µ)∗ = S(µ)∗ S(µ)S(µ)∗ and hence S(µ)∗ qx1 . . . qxn S(µ) = S(µ)∗ qx1 S(µ)S(µ)∗ qx2 . . . qxn−1 S(µ)S(µ)∗ qxn S(µ).

Partial Dynamical Systems and the KMS Condition

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Since each S(µ)∗ qxi S(µ) belongs to Cqz , the same holds for S(µ)∗ qx1 . . . qxn S(µ).
is linearly spanned by the set of products qx1 . . . qxn , we have proven (iii). Since Q With respect to (iv) write ν = ν  ν  , with |ν  | = |µ| and notice that S(µ)∗ S(ν) = S(µ)∗ S(ν  )S(ν  ) vanishes by (i) if µ = ν  . So, assuming that µ = ν  , we have    ∗ ∗

S(ν)QS(ν) S(µ)QS(µ) ∗ ∗ ∗


 )QS(ν)
= S(µ)QS(µ) S(µ)S(ν  )QS(ν) = S(µ)QS(ν ∗ ∗ ∗
 )QS(ν)
 )QS(ν)


= S(µ)S(ν  )S(ν  )∗ QS(ν = S(ν)S(ν  )∗ QS(ν ⊆ S(ν)QS(ν) ,

where we have used (iii) in the last step.

 


and S(F+ ).
TA ) in terms of Q The next result gives a total set for C(
TA ) coincides with the closed linear span of the set 6.4. Proposition. C(
{S(µ)aS(µ)∗ : µ ∈ F+ , a ∈ Q}. Proof. By (6.1.ii) we have that pg = 15τg . The set of all pg ’s therefore separates points
TA and hence generates C(
TA ) as a C ∗ -algebra. We claim that every nonzero pg of  belongs to the set in the statement. To see this note that S(g) = 0 = pg unless g = µν 1 , where µ, ν ∈ F+ are admissible words such that |g| = |µ| + |ν|, because S is semi-saturated and orthogonal (see [EL, 3.1]). Let us therefore suppose that g is of this form. If |ν| = 0 then the claim is obvious. Otherwise let z be the last generator in the reduced decomposition of ν and observe that pg = S(g)S(g)∗ = S(µ)S(ν)∗ S(ν)S(µ)∗ = S(µ)qν S(µ)∗ = S(µ)qz (µ)∗ , by (6.3.ii). It is now enough to show that the set in the statement is closed under multiplication, but this follows immediately from (6.3.iv).   7. Unital and Non-Unital Algebras In this section we propose to extend part of the discussion about units found in A and T
A . Recall that TO A , T
A , and O
A were defined via universal [EL, Sect. 8] to TO ∗ properties in the category of unital C -algebras and hence they are obviously unital. However all of them have possibly non-unital counterparts which are also of interest.
denotes any one of these algebras and {sx }x∈G is the canonical set of generating If B partial isometries we shall also consider the (non-necessarily unital) sub-C ∗ -algebra B
generated by the set {sx : x ∈ G}. B will be denoted, respectively, by TOA , TA , of B and OA , filling the third column of the following table: Table 7.1. Relations

CK1−3 CK1−3 + CK04 CK1−4

algebras

spaces


B

B






A TO
A T
A O

TOA TA OA


TO  A
T  A
O  A

TOA TA OA

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but it is nevertheless true that B ∪ {1} generates It may or may not happen that B = B

B. Therefore B can be seen as the unitization of B. It also follows that the codimension
is at most one. of B in B

TOA , 
TA , or 
OA according to the fourth column of Table 7.1 Let  be either one of 

so that B " C()  F as seen above. Let C = {e} be seen as a subset of F and hence as
TOA . It may or may not happen an element of 2F , which one may easily show lies in 

that C ∈  but we shall nevertheless let  =  \ {C}, leading to the space indicated in the last column of Table 7.1.  is then a locally compact topological space which is
TOA . clearly invariant under the partial action α of F on 
B, 
and  be accordingly 7.2. Proposition. Choose a row in Table 7.1 and let B,
= C(
)  F and B = C0 ()  F. chosen. Then B
have already been dealt Proof. The three cases corresponding to the column labeled B with, whereas the case of OA is treated in [EL, 8.4.iii]. We therefore focus on the remaining cases, i.e. TOA and TA .
) so that we may use [ELQ, 3.1] to conclude Clearly C0 () is an invariant ideal in C(
)F. Given x ∈ G let 1x := 15 ∩
) and observe that C0 ()F is an ideal in C( ∈ C( x
  that 1x (C) = x ∈ C = 0. Therefore 1x ∈ C0 () and hence sx = 1x δx ∈ C0 ()  F. So

)  F = B. B ⊆ C0 ()  F ⊆ C( Suppose by contradiction that B is a proper subset of C0 ()  F. As observed above the
is at most one, hence C0 ()  F = C(
)  F. Applying [ELQ, codimension of B in B
and hence that C ∈
. In the case of TOA this is already 3.1] we conclude that  =  /
TOA as already mentioned. So it remains to a contradiction because C does belong to 
consider TA . The characterization of TA given in 5.3 says that RC (e), namely the zero vector, is not in the closure of the set of columns of A. This implies that there exists a finite set Y ⊆ G such that the basic neighborhood of the zero vector in 2G given by / c, ∀y ∈ Y } V (∅, Y ) = {c ∈ 2G : y ∈ contains no column of A. It follows that A(∅, Y, · ) ≡ 0 and hence we have by CK04 that  0 = q(∅, Y ) = (1 − qy ). y∈Y

Upon expanding the right hand side above we discover that 1 is in the algebra generated
leading to a contradicby the qy ’s and hence also that 1 ∈ B. This implies that B = B tion.   Table 7.1 therefore displays six algebras (for each matrix A) which admit a crossedproduct structure and hence we may use the results of Sect. 2 to study their KMS states, or rather at least those which factor through the conditional expectation. However we wish to be able to apply the much stronger Theorem 4.3 which requires the corresponding partial actions to be semi-saturated and orthogonal. 7.3. Proposition. The partial action of F on each one of the six spaces appearing in the last two columns of Table 7.1 is semi-saturated and orthogonal.

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TOA . Beginning with Proof. We start by verifying conditions 4.1.a-b for the case of  (4.1.a) let g, h ∈ F be such that |gh| = |g| + |h|. In this case the shortest path from e to gh in the Cayley graph of F must pass through g. If ξ ∈ 5gh then both e and gh lie in ξ and hence so does g by convexity [EL, 4.4]. Therefore g ∈ ξ and ξ ∈ 5g . This proves that 5gh ⊆ 5g . Let us now check (4.1.b). Suppose that x, y ∈ G, x = y, and ξ ∈ 5x ∩ 5y . Then
TOA in (5.1). There{e, x, y} ⊆ ξ which contradicts the penultimate property defining  fore 5x ∩ 5y = ∅. Considering the other five partial actions under analysis observe that they are all A to invariant subsets. Properties (4.1.a-b) immeobtained by restricting the one for TO diately follow and so the proof is concluded.   8. Scaling States and the Partition Function Z(β) We are already working under the choice of a fixed 0–1 matrix A and now we are about to make other important standing hypotheses. For ease of reference we record them here. 8.1. Standing hypothesis. From now on and throughout the rest of this work we will let G be a countable set (meaning finite or countably infinite), A = {A(x, y)}x,y∈G be a 0–1 matrix having no identically zero rows, {N(x)}x∈G be a collection of real numbers in the interval (1, ∞), σ be the unique one-parameter group of automorphisms of each one of the algebras in Table 7.1 (by abuse of notation) satisfying σt (sx ) = N (x)it sx , and (v) all references to KMS states will be with respect to the one-parameter group σ above.

(i) (ii) (iii) (vi)

The existence of σ , in any one of its versions, may be deduced either from (4.3) or from the universal properties of our algebras. We begin with an important consequence of (4.3) and (7.3): 8.2. Corollary. Under (8.1) let B be any one of the C ∗ -algebras: A , TOA , T
A , TA , O
A , and OA , TO and let  be the respective space chosen from
TA , TA , 
OA , and OA ,
TOA , TOA ,   so that B " C0 ()F as seen above. Given β ∈ (0, ∞] the correspondence φ → φ ◦E is an affine homeomorphism between the set of states φ on C0 () satisfying (i) if β < ∞ : φ(θx (a)) = N (x) β φ(a) for all x ∈ G and for all a ∈ C0 (5x 1 ∩ ), (ii) if β = ∞ : φ(C0 (5x ∩ )) = {0} for all x ∈ G, and the KMSβ states on B. If λ is the probability measure on  corresponding via the Riesz Representation Theorem to φ then (i–ii) are respectively equivalent to: (i ) if β < ∞ : λ(αx (S)) = N (x) β λ(S) for all Borel subsets S ⊆ 5x 1 ∩ . (ii ) if β = ∞ : λ(5x ∩ ) = {0} for all x ∈ G. The states and measures of (8.2) will evidently become the main players in this theory and hence they deserve a name:

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8.3. Definition. Let  be as in (8.2), let φ be a state on C0 () and let λ be the probability measure on  associated to φ via the Riesz Representation Theorem. Then φ will be called a β-scaling state, and λ will be called a β-scaling measure, where β ∈ (0, ∞], if the conditions of (8.2) are satisfied. It is now perhaps the right time for us to make a choice among the six algebras of Theorem 8.2. From now on we shall concentrate our study on TA for several reasons, namely: • because the study of KMS states is most interesting in this case, • because TA is the Toeplitz extension of OA , viewing the latter as a Cuntz–Pimsner algebra [Sz], and • because every KMS state on OA gives a KMS state on TA , by composing with the canonical quotient map, and hence we include OA in the process. The KMS states of TOA are likely to be interesting as well, since they include everything else, again by considering the quotient maps. Moreover they could be studied with much the same tools we shall use here. But, alas, we won’t be looking at them in this work.
TA \ {C}. By definiRegarding T
A recall from the beginning of Sect. 7 that TA = 
TA if and only if RC (e) (which is clearly the
TA (see (5.3)) one has that C ∈  tion of  empty set, or the zero vector in 2F ) belongs to the closure of the set of columns of A,
TA and hence namely 9A . So when the zero vector is not in 9A one has that TA = 
also TA = TA . Nevertheless it is quite possible that the zero vector lies in 9A , and hence to study
TA . β-scaling measures on TA is, strictly speaking, not the same as to do so for 
TA The difference however is not very deep in the sense that a probability measure on  is given, in an essentially unique way, by a convex combination of a probability measure
TA assigning mass one to the on TA and the Dirac measure δC , i.e. the measure on  point C.
TA . Then: 8.4. Proposition. Suppose that C ∈  (i) δC is β-scaling for all β in (0, ∞],
TA consist precisely of δC and the convex (ii) for β in (0, ∞] the β-scaling measures on  combinations of a β-scaling measure on TA and δC . Proof. Part (i) follows easily from the fact that δC (5τg ) = 0 for all g ∈ F \ {e}. Part (ii) is then evident.   Therefore, once we classify all β-scaling measures on TA , we will be able to transfer
TA . By 8.2 we will therefore have classified the KMS states on T
A . that knowledge to  This said we shall now restrict our attention to studying the case of TA . A property of β-scaling states which will be often used is described in our next: 8.5. Proposition. Let β ∈ (0, ∞) and let φ be a β-scaling state on C0 (TA ). Then for every µ ∈ F+ one has that φ(pµ ) = N (µ) β φ(qµ ). Proof. By (6.1.vi) we have φ(pµ ) = φ(θµ (qµ )) = N (µ) β φ(qµ ).

 

We shall now collect some notations to be used sooner or later in this and the following sections.

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8.6. Definition. We will denote by: (i) µ the subset of TA formed by the ξ ∈ TA whose stem coincides with µ for µ ∈ F+ , µ (ii) e the intersection e ∩ 5τµ 1 for µ ∈ F+ , (iii) f the set of bounded elements of TA , i.e. elements with finite stem, (vi) ∞ the set formed by the unbounded elements of TA , (v) PA the subset of F+ formed by all admissible words, and (vi) PAn the subset of PA formed by the admissible words of length n.
TOA , when µ is a non-trivial Regarding (8.6.ii) note that, again by definition of  τ τ admissible word one has that 5µ 1 = 5x 1 , where x is the last generator in the reduced µ

decomposition of µ. So e = e ∩ 5τx 1 .
e is the set of all ξ ∈ 
TA whose stem is equal to e. Therefore, Recall from (5.4) that 
TA \ {C}, we have that e = 
e \ {C}. Also observe that by definition of since TA = 
TOA , if µ is not admissible then µ = ∅ (see [EL,
TOA , and since µ ⊆ TA ⊆   5.4]). The following are a few easy consequences of the definition: ·

8.7. Proposition. Indicating by ∪ the disjoint union of sets we have: ·

(i) TA = ∞ ∪ f , · ·   (ii) f = µ∈F+ µ = µ∈PA µ ,  · ·  τ (iii) TA = e ∪ x∈G 5x , µ

(vi) If µ ∈ F+ is admissible then µ = αµ (e ). 8.8. Definition. Let φ be a state on C0 (TA ) and let λ be the probability measure on TA associated to φ via the Riesz Representation Theorem. Then both φ and λ will be said to be of (i) finite type if λ(f ) = 1, (ii) infinite type if λ(∞ ) = 1. Observe that both f and ∞ are invariant under α. Therefore, given any β-scaling measure λ, where β ∈ (0, ∞], the restriction of λ to either one of f and ∞ satisfies (i ) or (ii ). This yields: 8.9. Proposition. Every β-scaling measure λ on TA which is not of finite nor of infinite type can be written in a unique way as a convex combination of a finite type β-scaling measure λf and an infinite type β-scaling measure λ∞ . It is easy to characterize the infinite type β-scaling measures: 8.10. Proposition. Let β ∈ (0, ∞) and let λ be a β-scaling measure on TA . Then the following are equivalent: (i) λ(e ) = 0. (ii) λ is of infinite type. Proof. (i)⇒(ii): By (8.7.iv) one has that

248

R. Exel, M. Laca τ β λ(µ ) = λ(αµ (µ e )) = N (µ) λ(e ∩ 5x 1 ) = 0

for all admissible words µ ∈ F+ ending in x. So λ(f ) = 0 by (8.7.ii) and the assumption that G is countable. That (ii)⇒(i) follows from: λ(e ) ≤ λ(f ) = 1 − λ(∞ ) = 0.   The appropriate form of the above result for the case β = ∞ is given by: 8.11. Proposition. A probability measure λ on TA is an ∞-scaling measure if and only if λ(e ) = 1. In particular every ∞-scaling measure is of finite type. Proof. Follows immediately from (8.7.iii).

 

Given a β-scaling measure, regardless of it being of finite or infinite type, it is possible to compute the measure of ∞ as follows: 8.12. Lemma. Let β ∈ (0, ∞) and let φ be a β-scaling state on C0 (TA ) with associated measure λ. Then  λ(∞ ) = lim N (µ) β φ(qµ ). n→∞

µ∈PAn

Proof. For each integer n consider the subset Sn of TA formed by all those ξ ∈ TA whose stem has length bigger than or equal to n. Observing that Sn =

·  µ∈PAn

5τµ ,

that 5τµ = ∅ unless µ is in PA , and using (8.5), we have    λ(Sn ) = λ(5τµ ) = φ(pµ ) = N (µ) β φ(qµ ). Clearly ∞ =



µ∈PAn

µ∈PAn

µ∈PAn

n∈N Sn

and the Sn are decreasing. Therefore  λ(∞ ) = lim λ(Sn ) = lim N (µ) β φ(qµ ). n→∞

n→∞

µ∈PAn

  Perhaps the most important consequence to be drawn from (8.12) is the following: 8.13. Proposition. Let β ∈ (0, ∞) and suppose that  N (µ) β = 0. lim n→∞

µ∈PAn

Then every β-scaling state on C0 (TA ) is of finite type. Proof. Let φ be a β-scaling state with associated measure λ. From (8.12) we have   N (µ) β φ(qµ ) ≤ lim N (µ) β = 0, λ(∞ ) = lim n→∞

n→∞

µ∈PAn

and hence λ(f ) = 1 − λ(∞ ) = 1.

 

µ∈PAn

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8.14. Definition. The partition function for the dynamical system (TA , σ, R) is the function Z(β) given by the Dirichlet series  Z(β) = N (µ) β . µ∈PA ·    Since PA = n∈N PAn , it is clear that Z(β) = n∈N µ∈P n N (µ) β . Therefore the A convergence of the series for Z(β) implies the hypothesis of (8.13). We therefore get the following special case of (8.13):

8.15. Corollary. Suppose that β ∈ (0, ∞) is such that the series for Z(β) converges. Then every β-scaling state on C0 (TA ) is of finite type. Observe that, by (8.11), the ∞-scaling states are also all of finite type. This may be seen as a generalization of the above result if one adopts the convention that N (x) ∞ = 0. The convergence of the series for Z(β) is not an extremely rare phenomena. For example:  8.16. Proposition. If β ∈ (0, ∞) and x∈G N (x) β < 1 then Z(β) ≤

1−



1 x∈G

N (x)

β

.

· n , where Fn denotes the subset of F consistProof. Observe that PA ⊆ F+ = n∈N F+ + + ing of words of length n. Therefore Z(β) ≤



N (µ)

β

=

∞   n=0

µ∈F+



n ∞    = N (x) β  = n=0

x∈G

N (µ)

β

µ∈Fn+

1−



1 x∈G

N (x)

β

.

  For every Dirichlet series there exists a critical value β¯ such that the series converges ¯ The behavior for β = β¯ depending on further for β > β¯ and diverges for β < β. analysis of the series under consideration. This critical value is often referred to as the abscissa of convergence. 8.17. Definition. The abscissa of convergence of Z(β) will be called the critical inverse temperature and will be denoted βc . The set Ic = {β ∈ (0, ∞) : Z(β) < ∞} ∪ {∞} will be called the interval of super-critical inverse temperatures. The possibilities for Ic are therefore (βc , ∞] or [βc , ∞] when βc < ∞. If βc = ∞ then we must necessarily have Ic = {∞}. We therefore obtain: 8.18. Corollary. For β ∈ Ic every β-scaling state on C0 (TA ) is of finite type.

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9. Existence of Finite Type Scaling States So far we have studied scaling states, and therefore KMS states on TA , under the assumption that they exist. In this section we shall obtain our first nontrivial existence results. Our main tool will be a parametrization of finite type scaling measures by means of their restriction to e . 9.1. Proposition. Let β ∈ (0, ∞) and let λ be a β-scaling measure on TA . Then  λ(f ) = N (µ) β λ(µ e ). µ∈PA

µ

µ

Proof. Given µ ∈ PA we have by (8.7.iv) that λ(µ ) = λ(αµ (e )) = N (µ) β λ(e ). The conclusion then follows from (8.7.ii).   The right-hand side expression in (9.1) will be of crucial importance both for meaµ sures defined in e (observe that e ⊆ e for all µ) and for measures on TA . This motivates the following: 9.2. Definition. For a measure3 γ defined on some measure space containing e we let  N (µ) β γ (µ Z(β, γ ) = e ), β ∈ (0, ∞). µ∈PA

Recall from our discussion immediately after (8.6) that, for a non-trivial admissible µ word µ, one has e = e ∩ 5τx 1 , where x is the last generator in the reduced decomµ position of µ. So e depends only on x. This said, given a measure γ on some measure space containing e , observe that      N (µ) β  γ (xe ), β ∈ (0, ∞), Z(β, γ ) = γ (e ) + x∈G

µ∈PAx

where, for each x ∈ G, PAx is the set of all admissible words ending in x. This motivates the introduction of our second (family of) partition function: 9.3. Definition. Let x ∈ G. The fixed-target partition function relative to the generator x for the dynamical system (TA , σ, R) is the function Zx (β) given by the Dirichlet series  N (µ) β , β ∈ (0, ∞). Zx (β) = µ∈PAx

For future reference we record the following: 9.4. Proposition. For every measure γ defined on some measure space containing e we have:  Z(β, γ ) = γ (e ) + Zx (β)γ (xe ), β ∈ (0, ∞). x∈G

Regarding (9.1), the observation that the series there converges and that the summands only depend on the restriction of λ to e lead us to our next step. 3

We assume all measures are positive regular Borel measures.

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9.5. Proposition. Let β ∈ (0, ∞) and let γ be a measure on e such that Z(β, γ ) = 1. Let λ be the measure on TA given for every measurable subset S ⊆ TA by    λ(S) = N (µ) β γ αµ 1 (S ∩ µ ) . µ∈PA

Then λ is a finite type β-scaling probability measure on TA . The correspondence γ → λ gives a one-to-one affine map from the set of all measures on e such that Z(β, γ ) = 1 onto the set of finite type β-scaling measures on TA . Proof. Given a measurable S ⊆ TA observe that S ∩ µ ⊆ µ ⊆ 5τµ , which is the µ µ domain of αµ 1 . Also αµ 1 (S ∩ µ ) ⊆ αµ 1 (µ ) = e by (8.7.iv). Since e ⊆ e and γ is defined on e we see that each summand in the definition of λ above is indeed well defined. Moreover     γ αµ 1 (S ∩ µ ) ≤ γ αµ 1 (µ ) = γ (µ e ), which implies that the series defining λ(S) is dominated by Z(β, γ ) and hence converges. For S = TA one has that      λ(S) = N (µ) β γ αµ 1 (µ ) = N (µ) β γ µ e ) = Z(β, γ ) = 1, µ∈PA

µ∈PA

and hence λ is indeed a probability measure. It is clearly of finite type. In order to show that λ is β-scaling we must show that λ satisfies λ(αx (S)) = N (x) β λ(S) for all x ∈ G and all Borel subsets S ⊆ 5τx 1 . Observing that both f and ∞ are invariant under α, and that λ vanishes on ∞ , we may suppose that S ⊆ f . By (8.7.ii) we may in fact assume that S ⊆ µ for some µ ∈ PA . Discarding the trivial case “S = ∅” we have that ∅ = S ⊆ 5τx 1 ∩ µ , and hence xµ is admissible. Moreover αx (S) ⊆ xµ and   λ(αx (S)) = N (xµ) β γ α(xµ) 1 (αx (S))   = N (x) β N (µ) β γ αµ 1 (S) = N (x) β λ(S). It is clear that the restriction of λ to e coincides with γ and hence our correspondence is injective. On the other hand given any finite type β-scaling measure λ on TA it is an easy exercise to show that the restriction of λ to e , say γ , is a measure that satisfies Z(β, γ ) = 1 and is mapped to λ under our correspondence. This proves surjectivity. Finally it is clear that we have an affine map.   Suppose we are given a nonzero measure γ on e such that Z(β, γ ) < ∞. Note that such a measure must necessarily be finite because γ (e ) ≤ Z(β, γ ). By normalizing it we obtain a measure 1 γ γ = Z(β, γ )

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that satisfies the hypothesis of (9.5) and hence gives rise to a β-scaling measure. Of course many different γ ’s are mapped to the same λ but this happens if and only if the γ ’s involved are multiples of each other. Even if Z(β, γ ) = ∞, actually even if γ is an infinite measure, one could attempt to define a β-scaling infinite measure on TA using the method of (9.5). This combined with a likely generalization of (8.2) for infinite measures would perhaps lead to interesting KMS weights on TA . However we will not pursue these ideas in the present work. 9.6. Definition. Given β ∈ (0, ∞) and a nonzero measure γ on e such that Z(β, γ ) < ∞ we will denote by Tβ (γ ) the finite type β-scaling (probability) measure λ obtained by applying the construction of (9.5) to γ /Z(β, γ ). If β = ∞ and γ is any nonzero finite measure on e we will let Tβ (γ ) be the measure on TA given simply by Tβ (γ )(S) =

γ (S ∩ e ) γ (e )

for all Borel subsets S ⊆ TA .

 Recall that Ic is the interval of convergence of the Dirichlet series µ∈PA N (µ) β . Given β ∈ Ic note that the convergence of that series implies the convergence of  N (µ) β γ (µ e ), µ∈PA

which defines Z(β, γ ), irrespective of which finite measure γ we have in mind. This says that Z(β, γ ) < ∞ and hence that Tβ (γ ) is defined for every nonzero finite measure γ on e . Combining what we have just found with (8.18) we obtain a complete characterization of β-scaling measures on TA (and hence also of KMSβ states on TA by (8.2)) in the interval of super-critical inverse temperatures: 9.7. Theorem. Under (8.1) let β ∈ Ic . Then the correspondence γ → Tβ (γ ) establishes a surjective map from the set of nonzero finite measures γ on e to the set of β-scaling measures on TA , all of which are of finite type. This correspondence is not injective but Tβ (γ1 ) = Tβ (γ2 ) if and only if γ1 is a multiple of γ2 . 10. Irreducible Matrices and the Fixed-Target Partition Function Zy (β) From now on we shall occasionally make a few other hypotheses, in addition to (8.1), which should perhaps be listed here for ease of reference: 10.1. Occasional hypotheses. (IRR) A is irreducible, i.e. for every x and y in G there exists an admissible word µ with µ1 = x and µ|µ| = y. (COL) A has no identically zero columns, (FTS) There exists a finite target set, i.e. a finite set {y1 , · · · , yn } ⊆ G such that for every x ∈ G one has A(x, yi ) = 1 for at least one i. (INF) The N (x)’s are bounded away from 1 in the sense that inf x∈G N (x) > 1. Except for the implication “(irr) ⇒ (col)”, which is easy to verify, there are no other logical relations between the conditions of (10.1).

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253

It should be stressed that we are assuming (8.1) throughout, often without notice, but we will be explicit when assuming any one of the “occasional hypotheses” above. In some cases there is a close relationship between the convergence of the series for the various Zx (β) which we would like to present now. 10.2. Proposition. Let x, y ∈ G. Suppose that there exists an admissible word ν ∈ PA beginning in x and ending in y. Then for every β ∈ (0, ∞) one has that Zx (β) ≤ N (x 1 ν)β Zy (β). y

Proof. Considering the (obviously injective) map µ ∈ PAx → µx 1 ν ∈ PA we have   N (µ) β ≥ N (µx 1 ν) β Zy (β) = y

µ∈PA

= N (x 1 ν)

β



µ∈PAx

N (µ)

β

= N (x 1 ν) β Zx (β).

µ∈PAx

  In particular, under the conditions above, if the series for Zy (β) converges then so does the series for Zx (β). The case in which this relationship is richest is when A is irreducible (see (10.1) (IRR)), in which case we get the following “solidarity” result for our Dirichlet series: 10.3. Proposition. Let A be an irreducible matrix. Then for every β ∈ (0, ∞) one has that either • Zx (β) < ∞ for all x ∈ G, or • Zx (β) = ∞ for all x ∈ G. Proof. Follows immediately from (10.2).

 

We will therefore assume throughout this section that A is an irreducible matrix. It follows that the set of β’s for which Zx (β) < ∞ does not depend on x, motivating the following: 10.4. Definition. If A is irreducible the abscissa of convergence for each and every one of the Dirichlet series Zx (β) will be called the fixed-target critical inverse temperature and will be denoted β˙c . The set of β’s where each and every one of these series converge, including β = ∞, will be called the interval of fixed-target super-critical inverse temperatures and will be denoted I˙c . As before I˙c can be either one of (β˙c , ∞] or [β˙c , ∞] when β˙c < ∞, and I˙c = {∞} when β˙c = ∞. Since Zx (β) is defined as a subseries of Z(β) it is obvious that the convergence of the latter implies the convergence of the former. This gives: 10.5. Proposition. One has that β˙c ≤ βc and I˙c ⊇ Ic . Recall that (8.18) says that for β ∈ Ic every β-scaling measure is of finite type. Our next result goes in the opposite direction stating that there are no finite type β-scaling measure for β ∈ / I˙c .

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10.6. Theorem. Under (8.1) assume that A is irreducible and let β ∈ / I˙c . Then every β-scaling measure on TA is of infinite type. Consequently there are no finite type β-scaling measures. Proof. Let λ be a β-scaling measure on TA . Then by (9.1) and (9.4) we have λ(f ) = λ(e ) +

 x∈G

Zx (β)λ(xe ).

Given that β ∈ / I˙c we have that Zx (β) = ∞ for all x implying that λ(xe ) = 0 and hence that λ(f ) = λ(e ). As we are working under the assumption that G is countable we also have that          0 = λ xe  = λ  e ∩ 5τx 1  = λ e ∩ 5τx 1  . x∈G

x∈G

x∈G

 We claim that e ⊆ x∈G 5τx 1 . To prove it assume by contradiction that ξ ∈ e but / ξ for all x, which gives Rξ (e) = ∅. Since ξ ∈ e ξ ∈ / 5τx 1 for all x in G. Then x 1 ∈ we have that the stem of ξ is e. Using [EL, 5.12] we conclude that ξ = C. But this is
a contradiction since   e ⊆ TA = TA \ {C}. This proves our claim and hence that  0 = λ (e ) = λ f .  The following diagram subsumes the information about β-scaling states on C0 (TA ), and hence also about KMSβ states on TA , that we have gathered so far in the case of an irreducible matrix A. β˙c

0

βc

Only infinite type (Theorem 10.6)

∞ Only finite type, one for each measure on e (Corollary 9.7)

Diagram 10.7

With this we essentially exhaust the conclusions that can be drawn from the techniques developed so far. In order to proceed further we need a characterization of β-scaling measures which, unlike (9.5), includes both finite and infinite type measures.
A 11. The Structure of T We retain, as always, the hypotheses listed in (8.1). In this section it will be convenient to deal with unital algebras and hence we will mainly consider T
A as opposed to TA . Our major desire is to describe, for each inverse temperature β, the KMSβ states of T
A , which we will do by characterizing the simplex formed by all β-scaling proba
TA . As an intermediate goal we will show that these measures are bility measures on 
defined in (6.2). In preparation for this parametrized by certain states on the algebra Q we will now dive into the study of this and other subalgebras of T
A . Recall that the space 9A , introduced shortly before (5.3), is the closure of the set {cx : x ∈ G}, formed by the columns cx of A, within the topological Cantor space 2G .

Partial Dynamical Systems and the KMS Condition

255


TA → 9A given by R(ξ ) = Rξ (e) and let 11.1. Proposition. Consider the map R : 
e be given by (5.4). Then:
TA →  r:
e such that the diagram (i) There exists a homeomorphism h : 9A →  r
e
TA −→    R! % h 9A commutes.
TA ) and rˆ : C(
e ) → C(
TA ) be obtained by transposing (ii) Let Rˆ : C(9A ) → C(
R and r, respectively. Then the range of both Rˆ and rˆ coincide with Q.
ˆ (iii) Both R and rˆ are isomorphisms onto Q and hence
" C(9A ) " C(
e ). Q
and every ξ ∈ 
TA one has that a(ξ ) = a(r(ξ )). (iv) For every a ∈ Q
TA one has that Proof. We claim that for ξ, η ∈  R(ξ ) = R(η) ⇔ r(ξ ) = r(η). On the one hand by definition of r one has that R(ξ ) = R(η) ⇒ r(ξ ) = r(η). On the other hand, observing that R(r(ξ )) = R(ξ ), we see that r(ξ ) = r(η) ⇒ R(ξ ) = R(η). Since both R and r are clearly surjective, a bijection h exists such that h ◦ R = r. By
TA both R and r are quotient maps and hence h is a homeomorphism. compactness of  By (6.1.iii) we have qx = 15τ 1 so that x

 qx (ξ ) = x

1

 ∈ξ ,


TA . ∀ξ ∈ 


it follows that the value of a(ξ ) depends only on {x ∈ G : x For a ∈ Q
TA , Rξ (e) = R(ξ ) in the sense that for ξ and η in   
a(ξ ) = a(η) . R(ξ ) = R(η) '⇒ ∀a ∈ Q

1

∈ ξ} =

This immediately implies (iv) in view of the fact that R(r(ξ )) = R(ξ ). The converse of the above implication also holds, as it can be proved by considering
TA by R (i.e. having the same a = qx . Therefore the equivalence relation defined on 
(i.e. having the image under R) coincides with the equivalence relation defined by Q
same image under every a ∈ Q). These in turn also coincide with the relation defined by r, whence (ii). Since both R and r are surjective we have that both Rˆ and rˆ are injective therefore proving (iii).  
We will later need a technical result about approximating positive elements of Q
TA correspond to c ∈ 9A under R (that which we would now like to present. Let ξ ∈     
is R(ξ ) = c) and observe that for all x ∈ Gqx (ξ ) = x 1 ∈ ξ = x ∈ c . Identifying Q ˆ with C(9A ) via R we may therefore think of qx as the function qx (c) = [x ∈ c].

(11.2)

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Given finite subsets X and Y of G it follows that q(X, Y ) is in turn identified with the characteristic function of the set V (X, Y ) = {c ∈ 9A : x ∈ c, y ∈ / c, ∀x ∈ X, ∀y ∈ Y }. Observe also that these sets form a basis for the product topology on 2G , consisting of clopen sets.
and any ε > 0 there are finite subsets X1 , · · · , Xn 11.3. Lemma. For each a ≥ 0 in Q and Y1 , · · · , Yn of G and positive real numbers λ1 , · · · , λn such that the element b=

n 

λi q(Xi , Yi )

i=1

satisfies (i) 0 ≤ b ≤ a, and (ii) a − b < ε. Proof. Let K = {c ∈ 9A : a(c) ≥ 2a/3} and U = {c ∈ 9A : a(c) > a/3} so that K is compact, U is an open set, and K ⊆ U ⊆ 9A . Choose a finite covering of K consisting of sets V (Xi , Yi ) ⊆ U . It is not hard to show that such a covering can be found so that the V (Xi , Yi ) are pairwise disjoint. Let n

b1 =

a  q(Xi , Yi ). 3 i=1

It is now easy to show that 0 ≤ b1 ≤ a and that a − b1  ≤ 2a/3. We may then repeat this procedure starting with a − b1 and, after n steps, we will have obtained a sequence
each of which is a scalar multiple of a sum of q(X, Y )’s, b1 , · · · , bn of elements of Q, and such that 0 ≤ bn ≤ a −b1 −· · ·−bn−1 and a −b1 −· · ·−bn−1 −bn  ≤ (2/3)n a. After a finite number of steps the element b = b1 + · · · + bn will satisfy the required properties.   We would now like to study other subalgebras of T
A . ∗ . Then
11.4. Proposition. For each µ ∈ F+ let I µ = S(µ)QS(µ)

(i) If µ, ν ∈ F+ are such that |µ| = |ν| but µ = ν then I µ I ν = {0}. (ii) If µ, ν ∈ F+ are such that |µ| ≤ |ν| then I µ I ν ⊆ I ν . (iii) Each I µ is a closed unital *-subalgebra of T
A and S(µ)S(µ)∗ is its unit. Proof. Part (i) follows at once from (6.3.i), while (ii) is just a restatement of (6.3.iv). That each I µ is a *-algebra follows from (ii) and it is obvious that S(µ)S(µ)∗ serves as a unit for it. It therefore remains to show that I µ is closed. So let a sequence {S(µ)an S(µ)∗ }n ,
converge to some b in T
A . Then with an ∈ Q, b = lim S(µ)an S(µ)∗ = lim S(µ)S(µ)∗ S(µ)an S(µ)∗ S(µ)S(µ)∗ n

n

= S(µ)S(µ)∗ bS(µ)S(µ)∗ ,


But and hence it suffices to show that S(µ)∗ bS(µ) is in Q. S(µ)∗ bS(µ) = lim S(µ)∗ S(µ)an S(µ)∗ S(µ), n


by (6.3.ii). which belongs to Q

 

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n will denote the subset of F consisting of elements µ with |µ| = n. From now on F+ + 0 1 = G. Thus F+ = {e} and F+  µ 11.5. Proposition. For each integer n ≥ 0 let In be the closure of µ∈Fn+ I within T
A . Then In is a closed *-subalgebra of T
A which is *-isomorphic to the c0 direct sum of the I µ , that is, the C ∗ -algebra consisting of families (aµ )µ∈Fn+ such that limµ aµ  = 0.  In particular the net of idempotents { µ∈J pµ }J , where J ranges in the collection of n , forms an approximate unit for I . finite subset of F+ n

Proof. The statement follows easily from the fact that the I µ considered form a collection of pairwise orthogonal C ∗ -algebras by (11.4.i).   11.6. Proposition. (i) For every n and m one has that In Im ⊆ Imax{n,m} . (ii) For n ≥ 1 and x ∈ G one has that sx∗ In sx ⊆ In−1 . (iii) For n ≥ 0 and x ∈ G one has that sx In sx∗ ⊆ In+1 . n . We
and µ ∈ F+ Proof. The first statement follows from (11.4.ii). As for (ii) let a ∈ Q ∗ ∗ then need to show that sx S(µ)aS(µ) sx lies in In−1 . Let y be the first generator in the reduced decomposition of µ so that µ = yµ , where µ ∈ F+ . Observe that unless x = y we have that sx∗ S(µ) = 0. So assume that x = y. The case n = 1 is somewhat special so let us treat it first. We then have that µ = x and thus
= I0 . sx∗ S(µ)aS(µ)∗ sx = qx aqx ∈ Q

If n ≥ 2 then |µ | ≥ 1 and hence sx∗ S(µ) = qx S(µ ) = εS(µ ), where ε ∈ {0, 1} by CK3 . Therefore sx∗ S(µ)aS(µ)∗ sx = εS(µ )aS(µ )∗ ∈ In−1 . The third assertion is obvious.

 

11.7. Proposition. For each integer n ≥ 0 let An be the closure of I0 + · · · + In within T
A . Then (i) An is a C ∗ -algebra. (ii) In is an ideal in An . (iii) An+1 = An + In+1 .
TA ) is the closure of ∪n An . (iv) C( (v) For n ≥ 1 and x ∈ G one has that sx∗ An sx ⊆ An−1 . (vi) For n ≥ 0 and x ∈ G one has that sx An sx∗ ⊆ An+1 . Proof. Clearly (i) and (ii) follow from (11.6.i). Using [Pe, 1.5.8] one gets (iii). As for (iv), it follows from (6.4). Finally (v) follows from a combination of (11.6.ii) and (6.3.iii) while (vi) is a direct consequence of (11.6.iii).   One more technical result is in order: 11.8. Proposition. For each n ∈ N one has that An ∩ In+1 = {0}.

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Let a ∈ Q
∩ I1 . Using (11.5) Proof. Assume that A0 = Q.  first that n = 0, observing z write a = z∈G az , where az ∈ I and limz az  = 0. We claim that each az is a scalar
we have by (6.3.iii) that sz∗ asz = λz qz multiple of pz . In fact observe that, since a ∈ Q, for some λz ∈ C. Therefore az = pz apz = sz (sz∗ asz )sz∗ = λz sz qz sz∗ = λz pz .  We then have that a = z∈G λz pz with limz λz = 0. Suppose by way of contradiction that a = 0. Then there exists at least one λz0 pz0 which is nonzero. Given that limz λz = 0 we see that λz0 is an isolated point in the set of all λz ’s. One may therefore take a continuous function f : C → C such that f (λz0 ) = 1 and f (λz ) = 0 whenever λz = λz0 . It follows that f (a) = z∈Z pz , where Z is the
∩ I1 . (necessarily finite) set Z = {z ∈ G : λz = λz0 }. It is also clear that 0 = f (a) ∈ Q
TA we have, using (11.1.iv), that Given ξ ∈     pz (r(ξ )) = z ∈ r(ξ ) = 0, f (a) ξ = f (a) r(ξ ) = z∈Z

z∈Z

because the stem of r(ξ ) is trivial. It follows that f (a) = 0, a contradiction. n we have that Now assume that n ≥ 1 and let a ∈ An ∩ In+1 . Given ν ∈ F+ ∗ ∗ S(ν) aS(ν) ∈ A0 ∩ I1 by (11.6.ii) and (11.7.v) and hence S(ν) aS(ν) = 0. With more n+1 . Therefore reason S(µ)∗ aS(µ) = 0 for µ ∈ F+ pµ a = pµ apµ = S(µ)S(µ)∗ aS(µ)S(µ)∗ = 0 n+1 for every µ ∈ F+ . The conclusion then follows from the last sentence in (11.5).

 


12. Invariant and Subinvariant States on Q
TA ), and hence also Our goal in this section is to show that β-scaling states on C(

These states KMSβ states on TA , are in 1–1 correspondence with certain states on Q. are best motivated by the following:
TA ). Denote 12.1. Proposition. Let β ∈ (0, ∞) and let φ be a β-scaling state on C(
Then, for every pair of finite subsets X and Y of G, we by ρ the restriction of φ to Q. have  A(X, Y, z)N(z) β ρ(qz ) ≤ ρ(q(X, Y )). z∈G

Proof. Recall from CK3 that qx sz = A(x, z)sz and hence also (1 − qy )sz = (1 − A(y, z))sz so that     q(X, Y )sz =  qx (1 − qy ) sz x∈X

 =



x∈X

y∈Y

A(x, z)

 y∈Y

 (1 − A(y, z)) sz = A(X, Y, z)sz

Partial Dynamical Systems and the KMS Condition

259

for all z in G. By multiplying this on the right-hand side by sz∗ we have that q(X, Y )pz = A(X, Y, z)pz and hence that A(X, Y, z)pz ≤ q(X, Y ) in the usual order of projections. Since the pz are pairwise orthogonal by CK2 we conclude that any finite sum  z∈Z A(X, Y, z)pz (Z a finite set) gives a projection dominated by q(X, Y ). It follows that

 φ A(X, Y, z)pz ≤ φ(q(X, Y )). z∈Z

On the other hand recall from (8.5) that φ(pz ) = N (z) β φ(qz ). Therefore  A(X, Y, z)N(z) β ρ(qz ) ≤ ρ(q(X, Y )). z∈Z

Since Z is arbitrary, the proof is concluded.

 

It should be noted that the result above covers the case X = Y = ∅, in which case it says that  N (z) β ρ(qz ) ≤ 1. (12.2) z∈G

The states ρ appearing above will acquire a crucial importance from this point on and hence we make the following:
is said to be 12.3. Definition. Let β ∈ (0, ∞). A state ρ on Q (i) β-subinvariant when the inequality in (12.1) holds for all finite subsets X, Y ⊆ G. (ii) β-invariant when the inequality in (12.1) becomes an equality for all finite subsets X, Y ⊆ G. A probability measure on 9A is said to be β-subinvariant (resp. β-invariant) if integra
Every tion against it leads to a β-subinvariant (resp. β-invariant) state on C(9A ) = Q. state or measure will be considered ∞-subinvariant by default. Our last result therefore says that the correspondence φ → φ|Q
maps the set of
β-scaling states on C(TA ) to the set of β-subinvariant states on 9A . We will now seek to prove that this is in fact a bijective correspondence, thus obtaining a new characterization of KMS states which is significantly better than the one obtained in (8.2) in the sense
TA . that 9A is a much more tractable space than  We begin by proving that φ → φ|Q
defines an injective map.
TA ) such 12.4. Proposition. Let β ∈ (0, ∞] and let φ and φ  be β-scaling states on C(   that φ|Q
= φ |Q
. Then φ = φ . Proof. We first claim that φ and φ  coincide on elements of the form S(µ)aS(µ)∗ , where
Using (6.1.v) we have that µ ∈ F+ and a ∈ Q.     φ S(µ)aS(µ)∗ = φ θµ (qµ a) = N (µ) β φ(qµ a).
by (6.3.ii) the claim is proven. By (6.4) it follows that φ and φ  coincide Since qµ a is in Q
TA ).  on C( 

260

R. Exel, M. Laca

In order to prove that the correspondence φ → φ|Q
is surjective we need the following general result about states. 12.5. Proposition. Let B be a unital C ∗ -algebra containing a closed two sided ideal I and a sub-C ∗ -algebra A such that 1 ∈ A and B = A + I . Also let φ be a state on A and
the canonical extension of ψ to a ψ be a positive linear functional on I . Denote by ψ
positive functional on B (that is, ψ (b) = limi ψ(bui ), where {ui }i is an approximate unit for I ). Suppose that
on A, and (i) φ ≥ ψ (ii) φ = ψ on A ∩ I . Then there exists a state ρ on B such that ρ|A = φ and ρ|I = ψ. Proof. Given b in B write b = a + x, where a ∈ A and x ∈ I , and put ρ(b) = φ(a) + ψ(x). It follows from (ii) that ρ is a well defined linear functional on B. In order to show that ρ is positive let b = a + x ∈ B and observe that ρ(b∗ b) = ρ(a ∗ a + a ∗ x + x ∗ a + x ∗ x) = φ(a ∗ a) + ψ(a ∗ x + x ∗ a + x ∗ x)
(a ∗ a) + ψ(a ∗ x + x ∗ a + x ∗ x) = ψ
(b∗ b) ≥ 0. ≥ψ Since 1 ∈ A we have ρ = ρ(1) = φ(1) = 1 and hence ρ is indeed a state.

 

With the following result we complete the announced parametrization of β-scaling

TA ) by means of β-subinvariant states on Q. states on C(
Then there 12.6. Proposition. Let β ∈ (0, ∞] and let ρ be a β-subinvariant state on Q.
exists a (necessarily unique) β-scaling state φ on C(TA ) such that φ|Q
= ρ. Proof. We begin with the case β < ∞. For each n ∈ N we will construct a state ρn on the algebra An (see (11.7)) such that for every n ≥ 1, (i) (ii) (iii) (iv)

ρ0 = ρ, ρn |An−1 = ρn−1 , ρn (sx asx∗ ) =N (x) β ρn−1 (asx∗ sx ) for a ∈ An−1 and x ∈ G, N(x) β ρn−1 sx∗ asx = ρn (asx sx∗ ) for a ∈ An and x ∈ G.

We shall proceed by induction and hence let us suppose we are given m ≥ 0 and {ρn }0≤n≤m satisfying (i–iv) for all n = 1, · · · , m. Define a linear functional χm+1 on Im+1 by  χm+1 (a) = N (x) β ρm (sx∗ asx ), a ∈ Im+1 . x∈G

In order to verify that this is well defined observe that, by (11.6.ii), for any a ∈ Im+1 we have sx∗ asx ∈ sx∗ Im+1 sx ⊆ Im ⊆ Am so that ρm (sx∗ asx ) is defined. To see that the sum converges it is enough to consider a positive a, in which case we have    N(x) β ρm (sx∗ asx ) ≤ N (x) β ρm (aqx ) = a N (x) β ρ(qx ) ≤ a, x∈G

x∈G

x∈G

where the last step follows from (12.2). It is then clear that χm+1 is a well defined positive linear functional on Im+1 . By (11.8) we have that Am ∩ Im+1 = {0} and hence the expression ρm+1 (a + b) = ρm (a) + χm+1 (b),

a ∈ Am , b ∈ Im+1

gives a well defined linear functional ρm+1 on Am + Im+1 = Am+1 .

Partial Dynamical Systems and the KMS Condition

261

We will now prove that (ii–iv) hold for n = m + 1. By definition ρm+1 |Am = ρm , taking care of (ii). In order to check (iii), that is ρm+1 (sx asx∗ ) = N (x) β ρm (asx∗ sx ),

a ∈ Am , x ∈ G,

(†)


so that sx asx∗ ∈ Im+1 and let us first suppose that m = 0. Then a ∈ Am = Q  ρm+1 (sx asx∗ ) = χm+1 (sx asx∗ ) = N (y) β ρm (sy∗ sx asx∗ sy ) y∈G

= N (x)

β

ρm (sx∗ sx asx∗ sx )

= N (x) β ρm (asx∗ sx ).

Let us suppose now that m ≥ 1 and, given that a ∈ Am = Am−1 + Im , it is enough to verify (†) separately for a ∈ Am−1 and for a ∈ Im . If a ∈ Im then sx asx∗ ∈ Im+1 and the exact same calculation used to deal with the case m = 0 just above gives the conclusion. If a ∈ Am−1 then sx asx∗ ∈ Am and, by induction, ρm+1 (sx asx∗ ) = ρm (sx asx∗ ) = N (x) β ρm−1 (asx∗ sx ) = N (x) β ρm (asx∗ sx ). This concludes the proof of (†). To prove (iv), that is   N (x) β ρm sx∗ asx = ρm+1 (asx sx∗ ), a ∈ Am+1 , x ∈ G,

(‡)

let us again first suppose that m = 0. Given that sx sx∗ ∈ I1 , which is an ideal in A1 by (11.7.ii), we have that asx sx∗ ∈ I1 and then  ρ1 (asx sx∗ ) = χm+1 (asx sx∗ ) = N (y) β ρ(sy∗ asx sx∗ sy ) = N (x) β ρ(sx∗ asx sx∗ sx ) y∈G

= N (x)

β

ρ(sx∗ asx ),

proving (‡) for m = 0. Assume now that m ≥ 1. Given a ∈ Am+1 we have by (11.7.v) that sx∗ asx ∈ Am . Therefore, plugging a := sx∗ asx into (†) gives ρm+1 (sx sx∗ asx sx∗ ) = N (x) β ρm (sx∗ asx sx∗ sx ) which implies that

ρm+1 (asx sx∗ ) = N (x) β ρm (sx∗ asx ),

concluding the proof of (‡) in the general case. It remains to prove that ρm+1 is a state and we shall derive this from (12.5) applied to the pair (ρm , χm+1 ). Clearly (12.5.ii) holds by (11.8). With respect to checking (12.5.i) let us use the approximate unit for Im+1 provided by (11.5). We then have for any a ≥ 0 in Am that     χ
m+1 (a) = χm+1 lim pµ a  = χm+1 (pµ a) J

=

  x∈G µ∈Fm+1 +

µ∈J

µ∈Fm+1 +

N (x) β ρm (sx∗ pµ asx ) ≤

 x∈G

N (x) β ρm (sx∗ asx ),

where the last inequality follows from the fact that the pµ are pairwise orthogonal (see (6.3.i)).

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Our present goal is to prove that χ
m+1 (a) ≤ ρm (a) for all a ∈ Am . In order to accomplish this let us first suppose that m = 0 and that a = q(X, Y ), where X and Y are finite subsets of G. By CK3 we have for any z ∈ G that sz∗ asz = sz∗ q(X, Y )sz = A(X, Y, z)qz (see also the beginning of the proof of (12.1)). Therefore,   χ
m+1 (a) ≤ N (z) β ρm (sz∗ asz ) = N (z) β A(X, Y, z)ρ(qz ) z∈G

z∈G

≤ ρ(q(X, Y )) = ρ(a), where the last inequality is from our hypothesis that ρ is β-subinvariant. Of course it also follows that χ
m+1 (a) ≤ ρ(a) whenever a is a linear combination of the q(X, Y )
= A0 . with positive coefficients. Thus by (11.3) the same holds for any a ≥ 0 in Q Assume now that m ≥ 1. By (iv) applied for n = m + 1 (i.e. by (‡)) we have that   χ
m+1 (a) ≤ N (x) β ρm (sx∗ asx ) = ρm (asx sx∗ ) ≤ ρm (a). x∈G

x∈G

This concludes the construction of the ρn so let us now take up the task of constructing
TA ) is the closure of ∪n An the β-scaling state φ mentioned in the statement. Since C(
TA ) simultaneously extending by (11.7.iv), and ρn+1 |An = ρn , there is a state φ on C(
TA ) note that all of the ρn . If a ∈ C( φ(sx asx∗ ) = N (x) β φ(asx∗ sx ) for all x ∈ G by (iii). If moreover a ∈ C0 (5τx 1 ) then we have φ(θx (a))

(6.1.iv)

φ(sx asx∗ ) = N (x) β φ(asx∗ sx )

(6.1.iii)

N (x) β φ(a15τ 1 ) = N (x) β φ(a),

=

=

x


and hence φ is β-scaling. Obviously φ coincides with ρ on Q. All of this is meant to work for β < ∞ but, with the usual interpretation of N (x) β , the argument above works also for β = ∞. Alternatively there is a more straightforward way to prove our statement for β = ∞ which we would now like to present.
under rˆ . Identifying these algebras we have
e ) " Q Recall from (11.1.iii) that C(
e ) and hence a probability measure on 
e . Extend this to that ρ defines a state on C(
TA by declaring that 
TA \ 
e has measure zero. This in turn gives the a measure on 
TA ) we are looking for. Precisely, φ is defined as follows: consider the state φ on C(
TA and let ιˆ : C(
TA ) → C(
e ) be the transposed map. φ is then
e →  inclusion ι :  the result of the composition ιˆ rˆ
ρ
e ) −→
TA ) −→ C( Q −→ C. C(
and ξ ∈ 
TA observe that For every a ∈ Q rˆ (ˆι(a))

ξ

= a(r(ξ ))

(11.1.iv)

=

a(ξ ),

so that rˆ (ˆι(a)) = a and hence φ(a) = ρ(ˆr (ˆι(a))) = ρ(a) proving that φ extends ρ.

Partial Dynamical Systems and the KMS Condition

263


e , Observe that for all x ∈ G one has that ιˆ(15τx ) is the characteristic function of 5τx ∩  τ
e which is the empty set because every ξ in 5x contains x while the stem of every ξ ∈  is trivial. So φ(15τx ) = 0 for all x ∈ G. It follows that φ(C0 (5τx )) = {0} and hence that φ is an ∞-scaling state.   Putting together (12.1), (12.4), and (12.6) we arrive at one of our main results: 12.7. Theorem. Under (8.1) let β ∈ (0, ∞]. Then the correspondence φ → φ|Q
defines
a bijection from the set of β-scaling states on C(TA ) to the set of β-subinvariant states
on Q. 13. States on Q
is defined to be the unital C ∗ -subalgebra of T
A generated by the qx . When Recall that Q
TA , it is convenient to work with the algethe emphasis is on TA , rather than on  bra Q ⊆ C0 (TA ) defined to be the (not necessarily unital) C ∗ -algebra generated by
TA ) in relation to their restric{qx : x ∈ G}. In the last section we studied states on C(
tion to Q. In order to extend these results to C0 (TA ) and Q it would be convenient to know whether states on C0 (TA ) restrict to states on Q, a fact which is no longer automatic as we are now working with non-necessarily unital C ∗ -algebras. Recall that our matrix A is assumed not to have identically zero rows. We will now need to assume (10.1) (COL) i.e. that there are no identically zero columns. 13.1. Proposition. Suppose that no column of A is identically zero. Then Q is an essential subalgebra of C0 (TA ) in the sense that an approximate identity for Q is always an approximate identity for C0 (TA ). Therefore the restriction to Q of any state on C0 (TA ) is a state on Q. Proof. It is clearly enough to show that there is no ξ ∈ TA such that a(ξ ) = 0 for all a ∈ Q. Suppose by contradiction that such a ξ exists. Given x ∈ G let a = qx so that   0 = qx (ξ ) = x 1 ∈ ξ , which implies that Rξ (e) = ∅. Suppose first that the stem of ξ is not trivial. In this case there exists y ∈ G such that y ∈ ξ . Given that no column of A is zero, pick an x ∈ G such that A(x, y) = 1. Then by Definition 5.1 we have that x 1 ∈ ξ which is a contradiction. The only alternative is then that the stem of ξ is trivial. By [EL, 5.12] we conclude that ξ = C which is again a contradiction since C was explicitly removed from TA .   We will therefore assume, throughout this section, that no column of A is identically zero keeping, of course, all the other hypotheses in (8.1). Given a state ρ on Q it is well known that there exists a unique extension of ρ to a
state ρ
on Q. 13.2. Definition. We will say that a state ρ on Q is β-invariant (resp. β-subinvariant)
if its canonical extension ρ
is a β-invariant (resp. β-subinvariant) state on Q. The next result is a generalization of (12.7) to the present context: 13.3. Theorem. Assuming (8.1) and (10.1) (COL) let β ∈ (0, ∞]. Then the correspondence φ → φ|Q defines a bijection from the set of β-scaling states on C0 (TA ) to the set of β-subinvariant states on Q.

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Proof. The result follows from (12.7) and (13.1) on noting that states on C0 (TA ) cor
TA ) whose restriction to C0 (TA ) is a state (i.e. of norm one), and respond to states C(
 similarly with respect to Q and Q.  We are now able to give two new characterizations of infinite type states on C0 (TA ), extending the result obtained in (8.10). We repeat here the conditions of (8.10): 13.4. Proposition. Let β ∈ (0, ∞) and let φ be a β-scaling state on C0 (TA ) corresponding to a measure λ on TA . Denote by ρ the restriction of φ to Q which is a state by (13.1). Then the following are equivalent: (i) λ(e ) = 0, (ii) λ is of infinite type, (iii) ρ is a β-invariant state, (iv) x∈G N (x) β ρ(qx ) = 1 (see 12.2). Proof. That (i)⇒(ii) was proved in (8.10). (ii)⇒(iii): Let X and Y be finite subsets of G and consider the sets S = {ξ ∈ TA : x

1

∈ ξ, y

1

∈ / ξ, ∀x ∈ X, ∀y ∈ Y },

and

T = {ξ ∈ TA : ∃z ∈ G, z ∈ ξ ∧ A(X, Y, z) = 1}. S and T are not necessarily equal but they have exactly the same unbounded elements, as a moment’s reflexion based on Definition 5.1 will reveal. Given that λ is of infinite type, and hence supported in the set of unbounded elements, we must therefore have that λ(S) = λ(T ). Observe that the characteristic function of S is precisely q(X, Y ) while the characteristic function of T is the infinite sum  A(X, Y, z)pz . z∈G

It follows from countable additivity that φ(q(X, Y )) =



A(X, Y, z)φ(pz ).

z∈G

By (8.5) we have ρ(q(X, Y )) =



A(X, Y, z)N (z) β ρ(qz ),

z∈G

which means that ρ is β-invariant. (iii)⇒(iv): Take X = Y = ∅ above. (iv)⇒(i): By (8.7.iii) we have that    λ(e ) = 1 − λ(5τx ) = 1 − φ(px ) = 1 − N (x) β φ(qx ) = 0. x∈G

x∈G

x∈G

  Given a β-invariant state ρ on Q observe that the identity ρ(qz ) = ρ(q(X, Y )) with X = {x} and Y = ∅ becomes  A(x, y)N (y) β ρ(qy ) = ρ(qx ) y∈G



z∈G

A(X, Y, z)N (z) β

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265

  for each x ∈ G. The vector ρ(qx ) x∈G is therefore a fixed point (i.e. an eigenvector with eigenvalue 1) for the matrix AN

β

= {A(x, y)N (y) β }x,y∈G .

We may take the above expression for AN β as no more than just a definition but note that if N is the diagonal matrix with N(x, x) := N (x) and we interpret N β in the only reasonable way then AN β can be also thought of as the product of A and N β .   13.5. Theorem. Under (8.1) let β ∈ (0, ∞). Then the correspondence ρ → ρ(qx ) x∈G defines onto the set of fixed points  a bijection from the set of β-invariant states on Q v = vx x∈G for AN β with vx ≥ 0 for all x and such that x∈G N (x) β vx = 1. Proof. Let ρ1 and ρ2 be β-invariant states such that ρ1 (qx ) = ρ2 (qx ) for all x ∈ G. By definition of β-invariant states it follows that ρ1 (q(X, Y )) = ρ2 (q(X, Y )) for all finite sets X and Y of G. So ρ1 = ρ2 , showing our correspondence to be injective. In order to show that it is also surjective let v be a fixed point as in the statement.
it is easy to see that the spectrum of Q is given by Viewing Q as an ideal in Q * 9Q = 9A \ {0}, where 0* is the zero vector in 2G . From our assumption that no column of A is identically zero it then follows that every column of A lies in 9Q . Define a probability measure on 9Q by λ(S) =



 cz ∈ S N (z) β vz

z∈G

for all Borel subsets S ⊆ 9Q , where cz refers to the zth column of A. By definition λ is an atomic measure with atoms the columns of A. Each column cx of A therefore has  mass equal to N (z) β vz , where the sum is over the set of z’s such that cz = cx . Let ρ be the state on Q given by integration against λ. of each qx as a   Thinking
TA , as in (11.2), and observing that qx (cz ) = x ∈ cz = A(x, z), we have function on  ρ(qx ) =



qx (cz )N (z) β vz =

z∈G



A(x, z)N (z) β vz = vx .

z∈G

We now wish to show that ρ is β-invariant. Let therefore X and Y be finite subsets of G and observe that q(X, Y ) c = A(X, Y, z). Therefore z

ρ(q(X, Y )) =



A(X, Y, z)N(z) β vz =

z∈G

proving that ρ satisfies the required properties.



A(X, Y, z)N (z) β ρ(qz ),

z∈G

 

The following summarizes much of what we have discovered so far:

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13.6. Theorem. Assuming (8.1) and (10.1) (COL) let β ∈ (0, ∞]. Then the vertical correspondences below are bijective σ –KMS states on TA at inverse temperature β  restriction ! to C0 (TA ) β-scaling states on C0 (TA ) ⊇ infinite type β-scaling states  restriction ! to Q

 restriction ! to Q

β-subinvariant states on Q

⊇

β-invariant states on Q    maps ρ to ! v = ρ(qx ) x∈G nonnegative fixed points 

v for AN β with

x∈G

N(x) β vx = 1.

Proof. We restrict ourselves to pointing out the result relating to each one of the above arrows. The uppermost arrow corresponds to (8.2). The arrow following that is (13.3). On the second column the uppermost arrow is (13.4) and the last one is (13.5).   14. The Fixed-Source-and-Target Partition Function Zxy (β) In this section we will introduce the third family of Dirichlet series associated to our context. 14.1. Definition. Let x, y ∈ G. The fixed-source-and-target partition function relative to the pair of generators x and y for the dynamical system (TA , σ, R) is the function Zxy (β) given by the Dirichlet series  Zxy (β) = N (µ) β , β ∈ (0, ∞), xy

µ∈PA xy

where PA is the set of all admissible words beginning in x and ending in y. As in (10.2) we have: 14.2. Proposition. Let x1 , x2 , y1 , y2 ∈ G. Suppose that there are admissible words ν, γ ∈ PA such that • ν1 = x1 , ν|ν| = x2 , • γ1 = y2 , γ|γ | = y1 , then for every β ∈ (0, ∞) one has that Zx2 y2 (β) ≤ KZx1 y1 (β), where K = N (νγ )β N(x2 y2 ) β .

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267 x y

x y1

Proof. Considering the injective map µ ∈ PA2 2 → νx21 µy21 γ ∈ PA1   Zx1 y1 (β) = N (µ) β ≥ N (νx21 µy21 γ ) β x y1

we have

x y2

µ∈PA1

µ∈PA2

= N (νγ ) β N (x2 y2 )β



N (µ)

β

= N (νγ ) β N (x2 y2 )β Zx2 y2 (β).

x y2

µ∈PA2

  Assuming that A is irreducible we have another “solidarity” property (see (10.3)) among these Dirichlet series: 14.3. Proposition. Let A be irreducible. Then for every β ∈ (0, ∞) one has that either • Zxy (β) < ∞ for all x, y ∈ G, or • Zxy (β) = ∞ for all x, y ∈ G. Proof. Follows immediately from (14.2).

 

14.4. Definition. Under the hypothesis that A is irreducible the abscissa of convergence for each and every one of the Dirichlet series Zxy (β) will be called the fixed-sourceand-target critical inverse temperature and will be denoted β¨c . The set of β’s where each and every one of these series converge, including β = ∞, will be called the interval of fixed-source-and-target super-critical inverse temperatures and will be denoted I¨c . As before, it is obvious that β¨c ≤ β˙c ≤ βc ,

and

I¨c ⊇ I˙c ⊇ Ic .

The relevance of these concepts lies in the following: 14.5. Theorem. Suppose (8.1) and ((10.1) (IRR) and let β < β¨c . Then there are no β-scaling states at all on C0 (TA ) and hence neither are there KMSβ states on TA . Proof. Assume by contradiction that φ is a β0 -scaling state on C0 (TA ) for some β0 < β¨c . Then by (13.3) the restriction ρ of φ to Q is a β0 -subinvariant state. Let x ∈ G and plug X = {x} and Y = ∅ in the definition of subinvariant states (see 12.3) to get  A(x, y)N (y) β0 ρ(qy ) ≤ ρ(qx ). y∈G

  This says that the nonnegative vector v = ρ(qx ) x∈G is a right 1-subinvariant vector for the irreducible matrix AN β0 in the sense of [V, Sect. 4]. We will now proceed to show that such a vector cannot exist, therefore arriving at a contradiction. Unfortunately we cannot just quote the result we need from [V, Corollary 1] because of the incompatibility between our point of view which emphasizes Dirichlet series in the variable β, and Vere-Jones’s point of view which emphasizes power series. Nevertheless, proceeding with the necessary care, we may still derive our conclusions from [V]. (n) Set the matrix T of [V] to be AN β and, according to [V, Section 2], let fij be the “first-entrance probabilities” for T , and Fij (z) be the corresponding generating function.

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Observe that Fij (z) depends on β, as T definitely does. Accordingly let us denote by Fij (β) the value of Fij at z = 1. By the right-hand-sided version of [V, Lemma 4.1], applied for β = β0 , r = 1, and i = j taken to be any fixed element in G, we have that Fii (1) = Fii (β0 ) ≤ 1. It is easy to see that Fii (β) is a strictly decreasing function of β and hence Fii (β) < 1 for all β > β0 . Observe that the generating function Tij , defined near the bottom of p. 362 of [V], is related to our partition function Zij by Tij (1) = N (i)β Zij (β). Using Eq. (3) in [V], namely Tii (z) = 1/(1 − Fii (z)), for z = 1 we therefore have that N (i)β Zii (β) =

1 . 1 − Fii (β)

Inspired by the idea of the proof of Lemma 2.1 in [V], we conclude that Zii (β) has no singularities in the interval (β, ∞) because, as seen above, Fii < 1 there. Since β < β¨c we have a contradiction.   We may now throw some more conclusions into Diagram 10.7 getting the following information about β-scaling states on C0 (TA ), and hence also about KMSβ states on TA , again in the case that A is irreducible. β¨c

0 None at all (Theorem 14.5)

β˙c

βc

Only infinite type (Theorem 10.6)

∞ Only finite type, one for each measure on e (Corollary 9.7)

Diagram 14.6

15. Energy Bounded Below In this section we will prove that there are no β-scaling states for β < β˙c under the hypothesis that the “energy” parameters N (x) satisfy inf x∈G N (x) > 1 (see (10.1) (INF)). The main tool to be used is the following lemma which takes advantage of the fact that a single state ρ on Q may be used to determine scaling states for different values of β, as long as ρ remains β-subinvariant. In particular note that if ρ is β-subinvariant for some β then the same holds for any β  > β, i.e. when the “temperature” 1/β decreases. 15.1. Lemma. (Cooling Lemma). Assume (10.1) (COL+INF) and let β ∈ (0, ∞). Given a β-scaling state φ on C0 (TA ) set ρ = φ|Q , so that ρ is a β-subinvariant state on Q. Let β  > β and observe that ρ is clearly also β  -subinvariant. Let φ  be the unique β  -scaling state on C0 (TA ) whose restriction to Q coincides with ρ, by (13.3). Then φ  is of finite type. Proof. Let λ and λ be the measures on TA corresponding to ρ and ρ  , respectively. Recall from (8.12) that   N (µ) β φ  (qµ ). λ (∞ ) = lim n→∞

µ∈PAn

Partial Dynamical Systems and the KMS Condition

269

However φ  (qµ ) = ρ(qµ ) = φ(qµ ) so that the above expression for λ (∞ ) actually depends only on β  . Let R = inf x∈G N (x) and let δ = β  − β. Observe that for all x ∈ G one has  N (x) β = N (x) δ N (x) β ≤ R δ N (x) β . It follows that for all µ ∈ PAn we have N (µ) λ (∞ ) ≤ lim R n→∞

nδ

β

≤ R nδ N (µ) β and hence that  N (µ) β φ(qµ ). µ∈PAn

Observe that for all n one has, using (8.5), that   N (µ) β φ(qµ ) = φ(pµ ) ≤ 1 µ∈PAn

µ∈PAn

because the pµ are pairwise orthogonal projections. Since R nδ → 0 as n → ∞, we conclude that λ (∞ ) = 0 and hence that λ is of finite type.   As a conclusion we may boost the result obtained in (10.6): 15.2. Theorem. Assume (10.1) (IRR+INF) and let β < β˙c . Then there are no β-scaling states at all on C0 (TA ) and hence neither are there KMSβ states on TA . Proof. Suppose by contradiction that φ is a β-scaling state on C0 (TA ). Choose δ > 0 such that β  := β + δ < β˙c and, using (13.3), let φ  be the unique β  -scaling state such that φ  |Q = φ|Q . Then φ  is of finite type by (15.1) contradicting (10.6).   The following diagram gives information about β-scaling states on C0 (TA ), and hence also about KMSβ states on TA , under the hypothesis of (15.2) improving upon Diagram 10.7: β˙c

0

βc

None at all (Theorem 15.2)

∞ Only finite type, one for each measure on e (Corollary 9.7)

Diagram 15.3

Unfortunately we don’t have much more to say about the case in which β lies in the interval between β˙c and βc . Nevertheless this mysterious interval sometimes collapses, as in the following situation: 15.4. Proposition. Assume (10.1) (IRR+FTS). Then β˙c = βc and I˙c = Ic . Proof. Let {y1 , · · · , yn }  ⊆ G be a finite target set as in (10.1) (FTS). Decompose G in a disjoint union G = ni=1 Gi such that for every x ∈ Gi one has A(x, yi ) = 1.  y Consequently PA decomposes as the disjoint union PA = {e} ∪ ni=1 PAi and so for every β, Z(β) =

 µ∈PA

N (µ)

β

=1+

n   i=1 µ∈P yi A

N (µ)

β

=1+

n 

Zyi (β).

i=1

Therefore if β ∈ I˙c we have that Zyi (β) < ∞ for all i and hence Z(β) < ∞ so that β ∈ Ic .  

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Under the hypotheses of (15.2) and (15.4), i.e. all of the hypotheses listed in (10.1), Diagram 15.3 therefore gives as much information as we could possibly want about β-scaling states throughout the whole interval (0, ∞], except perhaps at the critical point. 16. An Example of Behavior at the Critical Point In this section we will show that, even if one assumes all of the hypotheses listed in (10.1), there is not much more that can be said in general about the nature of KMS states at the critical inverse temperature βc . We will eventually prove that the following antagonistic situations may occur: (a) The KMSβc state may be unique and of infinite type. (b) There may be infinitely many KMSβc states all of which are of finite type. In fact in this section we will just give an example of situation (b) since we will later show that situation (a) is the rule for finite irreducible matrices. Let ζ (β) =

∞  k=1

Nk

β

¯ with be any Dirichlet series which converges at its abscissa of convergence, say β, β¯ ∈ (0, ∞). Put G = N. It is relevant to us that G consist of one element for each term of the above series, plus one more element, namely zero. Accordingly we will write G∗ = N \ {0}. Consider the matrix   0 1 1 1 ... 1 0 0 0 ...  1 0 0 0 ...   A= 1 0 0 0 ... ,   .. .. .. .. . . . . . . . whose index set is G × G. In other words the 0th column and the 0th row of A consist of ones, except for A(0, 0) which is zero. All other entries are zero. Clearly A is irreducible and satisfies (10.1) (COL). Observe that A also satisfies (10.1) (FTS) since for every x ∈ G∗ one has that A(x, 0) = 1, while A(0, 1) = 1. That is, the set {0, 1} is a finite target set. Discarding a finite number of terms and relabeling we may suppose that ¯ = ζ (β)

∞  k=1

β¯

¯

Nk < 2 β .

(†)

The convergence of the above Dirichlet series implies that limk→∞ Nk = ∞ and hence, discarding another finite set of terms, we may suppose that Nk ≥ 2 for all k. Set N(0) = 2 and N (k) = Nk for all k ∈ G∗ , so that (10.1) (INF) holds. We are therefore under a situation in which everything in (10.1) holds. We would now like to compute the partition function Z0 (β). In order to do this observe that the admissible words ending in 0 are precisely of the form  if |µ| = 2n, or  x1 0 x2 0 . . . 0 xn 0 µ=  0 x 0 x 0 . . . 0 x 0 if |µ| = 2n + 1, 1 2 n

Partial Dynamical Systems and the KMS Condition

271

where x* = (x1 , x2 , . . . , xn ) is an arbitrary element of G∗n . Therefore Z0 (β) =

∞ 

2

n=0



nβ

N (x1 ) β · · · N (xn )

β

x*∈G∗n

+

∞  n=0

2

(n+1)β



N (x1 ) β · · · N (xn ) β .

x*∈G∗n

By considering the summands corresponding to n = 1 above we see that Z0 (β) diverges ¯ ∞). when ζ (β) diverges. Therefore the convergence interval for Z0 (β) is contained in [β, Moreover notice that for all n ∈ N we have n    N (x1 ) β · · · N (xn ) β =  N (x) β  = ζ (β)n . x*∈G∗n

x∈G∗

So, Z0 (β) =

∞ 

2

nβ

ζ (β)n +

n=0

∞ 

2

(n+1)β

∞   β  n 2 ζ (β) . ζ (β)n = 1 + 2 β

n=0

n=0

¯ is a converging geometric series by (†). It follows that We therefore see that Z0 (β) ¯ ∞]. By (15.4) we also have Ic = [β, ¯ ∞]. β˙c = β¯ and I˙c = [β, Combining (15.2) with (9.7) we therefore obtain: 16.1. Proposition. Let A, N , and β¯ be given as above. Then (i) For β < β¯ there are no KMSβ states on TA , (ii) For β ≥ β¯ the simplex of KMSβ states on TA is affine-homeomorphic to the simplex of finite measures on e such that Z(β, γ ) = 1. In order to best appreciate this result it is important to observe that for all measures γ on e one has  Zx (β)γ (xe ) ≤ Z(β)γ (e ), Z(β, γ ) = γ (e ) + x∈G

¯ for any finite measure γ and hence, as long as Z(β) is finite, that is as long as β ≥ β, on e the measure γ /Z(β, γ ) fits into (16.1.ii). If follows that there are infinitely many KMS states at the critical inverse temperature. 17. KMS States on OA
A is the quotient of T
A obtained by imposing relation CK4 in addition to Recall that O CK1−3 . Clearly the quotient map
A N : T
A → O is then covariant for the respective one-parameter automorphism groups. For every KMS
A one therefore has that ψ ◦ N is a KMS state on T
A and hence the simplex state ψ on O
A may be seen as a subset of the KMS states on T
A . This section is of KMS states on O dedicated to giving a characterization of this subset.
A (and Nevertheless it should be observed that occasionally it happens that T
A = O hence also TA = OA ) and we start by characterizing when exactly this is the case.

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17.1. Proposition. For any 0–1 matrix A = {A(x, y)}x,y∈G having no identically zero rows the following are equivalent: (i) Given any neighborhood V of any point c ∈ 9A there are infinitely many j ∈ G such that the column cj of A lies in V ,
A . (ii) T
A = O Proof. Suppose that (i) holds. Then the closure of the set of columns of A within 2G , namely 9A , coincides with the set of accumulation points of the columns of A. Therefore
A . The converse is proven by running this
TA = 
OA by [EL, 7.7] and hence T
A = O  argument backwards.   It should be remarked that condition (17.1.i) comes close to saying that 9A is a perfect topological space (i.e. that it has no isolated points) except that when one “counts” how many columns there are in a neighborhood one should look at the set of indices rather than at the set of columns itself. When all columns are distinct (or repeated at most finitely often) one has that (17.1.i) is therefore equivalent to 9A being perfect. Regardless of the columns being distinct, if 9A is perfect one clearly has that (17.1.i) holds. Under the above circumstances the study of KMS states on OA is therefore identical to the corresponding study for TA . In the opposite case, however, it is useful to obtain criteria to distinguish, among the KMS states on TA , which ones factor through OA . 17.2. Theorem. Assuming (8.1) let β ∈ (0, ∞]. Also let • ψ be a KMSβ state on TA , • φ be the restriction of ψ to C0 (TA ), • λ be the measure on TA representing φ, and • ρ be the restriction of φ to Q. Then the following are equivalent: (i) there exists a KMSβ state ψ  on OA such that ψ = ψ  ◦ N,  (ii) ρ(q(X, Y )) = z∈G A(X, Y, z)N (z) β ρ(qz ) whenever X, Y ⊆ G are finite and A(X, Y, z) is finitely supported as a function of z, (iii) the support of λ is contained in the closure of ∞ .  Proof. Assume (i). If X and Y are as in (ii) then N(q(X, Y )) = z∈G A(X, Y, z)N(pz ) by CK4 and hence      ρ(q(X, Y )) = ψ(q(X, Y )) = ψ  N(q(X, Y )) = ψ   A(X, Y, z)N(pz ) =

 z∈G

A(X, Y, z)ψ(pz ) =



z∈G

A(X, Y, z)N (z) β ρ(qz ),

z∈G

proving (ii). Assume (ii) and let ξ ∈ TA \ ∞ . Pick a neighborhood V of ξ disjoint from ∞ which, by [EL, 6.2], may be chosen so as to have the form   η ∈ TA : ω ∈ η,     ωx 1 ∈ η, for x in X V = , ωy 1 ∈ / η, for y in Y     ωz ∈ / η, for z in Z

Partial Dynamical Systems and the KMS Condition

273

where ω is the (finite) stem of ξ and X, Y , and Z are finite subsets of G, with X ⊆ Rξ (ω) and Y ∩ Rξ (ω) = ∅. We wish to show that λ(V ) = 0 from which it will follow that ξ is not in the support of λ thus proving (iii). Let U = αω 1 (V ) so that U=

  η ∈ TA : 

x y z

1 1

 for x in X  for y in Y .  for z in Z

∈ η, ∈ / η, ∈ / η,

Because λ is β-scaling we have that λ(V ) = λ(αω (U )) = N (ω) β λ(U ) (when β = ∞ and ω = e this should be interpreted as zero), so it is enough to show that λ(U ) = 0. Note that the characteristic function of U is given precisely by 1U = q(X, Y )



(1 − pz ) = q(X, Y ) 1 −

z∈Z

= q(X, Y ) −





pz

= q(X, Y ) −

z∈Z



q(X, Y )pz

z∈Z

A(X, Y, z)pz ,

z∈Z

where the last equality follows from CK3 as shown in the beginning of the proof of (12.1). We claim that A(X, Y, z) = 0 for all z ∈ / Z. Arguing by contradiction suppose that z0 ∈ / Z and A(X, Y, z0 ) = 1. Therefore A(x, z0 ) = 1 for all x ∈ X and A(y, z0 ) = 0 for all y ∈ Y . Pick an infinite admissible word ν beginning in z0 (which exists because no row of A is zero). By [EL, 5.13] there exists η ∈ ∞ whose stem coincides with ν. Inspecting
TOA , and noting that z0 ∈ η it is easy to show Definition 5.1, observing that η ∈  that η ∈ U . This contradicts the fact that U and ∞ are disjoint and hence we see that A(X, Y, z) = 0 for all z ∈ / Z as claimed. Using (ii) we therefore have λ(U ) = φ(1U ) = φ(q(X, Y )) − = ρ(q(X, Y )) −





A(X, Y, z)φ(pz )

z∈Z

A(X, Y, z)N (z) β ρ(qz ) = 0.

z∈Z

This proves that (ii) implies (iii). In order to prove that (iii) implies (i) let λ be the restriction of λ to a measure on ∞ = OA (see [EL, 7.3]) which is a probability measure by hypothesis. Obviously λ is β-scaling and hence by (8.2) there exists a KMSβ state ψ  on OA whose restriction to C0 (OA ) is given by integration against λ . We claim that ψ = ψ  ◦ N. Given that ψ = φ ◦ E by (8.2) and similarly for ψ  it is enough to verify that ψ and ψ  ◦ N coincide on C0 (TA ) but this is now obvious.   It should be remarked that every infinite type β-scaling measure λ on TA satisfies (17.2.iii) and hence is associated to a KMSβ state on OA . However, since ∞ is not necessarily closed, there may exist measures supported in ∞ which are not of infinite type.

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18. The Finite Dimensional Case Throughout this section we assume that G is a finite set and hence A is a finite matrix. Many simplifications take place under this hypothesis and the results can be stated a bit more conclusively. Being the closure of the set of columns of A within 2G , 9A is hence a finite space with d(A) points, where: 18.1. Definition. We denote by d(A) the number of distinct columns of A. Throughout this section we will assume that no column of A is zero. So the zero
TA that C ∈
TA vector does not belong to 9A and it follows from the definition of  / 
TA and OA = 
OA . This implies that TA = T
A as well as that and hence that TA = 
A . In other words all algebras and spaces with no tilde coincide with their tilde OA = O versions in the last two rows of Table 7.1 (the same not holding for the first row because
TOA always). C∈ Before we proceed we need the following consequence of the Perron–Frobenius Theorem: 18.2. Lemma. Let S be the set of all n × n irreducible (in the sense of [Se, Definition 1.6]) nonnegative matrices. Let S1 be the subset of S formed by the matrices M such n that ∞ n=0 M converges. Then S1 is open in S.  n Proof. Let M ∈ S1 . Then clearly ∞ n=0 (rM) converges for all r ∈ (0, 1]. Therefore 1 − rM is invertible for all such r and hence no eigenvalue of M lies in the interval [1, ∞). By the Perron–Frobenius Theorem [Se, 1.5] it follows that the spectral radius of M is strictly less than 1. Since the spectrum is lower semicontinuous there exists a neighborhood of M consisting solely of matrices whose spectral radius is less than 1. This neighborhood is therefore contained in S1 .   Let us first study the three critical inverse temperatures for a finite irreducible matrix. As before we will denote by N the diagonal matrix with N (x, x) := N (x). 18.3. Proposition. Under (8.1) let A be a finite irreducible matrix. Then (i) β¨c = β˙c = βc < ∞, (ii) I¨c = I˙c = Ic = (βc , ∞], and (iii) the spectral radius of AN βc is 1. Proof. Breaking the admissible words according to their final and initial letter we have that   Z(β) = 1 + Zy (β) = 1 + Zxy (β). y∈G

x,y∈G

Since G is finite we therefore have that Z(β) < ∞ if and only if Zy (β) < ∞ for all y ∈ G if and only if Zxy (β) < ∞ for all x, y ∈ G. Therefore I¨c = I˙c = Ic and hence also β¨c = β˙c = βc .  For a large enough β one clearly has that y∈G N (y) β < 1 and hence, using (8.16), we have that Z(β) < ∞. This shows that βc < ∞.

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For every x and y in G note that the (x, y) entry of the formal power series of matrices ∞  

AN

 β n

(†)

n=0

is precisely given by N (x)β Zxy (β). Therefore that series converges if and only if all Zxy (β) < ∞, which is the same as saying that β ∈ I¨c . In other words ) I¨c = β ∈ (0, ∞) : AN

β

* ∈ S1 ∪ {∞},

where S1 is as in (18.2). So I¨c is an open set (in the extended real line) and it follows that I¨c = (β¨c , ∞] or, equivalently, that Ic = (βc , ∞]. In order to prove (iii) let, for each β ∈ R, r(β) be the spectral radius of AN β . If β > βc we have seen that (†) converges and hence r(β) ≤ 1. Taking the limit as β → βc we conclude that r(βc ) ≤ 1. Suppose by contradiction that r(βc ) < 1. Then we would have that (†) converges for β = βc which was ruled out in (ii).   With this we may give a precise description of the KMS states on TA : 18.4. Theorem. Under (8.1) let A be a finite irreducible 0–1 matrix. Then: (i) For β > βc the KMSβ states on TA form a simplex of dimension d(A) − 1 which is affine homeomorphic to the simplex of all measures γ on the finite measure space e such that Z(β, γ ) = 1. (ii) For β = βc there exists precisely one KMSβ state ψ. Its restriction to C0 (TA ) is of infinite type and it is determined uniquely by the fact that (ψ(qx ))x∈G is the unique nonnegative normalized (in the sense that x∈G N (x) β vx = 1) eigenvector v of the matrix AN βc with (dominant) eigenvalue 1. (iii) For β < βc there are no KMSβ states on TA at all.
e = Proof. As observed above #9A = d(A) and hence by (11.1.i) one also has that #
d(A). By the remark following (8.6) we have e = e \ {C}. Since A is irreducible and
TA ⊇ 
e so actually e = 
e . Therefore hence (10.1) (COL) holds we have that C ∈ / #e = d(A). Given β > βc the set of (positive) measures γ on e with Z(β, γ ) = 1 therefore forms a simplex of dimension d(A) − 1. Point (i) then follows from (9.7). As for (iii), this follows from (15.2) given that (10.1) (IRR+INF) are granted. / I˙c . From (10.6) we then In order to prove (ii) observe that by (18.3) one has that βc ∈ conclude that all βc -scaling states are of infinite type. Using (13.6) we therefore have that the KMSβc states on TA correspond bijectively to the normalized (in the above sense) nonnegative fixed points for the matrix AN βc . By (18.3.iii) and the Perron–Frobenius Theorem [Se, Theorem 1.5] we have that there is exactly one such vector. This concludes the proof.   Our next result gives a precise description of the KMS states on OA in terms of the eigenvalues of AN β , even if A is not irreducible. This was first proved in [EFW] under the special case that the N (x) are all the same.

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18.5. Theorem. Under (8.1) let A be a finite matrix without identically zero columns. Then the KMSβ states on OA occur exactly at the values of β for which there exists a nonnegative vector v = 0 satisfying AN β (v) = v. Given such a β the correspondence ψ → (ψ(qx ))x∈G defines an affine bijection from the simplex of all KMSβ states ψ on OA to the simplex  of all normalized (i.e. x∈G N (x) β vx = 1) nonnegative solutions of the equation AN β (v) = v. Proof. By (17.2) we know that the KMS states on OA , equivalently the KMS states on TA which factor through OA , correspond to the β-scaling measures λ on TA supported in the closure of ∞ . Under the present hypothesis that G is finite we claim that ∞ is closed in TA . To see this let ξ be a bounded element of TA with stem ω. Then the set / η for all z ∈ G} V = {η ∈ TA : ω ∈ η and ωz ∈ is a neighborhood of ξ not intersecting ∞ . Therefore the measures λ mentioned above consist precisely of the infinite type βscaling measures. The conclusion then follows from (13.6).   References [BR]

Bratteli, O., Robinson, D.W.: Operator algebras and quantum statistical mechanics. 2. Texts and Monographs in Physics, Berlin-Heidelberg-New York: Springer-Verlag, 1997 [BMS] Brown, L.G., Mingo, J.A., Shen, N.T.: Quasi-multipliers and embeddings of Hilbert C ∗ -bimodules Canad. J. Math. 46, 1150–1174 (1994) [C] Cuntz, J.: Simple C ∗ -algebras generated by isometries. Commun. Math. Phys. 57, 173–185 (1977) [CK] Cuntz, J., Krieger, W.: A Class of C ∗ -algebras and Topological Markov Chains. Invent. Math. 56, 251–268 (1980) [EFW] Enomoto, M., Fujii, M., Watatani, Y.: KMS states for gauge action on OA . Math. Japon. 29, 607–619 (1984) [Ev] Evans, D.E.: On On . Publ. Res. Inst. Math. Sci., Kyoto Univ. 16, 915–927 (1980) [E1] Exel, R.: Circle Actions on C*-Algebras, Partial Automorphisms and a Generalized Pimsner– Voiculescu Exact Sequence. J. Funct. Analysis 122, 361–401 (1994) [funct-an/9211001] [E2] Exel, R.: Amenability for Fell Bundles. J. reine angew. Math. 492, 41–73 (1997) [functan/9604009] [E3] Exel, R.: Twisted Partial Actions, A Classification of Regular C*-Algebraic Bundles. Proc. Lond. Math. Soc. 74, 417–443 (1997) [funct-an/9405001] [E4] Exel, R.: Partial actions of groups and actions of inverse semigroups. Proc. Am. Math. Soc. 126, 3481–3494 (1998) [funct-an/9511003] [EL] Exel, R., Laca, M.: Cuntz–Krieger Algebras for Infinite Matrices. J. reine angew. Math. 512, 119–172 (1999) [funct-an/9712008] [ELQ] Exel, R., Laca, M., Quigg, J.: Partial Dynamical Systems and C*-Algebras generated by Partial J. Operator Theory 47, 169–186 (2002) [funct-an/9712007] [FLR] Fowler, N., Laca, M., Raeburn, I.: The C*-algebras of infinite graphs. Proc. Am. Math. Soc. 128, 2319–2327 (2000) [L] Laca, M.: Semigroup of *-endomorphisms, Dirichlet series, and phase transitions. J. Funct. Anal. 152, 330–378 (1998) [M] McClanahan, K.: K-theory for partial crossed products by discrete groups. J. Funct. Anal. 130, 77–117 (1995) [OP] Olesen, D., Pedersen, G.K.: Some C ∗ -dynamical systems with a single KMS state. Math. Scand. 42, 111–118 (1978) [Pe] Pedersen, G.K.: C ∗ -Algebras and their automorphism groups. London-New York: Acad. Press, 1979

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Pimsner, M.: A class of C*-algebras generalizing both Cuntz–Krieger algebras and crossed products by Z. In: Free Probability Theory, D.-V. Voiculescu, ed., Fields Inst. Commun. 12, Toronto: Fields Distitute Publ. 1997, pp. 189–212 [PWY] Pinzari, C., Watatani, Y., Yonetani, K.: KMS States, Entropy and the Variational Principle in full C*-dynamical systems. Preprint, 1999 [math.OA/9912151] [Se] Seneta, E.: Non-negative Matrices and Markov Chains. Springer Series in Statistics, BerlinHeidelberg-New York: Springer–Verlag, 1981 [Sz] Szyma´nski, W.: Bimodules for Cuntz–Krieger Algebras of Infinite Matrices. Preprint, University of Newcastle, 1999 [V] Vere-Jones, D.: Ergodic properties of nonnegative matrices. Pacific J. Math. 22, 361–386 (1967) [Z] Zacharias, J.: Quasi-Free Automorphisms of Cuntz–Krieger–Pimsner Algebras. Preprint, University of Notingham, 2000 Communicated by A. Connes

Commun. Math. Phys. 232, 279–302 (2003) Digital Object Identifier (DOI) 10.1007/s00220-002-0749-5

Communications in

Mathematical Physics

Einstein Relation for a Class of Interface Models Roberto H. Schonmann∗ Department of Mathematics, UCLA, Los Angeles, CA 90095, USA. E-mail: [email protected] Received: 16 April 2002 / Accepted: 3 July 2002 Published online: 22 November 2002 – © Springer-Verlag 2002

Abstract: A class of SOS interface models which can be seen as simplified stochastic Ising model interfaces is studied. In the absence of an external field the long-time fluctuations of the interface are shown to behave as Brownian motion with diffusion coefficient (σ GK )2 given by a Green-Kubo formula. When a small external field h is applied, it is shown that the shape of the interface converges exponentially fast to a stationary distribution and the interface moves with an asymptotic velocity v(h). The mobility is shown to exist and to satisfy the Einstein relation: (dv/dh)(0) = β(σ GK )2 , where β is the inverse temperature. 1. Introduction 1.1. Motivation and overview. In this paper we are concerned with the movement of an interface between two phases of a system subject to a small external field which favors one of them. This is a well known problem, with a large number of applications (see, e.g., [Spo, RK] and references therein). In particular, our interest on this issue arose from the analysis of the nucleation phenomenon in the context of stochastic Ising models below the critical temperature, and under a small external field h (see, [SS, RTMS] and references therein). In this regime, assuming h > 0, droplets of the (+)-phase which are sufficiently large, grow and invade a background given by the (−)-phase. The order of magnitude of the velocity v(h) with which the linear dimensions of such a droplet grow is of great interest. In particular information on this is needed in order to estimate the time required for the system to relax from the metastable (−)-phase to the stable (+)-phase (see references quoted above, and especially Sect. 1.5 of [SS]). In two dimensions, at any subcritical temperature, and in higher dimensions above the roughening transition temperature, it is believed that v(h) is asymptotically linear in h for small values of h. In contrast, in higher dimensions and below the roughening transition temperature, there ∗

Work partially supported by the N.S.F. through grants DMS-0071766 and DMS-0074152.

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does not seem to be a consensus on what to expect. Some papers (see e.g., [RTMS] and references therein) suggest that v(h) is still asymptotically linear in h for small values of h. Other papers (see, e.g., Sect. 8 of [Spo] and references therein) propose that in this regime v(h) goes to 0 substantially faster than linearly in h (actually as exp(−C/ hd−2 )), as h → 0, due to the rigidity of the interface, with nucleation on the interface playing a major role in its movement. The rigorous results in [SS] on the growth of the (+)-phase into the (−)-phase in the two-dimensional case were sufficient to precisely estimate the relaxation time at the level of its logarithmic dependence on 1/ h, as h → 0. Nevertheless, the estimate in that paper on the behavior of the velocity v(h) in this limit, was quite modest. That estimate basically stated that for arbitrary  > 0, v(h) > exp(−/ h), for small h > 0. (No precise definition of v(h) was provided or needed in that paper, but the estimates on droplet growth, in Sect. 3.2 of that paper, imply this bound for any reasonable definition of v(h).) While such a bound is very far from a conjectured linear behavior, it seems very hard to improve it much in the context of stochastic Ising models. Here we will follow a common route in Mathematical Physics. Having as motivation the problem discussed above, and not being able to solve it, we will analyze some simplified interface models, in the hope that their behavior and the techniques used may shed light on the original problem. The idealized models to be considered correspond to finite interfaces, confined to move inside tubes. The movement will be described by a reversible Markov process, obtained by considering that the interface separates spins −1 and +1 and that these spins flip like a stochastic Ising model, but with suppression of flips that would create more than one interface, or violate a restriction on the interface. This restriction will be that the interface is of the SOS type, i.e., described by a function on a set of sites, perpendicular to the direction of movement. This restriction separates the idealized models further from the original models, but should be of lesser relevance at low temperatures. While the SOS interface models in a tube that we will consider are not new (see, e.g., Sect. 6 of [Spo]), our results are, to the best of our knowledge, new. We will relate the velocity with which the interface moves under a small external field h to the fluctuations of the same interface in the absence of an external field. The specific relation that will be derived, and which is stated in the abstract above, is a sort of Einstein or Einstein-Green-Kubo relation (see, e.g., [LR], or Sect. 8 of [LeS], and references therein for introductions to this relation and rigorous results in different contexts). In the context of interface models, such a relation was proposed based on non-rigorous arguments (for more ambitious models) in Sect. 3 of [Spo]. In this connection, we point out that in [But] rigorous results on the Einstein relation for interfaces in certain interacting particle systems were obtained. That paper is, nevertheless, completely different from the present one; there a Kac-type interaction was considered in infinite volume, and the definitions of the quantities appearing in the Einstein relation were different from those here. One corollary to the Einstein relation is that in our models the interface moves with velocity v(h) which is asymptotically linear in a small external field h. In other words, the mobility M = (dv/dh)(0) is positive and finite. While this may not be a surprise, since we are studying an interface confined to a tube of finite cross section, we could only prove it using some rather involved arguments. This is so because the tube itself is infinite. In particular, it does not seem obvious that even a small external field will not produce a major change in the behavior of the interface. To rule out this possibility, we considered the Markov chain that describes the shape of the interface, and showed that when h = 0 it is reversible and has positive spectral gap. This then allowed us to treat

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the case with small h as a perturbation. As a by-product, we learned that also under a small h = 0, the Markov chain which describes the shape of the interface, in spite of no longer being reversible, still has a single invariant distribution, to which it converges exponentially fast. Our results then raise the question of how the mobility M depends on the size of the interface, and how the answer to this question depends on the dimension of the interface and on the temperature and the corresponding rigidity or roughness of the interface in the absence of external field. These are fundamental questions, which should help clarify the issues discussed at the end of the first paragraph of this introduction. Since their solution seems to require substantial additional techniques on top of those used in the current paper, we are postponing these issues to a further investigation.

1.2. The models and basic notation. For reasons that will become clear soon, our interface models will be described as functions from the vertices of a finite graph to the integers. For this purpose, let G = (V , E) be a finite graph, with vertex set V and edge set E. For a technical reason, we will suppose that |V | ≥ 3. (The case |V | = 1 is trivial, and will be discussed at the end of the introduction. The case |V | = 2 can also be treated by elementary methods.) In order to present some basic examples, we recall first some terminology from graph theory. Given two graphs, G = (V , E ) and G = (V , E ), the Cartesian product G × G is defined as the graph G = (V , E) which has as vertex set V = V × V = {{v , v } : v ∈ V , v ∈ V }, and as edge set E = {{(v , v ), (u , u )} : ({v , u } ∈ E and v = u ) or (v = u and {v , u } ∈ E )}. Up to relabeling, the Cartesian product is associative and commutative. The basic examples of graphs on which our interface models are defined
fr are: fr  (1) 1 dimensional interface, free boundary condition: Gfr 1 (l) = V1 (l), E1 (l) , where V1fr (l) = {1, 2, . . . , l},

E1fr (l) = {{1, 2}, {2, 3}, . . . , {l − 1, l}}.
per per per (2) 1 dimensional interface, periodic boundary conditions: G1 (l) = V1 (l), E1  (l) , where per

V1 (l) = {1, 2, . . . , l},

per

E1 (l) = {{1, 2}, {2, 3}, . . . , {l − 1, l}, {l, 1}}.

(3) d dimensional interface, free boundary conditions: fr fr fr Gfr d (l1 , l2 , . . . , ld ) = G1 (l1 ) × G1 (l2 ) × · · · × G1 (ld ).

(4) d dimensional interface, periodic boundary conditions: per

per

per

per

Gd (l1 , l2 , . . . , ld ) = G1 (l1 ) × G1 (l2 ) × · · · × G1 (ld ). An SOS interface is defined as a function η : V → Z, and the set of all such functions will be denoted, as usual, by ZV . Of particular importance will be the flat interfaces, φi , i ∈ Z, defined by φi (v) = i, for each v in V . For η ∈ ZV and i ∈ Z, we will denote by η + i the interface obtained by shifting η by i, i.e., (η + i)(x) = η(x) + i, for each x in V .

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We will write η ≤ ζ if η(v) ≤ ζ (v), for each v in V (and similarly for ≥). We will write η < ζ if η ≤ ζ and η(v) < ζ (v) for some v in V (and similarly for >). To motivate the definition of our interface models, we describe now Ising models on the set of sites: V¯ = V × Z. Their configurations lie in the set ¯

 Ising = {−1, +1}V . In order to define the Hamiltonian, we declare sites in V¯ to be neighbors if they have the first coordinate in common and their second coordinates differ by one unit, or if they have the second coordinate in common and their first coordinates are endpoints of an ¯ In other words, edge in E. The set of pairs of neighbors of V¯ will be denoted by E. ¯ ¯ (V , E) = (V , E) × (Z, EZ ), in the sense of the Cartesian product of graphs, where EZ = {{i, i + 1} : i ∈ Z}. The formal Ising model Hamiltonian is defined by Ising

Hh

(σ ) = −

1  h σ (x)σ (y) − σ (x), 2 2 {x,y}∈E¯

(1.1)

x∈V¯

where h ∈ R is the external field. To each interface η ∈ ZV we associate the spin configuration σ [η] ∈  Ising , given by  +1 if i ≤ η(v), (σ [η])(v, i) = −1 if i > η(v). The set of Ising model configurations which correspond to SOS interfaces will be denoted by  SOS = {σ [η] : η ∈ ZV }. Note that σ [·] establishes a one-to-one correspondence between ZV and  SOS . In terms of the SOS interface, the formal Hamiltonian (1.1) restricted to  SOS can be rewritten, up to a constant (infinite) term as   Ising |η(v) − η(u)| − h η(v). (1.2) HhInt (η) = Hh (σ [η]) = {v,u}∈E

v∈V

Note that HhInt (·) is not just a formal Hamiltonian, but rather a well defined function ZV → R. For later use, note also that for i ∈ Z, HhInt (η + i) = HhInt (η) − |V |hi.

(1.3)

We introduce the unnormalized Gibbs measure at inverse temperature β ∈ (0, ∞) on ZV by
 Int µInt β,h (η) = exp − βHh (η) . Note that since ZV is countable, this defines a measure on this set. But   µInt µInt β,h (η) ≥ β,h (φi ) = ∞, η∈ZV

i∈Z

so that this measure cannot be normalized to a probability measure.

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We will now introduce stochastic dynamics on the interface space ZV . These stochastic dynamics will be introduced in discrete time, in order to simplify certain arguments later on, but they are motivated by continuous time spin flip dynamics for the underlying Ising model (stochastic Ising models), in which we suppress flips which would move the configuration outside of  SOS . Given η ∈ ZV and v ∈ V , define the interfaces ηv,↑ and ηv,↓ , via  η(u) if u = v, ηv,↑ (u) = η(u) + 1 if u = v. η

v,↓

 (y) =

η(u) η(u) − 1

if if

u = v, u = v.

We will write η ∼ ζ if η = ζ v, , for some v ∈ V and  ∈ {↑, ↓}. Note that this is a symmetric relation and that if η ∼ ζ then η < ζ or η > ζ . Suppose given a function F : R → (0, 1] which satisfies F (−r) = er F (r), For instance, we can have
 F (r) = exp −r + ,

where r + = r for r ≥ 0 and r + = 0, for r < 0,

or, F (r) = Set

for all r ∈ R.

1 . 1 + er

(1.4)

(1.5)

(1.6)



 p˜ β,h (η, ζ ) = F β HhInt (ζ ) − HhInt (η) .

Condition (1.4) was imposed in order for the detailed balance (reversibility) condition to hold for p˜ β,h (·, ·): Int µInt β,h (η) p˜ β,h (η, ζ ) = µβ,h (ζ ) p˜ β,h (ζ, η),

(1.7)

for every η, ζ ∈ ZV . Our discrete time dynamics on ZV can now be defined as follows. Suppose that at time n the process is in the configuration η. Then at time n+1 a vertex v ∈ V is randomly chosen, and then, independently, one of the two symbols ↑ or ↓ is also chosen at random; denote by  the chosen one. Given these choices, the state of the process at time n + 1 will be either ηv, or η, the former case happening with probability p˜ β,h (η, ηv, ) independently of anything else. In other words, we are defining a Markov chain (ηn )n=0,1,... on ZV , with transition probabilities, when η = ζ , given by  1 if η ∼ ζ , p˜ β,h (η, ζ ) Int pβ,h (η, ζ ) = (1.8) 2|V | 0 otherwise, while

Int pβ,h (η, η) = 1 −

 ζ =η

Int pβ,h (η, ζ ).

(1.9)

Int Pβ,h;µ will denote a probability measure under which (ηn )n=0,1,... is such a Markov chain, started from η0 which has probability distribution µ. The corresponding

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Int . When µ is concentrated on a single configuexpectation will be denoted by Eβ,h;µ ration ζ , we will also write ζ instead of µ in this notation. When F (·) is given by (1.5), (ηn )n=0,1,... evolves according to Metropolis dynamics, while when F (·) is given by (1.6), it evolves according to the Gibbs sampler dynamics. For each β and h,   
Int (1.10) β Hh (ζ ) − HhInt (η) : η ∼ ζ

˜ is a finite set. Therefore there exists C(β, h) > 0 such that Int ˜ pβ,h (η, ζ ) > C(β, h)

for all η ∼ ζ .

(1.11)

Lemma 1.1. For every β ∈ (0, ∞), h ∈ R and η ∈ ZV , Int pβ,h (η, η) ≥

1 − e−β . 2|V |

(1.12)

Proof. Suppose, with no loss of generality, that h ≥ 0. Let v ∈ V be a point of minimum of η(v). Then HhInt (ηv,↓ ) = HhInt (η) + DegG (v) + h, where DegG (v) = |{u ∈ V : {v, u} ∈ E}| is the degree of v in the graph G. So p˜ β,h (η, ηv,↓ ) = F (β(DegG (v)+h)) = e−β(DegG (v)+h) F (−β(DegG (v)+h)) ≤ e−β , since F takes values in (0, 1] and DegG (v) ≥ 1. Therefore, Int pβ,h (η, η) ≥


 1 − e−β 1
1 − p˜ β,h η, ηv,↓ ≥ , 2|V | 2|V |

where the expression in the middle is the probability that the vertex v and the symbol ↓ are chosen, but the jump from η to ηv,↓ does not occur.   For later convenience we introduce the notation η ≈ ζ when η = ζ or η ∼ ζ . From (1.11) and (1.12), we learn that there exists C(β, h) > 0 such that Int pβ,h (η, ζ ) > C(β, h) for all η ≈ ζ.

(1.13)

Int , it follows that the Markov chain (η ) From (1.7) and the definition of pβ,h n n=0,1,... , i.e., is reversible with respect to µInt β,h Int Int Int µInt β,h (η) pβ,h (η, ζ ) = µβ,h (ζ ) pβ,h (ζ, η),

(1.14)

for every η, ζ ∈ ZV . It is clear that (ηn )n=0,... is irreducible, and from (1.12) it is also aperiodic. Since µInt β,h has infinite mass, (ηn )n=0,... is either transient or null recurrent.

It is natural to describe an interface η ∈ ZV through its shape and its position. This “decomposition” will be crucial in this paper. In the next two subsections, we introduce each one of these “components” in technically convenient fashions.

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1.3. The shape Markov chain. To define the shape of an interface, first define the relation η∼ =ζ

if

η = ζ + i,

for some i ∈ Z.

Note that this is an equivalence relation, and define the shape of η ∈ ZV , denoted by Sh[η], as the equivalence class to which η belongs. The set of all shapes will be denoted by   S = Sh[η] : η ∈ ZV . The class of the flat interfaces will be denoted by φ = {φi : i ∈ Z}. We will write s ∼ s (resp. s ≈ s ) if s = Sh[η ], s = Sh[η ], for some η ; η ∈ ZV , η ∼ η (resp. η ≈ η ). The interface Markov chain (ηn )n=0,1,... , defined above, gives rise to the process (Sn )n=0,1,... = (Sh[ηn ])n=0,1,... . It is not hard to see that this process is also a Markov chain, since Int Int pβ,h (η, ζ ) = pβ,h (η + i, ζ + i),

for any η, ζ ∈ ZV , i ∈ Z, thanks to (1.3). We will call the process (Sn )n=0,1,... the Sh (·, ·). In order to shape Markov chain. Its transition probabilities will be denoted by pβ,h Sh (·, ·) in terms of p Int (·, ·), we state the following simple fact. express pβ,h β,h Lemma 1.2. Suppose Sh[η ] = Sh[ζ ], Sh[η ] = Sh[ζ ], η ∼ η , and ζ ∼ ζ . Then there is i ∈ Z such that ζ = η + i, ζ = η + i. In particular, if η = ζ , then η = ζ . Proof. Since η ∼ η , η and η differ at a single vertex v ∈ V . Since Sh[η ] = Sh[ζ ] and Sh[η ] = Sh[ζ ], there are i , i ∈ Z such that η = ζ + i , η = ζ + i . If it were the case that i = i , then ζ and ζ would differ at every vertex where η and η take the same value, i.e., at every vertex u ∈ V \{v}. Since |V | ≥ 3, this is in contradiction with the hypothesis that ζ ∼ ζ .   This lemma is the only place in this paper where the assumption |V | ≥ 3 plays a role. It is easy to see that the lemma would be false without this assumption. It is not hard to see that Lemma 1.2 implies that for s ≈ s , Sh Int pβ,h (s , s ) = pβ,h (η , η ) if s = Sh[η ], s = Sh[η ], η ≈ η ,

while, for s ≈ s ,

Sh pβ,h (s , s ) = 0.

(1.15) (1.16)

In particular, it follows from (1.13) that Sh (s , s ) > C(β, h) pβ,h

for all s ≈ s .

(1.17)

The shape Markov chain (Sn )n=0,1,... was defined above on the same probability space where the interface Markov chain (η)n=0,1,... was defined. For certain purposes in this paper, we will be interested in considering the shape Markov chain with transition probabilities given by (1.15) and (1.16) without making reference to the interface Sh Markov chain. Pβ,h;ν will denote a probability measure under which (Sn )n=0,1,... is such

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a Markov chain, started from S0 which has probability distribution ν. The corresponding Sh . When ν is concentrated on a single configuration expectation will be denoted by Eβ,h;ν s, we will also write s instead of ν in this notation. When h = 0, (1.3) states that HhInt (η) only depends on η through its shape. This allows us to define, for each s ∈ S, H0Sh (s) = H0Int (η), where η is such that Sh(η) = s. The unique element of S which minimizes H0Sh (·) is the flat shape φ. Lemma 1.3. For any β ∈ (0, ∞), 
 Sh Zβ,0 = exp − βH0Sh (s) < ∞. s∈S

Proof. For each s ∈ S choose a representative Rep[s] ∈ ZV as follows. Rep[s] is the unique η ∈ ZV that satisfies min η(v) = 0.

Sh[η] = s,

v∈V

Define Range(s) = max(Rep[s])(v). v∈V

Note that

H0Sh (s)

≥ Range(s). Therefore

Sh Zβ,0 ≤

∞ 



k=0

s∈S

Range(s)=k

e−βk ≤

∞ 

(k + 1)|V | e−βk < ∞.

 

k=0

We can now introduce the h = 0 Gibbs probability distribution on S as
 exp − βH0Sh (s) . νβ,0 (s) = Sh Zβ,0 From (1.17), we learn that for any value of β and h, the shape Markov chain (Sn )n=0,1,... is irreducible and aperiodic. From (1.14), (1.15) and (1.16), we learn that, when h = 0, (Sn )n=0,1,... is also reversible with respect to νβ,0 , i.e., Sh Sh νβ,0 (s ) pβ,h (s , s ) = νβ,0 (s ) pβ,h (s , s ),

(1.18)

for every s , s ∈ S. In particular, νβ,0 is then the unique invariant probability distribution for the shape Markov chain, which is, therefore positive recurrent in this case. In contrast, when h = 0, the shape Markov chain is not reversible. To see this, one can use a cycle argument (see Theorem 4.2 of Chapter 5 of [Dur], p. 303). (To show that the cycle condition is violated, one can consider a cycle of length |V |, starting and ending at φ, obtained by modifying the shape at each v ∈ V in exactly one of the steps. Since this negative result will not be needed here, the simple computation is left for the interested reader. Interestingly, this depends on the assumption |V | ≥ 3; one can also check that if |V | = 2, then even when h = 0, the shape Markov chain is reversible.)

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1.4. The position of the interface.It will be convenient to determine the position of an interface η ∈ ZV by means of x∈V η(v), so that the displacement of the interface process (ηn )n=0,... from time 0 to time n will be determined by  (ηn (v) − η0 (v)). (1.19) M(n) = v∈V

This quantity is proportional to the variation in magnetization from time 0 to time n in the spin process (σ [ηn ])n=0,... , since 1 {(σ [ηn ])(x) − (σ [η0 ])(x)} . M(n) = 2 x∈V¯

We will also consider the mean displacement of the interface, m(n) = Set so that

M(n) . |V |

Jn = M(n + 1) − M(n), M(n) = J0 + · · · + Jn−1 . n

(1.20)

n ,

Thanks to Lemma 1.2, for any 0 ≤ < the sequence of shapes Sh[ηn ], . . . , Sh[ηn ] determines the sequence of interfaces ηn , . . . , ηn up to a common additive constant. Therefore this sequence of shapes determines the sequence Jn , . . . , Jn −1 in a time invariant fashion, i.e., for n = n , . . . , n − 1,
 (1.21) Jn = g Sh[ηn ], Sh[ηn+1 ] , for a fixed function g : S 2 → {−1, 0, 1}. Given the shape Markov chain (Sn )n=0,1,... , we can use (1.21) to define (Jn )n=0,1,... via 
(1.22) Jn = g Sn , Sn+1 , and then use (1.20) to define (M(n))n=0,1,... (with M(0) = 0). The processes defined in this way have the same law as the ones defined in terms of the interface Markov chain (ηn )n=0,1,... . For values of β and h for which the shape Markov chain (Sn )n=0,... is recurrent, it is natural to look at the return times to the flat shape φ. For this purpose set N0 = 0, Nk+1 = inf{n > Nk : Sn = φ},

k = 1, 2, . . . .

From the observation in the last paragraph above, (Sn )n=0,... determines also the sequence of random variables M(Nk ), k = 0, 1, . . .. Set (+N )k = Nk+1 − Nk ,

and (+M)k = M(Nk+1 ) − M(Nk ).

Note that, regardless of the distribution of S0 , the sequences (+N )1 , (+N )2 , . . .

and

(+M)1 , (+M)2 , . . .

are both i.i.d. For the sake of notation, it is natural to introduce two random variables, +N and +M on a probability space with probability measure denoted by Pβ,h , and the

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R.H. Schonmann

corresponding expectation denoted by Eβ,h , so that the law of +N coincides with the law of each of the (+N )k in the first sequence above, while the law of +M coincides with the law of each of the (+M)k in the second sequence above. Note that +M is stochastically bounded above by +N : For any k ∈ Z, Sh Sh (M(N1 ) > k) ≤ Pβ,h;φ (N1 > k) Pβ,h (+M > k) = Pβ,h;φ

= Pβ,h (+N > k).

(1.23)

1.5. Results. Suppose that β and h are such that the shape Markov chain (Sn )n=0,... has a unique invariant probability distribution νβ,h (as we already know to be the case when h = 0). We will say that a probability distribution µ on the set of interfaces ZV is shape-stationary, if  µ(η) = νβ,h (s), η : Sh[η]=s for each s ∈ S. Note that if the distribution µ of η0 is shape-stationary, then the distribution of Sh[ηn ] is νβ,h for any time n. Theorem 1.1 (Long-time fluctuations of the interface when h = 0). For any β ∈ (0, ∞), if h = 0 and the distribution µ of η0 is shape-stationary, then M(n·) → σβGK B(·), √ n

(1.24)

in distribution, as n → ∞, where M(·) : [0, ∞) → R is obtained from M(n), n ∈ {0, 1, . . .} by linear interpolation, (B(t))t≥0 is standard Brownian motion started from 0, and

σβGK

2

∞ 
 Sh 2 Sh = Eβ,0;ν ) Eβ,0;ν (J0 Jn ) (J + 2 0 β,0 β,0 n=1


 Int = Eβ,0;µ (J0 )2 + 2

∞  n=1

Int Eβ,0;µ (J0 Jn )


 Eβ,0 (+M)2 = ∈ (0, ∞). Eβ,0 (+N )

(1.25)

The superscript GK above refers to the fact that the first two expressions above for σβGK in terms of the Jn are sometimes referred to as Green-Kubo formulas (see, e.g., Sect. 3 of [Spo] and Sect. 8 of [LeS]). We turn now to the case when possibly h = 0. We want to consider the case in which |h| is small as a perturbation of the case h = 0. This can only make sense if we add one more condition to our assumptions on the dynamics. We will suppose that F (·) is Int (η, ζ ) and p Sh (s , s ) are continuous. This implies that the transition probabilities pβ,h β,h continuous functions of h. This extra assumption is verified for the examples (1.5) and (1.6). Recall the definition of the total variation distance between two probability measures ν and ν on S:  |ν (s) − ν (s)|. ν − ν TV = s∈S

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Theorem 1.2 (The shape Markov chain under a small external field). Suppose that F is continuous. For each β ∈ (0, ∞) there is h(G, β) > 0 such that the following statements hold. (a) There is γβ > 0 such that for all s ∈ S there exists Cβ,s ∈ (0, ∞) such that for all |h| < h(G, β), Sh Pβ,h;s (N1 > n) ≤ Cβ,s e−γβ n , (1.26) for n = 0, 1, . . .. In particular, when |h| < h(G, β) the shape Markov chain (Sn )n=0,... is positive recurrent, with unique invariant probability distribution denoted by νβ,h . (b) For each β ∈ (0, ∞) and |h| < h(G, β), (Sn )n=0,1,... converges exponentially fast to equilibrium in the sense that there is γβ,h > 0 such that for each s ∈ S there is Cβ,h,s < ∞ such that  Sh  −γβ,h n P  , β,h;s (Sn ∈ ·) − νβ,h (·) TV ≤ Cβ,h,s e for n = 0, 1, . . .. (c) For each β ∈ (0, ∞), the distribution νβ,h is weakly continuous in h ∈ (−h(G, β), h(G, β)), i.e., for all s ∈ S, lim νβ,h (s) = νβ,h (s).

h →h

(d) For each β ∈ (0, ∞) and uniformly in h ∈ (−h(G, β),h(G, β)), νβ,h has exponentially decaying tails, in the sense that there are C¯ β , γ¯β ∈ (0, ∞) such that νβ,h ({s : Range(s) ≥ k}) ≤ C¯ β e−γ¯β k ,

(1.27)

for k = 0, 1, . . .. Theorem 1.3 (Movement of the interface under a small external field). Suppose that F is continuous. If |h| < h(G, β), then for any initial configuration η0 of the interface process (ηn )n=0,1,... , M(n) lim = vβM (h) a.s., (1.28) n→∞ n where  Eβ,h (+M) Eβ,h (|+M| tanh(β h |+M| / 2)) Sh vβM (h) = νβ,h (s) Eβ,h;s (J0 ) = = . Eβ,h (+N ) Eβ,h (+N ) s∈S (1.29) Moreover dvβM β GK 2 . (1.30) (0) = σβ dh 2 Also, vβM (h) maxv∈V {ηn (v)} minv∈V {ηn (v)} lim = lim = n→∞ n→∞ n n |V |

a.s.

(1.31)

The quantity vβM (h) is the long-time average velocity with which M(n) changes with time n, while (σβGK )2 is the diffusion coefficient which gives the long-time fluctuations

290

R.H. Schonmann

of the interface in equilibrium in the absence of an external field. Therefore (1.30), which shows that for small |h| this average velocity is roughly linear in h, with proportionality constant given by (β/2)(σβGK )2 , is a version of the “Einstein relation” (see, e.g., [LR] and references therein). There is an alternative way to express (1.24) and (1.30), which may seem more natural. Recall that while the interface process (ηn )n=0,... was introduced in discrete time for later convenience, it is related to a continuous time process in which spins on V¯ try to flip at rate 1, but flips out of  SOS are suppressed. Because there are exactly 2|V | spins that are allowed to flip at any time, if we define t = n/(2|V |), then t approximates the time in the continuous time process. Note that (1.24) can be restated as √ M(n·) → 2 σβGK B(·) = σβ B(·), √ √ |V | n/(2|V |) √ in distribution, as n → ∞.√The factor n/(2|V |) corresponds to the transformation of n into t and the extra factor |V | which appears now makes √ sense, in that the fluctuations of an interface of size of order |V | should grow with |V |, as |V | grows. As for the velocity of the interface, it is natural to locate the interface at m(n) = M(n)/|V |, and define its velocity then, when time is measured by t, as the following a.s. limit: m(n) vβ (h) = lim = 2 vβM (h). n→∞ n/(2|V |) (Thanks to (1.31), the position of the interface in this definition could also be taken as maxv∈V {ηn (v)} or minv∈V {ηn (v)}, rather than m(n).) Therefore (1.30) yields the Einstein relation for vβ (h): 2
dvβ β 2 (0) = β σβGK = σ . dh 2 β Mβ is called the mobility of the interface. Note that from Theorem 1.1 we learn that Mβ ∈ (0, ∞). Mβ =

The Einstein relation (1.30) may seem mysterious at first sight, and the following simple observation may shed some light on its meaning. Consider the case (ruled out before) in which |V | = 1. In this trivial case, (ηn )n=0,1,... is a simple random walk and direct computations give the following. When h = 0, using the same notation as in Theorem 1.1, M(n·) → F (0) B(·), √ n in distribution, as n → ∞. Also, for arbitrary h, M(n) F (−βh) − F (βh) = a.s. n 2 But if F (·) is continuous, then an application of condition (1.4) yields,

  β F (−βh) − F (βh) eβh − 1 β d = lim F (βh) = F (0). h→0 dh 2 2 βh 2 h=0 lim

n→∞

This is precisely the Einstein relation (1.30) in this case. Theorem 1.1 is proved in Sect. 2, while Theorem 1.2 and Theorem 1.3 are proved in Sect. 3.

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2. The Interface in the Absence of External Field 2.1. L2 theory of the shape Markov chain when h = 0. When h = 0, the shape Markov chain (Sn )n=0,1,... is reversible and therefore it is natural to exploit its L2 theory. For an introduction to the use of such techniques, see, e.g., [ST2] and references threrein. Define    2 2 L (S, νβ,0 ) = f : S → C : |f (s)| νβ,0 (s) < ∞ , s∈S

where C is the set of complex numbers and for z ∈ C, |z| denotes its modulus. L2 (S, νβ,0 ) is a Hilbert space with inner product  f, g = f¯(s)g(s)νβ,0 (s), s∈S

where for z ∈ C, z¯ denotes its complex conjugate. For f ∈ L2 (S, νβ,0 ), set  Sh (Pβ,0 f )(s) = pβ,0 (s, s ) f (s ). s ∈S

Then Pβ,0 is a self-adjoint contraction, so that in particular its spectrum lies in the real number interval [−1, 1]. Since the shape Markov chain is irreducible, the number 1 is a simple eigenvalue, corresponding to the eigenspace of constant functions. Define  (2β,0 f )(s) = f (s)νβ,0 (s). s ∈S

Then 2β,0 is a self-adjoint projection on the space of constant functions. The operator Pβ,0 − 2β,0 is identically 0 on the space of constant functions and is identical to Pβ,0 on the orthogonal complement of that space. Set 
  
 Rβ = sup |r| : r ∈ spec Pβ,0 \{1} = sup |r| : r ∈ spec Pβ,0 − 2β,0 , where spec(Q) denotes the spectrum of the operator Q. The main technical step in this paper will be the proof of the following theorem. This result will be used in the proofs of Theorem 1.1 and Theorem 1.2. In particular this is the result which will allow us to consider the shape Markov chain under a small external field as a perturbation of the case in which h = 0. Theorem 2.1. For any β > 0,

Rβ < 1.

Proof. The separation of the spectrum from −1 is related to the aperiodicity of the shape Markov chain in a standard way. Thanks to Lemma 1.1, we can write Pβ,0 = αI + (1 − α)Pβ,0 , is also a self-adjoint contraction (so that its spectrum where I is the identity operator, Pβ,0 is also contained in [−1, +1]) and 0 < α ≤ 1. Therefore

inf spec(Pβ,0 ) > −1.

(2.1)

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R.H. Schonmann

Set


 gap(Pβ,0 ) = sup spec(Pβ,0 )\{1} .

gap(Pβ,0 ) is usually referred to as the “spectral gap” of Pβ,0 . Thanks to (2.1) the proof of the theorem will be complete once we show that gap(Pβ,0 ) > 0.

(2.2)

For this purpose we use the following Cheeger type inequality from [LS] (Theorem 2.1 and remark on first paragraph of p. 560 about the discrete time case):
2 gap(Pβ,0 ) ≥ C Cβ , (2.3) where C is a positive constant and Cβ =



Sh (s , s ) νβ,0 (s ) pβ,0

s ∈A s ∈Ac

inf

νβ,0 (A) νβ,0 (Ac )

A⊂S

0<νβ,0 (A)<1

is usually referred to as the “Cheeger constant”. To estimate Cβ , we will use a technique from Jerrum and Sinclair based on choosing paths in S (see, e.g., the proof of Theorem 4.2 of [JS], or Proposition 7 of [DS]). This will take us on a long sequence of definitions and estimates. A path in S from s to s is a sequence (s1 , s2 , . . . , sn ) of elements of S, with si ∼ si+1 for i = 1, . . . , n − 1, s1 = s , sn = s . Given a path π = (s1 , s2 , . . . , sn ) , we will de
 ← and note by
π = (sn , sn−1, . . . , s1 ) the reversed path. Given paths π = s
1 , s2 , . . . , sm = s , we will denote by π ◦ π = s , s , . . . , s π = s1 , s2 , . . . , sn with sm 1 1 2 m−1 ,  sm , s2 , s3 , . . . sm the concatenated path. Set E = {{s , s } : s , s ∈ S, s ∼ s }. Given a path π = (s1 , s2 , . . . , sn ) and e ∈ E, we will, in an abuse of notation, write e ∈ π in case e = {si , si+1 } for some i ∈ {1, . . . , n − 1}. Given {s , s } ∈ E, set Sh Sh Q({s , s }) = νβ,0 (s ) pβ,h (s , s ) = νβ,0 (s ) pβ,h (s , s ),

where the second equality is the reversibility equation (1.18). Suppose that for any pair of distinct s , s ∈ S, we have chosen a path πs ,s , from ← s to s , with the only restriction that πs ,s = π s ,s . For e ∈ E set then  w(e) = νβ,0 (s ) νβ,0 (s ). {s ,s } : e∈πs ,s

(In this sum each unordered pair s , s is taken only once. Note that the condition e ∈ πs ,s is equivalent to e ∈ πs ,s .) Define now w(e) . Q(e) e∈E

Jβ = sup We will show next that Cβ ≥

1 . Jβ

Indeed, for A ⊂ S, set ∂A = {{s , s } ∈ E : s ∈ A, s ∈ Ac }, and write

(2.4)

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293

νβ,0 (A) νβ,0 (Ac ) =

 s ∈A s ∈Ac

=







νβ,0 (s ) νβ,0 (s ) ≤

νβ,0 (s ) νβ,0 (s )

e∈∂A s ,s :

w(e) ≤ Jβ

e∈∂A

e∈πs ,s



Q(e) = Jβ

 s ∈A s ∈Ac

e∈∂A

Sh νβ,0 (s ) pβ,0 (s , s ),

which implies (2.4). To use (2.4), we will choose the paths πs ,s in a convenient way, that we present next. Recall the definitions of Rep[s] and of Range(s), from the proof of Lemma 1.3, and define also the volume  Vol(s) = (Rep[s])(v). v∈V

Note that for each s ∈ S, Vol(s) ≥ 0, and that Vol(s) = 0 iff s = φ. We will define a tranformation D : S\{φ} → S which decreases the volume of a shape by one unit “moving it towards φ” in a convenient fashion. (D is for “down”.) For this purpose, order the vertices in V in an arbitrary fashion. Given s ∈ S, set V top (s) = {v ∈ V : (Rep[s])(v) = Range(s)}, and let v1 (s) be the first element of V top (s). Define now   D(s) = Sh (Rep[s])v1 (s),↓ . Observe that for any s ∈ S\{φ}, Vol(D(s)) = Vol(s) − 1, Therefore

D(s) ∼ s.

πs,φ = (s, D(s), D 2 (s), . . . , DVol(s) (s))

defines a path in S from s to φ. Define also ←

πφ,s = π s,φ . And for each pair of distinct s , s ∈ S\{φ}, define πs ,s = πs ,φ ◦ πφ,s . Set

E D = {{s , s } ∈ E : s = D(s )},

and observe that w(e) = 0, unless e ∈ E D . Therefore, Jβ = sup

e∈E D

w(e) . Q(e)

(2.5)

Given e = {s , s } ∈ E D , such that s = D(s ), set e↑ = s and e↓ = s . (Since D reduces the volume, there is no possibility that also s = D(s ) here.) Define also Ae = {s ∈ S : D k (s) = e↑ for some k ≥ 0}.

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R.H. Schonmann

(We are using the standard notation in which D 0 is the identity. A is for “above”.) Note that if e ∈ πs ,s , then s or s must be in Ae . Hence, for e ∈ E D ,   w(e) = νβ,0 (s ) νβ,0 (s ) ≤ νβ,0 (s ) νβ,0 (s ) = νβ,0 (Ae ). (2.6) {s ,s } : e∈πs ,s

s ∈Ae s ∈S

It follows from (1.17) that for some C(β) > 0, Q(e) ≥ C(β) νβ,0 (e↑ ), for all e ∈ E D . Combining this bound with (2.5) and (2.6), we obtain Jβ ≤

νβ,0 (Ae ) 1 sup . C(β) e∈E D νβ,0 (e↑ )

(2.7)

To estimate νβ,0 (Ae ) =


 1  exp − βH0Int (Rep[s]) , Sh Zβ,0 s∈A

(2.8)

e

we will show that for any s ∈ Ae , H0Int (Rep[s]) ≥ H0Int (Rep[e↑ ]) − |E| + Range(s) − Range(e↑ ).

(2.9)

(Recall that E is the edge set of G = (V , E).) To this end, we first observe that if s ∈ Ae , then for each v ∈ V , (Rep[s])(v) = e↑ (v)

if

0 ≤ e↑ (v) ≤ Range(e↑ ) − 2,

(2.10)

and (Rep[s])(v) ∈ {e↑ (v), . . . , Range(s)} For an arbitrary η ∈ ZV , write H0Int (η) =

e↑ (v) ≥ Range(e↑ ) − 1.

if



(2.11)

H0Int,i (η),

i∈Z

where

H0Int,i (η) =



?{η(v )
{v ,v }∈E

(Here ?. is the indicator function.) Using (2.10) and (2.11), we see that H0Int,i (Rep[s]) = H0Int,i (e↑ ) if

i ≤ Range(e↑ ) − 1 or i ≥ Range(s) + 1,

and H0Int,i (Rep[s]) ≥ 1 = H0Int,i (e↑ ) + 1

if

Range(e↑ ) + 1 ≤ i ≤ Range(s).

The inequality (2.9) follows from these two statements and the trivial bound H0Int,i (Rep[s]) ≥ H0Int,i (e↑ ) − |E|

if

i = Range(e↑ ).

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295

From (2.8) and (2.9), we obtain 
 νβ,0 (Ae ) ≤ νβ,0 (e↑ ) eβ|E| exp − β(Range(s) − Range(e↑ )) = νβ,0 (e↑ ) eβ|E|

s∈Ae ∞ 

e−βk |{s ∈ Ae : Range(s) − Range(e↑ ) = k}|

k=0

≤ νβ,0 (e↑ ) eβ|E|

∞ 

e−βk (k + 1)|V | ≤ C(|V |, |E|, β) νβ,0 (e↑ ),

(2.12)

k=0

where C(|V |, |E|, β) < ∞. Here we used that (2.10) and (2.11) imply {s ∈ Ae : Range(s) − Range(e↑ ) = k}

⊂ {s ∈ S : (Rep[s])(v) ∈ {(Rep[e↑ ])(v), . . . , (Rep[e↑ ])(v) + k} for all v ∈ V },

and the cardinality of that latter set is clearly bounded above by (k + 1)|V | . Combining (2.3), (2.4), (2.7) and (2.12), we obtain (2.2), finishing the proof.

 

It may be of interest to point out that for certain contour models which were also motivated by stochastic Ising model interfaces, it was shown in [ST1] that the analogue of Theorem 2.1 fails. The main reason for this difference is the fact that our interfaces are constrained to move inside the “tube” V¯ of constant “width” |V |, while the contours of [ST1] can grow in all directions. Denote the L2 (S, νβ,0 )-norm of f ∈ L2 (S, νβ,0 ) by  f L2 (S ,νβ,0 ) = f, f  = |f (s)|2 νβ,0 (s). s∈S

Then for arbitrary n,       Sh n  f (s)νβ,0 (s) (Pβ,0 ) (f ) −   s∈S L2 (S ,νβ,0 )        Sh  = (Pβ,0 − 2β,0 )n f − f (s)νβ,0 (s)    s∈S L2 (S ,νβ,0 )        ≤ f − f (s)νβ,0 (s) (Rβ )n .   s∈S

(2.13)

L2 (S ,νβ,0 )

Therefore, Theorem 2.1 implies that the shape Markov chain converges to equilibrium exponentially fast in L2 . It is known (see [RT] and references therein) that this is equivalent to exponentially fast convergence to equilibrium in the total variation norm, ∈ (0, ∞) such that i.e., there is γβ ∈ (0, ∞) such that for all s ∈ S there is Cβ,s    Sh  ≤ Cβ,s e−γβ n , (2.14) Pβ,0;s (Sn ∈ ·) − νβ,0 (·) TV

for n = 0, 1, . . ..

296

R.H. Schonmann

Furthermore, it is also known (see Proposition 2.11 of [Twe], or Theorem 15.0.1 in [MT] for a more general result) that (2.14) is equivalent to the statement that the time of the first visit to an arbitrary state of the shape Markov chain has exponentially decaying tails. More precisely, applied to the state φ, this statement guarantees that there is ∈ (0, ∞) such that γβ ∈ (0, ∞) such that for all s ∈ S there is Cβ,s

Sh (N1 > n) ≤ Cβ,s e−γβ n , Pβ,0;s

(2.15)

for n = 0, 1, . . .. (For later use, we point out that this equivalence does not require reversibility.) 2.2. Proof of Theorem 1.1. Let (Sn )n∈Z be a two-sided stationary version of the shape Markov chain, so that, in particular, Sn has law νβ,0 for each n ∈ Z. We will denote by Sh the law of this two-sided process, and by ESh the corresponding expectation. Pβ,0 β,0 Because the shape Markov chain is irreducible and aperiodic, the stationary sequence (Sn )n∈Z is ergodic (see [Dur], Example 4.2 of Chapter 6, p. 352). Note that if we define Jn , n ∈ Z, via (1.22), then (. . . , J−1 , J0 , J1 , . . .) is also stationary and ergodic Sh (J ) = 0. Set (see [Dur], Theorem 1.3 of Chapter 6, p. 340), and, by symmetry, Eβ,0 n Fn = σ (. . . , Sn , Sn+1 ). Then for each n, Jn is Fn -measurable. We want to use Theorem 7.6 of Chapter 7 of [Dur] (p. 419) applied to the sequence (. . . , J−1 , J0 , J1 , . . .) in order to obtain a functional central limit theorem for M(n) = J1 + · · · + Jn . For this purpose we need to show that 

2  Sh E Sh (J |F ) Eβ,0 < ∞. (2.16) 0 −n β,0 n≥1

Sh (J ). We can use the Markov property of (S ) Set f (s) = Eβ,0;s 0 n n∈Z to write



Sh Sh Sh Sh Sh Eβ,0 (J0 |F−n ) = Eβ,0 (J0 |F−1 )|F−n = Eβ,0 (J0 |S0 )|F−n Eβ,0 Eβ,0 



n−1 Sh Sh = Eβ,0 (f ) (S−n+1 ). Pβ,0 (f (S0 )|S−n+1 ) =

Since S−n+1 has law νβ,0 ,  

2    Sh n−1  Sh E Sh (J |F ) Eβ,0 = ) (f ) (P  0 −n β,0 β,0 n≥1



n≥1

L2 (S ,νβ,0 )

≤



(Rβ )n−1 ,

n≥1

from (2.13), since s∈S f (s)νβ,0 (s) = 0 and f L2 (S ) ≤ 1. Therefore (2.16) follows from Theorem 2.1. We can now apply the aforementioned Theorem 7.6 of Chapter 7 of [Dur], combined with the remark in the paragraph with display (1.22), to conclude that (1.24) holds, with

σβGK

2

=

 n∈Z

∞ 
 Sh Sh 2 Sh (J + 2 Eβ,0 (J0 Jn ) = Eβ,0;ν ) Eβ,0;ν (J0 Jn ) 0 β,h β,h


 Int (J0 )2 + 2 = Eβ,0;µ

n=1

∞  n=1

Int Eβ,0;µ (J0 Jn ) < ∞.

Einstein Relation

297

To prove that also



σβGK


 Eβ,0 (+M)2 = , Eβ,0 (+N )

2

it is now enough to show that M(n) √ → n



(2.17)


 Eβ,0 (+M)2 Z, Eβ,0 (+N )

(2.18)

in distribution, as n → ∞, where Z has a standard normal distribution. To derive (2.18), recall the definitions of Nk , (+N )k and (+M)k , from Subsect. 1.4, set K(n) = sup{k : Nk ≤ n}, and write 
n = (+N )0 + (+N )1 + · · · + (+N )K(n)−1 + n − NK(n) ,

 M(n) = (+M)0 + (+M)1 + · · · + (+M)K(n)−1 + M(n) − M NK(n) . Because the shape Markov chain is recurrent, |(+M)0 | ≤ (+N )0 < ∞ a.s. Recall that (+N )1 , (+N )2 , . . .

and

(+M)1 , (+M)2 , . . .

are both i.i.d. sequences. In the former the common law is that of +N and in the latter that of +M. From (2.15) and (1.23), we learn that +N and +M have finite second moments. By symmetry, Eβ,0 (+M) = 0. The proof of (2.18) from these facts is standard, being based on a random index central limit theorem and on the estimates n

|M(n) − M(NK(n) )| ≤ n − NK(n) ≤ sup(+N )k , k=0

and

supnk=0 (+N )k → 0, √ n in probability, as n → ∞. (See exercise 5.6 of Chapter 5 of [Dur], p. 324, for details.) Finally note that since, clearly, Eβ,0 ((+M)2 ) > 0, the claim that σβGK > 0 follows from (2.17) and the finiteness of Eβ,0 (+N ). 3. The Interface Under a Small External Field 3.1. Proof of Theorem 1.2. First we claim that for fixed β ∈ (0, ∞) and h ∈ R, Int Int lim pβ,h (η, ζ ) = pβ,h (η, ζ ),

h →h

(3.1)

uniformly over η, ζ ∈ ZV . To see this, first note that for each η, ζ ∈ ZV (3.1) is a consequence of the assumed continuity of F . Uniformity over η ∼ ζ follows from the finiteness of the set in display (1.10). Uniformity over all pairs η, ζ ∈ ZV now follows Int (η, ζ ) = 0, unless η ∼ ζ or η = ζ, from (1.9), and from the fact from the fact that pβ,h V that for each η ∈ Z , |{ζ ∈ ZV : ζ ∼ η}| = 2|V |. Thanks to (1.15) and (1.16), a similar statement holds for the shape Markov chain: For fixed β ∈ (0, ∞) and h ∈ R, Sh Sh lim pβ,h (s , s ) = pβ,h (s , s ),

h →h

uniformly over s , s ∈ S.

(3.2)

298

R.H. Schonmann

As a consequence of (3.2), we have for every n ∈ {1, 2, . . .} and A ⊂ S n , Sh lim Pβ,h ;s ((S1 , . . . , Sn ) ∈ A) 
1  Sh
1 2  Sh Sh n−1 n = lim pβ,h s , s · · · pβ,h p ,s ) s, s (s β,h

h →h

h →h

(s 1 ,...,s n )∈A

Sh ((S1 , . . . , Sn ) ∈ A), = Pβ,h;s

(3.3)

where we used the fact that the sum over (s 1 , . . . , s n ) is a finite sum (with at most (2|V | + 1)n terms). Also, from (3.2) and (1.17), lim

h →h

Sh (s , s ) pβ,h Sh (s , s ) pβ,h

= 1,

(3.4)

uniformly, over s ≈ s . Given  > 0, (3.4) and (1.16) guarantee that for small enough |h|, 
1  Sh
1 2 
n−1 n  Sh Sh Sh Pβ,h;s s, s pβ,h s , s · · · pβ,h s ,s (N1 > n) = pβ,h s 1 ,...,s n =φ

≤




1  Sh
1 2 
n−1 n  Sh Sh s, s pβ,0 s , s · · · pβ,0 s ,s (1 + )n pβ,0

s 1 ,...,s n =φ

Sh = (1 + )n Pβ,0;s (N1 > n) ≤ (1 + )n Cβ,s e−γβ n ,

where, in the last step, we used (2.15). A proper choice of  leads to (1.26), proving the statement in (a). As pointed out when (2.15) was derived, it is known that (1.26) implies the statement in (b). To prove the weak convergence claimed in (c), we use a “cycle trick” representation of νβ,h (see, e.g., Sect. 5.4 of [Dur]). For each s ∈ S, ∞ Sh n=0 Pβ,h;φ (Sn = s, N1 > n) νβ,h (s) = . (3.5) ∞ Sh n=0 Pβ,h;φ (N1 > n) From (3.3) we know that each term in the series in the numerator or the denominator is continuous in h. From (1.26) we learn that these series converge uniformly over h ∈ (−h(G, β), h(G, β)). Therefore numerator and denominator are continuous functions of h ∈ (−h(G, β), h(G, β)), and since the denominator is bounded below by 1 and hence does not vanish, we conclude that νβ,h (s) is also continuous in h ∈ (−h(G, β), h(G, β)), as claimed. To prove (1.27), we use (3.5) again. Note that if S0 = φ, then Range(Sn ) ≥ k can only occur when n ≥ k. Hence, using (1.26), ∞ Sh n=k Pβ,h;φ (Range(Sn ) ≥ k, N1 > n) νβ,h ({s : Range(s) ≥ k}) = ∞ Sh n=0 Pβ,h;φ (N1 > n) ≤

∞  n=k

Sh Pβ,h;φ (N1 > n) ≤

∞  n=k

Cβ,φ e−γβ n ≤ C¯ β e−γ¯β k .

Einstein Relation

299

3.2. Proof of Theorem 1.3. The claim (1.28), with Eβ,h (+M) , Eβ,h (+N )

vβM (h) =

can be proved in a standard way, by methods similar to, but simpler than those used to prove (2.18) (see Exercise 5.5 of Chapter 5 of [Dur], p. 323). Since |M(n)| ≤ n, (1.28), the dominated convergence theorem and (1.20) now imply vβM (h)

=

lim ESh n→∞ β,h;νβ,h

M(n) n

 = lim

n→∞

n−1 1  Sh Eβ,h;νβ,h (Jk ) n k=0

n−1 1   Sh Sh = lim Pβ,h;νβ,h (Sk = s) Eβ,h;s (J0 ) n→∞ n k=0 s∈S

n−1  1  Sh Sh = lim νβ,h (s) Eβ,h;s (J0 ) = νβ,h (s) Eβ,h;s (J0 ). n→∞ n k=0 s∈S

s∈S

To obtain the last equality in (1.29), it suffices to show Eβ,h (+M) = Eβ,h (|+M| tanh(β h |+M| / 2)).

(3.6)

Note that the l.h.s. of (3.6) can be written as   Eβ,h (+M) = k Pβ,h (+M = k) = k {Pβ,h (+M = k) − Pβ,h (+M = −k)}, k≥1

k∈Z

(3.7) since this series converges absolutely, thanks to (1.23) and (1.26). Obviously Pβ,h (+M = k) = 0 only if k = j |V | for some j ∈ Z. In this case we can use (1.14) to write: µInt β,h (φ0 )Pβ,h (+M = j |V |) Int = µInt β,h (φ0 ) Pβ,h;φ0 (M(N1 ) = j |V |)

= = = =

µInt β,h (φ0 ) ∞ 

∞  n=1

Int Pβ,h;φ (N1 = n, M(n) = j |V |) 0



n=1 η1 ,...,ηn−1 ∈φ ∞   n=1 η1 ,...,ηn−1 ∈φ ∞   n=1 η1 ,...,ηn−1 ∈φ

= ··· ∞  =



n=1 η1 ,...,ηn−1 ∈φ



1 2
n−1  Int 1 Int Int µInt , φj β,h (φ0 ) pβ,h φ0 , η pβ,h η , η · · · pβ,h η
1 
1  Int
1 2 
n−1  Int Int η , φ0 µInt pβ,h , φj β,h η pβ,h η , η · · · pβ,h η
1  Int
2 1  Int
2 
n−1  Int Int pβ,h η , φ0 pβ,h η , η µβ,h η · · · pβ,h η , φj
1  Int
2 1 

 Int Int pβ,h η , φ0 pβ,h η , η · · · pβ,h φj , ηn−1 µInt β,h φj

300

R.H. Schonmann

=

∞ 



n=1 η1 ,...,ηn−1 ∈φ


 Int


2 1 n−1 Int Int µInt pβ,h ηn−1 , ηn−2 · · · pβ,h η ,η β,h φj pβ,h φj , η


1  Int η , φ0 × pβ,h

∞
 
 Int φ N1 = n, M(n) = −j |V | Pβ,h;φ = µInt β,h j j n=1


 Int
Int = µInt β,h φj Pβ,h;φj M(N1 ) = −j |V | = µβ,h (φj ) Pβ,h (+M = −j |V |). Therefore, using (1.3), Pβ,h (+M = j |V |) =

µInt β,h (φj ) µInt β,h (φ0 )

Pβ,h (+M = −j |V |) = eβj |V |h Pβ,h (+M = −j |V |).

In case j ≥ 0, the combination of the last display with Pβ,h (|+M| = j |V |) = Pβ,h (+M = j |V |) + Pβ,h (+M = −j |V |) leads to Pβ,h (+M = j |V |) =

eβj |V |h/2 Pβ,h (|+M| = j |V |), eβj |V |h/2 + e−βj |V |h/2

and e−βj |V |h/2 Pβ,h (|+M| = j |V |). eβj |V |h/2 + e−βj |V |h/2 Applying these two equalities to (3.7), yields (3.6), and completes the proof of (1.29). We can now use (1.29) to write

Eβ,h |+M| tanh(β h h|+M| / 2) dvβM (0) = lim h→0 dh Eβ,h (+N )  tanh(β h k / 2) k≥1 Pβ,h (|+M| = k) k h  = lim h→0 P (+N = n) n β,h n≥1    tanh(β h k / 2) n Sh n≥1 k=1 Pβ,h;φ (N1 = n, |M(N1 )| = k) k h . = lim  Sh (N = n) n h→0 P 1 n≥1 β,h;φ Pβ,h (+M = −j |V |) =

And since | tanh(βhk/2)/ h| ≤ βk/2, one can use the same argument as in the proof of Part(c) of Theorem 1.2, based on (1.26), and (3.3), to conclude that    tanh(β h k / 2) n Sh dvβM n≥1 k=1 Pβ,0;φ (N1 = n, |M(N1 )| = k) k lim h→0 h (0) =  Sh dh n≥1 Pβ,0;φ (N1 = n) n    n Sh (N = n, |M(N )| = k) k 2 β/2) P 1 1 n≥1 k=1 β,0;φ =  Sh n≥1 Pβ,0;φ (N1 = n) n
  2 β Eβ,0 (+M)2 β GK 2 k≥1 Pβ,0 (|+M| = k) k β/2  σβ , = = = 2 Eβ,0 (+N ) 2 n≥1 Pβ,0 (+N = n) n where in the last step we used (1.25).

Einstein Relation

301

Finally, to prove (1.31), set Rn = Range(Sh[ηn ]). We will first prove that lim

n→∞

Rn = 0 n

a.s.

(3.8)

Given  > 0, Int (Rn > n) Pβ,h;η 0 Sh = Pβ,h; Sh[η ] (Range(Sn ) > n) ≤

Sh Pβ,h;ν (Range(Sn ) > n) β,h

νβ,h (Sh[η0 ]) − γ ¯ n β ¯ Cβ e νβ,h ({s : Range(s) > n}) ≤ , = νβ,h (Sh[η0 ]) νβ,h (Sh[η0 ]) 0

by (1.27). The claim (3.8) now follows from Borel-Cantelli and the arbitrariness of . The claim (1.31) follows from (1.28), (3.8), and the bounds M(n) M(n) ≤ max {ηn (v)} − min {η0 (v)} ≤ + R n + R0 , v∈V v∈V |V | |V | and

M(n) M(n) − Rn − R0 ≤ min {ηn (v)} − max {η0 (v)} ≤ . v∈V v∈V |V | |V |

Acknowledgements. I am grateful to R. Caflisch for having stimulated my interest on interface movement. Thanks go to D. Aldous, J. Fill, and N. Madras for having suggested reference [MT] for equivalent formulations of exponential ergodicity. It is also a pleasure to thank S. Shlosman for a careful reading of the manuscript and several comments which helped improve the presentation. Finally, thanks go to two anonymous referees, for their comments.

References [But]

Butta, P.: On the validity of an Einstein relation in models of interface dynamics. J. Stat. Phys. 72, 1401–1406 (1993) [DS] Diaconis, P., Stroock, D.: Geometric bounds for eigenvalues of Markov chains. Ann. Appl. Probab. 1, 36–61 (1991) [Dur] Durrett, R.: Probability: Theory and Examples, Second edition, Duxbury Press, 1996 [JS] Jerrum, M., Sinclair, A.: Approximating the permanent. SIAM J. Computing 18, 1149–1178 (1989) [LS] Lawler, G.F., Sokal, A.D.: Bounds on the L2 spectrum for Markov chains and Markov processes: A generalization of Cheeger’s inequality. Trans. Am. Math. Soc. 309, 557–580 (1988) [LR] Lebowitz, J.L., Rost, H.: The Einstein relation for the displacement of a test particle in a random environment. Stoch. Proc. Appl. 54, 183–196 (1994) [LeS] Lebowitz, J.L., Spohn, H.: A Gallavotti-Cohen type symmetry in the large deviation functional for stochastic dynamics. J. Stat. Phys. 95, 333–365 (1999) [MT] Meyn, S.P., Tweedie, R.: Markov Chains and Stochastic Stability. London: Springer Verlag, 1993 [RK] Rikvold, P.A., Kolesik, M.: Analytic approximations for the velocity of field-driven Ising interfaces. J. Stat. Phys. 100, 377–403 (2000) [RTMS] Rikvold, P.A., Tomita, H., Miyashita, S., Sides, S.W.: Metastable lifetimes in a kinetic Ising model: Dependence on field and system size. Phys. Rev. E 49, 5080–5090 (1994) [RT] Roberts, G.O., Tweedie, R.L.: Geometric L2 and L1 convergence are equivalent for reversible Markov chains. J. Appl. Probab. (to appear) [SS] Schonmann, R.H., Shlosman, S.B.: Wulff droplets and the metastable relaxation of kinetic Ising models. Commun. Math. Phys. 194, 389–462 (1998)

302 [ST1] [ST2] [Spo] [Twe]

R.H. Schonmann Sokal, A.D., Thomas, L.E.: Absence of mass gap for a class of stochastic contour models. J. Stat. Phys. 51, 907–947 (1988) Sokal, A.D., Thomas, L.E.: Exponential convergence to equilibrium for a class of random-walk models. J. Stat. Phys. 54, 797–828 (1989) Spohn, H.: Interface motion in models with stochastic dynamics J. Stat. Phys. 71, 1081–1132 (1993) Tweedie, R.L.: Markov Chains: structure and applications. In: Stochastic Processes: Theory and Methods. Handbook of Statistics. vol 19, D.N. Shanbhag, C.R. Rao (eds), Amsterdam: Elsevier 2001, pp. 817–851

Communicated by J.L. Lebowitz

Commun. Math. Phys. 232, 303–317 (2003) Digital Object Identifier (DOI) 10.1007/s00220-002-0747-7

Communications in

Mathematical Physics

Another Look at the Invariance Principle for Wave Operators∗ Jingbo Xia Department of Mathematics, State University of New York at Buffalo, Buffalo, NY 14260, USA. E-mail: [email protected] Received: 18 March 2002 / Accepted: 5 July 2002 Published online: 22 November 2002 – © Springer-Verlag 2002

Abstract: We give a new proof of the invariance principle for wave operators based on certain well-known properties of Calder´on commutators. This method extends the invariance principle to a larger class of admissible functions. In particular, admissible functions need not be monotonic on any interval. Introduction Let A and A
be self-adjoint operators on Hilbert spaces H and H
respectively and let J : H → H
be a bounded operator. Recall that the wave operators W+ (A
, A; J ) and W− (A
, A; J ) are defined by the formula


W± (A
, A; J ) = s- lim e−iλA J eiλA Pac (A) λ→±∞

if they exist. To state the invariance principle for wave operators, one begins with a class of real-valued functions ϕ called admissible functions. The invariance principle is the assertion that, under certain conditions on A, A
and J , the wave operators W± (ϕ(A
), ϕ(A); J ) exist for any function ϕ in the admissible class and W± (ϕ(A
), ϕ(A); J ) = W± (A
, A; J )EA ( + ) + W∓ (A
, A; J )EA ( − ), where EA is the spectral resolution for A and + , − are disjoint Borel sets determined by ϕ. See, e.g., [5, Theorem X.4.7], [10, Theorems XI.11 and XI.23], and [3, 4, 6, 7]. To describe which ϕ is admissible, one begins with an open set U in R such that σ (A)\U has Lebesgue measure 0. The usual admissibility requires that ϕ be monotonic on every component interval in U , that ϕ be absolutely continuous on every [a, b] contained in U , and some additional conditions such as ϕ
∈ L2local . See the references ∗

This work was supported in part by National Science Foundation grant DMS-0100249

304

J. Xia

cited above. In this paper we show that the invariance principle for wave operators can be deduced from certain well-known properties of Calder´on commutators. This method extends the invariance principle to a larger class of admissible functions than those considered previously. Most noticeably, the functions in our admissible class are not required to be monotonic on any interval. The usual approach to the proof of the invariance principle for wave operators is to divide it into the proofs of the following two natural parts: (a) The identity


s- lim e−iλϕ(A ) J eiλϕ(A) Pac (A) = W+ (A
, A; J )EA ( + ) + W− (A
, A; J )EA ( − ). λ→∞

(b) The identity Pac (A) = Pac (ϕ(A)). As it turns out, to prove (a) under the condition J (A − z)−1 − (A
− z)−1 J ∈ C1

for every z ∈ C\R,

(1)

where C1 denotes the trace class, we only need to assume that ϕ
(x) exists and is not 0 for a.e. x ∈ U . Even though (a) is the more difficult part of the invariance principle, its proof requires very little as far as ϕ is concerned. We will show that the condition ϕ
(x) = 0 a.e. on U also implies Hac (A) ⊂ Hac (ϕ(A)) or, equivalently, Pac (A) ≤ Pac (ϕ(A)). But the inclusion Hac (A) ⊂ Hac (ϕ(A)) in general can be proper, and, as we will see in Sect. 4, a very undesirable situation can occur on Hac (ϕ(A))  Hac (A). On the other hand, if ϕ is absolutely continuous on every [a, b] contained in U and if σ (A)\U is countable, then Pac (A) ≥ Pac (ϕ(A)). Thus, ironically, it is the proof of (b), by far the easier of the two parts, that requires additional assumptions. Our approach stems from the observation that the analytical difficulties in the proof of the invariance principle can be relegated to the proof of the following seemingly basic fact: If ϕ
(x) = 0 a.e. on U , then on the Hilbert space L2 (R) we have s- lim eiλϕ(M) H e−iλϕ(M) MχU = Mχ − − Mχ + , λ→∞

(2)

where H is the Hilbert transform, (Mg f )(x) = g(x)f (x) and (Mf )(x) = xf (x) for f ∈ L2 (R), + = {x ∈ U : ϕ
(x) > 0}, and − = {x ∈ U : ϕ
(x) < 0}. But (2) can be derived from certain well-known results about Calder´on commutators [1]. Although these results themselves are highly non-trivial, it does not require any hard analysis to derive (2) from these results. Indeed the proofs in this paper are all rather soft. The rest of the paper is organized as follows. The main results are stated in Sect. 1. Technically, the focus of the paper is Sect. 2, which contains the proof of (2). In Sect. 3 we give a proof of the invariance principle where, instead of (1), the condition of Cook’s method is assumed. Note that, under the condition of Cook’s method, the invariance principle was announced in [6] for admissible functions which are monotonic on every component interval of U , even though we have not been able to locate its full proof in the literature. But Sect. 3 removes the monotonicity requirement. In Sect. 4 we give an example to show that, if we only assume what we need for the proof of (a), then in general we have no control over what happens on Hac (ϕ(A))  Hac (A). 1. The Invariance Principle Throughout the paper, we will denote the trace class by C1 . As usual, the operator of multiplication by a function g will be denoted by Mg . But the operator of multiplication

Invariance Principle

305

by the coordinate function on L2 (R) will simply be denoted by M. That is, (Mf )(x) = xf (x),

f ∈ L2 (R).

The symbol H will be reserved for the Hilbert transform. Recall that
f (y) 1 p.v. dy, f ∈ L2 (R). (Hf )(x) = πi y−x Let D denote the differential operator (1/ i)(d/dx) on its usual domain in L2 (R). Since eiλM De−iλM = D − λ, we have eiλM H e−iλM = χ(0,∞) (D − λ) − χ(−∞,0) (D − λ). Thus s- lim eiλM H e−iλM = −1 λ→∞

and

s- lim eiλM H e−iλM = 1. λ→−∞

(1.1)

Definition 1.1. Suppose that U is an open set in R and that is a Borel subset of U . A sequence {ϕn } of Borel functions on R is said to be in the class #(U ; ) if and only if it satisfies the following conditions: (i) For every n, ϕn is a uni-modular function on R, i.e., |ϕn | = 1 on R. (ii) s-limn→∞ ϕn (M)H ϕn∗ (M)MχU = MχU \ − Mχ on L2 (R). Obviously, this definition is motivated by (1.1). If {λn } is a sequence of real numbers such that limn→∞ λn = ∞, then the sequence {eiλn x } belongs to the class #(R;R) whereas the sequence {e−iλn x } belongs to #(R;∅). Let A and A
be self-adjoint operators on Hilbert spaces H and H
respectively and let J : H → H
be a bounded operator. We write


W± (A
, A; J ) = s- lim e−iλA J eiλA Pac (A) λ→±∞

(1.2)

if the strong limits exist. Throughout the paper, m denotes the Lebesgue measure on R. Proposition 1.2. Let A and A
be self-adjoint operators on H and H
respectively and let J : H → H
be a bounded operator. Suppose that there is a z ∈ C\R such that J (A − w)−1 − (A
− w)−1 J ∈ C1

f or w ∈ {z, z¯ }.

(1.3)

Let U be an open set in R such that m(σ (A)\U ) = 0 and let be a Borel subset of U . Let {ϕn } be a sequence in the class #(U ; ). Then the strong limit W = s- lim ϕn∗ (A
)J ϕn (A)Pac (A) n→∞

exists and we have W = W+ (A
, A; J )EA ( ) + W− (A
, A; J )EA (U \ ), where EA is the spectral resolution of A and W± (A
, A; J ) are given by (1.2). Proof. We may assume that H = (⊕β∈B L2 (Eβ )) ⊕ Hs , where each summand is invariant under A, A|Hs is purely singular, and A|(⊕β∈B L2 (Eβ )) = M, the operator of multiplication by the coordinate function. Here, each Eβ is a bounded Borel subset of R and L2 (Eβ ) = {f ∈ L2 (R) : f = 0 on R\Eβ }. The condition m(σ (A)\U ) = 0 allows
us to further assume that Eβ ⊂ U for every β ∈ B. Let Wλ = e−iλA J eiλA , λ ∈ R. Denote K = J (A − z)−1 − (A
− z)−1 J and L = J (A − z¯ )−1 − (A
− z¯ )−1 J . Because
µ
(A
− z)−1 (Wµ − Wλ )(A − z)−1 = −i e−itA KeitA dt ∈ C1 λ

and (A
− z)−1 Wt − Wt (A − z)−1 ∈ C1 , we have (Wµ − Wλ )(A − z)−2 ∈ C1 .

306

J. Xia

Fix a β0 ∈ B for the moment. For each bounded function ξ ∈ L2 (Eβ0 ), define the operator Mξ : H → L2 (R) by the formula Mξ f = ξ¯ fβ0 , where f = (⊕β∈B fβ ) ⊕ hs , hs ∈ Hs and fβ ∈ L2 (Eβ ), β ∈ B. Recall that H denotes the Hilbert transform. Define Yξ± = πi(A − z)M∗ξ (H ± 1)Mξ (A − z).  

We have Wµ∗ Wλ , (A − z)−1 = Wµ∗ e−iλA KeiλA − e−iµA L∗ eiµA Wλ ∈ C1 and              tr Wµ∗ Wλ , (A − z)−1 Yξ±  ≤ KeiλA Yξ±  + Yξ± e−iµA L∗  . 1

Because Eβ0 is a bounded set, the operators (A − z)2 Yξ+ and bounded. Thus (Wµ − Wλ )Yξ± = (Wµ − Wλ )(A − z)−2 · (A  (A − z)−1 , Yξ± = ξ ⊗ ξ by the design of Yξ± . Therefore

1

(A − z)2 Yξ− are also − z)2 Yξ± ∈ C1 . Now

  

 Wµ∗ Wµ − Wλ ξ, ξ = tr Wµ∗ (Wµ − Wλ ) (A − z)−1 , Yξ±    = tr Wµ∗ (Wµ − Wλ ), (A − z)−1 Yξ± ,



  where the second = is due to the fact that tr (A − z)−1 , Wµ∗ (Wµ − Wλ )Yξ± = 0. Combining the above with the identities (Wµ − Wλ )∗ (Wµ − Wλ ) = Wµ∗ (Wµ − Wλ ) +     Wλ∗ (Wλ − Wµ ) and KeiλA Yξ± 1 = Yξ±∗ e−iλA K ∗ 1 , we find that           ±∗ −isA ∗   (Wµ − Wλ )ξ 2 ≤ (1.4) K  + Yξ± e−itA L∗  . Yξ e s,t∈{λ,µ}

1

1

+(∗) −iλA e

By (1.1) and the identity Mξ e−iλA = e−iλM Mξ , we have s-limλ→∞ Yξ

=0=

 −(∗) s-limλ→−∞ Yξ e−iλA . Since K, L ∈ C1 , it follows from (1.4) that limmin{λ,µ}→∞  Wµ  2 2 − Wλ ξ  = 0 = limmax{λ,µ}→−∞  Wµ − Wλ ξ  . Setting λ = 0 and letting µ → ∞ and µ → −∞ in (1.4), we respectively obtain          W+ (A
, A; J )ξ − J ξ 2 ≤ 2 Yξ+∗ K ∗  + Yξ+ L∗  , 1  1        W− (A
, A; J )ξ − J ξ 2 ≤ 2 Yξ−∗ K ∗  + Yξ− L∗  . 1

1

(1.5) (1.6)

Now let ξ + = ξ χ . Replacing ξ by ϕn (A)ξ + = ϕn (M)ξ + in (1.5), we obtain        + ∗ ∗ W+ (A
, A; J )ϕn (A)ξ + − J ϕn (A)ξ + 2 ≤ 2 Yϕ+∗ . + K  + Yϕ (A)ξ + L  (A)ξ n n 1 1 (1.7) Condition (1.3) also implies W+ (A
, A; J )f (A) = f (A
)W+ (A
, A; J ) for any bounded Borel function f . (This is trivial when f ∈ C0 (R), and the general case follows from a routine approximation.) Combining this with the fact that |ϕn | = 1, we see that W+ (A
, A; J )ϕn (A)ξ + −J ϕn (A)ξ +  = W+ (A
, A; J )ξ + −ϕn∗ (A
)J ϕn (A)ξ + . Hence it follows from (1.7) that        + ∗ ∗ W+ (A
, A; J )ξ + − ϕn∗ (A
)J ϕn (A)ξ + 2 ≤ 2 Yϕ+∗ . + K  + Yϕ (A)ξ + L  (A)ξ n n 1 1 (1.8)

Invariance Principle

307

Since Yϕ+∗ ¯ )M∗ξ + ϕn (M)(H + 1)ϕn∗ (M)Mχ Mξ (A − z¯ ), it follows + = −πi(A − z n (A)ξ

+ from Definition 1.1 that s-limn→∞ Yϕ+∗ + = 0. Similarly, s-lim n→∞ Yϕ (A)ξ + = 0. n (A)ξ n Because K, L ∈ C1 , the right-hand side of (1.8) tends to 0 as n → ∞. Thus we have the convergence ϕn∗ (A
)J ϕn (A)ξ + → W+ (A
, A; J )ξ + in the Hilbert-space norm as n → ∞. Let ξ − = ξ χU \ . The condition {ϕn } ∈ #(U ; ) also ensures s-limn→∞ Yϕ−∗ − = n (A)ξ

0 and s-limn→∞ Yϕ−n (A)ξ − = 0. Replacing ξ by ϕn (A)ξ − = ϕn (M)ξ − in (1.6) and repeating the above argument, we have the convergence ϕn∗ (A
)J ϕn (A)ξ − → W− (A
, A; J )ξ − in the Hilbert-space norm as n → ∞. Since Eβ0 ⊂ U , we have ξ = ξ + + ξ − . Hence the limit limn→∞ ϕn∗ (A
)J ϕn (A)ξ = W ξ exists and we have W ξ = W+ (A
, A; J )ξ + + W− (A
, A; J )ξ − = W+ (A
, A; J ) EA ( )ξ +W− (A
, A; J )EA (U \ )ξ . Letting the subscript β0 vary in B, the linear span of such ξ ’s is dense in ⊕β∈B L2 (Eβ ) = Hac (A). This proves the proposition.   Remark. The above proof is just an outgrowth of the ideas used in [12]. Definition 1.3. For an open set U in R, let A(U ) be the collection of real-valued Borel functions ϕ on R such that ϕ
(x) exists for a.e. x ∈ U and ϕ
(x) = 0 for a.e. x ∈ U . Proposition 1.4. For any open set U in R and any ϕ ∈ A(U ), we have s- lim eiλϕ(M) H e−iλϕ(M) MχU = Mχ − − Mχ + λ→∞

on L2 (R), where + = {x ∈ U : ϕ
(x) > 0} and − = {x ∈ U : ϕ
(x) < 0}. This is the most crucial proposition of the paper. But since its proof requires singular integral operators and is somewhat technical, we will present it in the next section, which also contains the proof of our next proposition. Proposition 1.5. Let U be an open set in R and let A be a self-adjoint operator on H such that m(σ (A)\U ) = 0. Then Hac (ϕ(A)) ⊃ Hac (A) for every ϕ ∈ A(U ). Theorem 1.6. Let A and A
be self-adjoint operators on H and H
respectively and let J : H → H
be a bounded operator. Suppose that there is a z ∈ C\R such that J (A − w)−1 − (A
− w)−1 J ∈ C1

for w ∈ {z, z¯ }.

(1.9)

Let U be an open set in R such that m(σ (A)\U ) = 0. If ϕ ∈ A(U ), then the strong limit


W = s- lim e−iλϕ(A ) J eiλϕ(A) Pac (A) λ→∞

exists and we have W = W+ (A
, A; J )EA ( + ) + W− (A
, A; J )EA ( − ), where EA is the spectral resolution of A, + = {x ∈ U : ϕ
(x) > 0}, − = {x ∈ U : ϕ
(x) < 0}, and W± (A
, A; J ) are given by (1.2).

308

J. Xia

Proof. Since ϕ
= 0 a.e. on U , U \ + differs from − by at most a set of Lebesgue measure zero. Let {λn } be a sequence of real numbers such that limn→∞ λn = ∞. Proposition 1.4 tells us that the sequence {eiλn ϕ } belongs to the class #(U ; + ). Applying Proposition 1.2 to the case where ϕn = eiλn ϕ , we obtain


s- lim e−iλn ϕ(A ) J eiλn ϕ(A) Pac (A) = W+ (A
, A; J )EA ( + )+W− (A
, A; J )EA ( − ). n→∞

Since this is true for any sequence of real numbers {λn } with limn→∞ λn = ∞, the conclusion of the theorem follows.  


Remark. Theorem 1.6 guarantees the convergence of e−iλϕ(A ) J eiλϕ(A) only on Hac (A), not on the entire Hac (ϕ(A)). Under the conditions of Theorem 1.6, Proposition 1.5 provides Hac (A) ⊂ Hac (ϕ(A)). But this inclusion can be proper. See Sect. 4. Lemma 1.7. Let U be an open set in R and let A be a self-adjoint operator on a Hilbert space H such that σ (A) ⊂ U ∪ C, where C is a countable set. Let f be a real-valued Borel function on R such that f is absolutely continuous on every closed interval [a, b] contained in U . Then Hac (f (A)) ⊂ Hac (A).  Proof. If µ is a regular Borel measure on R such that µ ⊥ m, then µ = i∈I µi , where each µi has a compact support Ki with m(Ki ) = 0 and Ki ∩ Ki
= ∅ for all i = i
in I . Thus under the orthogonal decomposition H = Hac (A) ⊕ Hs (A), we can write A|Hs (A) = ⊕j ∈J Aj , where each Aj is a self-adjoint operator such that m(σ (Aj )) = 0 and such that either σ (Aj ) ⊂ C or σ (Aj ) ⊂ U . Since C is countable, f (Aj ) has no continuous spectrum if σ (Aj ) ⊂ C. The absolute continuity of f on every [a, b] in U ensures  that if σ (Aj ) ⊂ U , then m(f (σ (Aj ))) = 0 and σ (f (Aj )) = f (σ (Aj )). Hence f Aj is purely singular in either case. Thus f (A)|Hs (A) is also purely singular, which  implies Hac (f (A)) ⊂ Hac (A).  Theorem 1.8. Let A and A
be self-adjoint operators on H and H
respectively and let J : H → H
be a bounded operator. Suppose that there is a z ∈ C\R such that J (A − w)−1 − (A
− w)−1 J ∈ C1

for w ∈ {z, z¯ }.

Let U be an open set in R such that σ (A)\U is countable. Let f be a real-valued Borel function on R such that f is absolutely continuous on every closed interval [a, b] contained in U and such that f
(x) = 0 for a.e. x ∈ U . Then the wave operator


W+ (f (A
), f (A); J ) = s- lim e−iλf (A ) J eiλf (A) Pac (f (A)) λ→∞

exists in the strong operator topology and we have W+ (f (A
), f (A); J ) = W+ (A
, A; J )EA ( + ) + W− (A
, A; J )EA ( − ),

(1.10)

where EA is the spectral resolution of A, + = {x ∈ U : f
(x) > 0}, − = {x ∈ U : f
(x) < 0}, and W± (A
, A; J ) are given by (1.2).


Proof. Clearly, f ∈ A(U ). Thus, by Theorem 1.6, W = s-limλ→∞ e−iλf (A ) J eiλf (A) Pac (A) exists and equals the right-hand side of (1.10). But Proposition 1.5 and Lemma 1.7 tell us that Pac (f (A)) = Pac (A). Hence W and W+ (f (A
), f (A); J ) are the same.  

Invariance Principle

309

2. Singular Integral Operators The purpose of this section is to prove Propositions 1.4 and 1.5. Because this involves the Lp -norm for various p’s, in this section we will denote the norm on Lp (R) by .p . We begin with a recollection of certain well-known properties of Calder´on commuators. Recall that for any Lipschitz function B on R, the Calder´on commutator
1 B(y) − B(x) (C(B)f )(x) = p.v. f (y)dy πi (y − x)2 is a bounded operator on L2 (R). Indeed there is an absolute constant C such that C(B)f 2 ≤ CB
∞ f 2 ,

f ∈ L2 (R).

(2.1)

See [1]. Numerous proofs of (2.1) can also be found in [9]. But for this paper we need another bound. Calder´on showed in [1, Theorem 2] that, if 1 < p < ∞, 1 < r < ∞ and 2−1 = p −1 + r −1 , then there is a constant Cp,r which depends only on p, r such that (2.2) C(B)g2 ≤ Cp,r B
r gp , g ∈ Lp (R). This will be crucial to the proof of Proposition 1.4. The proof of Proposition 1.4 also requires the class L of functions defined as follows. A function ψ on R is said to be in the class L if it satisfies the conditions (i) ψ is strictly increasing on R. (ii) There exist 0 < α = α(ψ) ≤ β = β(ψ) < ∞ such that α ≤ (ψ(x) − ψ(y))/(x − y) ≤ β for all x = y in R. (iii) There is a positive number N = N (ψ) such that ψ(x) = x when |x| ≥ N . For each ψ ∈ L, let Cψ denote the operator of composition with ψ on L2 (R). That is, (Cψ f )(x) = f (ψ(x)),

f ∈ L2 (R).

Obviously, Cψ and Cψ−1 = Cψ −1 are bounded operators on L2 (R) when ψ ∈ L. Lemma 2.1. Let B be a Lipschitz function on R and let ψ ∈ L. Then we have the strong convergence s-lim|λ|→∞ (C(B) − H MB
)Cψ eiλM = 0 on the Hilbert space L2 (R). Proof. For each N ∈ N, let XN = {ξ ∈ L∞ (R) : ξ ∞ ≤ 1 and ξ = 0 on R\(−N, N )}. 2 Since the linear span of ∪∞ N=1 XN is dense in L (R), it suffices to show that lim (C(B) − H MB
)Cψ eiλM ξ 2 = 0

|λ|→∞

for all ξ ∈ XN and N ∈ N.

(2.3)

For this purpose we fix an N . Let I = [a, b] = ψ −1 ([−N, N ]), J = [a − 1, b + 1], and K = [a − 2, b + 2]. There obviously exists a Lipschitz function L on R such that L = B on J and L
= 0 on R\K. It follows from the condition L = B on J that MχJ (C(L)−H ML
)Cψ eiλM ξ = MχJ (C(B)−H MB
)Cψ eiλM ξ for all ξ ∈ XN . (2.4) On the other hand, since Cψ eiλM ξ = 0 on R\I when ξ ∈ XN , we have MχR\J (C(B) − H MB
)Cψ eiλM ξ = MχR\J (C(B)−H MB
)MχI Cψ eiλM ξ . It is obvious that MχR\J (C(B)− H MB
)MχI is a compact operator. Since eiλM → 0 weakly as |λ| → ∞, we have

310

J. Xia

lim|λ|→∞ MχR\J (C(B) − H MB
)Cψ eiλM ξ 2 = 0, ξ ∈ XN . Recalling (2.4), we see that (2.3) will follow once we establish lim (C(L) − H ML
)Cψ eiλM ξ 2 = 0,

|λ|→∞

ξ ∈ XN .

(2.5)

To prove (2.5), we use the fact that L
vanishes outside K and is bounded on K. Let 9 > 0 be given. There is an F ∈ C ∞ (R) such that F
has a compact support and L
−F
4 ≤ 9. If f is supported on I = [a, b] and f ∞ ≤ 1, then it follows from (2.2) that {C(L) − C(F )}f 2 = C(L − F )f 2 ≤ C4,4 L
− F
4 f 4 ≤ C4,4 (b − a)1/4 9. For such an f , {H ML
−H MF
}f 2 ≤ (L
−F
)f 2 ≤ L
−F
4 f 4 ≤ (b−a)1/4 9. Hence (C(L)−H ML
)Cψ eiλM ξ −(C(F )−H MF
)Cψ eiλM ξ 2 ≤ (1+C4,4 )(b−a)1/4 9 (2.6) for every ξ ∈ XN . Since the kernel of C(F ) − H MF
equals (π i)−1 {(y − x)−2 (F (y) − F (x)) − (y − x)−1 F
(y)}, the condition F ∈ C ∞ (R) guarantees that MχJ (C(F ) − H MF
)MχI is compact. Thus (C(F ) − H MF
)MχI is also compact. Since Cψ eiλM ξ = MχI Cψ eiλM ξ when ξ ∈ XN , the weak convergence w-lim|λ|→∞ eiλM = 0 again yields lim (C(F ) − H MF
)Cψ eiλM ξ 2 = 0,

|λ|→∞

ξ ∈ XN .

(2.7)

Because 9 > 0 is arbitrary, (2.5) follows from (2.6) and (2.7).     Lemma 2.2. If ψ ∈ L and h ∈ L2 (R), then lim|λ|→∞ (Cψ H Cψ−1 − H )eiλM h2 = 0. Proof. By assumption, there are 0 < α ≤ β < ∞ such that α ≤ (ψ(x) − ψ(y))/(x − y) ≤ β for all x = y. For each s ∈ [0, 1], define ψs (x) = sψ(x) + (1 − s)x. Then sα +(1−s) ≤ (ψs (x)−ψs (y))/(x −y) ≤ sβ +(1−s), x = y. Define G(x) = ψ(x)−x on R. The membership ψ ∈ L implies that G has a bounded support. Thus, if f belongs to the domain of D = (1/ i)(d/dx), then the derivative dψs  d d Cψ f = f ◦ ψ s = f
◦ ψs = iMG Cψs Df ds s ds ds exists in the norm topology of L2 (R). Also, dCψ−1s /ds = −Cψ−1s (dCψs /ds)Cψ−1s . Obviously, Cψ−1s MG Cψs = Cψs−1 MG Cψs = MG◦ψs−1 and [D, H ] = 0. Therefore

d Cψ H Cψ−1s f = i{MG Cψs DH − Cψs H Cψ−1s MG Cψs D}Cψ−1s f ds s = iCψs [MG◦ψs−1 , H ]DCψ−1s f. Write Gs = G ◦ ψs−1 , 0 ≤ s ≤ 1. Integration-by-parts yields    i MGs , H Dg = H MG
s − C(Gs ) g

(2.8)

(2.9)

if g is in the domain of D. Therefore it follows from (2.8) that
1
1  d Cψ H Cψ−1 − H = Cψs H MG
s − C(Gs ) Cψs−1 ds. (2.10) Cψs H Cψ−1s ds = i 0 ds 0

Invariance Principle

311

For any h ∈ L2 (R), we have
    −1 iλM  h ≤  Cψ H Cψ − H e 2

   iλM  Cψ  
h ds. s  H MGs − C(Gs ) Cψs−1 e

1 0

2

    Lemma 2.1 asserts that lim|λ|→∞  H MG
s − C(Gs ) Cψs−1 eiλM h = 0 for every s ∈ 2 [0, 1]. Thus the proof is complete upon invoking the dominated convergence theorem.   Remark. Identities (2.8–10) are well known. To our knowledge, these identities were first used by Semmes in [11]. Later, a unit-circle version of these identities was used in [8]. Lemma 2.3. If ϕ ∈ L, then on the Hilbert space L2 (R) we have the strong convergence s- lim eiλϕ(M) H e−iλϕ(M) = −1 λ→∞

and

s- lim eiλϕ(M) H e−iλϕ(M) = 1. λ→−∞

Proof. Let ψ = ϕ −1 , the inverse of ϕ. Then eiλϕ(M) = Cϕ eiλM Cϕ−1 = Cψ−1 eiλM Cψ . Hence   eiλϕ(M) H e−iλϕ(M) = Cψ−1 eiλM Cψ H Cψ−1 − H e−iλM Cψ + Cψ−1 eiλM H e−iλM Cψ .  By Lemma 2.2, we have s-lim|λ|→∞ Cψ−1 eiλM Cψ H Cψ−1 − H e−iλM Cψ = 0. Recalling

(1.1), we have the strong convergence Cψ−1 eiλM H e−iλM Cψ → Cψ−1 (−1)Cψ = −1 as

λ → ∞ and Cψ−1 eiλM H e−iλM Cψ → Cψ−1 Cψ = 1 as λ → −∞. This proves the lemma.  

Lemma 2.4. Let f be a real-valued Borel function on R. Suppose that there exist a ϕ ∈ L and a Borel set E ⊂ R such that f (x) = ϕ(x) for every x ∈ E. Then, on the Hilbert space L2 (R), we have the strong convergence s- lim eiλf (M) H e−iλf (M) MχE = −MχE and λ→∞

s- lim eiλf (M) H e−iλf (M) MχE = MχE . λ→−∞

 2 (R) = L2 (E), then e−iλf (M) g = e−iλϕ(M) g and M Proof. If g ∈ M L χ R\E E     H e−iλϕ(M) g 2 = MR\E eiλϕ(M) H e−iλϕ(M) g 2 = MR\E (eiλϕ(M) H e−iλϕ(M) g ± g)2 .    By Lemma 2.3, lim|λ|→∞ MR\E H e−iλϕ(M) g 2 = 0. But eiλf (M) H e−iλf (M) g −     eiλϕ(M) H e−iλϕ(M) g 2 =  eiλf (M) −eiλϕ(M) H e−iλϕ(M) g 2 ≤ 2 MR\E H e−iλϕ(M) g2 . Therefore     lim eiλf (M) H e−iλf (M) g − eiλϕ(M) H e−iλϕ(M) g  = 0. |λ|→∞

The desired conclusion now follows from this and Lemma 2.3.

2

 

Proposition 2.5. Let U be an open set in R and let ϕ ∈ A(U ). Let + = {x ∈ U : ϕ
(x) > 0} and − = {x ∈ U : ϕ
(x) < 0}. Then there exist Borel subsets {Ej : j ∈ N} of + and Borel subsets {Fj : j ∈ N} of − with the following properties:

312

J. Xia

  ∞ (i) The sets + \ ∪∞ j =1 Ej and − \ ∪j =1 Fj have Lebesgue measure 0. (ii) For each j ∈ N, there are ϕj , ψj ∈ L such that ϕ = ϕj on Ej and −ϕ = ψj on Fj . Proof. For each n ∈ N, let +,n = {x ∈ U : 1/n ≤ ϕ
(x) ≤ n and |x| ≤ n}. We will   ν(n) find compact subsets Kn,1 , . . . , Kn,ν(n) of +,n with m +,n \ ∪i=1 Kn,i ≤ 1/n ν(n) such that ϕ equals some ψn,i ∈ L on each Kn,i . Since m( + \(∪∞ n=1 ∪i=1 Kn,i )) = 0, a re-enumeration of {Kn,i : 1 ≤ i ≤ ν(n), n ∈ N} gives us the desired {Ej : j ∈ N}. Similarly, the desired {Fj : j ∈ N} are obtained from − by considering −ϕ instead of ϕ. Fix an n ∈ N. To find the Kn,i ’s promised above, we proceed as follows. Let r1 , . . . , rj , . . . be an enumeration of the rational numbers in (−1, 0) ∪ (0, 1). For each k ∈ N, define      k rj  ri  k    ϕ x+ ϕ x+ − ϕ(x) − − ϕ(x)  . gk (x) = sup  r k r k i,j ∈N

i

j

For each x ∈ +,n , the existence of ϕ
(x) implies limk→∞ gk (x) = 0. By Egoroff’s theorem, +,n has a compact subset Kn with m( +,n \Kn ) ≤ 1/n such that the uniform convergence limk→∞ supx∈Kn gk (x) = 0 holds. Therefore there is a κ ∈ N such that     κ    κ rj  ri   ϕ x+ ϕ x+ − ϕ(x) − − ϕ(x)  sup sup  κ rj κ x∈Kn i,j ∈N ri = sup gκ (x) ≤ 1/10n. x∈Kn

Since ϕ
(x) exists for every x ∈ Kn and since  {ri /κ : i ∈ N} is dense in (−1/κ, 0) ∪ (0, 1/κ), the above implies supx∈Kn supj ∈N ϕ
(x) − (κ/rj )(ϕ(x + (rj /κ)) − ϕ(x)) ≤ 1/10n. Because 1/n ≤ ϕ
≤ n on the set Kn , we obtain the inequality 1/2n ≤ (κ/rj )(ϕ(x + (rj /κ)) − ϕ(x)) ≤ 2n for all x ∈ Kn and j ∈ N.

(2.11)

Since Kn is bounded, there exist a finite number of closed intervals [a1 , b1 ], . . . , [aν(n) , ν(n) bν(n) ] such that Kn ⊂ ∪i=1 [ai , bi ] and 0 < bi − ai ≤ 1/4κ for every 1 ≤ i ≤ ν(n). Now let Kn,i = Kn ∩ [ai , bi ], 1 ≤ i ≤ ν(n). Given such an i, we need to find a ψn,i ∈ L such that ϕ = ψn,i on Kn,i . This is trivial if Kn,i contains fewer than two points. Thus, without loss of generality, we may assume that ai , bi ∈ Kn,i . As a function on R, ϕ is continuous at every x where ϕ
(x) exists. In particular, as a function on R, ϕ is continuous at every point in Kn . Combining this continuity with the fact that {rj /κ : j ∈ N} is dense in (−1/κ, 0) ∪ (0, 1/κ) and with (2.11), we find that 1/2n ≤ δ −1 (ϕ(x + δ) − ϕ(x)) ≤ 2n for x and δ satisfying the conditions x ∈ Kn , x + δ ∈ Kn , and 0 < |δ| < 1/κ. But |x − y| ≤ 1/4κ for all x, y ∈ Kn,i . Therefore 1/2n ≤ (ϕ(y) − ϕ(x))/(y − x) ≤ 2n

if x, y ∈ Kn,i and x = y.

We now define ψn,i . Obviously, we must define ψn,i (x) = ϕ(x) for x ∈ Kn,i . If (α, β) is a component of the open set [ai , bi ]\Kn,i , i.e., if α, β ∈ Kn,i and Kn,i ∩ (α, β) = ∅, then we define ψn,i to be the linear function (α, β) such that ψn,i is continuous on [α, β], i.e., ψn,i (x) = ϕ(α) + {(ϕ(β) − ϕ(α))/(β − α)}(x − α) if α < x < β.

Invariance Principle

313

This defines ψn,i as a function on the entire [ai , bi ]. It is an elementary exercise to verify that this definition results in a ψn,i satisfying the condition 1/2n ≤ (ψn,i (y) − ψn,i (x))/(y − x) ≤ 2n if x, y ∈ [ai , bi ] and x = y. It is now trivial to extend ψn,i to R\[ai , bi ] so that the extended function belongs to L. Thus we have found a ψn,i ∈ L such that ψn,i = ϕ on Kn,i . This completes the proof.   Proof of Proposition 1.4. Given a ϕ ∈ A(U ), let {Ej : j ∈ N} and {Fj : j ∈ N} be, respectively, the Borel subsets of + = {x ∈ U : ϕ
(x) > 0} and − = {x ∈ U : ϕ
(x) < 0} provided by Proposition 2.5. By that proposition, for each j ∈ N, ϕ coincides with a ϕj ∈ L on Ej . Thus it follows from Lemma 2.4 that, if ξ ∈ L2 (Ej ) = MχEj L2 (R) for some j , then     lim eiλϕ(M) H e−iλϕ(M) ξ + ξ  = 0. (2.12) λ→∞

2

2 2 Since the linear span of ∪∞ j =1 L (Ej ) is dense in L ( + ), (2.12) holds for all ξ ∈ 2 L ( + ). Similarly, since −ϕ = ψj on Fj for some ψj ∈ L, it follows from Lemma 2.4 that     lim eiλ(−ϕ(M)) H e−iλ(−ϕ(M)) η − η = 0 (2.13) λ→−∞

2

2 2 if η ∈ L2 (Fj ). Again, because the linear span of ∪∞ j =1 L (Fj ) is dense in L ( − ),(2.13)  holds for all η ∈ L2 ( − ). Therefore limλ→∞ eiλϕ(M) H e−iλϕ(M) (η+ξ )−(η−ξ )2 = 0 if ξ ∈ L2 ( + ) and η ∈ L2 ( − ). This proves Proposition 1.4.  

Proof of Proposition 1.5. First of all, it is clear that, if ψ ∈ L and if B is a self-adjoint operator which is purely absolutely continuous, then the operator ψ(B) is also purely absolutely continuous. Let A, U and ϕ ∈ A(U ) be given as in the statement of the proposition. Let {Ej : j ∈ N} and {Fj : j ∈ N} be, respectively, the Borel subsets of + = {x ∈ U : ϕ
(x) > 0} and − = {x ∈ U : ϕ
(x) < 0} provided by Proposition 2.5. Since m(σ (A)\U ) = 0 and ∞ m(U \{(∪∞ j =1 Ej ) ∪ (∪j =1 Fj )}) = 0, we have the decomposition Aac = ⊕ν∈N Bν such that, for each ν ∈ N , the spectrum σ (Bν ) of Bν is a compact subset of either some Ej or some Fi . This means either ϕ(Bν ) = ϕj (Bν ) for some ϕj ∈ L or ϕ(Bν ) = −ψi (Bν ) for some ψi ∈ L. Being a direct summand of Aac , each Bν is, of course, purely absolutely continuous. Thus we see from the preceding paragraph that ϕ(Aac ) = ⊕ν∈N ϕ(Bν ) is purely absolutely continuous. Since ϕ(A)|Hac (A) = ϕ(Aac ), this means Hac (A) ⊂ Hac (ϕ(A)).   3. Cook’s Condition Revisited The invariance principle for wave operators can also be proved for our admissible functions under the condition of Cook’s method. More precisely, condition (1.9) in Theorem 1.6 can be replaced by (3.1) below. The conventional view is that the invariance principle is more difficult to prove under (3.1) than it is under (1.9). But in this respect we benefit from Konstantinov’s insight in [6]: In fact, condition (3.1) is so strong that it implies a local version of (1.9), which is all that we need for the proof of the invariance principle.

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Theorem 3.1. Let A and A
be self-adjoint operators on Hilbert spaces H and H
respectively and let J : H → H
be a bounded operator. Let EA be the spectral resolution of A. Suppose that there exist a ξ ∈ Hac (A) and a z ∈ C\R such that
∞ {J (A − w)−1 − (A
− w)−1 J }eiλA ξ dλ < ∞ for w ∈ {z, z¯ }. (3.1) −∞

Let H(A; ξ ) be the smallest subspace of H which contains ξ and which is invariant under EA . Let U be an open set in R such that m(σ (A)\U ) = 0 and let ϕ ∈ A(U ). Then the limit
Wϕ ζ = lim e−iλϕ(A ) J eiλϕ(A) ζ λ→∞

exists in the norm topology of H
for every ζ ∈ H(A; ξ ). Furthermore, when ζ ∈ H(A; ξ ), Wϕ ζ = W+ ζ + + W− ζ − , where ζ ± = EA ( ± )ζ , + = {x ∈ U : ϕ
(x) > 0}, − = {x ∈ U : ϕ
(x) < 0}, and


W± η = lim e−iλA J eiλA η, λ→±∞

η ∈ H(A; ξ ),

which exist in the norm topology of H
. Remark. This result was announced in [6] under the condition ϕ is monotonic on every component interval of U and ϕ
> 0 a.e. on U . Clearly, our condition ϕ ∈ A(U ) is weaker. Proposition 3.2. Let A be a self-adjoint operator on H and let T : H → H
be a bounded operator. Suppose that there exists a ξ ∈ Hac (A) such that
∞   iλA  (3.2) T e ξ  dλ < ∞. −∞

Let Pξ : H → H(A; ξ ) be the projection, where H(A; ξ ) is the smallest subspace of H which contains ξ and which is invariant under the spectral resolution EA of A. Then there is a sequence { k } of Borel subsets of R such that s-limk→∞ EA ( k )Pξ = Pξ and such that T EA ( k )Pξ ∈ C1 for every k ∈ N.

Proof. We follow the ideas outlined in [6]. Define the measure µ( ) = EA ( )ξ, ξ on R. Since ξ ∈ Hac (A), there exists a non-negative function ρ ∈ L1 (R) such that dµ = ρdm, where m is the Lebesgue measure. It is elementary that there is a unitary operator V : H(A; ξ ) → L2 (R, dµ) = L2 (R, ρdm) such that Vf (A)ξ = f for every f ∈ L2 (R, ρdm). For each k ∈ N, let k = {t ∈ R : 1/k ≤ ρ(t) ≤ k}. It is easy to see that s-limk→∞ EA ( k )Pξ = Pξ . Also, keep in mind that Pξ and EA commute. Let F be a finite-rank projection on H
. Then there exist vectors {x1 , . . . , xn } ⊂ H
and {y , . . . , yn } ⊂ H such that "xi , xj # = 0 = "yi , yj # if i = j and such that F T = n 1 n iλA ξ 2 = iλA ξ, y #|2 x 2 . By the preceding j j j =1 xj ⊗ yj . We have F T e j =1 |"e paragraph, if we write yˆj = V Pξ yj , which is a function in L2 (R, ρdm), then
"eiλA ξ, yj # = eiλt yˆj (t)ρ(t)dt. (3.3)

Invariance Principle

315

Without loss of  generality, we may assume that xj = 0 for all j ∈ {1, . . . , n}. Condition (3.2) implies F T eiλA ξ 2 dλ < ∞. Thus the function λ %→ "eiλA ξ, yj # belongs to L2 (R). By (3.3), yˆj ρ ∈ L2 (R) and yˆj ρ2L2 (R) = (1/2π ) |"eiλA ξ, yj #|2 dλ < ∞. Thus Pξ yj belongs to the domain of ρ 1/2 (A) and ρ 1/2 (A)Pξ yj 2 = yˆj ρ2L2 (R) . Hence 1 2π




F T eiλA ξ 2 dλ =

n 

ρ 1/2 (A)Pξ yj 2 xj 2

j =1

= tr

  ∗  F T Pξ ρ 1/2 (A) F T Pξ ρ 1/2 (A)

For the same reason, this identity still holds if F T is replaced by any finite-rank operator S so long as the range of S ∗ is contained in the domain of ρ 1/2 (A)Pξ . Thus, by the familiar polarization formula, if X : H → H
is a finite-rank operator, then
  1 "F T eiλA ξ, XEA ( k ) eiλA ξ #dλ = tr (F T Pξ ρ 1/2 (A))(XEA ( k ) Pξ ρ 1/2 (A))∗ 2π  = tr F T Pξ ρ(A)EA ( k ) X ∗ .    Therefore |tr(F T Pξ ρ(A)EA k )X ∗ ) ≤ Xξ  T eiλA ξ dλ for any finite-rank operator X : H → H
and any finite-rank projection F on H
. It follows from this inequality and (3.2) that T Pξ ρ(A)EA ( k ) ∈ C1 . The definition of k ensures that ρ −1 (A)EA ( k ) is also bounded. Hence T EA ( k )Pξ = T Pξ ρ(A)EA ( k ) · ρ −1 (A)EA ( k ) ∈ C1 as desired.   Proof of Theorem 3.1. Let T = J (A−z)−1 −(A
−z)−1 J and S = J (A− z¯ )−1 −(A
− z¯ )−1 J . Let Pξ : H → H(A; ξ ) be the projection.By (3.1) and Proposition 3.2, there  exist Borel sets { k }, {Fk } such that s-limk→∞ EA k Pξ = Pξ = s-limk→∞ EA Fk Pξ and such that T EA ( k )Pξ ∈ C1 and SEA (Fk )Pξ ∈ C1 , k ∈ N. Let Pk = EA ( k ∩Fk )Pξ = EA ( k )EA (Fk )Pξ . Clearly, s-limk Pk = Pξ , T Pk ∈ C1 , and SPk ∈ C1 . Therefore J Pk (A − w)−1 − (A
− w)−1 J Pk = {J (A − w)−1 − (A
− w)−1 J }Pk ∈ C1 for w ∈ {z, z¯ }. Hence it follows from Theorem 1.6 that the strong limit





Wk = s- lim e−iλϕ(A ) J Pk eiλϕ(A) Pac (A) = s- lim e−iλϕ(A ) J eiλϕ(A) Pk λ→∞

λ→∞

exists and equals W+ (A
, A; J Pk )EA ( + ) + W− (A
, A; J Pk )EA ( − ). This proves the existence of Wϕ ζ and the identity Wϕ ζ = W+ ζ + + W− ζ − in the case where ζ ∈ Pk H for some k ∈ N. The proof is complete upon reciting the fact that s-limk→∞ Pk = Pξ .   4. An Example In this section we will construct an example of a pair of bounded self-adjoint operators A, A
on a Hilbert space H = {0} and a ϕ ∈ A(R) with the following properties: (i) A − A
∈ C1 . (ii) A is purely singular, i.e., Hac (A) = {0}. (iii) ϕ(A) is purely absolutely continuous, i.e., Hac (ϕ(A)) = H.
(iv) The limits s-limλ→±∞ e−iλϕ(A ) eiλϕ(A) Pac (ϕ(A)) fail to exist.

316

J. Xia

Indeed the failure of the above strong convergence is complete: (iv
) If ξ ∈ H and {λn } is a sequence of real numbers such that limn→∞ |λn | = ∞
and such that the sequence {e−iλn ϕ(A ) eiλn ϕ(A) ξ } weakly converges in H, then the
weak limit of {e−iλn ϕ(A ) eiλn ϕ(A) ξ } is 0. Before the construction, let us recall a definition. An operator T is said to be diagonal if it is unitarily equivalent to an operator diag(aj ) on G2+ defined by the formula  diag aj {c1 , . . . , cn , . . .} = {a1 c1 , . . . , an cn , . . .}. We begin the construction with a probability measure µ on R which is devoid of point masses. We further require that µ be supported on a compact subset X of [0, 1] such that m(X) = 0, which ensures that µ is singular to the Lebesgue measure. Let H = L2 (X, dµ) = L2 ([0, 1], dµ) and define the operator A by the formula (Af )(x) = xf (x),

f ∈ L2 ([0, 1], dµ).

Obviously, (ii) follows from the assumption µ ⊥ m. Since A is purely singular, a wellknown theorem of Carey and Pincus [2, p. 484] tells us that there is a K ∈ C1 such that A
= A + K is a self-adjoint diagonal operator. Thus A
has no continuous spectrum. Let ϕ(x) = x + µ((−∞, x)), x ∈ R. (4.1) It is obvious that ϕ ∈ A(R). To establish (iii), we need the condition that µ has no point masses, which implies that ϕ is continuous. Hence the range of ϕ equals the entire real line R. Let u ∈ L∞ ([0, 1], dµ). Let (a, b) be a finite open interval. Since ϕ(R) = R, there are α < β in R such that ϕ(α) = a and ϕ(β) = b. Therefore
"χ(a,b) (ϕ(A))u, u# = χ(a,b) (ϕ(t))|u(t)|2 dµ(t) ≤ u2∞ µ(ϕ −1 (a, b)) = u2∞ µ((α, β)) ≤ u2∞ (ϕ(β) − ϕ(α)) = u2∞ (b − a). It is easy to deduce from the above that "Eϕ(A) ( )u, u# ≤ u2∞ m( ) for any Borel set

⊂ R, where Eϕ(A) is the spectral resolution of ϕ(A). This proves (iii). To prove (iv
), let {λn } be a sequence of real numbers such that limn→∞ |λn | = ∞. Passing to a subsequence of {λn } if necessary, it suffices to show that if the weak limit


W = w- lim e−iλn ϕ(A ) eiλn ϕ(A) n→∞

exists on H, then W = 0. To this end we use the weak convergence w-limn→∞ eiλn ϕ(A) = 0, which is a consequence of (iii). Combining this with the fact that A − A
∈ C1 , we have A
W = W A and, therefore, A|W | = |W |A. Thus if W = V |W | is the polar decomposition of W , then (A
V − V A)|W | = 0. Let H1 (resp. H2 ) be the closure of the range of |W | (resp. W ). Then H1 (resp. H2 ) is invariant under A (resp. A
). The equation (A
V − V A)|W | = 0 implies that A
|H2 is unitarily equivalent to A|H1 . Since A
has no continuous spectrum whereas A has no eigenvalues, this is possible only if H1 = {0} = H2 . In other words, W = 0 as promised. This proves (iv
). Remark. In this example, because σ (A) = X, we have m(σ (A)) = 0. Let U = R\X. Then the function ϕ defined by (4.1) is locally absolutely continuous on U . In fact, if I is a component interval in U , then there is a cI ∈ [0, 1] such that ϕ(x) = cI + x for every x ∈ I . Because m(X) = 0, we also have U = R ⊃ σ (A). Thus this example shows that the requirement that σ (A)\U be countable in Theorem 1.8 cannot be replaced by m(σ (A)\U ) = 0 even in the case where U ⊃ σ (A).

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References 1. Calder´on,A: Commutators of singular integral operators. Proc. Nat.Acad. Sci. U.S.A. 53, 1092–1099 (1965) 2. Carey, R., Pincus, J.: Unitary equivalence modulo the trace class for self-adjoint operators. Am. J. Math. 98, 481–514 (1976) 3. Chandler, C., Gibson, A.: Invariance principle for scattering with long-range (and other) potentials. Indiana Univ. Math. J. 25, 443–460 (1976) 4. Kato, T.: Wave operators and unitary equivalence. Pacific J. Math. 15, 171–180 (1965) 5. Kato, T.: Perturbation Theory for Linear Operators. New York: Springer-Verlag, 1976 6. Konstantinov, A.: The invariance principle for wave operators. Soviet Math. Dokl. 31, 374–376 (1985) 7. Konstantinov, A.: A class of admissible functions in the invariance principle for wave operators. Ukrainian Math. J. 46, 236–239 (1994) 8. Muhly, P., Xia, J.: Calder´on-Zygmund operators, local mean oscillation and certain automorphisms of the Toeplitz algebra. Am. J. Math. 117, 1157–1201 (1995) 9. Murai, T.: A real variable method for the Cauchy transform, and analytic capacity. Lecture Notes in Mathematics, Vol. 1307. Berlin: Springer-Verlag, 1988 10. Reed, M., Simon, B.: Methods of Modern Mathematical Physics. III. Scattering theory. New York: Academic Press, 1979 11. Semmes, S.: The Cauchy Integral and Related Operators on Smooth Curves. Dissertation, Washington Unversity, 1983 12. Xia, J.: Trace-class perturbation and strong convergence: Wave operators revisited. Proc. Am. Math. Soc. 128, 3519–3522 (2000) Communicated by B. Simon

Commun. Math. Phys. 232, 319–326 (2003) Digital Object Identifier (DOI) 10.1007/s00220-002-0744-x

Communications in

Mathematical Physics

An Extension of the Beale-Kato-Majda Criterion for the Euler Equations Fabrice Planchon Laboratoire Analyse, G´eom´etrie & Applications, UMR 7539, Institut Galil´ee, Universit´e Paris 13, 99 avenue J.B. Cl´ement, 93430 Villetaneuse, France Received: 27 February 2002 / Accepted: 29 July 2002 Published online: 14 November 2002 – © Springer-Verlag 2002

Abstract: The Beale-Kato-Majda criterion asserts that smooth solutions to the Euler
T equations remain bounded past T as long as 0
ω
∞ dt is finite, ω being the vorticity.
T We show how to replace this by a weaker statement, on supj 0
j ω
∞ dt, where j is a frequency localization around |ξ | ≈ 2j . Introduction The incompressible Euler equations read  ∂u   + u · ∇u = −∇p, ∂t ∇ · u = 0,   u(x, 0) = u0 (x), x ∈ Rn , t ≥ 0.

(1)

These equations are known to be locally well-posed for data u0 ∈ H s , s > n2 + 1, or more generally Wps with s − pn > 1 (see [6] and references therein). In a celebrated paper, Beale-Kato-Majda gave the following criterion for blow-up: if blow-up occurs at time T , then necessarily,  T
ω
∞ dt = +∞, (2) 0

where ω = ∇ ∧ u is the vorticity. One may rephrase it as: if  T
ω
∞ dt < +∞,

(3)

0

then the solution can be continued past time T . For a generic quasilinear hyperbolic system with unknown φ, it is well-known that
T if one can control 0
∇φ
∞ dt, then the estimates can be closed (at least in Sobolev

320

F. Planchon

spaces). Hence, the difficulty in [2] was to replace ∇u by ω, overcoming the lack of continuity of Riesz transforms on L∞ , when using the Biot-Savard law, ∇u = ∇∇ ⊥ −1 ω. This was done using a logarithmic Sobolev inequality, yielding control of
∇u
∞ in terms of
ω
∞ and a logarithmic correction involving an higher order norm. It was recently observed by Kozono-Taniuchi ([8]) that one could replace the L∞ norm in (3) by the BMO norm. The same authors and T. Ogawa ([7]) then refined (3) to 

T


ω
B˙ 0,∞ dt < +∞.

(4)

∞

0 s,q

Here and thereafter, B˙ p stands for an homogeneous Besov space, see the Appendix for definition and [3, 9] for an extensive treatment. We will also use their inhomogeneous s,q s,q counterpart, Bp = Lp ∩ B˙ p for s ≥ 0. One of the advantages of (4) (besides obviously 0,∞ . One being a weaker statement than (3)) is that one may freely replace ∇u by ω in B˙ ∞ 0,∞ ˙ could think this refinement to be the best possible: certainly B∞ is the largest space above L∞ in the Besov scale. However, in several recent works on the Navier-Stokes equations, the following quantities appear to be of interest (here and thereafter j is a frequency localization operator at |ξ | ≈ 2j ): sup 2

j qs



j

T 0

q


j u
p dt < ∞,

(5)

where s = q2 + pn − 1 > 0. Indeed, whenever p < ∞ they control the equation. Note that for p = ∞ one is fairly close to (4), except the supremum over frequencies and the time integration have been reversed. We intend to relax (4) in a similar
T fashion. In order to achieve this, we will first prove an estimate involving supj 0
∇Sj u
∞ dt, where Sj is the projector over low frequencies |ξ |
2j . Then, via a space-time logarithmic inequality, we replace Sj by j to obtain the desired result. We remark that one could also obtain local well-posedn

+1,1

from the intermediate result, but we elected not to do it to keep this note ness in B˙ pp short. Such a result was announced in [10] where it is proved for n = 2, a particular case since the solution can be extended for all times. Finally, it should be said that the same type of estimate could be proved for a generic quasilinear hyperbolic system: thus,
T supj 0
∇j φ
∞ dt would control Sobolev norms of φ for s > n2 + 1. 1. Statements of Results and Proofs We intend to prove the following theorem, s,q

Theorem 1. Let u be a smooth solution to (1), u ∈ Ct (Bp ) with s > q < ∞. There exists an absolute constant M0 such that: • if

 lim sup

ε→0 j

T T −ε

n p

+ 1, 1 ≤ p,


j ω
∞ dt = ε0 < M0 , s,q

then ε0 = 0 and one may continue the solution in Ct (B˙ p ) past time T .

(6)

Extension of Beale-Kato-Majda Criterion

321

• if

 lim sup

ε→0 j

T T −ε


j ω
∞ dt ≥ M0 ,

(7)

then
∇u
∞ blows up at time T . Several remarks are in order. Firstly, one cannot simply state 

T

sup j

0


j ω
∞ dt < ∞,

as a non-blowup condition. Indeed, while at fixed j , the integral over (T − ε, T ) will go to zero, one cannot expect uniformity with respect to j with only an l ∞ (j ) condition. A sufficient condition to ensure that uniformity would be condition (4), or to strengthen the l ∞ (j ) to l r (j ) for some r < ∞,   j

T 0

r
j ω
∞ dt

< ∞.

Next, one may wonder why there is a limiting value M0 . When the limit defined above is non-zero, then necessarily
∇u
∞ (and
ω
∞ ) blows up: indeed, there exists a sequence jn such that  T
jn ω
∞ > ε0 /2, T −1/n

but
jn ω
∞

jn ∇u
∞

∇u
∞ , and therefore sup

[T −1/n,T ]


∇u
∞ ≥ nε0 /2,

which means blow-up. However, the a priori estimate we will get provides good control of ∇u if ε0 is small enough: hence, it can only be zero, otherwise one would have a contradiction. Next, we elected to measure the regularity in the Besov spaces scale. This is intimately tied to the type of estimates we will use, mainly frequency localization and paraproduct s,q techniques. It doesn’t seem possible to recover estimates in F˙p spaces, and therefore s we miss the usual Hp spaces, except for p = 2. Before proceeding with the proof, we state a proposition which might be of inde
T pendent interest. The importance of 0
∂Sj u
∞ dt has already been observed, see for instance [5]. s,q

Proposition 1. Let u be a solution to the Euler equation with data u0 ∈ B˙ p , s ≥ Define αj,T = sup[0,T ]
j u
p 2j s , and αT =
αj,T
l q . Then,

q αT

≤

q α0



T

+ C sup j

0

n p

+ 1.


∇Sj u
∞ dt s,q

q

αT .

Remark that the usual norm on L∞ ([0, T ], B˙ p ) is controlled by αT .

(8)

322

F. Planchon

We now prove the proposition. The techniques we rely on were extensively used in connection with the Euler equation, see [1, 4] and recent work of Vishik [10, 11]. Localize at frequency ξ ≈ 2j , uj = j u, ∂t uj + j ((u · ∇)u) = −∇pj .

(9)

We then project on divergence-free vector fields, ∂t uj + Pj ((u · ∇)u) = 0,

(10)

where P is a matrix operator, with entries δij − Ri Rj , where δij is the Kronecker symbol and Ri , Rj are Riesz transforms. Multiply (9) coordinate by coordinate with (l) (l) (l) |uj |p−2 uj , where uj is a coordinate of uj , and integrate over the space variable,  1 d (l) p (l) (l) (l) ∂t uj |uj |p−2 uj dx = (11)
u
p . p dt j x We then deal with the convective term: denote by vj the vector with coordinates (l) p−2 (l) u uj , we have to estimate j  (12) Pj ((u · ∇)u) · vj dx. We decompose (u · ∇)u as a paraproduct,   (Sk−1 u · ∇)uk + (uk · ∇)Sk−1 u + (uk  · ∇)uk . (u · ∇)u =

(13)

|k−k  |≤1

k

Of these three terms P1 , P2 , P3 , we need to preserve the vector structure only for the first one. For the remaining two one can just forget about the vectors and think of it as a scalar. On the second term we just use H¨older and boundedness of P on Lp to get   p−1
∇Sk−1 u
∞
uk
p
uj
p , (14) | P2 · vj dx|
k∼j

where k ∼ j stands for k = j − 1, j, j + 1 due to support conditions. The third term can be rewritten as  (l)    (l) (l) P3 · vj dx = j  P∇ · (uk  ⊗ uk ) |uj |p−2 uj dx, (15) l

j
k∼k 

on which one may integrate by parts and use H¨older,   (l) p−2
uk 
p
uk
p
uj
p
∇uj
∞ . | P3 · vj dx|
(p − 1)

(16)

Note that if one does not integrate by parts, one may simply write   p−1
∇uk 
∞
uk
p
uj
p . | P3 · vj dx|


(17)

j
k∼k 

j
k∼k 

Extension of Beale-Kato-Majda Criterion

323

Now we concentrate on the first term in the paraproduct, namely P1 : again from the support conditions, the sum reduces to   ((Sk−2 u · ∇)uk = Pj (Sj −2 u · ∇) uk (18) Pj k

+ Pj

k∼j



(uk  · ∇)uk = P1 + P1 .

k∼k  ∼j 

The second sum, P1 , is easy, and can be estimated like (14):    p−1 | P1 · vj dx| ≤
∇uk
∞
uk 
p
uj
p .

(19)

k∼k  ∼j

The first one, P1 , requires the commutator between 2 matrix operators, Cl = [Pj , Sj −2 u(l) I d], it writes      C l ∂l uk + (Sj −2 u · ∇) Pj uk = P11 + P12 (20) l

k∼j

k∼j



and the last sum is obviously k∼j Pj uk = uj since ∇ · uj = 0, thus exactly a sum over l of  (l) (l) (l) ((Sj −2 u · ∇)uj )|uj |p−2 uj dx, which by integration by parts, as   2 |b|p ∂a, a|b|p−2 b∂b = − p 

becomes |

 P12 · vj dx| ≤

C l

p




 · v dx is P12 j

(21)

(22) (l) p


∇Sj −2 u
∞
uj
p .

(23)

Note that this term is the one which vanishes if p = 2. For the commutator, one can forget again about the matrices and perform a scalar estimate. Hence we need an es˜ j , Sj −2 u(l) ], where  ˜ j stands for j or Rµ Rν j timate on the scalar commutator [ ˜ j is the convolution depending on which entry in the P matrix we pick. In either case,  by a smooth function. We then use the following lemma, Lemma 1. Let f ∈ Lp , ∇g ∈ L∞ , then ˜ j , g]f
p
2−j
∇g
∞
f
p .
[

(24)

We provide the (simple) proof in the Appendix.  in (20), becomes, estimating coorThus the term with the commutator, namely P11 dinate by coordinate,   (l) (l) p−1  · vj dx|

∇Sj −2 u
∞ 2−j
∇uj
p
uj
p | P11 l

p



∇Sj −2 u
∞
uj
p .

(25)

324

F. Planchon

Now putting everything together yields  1 d p p−1
∇Sj +1 u
∞
uk
p
uj
p (
uj
p )
p dt k∼j  p−2 +
uk 
p
uk
p
uj
p
∇uj
∞ ,

(26)

j
k  ∼k

where we put together all possible Sk u for k ∼ j , and one may replace the second sum by the alternate (17). Now, assume momentarily that q ≥ 2, then, after integrating in time and multiplying by 2j qs , we have q

q

2j qs
uj
p (t)
2j qs
uj
p (0)   t  q−1 +
∇Sj +1 u
∞ (s) ds  sup(2j qs

uj
p

uk
p ) k∼j [0,t]

0

+





k s

q−2

sup(22s(j −k) 2ks
uk 
p 2
uk
p 2j (q−2)s
uj
p ) . (27)

j
k  ∼k

[0,t]

Therefore, taking the supremum over [0, T ], and using the notations of Proposition 1, we get    T   q q q−1 q−2 αj,T
αj,0 +
∇Sj +1 u
∞ (s) ds  αk,T αj,T +αj,T 22s(j −k) αk  ,T αk,T  . 0

k∼j

j
k  ∼k

(28) The last sum is an l 1 −l q/2 convolution, hence one can sum over j , use H¨older for series, to get the final estimate,

  q

T

q

αT
α0 + sup j

0

q


∇Sj +1 u
∞ (s) ds αT .

(29)

When q < 2, one uses (17) to write  T  q q q−1
∇Sj +1 u
∞ (s) ds αk,T αj,T αj,T
αj,0 + 0

k∼j



q−1

+ αj,T sup k

T 0


∇k u
∞ (s) ds



2s(j −k) αk,T ,

(30)

j
k

and one proceeds similarly to get (29). This ends the proof of Proposition 1. We now prove Theorem 1. Firstly, we easily dispose of the low frequency part, S1 u and even of u in Lp , since one can crudely estimate
u
p :  sup
u
p

u(0)
p +

[0,T ]

T 0


∇u
∞ ds sup
u
p . [0,T ]

(31)

Extension of Beale-Kato-Majda Criterion

325

On the other hand, define γT = sup(αT , sup[0,T ]
u
p ). One has q



q

γT
γ0 +

T 0

q
∇u
∞ (s) ds γT .

(32)

One then needs some kind of logarithmic Sobolev inequality. Consider 

∇u = S−N u +



j ∇u +

|j |≤N

j ∇u.

(33)

j >N

Then, one may integrate in time,  0

T


∇u
∞ dt ≤

T 0


2

−N pn





T

2

j ( pn −s−1)





n p,s



T 0


j ∇u
∞ dt


j ∇u
p dt.

−1− 


∇u
∞ dt
(2N + 1) sup j

0

T


j ∇u
∞ )dt,

j >N

j

0

This reduces to, denoting by λ = inf T




S−N u
p dt + (2N + 1) sup

j >N




j ∇u
∞ +

|j |≤N

0

+



(
S−N u
∞ +

T 0

n p

 ,


j ∇u
∞ dt + T 2−λN γT ,

(34)

which can be optimized in N , to yield  0

T


∇u
∞ dt
sup j

T 0


j ∇u
∞ dt (1 + log(1 + T γT )).

(35)

Inserting this into (32), we have

q γT




q γ0



T

+ sup j

0

 q


j ∇u
∞ dt γT (1 + log(1 + T γT )).

(36)

Now, we remark that everything we wrote on the time interval [0, T ] can be written on [T − ε, T ]. Hence, under the hypothesis of the theorem, one will have the crude estimate q

q

q

γT
γT −ε + ηε γT log(1 + γT ),

(37)

for ηε as small as we want, which yields control of γT and therefore of
u
Bps,q .
T One may then freely pass from ∇u to ω in 0
j ∇u
∞ dt since the gradient of the 0,∞ Biot-Savard kernel is a composition of Riesz transforms, which are bounded on B˙ ∞ .

326

F. Planchon

Appendix We recall the usual definition of Besov spaces, using localization operators in Fourier space:  = 1 for |ξ | ≤ 1 and φ  = 0 for |ξ | > 2, Definition 1. Let φ ∈ S(Rn ) be such that φ φj (x) = 2nj φ(2j x), Sj = φj ∗ ·, j = Sj +1 − Sj . Let f be in S  (Rn ). We say f s,q belongs to B˙ p if and only if  • The partial sum m −m j (f ) converges to f as a tempered distribution, modulo polynomials. • The sequence 6j = 2j s
j (f )
Lp belongs to l q . We now prove Lemma 1. Recall f ∈ Lp , ∇g ∈ L∞ imply ˜ j , g]f
p
2−j
∇g
∞
f
p .
[ ˜ j f = 2nj φ(2j x) 7 f , one writes Since   [j , g]f (x) = 2nj φ(2j (x − y))(g(y) − g(x))f (y)dy. Then



2nj |φ|(2j (x − y))|x − y|
∇g
∞ |f |(y)dy,  ≤ 2−j
∇g
∞ 2nj ψ(2j (x − y))|f |(y)dy,

|[j , g]f (x)| ≤


[j , g]f
p ≤ 2−j
∇g
∞
ψ
1
f
p , since ψ(x) = |x||φ|(x) ∈ L1 . This ends the proof. References ´ 1. Bahouri, H., Chemin, J.-Y.: Equations de transport relatives a` des champs de vecteurs non-lipschitziens et m´ecanique des fluides. Arch. Rational Mech. Anal. 127(2), 159–181 (1994) 2. Beale, J.T., Kato, T., Majda, A.: Remarks on the breakdown of smooth solutions for the 3-D Euler equations. Commun. Math. Phys. 94(1), 61–66 (1984) 3. J¨oran Bergh, J¨orgen L¨ofstr¨om: Interpolation Spaces. An Introduction. Grundlehren der Mathematischen Wissenschaften, No. 223. Berlin: Springer-Verlag, 1976 4. Jean-Yves Chemin: Perfect Incompressible Fluids. New York: The Clarendon Press Oxford University Press, 1998. Translated from the 1995 French original by Isabelle Gallagher and Dragos Iftimie 5. Chemin, Jean-Yves, Masmoudi, Nader: About lifespan of regular solutions of equations related to viscoelastic fluids. SIAM J. Math. Anal. 33(1), 84–112 (electronic) (2001) 6. Tosio Kato, Gustavo Ponce: Commutator estimates and the Euler and Navier-Stokes equations. Comm. Pure Appl. Math. 41(7), 891–907 (1988) 7. Hideo Kozono, Takayoshi Ogawa,Yasushi Taniuchi: The critical sobolev inequalities in besov spaces and regularity criterion to some semi-linear evolution equations. Math. Z. 242, 251–278 (2002) 8. Hideo Kozono, Yasushi Taniuchi: Limiting case of the Sobolev inequality in BMO, with application to the Euler equations. Commun. Math. Phys. 214(1), 191–200 (2000) 9. Hans Triebel: Theory of Function Spaces. Basel: Birkh¨auser Verlag, 1983 10. Misha Vishik: Hydrodynamics in Besov spaces. Arch. Ration. Mech. Anal. 145(3), 197–214 (1998) 11. Misha Vishik: Incompressible flows of an ideal fluid with vorticity in borderline spaces of Besov ´ type. Ann. Sci. Ecole Norm. Sup. (4), 32(6), 769–812 (1999) Communicated by P. Constantin

Commun. Math. Phys. 232, 327–375 (2003) Digital Object Identifier (DOI) 10.1007/s00220-002-0739-7

Communications in

Mathematical Physics

A Stochastic Model Associated with Enskog Equation and Its Kinetic Limit Fraydoun Rezakhanlou∗ Department of Mathematics, University of California at Berkeley, Berkeley, CA 94720–3840, USA Received: 29 January 2002 / Accepted: 30 July 2002 Published online: 14 November 2002 – © Springer-Verlag 2002

Abstract: We examine a system of particles in which the particles travel deterministically in between stochastic collisions. The collisions are elastic and occur with probability εd when two particles are at a distance σ . When the number of particles N goes to infinity and N εd goes to a nonzero constant, we show that the particle density converges to a solution of the Enskog Equation. 1. Introduction The Boltzmann Equation provides a successful description for dilute gases and can be derived from a Hamiltonian particle system in a suitable scaling limit. In the hard sphere model, one starts with N balls of radius ε that travel according to their velocities and collide elastically. In a Boltzmann-Grad limit, we send N → ∞, ε → 0 in such a way that Nεd−1 → 1. If f (x, v, t) denotes the density of particles of velocity v, then f satisfies the Boltzmann Equation

ft + v · fx = (n · (v − v∗ ))+ [f (x, v  )f (x, v∗ ) − f (x, v)f (x, v∗ )]dn dv∗ , Rd

S

(1.1)

where S denotes the unit sphere, dn denotes the d − 1 dimensional Hausdorff measure on S, and v  = v − (n · (v − v∗ ))n, v∗ = v∗ + (n · (v − v∗ ))n. The Boltzmann Equation ceases to be valid when the density of the gas increases. In 1922, Enskog proposed a modification of the Boltzmann Equation (1.1) that provided a very useful description of moderately dense gases. In the Enskog Equation, the ∗

Research supported in part by NSF Grant DMS-0072666.

328

F. Rezakhanlou

expression in the brackets in (1.1) is replaced with   σ d−1 f (x, v  )f (x − σ n, v∗ ) − f (x, v)f (x + σ n, v∗ ) .

(1.2)

See for example Resibois and De Leener [RD] for more information on the Enskog Equation. Even though (1.2) suggests a model in which a collision occurs when particles are at distance σ , it is not known whether the Enskog Equation can be derived from a Hamiltonian particle system. In spite of the original motivation behind the Enskog Equation, we only regard it in this article as a mollification of the Boltzmann Equation. Technically speaking, the Enskog Equation is as singular as the one dimensional Boltzmann Equation (see Sect. 3 for more details). Because of this, it seems plausible that some of the known techniques for one–dimensional particle systems with kinetic behavior can be utilized for particle systems associated with the Enskog Equation. In Pulvirenti [P2], a stochastic particle system is introduced as a microscopic model associated with the Enskog Equation. In this model we have N particles in Rd that travel freely in between the collision times. When two particles are at distance σ , then with probability εd they collide elastically, and with probability 1−ε d they go ahead with no change in their velocities. This model should be regarded as a suitable smoothing of the hard sphere model because the interaction length does not go zero as the number of particles N goes to infinity. In this article we show that as N → ∞ and ε → 0 with Nεd → Z, the macroscopic particle density f (x, v, t) satisfies an Enskog type equation. The rigorous derivation of the Boltzmann Equation for models with deterministic collision rules was carried out by Lanford [L] and King [K]. Lanford established the kinetic limit for short times for the hard sphere model. King in his thesis utilizes Lanford’s method to treat the case of Hamiltonian systems. Later Illner and Pulvirenti [IP, P1] showed that Lanford’s restriction on time can be replaced by an assumption on the smallness of the initial density. Caprino and Pulvirenti [CP] derived a discrete Boltzmann Equation for stochastic particle systems on a line with four velocities. Their derivation is valid globally in time with no smallness condition on the initial data. Their approach, as in [L] and [K], is based on a detailed analysis of the hierarchy equations for the correlation functions. In Rezakhanlou [R1] and Rezakhanlou-Tarver [RT], a different approach for the derivation of Boltzmann-type equations was proposed. This approach explores the entropy bound and some microscopic bound on the total number of collisions. The key idea behind the latter is that some well-established techniques of Tartar [T] and Bony [B] have indeed microscopic counterparts that can be exploited for our purposes. The Enskog Equation differs from the Boltzmann Equation because of the presence of the interaction length σ , making the Cauchy problem associated with the Enskog Equation much easier to handle. We refer to Arkeryd [Ar] and Arkeryd and Cercignani [ArC] and the references therein for the existence and uniqueness of the initial value problem. As we will see in Sect. 3, one can establish the existence of solutions using the dispersive behavior of the free motion part of the equation and Tartar’s ideas [T]. This will allow us to follow [RT] in studying the collision term by obtaining bounds on the expected value of the total number of collisions. The organization of the paper is as follows. In Sect. 2 the main result is stated. In Sect. 3 the proof of the main result is sketched. Section 4 is devoted to an entropy bound. In Sect. 5 we establish a bound on the total number of collisions. In Sect. 6, a variant of

A Stochastic Model Associated with Enskog Equation and its Kinetic Limit

329

the Stosszahlensatz (Boltzmann’s molecular chaos principle) will be established. Section 7 is devoted to an improvement of the bounds of Sect. 5. In the last section we give a proof of the kinetic limit. 2. Notation and Main Result This section is devoted to the statement of our main result. We first describe the model for which the kinetic limit will be established. In our model, we have N particles in the d-dimensional Euclidean space Rd . Define the state space E = (Rd × Rd )N ; q ∈ E is the N -tuple, q = (x, v) = (q1 , . . . , qN ) , x = (x1 , . . . .xN ) , v = (v1 , . . . , vN ) , where qi = (xi , vi ). The process q(t) is a Markov process with the infinitesimal generator Aε , where ε denotes the length scale and Aε = A0 + Ac .

(2.1)

We have, for any smooth function g : E → R, A0 g(q) =

N 

vi ·

i=1

Ac g(q) =

1 2



∂g (q) , ∂xi

V σ,ε (|xi − xj |)B(vi − vj , −nij )(g(Sij q) − g(q)) ,

(2.2)

(2.3)

i,j

where V σ,ε(z) = εd−1 Vˆ (ε −1 (z − σ )) with Vˆ : R → [0, ∞) a continuous even function satisfying R V (z)dz = 1; B : Rd × S → R is defined by B(v, n) = (v · n)+ with S x −x denoting the unit sphere; nij = |xii −xjj | ; Sij q is the configuration obtained from q by j

replacing vi and vj with vi and vji defined by j

vi = vi − (nij · (vi − vj ))nij ,

vji = vj − (nj i · (vj − vi ))nj i = vj + (nij · (vi − vj ))nij .

(2.4)

Given a positive number β, define the measure νβ by   Nd N  β 2 2 νβ (dq) = exp −β |xi | + |vi | dq . π i=1

Here and below we use the notations dx = dx1 . . . dxN , dv = dv1 . . . dvN , dq = dx dv . Let µε be a family of probability measures on E and f 0 : Rd × Rd → [0, ∞) be an integrable function. Notation 2.1. We will say that µε ∼ f 0 if the following conditions hold: (i) For every bounded continuous function J : Rd × Rd → R,

330

F. Rezakhanlou

lim

ε→0



N

d J (xi , vi ) −

ε i=1


Rd

Rd



J (x, v)f 0 (x, v)dx dv µε (dq) = 0 .

(2.5)

(ii) There exist a function F ε : E → [0, ∞) and constants p > 1 and b > 0, such that sup e

−bε−d




ε>0

µε (dq) = F ε (q)νβ (dq), (F ε (q))p νβ (dq) ≤ 1 .

Note that if we choose J ≡ 1 in (2.5), we deduce that

f 0 (x, v)dxdv . εd N → Z =:

(2.6)

Also, if we define Gε (q) =

N 1  0 f (xk , vk ) ZN k=1

µε (dq) =

satisfies µε ∼ f 0 if and only if with Z as in (2.6), then



(f 0 (x, v))p exp β(p − 1) |x|2 + |v|2 dxdv < ∞ . Rd

Gε (q)dq

Rd

We assume that q(0) is distributed according to µε and we denote the expectation associated with q(·) by Eε . We are now ready to state the main result. Theorem 2.2. Suppose µε ∼ f 0 . Then for every bounded continuous function J : Rd × Rd → R,





N



d

J (xk (t), vk (t)) − J (x, v)f (x, v, t)dx dv = 0, lim Eε ε



d d ε→0 R R k=1

where f (x, v, t) is the unique solution to  ft + v · fx = Q(f, f ) f (x, v, 0) = f 0 (x, v)

,

(2.7)

where Q = Q(f, f ) is defined by

  d−1 B(v − v∗ , n) f (x, v  )f (x −σ n, v∗ )−f (x, v)f (x + σ n, v∗ ) dn dv∗ . Q=σ Rd

S

A solution to (2.7) is understood in the following sense:  f log+ f dxdv < ∞, (i) f ∈ C([0, T ], L1 (Rd × Rd )) and supt∈[0,T ]  T  (ii) 0 (x + σ n, v∗ )dn dv dv∗ dx dt =: X(T ) < ∞, S B(v−v∗ , n)f (x, v)f t (iii) f (x, v, t) = f 0 (x − vt, v) + 0 Q(x − (t −s)v, v, s)ds for all t ∈ [0, T ] and almost all (x, v), (iv) limT →0 X(T ) = 0,

A Stochastic Model Associated with Enskog Equation and its Kinetic Limit

331

where Q(x, v, t) := Q(f, f )(x, v, t). It turns out that any such a solution satisfies

F (x, v, T )|v|r dxdv < ∞ (2.8) for every nonnegative integer r, where F (x, v, T ) denotes the essential supremum of f (x + vt, v, t) over the t-interval [0, T ]. The proof of this follows the arguments of [Ar]. For example, a slight modification of Lemma 4.2 of [Ar] can be used to establish (2.8) for small T . The proof of (2.8) for general T is achieved by a bootstrap. We can then appeal to Theorem 4.1 of [Ar] to assert that the above solution is in fact unique. Remark 2.3. Most of our arguments are valid for an arbitrary collision rate B that is merely Lipschitz continuous. It is only in Sect. 4 that the special form of B(v, n) = (v · n)+ is used to construct an invariant measure for the process q(t). 3. Sketch of Proofs A well-known proof of the existence of solutions to the Boltzmann-type equation in dimension one is due to Tartar [T]. Even though we are dealing with the Enskog Equation in arbitrary dimension, technically speaking, the Enskog Equation is as singular as the one-dimensional Boltzmann Equation. The reason is that in the collision term of the Enskog Equation we have something like
f (x, v, t)f (y, v∗ , t)γ (dx, dy), (3.1) where y belongs to a sphere of diameter σ about the point x. Hence the support of the measure γ in (3.1) is a set of codimension one. In the Boltzmann Equation the measure γ (dx, dy) is the Dirac measure δ(x − y). Hence the support is of codimension d. As a result the codimension of the support of γ in the case of the Enskog Equation and one-dimensional Boltzmann Equation are equal and this is responsible for the fact that a Tartar-type argument can be used to establish bounds on the collision term in both cases. To explain our method of proof in this paper, we start with a bound on the collision term. Suppose f is a solution to (2.7). A straightforward calculation yields that the expression



d |x − vt|2 f (x, v, t)dxdv = |x − vt|2 Q(x, v, t)dxdv , dt equals to




1 d−1 B(v − v∗ , n)f (x, v, t)f (x + σ n, v∗ , t) σ 2 S

 · |x − v  t|2 + |x + σ n − v∗ t|2 − |x − vt|2 − |x + σ n − v∗ t|2 dndxdv


d = −tσ B(v − v∗ , n)2 f (x, v, t)f (x + σ n, v∗ , t)dn dxdv . S

Since the right–hand side is nonpositive, we deduce



sup |x − vt|2 f (x, v, t)dxdv ≤ |x|2 f 0 (x, v)dxdv . t

332

F. Rezakhanlou

This and the conservation of the energy

d |v|2 f (x, v, t)dxdv = 0 dt imply that



 2|x|2 + (1 + 2t 2 )|v|2 f 0 (x, v)dxdv . (|x|2 + |v|2 )f (x, v, t)dxdv ≤ (3.2) Again, it is straightforward to show

d x · v f (x, v, t)dxdv dt


σd = B(v − v∗ , n)2 f (x, v, t)f (x + σ n, v∗ , t)dndxdv 2 S

+ |v|2 f (x, v, t)dxdv . From this and (3.2) we deduce that
T


B(v − v∗ , n)2 f (x, v, t)f (x + σ n, v∗ , t)dn dxdv S 0

≤ c0 (|x|2 + |v|2 )f 0 (x, v)dxdv ,

(3.3)

for a constant c0 that depends on T only. From now on we assume that the right–hand side of (3.3) is finite. Put B1 (v, n) = B(v, n)?(B(v, n) ≤ 1) and define
T



X1 (T ) = B1 (v−v∗ , n)f (x, v, t)f (x + σ n, v∗ , t)dn dv dv∗ dx dt . 0

S

From (3.3) we learn X∞ (T ) − X1 (T ) ≤

c1 1

(3.4)

for a constant c1 that depends on T only. As a result, a bound on X1 for any finite 1 implies a bound on X∞ . We certainly have f (x, v, t)f (x + σ n, v∗ , t) = f 0 (x − vt, v)f 0 (x + σ n − v∗ t, v∗ )
t d  f (x − v(t −s), v, s) + ds 0  × f (x + σ n − v∗ (t −s), v∗ , s) ds = f 0 (x − vt, v)f 0 (x + σ n − v∗ t, v∗ )
t + Q(x − v(t −s), v, s)f (x +σ n−v∗ (t −s), v∗ , s)ds
0 t + f (x − v(t −s), v, s)Q(x +σ n−v∗ (t −s), v∗ , s)ds. 0

A Stochastic Model Associated with Enskog Equation and its Kinetic Limit

333

Hence X1 = 21 + 22 + 23 , where 2i is obtained by multiplying the i th term on the right-hand side of (3.2) by B1 (v−v∗ , n) and integrating with respect to n, v, v∗ , x and t. We start with the term 22 . After a translation we obtain that 22 equals to


T





0

Q(x, v, s) 




×

T

S s

 B1 (v−v∗ , n)f (x + σ n + (v−v∗ )(t −s), v∗ , s)dtdndv∗ dvdxds .

A straightforward calculation yields dz = σ |w · n| dn dt for z = x + σ n + (t − s)w. As a result 22 is bounded above by 




1 T + Q (x, v, s) f (z, v∗ , s)dz dv∗ dvdxds, σ 0 A where +

Q (x, v, t) = σ

d−1





Rd

S

(3.5)

B(v − v∗ , n)f (x, v  , t)f (x − σ n, v∗ , t)dn dv∗ ,

and A = {z : z = x + (v − v∗ )τ + σ n for some (n, τ ) ∈ S × [0, T ] with B(v − v∗ , n) ∈ [0, 1]} . Note that the volume of A can be estimated as
T

|A| ≤ B1 (v − v∗ , n)dndt ≤ 1T dn . 0

S

S

As a result, 22 ≤ σ d−2 γ X∞ (T ),

(3.6)



γ (v, v∗ ) dv∗ and  

f (z, v∗ , s)dzdv∗ : s ∈ [0, T ], |A| ≤ c2 1T , γ = sup

where γ = supv

A

with c2 the d − 1 dimensional measure of the sphere S. The term 23 can be treated likewise. Moreover, it is not hard to show 

2 0 21 ≤ c3 f (x, v)dx dv ,

334

F. Rezakhanlou

for some constant c3 . From this, (3.4) and (3.6) we deduce 

2 0 X1 (T ) ≤ c3 f (x, v)dx dv + 2σ d−2 γ X∞ (T ) ≤ c4 + c4 γ X1 (T ) . A bound on the entropy




H (T ) = sup



 f log+ f (x, v, t)dx dv < ∞ ,

(3.7)

(3.8)

0≤t≤T

 2  2 0 the conservation of the kinetic energy |v| f (x, v, t)dxdv = |v| f (x, v)dxdv, and



f (z, v∗ , s)dzdv∗ ≤ f (z, v∗ , s)?(|v∗ | ≤ 1 )dzdv∗ A A

|v∗ |2 f (z, v∗ , s)dzdv∗ , + (1 )−2 imply that there exists a positive constant c5 such that γ ≤ c5 [log((1 )d 1T )]−1 + c5 (1 )−2 ,

(3.9)

for every positive 1 . Hence, if T is sufficiently small, then we can choose an appropriate 1 so that c4 γ < 1. This in turn implies a bound of the form X1 (T ) ≤

c4 . 1 − c4 γ

(3.10)

In Sect. 4 we establish a microscopic analog of the entropy bound (3.8) and its consequence (3.9). The bound (3.9) will be used in Sect. 5 to establish a microscopic analog of the collision bound (3.10). For our main result Theorem 2.2, we need to show that the number of collisions near a point (x, v, t) converges to the quadratic functional Q as N diverges to infinity. Formally, this is the Stosszahlensatz of Boltzmann. This principle asserts that the particles before a collision are almost (stochastically) independent. We establish a variant of this principle in√Sect. 6 that macroscopically has the following flavor: If √ −d ζ (z/ δ, w/ δ) for a nonnegative smooth function ζ of compact supδ ζ δ (z, w) =  port with ζ (z, w)dz dw = 1, and if J is a bounded continuous function of compact support, then
T

lim (Q1 (f ∗ ζ δ , f ∗ ζ δ ) − Q1 (f, f ))J dx dv dt = 0, (3.11) δ→0 0

where Q1 denotes the collision term when B is replaced with B1 . Recall that by (3.4), the replacement of Q with Q1 causes an error of order O(1/1). Note that (3.11) would follow from a stronger statement
T

(Q1 (τz,w f, f ) − Q1 (f, f ))J dx dv dt = 0, lim sup (3.12) δ→0 |z|,|w|<δ 0

where (τz,w f )(x, v, t) = f (x + z, v + w, t) .

A Stochastic Model Associated with Enskog Equation and its Kinetic Limit

335

The limit (3.12) can be established by starting from (3.2) but now (x, v) in f (x, v, t) and f 0 (x − vt, v) is replaced with (x + z, v + w). We then multiply both sides by J , integrate, compare with the case z = w = 0, and repeat our previous arguments. The corresponding set A in (3.5) now satisfies |A| ≤ c6 | log δ|−1 . Again the entropy inequality can be used to conclude
T

(Q1 (τz,w f, f ) − Q1 (f, f ))J dx dv dt ≤ c7 | log δ|−1 .

(3.13)

0

A microscopic analog of (3.13) will be established in Sect. 6 and this will allow us to replace the microscopic collision term with Q(f ∗ ζ δ , f ∗ ζ δ ) + o(1), where f represents the microscopic density. Here o(1) represents a random term that goes to zero as ε and δ go to zero. From this we deduce that the macroscopic density satisfies ft + v · fx = Q(f ∗ ζ δ , f ∗ ζ δ ) + o(1)

(3.14)

(after sending ε → 0) where o(1) now represents an error term that goes to zero as δ → 0. To complete the proof, we now need to establish some kind of uniform integrability of the collision term so that we can replace back f ∗ ζ δ with f . This will be carried out in two steps. The first step is carried out before sending ε → 0, and is only a uniform integrability in the space variable. More precisely, we establish a microscopic version of the bound 


T + ; Q (f, f )(x + vt, v, t)dt dx dv < ∞, (3.15) 0

where ;(z) = z log+ log+ z and

Q+ (f, f ) = σ d−1 B(v − v∗ , n)f (x, v  )f (x − σ n, v∗ ) dn dv∗ .

(3.16)

It is more convenient to establish (3.15) for Q− = Q+ − Q; 


T ; Q− (f, f )(x + vt, v, t)dt dx dv < ∞.

(3.17)

Rd

S

0

In fact (3.8) and (3.17) imply (3.15) because (2.7) implies
T Q+ (f, f )(x + vt, v, t)dt = f (x + vT , v, T ) − f 0 (x, v) 0


+

T

Q− (f, f )(x + vt, v, t)dt .

(3.18)

0

A microscopic analog of (3.17) will be established in Sect. 7. This will be used in Sect. 8 to show that if f is the macroscopic density, then

336

F. Rezakhanlou


T

sup

δ>0 0

;(Q+ (f ∗ ζ δ , f ∗ ζ δ ))dx dv dt < ∞ .

This will be used to replace f ∗ ζ δ with f in (3.14). As we will see in Sect. 7, (3.17) follows if we can show that for some constant c8 ,


U (x, v)?(U (x, v) > 1)dx dv ≤ c8 (log log 1)−1 ,

(3.19)

where U (x, v) is the argument of ; in (3.17). Put A = {(x, v) : U (x, v) > 1}. After a translation we can rewrite (3.17) as


T





S

0

B(v − v∗ , n)f (x, v)f (x + σ n, v∗ )? ((x − vt, v) ∈ A) dndxdv dv∗ dt

≤ c8 σ d−1 (log log 1)−1 .

(3.20)

It seems plausable that we can establish (3.18) using the same approach we utilized for (3.10) and (3.13). Instead we prefer to appeal to a new trick that has the same flavor as the method used in Sect. 6 of [R2]. Define Y (t) to be





f (x, v, t)f (y, v∗ , t)?(|x − y| ≤ σ )[(x − y) · (v∗ − v)]+

×H (x, y, t, v, v∗ )dx dy dv dv∗ for a suitable nonnegative function H that will be determined later. We have that dY = 21 + 22 + 23 + 24 , dt where



21 = −



 v · fx (x, v, t)f (y, v∗ , t) + v∗ · fy (y, v∗ , t)f (x, v, t)

?(|x − y| ≤ σ )[(x − y) · (v∗ − v)]+ H (x, y, t, v, v∗ )dx dy dv dv∗ ,



22 = f (x, v, t)Q(y, v∗ , t)?(|x − y| ≤ σ )[(x − y) · (v∗ − v)]+ ×H (x, y, t, v, v∗ )dx dy dv dv∗ ,



23 = Q(x, v, t)f (y, v∗ , t)?(|x − y| ≤ σ )[(x − y) · (v∗ − v)]+ × H (x, y, t, v, v∗ )dx dy dv dv∗ ,



24 = f (x, v, t)f (y, v∗ , t)?(|x − y| ≤ σ )[(x − y) · (v∗ − v)]+ × Ht (x, y, t, v, v∗ )dx dy dv dv∗ .

A Stochastic Model Associated with Enskog Equation and its Kinetic Limit

337

After an integration by parts,



|v − v∗ |2 ? ((x − y) · (v∗ − v) > 0) f (x, v, t)f (y, v∗ , t) 21 + 2 4 = − ?(|x − y| ≤ σ )H (x, y, t, v, v∗ )dx dy dv dv∗



 2 f (x, v, t)f (x + σ n, v∗ , t) ((v − v∗ ) · n)+ +σ d S

H (x, x − σ n, t, v, v∗ )dx dn dv dv∗



+ f (x, v, t)f (y, v∗ , t)(Ht + v · Hx + v∗ · Hy )(x, y, t, v, v∗ ) ?(|x − y| ≤ σ ) [(x − y) · (v∗ − v)]+ dx dy dv dv∗ =: 211 + 212 + 213 . In view of (3.20) we would like to choose H (x, y, t, v, v∗ ) = ? ((x − vt, v) ∈ A) .

(3.21)

Ht + v · Hx + v∗ · Hy = 0

(3.22)

Note that H satisfies weakly, and as a result 213 = 0. In order to obtain
T



 2 ((v − v∗ ) · n)+ f (x, v)f (x + σ n, v∗ ) S

0

? ((x − vt, v) ∈ A) dndxdv dv∗ dt ≤ c9 (log 1)−1 ,

it suffices to verify





T





22 dt

+

Y (T ) +

0

T 0






23 dt

+

T

211

0



dt

≤ c10 (log |A|)−1

(3.23)

(3.24)

because by Chebyshev’s inequality,




1 T B(v − v∗ , n)f (x, v, t)f (x + σ n, v∗ , t)dn dx dv dv∗ dt . |A| ≤ 1 0 S In fact one can establish (3.24) in just the same way we obtained (3.9). To complete the proof of (3.15), we need to explain why (3.23) implies (3.20). ˆ1+2 ˆ 2 , where Observe that the left-hand side of (3.20) is bounded above by 2
T



ˆ1 = B(v − v∗ , n) ? (|(v − v∗ ) · n| ≤ δ) f (x, v)f (x + σ n, v∗ ) 2 S

0

ˆ2 = 2




T 0





S

? ((x − vt, v) ∈ A) dndxdv dv∗ dt, B(v − v∗ , n) ? (|(v − v∗ ) · n| ≥ δ) f (x, v)f (x + σ n, v∗ ) ? ((x − vt, v) ∈ A) dndxdv dv∗ dt .

We can show ˆ 1 ≤ c11 | log δ|−1 , 2

(3.25)

338

F. Rezakhanlou

in just the same way we obtained (3.9). On the other hand, (3.23) implies ˆ 2 ≤ c12 (δ log 1)−1 . 2

(3.26)

This and (3.25) imply (3.20) if we choose δ = (log 1)−1/2 . 4. Entropy In this section we establish an exponential bound on an entropy-like functional of the microscopic particle density. This estimate will play the same role as the bound (3.8) played in the previous section, and will prove to be the key to demonstrating the boundedness of the number of collisions. Before discussing our entropy bound and some of its consequences, we need to make some definitions. We partition Rd into sets of the form d 

[ar , br ) ,

r=1

each of which is of side length δ. This partition will be denoted by J δ . We also partition the set Rd × Rd into sets of the form d 

[ar , br ) ×

r=1

d 

[ar , br ),

r=1

each of which is of side length at most δ. This partition will be denoted by Jˆ δ . We then define N (q; K) = N (x1 , v1 , . . . , xN , vN ; K) =

N 

?((xi , vi ) ∈ K)

(4.1)

i=1

for every set K ⊆ Rd × Rd and define  @ε (q) = ϕ(N (q; I × Rd )) + ϕ(N (q; Rd × I )) , I ∈J ε

ˆ ε (q) = @



I ∈Jˆ

√

ϕ(N (q; I )) ,

(4.2)

ε

where ϕ(z) = z log z. Recall the constant p that appeared in Notation 2.1. The main result of this section is Theorem 4.1. Theorem 4.1. There exists a constant C0 (T ) such that   p−1 ε Eε sup exp @ (q(s)) ≤ exp(C0 (T )ε −d ) , 2p 0≤s≤T   p−1 ε ˆ (q(s)) ≤ exp(C0 (T )ε −d ) . Eε sup exp @ 2p 0≤s≤T

(4.3)

As the first step, we drop the supremum in (4.3) and replace the expectation with the integration with respect to the measure νβ .

A Stochastic Model Associated with Enskog Equation and its Kinetic Limit

339

Lemma 4.2. There exists a constant c0 such that



exp[@ε (q)]νβ (dq) ≤ exp(c0 ε −d ) ,

(4.4)

ˆ ε (q)]νβ (dq) ≤ exp(c0 ε −d ) . exp[@

(4.5)

Proof. We only prove (4.5) because√the proof of (4.4) is identical. Let us write I1 , I2 , . . . ε and write a for N (q; I ). Given a collection of for the elements of the partition Jˆ  r r nonnegative integers k1 , k2 , . . . with r kr = N , put nj = k1 + . . . + kj . Note that only finitely many kr ’s are nonzero. Given i ∈ {1, . . . , N}, we can find a unique r = r(i) such that nr < i ≤ nr+1 . Given r, put 1(r) = min{|q| : q ∈ Ir }. For each nonnegative integer n, define Jn = {r : n ≤ 1(r) < n + 1},

Nn =



kr .

r∈Jn

We also write γn for the cardinality of the set Jn . We have that νβ (a1 = k1 , a2 = k2 , . . . ) is equal to N! ∞

r=1 kr !

νβ (q1 , . . . , qn1 ∈ I1 , qn1 +1 , . . . , qn2 ∈ I2 , . . . )

N! = ∞

r=1 kr !

N! = ∞

νβ (qi ∈ Ir(i) for i = 1, . . . , N ) N 

r=1 kr ! i=1

νβ (qi ∈ Ir(i) )

  dN N  β d N 2 ≤ ∞ (ε ) exp −β 1(r(i)) r=1 kr ! π i=1   dN ∞  β N! d N 2 ≤ ∞ (ε ) exp −β n Nn , r=1 kr ! π N!

(4.6)

n=0

because  d  d
 d 

β β β exp −β|q|2 dq ≤ |Ir | exp(−β1(r)2 ) = ε d exp(−β1(r)2 ) . π π Ir π As a consequence of Stirling’s theorem, we have ec1 k ≤

1 k log k e ≤ ec2 k k!

(4.7)

for some positive constants c1 and c2 . Also, ε d ≤ c3 /N for some constant c3 . From this and (4.6) we learn

340




F. Rezakhanlou

ˆ ε (q))νβ (dq) exp(@ =


 ∞

ear log ar νβ (dq)

r=1

=

∞ 

 

ekr log kr νβ (a1 = k1 , a2 = k2 , . . . )

kr =N r=1

  dN ∞  β dN 2 ∞ ≤ e ε exp −β n Nn π kn !  r=1 r=1 n=1 kr =N   dN  ∞  β ≤ c3N e−c1 N ec2 N exp −β n2 Nn π  n=1 kr =N  ∞   = ec4 N exp −β n2 Nn 

N!



= ec4 N =e

kr =N

 





kr log kr

n=1

exp −β

Nn =N



c4 N

∞ 

 exp −β

∞ 

n2 Nn

n=1 ∞ 

n Nn 2

n=1

Nn =N



 ?



 kr = Nn for n = 0, 1, . . . 

r∈Jn

 ∞   γn + N n − 1 . Nn

n=0

It is not hard to see that γn ≤ c5 n2d−1 N for some constant c5 . Also N    (c5 n2d−1 + 1)N n γn (γn + 1) . . . (γn + Nn − 1) γn + Nn − 1 = ≤ . Nn Nn ! Nn ! As a result, ∞  ∞   Nn log n) γn + N n − 1 N exp(c6 ∞n=1 . ≤N Nn n=0 Nn !

n=0

From this and (4.7) we deduce  ∞  ∞     γn + N n − 1 c2 N ≤e exp c6 Nn log n + N log N − Nn log Nn . Nn

n=0

n=1

Let us write α for



nn

−2 .

n

Recall that ϕ(z) = z log z. We certainly have

  N 1 1 2 ϕ(N ) = αϕ Nn n + N log α + N log α = αϕ α α n n2  1   ≤ ϕ(Nn n2 ) + N log α = Nn log Nn + 2 Nn log n + N log α . 2 n n n 

A Stochastic Model Associated with Enskog Equation and its Kinetic Limit

Hence

341

 ∞  ∞    γn + N n − 1 c7 N ≤e exp c7 Nn log n . Nn

n=0

n=1

As a result
ˆ ε (q))ν β (dq) ≤ ec8 N exp(@ ≤e

 



exp −β

Nn =N



c9 N



Nn =N



β exp − 2

∞ 

n Nn + c 7

n=1 ∞ 

2

n Nn

Nn log n

n=1

2

∞ 

.

n=1

¯ Let  A1 denote the set of N = (N0 , N2 , . . . ) such that each Nn is a nonnegative integer, N = N , N > 0, and Nn = 0 for every n > 1. Using (4.7), it is not hard to show 1 n n that   1+N −1 ≤ ec10 (N+1) , N for some constant c10 . Hence


∞ β 2 ˆ (q))ν (dq) ≤ e exp − n Nn exp(@ 2 n=1 ¯ 1=1 N∈A 1   ∞   β exp − 12 ≤ ec11 N 2 ¯ 1=1 N∈A 1     ∞  β 2 1+N −1 c11 N exp − 1 ≤e N 2 1=1   ∞  β ≤ ec11 N ec10 (N+1) exp − 12 ≤ ec12 N , 2 ε

β

c11 N

∞  



1=1

for some constant c12 and this completes the proof.

 

Recall that initially the configuration q is distributed according to the measure µε (dq) = F ε (q)νβ (dq) = Gε (q)dq ,  where Gε (q) = F ε (q) exp(−βK(q)) with K(q) = i |xi |2 + |vi |2 . Since dq is not an invariant measure for the process q(t), let us first find an invariant measure to replace dq. To this end, choose a positive constant c0 such that the support of Vˆ is contained in the interval (−c0 , c0 ). We take Wˆ : R → R to be a function such that Wˆ  = −Vˆ , Wˆ (z) = 0 for z > εc0 and Wˆ (z) = 1 for z < −εc0 . Define W σ,ε (z) = εd Wˆ (ε −1 (z − σ )), and R(q) =

1  σ,ε W (|xi − xj |) , 2 i,j

m(dq) = exp(−R(q))dq .

342

F. Rezakhanlou

Lemma 4.3. The adjoint of the operator Aε with respect to the measure m(dq) is −A0 + Aˆ c , where  Aˆ c g(q) = 21 V σ,ε (|xi − xj |)B(vi − vj , nij )(g(Sij q) − g(q)) . (4.8) i,j

Moreover, the measure m is invariant. Proof. Let η1 (q) and η2 (q) be two continuously differentiable functions of compact support. We have

 1 A0 η1 η2 dm = − V σ,ε (|xi − xj |) (vi − vj ) · ni,j m(dq) η1 (q)η2 (q) 2 i,j
− η1 A0 η2 dm . In other words, A∗0 = −A0 −

1  σ,ε V (|xi − xj |) (vi − vj ) · ni,j . 2

(4.9)

i,j

On the other hand, since R is independent of v and the Jacobian of the transformation (v, v∗ ) → (v  , v∗ ) is equal to one, we have
Ac η1 η2 dm
1  σ,ε = V (|xi − xj |)B(vi − vj , −nij )η1 (Sij q))η2 (q)m(dq) 2 i,j
1  σ,ε − V (|xi − xj |)B(vi − vj , −nij )η1 (q)η2 (q)m(dq) 2 i,j
1  σ,ε j = V (|xi − xj |)B(vi − vji , nij )η1 (Sij q)η2 (q)m(dq) 2
i,j 1  σ,ε − V (|xi − xj |)B(vi − vj , −nij )η1 (q)η2 (q)m(dq) 2 i,j
 1  σ,ε = V (|xi − xj |)η1 (q) B(vi − vj , nij )η2 (Sij q) 2 i,j  − B(v − v , −n )η (q) m(dq) i j ij 2

 1 = η1 Aˆ c η2 dm + η1 (q)η2 (q) V σ,ε (|xi − xj |) (vi − vj ) · nij m(dq) . 2 i,j

This and (4.9) imply (4.8).  An immediate consequence of (4.8) is Aε ηdm = 0 for every continuously differentiable function of compact support η. This implies the invariance of m.   As our next step, we ˆ = exp(−βK(q))dm(q) for the initial distribu choose m(dq) tion where K(q) = i |xi |2 + |vi |2 . Let us write m(t, ˆ dq) = Z(t, q)m(dq) for the  distribution of q(·) at time t. Define K(t, q) = i |xi − tvi |2 + |vi |2 .

A Stochastic Model Associated with Enskog Equation and its Kinetic Limit

343

Lemma 4.4. We have K(t, q(t)) ≤ K(0, q(0)) and Z(t, q) ≤ exp(−βK(t, q)) for every t. Proof. The proof of K(t, q(t)) ≤ K(0, q(0)) follows from two straightforward facts: K(t, q(t)) does not change between the collision times and K(t, q(t)) decreases each time a collision occurs. The former is obvious and the latter follows from j

|xi − vi t|2 + |xj − vji t|2 − |xi − vi t|2 − |xj − vj t|2 j

= 2t (xi · vi + xj · vj − xi · vi − xj · vji ) j

= 2t (xi − xj ) · (vi − vi ) = 2t (vi − vj ) · (xi − xj ) ≤ 0 , q

whenever B(vi − vj , −nij ) = 0. Let us write Eε for the expectation when the process q(t) is q at t = 0. If η is a nonnegative test function, then

η(q)Z(t, q) m(dq) = Eεq η(q(t)) exp(−βK(q)) m(dq)
≤ Eεq η(q(t)) exp(−βK(t, q(t))) m(dq)
= η(q) exp(−βK(t, q)) m(dq) , where for the last equality we used the fact that m is an invariant measure. This evidently completes the proof.   We now write µε (dq) = H0 (q)m(dq) , where H0 (q) = F ε (q) exp(R(q) − βK(q)). This implies that the configuration q(t) is distributed according to the measure µε (t, dq) = H (t, q)m(dq) , where H satisfies H (0, q) = H0 (q) and the backward equation ∂H = Aε,∗ H , ∂t where Aε,∗ = −A0 + Aˆ c is the adjoint of the operator Aε with respect to the measure m. Lemma 4.5. There exists a constant c such that 
 H (t, q) p −d Z(t, q) m(dq) ≤ ecε , Z(t, q) for every t.

344

F. Rezakhanlou

 p Proof. Define ψ(a, b) = ab b. It is not hard to show that ψ is a convex function. We now show that the convexity of ψ implies
d ψ(H (t, q), Z(t, q)) m(dq) ≤ 0 . (4.10) dt To see this, first observe that the left–hand side of (4.10) equals to
  ψa (H, Z)Aε,∗ H + ψb (H, Z)Aε,∗ Z m(dq) , where ψa and ψb are the partial derivatives of ψ. To ease the notation, let us write B for Aε,∗ . Note that B is an infinitesimal generator for a Markov process. We now claim that the Maximum Principle and the convexity of ψ imply Bψ(H, Z) ≥ ψa (H, Z)BH + ψb (H, Z)BZ .

(4.11)

To show this, define ¯ = ψ(H (t, q), ¯ Z(t, q)) ¯ − ψ(H (t, q), Z(t, q)) k(q) ¯ − H (t, q)) − ψa (H (t, q), Z(t, q))(H (t, q) ¯ − Z(t, q)) . − ψb (H (t, q), Z(t, q))(Z(t, q) This function attains its minimum value at q. By Maximum Principle, (Bk)(q) ≥ 0.  This and Bψ(H (t, ·), Z(t, ·)) dm = 0 imply (4.11). This in turn implies (4.10). As a consequence,

ψ(H (t, q), Z(t, q)) m(dq) ≤ ψ(H (0, q), Z(0, q)) m(dq)
 ε p −d = F (q) exp(R(q)) exp(−R(q)) νβ (dq) ≤ ecε , because of our condition on the initial distribution and the fact that R(q) = O(ε −d ).   We are now ready to prove Theorem 4.1. Proof of Theorem 4.1. We only prove the first bound in (4.3) because the proof of the second bound is identical. By H¨older inequality and Lemma 4.5,  
  p−1 ε p−1 ε Eε exp @ (q(t)) ≤ exp @ (q(t)) µε (t, dq) p  p 
p−1 ε H (t, q) = exp @ (q(t)) Z(t, q)m(dq) p Z(t, q)  p−1 
p   ε −d ecε /p . ≤ exp @ (q(t)) Z(t, q)m(dq) From this, Lemma 4.4 and Lemma 4.2 we deduce   p−1 ε −d Eε exp @ (q(t)) ≤ ec1 ε p

(4.12)

for a constant c1 . We can now follow the proof of Theorem 2.1 of [RT] to deduce (4.3) from (4.12). (See Steps 2 and 3 of that proof.)  

A Stochastic Model Associated with Enskog Equation and its Kinetic Limit

345

Corollary 4.6 is the main consequence of Theorem 4.1 that is used in this paper. The straightforward proof of this corollary is omitted. Corollary 4.6. There exists a constant C1 (T , r) such that r Eε sup ε d @ε (q(s)) ≤ C1 (T , r) , 0≤s≤T r ˆ ε (q(s)) ≤ C1 (T , r) . Eε sup ε d @ 0≤s≤T

We end this section with four lemmas that will prepare us to use Corollary 4.4 in the proceeding sections. Let  |1 + log δ|−1 if δ < 1 , h(δ) = 1 otherwise . We also fix a continuous function η : Rd × Rd → [0, ∞) of compact support. Given a measurable function ρ, we set

ε ρ (x, v) = ηε (x − z, v − w)ρ(z, w)dzdw , where

ηε (x, v)

=

ε−d η



Rd

Rd

√ √  x/ ε, v/ ε .

Lemma 4.7. There exists a constant C1 (η) such that for every nonnegative ρ, ε

d

N 

ˆ ε (q)) , ρ ε (xi , vi ) ≤ C1 (η)ρL∞ h(ρL1 )(1 + ε d @

i=1

where  · 

Lp

denotes the Lp norm with respect to the set Rd × Rd .

The proof of Lemma 4.7 follows Lemma 5.2 of [RT] and is omitted. Given a measurable set K ⊆ Rd × Rd , we choose ρ(x, v) = ? ((x, v) ∈ Bε K), where √ Bε K = K + ε[0, 1]2d .   We also choose η(x) = ? (x, v) ∈ [0, 1]2d . It is not hard to see that for such a pair (ρ, η), we have ρ ε (x, v) ≥ ? ((x, v) ∈ K) . This and Lemma 4.7 yields Lemma 4.8. There exists a constant Cˆ 1 such that for every measurable set K, ˆ ε (q)) . εd N (q; K) ≤ Cˆ 1 h(|Bε K|)(1 + ε d @ Similarly we define Bˆ ε K = K + ε[0, 1]d , for every set K ⊆ Rd . In the same way, we can show Lemma 4.9. There exists a constant C˜ 1 such that for every measurable set K ⊆ Rd , 

εd N q; K × Rd ≤ C˜ 1 h(|Bˆ ε K|)(1 + ε d @ε (q)),

 εd N q; Rd × K ≤ C˜ 1 h(|Bˆ ε K|)(1 + ε d @ε (q)) .

346

F. Rezakhanlou

5. Collision Bound In this section we establish a microscopic analog of (3.10). Define  V σ,ε (|xi − xj + z|)B1 (vi − vj , −nˆ ij (z)) , Aε1 (q; z) = εd i,j

where B1 = B?(B ≤ 1) and

xi − xj + z . |xi − xj + z|

nˆ ij (z) =

We write Aε1 (q) for Aε1 (q; z) when z = 0. We also write Aε (q; z) (respectively Aε (q)) for Aε1 (q; z) when 1 = ∞ (respectively z = 0 and 1 = ∞). Theorem 5.1. There exists a constant C2 (T ) such that
T Eε Aε (q(t))dt ≤ C2 (T ) .

(5.1)

0

As in Sect. 3, we first find a bound on the number of collisions when the collision rate is not small. Lemma 5.2. There exists a constant C¯ 2 (T ) such that
T  εd V σ,ε (|xi (s) − xj (s)|)B(vi (s) − vj (s), −nij (s))2 ds ≤ C¯ 2 (T ) 0

i,j


Eε

T 0

(Aε (q(t)) − Aε1 (q(t)))dt ≤

C¯ 2 (T ) . 1

Proof. By Lemma 4.4 and the conservation of the kinetic energy,     |xi (t) − vi (t)t|2 ≤ |xi (0)|2 , |vi (t)|2 = |vi (0)|2 . i

i

i

i

From this and our condition on the initial distribution µε we deduce









|vi (t)|2 + sup ε d (xi (t) · vi (t)) ≤ c0 Eε sup



0≤t≤T 0≤t≤T i

(5.2)

i

for a constant c0 that  depends on T only. Set F (q) = εd i (xi · vi ). By the semigroup property,
T Eε F (q(T )) = Eε F (q(0)) + Aε F (q(s))ds . 0

Using this and j

j

xi · vi + xj · vji − xi · vi − xj · vj = (xi − xj ) · (vi − vi ) = (xi − xj ) · (vi − vj ) , we deduce




Eε F (q(T )) = Eε F (q(0)) +

εd 0




|vi (s)|2 ds +

i

T 0

εd



V σ,ε (|xi (s)

i,j

− xj (s)|)B(vi (s) − vj (s), −nij (s)) |xi (s) − xj (s)| ds . 2

This, (5.2), and the trivial inequality B − B1 ≤ B 2 /1 complete the proof of the lemma.  

A Stochastic Model Associated with Enskog Equation and its Kinetic Limit

347

q

Let us write Eε for the expectation when the process q(·) satisfies q(0) = q. To prepare for the proof of Theorem 5.1, we start with an elementary identity that is a consequence of the semigroup property. For any continuous function F : E × R → R,
Eεq F (q(t))

= F (St q) +

t 0

Eεq Ac (F oSt−s )(q(s))ds ,

(5.3)

where for q = (x1 , v1 , . . . , xN , vN ) = (x, v), Sθ q = (x1 + v1 θ, v1 , . . . , xN + vN θ, vN ) = (x + vθ, v), F oSθ (q) = F (Sθ q) . To show (5.3), fix t and define G(q, s) = F (St−s q). By the semigroup property,
Eεq G(q(t), t) = G(q, 0) +

t 0

 Eεq

 ∂ + Aε G(q(s), s)ds . ∂s

∂  This implies (5.3) because ∂s + Aε G = Ac F . For Theorem 5.1 and Theorem 6.1 of the next section, we apply (5.2) with F (q) = F ε (q; J, z, 1) = ε d



V σ,ε (|xi − xj + z|)B1 (vi − vj , −nˆ ij (z))J (xi , vi ) .

i,j j

Let us write vi (s) and vji (s) for the outgoing velocities where the ingoing velocities are vi (s) and vj (s). Define M(a) = a/|a| and nij (s, θ ; z) = M(xi (s) − xj (s) + (vi (s) − vj (s))θ + z) , j

nij (s, θ ; z) = M(xi (s) − xj (s) + (vi (s) − vji (s))θ + z) , nkij (s, θ ; z) = M(xi (s) − xj (s) + (vik (s) − vj (s))θ + z) ,

(5.4)

n˜ kij (s, θ ; z) = M(xi (s) − xj (s) + (vi (s) − vjk (s))θ + z) . We simply write nij (s, θ ), nij (s, θ ), nkij (s, θ ), and n˜ ij (s, θ ) if z is zero in (5.4). We also write nij (s) for nij (s, 0). We have,
Eεq

T

F (q(t); J, z)dt = 21 (z, J ) + 22 (z, J ) + 23 (z, J ) + 24 (z, J ) ,

(5.5)

0

where


T

21 (z, J ) = 0

εd



V σ,ε (|xi − xj + (vi − vj )t + z|)B1 (vi − vj , −nij (0, t; z))

i,j

·J (xi + vi t, vi ) dt ,

(5.6)

348

F. Rezakhanlou

22 (z, J ) =



1 q T t  σ,ε V (|xi (s) − xk (s)|)B(vi (s) − vk (s), −nik (s)) Eε 2 0 0 i,j,k

· V σ,ε (|xi (s) − xj (s) + (vik (s) − vj (s))(t − s) + z|) × B1 (vik (s) − vj (s), −nkij (s, t − s; z))J (xi (s) + vik (s)(t − s), vik (s))

− V σ,ε (|xi (s) − xj (s) + (vi (s) − vj (s))(t − s) + z|)

 × B1 (vi (s) − vj (s), −nij (s, t − s; z))J (xi (s) + vi (s)(t − s), vi (s)) ds dt , (5.7)

23 (z, J ) =



1 q T t d  σ,ε ε V (|xj (s) − xk (s)|)B(vj (s) − vk (s), −nj k (s)) Eε 2 0 0 i,j,k

· V σ,ε (|xi (s)−xj (s) + (vi (s)−vjk (s))(t −s) + z|) × B1 (vi (s) − vjk (s), −n˜ kij (s, t − s; z))J (xi (s)+vi (s)(t −s), vi (s)) − V σ,ε (|xi (s) − xj (s) + (vi (s) − vj (s))(t − s) + z|)

 × B1 (vi (s) − vj (s), −nij (s, t − s; z))J (xi (s)+vi (s)(t −s), vi (s)) ds dt , (5.8)

24 (z, J ) =



1 q T t d  σ,ε ε V (|xi (s)− xj (s)|)B(vi (s) − vj (s), −nij (s)) Eε 2 0 0 i,j

j × V σ,ε (|xi (s)−xj (s) + (vi (s)−vji (s))(t −s) + z|) j

j

j

× B1 (vi (s) − vji (s), −nij (s, t − s; z))J (xi (s) + vi (s)(t − s), vi (s)) − V σ,ε (|xi (s)−xj (s) + (vi (s)−vj (s))(t −s) + z|)

 × B1 (vi (s) − vj (s), −nij (s, t − s; z))J (xi (s) + vi (s)(t − s), vi (s)) . (5.9) Remark 4.2. By the Optimal Sampling Theorem, we have that (5.5) holds when T is replaced by a stopping time τ . By averaging over all configurations with the weights q given by the initial distribution µε , we also have (5.5) when Eε is replaced with Eε . Proof of Theorem 5.1. Step 1. We apply (5.4) with J ≡ 1, z = 0, and denote the corresponding terms by 21 , 22 , 23 , 24 . We start with 21 . We have


T

21 = 0

εd



V σ,ε (|xi − xj + (vi − vj )t|)B1 (vi − vj , −nij (0, t))dt .

(5.10)

i,j

Define H ε (v, T ) = {x : (x, v) ∈ Gε (T )} ,

(5.11)

A Stochastic Model Associated with Enskog Equation and its Kinetic Limit

349

where 
Gε (T ) = (x, v) :

T

0

   x + θv dθ = 0 . V σ,ε (|x + θ v|)B1 v, − |x + θv|

(5.12)

Recall Bˆ ε K = K + ε[0, 1]d . Since V is of compact support, there exists a constant c0 such that V σ,ε (|z|) = 0 implies σ − c0 ε ≤ |z| ≤ σ + c0 ε. Hence, there exists a constant c1 such that if x ∈ Bˆ ε H ε (v, T ), then σ − c1 ε ≤ |x + θv| ≤ σ + c1 ε for some θ ∈ [0, T ]. Note that if σ − c1 ε ≤ |x + θv| ≤ σ + c1 ε for some θ ∈ [0, T ] and |x| ∈ / [σ − c1 ε, σ + c1 ε], then the line segment {x + θv : θ ∈ [0, T ]} intersects either the sphere {z : |z| = σ + c1 ε} or the sphere {z : |z| = σ − c1 ε}. Moreover, the function a M(a) = |a| is uniformly Lipshitz away from the origin a = 0. As a result, there exists a constant c2 such that Bˆ ε H ε (v, T ) ⊆ A0 ∪ A− ∪ A+ , where A0 = {x : σ − c1 ε ≤ |x| ≤ σ + c1 ε} and   x + θv A± = x : |x + θ v| = σ ± c1 ε, v · ∈ [−1 − c2 ε, c2 ε] for some θ ∈ [0, T ] . |x + θ v| Observe that if x ∈ A± , then x = −θ v + (σ ± c1 ε)n for some (θ, n) with θ ∈ [0, T ] and v · n ∈ [−1 − c2 ε, c2 ε]. From this and the fact that dx = |v · n|dθ dn we deduce

T
|A± | = dx = |v · n|?(v · n ∈ [−1 − c2 ε, c2 ε])dn dθ ≤ αT (1 + c2 ε) , A±

0

S

(5.13) where α denotes the d − 1-dimensional measure of the sphere S. As a result, |Bˆ ε H ε (v, T )| ≤ c3 T (1 + ε) = c4 T for some constants c3 and c4 . Moreover,  
a + θv V σ,ε (|a + θ v|)B v, − dθ |a + θv| R 


 a + θv − d−1 ˆ −1 ε V ε (|a + θ v| − σ ) v · dθ = |a + θv|
R

 = ε d−1 Vˆ ε−1 (η − σ ) dη = ε d . R

(5.14)

(5.15)

From (5.10–15) we deduce 21 ≤ ε 2d

   ? (xi − xj , vi − vj ) ∈ Gε (T ) i,j

≤ε

2d

   ? xi ∈ H ε (vi − vj , T ) + xj i,j

≤ε N 2d

sup |Bˆ ε K|≤c4 T

N (q; K × Rd ) .

(5.16)

350

F. Rezakhanlou

From this and Lemma 4.9 we deduce that 21 ≤ c5 h (c4 T ) (1 + ε d @ε (q)) ,

(5.17)

for some constant c5 . Step 2. We now consider 22 . First observe that 22 is bounded above by

1 q T t  σ,ε V (|xi (s) − xk (s)|)B(vi (s) − vk (s), −nik (s)) Eε 2 0 0 i,k  · V σ,ε (|xi (s) − xj (s) + (vik (s) − vj (s))(t − s)|) j

× B1 (vik (s) − vj (s), −nkij (s, t − s; z))ds dt
1 q T  σ,ε V (|xi (s) − xk (s)|)B(vi (s) − vk (s), −nik (s)) = Eε 2 0 i,k
T · V σ,ε (|xi (s) − xj (s) + (vik (s) − vj (s))(t − s)|) s

j

× B1 (vik (s) − vj (s), −nkij (s, t − s; z))dt ds . We concentrate upon the t-integral. As in Step 1, we have that the t-integral is bounded by   c6 ε d ? (xi (s) − xj (s), vik (s) − vj (s)) ∈ Gε (T − s) , j

for some constant c6 . This, a repetition of (5.16) and Lemma 4.9 imply
T q Aε (q(s)) ε d sup N (q(s); K) ds |22 | ≤ c6 Eε
≤ c7 Eεq

|Bˆ ε K|≤c4 T

0

T 0

0≤s≤T




+c7 Eεq

Aε1 (q(s))ds h (c4 T ) sup (1 + ε d @ε (q(s))) T

0

(Aε (q(s)) − Aε1 (q(s)))ds .

The same estimate holds for 23 because 23 = 22 . Step 3. As for 24 , note that |24 | is bounded above by
T  Eεq εd V σ,ε (|xi (s) − xj (s)|)B(vi (s) − vj (s), −nij (s)) 0


·

s

i,j

T

j

V σ,ε (|xi (s) − xj (s) + (vi (s) − vji (s))(t − s)|) j

× B1 (vi (s) − vji (s), −nij (s, t − s))dt ds
T d q Aε (q(s))ds , ≤ ε Eε 0

where for the last inequality we used (5.15).

(5.18)

A Stochastic Model Associated with Enskog Equation and its Kinetic Limit

351

Step 4. After putting all the pieces together, we see that for some constant c8 ,
T Aε1 (q(s))ds ≤ c8 h(c4 T )(1 + ε d @ε (q)) Eεq 0
T q + c 8 Eε Aε1 (q(s))ds h (c4 T ) sup (1 + ε d @ε (q(s))) 0 0≤s≤T
T (Aε (q(s)) − Aε1 (q(s)))ds . + c8 Eεq 0

Since (5.5) holds whenever T is replaced by a stopping time, a similar argument shows that
τ Eε Aε1 (q(s))ds ≤ c9 Eε h(c4 τ ) + c9 /1 0
τ +c8 Eε Aε1 (q(s))ds h (c4 τ ) sup (1 + ε d @ε (q(s)))ds , 0≤s≤τ

0

(5.19) where τ is any stopping time with τ ≤ T . Here we are using Lemma 5.2 and the fact that εd Eε @ε (q(0)) is uniformly bounded in ε. We now choose   τ = τ ε := T ∧ inf t : c8 h(c4 t) sup (1 + ε d @ε (q(s))) ≤ 0≤s≤t

1 2

.

For such a stopping time τ we deduce
τ Eε Aε (q(s))ds ≤ 2c9 h(c4 T ) + 2c9 /1 . 0

From this and Lemma 5.2 we deduce
τ Eε Aε (q(s))ds ≤ 2c9 h(c4 T ) + c10 /1 .

(5.20)

0

Final Step. From (5.2), it is not hard to show
T Eε Aε (q(s))ds ≤ c11 ε −1 . 0

As a result,
Eε

T 0




ε

A (q(s))ds ≤ Eε ≤ Eε

τ


0 τ

ε




A (q(s))ds + Eε Aε (q(s))ds + c12 ε

0

T

0 −1

Aε (q(s))?(τ < T )ds Pε (τ < T ) .

We can now repeat the proof of Theorem 3.1 of [RT] and use Theorem 4.1 to argue that for T sufficiently small, Pε (τ < T ) ≤ exp(−c13 ε −d ) . This and (5.20) imply (5.1) for sufficiently small T . The proof of general T is done by a bootstrap.   As in [RT], we can readily deduce

352

F. Rezakhanlou

Corollary 5.3. There exists a function C2 (T , k) such that limT →0 C2 (T , k) = 0 and for every integer k, 
T k ε Eε A (q(s))ds ≤ C2 (T , k) , 0


sup Eε τ

T +τ τ

Aε (q(s))ds ≤ C2 (T , 1) ,

where the supremum is over all stopping times. We continue with two more corollaries that will be needed in Sects. 6 and 7. Corollary 5.4. There exists a constant C˜ 2 (T ) such that for any 1 ∈ (0, 1],
0

T

εd



V σ,ε (|xi (s) − xj (s)|)B1 (vi (s) − vj (s), −nij (s))ds

i,j

≤ C˜ 2 (T )| log(1 + ε)|−1 .

(5.21)

Proof. The proof is similar to the proof of Theorem 5.1 and we only sketch it. As in the proof of Theorem 5.1, the right-hand side of (5.21) can be written as 21 + . . . + 24 and it suffices to show that |2r | ≤ const.| log(1 + ε)|−1 ,

(5.22)

for each r. For example, we can readily show
|22 | ≤ c6 Eε
≤ c7 Eε

T

Aε (q(s)) ε d

0 T

sup |Bˆ ε K|≤c3 T (1+ε)

N (q(s); K) ds

Aε (q(s))ds h (c3 T (1 + ε)) sup (1 + ε d @ε (q(s)))

0

0≤s≤T

in just the same way we showed (5.18). See also (5.14). We can now apply Schwartz’s inequality, Corollary 5.3 with k = 2, and Corollary 4.6 with r = 2 to deduce (5.22) for r = 2. The other cases are treated likewise.   Define Aˆ ε (q) = εd



V σ,ε (|xi − xj |)B(vi − vj , −nij )|vi | .

i,j

Corollary 5.5. There exists a constant Cˆ 2 (T , k) such that, 
Eε for every positive integer k.

T 0

Aˆ ε (q(s))ds

k

≤ Cˆ 2 (T , k) ,

(5.23)

A Stochastic Model Associated with Enskog Equation and its Kinetic Limit

353

Proof. The proof of this corollary is also similar to the proof of Theorem 5.1 and we only sketch it. First assume that k = 1. As in the proof of Theorem 5.1, we apply (5.5) with z = 0 and J (x, v) = |v| to say that the left-hand side of (5.23) can be written as 21 + . . . + 24 . For example, we can readily show
T 1 22 ≤ Eεq V σ,ε (|xi (s) − xk (s)|)B(vi (s) − vk (s), −nik (s))|vik (s)| 2 0 i,k
T V σ,ε (|xi (s) − xj (s) + (vik (s) − vj (s))(t − s)|) · s

j

× B(vik (s) − vj (s), −nkij (s, t − s; z))dt ds . Using |vik | ≤ |vi | + |(vi − vk ) · nik | and Lemma 5.2 we deduce that for some constant c,
T 22 ≤ c6 Eε Aˆ ε (q(s)) ε d sup N (q(s); K) ds + c
≤ c7 Eε

|Bˆ ε K|≤c4 T

0

T

Aˆ ε (q(s))ds h(c4 T ) sup (1 + ε d @ε (q(s))) + c ,

0

0≤s≤T

in just the same way we showed (5.18). We now replace T with the stopping time τ as in the proof of Theorem 5.1 to deduce that for some constant c ,
τ Eε Aˆ ε (q(s))ds ≤ c . 0

A repetition of the final step of the proof of Theorem 5.1 would complete the proof of (5.23) when k = 1. The proof for the general k is achieved as in [RT].   6. Spatial Regularity of Collision As we mentioned in the introduction, the main hypotheses behind a kinetic derivation is the Stosszahlensatz of Boltzmann that asserts that a pair of particles before a collision are uncorrelated. In our version of this principle, we show that shifting particles around somewhat does not dramatically alter the value of the collision term. In the last section we show that such a regularity of the collision term implies a weak form of Stosszahlensatz. See also Sect. 3 where we argued that a statement like (3.12) implies (3.11). The next theorem is our regularity claim. Define  F ε (q; J, z) = εd V σ,ε (|xi − xj + z|)B(vi − vj , −nˆ ij (z))J (xi , vi ) , i,j

F˜ ε (q; J, z) = εd

 i,j

j

V σ,ε (|xi − xj + z|)B(vi − vj , −nˆ ij (z))J (xi , vi ) .

(6.1)

Theorem 6.1. For every continuously differentiable function J of compact support, there exists a constant C3 (T , J ) such that


T



 ε 

sup sup Eε

F (q(t); J, z) − F ε (q(t); J, 0) dt

≤ C3 (T , J ) h(δ + ε) , (6.2) ε>0 |z|<δ 0


T 



sup sup Eε

F˜ ε (q(t); J, z) − F˜ ε (q(t); J, 0) dt

≤ C3 (T , J ) h(δ + ε) . (6.3) ε>0 |z|<δ

0

354

F. Rezakhanlou

To prepare for the proof of Theorem 6.1, we start with some definitions and preliminary lemmas. Define G = {(x, v) ∈ Rd × Rd : |x + vt| = σ for some t ∈ R}  = (x, v) ∈ Rd × Rd : (x · v)2 − |x|2 |v|2 + σ 2 |v|2 ≥ 0 .

(6.4)

Given (x, v) ∈ G, the line {x + vt : t ∈ R} intersects the sphere Sσ = {σ n : n ∈ S} at at most two points. For such an (x, v), we define t − (x, v), t + (x, v) ∈ R, n− (x, v), n+ (x, v) ∈ S so that {x + vt : t ∈ R} ∩ Sσ = {σ n− (x, v), σ n+ (x, v)} t − (x, v) ≤ t + (x, v) .

(6.5)

More explicitly, 1/2 x·v 1 2 2 2 2 2 (x · v) ± − |x| |v| + σ |v| |v|2 |v|2 ± x + vt (x, v) 1 n± (x, v) = = (x + vt ± (x, v)) . ± |x + vt (x, v)| σ t ± (x, v) = −

(6.6)

We define Gδ = {(σ n − vt, v) ∈ G : t ∈ R, |n · v| ≤ δ} = {(x, v) ∈ G : |(n+ (x, v) · v| or |(n− (x, v) · v| ≤ δ} .

(6.7)

We continue with an elementary lemma. Lemma 6.2. We have,

±

±

∂t

∂n

1 + |v|





∂x (x, v) + ∂x (x, v) ≤ |v · n± (x, v)| .

(6.8)

Moreover, v · n± (x, v) = ±

1 1 2 (x · v)2 − |x|2 |v|2 + σ 2 |v|2 . σ

(6.9)

Proof. Observe that if we put g(x, v, t) = |x + vt|2 , then g(x, v, t ± (x, v)) = σ 2 . By an implicit differentiation we obtain ∂t ± x + vt ± (x, v) n± (x, v) (x, v) = − = − . ∂x (x + vt ± (x, v)) · v n± (x, v) · v

(6.10)

Moreover, n± (x, v) = (x + t ± (x, v)v)/σ . From this and (6.10) we deduce   v ⊗ n± ∂n± 1 I− ± (x, v) = , σ n ·v ∂x where I denotes the identity matrix and a ⊗ b means the tensor product of vectors a, b. From this and (6.10) we deduce (6.8). The proof of (6.9) is straightforward and follows from (6.6).  

A Stochastic Model Associated with Enskog Equation and its Kinetic Limit

355

Given a continuously differentiable function J of compact support, define fε (x, v, w; T , J ) = fε (x, v, w) by  
T x + vt − −d σ,ε fε (x, y, v, w) := ε V (|x + vt|) v · J (y + wt, w)dt . |x + vt| 0 We also define ;(α) = {(x, v) ∈ G : t − (x, v) ∈ [−α, α] ∪ [T − α, T + α]} . Lemma 6.3. There exist constants C4 (J ), C5 and Cˆ 5 such that if (x, v) ∈ G − Gδ , (x, v) ∈ / ;(C5 ε/δ), and |v| ≤ Cˆ 5 δε −1/2 , then ε |fε (x, y, v, w) − J (y + wt − (x, v), w)?(t − (x, v) ∈ [0, T ])| ≤ C4 (J ) . δ

(6.11)

Proof. Choose a number c0 so that if V σ,ε (|a|) = 0, then σ − c0 ε ≤ |a| ≤ σ + c0 ε. We then choose a constant C˜ 5 such that (x · v)2 − |x|2 |v|2 + η2 |v|2 = σ 2 (v · n+ )2 + (η2 − σ 2 )|v|2 1 ≥ σ 2 δ 2 − |v|2 |η2 − σ 2 | ≥ σ 2 δ 2 , 2 whenever η ∈ [σ − c0 ε, σ + c0 ε] and |v| ≤ C˜ 5 δε −1/2 . As a result, if |x + vt| = η ∈ [σ − c0 ε, σ + c0 ε], then t = tˆ+ or tˆ− , where 1/2 x·v 1 . (6.12) tˆ± = − 2 ± 2 (x · v)2 − |x|2 |v|2 + η2 |v|2 |v| |v| For t ∈ Iε we √ have η = √ σ + cε for √ some c ∈ [−c0 , c0 ]. From this and the elementary inequality | a + b − a| ≤ |b|/ a we deduce

1/2

± x·v 1 2 2 2 2 2

tˆ + ∓ 2 (x · v) − |x| |v| + σ |v|



|v|2 |v|

 −1/2 ≤ 2|c|σ ε + c2 ε 2 (x · v)2 − |x|2 |v|2 + σ 2 |v|2 . This and (6.9) imply |tˆ± − t ± (x, v)| ≤

C5 ε C5 ε ≤ , |v · n± (x, v)| δ

for some constant C5 . Furthermore, if |v| ≤ Cˆ 5 δε −1/2 with Cˆ 5 = min(C˜ 5 , and (x, v) ∈ / Gδ , then we can use (6.6) and (6.9) to assert t + (x, v) − t − (x, v) =

(6.13) √ σ/C5 ),

C5 ε 2σ 2σ ε ≥2 . v · n+ (x, v) ≥ 2 ˆ |v|2 δ δ C5

This and (6.13) imply that the set Iε =: {t : V σ,ε (|x + vt|) = 0} can be written as a union of two disjoint intervals (aε− (x, v), bε− (x, v)) and (aε+ (x, v), bε+ (x, v)) such that t ± (x, v) ∈ (aε± (x, v), bε± (x, v)) .

356

F. Rezakhanlou

We now write fε (x, y, v, w) = fε− (x, y, v, w) + fε+ (x, y, v, w),

(6.14)

where fε± (x, y, v, w) = ε −d




bε± (x,v)

V σ,ε (|x + vt|)J (y + wt, w)   x + vt − × ?(t ∈ [0, T ]) v · dt . |x + vt| aε± (x,v)

As in (6.9) we have that if t ∈ [aε± , bε± ] and |x + vt| = η, then v·

1 1 x + vt 2 = ± (x · v)2 − |x|2 |v|2 + η2 |v|2 . |x + vt| η

As a result fε+ = 0. Define 

 x + vt − V (|x + vt|) v · ?(t ∈ [0, T ])dt |x + vt| aε− (x,v) · J (y + wt − (x, v), w),  
bε− (x,v) x + vt − f˜ε− (x, y, v, w) := ε −d V σ,ε (|x + vt|) v · dt |x + vt| aε− (x,v) · ?(t − (x, t) ∈ [0, T ])J (y + wt − (x, v), w) = ?(t − (x, t) ∈ [0, T ])J (y + wt − (x, v), w) .

fˆε− (x, y, v, w) := ε −d




bε− (x,v)

σ,ε

From (6.13) and the Lipschitzness of the function J we learn c1 ε |fε− (x, y, v, w) − fˆε− (x, y, v, w)| ≤ δ

(6.15)

for some constant c1 . Once more we use (6.13) to claim that if (x, v) ∈ / ;(C5 ε/δ), then the term ?(t ∈ [0, T ]) in fˆε can be replaced with the term ?(t − (x, t) ∈ [0, T ]). This means fˆε− = f˜ε− . From this, (6.14) and (6.15) we conclude (6.11).   + Lemma 6.4. There exists a constant √ C6 = C6 (T ) such that if (x, v) ∈ G, |t (x, v)| > − C6 , |t (x, v)| > C6 and |v| ≥ ε, then fε (x, y, v, w; T , J ) = 0 for every y and w. Moreover, fε (x, y, v, w; T , J ) = 0 whenever (x, v) ∈ / G. σ,ε (|x + vt|) = 0}. Given (x, v) ∈ G, we can use (6.12) Proof. Recall that Iε = {t : V √ √ √ and the elementary inequality a + b − a ≤ b to deduce that if t ∈ Iε , then either √ |t − t − | or |t − t + | is bounded above by |v|−1 ((σ + c0 ε)2 − σ 2 )1/2 . For v with |v| ≥ ε we deduce

min(|t − t − |, |t − t + |) ≤ c1 , for some constant c1 . From this and the definition of fε we deduce the lemma. Finally, it is not hard to show that if Iε = ∅, then (x, v) ∈ G. This implies that fε (x, y, v, w; T , J ) = 0 if (x, v) ∈ / G.    Define K(q) = i |vi |2 .

A Stochastic Model Associated with Enskog Equation and its Kinetic Limit

357

Lemma 6.5. There exists a constant C7 (J ) such that





N

d

ε

(f (x − x , y, v − v , w) − f (x + z − x , y, v − v , w)) ε j j ε j j



j =1

 ˆ ε (q) + ε d @ε (q) + ε d K(q) + |v| , (6.16) ≤ C7 (J ) h(|z| + ε) 1 + ε d @ for every ε, |z| ∈ (0, 1]. Proof. Step 1. Using Lemma 6.4, we have that the left-hand side of (6.16) is bounded  above by 8r=1 2r , where





N

d 2r =

ε (fε (x − xj , y, v − vj , w) − fε (x − xj + z, y, v − vj , w)) ξjr

,

j =1 where  ξj1 = ? |v − vj | ≤ Cˆ 5 δε −1/2 , (x − xj , v − vj ),  × (x − x + z, v − v ) ∈ G − (G ∪ ;(C ε/δ)) j j δ 5 ,

ξj2 = ? (x − xj , v − vj ) ∈ ;(C5 ε/δ), |v − vj | ≤ ε−1/2 , 

ξj3 = ? (x − xj + z, v − vj ) ∈ ;(C5 ε/δ)), |v − vj | ≤ ε−1/2 , 

√ ξj4 = ? (x − xj , v − vj ) ∈ Gδ , ε ≤ |v − vj | ≤ δ −1/2 ,

 √ ξj5 = ? (x − xj + z, v − vj ) ∈ Gδ , ε ≤ |v − vj | ≤ δ −1/2 , 

√ ξj6 = ? (x − xj + z, v − vj ) ∈ G, (x − xj , v − vj ) ∈ / G, ε ≤ |v − vj | ≤ δ −1/2 ,

 √ ξj7 = ? (x − xj + z, v − vj ) ∈ / G, (x − xj , v − vj ) ∈ G, ε ≤ |v − vj | ≤ δ −1/2 ,

 ξj8 = ? δ −1/2 or ε −1/2 or Cˆ 5 δε −1/2 ≤ |v − vj | ,  √  ξj9 = ? |v − vj | ≤ ε . Step 2. By Lemmas 6.2 and 6.3, (6.8) and the Lipschitzness of J we learn that if (x, v), (x + z, v) ∈ G, (x, v), (x + z, v) ∈ Gδ ∪ ;(C5 ε/δ), |v| ≤ Cˆ 5 δε −1/2 , then  |fε (x, y, v, w) − fε (x + z, y, v, w)| ≤ c0

ε |z| + δ δ

 ,

for some constant c0 . This implies  21 ≤ c1 for some constant c1 .

ε |z| + δ δ

 ,

(6.17)

358

F. Rezakhanlou

Step 3. Note that by (5.15), the function fε is uniformly bounded. By Lemma 4.9, 29 ≤ c2 h(ε)(1 + ε d @ε (q))

(6.18)

for some constant c2 . Also, from ?(|v − vj | ≥ 1) ≤ |v − vj |/1 ≤ (|v| + 1 + |vj |2 )/1 we deduce √ √ (6.19) 28 ≤ c3 ( δ + ε δ −1 )(|v| + 1 + K(q)) , for some constant c3 . It is not hard to show √ |Bε (;(C5 ε/δ))| ≤ c4 ε/δ , for some constant c4 . We then use (5.15) and Lemma 4.8 to deduce √ ˆ ε (q)) . 22 + 23 ≤ c5 h( ε/δ)(1 + ε d @ Step 4. Define Rδ,ε = {x : (x, v) ∈ Kδ,ε for some v}, where  √ Kδ,ε = (x, v) ∈ Gδ : |t + (x, v)| or |t − (x, v)| ≤ C6 , and ε ≤ |v| ≤ δ −1/2

(6.20)

.

We certainly have 1 , Bˆ ε Rδ,ε ⊆ R 0 ∪ Rδ,ε

(6.21)

where R 0 is the set of points x + a such that (x, v) ∈ G, |t + (x, v)| or |t − (x, v)| ≤ C6 , |v| ≤ δ −1/2 , but (x + a, v) ∈ / G for some a ∈ ε[0, 1]d , and 1 = {x + a : (x, v) ∈ Kδ,ε and (x + a, v) ∈ G for some (a, v) ∈ ε[0, 1]d × Rd } . Rδ,ε

It is not hard to show that |R 0 | ≤ c6 εδ (1−d)/2 .

(6.22)

Also, we can find a constant c7 such that if (x, v) ∈ Kδ,ε , then |x| ≤ c7 δ −1/2 . We use 1 with (x, v) ∈ K , (x + a, v) ∈ G, a ∈ ε[0, 1]d , this and (6.9) to show that if x ∈ Rδ,ε δ,ε then √ |v · n± (x + a, v) − v · n± (x, v)| ≤ c7 ε δ −3/4 . This implies that √ |v · n± (x + a, v)| ≤ δ + c7 ε δ −3/4 . Moreover, from |x + vt ± (x, v)| = σ we deduce that if η = |x + a + vt ± (x, v)|, then |η − σ | ≤ c8 ε. We can now repeat the proof of Lemma 4.6 and use |t + (x, v)| or |t − (x, v)| ≤ C6 to deduce that |t + (x, v)| or |t − (x, v)| ≤ c9 for some constant c9 . In 1 , then summary, if z ∈ Rδ,ε √ |v · n± (z, v)| ≤ δ + c7 ε δ −3/4 ,

|t + (z, t)| or |t − (z, v)| ≤ c9 .

(6.23)

A Stochastic Model Associated with Enskog Equation and its Kinetic Limit

359

1 | ≤ Recall that dz = |v · n|dvdt if z = σ n − vt. This and (6.23) imply that |Rδ,ε √ −3/4 c10 (δ + ε δ ). From this, (6.22) and (6.21) we deduce

|Bˆ ε (Rδ )| ≤ c11 (δ +

√

ε δ −3/4 + εδ (1−d)/2 ) ,

for some constant c11 . This and Lemma 4.9 imply 

√ 24 + 25 ≤ c12 h δ + ε δ −3/4 + εδ (1−d)/2 (1 + ε d @ε (q)) .

(6.24)

/ G for some Moreover, if Aδ (z) is the set of points (x, v) ∈ G such that (x + z, v) ∈ v with |t − (x, v)| or |t + (x, v)| ≤ C6 and |v| ≤ δ −1/2 , then we can show that for some constant c13 , |Bˆ ε Aδ (z)| ≤ c13 |z|δ (1−d)/2 , in just the same way we established (6.22). This and Lemma 4.9 imply 

27 + 28 ≤ c13 h |z|δ (1−d)/2 (1 + ε d @ε (q)) .

(6.25)

Final Step. Now (6.16) follows from (6.17–19), (6.21), (6.24–25) if we choose δ =  (|z| + ε)α with α any positive number strictly less than min(1/2, 2/(d − 1)).  Proof of Theorem 6.1. Step 1. We only prove (6.1) because the proof of (6.2) is similar. Let T0 ∈ [0, T ]. Using formula (5.5), we write
X(q) :=

Eεq

T0


R(q(t))dt :=

0

Eεq

T0

[F (q(t); J, z) − F (q(t); J, 0)]dt

0

ˆ1+2 ˆ2+2 ˆ3+2 ˆ 4, =2 ˆ r = 2r (z, J ) − 2r (0, J ) for r = 1, 2, 3, 4 and 2r (z, J ) are defined by where 2 ˆ 1, (5.6),. . .,(5.9). We first consider 2  ˆ 1 = εd gε (xi , xi , vi ; q), 2 i

where gε (x, y, v; q) = εd



(fε (x − xj , y, v − vj , v) − fε (x + z − xj , y, v − vj , v)) .

j

By Lemma 6.5, ˆ ε (q) + ε d @ε (q) + ε d K(q) + |v|) , |gε (x, y, v; q)| ≤ c0 h(|z| + ε)(1 + ε d @ for some constant c0 . Hence, ˆ ε (q) + ε d @ε (q) + ε d K(q)) , ˆ 1 | ≤ c1 h(|z| + ε)(1 + ε d @ |2 for some constant c1 . Here we are using |v| ≤ 1 + |v|2 .

(6.26)

360

F. Rezakhanlou

ˆ2 =2 ˆ 21 − 2 ˆ 22 where Step 2. We may write 2 ˆ 21 = 2


1 q T0 d  σ,ε Eε ε V (|xi (s) − xk (s)|)B(vi (s) − vk (s), −nik (s)) 2 0 i,k  · εd fε (xi (s) − xj (s) + z, xi (s), vik (s) − vj (s), vik (s); T0 − s, J ) j

ˆ 22 2

 − fε (xi (s) − xj (s), xi (s), vik (s) − vj (s), vik (s); T0 − s, J ) ds,
T0  1 = Eεq V σ,ε (|xi (s) − xk (s)|)B(vi (s) − vk (s), −nik (s)) 2 0   i,k d ·ε fε (xi (s) − xj (s) + z, xi (s), vi (s) − vj (s), vi (s); T0 − s, J ) j  − fε (xi (s) − xj (s), xi (s), vi (s) − vj (s), vi (s); T0 − s, J ) ds .

Note that by the conservation of the energy, Kˆ ε = εd K(q(s)) is independent of s. From applying Lemma 6.5 we obtain ˆ 2 | ≤ c2 h(|z| + ε)Eεq |2




T 0

+ c2 h(|z| + ε)Eεq



ˆ ε (q(s)) + ε d @ε (q(s)) + Kˆ ε ds Aε (q(s)) 1 + ε d @




T

Aˆ ε (q(s)) ds .

0

(6.27) ˆ 3 | is treated likewise. The term |2 Step 3. As we saw in the previous section, ˆ 4 | ≤ c3 ε d Eεq |2




T

Aε (q(s))ds .

0

(See Step 3 of the proof of Theorem 5.1.) From this, (6.26) and (6.27) we deduce ˆ ε (q) + ε d @ε (q) + Kˆ ε ) |X(q)| ≤ c2 h(|z| + ε)(1 + ε d @
T

 ˆ ε (q(s)) + ε d @ε (q(s)) + Kˆ ε ds + c4 h(|z| + ε)Eεq Aε (q(s)) 1 + ε d @
+ c4 h(|z| + ε)Eεq

0

T

Aˆ ε (q(s)) ds .

0

(6.28) We certainly have 




Eε

T 0

2
T
T −s

q(s)

R(q(s))ds = 2Eε R(q(s)) Eε R(q(t))dt ds 0 0




T


q(s) T −s



≤ 2Eε |R(q(s))| Eε R(q(t))dt

ds . 0

0

(6.29)

A Stochastic Model Associated with Enskog Equation and its Kinetic Limit

361

From (6.28) we deduce that the left-hand side of (6.29) is bounded above by
T 

ˆ ε (q(s)) + ε d @ε (q(s)) + Kˆ ε |R(q(s))| 1 + ε d @ 2c4 h(|z| + ε)Eε 0


T
T −s  + 2c4 h(|z| + ε)Eε |R(q(s))| Eεq(s) Aε (q(t)) + Aˆ ε (q(t)) 0 0

 d ˆε d ε d · 1 + ε @ (q(t)) + ε @ (q(t)) + ε K(q(t)) dt ds  
T
T ε ˆ ≤ c5 h(|z| + ε)Eε |R(q(s))|ds · T + A(q(t)) + A (q(t))dt 0 0

 ˆ ε (q(t)) + ε d @ε (q(t)) + Kˆ ε · sup 1 + ε d @ 0≤t≤T

≤ c6 h(|z| + ε) , where for the last inequality, we used H¨older’s inequality, our conditions on the initial distribution, and Corollaries 4.6, 5.3 and 5.5 for the last inequality. This completes the proof of (6.1).   7. Uniform Integrability Define


Xt (x, v) =

t

εd

0



V σ,ε (|xi (s) − xj (s)|)B(vi (s) − vj (s), −nij (s))

i,j

ζ ε (xi (s) − x − svi (s), vi (s) − v) ds , where ζ : Rd ×Rd → [0, ∞) is a smooth function of compact support with √ √ 1 and ζ ε (x, v) = ε−d ζ (x/ ε, v/ ε). Put



(7.1) ζ dxdv =

τ (x, v) = τ1 (x, v) := inf{t : Xt (x, v) ≥ 1} . Theorem 7.1. There exists a constant C8 (T ) such that

sup Eε ψ(XT (x, v))dxdv ≤ C8 (T ), ε>0

where

(7.2)

 ψ(α) =

α(log log α)1/2 α ≥ ee . ee α < ee

The next lemma holds the key to verifying (7.2). Lemma 7.2. There exists a constant C9 (T ) such that

 Eε XT (x, v) − XT ∧τ1 (x,v) (x, v) dxdv

  ≤ C9 (T ) (log log 1)−1 + ε d log 1 + | log ε|−1 .

(7.3)

362

F. Rezakhanlou

We prove Lemma 7.2 below and omit the rest since the remainder of the proof is essentially identical to the proof of Theorem 7.1 in [R1]. Proof of Lemma 7.2. Step 1. Suppose that the support of Vˆ is contained in the interval (−c0 , c0 ). We take Wˆ : R → R to be a function such that Wˆ  = −Vˆ , Wˆ (z) = 0 for z > εc0 and Wˆ (z) = 1 for z < −εc0 . Define W σ,ε (z) = εd Wˆ (ε −1 (z − σ )) and   + F ε (q, t) = ε d W σ,ε (|xi − xj |) (xi − xj ) · (vj − vi ) H (xi , vi , t) , (7.4) i,j

where H (x, v, t) = ? (τ1 (x − vt, v) < T , |v| ≤ 11 ) , for a given constant 11 . The randomness of τ1 (x, v) and the nonsmoothness of the indicator function does not allow us to apply (5.3). Because of this we introduce two approximation procedures. Let δ be a small positive number such that k0 = δ −1 is an integer. We then divide the interval [0, T ) into smaller subintervals [λk , λk+1 ) of length δT and put g(a, v; k, δ) = ? (τ1 (a, v) ∈ [λk , λk+1 ) , |v| ≤ 11 ) , g(a, ˜ v, t) = ? (τ1 (a, v) < t < T , |v| ≤ 11 ) , g(a, v) = ? (τ1 (a, v) < T , |v| ≤ 11 ) . Using this we define
gˆ ε (a, v; k, δ) = gε (a, v) = g˜ ε (a, v, t) =





g(a − a  , v − w; k, δ)ζ ε (a  , w)da  dw, g(a − a  , v − w)ζ ε (a  , w)da  dw, g(a ˜ − a  , v − w, t)ζ ε (a  , w)da  dw .

We then define Hkδ (x, v, t) = gˆ ε (x − vt, v; k, δ), Hε (x, v, t) = gε (x − vt, v), Kε (x, v, t) = g˜ ε (x − vt, v, t) .

(7.5)

We replace H in (7.4) with Hkδ . The resulting expression is denoted by Fkδ (q, t). We finally put Gδ (q, t) =

k 0 −2 k=0

Fkδ (q, t)?(t ≥ λk+1 ) .

(7.6)

To understand the motivation behind (7.6), observe that lim

δ→0

k 0 −2 k=0

?(τ1 ∈ [λk , λk+1 ), t ∈ [λk+1 , T )) = ?(τ1 < T , t ∈ (τ1 , T )) .

(7.7)

A Stochastic Model Associated with Enskog Equation and its Kinetic Limit

363

Also note that the time intergration in the expression XT (x, v) − XT ∧τ1 (x,v) (x, v) is over the interval (T ∧ τ1 , T ). To be able to use the Markov property of the process q(·), we would rather integrate over (λk+1 , T ) whenever τ1 (x, v) ∈ [λk , λk+1 ), and sum over k. Because of (7.7), the resulting error is small. If Ft is the σ –field generated by (q(s) : s ≤ t), then the event τ1 ∈ [λk .λk+1 ) is measurable with respect to the σ -field Fλk+1 . Because of this, we can use the semigroup property (5.3) to have Eε Fkδ (q(T ), T )

=




Eε Fkδ (q(λk+1 ), λk+1 ) + Eε

T λk+1



∂Fkδ ε δ + A Fk (q(s), s)ds . ∂t

As a result, Eε Gδ (q(T ), T ) = Eε

k 0 −2 k=0

+Eε +Eε

Fkδ (q(λk+1 ), λk+1 )

k 0 −2
T k=0

λk+1

k=0

λk+1

k 0 −2
T

=: 2δ1 +

k 0 −2



∂Fkδ δ + A0 Fk (q(s), s)ds ∂t

(Ac Fkδ )(q(s), s)ds

2δ2,k +

k=0

k 0 −2

2δ3,k .

(7.8)

k=0

Note that the function Fkδ is not differentiable and we can not apply (5.3) directly. But now the term [(xi − xj ) · (vi − vj )]+ is the only nonsmooth component and after replacing this with a smooth approxiamtion, applying (5.3) and passing to the limit, we d + obtain (7.8) where ?(z ≥ 0) is used for dz z . Step 2. A straightforward calculation yields, ∂Fkδ + A0 Fkδ ∂t  = εd V σ,ε (|xi − xj |)[nij · (vj − vi )][(xi − xj ) · (vj − vi )]+ Hkδ (xi , vi , t) i,j

+ε

d



  W σ,ε (|xi − xj |)[(vi − vj ) · (vj − vi )] ? (xi − xj ) · (vj − vi ) ≥ 0

i,j

·Hkδ (xi , vi , t)  +ε d W σ,ε (|xi − xj |)[(xi − xj ) · (vj − vi )]+ 

i,j

∂ ∂ + vi · ∂t ∂xi =: X1 + X2 + X3 . ·



Hkδ (xi , vi , t)

364

F. Rezakhanlou

We certainly have X3 = 0. Hence k 0 −2

2δ2,k = Xˆ 1 (δ) + Xˆ 2 (δ),

k=0

where Xˆ 1 (δ) = Eε

k 0 −2
T k=0

λk+1



εd

2  V σ,ε (|xi (s) − xj (s)|) [nij (s) · (vj (s) − vi (s))]+

i,j

|xi (s) − xj (s)|Hkδ (xi (s), vi (s), s)ds ,

Xˆ 2 (δ) = −Eε

k 0 −2
T λk+1

k=0

εd



(7.9)

W σ,ε (|xi (s) − xj (s)|)|vi (s) − vj (s)|2

i,j

Hkδ (xi (s), vi (s), s)ds

.

(7.10)

Since Fkδ ≥ 0, we deduce Xˆ 1 (δ) ≤ Eε Gδ (q(T ), T ) − Xˆ 2 (δ) − Xˆ 3 (δ) ,

(7.11)

for Xˆ 3 (δ) = Eε

k 0 −2

2δ3,k .

k=0

Note that k 0 −2

?(τ1 ∈ [λk , λk+1 ), t ∈ [λk+1 , T )) ≤ ?(τ1 < T , t ∈ [0, T )) .

(7.12)

k=0

From this, (7.7) and the dominated convergence theorem we deduce lim Xˆ 1 (δ) = Eε

δ→0




T 0

εd



2  V σ,ε (|xi (s) − xj (s)|) [nij (s) · (vj (s) − vi (s))]+

i,j

|xi (s) − xj (s)|Kε (xi (s), vi (s), s)ds . Step 3. We now turn to Xˆ 2 (δ). Recall K(q) = −Xˆ 2 (δ) ≤ c1 Eε ε d ≤ c2 Eε ε d

k 0 −2
T λk+1

k=0 k 0 −2
T k=0

λk+1

εd

 i,j

εd

 i,j



i

(7.13)

|vi |2 and Kˆ ε = εd K(q(s)). We have

|vi (s) − vj (s)|2 Hkδ (xi (s), vi (s), s)ds Hkδ (xi (s), vi (s), s)(N 11 + K(q(s)))ds ,

A Stochastic Model Associated with Enskog Equation and its Kinetic Limit

365

because |vi − vj |2 ≤ 2|vi |2 + 2|vj |2 and if Hkδ (x, v, t) = 0, then |v| ≤ 211 for small δ and large 11 . This and (7.12) imply −Xˆ 2 (δ) ≤ c2 Eε




T

ε 2d



0

Hε (xi (s), xj (s), s, vi (s), vj (s))(N 11 + K(q(s))ds

i,j

≤ c3 Eε sup ε d 0≤s≤T



Hε (xi (s), vi (s), s)(11 + Kˆ ε ) .

(7.14)

i

Observe
Hε (x, v, t) =

g(x − vt − a  , v − w)ζ ε (a  , w)da  dw




=

g(x − (v − w)t − a  , v − w)ζ ε (a  + wt, w)da  dw




= =ζ¯




H (x − a  , v − w, t)ζ ε (a  + wt, w)da  dw H (x − a  , v − w, t)ηε (a  , w)da  dw,

 where ζ¯ = ζ (a + wt, w)dadw and ηε (a, w) = ζ ε (a + wt, w)/ζ¯ . We now apply Lemma 4.7 and (7.14) to assert 


 ˆ ˆ ε (q(s)) (11 + Kˆ ε ) . −X2 (δ) ≤ Eε c4 h sup H (x, v, s)dxdv sup 1 + ε d @ 0≤s≤T

0≤s≤T

(7.15) For this, we need to estimate the intergal in the argument of h. By the Chebyshev’s inequality,


H (x, v, s)dxdv ≤ ? (τ1 (x − vs, v) < T ) dxdv = ?(τ1 (x, v) < T )dxdv


1 1 T ε ≤ A (q(t))dt . XT (x, v)dxdv ≤ c4 1 1 0 From this and (7.15) we deduce −Xˆ 2 (δ) ≤ c3 Eε h



c4 1




T

Aε (q(t))dt

0



 ˆ ε (q(t)) (11 + Kˆ ε ) . (7.16) 1 + εd @

sup 0≤t≤T

In the same fashion we can show, lim EGδ (q(T ), T ) ≤ c5 Eε ε 2d

δ→0



 i,j

c4 ≤ c6 Eε h 1

Hε (xi (T ), vi (T ), T )




T 0

ε

A (q(t))dt



 ˆ ε (q(T ) (11 + Kˆ ε ) . 1 + εd @ (7.17)

366

F. Rezakhanlou

Step 4. Once more we apply (7.12) to obtain,


lim (−Xˆ 3 (δ)) ≤ Eε

δ→0

T

εd

0



V σ,ε (|xi (s) − xk (s)|)B(vi (s) − vk (s), −nik (s))

i,j,k

× W σ,ε (|xi (s) − xj (s)|)|xi (s) − xj (s)||vi (s) − vj (s)| × Hε (xi (s), vi (s), s)ds
T  + Eε εd V σ,ε (|xi (s) − xj (s)|)B(vi (s) − vj (s), −nij (s)) 0

i,j

× W σ,ε (|xi (s) − xj (s)|)|xi (s) − xj (s)||vi (s) − vj (s)| × Hε (xi (s), vi (s), s)ds =: X31 + X32 . We have 
c4 T ε ≤ c7 Eε A (q(s))ds h A (q(t))dt 1 0 0

 ˆ ε (q(t)) (11 + Kˆ ε ) , × sup 1 + ε d @


X31

T



ε

0≤t≤T

in just the same way we obtained (7.14) and (7.15). Since W ε = O(ε d ), we have X32 ≤ c8 ε

d




T

Aε (q(s))ds .

0

Hence 
c4 T ε A (q(s))ds h A (q(t))dt 1 0 0


 ˆ ε (q(t)) (11 + Kˆ ε ) + c8 ε d · sup 1 + ε d @

lim (−Xˆ 3 (δ)) ≤ c7 Eε

δ→0




T



ε

T

Aε (q(s))ds .

0

0≤t≤T

From this, Theorem 5.1, (7.11), (7.16) and (7.17) we deduce

δ→0




 
T  c4 ε A (q(s))ds + 1 h A (q(t))dt 1 0 

0 ˆ ε (q(s)) (11 + Kˆ ε ) + c9 ε d · sup 1 + ε d @

lim Xˆ 1 (δ) ≤ c9 Eε

T

ε

0≤s≤T






≤ c9 Eε

T

Aε (q(s))ds + 1

 4  41 

0

ˆ ε (q(s)) Eε 1 + ε d sup @




T   41  1 · E ε h4 Aε (q(s))ds Eε (11 + Kˆ ε )4 1 0 1

1

1

4  41 

 0≤s≤T 1 4



+ c9 ε d

1

=: Y14 Y24 Y34 Y44 + c9 ε d .

(7.18)

A Stochastic Model Associated with Enskog Equation and its Kinetic Limit

367

By Corollary 4.4 and Theorem 5.1 we have Y1 ≤ c10 , Y2 ≤ c10 , Y4 ≤ c10 11 for some constant c10 . By the concavity of h4 and the Jensen’s inequality, 



1 4

Y3 ≤ h

4

c4 Eε 1




T

ε

A (q(s))ds

0

 41

≤h

c  11

1

≤ c12 (log 1)−1 .

This, (7.13) and (7.18) imply
T   2 εd V σ,ε (|xi (s) − xj (s)|) [nij (s) · (vi (s) − vj (s))]+ Eε 0

i,j

|xi (s) − xj (s)|Kε (xi (s), vi (s), s)ds ≤ c13 (log 1)−1 + c9 ε d . Hence


Eε

T

εd

0



2  V σ,ε (|xi (s) − xj (s)|) [nij (s) · (vi (s) − vj (s))]+

i,j

Kε (xi (s), vi (s), s)ds ≤ c14 11 (log 1)−1 + c9 ε d .

(7.19)

Step 5. By Corollary 5.4,
T  Eε εd V σ,ε (|xi (s) − xj (s)|)B(vi (s) − vj (s), −nij (s)) 0

i,j

  ×? |nij (s) · (vi (s) − vj (s))| < δ Kε (xi (s), vi (s), s)ds

≤ c15 | log(δ + ε)|−1 . From this with δ = (log 1)−1/2 and (7.19) we deduce
T  Eε εd V σ,ε (|xi (s) − xj (s)|)B(vi (s) − vj (s), −nij (s))Kε (xi (s), vi (s), s)ds 0

i,j

  ≤ c16 (log log 1)−1 + c16 ε d log 1 + c16 | log ε|−1 + c16 11 / log 1 , because

(7.20)

 2 (v · n)− ≤ δ −1 (v · n)− + (v · n)− ?(|v · n| ≤ δ) .

The left–hand side of (7.20) equals to




T  Eε εd V σ,ε (|xi (s) − xj (s)|)B(vi (s) − vj (s), −nij (s)) 0

i,j

× ? (τ1 (xi (s) − vi (s)s − x, vi (s) − v) < s, |vi (s) − v| ≤ 11 ) ζ ε (x, v)dsdxdv . After a change of variables (x, v) → (x − xi + vi s, v − vi ), we have that the left-hand side of (7.20) equals to




T  Eε εd V σ,ε (|xi (s) − xj (s)|)B(vi (s) − vj (s), −nij (s)) 0

i,j

? (τ1 (x, v) < s, |vi (s) − v| ≤ 11 ) ζ ε (xi (s) − vi (s)s − x, vi (s) − v)dsdxdv .

368

F. Rezakhanlou

This implies

 XT (x, v) − XT ∧τ1 (x,v) (x, v) ?(|v| ≤ 11 )dxdv

   ≤ c17 (log log 1)−1 + ε d log 1 + | log ε|−1 + 11 / log 1 .

Eε

(7.21)

Equation (7.3) now follows from this and Corollary 5.5 by choosing 11 = log log 1 because


T  2 Eε XT (x, v) − XT ∧τ1 (x,v) (x, v) ?(|v| ≥ 11 )dxdv ≤ Eε Aˆ ε (q(s)) ds . 11 0   8. The Kinetic Limit Let ζ be a nonnegative smooth function of compact support such that √ √ and define ζ ε (x, v) = ε−d ζ (x/ ε, v/ ε). Define a process F ε to be F ε (x, v, t; q) = F ε (x, v, t) := ε d

N 



ζ dxdv = 1,

ζ ε (xi (t) − x, vi (t) − v) .

i=1

The map q → F ε (·; q) induces a probability measure on the Skorohod space D([0, T ]; L1 (Rd × Rd ), which we refer to as Pε . We now restate Theorem 2.1 as follows: Theorem 8.1. The sequence Pε converges to P, where P is concentrated upon the single function f that solves (2.7). The proof of Theorem 8.1 is omitted because it is a straightforward consequence of Lemmas 8.2–8.4 below. (See Theorem 6.1 of [RT].) The proofs of Lemmas 8.3 and 8.4 are very similar to the proof of Lemmas 6.2 and 6.4 of [RT]. The proof of Lemma 8.3 is only sketched. Lemma 8.2. Suppose J is a smooth function of compact support. Then  

∞
 

  ∂J ∂J lim lim sup

+v· f + J Q f ∗ ζ δ , f ∗ ζ δ dxdt δ→0 ε→0 ∂t ∂x 0




+ J (x, v, 0)f 0 (x, v)dxdv

Pε (df ) = 0. (8.1) Lemma 8.3. The family Pε is precompact. Lemma 8.4. Let P be any limit point of Pε . Then any P is concentrated on the set of f for which
T



sup ψ(f ∗ ζ d (x, v, t)f ∗ ζ δ (x + σ n, v∗ , t)) δ>0 0

S

Rd

× B(v − v∗ , n)dx dv dv∗ dn dt < ∞.

A Stochastic Model Associated with Enskog Equation and its Kinetic Limit

369

Proof of Lemma 8.2. Put N  J (xi , vi , t) , G1 (t, q) = ε d
i=1 G2 (t, q) = F ε (x, v, t)J (x, v, t)dxdv .

Since J is smooth, we clearly have

√ G2 (t, q) = G1 (t, q) + O( ε) .

(8.2)

It is well known that the processes


s



 ∂G1 ε + A G1 (t, q(t))dt , ∂t

M(s) = G1 (s, q(s)) − G1 (0, q(0)) − 0
s 2 ε 2 N(s) = (M(s)) − (A G1 − 2G1 Aε G1 )(t, q(t))dt 0

are Martingales. Choose T large enough so that J (x, v, t) = 0 for t ≥ T . It is straightforward to show that A0 G21 − 2G1 A0 G1 = 0 and  |Ac G21 − 2G1 Ac G1 | ≤ c0 ε 2d V σ,ε (|xi − xj |)B(vi − vj , −nij ) , (8.3) i,j

for some constant c0 . Using this, EN (T ) = 0, Doob’s inequality and Theorem 5.1 we deduce,
T 2 2 d Eε sup (M(s)) ≤ 4Eε (M(T )) ≤ c0 ε Eε Aε (q(t))dt ≤ c1 ε d . (8.4) 0

0≤s≤T

It is not hard to show 
T ∂G1 G1 (0, q(0)) + + A0 G1 (t, q(t)) ∂t 0 

∞
 √ ∂J ∂J = J (x, v, 0)F ε (x, v, 0)dxdv + +v· F ε dxdt + O( ε) . ∂t ∂x 0 (8.5) Note that
0

T

Ac G1 (q(t))dt = Xˆ ε − Xε ,

where
Xε = Xˆ ε =

T

εd

0




0

T



V σ,ε (|xi (t) − xj (t)|)B(vi (t) − vj (t), −nij (t))J (xi (t), vi (t), t)dt,

i,j

εd

 i,j

j

V σ,ε (|xi (t) − xj (t)|)B(vi (t) − vj (t), −nij (t))J (xi (t), vi (t), t)dt .

370

F. Rezakhanlou

On the other hand, Theorem 6.1 and the Lipschitzness of B and J imply  Eε |Xε − Yε | + |Xˆ ε − Yˆε | ≤ c2 h(δ + ε) , where








Yε =

T

0

εd



(8.6)

V σ,ε (|xi (t) − z1 − xj (t) + z2 |)

i,j

× B(vi (t) − vj (t) + w1 − w2 , −nij (t, z2 − z1 ))J (xi (t) − z1 , vi (t) − w1 , t) × ζ δ (z1 , w1 )ζ δ (z2 , w2 )dz1 dw1 dz2 dw2 dt ,




T  Yˆε = εd V σ,ε (|xi (t) − z1 − xj (t) + z2 |) 0

i,j

j

× B(vi (t) − vj (t) + w1 − w2 , −nij (t, z2 − z1 ))J (xi (t) − z1 , vi (t) − w1 , t) × ζ δ (z1 , w1 )ζ δ (z2 , w2 )dz1 dw1 dz2 dw2 dt . After a change of variables we deduce,  




T  z1 − z2 Yε = εd V σ,ε (|z1 − z2 |)J (z1 , w1 , t)B w1 − w2 , − |z1 − z2 | 0 i,j

× ζ δ (xi (t) − z1 , vi (t) − w1 )ζ δ (xj (t) − z2 , vj (t) − w2 )dz1 dz2 dw1 dw2 dt  
T



z1 − z2 = ε−d V σ,ε (|z1 − z2 |)J (z1 , w1 , t)B w1 − w2 , − |z1 − z2 | 0 × (F ε ∗ ζ δ )(z1 , w1 , t)(F ε ∗ ζ δ )(z2 , w2 , t)dz1 dz2 dw1 dw2 dt . If we replace (F ε ∗ζ δ )(z2 , w2 , t) with (F ε ∗ζ δ )(z1 +σ n, w2 , t) with V σ,ε (|z1 −z2 |) = 0 −z2 and n = |zz11 −z , then we cause an error of order εδ −d−1/2 . Hence 2|
Yε =

T





ε −d V σ,ε (|z1 − z2 |)J (z1 , w1 , t)B(w1 − w2 , −n)(F ε ∗ ζ δ )(z1 , w1 , t)

0

× (F ε ∗ ζ δ )(z1 + nσ, w2 , t)dz1 dz2 dw1 dw2 dt + O(εδ −d−1/2 ) . In the same fashion we can show
T

ˆ Yε = ε −d V σ,ε (|z1 − z2 |)J (z1 , w1 , t)B(w1 − w2 , −n) 0

× (F ε ∗ ζ δ )(z1 , w1 , t)(F ε ∗ ζ δ ) × (z1 + nσ, w2 , t)dz1 dz2 dw1 dw2 dt + O(εδ −d−1/2 ) , where w1 = w1 − (n.(w1 − w2 ))n. The result follows after making a change of variables z2 − z1 → z2 and writing the z2 –integration in polar coordinates.   ˆ d × Rd ) denote the space of nonnegProof of Lemma 8.3. Put Zˆ = supε Z ε . Let M(R d d ˆ ˆ1 d ative measures µ(dx, dv) such that µ(Rd × R  ) ≤ Z. Let L (R × R ) be the space ˆ of measures µ(dx, dv) = f (x, v)dxdv with f (x, v)dxdv ≤ Z. Note that the space ˆ d ×Rd ) is a complete separable metric space with respect to the weak topology. We M(R ˆ d × Rd )) which is also a complete separable metric space. define D = D([0, T ]; M(R We regard Pε as a family of probability measures on D. Since D is a Polish space, we

A Stochastic Model Associated with Enskog Equation and its Kinetic Limit

371

can appeal to the Prohorov’s theorem to assert that the family Pε is relatively compact if it is tight. Observe that Lemma 4.4 implies,
sup Eε sup ϕ(F ε (x, v, t))dxdv < ∞. ε>0

0≤t≤T

Since the space of functions f with
sup ϕ(f (x, v, t))dxdv ≤ 1 0≤t≤T

is weakly closed in D for every 1, we deduce that any limit point of Pε is concentrated on the space D([0, T ], Lˆ 1 (Rd × Rd )). As in Lemma 8.2, one can readily show that for every smooth function J of compact support,





lim sup

J (x, v)(f (x, v, t) − f (x, v, s))dxdv ε→0

−

0≤t,s≤T


t
s

v·

∂J (x, v)f (x, v, θ )dθ + ∂x




t s



Y ε (θ )dθ

Pε (df ) = 0,

(8.7)

where by Corollary 5.3 we have




lim sup Eε

ε→0 0≤τ ≤T

τ +δ τ



YL (θ )dθ

≤ Cˆ 1 (δ),

where τ ranges over all stopping times taking values inthe interval [0, T ]. By the Aldous’ t theorem [Al] we deduce the tightness of the process 0 YL (θ )dθ . This and (8.7) imply the tightness of the family Pε .   Proof of Lemma 8.4. Define  d Q− V σ,ε (|xi (t) − xj (t)|)B(vi (t) − vj (t), nij (t)) ε (x, v, t) = ε i,j ε

Q+ ε (x, v, t)

×ζ (xi (s) − x, vi (s) − v) ,  = εd V σ,ε (|xi (t) − xj (t)|)B(vi (t) − vj (t), nij (t)) i,j ε

j

×ζ (xi (s) − x, vi (s) − v) ,

− Qε = Q + ε − Qε .

Let H : Rd × Rd → R, η : [0, ∞) → R be two smooth functions of compact support and put J (x, v, t) = H (x − vt, v)η(t). As in Lemma 8.2, it is not hard to show  

∞


∂J ∂J ε

lim Eε dx dv dt + v · F + J Q ε

ε→0 ∂t ∂x 0




+ F ε0 (x, v)J (x, v, 0)dxdv

= 0.

372

F. Rezakhanlou

We then make a change of variable (x, v, t) → (x + vt, v, t) and use Qε < Q+ ε to deduce that if both H and η are nonnegative, then 
∞

lim Eε H (x, v) (f (x + vt, v, t)η (t) + Q+ ε (x + vt, v, t)η(t))dt ε→0

0

 − +f 0 (x, v)η(0) dx =0.

Here {a}− = − min{a, 0}. Put ; ε (x, v, t) =

(8.8)




t 0

Q+ ε (x + vs, v, s)ds .

To take advantage of Theorem 7.1, we extend Pε (df ) to a probability measure P˜ε (df, d;) that is the distribution of the pair (F ε , ; ε ) ∈ D([0, T ]; Lˆ 1 (Rd × Rd )) × L1 (Rd × Rd ) with respect to Pε . Suppose η is nonincreasing and that the support of η is contained in [0, T ]. Then
∞
∞ + Qε (x + vt, v, t)η(t)dt = − ; ε (x, v, t)η (t)dt 0 0
∞ η (t)dt = ; ε (x, v, T )η(0) . ≤ −; ε (x, v, T ) 0

Hence (8.7) implies 


lim H (x, v) ε→0

∞

f (x + vt, v, t)η (t)dt + ;(x, v, T )η(0)

0

 − +f 0 (x, v)η(0) dx dv P˜ε (df, d;) = 0,

(8.9)

provided H ≥ 0, η ≤ 0 and the support of η is contained in [0, T ]. Theorem 7.1 says 

sup ψ (;(x, v, T )) P˜ ε (df, d;) ≤ C8 (T ) . (8.10) ε>0

Let us denote the integrand of (8.9) by Y(f, ;). The functional Y(f, ;) is a continuous functional of the pair (f, ;). Although Y is not bounded, by (8.10) we can approximate Y(f, ;) by a sequence of bounded continuous functionals. Therefore we can pass to the limit ε → 0. If P˜ is any limit point of P˜ε , then we have
˜ Y(f, ;)P(df, d;) = 0 . This means 

H (x, v)

∞

 f (x + vt, v, t)η (t)dt + ;(x, v, T )η(0) + f (x, v)η(0) dxdv ≥ 0 

0

0

(8.11)

A Stochastic Model Associated with Enskog Equation and its Kinetic Limit

373

˜ Since the space of nonnegative smooth functions with probability one with respect to P. is separable, we may take a countable dense set of H in (8.11) to obtain
f (x + vt, v)η (t)dt + ;(x, v, T )η(0) + f 0 (x, v)η(0) ≥ 0 , ˜ By taking another countfor almost all x, and with probability one with respect to P. able dense set of smooth η of compact support in [0, T ) with η ≥ 0, η ≤ 0, we can approximate functions of the form ?[0,s] (t) for s ∈ [0, T ], to deduce ˆ v) f (x + vt, v, t) ≤ f 0 (x, v) + ;(x, v, T ) =: ;(x,

(8.12)

˜ for almost all (x, v, t) and with probability one with respect to P. On the other hand, since ψ is convex, the functional ; → ψ(;)dx is lower semicontinuous with respect to the weak topology. Therefore (8.10) implies 

˜ ψ(;(x, v, T ))dxdv P(df, d;) < ∞ . This in turn implies,



We have
T



0


≤

S Rd

T



0

 ˜ ˆ ψ(;(x, v))dxdv P(df, d;) < ∞ .

(8.13)

ψ(f (x, v, t)f (x + σ n + z, v∗ , t))B(v − v∗ , n)dx dn dv dv∗ dt



S Rd

ˆ − vt, v);(x ˆ + σ n + z − v∗ t, v∗ )) ψ(;(x

× B(v − v∗ , n)dx dn dv dv∗ dt. Furthermore,


T



0

≤ ee



S Rd
T



ˆ − vt, v);(x ˆ + σ n + z − v∗ t, v∗ ))B(v − v∗ , n)dx dn dv dv∗ dt ψ(;(x



0

S Rd

0

S Rd

0

S Rd

ˆ − vt, v);(x ˆ + σ n + z − v∗ t, v∗ ) ≤ ee ) ?(;(x

× B(v − v∗ , n)dx dn dv dv∗ dt
T



1 ˆ − vt, v)| 2 ˆ − vt, v);(x ˆ + σ n + z − v∗ t, v∗ )| log log ;(x + ;(x ×B(v − v∗ , n)dx dn dv dv∗ dt
T



ˆ − vt, v);(x ˆ + σ n + z − v∗ t, v∗ )| + ;(x ˆ + σ n + z − v∗ t, v∗ )| 2 B(v − v∗ , n)dx dn dv dv∗ dt × log log ;(x := 21 + 22 + 23 . 1

374

F. Rezakhanlou

We then make a change of variable (x − vt, x + σ n + z − v∗ t) → (a, b) to find that 22 + 23 bounded above by a constant multiple of
 ˆ ˆ ˆ ˆ v)ψ(;(b, v∗ )) da db dv dv∗ . ψ(;(a, v));(b, v∗ ) + ;(a, This implies
T

sup ψ(f (x, v, t)f (x + σ n + z, v∗ , t))B(v − v∗ , n)dv dv∗ dx dn dt < ∞ z

0

(8.14) with probability one with respect to P. Finally by the convexity of ψ and Jensen’s inequality, we deduce
T



  ψ (f ∗ ζ δ )(x, v, t)(f ∗ ζ δ )(x + σ n, v∗ , t) S Rd

0

× B(v − v∗ , n)dx dn dv dv∗ dt
T



≤ sup ψ(f (x, v, t)f (x + σ n + z, v∗ , t))dx dn dv dv∗ dt . z

0

S Rd

This and (8.14) complete the proof.

 

Acknowledgements. I wish to thank Mario Pulvirenti for introducing me to the problem and many fruitful discussions. A part of this work was done when I was visiting Universit´a di Roma uno and Universit´a di Roma due during the Spring of 1998. I would like to thank both universities for their generous support. Special thanks to Rossana Marra and Mario Pulvirenti for their warm hospitality.

References [Al] [Ar]

Aldous, D.: Stopping times and tightness. Annals of Probability 6, 335–340 (1978) Arkeryd, L.: On the Enskog equation with large initial data. SIAM J. Math. Anal. 21, 631–646 (1990) [ArC] Arkeryd, L., Cercignani, C.: Global existence in L1 for the Enskog equation and convergence of the solutions to solutions of the Boltzmann equation. J. Statist. Phys. 59 no. 3–4, 845–867 (1990) [B] Bony, J.: Solutions globales born´ees pour les mod`eles discrets de l’equation de Boltzmann en dimension 1 d’espace. Actes Journ´ees E.D.P. St. Jean de Monts, XVI (1987) [IP] Illner, R., Pulvirenti, M.: Global validity of the Boltzmann equation for a two- and three-dimensional rare gas in vacuum. Commun. Math. Phys. 105, 189–203 (1986); Erratum and improved results, Commun. Math. Phys. 121, 143–146 (1989) [K] King, F.: BBGKY hierarchy for positive potentials. Ph.D. Thesis, Dept of Mathematics, University of California at Berkeley (1975) [L] Lanford, O.E.: Time evolution of large classical systems. In: Lecture Notes in Physics, ed. J. Moser, Vol. 38, Berlin, Heidelberg: Springer, pp. 1–111 [P1] Pulvirenti, M.: Global validity of the Boltzmann equation for a three-dimensional rare gas in vacuum. Commun. Math. Phys. 113, 79–85 (1987) [P2] Pulvirenti, M.: On the Enskog hierarchy: Analyticity, uniqueness and derivability by particle systems. Rendiconti del Circolo Matematica di Palermo, Serie II, Suppl. 45, 529–542 (1996) [R1] Rezakhanlou, F.: Kinetic limits for a class of interacting particle systems. Probab. Theory Related Fields 104, 97–146 (1996) [R2] Rezakhanlou, F.: Large deviation from a kinetic limit. Annals of Probability 26, 1259–1340 (1998) [RT] Rezakhanlou, F., Tarver III, J.L.: Boltzmann-Grad limit for a particle system in continuum. Ann. Inst. Henri Poincar´e 33, 753–796 (1997) [RD] Resibois, P., De Leener, M.: Classical Kinetic Theory of Fluids. New York: Wiley, 1977

A Stochastic Model Associated with Enskog Equation and its Kinetic Limit [T]

375

Tartar, L.: Some existence theorems for semilinear hyperbolic systems in one space variables. University of Wisconsin, MRS Technical Summary Report (1980)

Communicated by H. Spohn

Commun. Math. Phys. 232, 377–428 (2003) Digital Object Identifier (DOI) 10.1007/s00220-002-0748-6

Communications in

Mathematical Physics

Burgers Turbulence and Random Lagrangian Systems R. Iturriaga1,2 , K. Khanin2,3,4 1

Centro de Investigacion en Matematica, Apartado Postal 402, Guanajuato Gto., 36000 Mexico. E-mail: [email protected] 2 Isaac Newton Institute for Mathematical Sciences, University of Cambridge, 20 Clarkson Road, Cambridge CB3 OEH, UK. E-mail: [email protected] 3 Department of Mathematics, Heriot-Watt University, Edinburgh, UK 4 Landau Institute for Theoretical Physics, Moscow, Russia Received: 25 March 2002 / Accepted: 30 July 2002 Published online: 22 November 2002 – © Springer-Verlag 2002

Abstract: We consider a spatially periodic inviscid random forced Burgers equation in arbitrary dimension and the random time-dependent Lagrangian system related to it. We construct a unique stationary distribution for “viscosity” solutions of the Burgers equation. We also show that with probability 1 there exists a unique minimizing trajectory for the random Lagrangian system which generates a non-trivial ergodic invariant measure for the non-random skew-product extension of the Lagrangian system. 1. Introduction In this paper we discuss two closely related problems: stationary solutions for d-dimensional spatially periodic inviscid random forced Burgers equation and properties of minimizing trajectories for random time-dependent Lagrangian systems. The random forced Burgers equation was a subject of intensive study in the physical literature in the last 5 years (see [3, 5, 7, 8, 10, 24, 26, 27, 41]). The methods used vary from numerical simulations to quantum field theoretic closure techniques. Although the Burgers equation arises naturally in many different physical problems, recent interest was mostly motivated by the hydrodynamics applications and the theory of turbulence. The mathematical theory in the one-dimensional case was developed in [15, 16]. It was proven that there exists a unique stationary distribution for the solutions of the random inviscid Burgers equation, and typical solutions are piecewise smooth with a finite number of jump discontinuities corresponding to shocks. The analysis in [15, 16] was based on the study of geometric and dynamical properties of minimizing orbits. It was shown that there exists a unique global minimizer for the corresponding Lagrangian system. Moreover, the global minimizer is a hyperbolic orbit of the Lagrangian flow, and its unstable manifold is closely connected with the solutions of the inviscid Burgers equation. One can say that all shocks, except a special one which is called the main shock, are connected with local double-folds of the unstable manifold. On the contrary, the main shock has a topological nature. Its existence follows from spatial periodicity of the equation.

378

R. Iturriaga, K. Khanin

It is important to mention that the geometrical picture proved in [15] allows not only analyse the structure of singularities for typical stationary solutions, but also leads to quantitative predictions for universal scaling exponents related to the pdf (probability distribution function) for the velocity gradients ([16]). One of the main aims of the present paper is to study what happens in the ddimensional case. The methods which were used in [15, 16] were purely one-dimensional and in order to achieve our goals we use in a much more systematic way the Lagrangian formalism and the Hamilton-Jacobi equation. This brings us close to a connected line of research: study of minimizing orbits for Lagrangian systems. The theory of minimizing orbits for Lagrangian systems with autonomous or time-periodic potentials has a long history. It goes back to the works of M. Morse and G. Hedlund (see [28, 39])). In the last 15 years this subject became a very active research area in connection with Aubry-Mather theory (see [2, 13, 21, 35–38]). R. Ma˜ne´ has conjectured that for a generic time-periodic potential there exists a unique global minimizer which is a hyperbolic periodic trajectory of the Lagrangian system. This conjecture remains open. However, a related result on the uniqueness of a minimizing measure has been proved in [35]. It was also shown in [12] that the invariant measure is generically hyperbolic provided it is supported by a single periodic orbit. The work presented here can be considered as a theory of minimizing orbits for random Lagrangian systems. This problem is quite interesting on its own even without reference to the hydrodynamics and the Burgers equation. It is worthwhile to emphasise a connection between the random case and Ma˜ne´ ’s setting. Indeed, random Lagrangian systems are generic with probability 1 in the measure-theoretic sense, while Ma˜ne´ ’s conjecture is related to topologically generic Lagrangian systems. Not surprisingly, as we shall see there is also certain similarity in the results in both cases. We start with the d-dimensional Burgers equation: ∂t u + (u · ∇)u = νu + f (y, t),

(1)

where u(y, t) = (ui (y, t), 1 ≤ i ≤ d) is a velocity field, y = (yi , 1 ≤ i ≤ d) ∈ Rd , and f (y, t) = (fi (t, y), 1 ≤ i ≤ d) is an external force. As we have already mentioned above the Burgers equation is a prototype equation which naturally appears in very different physical problems. It can be obtained from the Navier-Stokes equations ∂t u + (u · ∇)u = νu − ∇p + f (y, t), div u = 0

(2)

by dropping the pressure term −∇p and the incompressibility condition div u = 0. Sometimes it is said that Burgers equation describes pressure-less turbulence. Correspondingly it is connected with the dynamics of low density gases. Since we are mostly interested in hydrodynamics motivation and a turbulent type behaviour, we shall assume that the equation is driven by a potential random force f (y, t) = −∇F (y, t), which is smooth in the space variable y and quite irregular in time. This basically means that the forcing term does not excite high harmonics. We shall also assume that the random potentials F (y, t) are spatially-periodic. The irregular behaviour in time will be captured by two main models. In the first model, which we call “white force”, the forcing term is similar to white noise in time. The second model corresponds to a “kicking force” for which the velocity field u changes discontinuously after kicks. We shall assume that the kicks occur at discrete moments of time. The forcing term is responsible for the constant pumping of energy into the system. The energy also dissipates through different dissipation mechanisms. The balance between pumping and dissipation should in principle

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lead to the establishment of a stationary probability distribution for solutions. This is a general philosophy and the first task of a mathematical theory of turbulence consists in establishing the existence and, preferably, uniqueness of this stationary distribution, or, in dynamical systems terminology, of an invariant measure. The difficulty of the problem depends on the mechanism of dissipation involved. This mechanism obviously depends on the dimension d. The difficulty of the problem increases when one passes from viscous case ν > 0 to inviscid limit ν → 0, and from dimension d = 1 to higher dimensions. It is important to mention that the construction of the stationary distribution or the invariant measure corresponds to the limit t → ∞. At the same time, the inviscid case, which is mostly interesting for turbulence theory, requires another limiting process ν → 0. It is tempting, first, to construct an invariant measure µν for ν > 0, that is first to take the limit as t → ∞ and then study the limit of µν as ν → 0. However, it is extremely difficult to control the last limiting process. Experience of the random Burgers equation, for which the invariant measure can be constructed for both viscous (see [43]) and inviscid cases, suggests that it is more productive to consider from the very beginning ν = 0, construct the unique invariant measure µ, and only later prove that µν → µ as ν → 0. As we have mentioned above, a mathematical theory starts with establishing the existence and uniqueness of an invariant measure. However, from the physical point of view the most interesting problem is the analysis of statistical properties of stationary solutions. In other words, how does a typical solution look? What are the leading singularities? How does the power spectrum decay? What are the asymptotic properties of pdf’s and the structure functions? Those questions are much harder to answer. They form the core of the problem of turbulence. It is probably fair to say that the problem of turbulence splits into a “soft part” of existence-uniqueness statements and a “hard part” of analysing properties of typical solutions. In this paper we show that for the d-dimensional spatially-periodic inviscid Burgers equation one can complete the “soft part” in a very general situation, and also answer some of the “hard part” questions. Notice that the difficulties of the multi-dimensional case are connected with a much more complicated structure of singularities for typical solutions. We shall discuss it in more detail in the last section of the paper. Coming back to the problem of Lagrangian minimizers we can roughly formulate our main results in the following way: with probability 1 there exists a unique global minimizer for a random Lagrangian system. This result holds for an arbitrary compact connected Riemannian manifold. However, in this paper we consider for simplicity only the case of the d-dimensional torus Td , which corresponds to the case of spatiallyperiodic potentials. As we shall see, the case of “kicking” force corresponds to a random generalisation of Aubry–Mather theory (see [2, 37]). The Lagrangian in this case can be considered as a random Frenkel-Kontorova type model, which describes an infinite chain of particles placed in periodic potentials and connected by elastic springs. In the random situation the potentials are chosen independently for each particle, contrary to the classical case when one periodic potential serves all of them. In the non-random case the structure of global minimizers is very complicated. It crucially depends on the arithmetic properties of the rotation vector. In the case of fixed Diophantine rotation number (d = 1) it is believed that one has transition from elliptic minimizers for small potentials to hyperbolic global minimizers when potentials are large enough. This transition is related to destruction of invariant KAM curves which correspond to elliptic minimizers. All this complicated picture should be compared with the stated above uniqueness of the global minimizer in the random situation. Quite naturally, the number-theoretical properties of the rotation vector play no role in the random case.

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The paper has the following structure. In Sect. 2 we define the random force and discuss the notion of invariant measure and stationary distribution for the random Burgers equation. In Sect. 3 we introduce the random Hamilton-Jacobi equation and discuss some properties of the minimizers. The main results are formulated in Sect. 4. In Sect. 5 we prove uniqueness statements for solutions of the random Hamilton-Jacobi equation in infinite domains. Section 6 is devoted to the stationary distribution for the random Burgers equation. We first construct the stationary distribution and then prove its uniqueness. In Sect. 7 we show that with probability 1 the global minimizer is unique. We finish with concluding remarks in Sect. 8. In the Appendix we prove technical lemmas on the stochastic Lagrangian flow. We have deliberately decided to state the results only after all the setting and preliminaries are completed. However, experts who are familiar enough with the background will be able to understand the results in Sect. 4 without introductory Sects. 2 and 3. 2. Random Setting To make the general discussion more precise we have to make concrete assumptions on the random force f (y, t) = −∇F (y, t). As we have already mentioned above, all the results of this paper can be extended to the Burgers equation and Lagrangian systems on a compact connected Riemannian manifold. However, for simplicity we have chosen to deal only with the case of the torus Td = Rd /Zd which corresponds to periodic potentials F (y, t) = F (y + m, t), m ∈ Z. We shall denote points of Td by x and points of the universal cover Rd by y. We start with the “white force” case: f (y, t) = −∇F (y, t), F (y, t) =

N


F k (y)Bk (t).

(3)

k=1

Here F k (y) are smooth periodic potentials, F k (y) = F k (y + m), m ∈ Z and Bk (t) = W˙ k (t) are independent white noises, corresponding to independent Wiener processes Wk (t). Summation in (3) can be finite or infinite. In the latter case one has to make sure that potentials F k (y) decay fast enough so that the series converges and defines a smooth forcing term f (y, t). For simplicity everywhere below we assume that N is finite. We also assume that all potentials F k (y) are C ∞ -smooth although all the results below hold true in the case when F k belong to C 5 . Burgers equation (1) with the random forcing (3) becomes the stochastic PDE. However, one can essentially eliminate the stochastic part by considering the “kicking force” model. Let
Fj (y)δ(t − tj ). (4) f (y, t) = −∇F (y, t), F (y, t) = j ∈Z

The force in (4) corresponds to the situation when the system evolves without any force between moments of kicks tj −1 and tj , so that u(y, tj −) is obtained from u(y, tj −1 +) as a result of free evolution of a system. Then, suddenly, we apply a kicking force which changes the solution discontinuously, i.e. u(y, tj +) = u(y, tj −) − ∇Fj (y).

(5)

Smooth periodic potentials {Fj (y), j ∈ Z} form a realization of some stationary potential-valued random process. The simplest situation corresponds to the Bernoulli

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process, when different Fj are chosen independently according to some probability distribution χ ∈ P(C 1 (Td )) in a space of smooth potentials. Here and below we denote by P(M) the set of all probability distributions on the measure space M. One can also make different assumptions on kicking times tj . We shall just indicate two natural cases. The first one corresponds to the periodic sequence tj = hj, j ∈ Z for some positive h in R. In the second situation kicking times are determined by the realization of a stationary random process which is independent from the random process Fj (y). For example, {tj , j ∈ Z} can be a realization of the Poisson process. In this paper we consider only the simplest case when kicking potentials form a Bernoulli sequence, and kicking times are periodic with h = 1. In both “white” and “kicked” settings the force is given by a stationary random process. Denote by (, F, P ) the probability space corresponding to this random process. We can assume without loss of generality that (, F, P ) is a complete measure space. Sometimes we shall use the notation f ω (y, t), F ω (y, t), Fjω , ω ∈ , for the forcing term and the potentials, indicating their randomness. The time shift t → t + τ corresponds to the flow of automorphisms of the probability space : θ τ :  → , θ τ P = P , F ω (y, t + τ ) = F θ

τω

(y, t), τ ∈ R.

(6)

Equality θ τ P = P in (6) expresses the stationarity of the random force. Obviously, in the case of periodic kicking times one should consider only the values τ = j, j ∈ Z in (6). We can define now a standard skew-product structure which leads to the definition of invariant measure. Consider the Cauchy problem for the random Burgers equation (1). Denote by U some natural functional space, which is invariant under the evolution given by (1), or an invariant set in such space. This means that u(y, t) ∈ U for all t > 0, provided u(y, 0) ∈ U. Denote by B ω (τ ) a nonlinear random transformation on U corresponding to the solution of the Cauchy problem on the time interval [0, τ ] with the random force f ω (y, t). In other words, u(y, τ ) = B ω (τ )u(y, 0) in the “white force” case and u(y, τ +) = B ω (τ )u(y, 0+), τ ∈ Z, in the “kicked” case. Clearly, solution of the equation for longer times implies iteration of B ω (τ ) for shifted ω. It is easy to see that this process corresponds to the non-random transformation Bˆ acting on the product space U × : ˆ )(u, ω) = (B ω (τ )u, θ τ ω). B(τ

(7)

Definition 1. A probability measure ν ∈ P(U × ) is called an invariant measure for the random equation if ˆ ) preserves ν : Bˆ ∗ (τ )ν = ν. 1. For any τ ∈ R (τ ∈ Z in the “kicked” case), B(τ 2. Marginal distribution of ν on  is given by P , i.e. ν(du, dω) = ν ω (du)P (dω), where ν ω (du) are the conditional distributions on U under the condition of fixed ω. 3. The conditional distributions ν ω are measurable with respect to the σ -algebra 0 F−∞ generated by the random process on the time interval (−∞, 0]. The last condition means that ν ω does not depend on the future behaviour of a random force f ω (y, t), t > 0. It may look a bit artificial but one may say that all physically relevant invariant measures always satisfy this condition.

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We shall describe now a slightly different point of view which also leads to the notion of a stationary distribution for solutions of the random equation. It is easy to see that B ω (τ ) defines a stationary Markov process on U with transition probabilities given by:   (8) pτ (A | u0 ) = P ω : u(y, τ ) = B ω (τ )u0 ∈ A . Definition 2. A probability measure µ ∈ P(U) is called a stationary distribution for the random equation if it is a stationary distribution for the Markov process (8), i.e. for all τ > 0 (τ ∈ N in the kicked case):  pτ (A | uo )µ(du0 ) . µ(A) = U

The notions of invariant measure and stationary distribution for a random equation are closelyconnected. Namely, if ν is an invariant measure then its marginal distribution µ(du) =  ν ω (du)P (dω) is a stationary distribution for the corresponding Markov process. Notice that this property holds only if an invariant measure ν satisfies condition 3 in Definition 1. The opposite statement is also correct. It follows from [1, 14] that any stationary distribution µ(du) can be obtained as the projection of a unique invariant measure ν(du, dω) = ν ω (du)P (dω) satisfying condition 3. 3. Random Hamilton-Jacobi Equation and Minimizers It is well known that due to the creation of shocks the inviscid Burgers equation has no strong, or, in other words, smooth solutions. However, there exists a unique “physical” weak solution for the Cauchy problem, which is called the viscosity solution. Given initial data u(x, 0) = u0 (x) one can construct the viscosity solutions by considering the viscous Burgers equation and taking the limit as ν → 0. We shall always assume that u0 is a gradient-like function, i.e. u0 = ∇φ0 , where φ0 is a Lipschitz function. It is easy to see that the Burgers equation with a potential force is an evolution equation in a space of gradient-like functions. This means that the set of gradient-like functions is an invariant set. Taking u(x, t) = ∇φ(x, t) we obtain the Hamilton-Jacobi equation for φ: 1 ∂t φ(y, t) + (∇φ(y, t))2 + F ω (y, t) = 0. (9) 2 As in the case of the Burgers equation there are many weak solutions to the HamiltonJacobi equation. However, there exists a unique viscosity solution corresponding to the Burgers viscosity solution. The exact expression for the viscosity solution is given by the Hopf-Lax-Oleinik variational principle. The variational principle was first proved in the one-dimensional case ([29, 33, 40]), and then extended to the multi-dimensional Hamilton-Jacobi equation by P. Lions ([34], see also A. Fathi [21] for the case of general Riemannian manifolds). It says that the viscosity solution of the Cauchy problem for (9) in Rd × [0, T ] with the initial condition φ(·, 0) = φ0 is given by:    t 1 2 ω (10) γ˙ − F (γ (τ ), τ ) dτ , t ∈ [0, T ], φ(y, t) = inf φ0 (γ (0)) + 2 0 where the infimum is taken over all absolutely continuous curves γ : [0, t] → Rd such that γ (t) = y. If the potentials F ω are Zd -periodic functions, and the initial value φ0 (y) can be written in the form: φ0 (y) = b · y + ψ0 (y), where b ∈ Rd and ψ0 (y) is Zd -periodic, then the solution (10) preserves the same form. Namely,

Burgers Turbulence and Random Lagrangian Systems

φ(y, t) = b · y + ψ(y, t), t > 0,

383

(11)

where ψ(y, t) is a Zd -periodic function of y. The linear form b · y can be considered as a first integral of the Hamilton-Jacobi equation. For the Burgers equation b corresponds to the average velocity  b= u(y, t) dx, Td

which is well known to be a first integral. From now one we shall assume that the value of the first integral b is fixed. Using formula (10) we get the variational principle for ψ(x, t), which we consider now as a function on the torus Td . For fixed b denote the action of a curve γ : [s, t] → Td by   t 1 b2 2 ω (γ ) = − F (γ (τ ), τ ) − ( γ ˙ (τ ) − b) dτ. (12) Aω,b s,t 2 2 s Then, ψ(x, t) =

 ψ0 (γ (0)) + Aω,b 0,t (γ ) ,

inf

γ ∈AC(0;x,t)

(13)

where AC(s; x, t) is the set of all absolutely continuous curves γ : [s, t] → Td on the torus Td such that γ (t) = x. We shall also use the following notations. Denote AC(x1 , s; x2 , t) the set of absolutely continuous curves γ : [s, t] → Td such that γ (s) = x1 and γ (t) = x2 ; AC(x, t) the set of absolutely continuous curves γ : (−∞, t] → Td such that γ (t) = x; AC the set of all absolutely continuous curves γ : (−∞, +∞) → ω,b Td . Finally, we define the Lax operator Ks,t :

 ω,b Ks,t ψ(γ (s)) + Aω,b ψ(x) = inf (14) s,t (γ ) . γ ∈AC(s;x,t)

Obviously, a function φ(y, t), y ∈ Td , t ∈ [t1 , t2 ] is a “viscosity” solution of the Hamilton-Jacobi equation if φ(y, t) = b · y + ψ(y, t), where ψ is Zd -periodic and for all t1 ≤ s < t ≤ t2 : ω,b ψ(·, s) = ψ(·, t). Ks,t

(15)

We next discuss an extension of the variational principle to the random case (3), (4) where dependence of the potentials on time is quite singular. In the case of “white force” the integral in (12) is in principle a stochastic integral and it is not immediately clear why it is well defined for all absolutely continuous curves with probability 1. However, using integration by parts we can effectively eliminate the stochastic nature of the integral:

 t N
1 b2 ω,b 2 k ˙ F (γ (τ ))Wk (τ ) − (γ˙ (τ ) − b) − dτ As,t (γ ) = 2 2 s k=1

 t N
1 b2 2 k = (γ˙ (τ ) − b) + ∇F (γ (τ )) · γ˙ (τ )(Wk (τ ) − Wk (s)) − dτ 2 2 s k=1

−

N
k=1

F k (γ (t))(Wk (t) − Wk (s)).

(16)

384

R. Iturriaga, K. Khanin

Everywhere below in the case of “white force” we shall use the action in the form (16). The proof of the fact that (11), (13), (16) indeed give the unique viscosity solution basically repeats the proof of the similar statement in [15], Appendix A. In the “kicked” case the extension of the variational principle is straightforward. Let φ(y, n±) = b · y + ψ(y, n±), ψ(y, n+) = ψ(y, n−) − F n,ω (y), u(y, n±) = d ∇φ(y, n±). Denote by Aω,b m,n (X) the “kicked” action of the sequence X = {xi ∈ T , m ≤ i ≤ n}: Aω,b m,n (X)

=

n−1 
1 i=m

2

ρb2 (xi+1 , xi ) − Fjω (xi ) −

b2 2

 ,

(17)

where ρb (x, x  ) = mink∈Zd (x − x  − b − k). Notice that ρ(x, x  ) = ρ0 (x, x  ) is the metric on a flat torus Td . Here and everywhere below we denote by y the Euclidean norm in Rd . Then,   ψ(x0 , 0−) + Aω,b (18) ψ(x, n−) = inf m,n (X) , X∈S(0;x,n)

where S(m; x, n) is the set of sequences X = {xi ∈ Td , m ≤ i ≤ n} such that xn = x. In analogy with the continuous case we shall also use the following notations: S(x  , m; x, n) is the set of all sequences {xi ∈ Td , m ≤ i ≤ n} such that xm = x  , xn = x; S(x, n) is the set of infinite sequences {xi ∈ Td , −∞ < i ≤ n} such that xn = x; S ω,b is the set of all double-infinite sequences {xi ∈ Td , −∞ < i − ∞}. Denote by Km,n the Lax operator in the “kicked” case; 

ω,b ψ(x) = inf (19) ψ(xm ) + Aω,b Km,n m,n (X) . X∈S(m;x,n)

Then, as in (15), for all m < n, m, n ∈ Z: ω,b ψ(·, m−) = ψ(·, n−), Km,n

(20)

if ψ(·, n±) defines the viscosity weak solution to the “kicked” Hamilton-Jacobi equation. Notice that the sequence ψ(·, n−) contains complete information about a solution since ψ(x, n+) = ψ(x, n−) − Fnω (x). ω,b ω,b Obviously, the Lax operators Ks,t , Km,n commute with an operation of addition of a constant. This reflects the well known fact that potentials are defined up to an additive constant. By this reason it is convenient to define the action of Lax operators on the corresponding factor space. Denote by C(Td )/R the space of equivalence classes of continuous functions on Td up to an additional constant and by ψ˜ ∈ C(Td )/R the equivalence class of the potential ψ: ψ˜ = {ψ1 ∈ C(Td ) : ψ1 = ψ + const}. ω,b ˜ ω,b ω,b ω,b ˜ We shall denote by K˜ s,t , Km,n the actions of Ks,t , Km,n on C(Td )/R. Let ψ(t), t ∈I d be a curve in C(T )/R, where I is a finite or infinite interval of time. We shall say that ˜ ψ(t), t ∈ I satisfies the factorized Hamilton-Jacobi equation for fixed b if for all s < t, s, t ∈ I : ω,b ˜ ˜ K˜ s,t ψ(s) = ψ(t).

(21)

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ω,b As above, in the “kicked case” one has to replace s, t by m < n, m, n ∈ Z, K˜ s,t by ω,b ˜ ˜ ˜ ˜ ˜ Km,n and ψ(s), ψ(t) by ψ(m−), ψ(n−). We next define an important notion of a minimizer.

Definition 3. Consider the action Aω,b . 1. A curve γ : [s, t] → Td is called a minimizer if Aω,b s,t (γ ) =

min

σ ∈AC(γ (s),s;γ (t).t)

Aω,b s,t (σ ).

2. A curve γ : [s, t] → Td is called a ψ-minimizer if

 min ψ(σ (s)) + Aω,b ψ(γ (s)) + Aω,b s,t (γ ) = s,t (σ ) . σ ∈AC(s;γ (t).t)

3. A curve γx0 ,t0 : (−∞, t0 ] → Td is called a one-sided minimizer if it is a minimizer for all (s, t), −∞ < s < t ≤ t0 and γx0 ,t0 (t0 ) = x0 . 4. A curve γ : (−∞, ∞) → Td is called a global minimizer if it is a minimizer for all (s, t), −∞ < s < t < ∞. Similar definitions apply to the “kicked” case. One has just to replace curves γ by ω,b sequences X, the action Aω,b s,t by the “kicked” action Am,n and pairs (s, t) by the pairs of integer moments of time (m, n). Clearly, minimizers are special trajectories of the Lagrangian flow Lωs , s ∈ R which is generated by solutions of the Euler-Lagrange system of equations: x˙ = v, v˙ = −∇F ω (x, t).

(22)

Although formally Eq. (22) is a stochastic differential equation, as above, after integration by parts we can regard it as a non-stochastic integro-differential equation. Let x(t), t ∈ [t1 , t2 ], be a C 1 curve on Td . We shall say that x(t) is a solution of (22) if for all s ≤ t, s, t ∈ [t1 , t2 ]:  t ∇F ω (x(τ ), τ )dτ v(t) = v(s) − s

= v(s) −

 t
N

∇F k (x(τ ))W˙ k (τ ) dτ

s k=1

= v(s) +

 t
N

(v(τ ) · ∇)∇F k (x(τ ))(Wk (τ ) − Wk (s)) dτ

s k=1

−

N


∇F k (x(t))(Wk (t) − Wk (s)),

(23)

k=1

where v(t) = x(t). ˙ It is well known (see [20, 32]) that the Euler-Lagrange system (22) defines a stochastic flow of diffeomorphisms of Td × Rd : Lωs : (x0 , v0 ) → (xs , vs ), s ∈ R.

(24)

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We call Lωs the Lagrangian flow. We shall also define the skew-product extension Lˆ s of the Lagrangian flow. Namely, we consider the flow of non-random transformations of Td × Rd ×  given by: Lˆ s : (x0 , v0 , ω) → (Lωs (x0 , v0 ), θ s ω) = (xs , vs , θ s ω), s ∈ R.

(25)

In the “kicked” case the Lagrangian dynamics is given by a random Standard-like map (instead of Euler-Lagrange equation):   xn+1 = xn + vn − ∇Fnω (xn ) mod Zd , vn+1 = vn − ∇Fnω (xn ).

(26)

Then the Lagrangian flow Lωn and its skew-product extension is defined exactly in the same way as above: Lωm : (x0 , v0 ) → (xm , vm ), Lˆ m : (x0 , v0 , ω) → (xm , vm , θ m ω), m ∈ Z.

(27)

An easy calculation shows that all minimizing sequences X are trajectories of the map (26). Remark 1. In order to show that a minimizing sequence X satisfies (26) one have to differentiate Aω,b m,n (X) with respect to xi , m ≤ i ≤ n. This may not be possible since Aω,b (X) is not everywhere differentiable. However, it is not a problem. Indeed, one can m,n easily show that Aω,b m,n (X) is differentiable if X is a minimizing sequence. ω,b Let Xx,n be a one-sided minimizing sequence which ends at point x ∈ Td at time n ∈ Z. It corresponds to a trajectory {(xi , vi ), −∞ < i ≤ n} of the random map (26). ω,b Denote by γx,n (t), t ∈ (−∞, n], the piecewise linear curve on Td which is formed by a particle moving from xi−1 to xi with the velocity vi : ω,b γx,n (t) = xi−1 + vi (t − (i − 1)), i − 1 ≤ t ≤ i, −∞ < i ≤ n.

(28)

ω,b (t), t ∈ (−∞, n], a one-sided minimizer in the “kicked” Sometimes we shall call γx,n case. Finally, we define the notion of a minimizing measure for the Lagrangian flow. Denote by T Td = Td × Rd the tangent bundle of Td and consider a probability measure χ((dx, dv), dω) on T Td × . We shall say that the measure χ is an invariant measure for the random Lagrangian flow if: 1. The probability measure χ is invariant under Lˆ s : (Lˆ s )∗ χ = χ , s ∈ R. 2. Marginal distribution of χ on  is given by P , i.e. χ = χ ω (dx, dv)P (dω), where ω χ (dx, dv) are the conditional distributions on T Td under the condition of fixed ω.

As usual in the “kicked case” one has to replace s by an integer m, and Lˆ s by Lˆ m . Notice that we don’t require invariant measures for the random Lagrangian flow to be “physical” in the sense of condition 3 of Definition 1. Namely, the conditional distributions mω (dx, dv) may depend on the future behaviour of the random potential F ω (·, t), t > 0. In fact, it will be exactly the case for invariant measures corresponding to global minimizers which we construct in the next section (see Remark 4).

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4. Formulation of the Main Results Consider the random Hamilton-Jacobi equation (9) with the “white force” potential (3) in a semi-infinite interval of time (−∞, t0 ]. It turns out that for any b ∈ Rd there exists a unique (up to a constant) “viscosity” solution. More precisely, the following theorem holds. Theorem 1. 1. For almost all ω and all b ∈ Rd , t0 ∈ R there exists a unique (up to an additive constant) function φbω (y, t), y ∈ Rd , t ∈ (−∞, t0 ], such that φbω (y, t) = b · y + ψbω (y, t), where ψbω is a Zd -periodic function and for all −∞ < s < t ≤ t0 , ω,b ω ψbω (y, t) = Ks,t ψb (y, s).

2. The function ψbω is Lipschitz in x : ψbω (·, t) ∈ Lip(Td ). If x ∈ Td is a point of ω,b differentiability of ψbω (x, t) then there exists a unique one-sided minimizer γx,t at (x, t) ω,b ω ω and its velocity is given by the gradient of φ: γ˙x,t (t) = ∇φb (x, t) = b + ∇ψb (x, t). 3. For almost all ω and all b ∈ Rd one has: lim

s→−∞

ω,b sup min max |Ks,t η(x) − ψbω (x, t) − C| = 0.

d η∈C(Td ) C∈R x∈T

Moreover, if x is a point of differentiability of φbω (x, t), so that there exists a unique ω,b one-sided minimizer at (x, t) and γ˙x,t (t) = ∇φbω (x, t), then lim

s→−∞

sup

ω,b ω,b sup |γ˙η,s;x,t (t) − γ˙x,t (t)| = 0,

η∈C(Td ) γ ω,b

η,s;x,t

ω,b is a η-minimizer for Aω,b on [s, t] at point x. where γη,s;x,t

Statement 3 of Theorem 1 means that for an arbitrary one-parameter family of the Cauchy problems with initial values ζs (y) = b · y + ηs (y) their solutions at time t converge up to an additive constant to φbω (y, t) as s → −∞. Under additional assumptions which we formulate below a similar theorem holds in the “kicked” case. Denote by supp(χ ) the support of the probability measure χ in the space of C 1 -smooth potentials: supp(χ ) = {F ∈ C 1 (Td ) : ∀  > 0, χ (O (F )) > 0}, where O (F ) is the -ball in C 1 (Td ) with the centre at F. We shall consider a probability distribution χ for the kicking potentials Fjω (x) which satisfy one of the following assumptions. Assumption 1. For any y ∈ Td there exists a potential Gy belonging to supp(χ ) such that Gy has a unique global maximum at y. For any ρ > 0 potentials Gy satisfy the following condition: a(ρ) = inf

min (Gy (y) − Gy (x)) > 0 .

y∈Td |x−y|≥ρ

(29)

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Assumption 2. Zero potential G0 ≡ 0 belongs to supp(χ ). Assumption 3. There exists a potential G belonging to supp(χ ) with a unique global maximum. Theorem 2. 1. Suppose probability distribution χ ∈ P(C 1 (Td )) satisfies Assumptions 1 or 2. Then all the statements of Theorem 1 remain true with replacement of s, t by ω,b m, n ∈ Z; φbω (y, t), ψbω (y, t), ψbω (y, s) by φbω (y, n−), psibω (y, n−), ψbω (y, m−); Ks,t , ω,b ω,b ω,b ω,b ω,b Aω,b by Km,n , Aω,b ; and γ˙x,t (t), γ˙η,s;x,t (t) by γ˙x,n (n−), γ˙η,m;x,n (n−). 2. In the case b = 0 all the statements above hold under Assumption 3. Remark 2. The first statement of Theorem 2 holds for all b ∈ Rd . The second statement says that in the case b = 0 Assumption 1 can be replaced by the much weaker Assumption 3. It is easy to see that Assumption 3 is equivalent to the following “mild” condition: the set of all C 1 -smooth potentials F with a unique global maximum has a positive χ -measure. We next formulate a very general statement which implies both Theorems 1 and 2. First we define an important notion of an -narrow place. Definition 4. An interval of time [t1 , t2 ] is called an -narrow place for an action A if there exist a subinterval [s1 , s2 ] ⊂ [t1 , t2 ] and sets M1 and M2 such that 1. For every minimizer γ = γ(x,t1 ;y,t2 ) on [t1 , t2 ] one has: γ (s1 ) ∈ M1 , γ (s2 ) ∈ M2 . 2. If x1 , x2 ∈ M1 and y1 , y2 ∈ M2 and γ1 = γ(x1 ,s1 ;y1 ,s2 ) , γ2 = γ(x2 ,s1 ;y2 ,s2 ) are two minimizers on [s1 , s2 ], then |As1 ,s2 (γ1 ) − As1 ,s2 (γ2 )| ≤ . The same definition applies in the “kicked” case for minimizing sequences X, “kicked” action A and integer time intervals [m1 , m2 ] ⊂ [n1 , n2 ]. Consider again the Banach space C(Td )/R of equivalence classes ψ˜ = {ψ1 ∈ ˜ = minC∈R ψ − CC(Td ) . C(Td ) : ψ1 = ψ + const} with the norm ψ Proposition 1. Fix b ∈ Rd . Suppose that for any  > 0 there exists an -narrow place for an action Ab (Ab ) belonging to (−∞, t0 ] ((−∞, n0 ]). Then ˜ ˜ 1. There exists a unique solution ψ(t), t ∈ (−∞, t0 ] (ψ(n±), n ∈ (−∞, n0 ]) of the factorized Hamilton-Jacobi equation (see (21)). 2. For any t ∈ (−∞, t0 ] (n ∈ (−∞, n0 ]) one has: sup

lim

s→−∞

η∈C( ˜ Td )/R

lim

m→−∞

˜ K˜ s,t η˜ − ψ(t) = 0 (in the “white f orce” case),

sup η∈C( ˜ Td )/R

˜ K˜ m,n η˜ − ψ(n) = 0 (in the “kicked” case).

We shall show in the next section that Theorems 1, 2 follow easily from Proposition 1 and the following result.

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Proposition 2. Consider the action Aω,b in the case of the “white force” potential (3) or Aω,b in the “kicked” case (4). For P-almost all ω the following statements hold: 1. In the “white force” case for all b ∈ Rd , t0 ∈ R and all  > 0 there exists an -narrow place for Aω,b in (−∞, t0 ]. 2. In the “kicked case” for all b ∈ Rd , n0 ∈ Z and all  > 0 there exists an -narrow place for Aω,b in (−∞, n0 ], provided Assumptions 1 or 2 hold. 3. If Assumption 3 holds then for b = 0, all n0 ∈ Z and all  > 0 there exists an -narrow place for Aω,0 in (−∞, n0 ]. The following corollary is a simple consequence of Theorems 1, 2. Corollary 1. Consider the random Burgers equation (1) in a semi-infinite interval of time (−∞, t0 ] ((−∞, n0 ]) with the “white force” (3) or with the “kicked” force (4) under the condition that it satisfies one of the Assumptions 1 or 2. Then for P-almost all ω the following statements hold: 1. For all b ∈ Rd there exists a unique “viscosity” solution uωb (x, t), t ∈ (−∞, t0 ] (or uωb (x, n±), n ∈ (−∞, n0 ]) with the average velocity b. 2. For Lebesgue-almost all x, ω,b (t), t ∈ (−∞, t0 ], uωb (x, t) = ∇φbω (x, t) = γ˙x,t ω,b where γx,t is the unique one-sided minimizer at (x, t). In the “kicked” case: ω,b uωb (x, n−) = ∇φbω (x, n−) = γ˙x,n (n−), n ∈ (−∞, n0 ].

3. For all all b ∈ Rd , t ∈ (∞, t0 ] and an arbitrary sequence of the Cauchy problems for the forced Burgers equation with the initial valueat time sk , sk → −∞ as k → ∞, given by gradient-like functions wk (x, sk ), such that Td wk (x, sk )dx = b, the following convergence holds for Lebesgue-almost all x ∈ Td : lim wk (x, t) = uωb (x, t),

k→∞

(30)

where wk (x, t) is the “viscosity” solution of the corresponding Cauchy problem. In the “kicked” case one has to replace t by n−, n ∈ Z in (30). 4. In the case b = 0 and “kicked” force all the statements above hold under Assumption 3. Corollary 1 implies the uniqueness of the stationary distribution for the random Burgers equation. Denote by Ub = {u(x) : u(x) = b + ∇ψ(x), ψ ∈ Lip(Td )}. Obviously, Ub is an invariant set for the Burgers dynamics. We shall regard Ub as a subset of a space Lp (Td , dx), 1 ≤ p < ∞, of vector-valued functions on Td . Since Lp (Td , dx) is equipped with the Borel σ -algebra B p , the embedding above defines a measurable structure on Ub . It is easy to see that Ub itself is a measurable subset of Lp (Td , dx) with respect to B p for all 1 ≤ p < ∞. Lemma 1. The σ -algebras generated on Ub by B p coincide for all 1 ≤ p < ∞. Denote by Bb the corresponding σ -algebra on Ub . Define the following mapping U :  → Ub , U (ω) = uωb (x, 0) (or U (ω) = uωb (x, 0+) in the “kicked” case).

(31)

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Lemma 2. The mapping U :  → Ub is measurable with respect to the σ -algebra Bb . Denote by δbω (du) the atomic measure on Ub with the atom at uωb (x, 0) (or at the uωb (x, 0+) in the “kicked case”), and define a probability measure νb on P(Ub × ) by νb (du, dω) = δbω (du)P (dω).

(32)

We shall also consider the marginal distribution of νb on Ub . Denote this marginal distribution by µb :  µb (du) = δbω (du)P (dω). (33) 

Theorem 3. Consider the evolution given by the random Burgers equation (1) with the “white force” (3) or with the “kicked” force (4). In the latter case assume that one of the Assumptions 1 or 2 is satisfied. Then the following statements hold: 1. For all b ∈ Rd the probability measure νb ∈ P(Ub × ) is the unique invariant measure for the skew-product dynamics (7) on Ub ×  with the marginal distribution on  equal to P (dω). 2. For all b ∈ Rd the probability measure µb ∈ P(Ub ) is the unique stationary distribution for the Markov process (8) on Ub . 3. In the case b = 0 and “kicked” force both statements above hold under Assumption 3. We next discuss the properties of minimizers. It follows from Theorem 1 that for a fixed time t and for Lebesgue-almost all points x there exists a unique one-sided minimizer. Points of non-uniqueness are called shock waves, or simply shocks. Every one-sided minimizer γx,t0 is a trajectory of the Lagrangian flow corresponding to the Euler-Lagrange system (22), or the random map (26). It can be continued as a trajectory of Lagrangian flow for all t. However, its continuation is likely not to be a one-sided minimizer for large enough t > t0 . Global minimizers correspond to exactly those onesided minimizers that can be continued as minimizers for all t. It turns out that in a very general situation there exists a unique global minimizer. We shall first formulate a theorem in the “white” force case. Theorem 4. Suppose that a mapping (F 1 (x), . . . , F N (x)) : Td → RN

(34)

is an embedding. Then for all b ∈ Rd and P-almost all ω there exists a unique global minimizer xgω,b (t), t ∈ R, for the action Aω,b . In the “kicked” case we shall assume that the kicking potentials Fjω (x) are given by linear combination of the smooth non-random potentials F i , 1 ≤ i ≤ M with random coefficients. Assumption 4. Fjω (x) =

M


ξji (ω)F i (x), j ∈ Z,

i=1

where the random vectors ξj (ω) = {ξji (ω), 1 ≤ i ≤ M} are independent identically distributed vectors in RM with an absolutely continuous distribution.

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Theorem 5. 1. Suppose that Assumption 4 and one of the Assumptions 1, 2 hold, and in addition the potentials F i (x), 1 ≤ i ≤ M, are such that the mapping (F 1 (x), . . . , F M (x)) : Td → RM

(35)

is one-to-one. Then for all b ∈ Rd and P-almost all ω there exists a unique global minimizing sequence Xgω,b (n), n ∈ Z for the action Aω,b . 2. In the case b = 0 one has uniqueness of a global minimizing sequence for the action Aω,0 for P-almost all ω, provided that Assumptions 3, 4 hold and the mapping (35) is one-to-one. Remark 3. 1. In Theorems 4, 5 the full P -measure set of ω for which one has uniqueness depends on b. The stronger statement of existence of a “universal” full P -measure set of ω for which uniqueness holds for all b ∈ Rd is probably incorrect. 2. The injectivity conditions (34), (35) are quite natural. They exclude the situation when all the potentials have an additional symmetry which results in non-uniqueness of the global minimizer. We shall discuss this in more detail in the last section. 3. It is easy to give examples of potentials for which the conditions of Theorems 4, 5 hold. Thus in the case d = 1 condition (34) holds if F 1 (x) = sin 2π x and F 2 (x) = cos 2π x. In the “kicked” case it is enough to consider Fjω (x) = ξj1 (ω) sin 2π x + ξj2 (ω) cos 2π x . Obviously Assumptions 2, 4 and condition (35) hold if (ξj1 (ω) , ξj2 (ω)) have positive density inside a disc Dr = {(ξ 1 )2 + (ξ 2 )2 ≤ r} , r > 0. It is easy to see that Assumption 1 is satisfied if (ξj1 (ω) , ξj2 (ω)) have positive density inside an annulus Ar1 ,r2 = {r1 ≤ (ξ 1 )2 + (ξ 2 )2 ≤ r2 } , r2 > r1 > 0. Denote by xgω,b the position of the unique global minimizer or minimizing sequence at time t0 = 0, or n0 = 0. For fixed t > 0 or n > 0 consider all one-sided minimizω,b ω,b ers γx,t or one-sided minimizing sequences Xx,n , x ∈ Td . Denote by E ω,b (t) the set of points at which one-sided minimizers or minimizing sequences cross Td at t = 0: ω,b ω,b E ω,b (t) = {x0 = γx,t (0)} in the “white” case, or E ω,b (t) = {x0 = Xx,n (0)} in the ω,b ω,b “kicked” case. Notice that xg ∈ E (t). The following statement follows immediately from the uniqueness of the global minimizer. Corollary 2. If there exists a unique global minimizer or minimizing sequence then diam(E ω,b (t)) → 0 as t → ∞. As we have mentioned above the unique global minimizer or global minimizing sequence gives rise to an invariant measure for the Lagrangian flow. To define this measure consider the velocity vgω,b of the global minimizer at time t0 = 0 : vgω,b = x˙gω,b (0): In the “kicked” case vgω,b = v0 , where {(xn , vn ), n ∈ Z} is the trajectory of the random map (26) corresponding to the global minimizing sequence Xgω,b (n), n ∈ Z. It is easy to see that v0 is the velocity at time n = 0-. Define a probability measure κ ∈ P(Td × Rd × ) by κ b (dx, dv, dω) = δx ω,b ,v ω,b (dx, dv)P (dω), g

g

(36)

where δx ω,b ,v ω,b (dx, dv) is an atomic measure in P(Td ×Rd ) with the atom at (xgω,b , vgω,b ). g

g

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Corollary 3. Suppose all the assumptions of Theorems 4, 5 are satisfied. Then for all b ∈ Rd the probability measure κ b (dx, dv, dω) is an ergodic invariant measure for the skew-product extension Lˆs (or Lˆm in the “kicked” case) of the Lagrangian flow. Remark 4. As we have already mentioned above the minimizing measures κ b (dx, dv, dω) although invariant under skew-product extension of the Lagrangian flow, does not satisfy Condition 3 of Definition 1. Indeed, the random variables xgω,b , vgω,b are measur∞ and are not measurable with respect to able only with respect to the full σ -algebra F−∞ 0 . the “past” σ -algebra F−∞ 5. Uniqueness of Solutions for Random Hamilton-Jacobi Equation We shall first prove Proposition 1, then derive Theorems 1, 2 and Corollary 1 from Propositions 1, 2. Finally we shall prove Proposition 2. Except for Proposition 2 the proofs in the “white” and the “kicked” cases are the same. So we only consider the “white force” case. Since for all the statements with the exception of Proposition 2 the dependence on b is not essential, we fix b and omit it from most of our notations. For the proof of Proposition 1 we need the following three lemmas. Let 1 ⊂  be the set of ω such that the realizations of the Wiener processes Wk (t), 1 ≤ k ≤ N, are continuous functions of t. Obviously P (1 ) = 1. ω is a weak Lemma 3. For all ω ∈ 1 and for all −∞ < s < t < ∞, the operator Ks,t contraction in the uniform topology, that is for every pair ψ1 , ψ2 ∈ C(Td ) we have: ω ω ψ1 − Ks,t ψ2 ||C ≤ ||ψ1 − ψ2 ||C . ||Ks,t

(37)

Proof. Recall that ω ψ1 (x) = Ks,t

inf

γ ∈AC(s;x,t)



 ψ1 (γ (s)) + Aωs,t (γ ) .

Let γ = γ(ψ1 ,s;x,t) be a ψ1 -minimizer, i.e. a curve such that the infimum in the above equation is realized. We have: ω ψ1 (x) = ψ1 (γ (s)) + Aωs,t (γ ), Ks,t

and ω ψ2 (x) ≤ ψ2 (γ (s)) + Aωs,t (γ ). Ks,t

Subtracting we get ω ω ψ2 (x) − Ks,t ψ1 (x) ≤ ψ2 (γ (s)) − ψ1 (γ (s)) ≤ ||ψ1 − ψ2 ||C . Ks,t

Interchanging ψ1 and ψ2 we obtain ω ω ψ2 (x) − Ks,t ψ1 (x)| ≤ ||ψ1 − ψ2 ||C , |Ks,t

which implies (37).

 

Burgers Turbulence and Random Lagrangian Systems

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Lemma 4 (Semigroup Property). For all ω ∈ 1 and for all −∞ < s < τ < t < ∞ one has: ω ω ω Ks,t = Kτ,t ◦ Ks,τ .

Moreover if γ = minimizer on [τ, t].

ω γ(ψ,s;x,t)

is a ψ-minimizer and ψτ =

Proof. By definition we have:

(38) ω ψ Ks,τ

then γ |[τ,t] is a ψτ -



 ω Ks,τ ψ(σ (τ )) + Aωτ,t (σ ) σ ∈AC(τ ;x,t)   = inf ψ(β(s)) + Aωs,τ (β) + Aωτ,t (σ ) β∈AC(s;σ (τ ),τ ),σ ∈AC(τ ;x,t)   ψ(γ (s)) + Aωs,t (γ ) = inf

ω ω Kτ,t ◦ Ks,τ ψ(x) =

inf

γ ∈AC(s;x,t)

ω = Ks,t ψ(x).

To prove the second assertion suppose that there exist σ1 such that Aωτ,t (σ1 ) + ψτ (σ1 (τ )) < Aωτ,t (γ ) + ψτ (γ (τ )). ω and by σ : [s, t] → Td the curve which coincides Denote by σ2 a minimizer γ(ψ,s;σ 1 (τ ),τ ) with σ1 on [τ, t] and with σ2 on [s, τ ]. We have:

ψτ (σ1 (τ )) = Aωs,τ (σ2 ) + ψ(σ2 (s)), ψτ (γ (τ )) = Aωs,τ (γ ) + ψ(γ (s)). Hence, Aωs,t (σ ) + ψ(σ (s)) = Aωs,τ (σ2 ) + Aωτ,t (σ1 ) + ψ(σ2 (s)) < Aωs,t (γ ) + ψ(γ (s)), which is a contradiction.

 

Remark 5. It follows immediately from Lemmas 3, 4 that ω ˜ ω ˜ ψ1 − K˜ s,t ψ2 || ≤ ||ψ˜ 1 − ψ˜ 2 || ||K˜ s,t

(39)

ω ω ω K˜ s,t = K˜ τ,t ◦ K˜ s,τ .

(40)

and

Lemma 5 ((Main Lemma)). If [t1 , t2 ] is an -narrow place then for every ψ1 , ψ2 ∈ C(Td ) there exists a constant c such that |Ktω1 ,t2 ψ1 (x) + c − Ktω1 ,t2 ψ2 (x)| ≤ 3.

(41)

Proof. Let [s1 , s2 ] ⊂ [t1 , t2 ] be the subinterval and M1 , M2 subsets of Td from Definition 4. Suppose γ is an arbitrary ψ-minimizer on [t1 , t2 ]. Then for all p ∈ M1 we have: ψ  (p) ≥ ψ  (γ (s1 )) − ,

(42)

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where ψ  = Ktω1 ,s1 ψ. To prove (42) consider α = γψ,t1 ;p,s1 , β = γ(p,s1 ;γ (s2 ),s2 ) and   α if τ ∈ [t1 , s1 ] γ¯ (τ ) = β if τ ∈ [s1 , s2 ] .  γ if τ ∈ [s , t ] 2 2 If (42) does not hold we get: ψ(γ¯ (t1 )) + Aωt1 ,t2 (γ¯ ) = ψ  (p) + Aωs1 ,t2 (γ¯ )

< ψ  (γ (s1 )) + Aωs1 ,s2 (β) + Aωs2 ,t2 (γ ) − 

≤ ψ  (γ (s1 )) + Aωs1 ,s2 (γ ) +  + Aωs2 ,t2 (γ ) − 

= ψ  (γ (s1 )) + Aωs1 ,t2 (γ ) = ψ(γ (t1 )) + Aωt1 ,t2 (γ ), which contradicts the fact that γ is a ψ-minimizer on [t1 , t2 ]. ω For any ψ ∈ C(Td ) denote by Mψ the set of points p = γ(ψ,t (s1 ) for some 1 ;x,t2 ) d x ∈ T . Clearly Mψ is a subset of M1 . Using (42) we get |ψ  (p, s1 ) − ψ  (q, s1 )| ≤ ,

(43)

for all p, q in Mψ . Let ψ1 , ψ2 ∈ C(Td ). The estimate (43) implies that there exists a constant c such that for all p ∈ Mψ1 and q ∈ Mψ2 we have |ψ1 (p) + c − ψ2 (q)| ≤ 2.

(44)

ω ω Finally, let γ1 , γ2 be ψ1 , ψ2 -minimizers at x ∈ Td : γ1 = γ(ψ , γ2 = γ(ψ . 1 ,t1 ;x,t2 ) 2 ,t1 ;x,t2 ) Denote p1 = γ1 (s1 ), p2 = γ1 (s2 ), q1 = γ2 (s1 ), q2 = γ2 (s2 ). Also denote by δ the ω minimizer γ(p . Then we have: 1 ,s1 ;q2 ,s2 )

Ktω1 ,t2 ψ1 (x) ≤ ψ1 (p1 ) + Aωs1 ,s2 (δ) + Aωs2 ,t2 (γ2 ),

Ktω1 ,t2 ψ2 (x) = ψ2 (q1 ) + Aωs1 ,s2 (γ2 ) + Aωs2 ,t2 (γ2 ). Hence Ktω1 ,t2 ψ1 (x) + c − Ktω1 ,t2 ψ2 (x) ≤ ψ1 (p1 ) + c − ψ2 (q1 ) + Aωs1 ,s2 (δ) − Aωs1 ,s2 (γ2 ) ≤ 2 +  ≤ 3. Interchanging ψ1 and ψ2 we get (41).

 

An easy but useful consequence of the above lemmas is the following corollary. Corollary 4. If the interval [s, t] contains an -narrow place then for every ψ˜ 1 , ψ˜ 2 ∈ ω ψ ω ψ ˜ 1 − K˜ s,t ˜ 2 || ≤ 3. C(Td )/R we have ||K˜ s,t Proof. Let [t1 , t2 ] be an -narrow place which is contained inside the interval [s, t]. Using Remark 5 we can decompose the operator K˜ s,t into three parts: ω ω K˜ s,t = K˜ tω2 ,t ◦ K˜ tω1 ,t2 ◦ Kˆ s,t . 1

It follows from (39) that the last part K˜ tω2 ,t does not increase the distance. Now it is enough to notice that (41) implies ||K˜ tω1 ,t2 η˜ 1 − K˜ tω1 ,t2 η˜ 2 || < 3 for all η˜ 1 , η˜ 2 ∈ C(Td )/R.  

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ω η˜ Proof of Proposition 1. We show that for all η˜ ∈ C(Td )/R and all t ∈ R fixed K˜ s,t satisfies the Cauchy property, i.e for all δ > 0 there exists S < t such that for all s1 , s2 < S:

˜ ≤ δ. ||K˜ sω1 ,t η˜ − K˜ sω2 ,t η||

(45)

Indeed, let [t1 , t2 ] be an 3δ -narrow place inside (−∞, t]. Put S = t1 . Then (45) follows immediately from Corollary 4 and the semigroup property 40. ω ψ ˜ converges as s → −∞. Denote the limit Since C(Td )/R is a Banach space K˜ s,t ω ˜ by ψ (t). It follows from Corollary 4 that the limit does not depend on the initial η. In fact, Corollary 4 implies the following statement: lim

s→−∞

sup η∈C( ˜ Td )/R

ω K˜ s,t η˜ − ψ˜ ω (t) = 0,

which proves the second part of Proposition 1. We next prove that ψ˜ ω (t), t ∈ R is a solution of the factorized Hamilton-Jacobi equation. Take arbitrary s < τ < t and consider an identity: ω ω ω K˜ s,t η˜ = K˜ τ,t ◦ K˜ s,τ η. ˜ Taking the limit as s → −∞ we obtain ω ˜ω ψ˜ ω (t) = K˜ τ,t ψ (τ ).

To complete the proof we just notice that Corollary 4 obviously implies the uniqueness of the solution ψ˜ ω (t).   For the proof of Theorem 1 we shall need the following lemmas. Lemma 6. For all ω ∈ 1 and all −∞ < s < t < ∞ there exists a constant C(ω, s, t) such that: sup x1 , x2

where |v| = v =

∈Td

sup ω γ(x 1 ,s;x2 ,t)

ω max |γ˙(x,s;y,t) (τ )| ≤ C(ω, s, t),

τ ∈[s,t]

(46)

√ v · v, v ∈ Rd .

Proof. From the formula (23) it follows that there exist positive constants K1 (s, t, ω), K2 (s, t, ω) such that  τ |v(τ )| ≤ |v(s)| + K1 (s, t, ω) + K2 (s, t, ω) |v(u)|du. s

Hence, using the Gronwall inequality we have: |v(τ )| ≤ (|v(s)| + K1 (s, t, ω))eK2 (s,t,ω)(τ −s) .

(47)

The estimate (47) implies that for any v0 > 0 there exists V ω (v0 ) such that if |v(τ0 )| ≥ V ω (v0 ) for some τ0 ∈ [s, t] then |v(τ )| ≥ v0 for all τ ∈ [s, t]. Denote by (v0 ) = {γ ∈ C 1 [s, t] : |γ˙ (τ )| ≥ v0 for all τ ∈ [s, t]}. Let Aω (v0 ) =

inf

γ ∈(v0 )

Aωs,t (γ ).

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It follows immediately from (16) that Aω (v0 ) → ∞ as v0 → ∞. Denote by δ(x1 ,s;x2 ,t) a curve with a constant velocity v ≡ δ˙(x1 ,s;x2 ,t) (τ ), τ ∈ [s, t] connecting points x1 and x2 . It is easy to √ see that for all x1 , x2 one can always choose δ(x1 ,s;x2 ,t) in such a way that |v| ≤ 21 (t − s) d. Let  be the set of all curves δ(x1 ,s;x2 ,t) for different x1 , x2 ∈ Td √ satisfying the condition |v| ≤ 21 (t − s) d. Denote by A¯ ω the supremum of actions of the curves γ¯ ∈ : A¯ ω = sup Aωs,t (δ). δ∈

Take now v0 so big that Aω (v0 ) > A¯ ω . Then, obviously, ω |γ˙(x,s;y,t) (τ )| ≤ V (v0 ).

 

Remark 6. In the “kicked” case one can give a more explicit estimate for velocities of minimizers. Suppose X = {xi ∈ Td , m ≤ i ≤ n} is a minimizing sequence for the j action Aω,b m,n , where b = (b , 1 ≤ j ≤ d). Let vi , m + 1 ≤ i ≤ n, be the velocities ω,b corresponding to the minimizer X : γ˙x,n (i−) = vi (see (26), (28)). It is easy to see j that for all m + 1 ≤ i ≤ n, 1 ≤ j ≤ d one has |vi − bj | ≤ 21 . Hence, √ d |vi | ≤ |b| + , m+1≤i ≤n. (48) 2 The next lemma is a very general existence statement which follows easily from estimates (46), (48). Lemma 7. Consider the action Aω,b corresponding to the “white force” potential (3) or Aω,b in the case of the “kicked” potential (4). In both cases there exists a subset of  of full P -measure such that the following statements hold: 1. For all b ∈ Rd , x ∈ Td , t ∈ R, or n ∈ Z in the “kicked” case, there exists at ω,b least one one-sided minimizer γx,t (τ ), τ ∈ (−∞, t], which ends at (x, t), or, in the ω,b “kicked” case, there exists at least one one-sided minimizing sequence Xx,n (m), m ∈ (−∞, n], which ends at (x, n). 2. For all b ∈ Rd there exists at least one global minimizer γ ω,b (τ ), τ ∈ R, or, in the “kicked” case, there exists at least one global minimizing sequence X ω,b (m), m ∈ Z. Proof. The proof is based on a general compactness argument. For an arbitrary (x, t) consider a sequence of finite-time minimizers γn (τ ) = γxn ,−n;x,t (τ ), τ ∈ [−n, t]. Denote vn = γ˙n (t). It follows from (46), (48) that |vn | are uniformly bounded. Hence there exists a convergent subsequence vnk → v as k → ∞. Denote by γv (τ ), τ ∈ (−∞, t] a solution of Euler-Lagrange equation such that γv (t) = x, γ˙v (t) = v . Since the set of all minimizers is closed one can easily show that γv is a one-sided minimizer at (x, t). To construct a global minimizer take again a sequence of finite-time minimizers γ¯n (τ ) = γxn ,−n;xn ,n (τ ). Denote yn = γ¯n (0), v¯n = γ˙¯ n (0). Again there exists a convergent subsequence ynk → y, v¯nk → v¯ as k → ∞. It is easy to see that a solution of the Euler-Lagrange equation γ0 (τ ), τ ∈ R with the initial data γ0 (0) = y, γ˙0 (0) = v¯ is a global minimizer.   In the following two lemmas we demonstrate the close connection between the notions of one-sided minimizer and ψ-minimizer, where ψ is a solution to the Hamilton-Jacobi equation. Both lemmas are formulated for the “white force” case. The generalization for the “kicked” case is straightforward.

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Lemma 8. Let φ(x, t) = b · y + ψ(x, t), t ∈ (−∞, t0 ] be an arbitrary solution to the Hamilton-Jacobi equation in (x ∈ Td , t ∈ [−∞, t0 ]) and γ¯ (τ ), τ ∈ [t1 , t0 ] be a ψ-minimizer on [t1 , t0 ]. Then γ¯ can be continued to a one-sided minimizer. Namely, there exists a one-sided minimizer γ (τ ), τ ∈ (−∞, t0 ] such that γ (τ ) = γ¯ (τ ), τ ∈ [t1 , t0 ]. In addition γ (τ ), τ ∈ [t, t0 ] is a ψ-minimizer on [t, t0 ] for any t < t0 . Proof. Let γ¯ (τ ), τ ∈ [t1 , t0 ] be a ψ-minimizer on [t1 , t0 ]. Denote by γ¯1 (τ ), τ ∈ [t1 − 1, t1 ] a ψ-minimizer on [t1 − 1, t1 ] which starts at a point (γ¯ (t1 ), t1 ) and by γ1 (τ ), τ ∈ [t1 − 1, t0 ] the curve which is the union of γ¯ and γ¯1 . It is easy to see that γ1 is a ψ-minimizer on [t1 − 1, t0 ]. Indeed for any other curve σ (τ ), τ ∈ [t1 − 1, t0 ] we have At1 ,t0 (γ¯ ) + ψ(γ¯ (t1 ), t1 ) = At1 −1,t0 (γ ) + ψ(γ (t1 − 1), t1 − 1) and

At1 ,t0 (σ ) + ψ(σ (t1 ), t1 ) ≤ At1 −1,t0 (σ ) + ψ(σ (t1 − 1), t1 − 1).

Since

At1 ,t0 (γ¯ ) + ψ(γ¯ (t1 ), t1 ) ≤ At1 ,t0 (σ ) + ψ(σ (t1 ), t1 )

we conclude that At1 −1,t0 (γ ) + ψ(γ (t1 − 1), t1 − 1) ≤ At1 −1,t0 (σ ) + ψ(σ (t1 − 1), t1 − 1). In the same way we can construct a ψ-minimizer γ2 (τ ), τ ∈ [t1 − 2, t0 ] which is a continuation of γ1 and so on. By induction we get a curve γ (τ ), τ ∈ (−∞, t0 ] which is a ψ-minimizer on [t, t0 ] for any t < t0 and coincides with γ¯ (τ ) for all τ ∈ [t1 , t0 ]. It is also obvious that γ is a one-sided minimizer.   Lemma 9. Suppose that for any t0 ∈ R and any  > 0 there exists an -narrow place for an action Aω,b belonging to (−∞, t0 ]. Let φbω (x, t) = b · y + ψbω (x, t), t ∈ (−∞, t0 ] be a solution to the Hamilton-Jacobi equation in (x ∈ Td , t ∈ [−∞, t0 ]). Then the following statements hold. ω,b (τ ), τ ∈ (−∞, t0 ], is also a ψ-minimizer for 1. Any one-sided minimizer γ0 (τ ) = γx,t 0 ω ψ = ψb (x, t) and all t ≤ t0 . 2. Suppose two one-sided minimizers γxω,b (τ ), γxω,b (τ ) intersect, i.e. there exists τ0 ≤ 1 ,t1 2 ,t2 ω,b min(t1 , t2 ) such that γxω,b (τ ) = γ (τ ). Then either x1 = x2 and t1 = t2 , or 0 x2 ,t2 0 1 ,t1 ω,b ω,b γx1 ,t1 (τ ) = γx2 ,t2 (τ ) for all τ ≤ min(t1 , t2 ).

Proof. We start with the proof of the first statement. Suppose γ0 is not a ψ-minimizer for some t¯ < t0 . Then for some  > 0, At¯,t0 (σ¯ ) + ψ(σ¯ (t¯), t¯) +  = At¯,t0 (γ0 ) + ψ(γ (t¯), t¯), where σ¯ is a ψ-minimizer on [t¯, t0 ] at point x. Let [t1 , t2 ] be a 8 -narrow place, with t2 < t¯, and [s1 , s2 ] ⊂ [t1 , t2 ] the corresponding subinterval (see Definition 4). Denote by σ a continuation of σ¯ to a ψ-minimizer on [t1 , t0 ]. Obviously, for any τ in [t1 , t¯] we have Aτ,t0 (γ0 ) = Aτ,t¯(γ0 ) + At¯,t0 (γ0 ) ≥ At¯,t0 (γ0 ) + ψ(γ0 (t¯), t¯) − ψ(γ0 (τ ), τ ) = At¯,t0 (σ ) + ψ(σ (t¯), t¯) +  − ψ(γ0 (τ ), τ ) = Aτ,t0 (σ ) + ψ(σ (τ ), τ ) +  − ψ(γ0 (τ ), τ ).

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In particular, we get for τ = s2 : As2 ,t0 (σ ) + ψ(σ (s2 ), s2 ) +  ≤ As2 ,t0 (γ0 ) + ψ(γ0 (s2 ), s2 ).

(49)

On the other hand, ψ(γ0 (s2 ), s2 ) − ψ(σ (s2 ), s2 ) ≤

 . 4

(50)

Indeed, since σ is a ψ-minimizer, σ (s1 ) ∈ M1 , σ (s2 ) ∈ M2 . Similarly, γ0 (s1 ) ∈ M1 , γ0 (s2 ) ∈ M2 . Consider now  γ = Then and

σ γσ (s1 ),s1 ;γ0 (s2 ),s2

for τ ∈ [t1 , s1 ] for τ ∈ [s1 , s2 ].

ψ(σ (s2 ), s2 ) = At1 ,s1 (σ ) + As1 ,s2 (σ ) + ψ(σ (t1 ), t1 ), ψ(γ0 (s2 ), s2 ) ≤ At1 ,s1 (σ ) + As1 ,s2 (γ ) + ψ(σ (t1 ), t1 ).

Subtracting and using |As1 ,s2 (σ )|, |As1 ,s2 (γ )| ≤

 , 8

one gets (50). Combining (49) and (50) we have As2 ,t0 (γ0 ) ≥ As2 ,t0 (σ ) +

3 . 4

Finally, using a minimizing curve γγ0 (s1 ),s1 ;σ (s2 ),s2 we construct a curve  γγ0 (s1 ),s1 ;σ (s2 ),s2 for τ ∈ [s1 , s2 ] γ˜ = σ for τ ∈ [s2 , t0 ]. Since |As1 ,s2 (γ0 )|, |As1 ,s2 (α)| ≤ 8 , we obtain As1 ,t0 (γ0 ) = As1 ,s2 (γ0 ) + As2 ,t0 (γ0 ) ≥ As1 ,s2 (α) + As2 , t0 (σ ) +  = As1 ,t0 (γ˜ ) + , 2

 2

which is a contradiction with the fact that γ0 is a one-sided minimizer. To prove the second statement suppose that γ1 (τ ) = γxω,b (τ ), γ2 (τ ) = γxω,b (τ ) 1 ,t1 2 ,t2 have a “proper” intersection. Assume without loss of generality that t1 ≥ t2 . Then there exists t < t1 such that γ1 (t) = γ2 (t). It was shown above that for any t¯ < t the curves γ1 , γ2 are ψ-minimizers on [t¯, t1 ] and [t¯, t1 ] respectively. This implies that a curve  γ2 (τ ) for τ ∈ [t¯, t] γ˜3 (τ ) = γ1 (τ ) for τ ∈ [t, t1 ] is also a ψ-minimizer on [t¯, t1 ]. However γ3 (τ ) is not a solution of the Euler-Lagrange equation. This contradiction finishes the proof.  

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Remark 7. The classical result in the non-random case states that two minimizers cannot intersect twice. It is interesting to compare this fact with the second statement of Lemma 9 which implies that in the presence of -narrow places two one-sided minimizers cannot have even one “proper” intersection. An intersection is possible only when both one-sided minimizers γ1 , γ2 start at the same point (x, t). It follows from the lemma above that both γ1 and γ2 cannot be continued as one-sided minimizers beyond time t. Points (x, t) of non-uniqueness of one-sided minimizers are called shocks. As we show below they correspond to singularities of ψbω (see Theorems 1, 2). The statement of the following lemma is a well-known general fact in the case when a force depends smoothly on time. We show that the same statement holds in the “white force” setting. Lemma 10. Let γ0 (τ ), τ ∈ [s, t], be an arbitrary solution to the Euler-Lagrange system of Eqs. (22) and γ (τ, ), τ ∈ [s, t], be a C 1 -smooth variation of γ0 with the fixed end point at time s: γ (τ, 0) = γ0 (τ ), τ ∈ [s, t] ; γ (s, ) = γ0 (s). Then

  ∂Aωs,t (γ (·, )  ∂γ (t, )   = (γ˙0 (t) − b) · .  ∂ ∂ =0 =0

(51)

(τ,) |=0 . From formula (16) for the action we have: Proof. Define η(τ ) = ∂γ ∂   t ∂Aωs,t (γ (·, )   = (γ˙0 (τ ) − b) · η(τ ˙ ) dτ  ∂ s =0  t
N   + (γ˙0 (τ ) · ∇)(∇F k (γ0 (τ )) · η(τ ))(Wk (τ ) − Wk (s)) dτ s k=1

+

 t
N 

 ˙ ))(Wk (τ ) − Wk (s)) dτ (∇F k (γ0 (τ )) · η(τ

s k=1

−

N


(∇F k (γ0 (t)) · η(t))(Wk (t) − Wk (s)).

(52)

k=1

Using (23) for v(τ ) = γ˙0 (τ ) we obtain:  s

t



t

(γ˙0 (τ ) − b) · η(τ ˙ ) dτ =

γ˙0 (s) · η(τ ˙ ) dτ +

s

 t  s

s

N τ


(γ˙0 (ρ) · ∇)

k=1



˙ ) dτ ×∇F (γ0 (ρ))(Wk (ρ) − Wk (s))dρ · η(τ k

− −

 t
N s k=1  t

(∇F k (γ0 (τ )) · η(τ ˙ ))(Wk (τ ) − Wk (s)) dτ

b · η(τ ˙ ) dτ.

s

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R. Iturriaga, K. Khanin

Integrating the first and the last integrals and using integration by parts in the second we get:  t (γ˙0 (τ ) − b) · η(τ ˙ ) dτ s

= γ˙0 (s) · η(t) −  +

 t
N 

 (γ˙0 (τ ) · ∇)(∇F k (γ0 (τ )) · η(τ ))(Wk (τ ) − Wk (s)) dτ

s k=1 N t


(γ˙0 (ρ) · ∇)∇F (γ0 (ρ))(Wk (ρ) − Wk (s))dρ · η(t) ˙ k

s k=1

−

 t
N

(∇F k (γ0 (τ )) · η(τ ˙ ))(Wk (τ ) − Wk (s)) dτ − b · η(t).

(53)

s k=1

Combining (52), (53) and using again (23) we finally get:    t
N ∂Aωs,t (γ (·, )   = η(t) · v(s) + (γ˙0 (ρ) · ∇)∇F k (γ0 (ρ))(Wk (ρ)  ∂ s =0 k=1

−Wk (s)) dρ −

N




(∇F k (γ0 (t))(Wk (t) − Wk (s)) − b

k=1

= η(t) · (γ˙0 (t) − b).

 

In the proof of the following lemma we shall use the following notation. For all −∞ < s < t < ∞ denote F ωs,t = max

s≤τ ≤t

N


F k C 2 (Td ) |Wk (τ ) − Wk (s)| .

(54)

k=1

Lemma 11. For all ω in 1 and every −∞ < s < t < ∞ there exists a constant ω ψ : c(ω, s, t) ≥ 0 such that for every continuous function ψ the function given by Ks,t Td → R is Lipschitz continuous and its Lipschitz constant is less than or equal to c(ω, s, t). Moreover there exists a constant c(ω, t) ≥ 0 such that, Ks,t,ω ψ has Lipschitz constant less than or equal to c(ω, t) for every continuous ψ and all s ≤ t − 1. ω (τ ). Proof. Take arbitrary x, y ∈ Td . Let γx be a minimizer defined by γx (τ ) = γ(ψ,s;x,t) Denote by γ a curve such that its lifting into universal cover γ˜ is equal to

γ˜ (τ ) = γ˜x (τ ) +

(y˜ − x)(τ ˜ − s) , τ ∈ [s, t], t −s

(55)

where γ˜x , x, ˜ y˜ are liftings of γx , x and y respectively. Obviously, γ (s) = γx (s), γ (t) = y. It follows from (55) that for all τ ∈ [s, t], dist (γx (τ ), γ (τ )) ≤ dist (y, x), |γ˙x (τ ) − γ˙ (τ )| ≤ Since

ω Ks,t ψ(y) ≤ ψ(γx (s))) + Aωs,t (γ )

dist (y, x) . t −s

(56)

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401

and ω Ks,t ψ(x) = ψ(γx (s))) + Aωs,t (γx )

we have ω ω Ks,t ψ(y) − Ks,t ψ(x) ≤ Aωs,t (γ ) − Aωs,t (γx ).

Using (16) one easily gets the following estimate:   |Aωs,t (γ ) − Aωs,t (γx )| ≤ (2 + (t − s) max |γ˙x |)F ωs,t + |b| dist (y, x). τ ∈[s,t]

(57)

(58)

By Lemma 6, maxτ ∈[s,t] |γ˙x | ≤ C(ω, s, t), which gives |Aωs,t (γ ) − Aωs,t (γx )| ≤ c(ω, s, t)d(y, x),

(59)

c(ω, s, t) = (2 + (t − s)C(ω, s, t))F ωs,t + |b|.

(60)

where

Interchanging x and y we obtain ω ω |Ks,t ψ(y) − Ks,t ψ(x)| ≤ c(ω, s, t)d(y, x).

(61)

To prove the second statement notice that the estimate (61) does not depend on ψ. Hence, for all s ≤ t − 1, ω ω |Ks,t ψ(y) − Ks,t ψ(x)| ≤ c(ω, t)d(y, x),

where c(ω, t) = c(ω, t − 1, t).

(62)

 

Proof of Theorem 1. To prove statement 1 fix arbitrary x0 ∈ T d . Obviously there exists a ˆ 0 ) = {ψ1 = ψ(x, t0 )+const}, unique function ψ(x, t0 ) such that ψ(x0 , t0 ) = 0 and ψ(t ˆ where ψ(t) is the unique solution of the factorized Hamilton-Jacobi equation. Notice ω,b that Kt,t commutes with an addition of a constant. Namely, 0 ω,b ω,b (ψ + c) = Kt,t ψ + c. Kt,t 0 0

(63)

ˆ = This implies that for any t < t0 there exists a unique function ψ(x, t) such that ψ(t) {ψ1 = ψ(x, t) + const} and ω,b Kt,t ψ(x, t) = ψ(x, t0 ). 0

(64)

Let us show that φ(x, t) = b · x + ψ(x, t) is a solution of the Hamilton-Jacobi equation. ˆ Indeed, since ψ(t) is the solution of the factorized Hamilton-Jacobi equation we have Ktω,b ψ(x, t1 ) = ψ(x, t2 ) + c(t1 , t2 ), 1 ,t2 where t1 < t2 < t0 . Using (63)–(65) one gets ψ(x, t0 ) = Ktω,b ψ(x, t1 ) = Ktω,b Kω,b ψ(x, t1 ) 1 ,t0 2 ,t0 t1 ,t2 = Ktω,b (ψ(x, t2 ) + c(t1 , t2 )) = ψ(x, t0 ) + c(t1 , t2 ), 2 ,t0

(65)

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R. Iturriaga, K. Khanin

which implies c(t1 , t2 ) = 0. Uniqueness (up to additive constant) of the solution ψ(x, t) follows easily from uniqueness for the factorized Hamilton-Jacobi equation and (63). We next prove statement 2. Lemma 11 implies that the solution of the HamiltonJacobi equation is Lipschitz. Let x be a point of differentiability of ψbω (x, t) and γx,t be a one-sided minimizer at (x, t). It follows from Lemma 9 that γx,t is a ψbω -minimizer for some s < t. Fix arbitrary s < t. Let x(),  ∈ (−0 , 0 ), be a smooth curve in Td such that x(o) = x, and γ (τ ),  ∈ (−0 , 0 ), be a smooth family of curves such that γ0 (τ ) = γx,t (τ ) for all τ ∈ [s, t], γ (s) = γx,t (s) and γ (t) = x(). Then ψbω (x(), t) ≤ ψbω (γx,t (s), s) + Aω,b s,t (γ ) and

ψbω (x, t) = ψbω (γx,t (s), s) + Aω,b s,t (γx,t ).

Hence for positive , ψbω (x(), t) − ψbω (x, t) 1 ≤ (Aω,b (γ ) − Aω,b s,t (γx,t )).   s,t

(66)

Taking the limit as  ↓ 0 and using differentiability of ψbω at x and Lemma 10 we get ∇ψbω (x, t) ·

dx dx (0) ≤ (γ˙x,t (t) − b) · (0). d d

(67)

Analogously, for negative  ↑ 0 we get ∇ψbω (x, t) · Since

dx d (0)

dx dx (0) ≥ (γ˙x,t (t) − b) · (0). d d

(68)

is an arbitrary vector it follows from (67), (68) that γ˙x,t (t) = b + ∇ψbω (x, t).

Finally, uniqueness of a one-sided minimizer at (x, t) follows from the fact that any minimizer is uniquely determined by its position and velocity at time t. It remains to prove statement 3. The convergence in lim

s→−∞

ω,b sup min max |Ks,t η(x) − ψbω (x, t) − C| = 0

d η∈C(Td ) C∈R x∈T

follows immediately from Proposition 1. To prove lim

s→−∞

sup η∈C(Td )

ω,b ω,b sup |γ˙η,s;x,t (t) − γ˙x,t (t)| = 0

ω,b γη,s;x,t

assume that convergence above does not hold. Then, for some δ > 0 there exists a sequence sn → −∞, ηn ∈ C(Td ) and γηω,b such that n ,sn ;x,t ω,b |γ˙ηω,b (t) − γ˙x,t (t)| ≥ δ. n ,sn ;x,t

(69)

Since by Lemma 6 the sequence γ˙ηω,b (t) is bounded, it follows from (69) that there n ,sn ;x,t

ω,b (t) different from γ˙x,t (t). Hence γ˙ is a velocity of exists a limiting point γ˙ for γ˙ηω,b n ,sn ;x,t another one-sided minimizer at (x, t), which contradicts the uniqueness.  

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403

We finish this section with the proof of Proposition 2. Since the “white force” case is similar to the case of “kicked force” when Assumption 2 holds, we shall consider the “kicked” case only. We start with two lemmas. Denote by Oδ = {F (x) ∈ C 1 (Td ) : F C 1 ≤ δ} and by Oδ (G) = {F (x) ∈ C 1 (Td ) : F − GC 1 ≤ δ}. Lemma 12. Let δn = x, y ∈ Td ,

1 n2

and Fj ∈ Oδn for all j ∈ [n1 , n2 ], n2 − n1 = n. Then, for all    b b2 n  6d A  n1 ,n2 (Xx,n1 ;y,n2 ) − 2  ≤ n ,

(70)

where Xx,n1 ;y,n2 is a minimizing sequence with end points at (x, n1 ), (y, n2 ). Proof. Since diam (Td ) ≤ such that

√

d ¯ 2 , we can connect any two points by a sequence Xx,n1 ;y,n2 ,

ρb (x¯j +1 , x¯j ) ≤

√ d for all j ∈ [n1 , n2 − 1]. 2n

Hence, 2 2 ¯ x,n1 ;y,n2 ) ≤ nδn + d − b n = d + 8 − b n . Abn1 ,n2 (X 8n 2 8n 2

(71)

It follows from (71) that for any minimizing sequence Xx,n1 ;y,n2 the following estimate holds: √ 16 + d + 2 ρb (xj +1 , xj ) ≤ , j ∈ [n1 .n2 − 1]. (72) 2n To prove (72) notice that |vj +1 − vj | ≤ δn , j ∈ [n1 , n2 − 1]. Hence, if ρb (xj0 +1 , xj0 ) > √ 16+d+2 2n

for some j0 then for all j ∈ [n1 , n2 − 1], √ √ 16 + d + 2 16 + d ρb (xj +1 , xj ) > − nδn = . 2n 2n

The last estimate implies that Abn1 ,n2 (Xx,n1 ;y,n2 ) > −nδn +

16 + d b2 n ¯ x,n1 ;y,n2 ). − ≥ Abn1 ,n2 (X 8n 2

This contradiction proves (72). It follows that √    b b2 n  ( 16 + d + 2)2 6d A + nδn ≤ .  n1 ,n2 (Xx,n1 ;y,n2 ) − 2  ≤ 8n n

 

Consider again an interval [n1 , n2 ], n2 − n1 = n. Let zj ∈ Td , j ∈ [n1 , n2 ] be a sequence of points such that zj +1 = zj + b¯ (mod 1). Denote by Gj , j ∈ [n1 , n2 − 1], a sequence of potentials which have a unique global maximum at zj . We shall also assume that Gj satisfy the condition (29).

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R. Iturriaga, K. Khanin

Lemma 13. Let βn = n1 , δn = n12 and Fj ∈ Oδn (Gj ) for all j ∈ (n1 , n2 ). Then for any ¯ = {b : |b − b| ¯ ≤ βn } α > 0 there exists N (α) such that for all n ≥ N (α), all b ∈ Oδn (b) b and for all minimizing sequences Xx,n1 ;y,n2 for the actions An1 ,n2 the following estimate holds: dist (xj , zj ) ≤ α, n1 + [n/3] < j < n2 − [n/3], where [·] stands for an integer part. Proof. Assume without loss of generality that maxx∈Td Gj (x) = 0, j ∈ (n1 , n2 ). For ¯ be the following sequence connecting (x, n1 ) with (y, n2 ): arbitrary x, y ∈ Td let X ¯ 1 ) = x, X(j ¯ ) = zj , n1 < j < n2 , X(n ¯ 2 ) = y. X(n Then (diam (Td ))2 b2 n (n − 2)βn2 + + (n − 1)δn − Fn1 (xn1 ) − 2 2 2 d b2 n 3 ≤ + − Fn1 (xn1 ) − . (73) 4 2n 2

¯ ≤2 Abn1 ,n2 (X)

Consider now a minimizing sequence Xx,n1 ;y,n2 . Fix ρ > 0 such that ρ 2 < a(α). (See condition (29)). Obviously, Fj (x) ≥ a(ρ) − δn for all x such that dist (x, zj ) ≥ ρ. Denote by R the number of j  s, j ∈ (n1 , n2 ) such that dist (xj , zj ) ≥ ρ. Then the following estimate holds: Abn1 ,n2 (Xx,n1 ;y,n2 ) ≥ R(a(ρ) − δn ) − (n − R − 1)δn − Fn1 (xn1 ) −

b2 n . 2

(74)

Let n be so large that n + d4 < . a(ρ) 3

5 2n

(75)

It follows from (73)–(75) that R≤

3 (n − 1)δn + 2n + a(ρ)

d 4

<

n 3

.

This implies that there exist m1 , m2 , n1 + 1 ≤ m1 ≤ n1 + [n/3], n2 − [n/3] ≤ m2 ≤ n2 − 1 such that dist (xm1 , zm1 ), dist (xm2 , zm2 ) ≤ ρ. Finally, we show that dist (xj , zj ) ≤ α for all j ∈ (m1 , m2 ) provided n is so large that   5 1 2 . + a(α) > ρ + n 2n

(76)

Indeed, if dist (xj , zj ) > α for some j ∈ (m1 , m2 ), then Abm1 ,m2 (Xx,n1 ;y,n2 ) ≥ (a(α) − δn ) − (m − 2)δn − Fm1 (xm1 ) −

b2 m , 2

(77)

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405

where m = m2 − m1 . At the same time for the sequence ˜ ) = zj , m1 < j < m2 , X ˜ = xm2 ˜ 1 ) = xm1 , X(j X(m one has ˜ ≤2 Abm1 ,m2 (X)

(ρ + βn )2 (m − 2)βn2 b2 m + + (m − 1)δn − Fm1 (xm1 ) − . 2 2 2

(78)

Using (76)–(78) we obtain ˜ < Abm ,m (Xx,n1 ;y,n2 ), Abm1 ,m2 (X) 1 2 which contradicts the fact that Xx,n1 ;y,n2 is a minimizer. Hence, dist (xj , zj ) ≤ α for all j ∈ (m1 , m2 ).   Proof of Proposition 2. We shall first deal with the case when Assumption 2 holds. Fix arbitrary n ∈ N. For an interval [n1 , n2 ], n2 − n1 = n consider the following event: Fj ∈ Oδn for all j ∈ [n1 , n2 ],

(79)

where δn = n12 . Since zero potential F0 (x) ≡ 0 belongs to the supp (χ ), an event (79) has positive probability. Hence, with probability 1 there exists an infinite sequence of (k) (k) (k) intervals [n1 , n2 ] satisfying (79) and such that n1 → −∞ as k → ∞. It follows that (k) (k) with probability 1 such a sequence of intervals [n1 , n2 ] exists for all n. Lemma 12 (k) (k) immediately implies that for a given n each interval [n1 , n2 ] form an -narrow place (k) (k) d d with  = 12d n and s1 = n1 , s2 = n2 , M1 = T , M2 = T . This proves statement 2 in the case of Assumption 2. We next consider the case of Assumption 1. Fix arbitrary x¯ ∈ Td . For all x ∈ d T denote by G = Gx a potential with unique global maximum in x which belongs to supp(χ ) and satisfies condition (29). Let B¯ be a countable dense set of vectors b¯ in Rd . For an arbitrary n ∈ N, b¯ ∈ B¯ and interval [n1 , n2 ], n2 − n1 = n put δn = n12 and consider the following event: Fj ∈ Oδn (Gzj ) for all j ∈ [n1 , n2 ],

(80)

where zj ∈ Td , j ∈ [n1 , n2 ] is a sequence of points such that zj +1 = zj + b¯ (mod 1) and zn1 +[n/2] = x. ¯ As above, this event has positive probability. Hence, with probability 1 for (k) (k) ¯ all n ∈ N, b ∈ B¯ there exists an infinite sequence of intervals [n1 , n2 ] satisfying (80) (k) and such that n1 → −∞ as k → ∞. Let the interval [n1 , n2 ], n2 − n1 = n satisfy (80) and n ≥ N(α), where N (α) is defined in Lemma 13. Take s1 = n1 + [n/2], s2 = s1 + 1, M1 = {x : dist (x, zs1 ) ≤ α}, M2 = {x : dist (x, zs2 ) ≤ α}. It follows from Lemma ¯ all minimizers for actions Abn ,n pass through M1 , M2 at times 13 that for all b ∈ Oβn (b) 1 2 s1 , s2 respectively. Notice that zs1 = x. ¯ Denote by c(α) = maxdist (x,x)≤α (Gx¯ (x) ¯ − ¯ Gx¯ (x)). It follows from Lemma 13 that {[n1 , n2 ], s1 , M1 , s2 , M2 } form an -narrow place with  = (2α + βn )2 + 2c(α).

(81)

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R. Iturriaga, K. Khanin 2

Indeed, for any x ∈ M1 , y ∈ M2 we have Abs1 ,s2 {x, y} = (ρb (x,y)) − Gx¯ (x) − 2 Hence,    b b2  (2α + βn )2 A + c(α), ¯ + ≤  s1 ,s2 {x, y} + Gx¯ (x) 2 2

b2 2 .

which gives (81). Since c(α) → 0 as α → 0, the estimate (81) implies the statement of Proposition 2. To construct an -narrow place for arbitrary  > 0, b ∈ Rn we first choose α so small that 2c(α) + (2α)2 ≤ /2, then take n ≥ N (α) so large that 4αβn + βn2 ≤ /2 and finally pick one of the intervals [n1 , n2 ], n2 − n1 = n satisfying (80) for some b¯ in the the βn -neighbourhood of b. In the case of Assumption 3 the proof is similar.   6. Stationary Distribution for the Random Burgers Equation In this section we prove Lemmas 1, 2 and Theorem 3. We consider only the “white force” case since in the “kicked case” the proofs are similar. p

Proof of Lemma 1. Denote by Bb the σ -algebra on Ub generated by (Lp (Td , dx), B p ), 1 ≤ p < ∞: p

Bb = {A ∈ Ub : A = Ub ∩ B, B ∈ B p },

(82)

where B p are Borel σ -algebras on Lp (Td , dx). We shall show that p

Bb = Bb1 for all p > 1.

(83)

Fix arbitrary p > 1. Denote by B¯ the σ -algebra on Lp (Td , dx) which is generated by L1 (Td , dx): B¯ = {A ∈ Lp (Td , dx) : A = Lp (Td , dx) ∩ B, B ∈ B 1 }. To prove (83) it is enough to show that B¯ = B p . Notice that B¯ is the minimal σ -algebra which contains all sets U¯ = Lp (Td , dx) ∩ U, where U is an open subset of L1 (Td , dx). It follows immediately that U¯ is open in Lp (Td , dx). Hence B¯ ⊆ B p .

(84)

On the other hand, since Lp (Td , dx) is a separable Banach space any open set U˜ ∈ Lp (Td , dx) is a countable union of closed balls: U˜ = ∪Bi . We next show that any closed ball Bi in Lp (Td , dx) is a closed set in L1 (Td , dx). Indeed, assume that a sequence fn ∈ Bi converges to g in L1 (Td , dx) : fn − gL1 → 0 as n → ∞. This implies that for any h ∈ L∞ (Td , dx),   (fn · h)dx → (g · h)dx as n → ∞. (85) Td

Td

Since a closed ball in Lp (Td , dx), p > 1 is compact in weak-topology, there exists a subsequence fnk which converges weakly to f ∈ Bi . It follows that for all h ∈ L∞ (Td , dx),   (fnk · h)dx → (f · h)dx as k → ∞. (86) Td

Td

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Using (85), (86) we get 

 Td

(g · h)dx =

Td

(f · h) dx,

which implies g = f (a.s.). Hence g ∈ Bi which proves that Bi is closed in L1 (Td , dx). ¯ Thus, It follows immediately that all Bi and, hence, U˜ belong to B. ¯ B p ⊆ B. Using (84), (87) we finally get B¯ = B p .

(87)

 

Proof of Lemma 2. For a sequence −Tn → −∞ as n → ∞ denote by uωb,−Tn (x) = uωb,−Tn (x, 0) the solution of the Cauchy problem for the Burgers equation on [−Tn , 0] with the initial value uωb,−Tn (x, −Tn ) ≡ 0. We shall regard uωb,−Tn (x) as a “random variable” taking values in Ub ⊂ L1 (Td ). Clearly, for every finite Tn , uωb,−Tn corresponds to a measurable mapping from (, F) into (Ub , Bb1 ), where Bb1 is defined in (82). It follows from Theorem 1 and Lemma 6 that for almost all ω uωb,−Tn (x) − uωb (x)L1 → 0 as n → ∞.

(88)

This relation means that “random variables” uωb,−Tn converge almost surely to uωb . Using (88) and completeness of a measure space (, F, P ) we conclude that uωb defines a measurable mapping from (, F) into (Ub , Bb1 ). To finish the proof it is enough to notice that Bb1 = Bb (see Lemma 1).   Proof of Theorem 3. We shall start with the proof of the first statement. It follows from the construction of δbω (du) that B∗ω (τ ) δbω (du) = δbθ

τω

(du), τ > 0,

which immediately implies invariance of the measure νb (du, dω) = δbω (du)P (dω). To prove uniqueness suppose that ν(du, dω) = ν ω (du)P (dω) ∈ P(Ub × ) is an arbitrary ˆ ), τ > 0 with the marginal distribution P (dω) on . Then invariant measure for B(τ for all τ > 0 and P -almost all ω ∈  we have B∗θ

−τ ω

(τ ) ν θ

−τ ω

(du) = ν ω (du).

It follows that for P -almost all ω ∈  there exists a subset U ω ⊆ Ub of full ν ω -measure such that for all n > 0 and all u ∈ U ω , Bθ

−n ω

(n)un (x) = u(x)

for some un (x) ∈ Ub . Using statement 3 of Theorem 1 we conclude that u(x) = uωb (x, 0) for Lebesgue-almost all x ∈ Td . Hence, ν ω (du) = δbω (du) and ν = νb . We next prove the second statement. Since δbω (du) is measurable with respect to 0 σ -algebra F−∞ and measure νb is invariant for the skew-product dynamics (7), it is easy to show that the measure  µb (du) = δbω (du)P (dω) 

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gives a stationary distribution for the Markov process (8). Let µ ∈ P(Ub ) be another stationary distribution. Consider the arbitrary bounded continuous function C(u) on L1 (Td , dx). Then,    −n C(u)µ(du) = C(B θ ω (n)u)P ¯ (dω) µ(d u). ¯ Ub

Ub



Theorem 1 and Lemma 6 imply that for all u¯ and P -almost all ω B θ

−n ω

(n)u¯ − uωb (x, 0)L1 → 0 as n → ∞.

Since C(u) is continuous in L1 (Td ) and bounded we get C(B θ and

 C(B θ

−n ω

−n ω

(n)u) ¯ → C(uωb (x, 0)) as n → ∞ 

(n)u)P ¯ (dω) →





C(uωb (x, 0))P (dω).

Hence, in a limit n → ∞ we have     θ −n ω ω C(B (n)u)P ¯ (dω) µ(d u) ¯ → C(ub (x, 0))P (dω) µ(d u) ¯ Ub  U   b = C(uωb (x, 0))P (dω)   = C(u)µb (du). Ub



It follows that

Ub

and µ = µb .

 C(u)µ(du) =

Ub

C(u)µb (du)

 

Remark 8. 1. Notice that in the proof of the first statement of Theorem 3 we did not use condition 3 of Definition 1. Hence the invariant measure νb is unique in a class of all invariant measures for the skew-product dynamics with the marginal distribution on  equal to P (dω). 2. We gave an independent proof of the second statement of Theorem 3 although it can be obtained from the first statement. Indeed, it follows from [1, 14] that there exists a one-to-one correspondence between stationary distributions µb (du) and invariant measures νb (du, dω) which satisfy condition 3 of Definition 1. 7. Uniqueness of the Global Minimizer In this section we shall prove Theorems 4 and 5. The idea of the proof is similar in both cases, but in the “white force” case the proof has some technical difficulties. We start with the extension of the results of Sect. 5 to the forward minimizers. More precisely, in Sect. 5 we were concerned with one-sided minimizers ending at some point x0 at

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time t0 . In this section we shall call such curves backward minimizers. It was shown in Sect. 5 that backward minimizers are related to a unique solution of the Hamilton-Jacobi equation in a semi-infinite interval of time (−∞, t0 ]. The following definition is similar to Definition 3. Definition 5. A curve γx+0 ,t0 : [t0 , ∞) → Td is called a forward minimizer at a point (x0 , t0 ) for the action Aω,b if it is a minimizer for all t0 ≤ s < t < ∞ and γx+0 ,t0 (t0 ) = x0 . The same definition applies to forward minimizing sequences for the action Aω,b . We shall also define the forward Lax operators

 +,ω,b ψ(x) = inf ψ(γ (t)) + Aω,b Ks,t s,t (γ ) , γ ∈AC(x,s;t)

where AC(x, s; t) is the set of all absolutely continuous curves γ : [s, t] → Td such that γ (s) = x. In the “kicked” case the definition of LAX operators is similar:

 +,ω,b ψ(xn ) + Aω,b ψ(x) = inf (X) , Km,n m,n X∈S(x,m;n)

where S(x, m; n) is the set of sequences X = {xi , m ≤ i ≤ n, xi ∈ Td } such that xm = x. As is explained below Theorems 1, 2 remain valid for forward minimizers ω,b ω,b +,ω,b +,ω,b and minimizing sequences if we replace Ks,t , Km,n by Ks,t , Km,n . In particular, this implies that for P -almost all ω and all b there exists a unique Lipschitz function ψb+,ω (x, t) such that for all s < t, +,ω,b +,ω ψb (x, t). ψb+,ω (x, s) = Ks,t

In the “kicked” case ψb+,ω (x, t) is replaced by ψb+,ω (x, n±), ψb+,ω (x, n−) = ψb+,ω (x, n+) − Fnω (x). Then for all m < n we have the following equation: +,ω,b +,ω ψb (x, n+). ψb+,ω (x, m+) = Km,n

As in the “backward” case, functions ψb+,ω (x, t), ψb+,ω (x, n±) are related to the velocities of the forward minimizers. Namely, if x is a point of differentiability of ψb+,ω (x, t) (ψb+,ω (x, n+)) then there exists a unique forward minimizer (minimizing sequence) at (x, t) ((x, n)) and its velocity at time t (n+) is given by b − ∇ψb+,ω (x, t) (b − ∇ψb+,ω (x, n+)). All the statements above follow from the results of Sect. 5. The best way to see this is to apply an automorphism R of the measure space (, F, P ) corresponding to the time-reversing transformation t → −t: F Rω (x, t) = F ω (x, −t). Obviously, R is a measure-preserving automorphism in both “white force” and “kicked” cases. It is easy to see that Rω Rω (x, −t), ψb+,ω (x, n±) = ψ−b (x, −n∓) ψb+,ω (x, t) = ψ−b

and (τ ) = γxRω,−b (−τ ), t0 ≤ τ < ∞, Xx+,ω,b (n) = XxRω,−b (−n), n0 ≤ n < ∞. γx+,ω,b 0 ,t0 0 ,−t0 0 ,−n0 0 ,n0 It follows immediately that all the statements of Sect. 5 remain valid with obvious changes: ω → Rω, b → −b, t → −t. Notice that φb+,ω (y, t) = −b · y + ψb+,ω (y, t) satisfy

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the “backward” Hamilton-Jacobi equation: 1 −∂t φ(y, t) + (∇φ(y, t))2 + F ω (y, t) = 0. 2 We shall also use the existence of -narrow places in the past and in the future. More precisely, Proposition 2 implies that there exists a subset 1 ⊂  of full P -measure such that for any ω ∈ 1 , any  > 0, b ∈ Rd and any t < t¯ there exist -narrow places [t1 , t2 ] ⊂ (−∞, t], [t¯1 , t¯2 ] ⊂ [t¯, ∞). To make notations more symmetric we shall denote ψbω (x, t), ψbω (x, n±) by ψb−,ω (x, t), ψb−,ω (x, n±). We also denote backward minimizers by γx−,ω,b (τ ), Xx−,ω,b 0 ,n0 (n). As 0 ,t0 ω,b before we denote by γ(x1 ,s;x2 ,t) (τ ), τ ∈ [s, t] any minimizer connecting points (x1 , s) and (x2 , t), s < t. In the “kicked” case we denote the minimizing sequence connecting ω,b (x1 , m) and (x2 , n), m < n by X(x . Below we also use the following notations: 1 ,m;x2 ,n)



 ω,b ω,b ω,b ω,b ω,b γ(x Aω,b (x .x ) = A , A X . 1 2 s,t (x1 .x2 ) = As,t m,n m,n ,s;x ,t) ,m;x ,n) (x 1 2 1 2 Obviously, Aω,b s,t (x1 .x2 ) =

min

γ ∈AC(x1 ,s;x2 ,t)

ω,b Aω,b s,t (γ ), Am,n (x1 .x2 ) =

min

X∈S(x1 ,m;x2 ,n)

Aω,b m,n (X).

Since b in this section is fixed we shall omit it in the notations whenever possible. The following proposition provides a useful condition for global minimizers and global minimizing sequences. Proposition 3. Suppose ω belongs to 1 . Fix arbitrary s ≤ t, s, t ∈ R and m ≤ n, m, n ∈ Z. Then the following statements hold. 1. A curve γ (τ ), τ ∈ R is a global minimizer for an action Aω if and only if γ (τ ), τ ∈ (−∞, s] and γ (τ ), τ ∈ [t, ∞) are backward and forward minimizers, γ (τ ), τ ∈ [s, t] is a minimizer connecting points (γ (s), s), (γ (t), t) and ψ −,ω (γ (s), s) + ψ +,ω (γ (t), t) + Aωs,t (γ (s), γ (t)) =

min

x1 ,x2 ∈Td ×Td

[ψ −,ω (x1 , s) + ψ +,ω (x2 , t) + Aωs,t (x1 , x2 )].

(89)

2. A sequence X = {xi , i ∈ Z} is a global minimizing sequence if and only if {xi , i ≤ m} and {xi , i ≥ n} are backward and forward minimizing sequences, {xi , i ∈ [m, n]} is a minimizing sequence connecting points (xm , m), (xn , n) and ψ −,ω (xm , m−) + ψ +,ω (xn , n−) + Aωm,n (xm , xn ) =

min

x1 ,x2 ∈Td ×Td

[ψ −,ω (x1 , m−) + ψ +,ω (x2 , n−) + Aωm,n (x1 , x2 )].

(90)

Remark 9. Intuitively one can think of ψ −,ω (x, s), ψ +,ω (x, t) as the action of the backward and forward minimizers at (x, s) and (x, t) for infinitely long intervals of time (−∞, s] and [t, ∞). The same interpretation is also helpful in the “kicked” case. Notice that as random variables ψ −,ω (x, s), ψ +,ω (x, t) and ψ −,ω (x, m±), ψ +,ω (x, n±) are statistically independent. Proof of Proposition 3. We shall give a proof for the case of “white force”. In the “kicked” case the proof is similar. Consider a global minimizer γ (τ ), τ ∈ R. Obviously, it consists of three minimizing pieces: γ (τ ), τ ∈ (−∞, s] and γ (τ ), τ ∈ [t, ∞) are

Burgers Turbulence and Random Lagrangian Systems

411

backward and forward minimizers, γ (τ ), τ ∈ [s, t] is a minimizer connecting points (γ (s), s), (γ (t), t). Suppose that γ does not satisfy (89). Then for some x, y ∈ Td and  > 0 we have ψ −,ω (x, s) + ψ +,ω (y, t) + Aωs,t (x, y) +  < ψ −,ω (γ (s), s) +ψ +,ω (γ (t), t)Aωs,t (γ (s), γ (t)).

(91)

Define σ (τ ), τ ∈ R by joining the following minimizers:  −,ω  for τ ∈ (−∞, s] γx,s (τ ) ω σ (τ ) = γ(x,s;y,t) (τ ) for τ ∈ [s, t]  γ +,ω (τ ) for τ ∈ [t, ∞). y,t  Let [t1− , t2− ] ⊂ (−∞, s] and [t1+ , t2+ ] ⊂ [t, ∞) be 10 -narrow places and [s1− , s2− ] ⊂ − − + + + + [t1 , t2 ], [s1 , s2 ] ⊂ [t1 , t2 ] be corresponding subintervals from Definition 4. Using Lemma 9 and (42) we obtain  |ψ −,ω (σ (s1− ), s1− ) − ψ −,ω (γ (s1− ), s1− )| ≤ , 10 and  |ψ +,ω (σ (s2+ ), s2+ ) − ψ +,ω (γ (s2+ ), s2+ ) ≤ . 10 Hence, 3 |ψ −,ω (σ (s2− ), s2− ) − ψ −,ω (γ (s2− ), s2− )| ≤ , 10 and 3 |ψ +,ω (σ (s1+ ), s1+ ) − ψ +,ω (γ (s1+ ), s1+ )| ≤ . 10 Moreover by Lemma 9 the following relations hold

ψ −,ω (γ (s), s) = ψ −,ω (γ (s2− ), s2− ) + Aωs− ,s (γ ), 2

ψ +,ω (γ (t), t) = ψ +,ω (γ (s1+ ), s1+ ) + Aωt,s + (γ ), 1

ψ −,ω (σ (s), s) = ψ −,ω (σ (s2− ), s2− ) + Aωs− ,s (σ ), 2

ψ +,ω (σ (t), t) = ψ −,ω (σ (s1+ ), s1+ ) + Aωt,s + (σ ). 1

Using inequality (91) we conclude Aωs− ,s + (γ ) = Aωs− ,s (γ ) + Aωs,t (γ (s), γ (t)) + Aωt,s + (γ ) 2

1

2

1

= ψ −,ω (γ (s), s) − ψ −,ω (γ (s2− ), s2− ) + Aωs,t (γ (s), γ (t)) +ψ +,ω (γ (t), t) − ψ +,ω (γ (s1+ ), s1+ )

> ψ −,ω (σ (s), s) − ψ −,ω (γ (s2− ), s2− ) + Aωs,t (x, y) +ψ +,ω (σ (t), t) − ψ +,ω (γ (s1+ ), s1+ ) + 

≥ ψ −,ω (σ (s), s) − ψ −,ω (σ (s2− ), s2− ) + Aωs,t (σ (s), σ (t)) 3 +ψ +,ω (σ (t), t) − ψ +,ω (σ (s1+ ), s1+ ) +  − 5 2 = Aωs− ,s + (σ ) + . 5 2 1

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R. Iturriaga, K. Khanin

Finally, taking a curve   (τ ) for τ ∈ [s1− , s2− ] γω   (γ (s1− ),s1− ;σ (s2− ),s2− ) σ¯ (τ ) = σ (τ ) for τ ∈ [s2− , s1+ ]   γ ω + + + + (τ ) for τ ∈ [s1+ , s2+ ] (σ (s ),s ;γ (s ),s ) 1

1

2

2

we have 2 2 2 < Aωs− ,s + (γ ) − + 10 5 10 2 1 2 1 4 2 ω ω ≤ As − ,s + (γ ) − + = As − ,s + (γ ), 5 10 1 2 1 2

Aωs− ,s + (σ¯ ) ≤ Aωs− ,s + (σ ) + 1

2

which is a contradiction since γ is a global minimizer. We next prove that (89) provides a sufficient condition for a curve to be a global minimizer. Suppose γ is not a global minimizer although it consists of three minimizing pieces and satisfies (89). Hence, there exists a curve σ˜ (τ ), τ ∈ [˜s , t˜] such that s˜ < s, t˜ > t, γ (˜s ) = σ˜ (˜s ), γ (t˜) = σ˜ (t˜) and Aωs˜,t˜(σ˜ ) < Aωs˜,t˜(γ ). Denote x = σ˜ (s), y = σ˜ (t). Since ψ −,ω (x, s) ≤ Aωs˜,s (σ˜ ) + ψ −,ω (γ (˜s ), s˜ ), ψ +,ω (y, t) ≤ Aωt,t˜(σ˜ ) + ψ +,ω (γ (t˜), t˜), we have ψ −,ω (x, s) + ψ +,ω (y, t) + Aωs,t (x, y) ≤ Aωs˜,s (σ˜ ) + ψ −,ω (γ (˜s ), s˜ ) + Aωt,t˜(σ˜ ) +ψ +,ω (γ (t˜), t˜) + Aωs,t (σ˜ ) = ψ −,ω (γ (˜s ), s˜ ) + ψ +,ω (γ (t˜), t˜) + Aωs˜,t˜(σ˜ ) < ψ −,ω (γ (˜s ), s˜ ) + ψ +,ω (γ (t˜), t˜) + Aωs˜,t˜(γ )

= ψ −,ω (γ (s), s) + ψ +,ω (γ (t), t) + Aωs,t (γ (s), γ (t)), which contradicts (89).

 

Remark 10. Notice that Proposition 3 remains valid if s = t or m = n. In this case we have ψ −,ω (γ (t), t) + ψ +,ω (γ (t), t) = min [ψ −,ω (x, t) + ψ +,ω (x, t)] x∈Td

(92)

instead of (89) and ψ −,ω (xn , n−) + ψ +,ω (xn , n+) − Fnω (xn )

= min [ψ −,ω (x, n−) + ψ +,ω (x, n+) − Fnω (x)] x∈Td

instead of (90). Below we use Proposition 3 in the “kicked” case in the form (93). Next we prove the following general result.

(93)

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413

Lemma 14. Let V (x, c), x ∈ Td , c ∈ Rk be a real differentiable function and ψ(x) a continuous function on Td . Define G(c) = min [ψ(x) + V (x, c)]. x∈Td

Then the following statements hold. 1. G(c) is a local Lipschitz function. 2. If G(c) is differentiable at c0 and x0 is a point of minimum corresponding to c0 then ∂V dG (c0 ) = (x0 , c0 ). dc ∂c

(94)

Proof. Let x1 , x2 be points of minimum corresponding to c1 , c2 respectively. Then G(c1 ) = ψ(x1 ) + V (x1 , c1 ) and G(c2 ) ≤ ψ(x1 ) + V (x1 , c2 ). Subtracting the first equation from the second one we get G(c2 ) − G(c1 ) ≤ V (x1 , c2 ) − V (x1 , c1 ). Similarly, G(c1 ) − G(c2 ) ≤ V (x2 , c1 ) − V (x2 , c2 ). For any C > 0 define    ∂V   D(C) = max max  (x, c) . d c:|c|≤C ∂c x∈T Then for all c1 , c2 such that |c1 |, |c2 | ≤ C, |G(c1 ) − G(c2 )| ≤ D(C)|c1 − c2 |. Suppose now that G is differentiable at c0 and x0 is a point of minimum. As above, for all h ∈ Rk and all  ∈ R, G(c0 + h) − G(c0 ) ≤ V (x0 , c0 + h) − V (x0 , c0 ). Dividing by  > 0 and taking the limit as  tends to zero we obtain: ∂V dG (c0 ) · h ≥ (x0 , c0 ) · h. dc ∂c In the same way taking the limit for negative  we get: ∂V dG (c0 ) · h ≤ (x0 , c0 ) · h, dc ∂c which immediately implies (94).

 

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Corollary 5. Suppose that for every fixed c ∈ Rk the mapping ∂V (·, c) : Td → Rk ∂c is one-to-one. Then for Lebesgue-almost all c there is a unique point of minimum x(c). Proof. It follows from Lemma 14 that G(c) = minx∈Td [ψ(x) + V (x, c)] is a Lipschitz function. Hence, by the Rademacher Theorem it is differentiable almost everywhere. It is easy to see that for any point of differentiability there exists a unique point of minimum. Indeed, if c is a point of differentiability and x1 (c), x2 (c) are two points of minimum we have dG ∂V ∂V (c) = (x1 (c), c) = (x2 (c), c), dc ∂c ∂c which implies x1 (c) = x2 (c).   Proof of Theorem 5. Denote c = (ci , 1 ≤ i ≤ M) = ξ0 (ω) = {ξ0i (ω), 1 ≤ i ≤  i M}, ψ ω (x) = ψ −,ω (x, 0−) + ψ +,ω (x, 0+), V (x, c) = − M i=1 ci F (x). Let X = {xj , j ∈ Z} be a global minimizing sequence. It follows from (93) that x0 is a point of minimum for ψ ω (x) + V (x, c) = ψ −,ω (x, 0−) + ψ +,ω (x, 0+) −

M


ci F i (x).

i=1

Assumption (35) of Theorem 5 implies that ∂V (·, c) = −(F 1 (·), F 2 (·), . . . , F M (·)) ∂c is a one-to-one mapping from Td to RM . Hence, using Corollary 5 we conclude that for Lebesgue-almost all c a point of minimum x0 is unique. According to Lemma 9 two global minimizing sequences cannot intersect. This implies that for Lebesgue-almost all c there exists a unique global minimizer. It is easy to see that the set of c for which uniqueness holds depends only on {ξj , j = 0}. To finish the proof it is enough to notice that the random vector c = ξ0 (ω) is independent from {ξj , j = 0} and has absolutely continuous distribution with respect to the Lebesgue measure.   We next consider the “white force” case. The strategy is similar to the one used in the “kicked” case. However in the “white force” case there are two serious difficulties. The first one is connected with the use of the Rademacher Theorem which is not immediately available in the case of “white” noises. Secondly, the control of the term Aωs,t (γ (s), γ (t)) in Proposition 3 is not as straightforward as in the “kicked” case. To overcome these difficulties we explicitly add drift terms to the the Brownian motions. Namely, for any vector c = (c− , c+ ) = (ci− , 1 ≤ i ≤ N, ci+ , 1 ≤ i ≤ N ) ∈ R2N we define   W ω (t) + ci+ for t ∈ [, ∞)   i  W ω (t) + ci+ t for t ∈ [0, ] ω i  (95) Wi (t; c, ) = − W ω (t) + ci t for t ∈ [−, 0]   i    ω Wi (t) − ci− for t ∈ (−∞, −]. Denote by Tc, the measurable transformation of the measurable space (, F) corresponding to (95):

Burgers Turbulence and Random Lagrangian Systems Tc, (ω)

Tc, :  → , Wi

415

= Wiω (t; c, ), 1 ≤ i ≤ N.

Let Pc, be the probability measure on (, F) which is the image of measure P under Tc, . Then the following simple consequence of the Cameron-Martin-Girsanov Theorem holds. Proposition 4. The probability measure Pc, is absolutely continuous with respect to the measure P and the corresponding density is given by
Pc, (dω) = exp P (dω) N



ci+ Wiω () − ci− Wiω (−) −

i=1

 1 + 2 1 − 2 (c ) − (ci ) . 2 i 2

The proof of Proposition 4 is standard (see, for example, [42]). To simplify notations we shall not indicate dependence on  which is everywhere below assumed to be a fixed sufficiently small positive number. We shall denote by Aω,c the action ATc (ω) : 1
 N

A

ω,c

(γ ) =

Aωs,t (γ ) −



t

k=1 s

F k (γ (τ ))(ck− χ− (τ ) + ck+ χ+ (τ )) dτ,

(96)

where Aωs,t (γ ) is given by (16) and χ− (τ ), χ+ (τ ) are the indicator functions of the intervals [−, 0], [0, ]:   1 for τ ∈ [−, 0] 1 for τ ∈ [0, ] χ− (τ ) = χ+ (τ ) = 0 for τ ∈ R − [−, 0], 0 for τ ∈ R − [0, ]. T (ω)

The corresponding Lagrangian flow Ls c is denoted below by Lω,c s . As in (23) we shall say that (x(t), v(t)), t ∈ [t1 , t2 ] is a trajectory of the Lagrangian flow Lω,c if s x(t) ˙ = v(t) and for any s ≤ t, s, t ∈ [t1 , t2 ], v(t) = v(s) +

 t
N

(v(τ ) · ∇)∇F k (x(τ ))(Wk (τ ) − Wk (s)) dτ

s k=1

1
−  N



k=1 s

−

N


t

∇F k (x(τ ))(ck− χ− (τ ) + ck+ χ+ (τ )) dτ

∇F k (x(t))(Wk (t) − Wk (s)).

(97)

k=1 ω Finally, for any t1 < t2 we shall denote by c,t the set of all minimizers on [t1 , t2 ] for 1 ,t2 ω,c ω the action A and by c,t the set of all backward minimizers:     T (ω) T (ω) ω ω = γ(xc1 ,t1 ;x2 ,t2 ) , x1 , x2 ∈ Td , c,t = γx,tc , x ∈ Td . c,t 1 ,t2

For any x, y ∈ Td , c ∈ RN define T (ω)

c Vω (x, y, c) = A−, (x, y).

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Below we assume that c belongs to E = {c ∈ R2N : |ci− |, |ci+ | ≤ 1, 1 ≤ i ≤ N }. c±

The idea is that if we add large drift terms i , 1 ≤ i ≤ N on a small interval of time  − + k [−, ] then Vω (x, y, c) converges to −δ(x − y) N k=1 F (x)(ck + ck ) in the limit  → 0. More precisely we have the following proposition. Proposition 5. For P -almost all ω there exist 0 = 0 (ω) > 0 and a constant K = K(ω) > 0 such that for all  ≤ 0 (ω) and all c ∈ E the following statements hold: 1. V (x, y, c) is a C ∞ function on ω × E, where ω = {(x, y) ∈ Td × Td : ρ(x, y) ≤ K(ω)}.

(98)

2. If γ is a global minimizer for Aω,c then (x = γ (−), y = γ ()) belongs to the interior of ω .  − + k 3. The function Vω (x, y, c) converges to −δ(x − y) N k=1 F (x)(ck + ck ). Namely,   N  
 − +  k lim max ω max V (x, y, c) + F (x)(ck + ck ) = 0. (99)  →0 (x,y)∈ c∈E  k=1

4. If condition (34) of Theorem 4 holds then the mapping of ω into R2N .

∂V ∂c (·, ·, c)

is an embedding

Proposition 5 is proven in the Appendix. We can now finally prove Theorem 4. Proof of Theorem 4. Consider the random variable 0 (ω). Denote by  = {ω ∈  : (ω) ≥ }. Obviously, there exists a constant ¯ such that P (¯ ) ≥ 21 . Fix  = ¯ . It follows from Propositions 3, 5 that for any c ∈ E and any global minimizer γ for the action Aω,c a point (γ (−¯ ), γ (¯ )) ∈ ω¯ and corresponds to a minimum of ψ −,ω (x, −¯ )(x) + ψ +,ω (y, ¯ ) + V¯ (x, y, c).

(100)

∂V ω ∂c (·, ·, c) is an embedding of ¯

we conclude that for Since Proposition 5 implies that P -almost every ω ∈ ¯ and Lebesgue-almost every c ∈ E there exists a unique global minimum in (100). Hence, by Lemma 9 there exists a unique global minimizer for the action Aω,c . The Fubini theorem implies that for Lebesgue-almost every c ∈ E there exists a subset of ¯ of full P -measure for which uniqueness of the global minimizer holds. Let c¯ ∈ E be such a vector. Then there exists a subset ¯ ,c¯ ⊂ ¯ , P (¯ ,c¯ ) ≥ 21 such that for all ω ∈ ¯ ,c¯ there exists a unique global minimizer for Aω,c . Since ¯  (ω) we get the uniqueness of the global minimizer for the original action Aω,c = ATc,¯ Aω for all ω ∈ Tc,¯ ¯  (¯ ,c¯ ). Notice that   1 Pc,¯ (101) ¯  Tc,¯ ¯  (¯ ,c¯ ) = P (¯ ,c¯ ) ≥ . 2   (u) the set ω ∈  for Using Proposition 4 we have P Tc,¯ ¯  (¯ ,c¯ ) > 0. Denote by  ω which the uniqueness of the global minimizer for the action A holds. It follows from (101) that P ((u) ) > 0. Since the set (u) is invariant under time-translations θ τ , τ ∈ R and (, P , θ τ , τ ∈ R) is an ergodic flow, we finally get P ((u) ) = 1.   We finish with the proof of Corollaries 2, 3 from Theorems 4, 5. Since the proofs are similar in the “white force” and “kicked” cases we consider only the former one.

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417

Proof of Corollary 2. Suppose diam(E ω,b (t)) does not tend to 0 as t → ∞. This implies that there exists δ > 0 and an infinite sequence of backward minimizers γxω,b (τ ) n ,tn such that tn → ∞ as n → ∞ and ρ(γxω,b (0), xgω,b ) ≥ δ, n ,tn where xgω,b is the position of the global minimizer at time t = 0. Since velocities γ˙xω,b (0)of backward minimizers at time t = 0 are bounded there exists a subsequence n ,tn (0), γ˙xω,b (0)) converges to some (x, ¯ v), ¯ where x¯ ∈ Td and nk such that (γxω,b n ,tn n ,tn k

k

k

k

¯ v¯ (τ ), τ ∈ R such that γ x, ¯ v¯ (0) = ρ(x, ¯ xgω,b ) ≥ δ. Then the Lagrangian trajectory γ x, x, ¯ v ¯ x, ¯ γ˙ (0) = v¯ has to be a global minimizer which contradicts the uniqueness.  

Proof of Corollary 3. Obviously Lωs δx ω,b ,v ω,b (dx, dv) = δx θ s ω,b ,v θ s ω,b (dx, dv). g

g

g

g

It follows immediately that κ b (dx, dv, dω) is an invariant measure for Lˆs , s ∈ R. Notice that under the condition of fixed ω the conditional distributions κ b (dx, dv|ω) = δx ω,b ,v ω,b (dx, dv) are atomic measures. This together with the ergodicity of P (dω) for g

g

θ s , s ∈ R implies the ergodicity of κ b (dx, dv, dω) for Lˆs , s ∈ R.

 

8. Conclusions and Open Problems We finish the paper with a discussion of the results. We also discuss related problems. Solutions to some of those problems are pretty straightforward and will be published elsewhere in the near future; others remain open. Below we discuss open problems very briefly aiming to convince the reader that they are indeed quite challenging and of significant importance for the future development of the theory. 1. The results of the paper can be roughly split into two parts. The first group of results are connected with existence and uniqueness of the stationary distributions and related results on uniqueness of solutions to the random Hamilton-Jacobi and Burgers equations for semi-infinite intervals of time (−∞, t0 ). Those results are of an extremely general nature and are obtained under very mild conditions. The uniqueness of the “viscosity” solutions demonstrates the extreme stability of the random Hamilton-Jacobi and Burgers equations. In short this stability can be formulated as the “one force – one solution” principle. We just mention that in the case of the Navier-Stokes equation the situation is much more complicated. The “one force – one solution” principle is valid only for large enough values of the viscosity (see [9, 17, 23, 31]). The second group of results concerns the uniqueness of the global minimizer which is a fundamental property of random Lagrangian systems. The conditions here are more restrictive. They are quite satisfactory in the “white force” case where we just need to have enough “pumping” potentials F i (x). In the “kicked” case we have to make a more technical assumption on absolute continuity for the distribution of the random vectors {ξji (ω), 1 ≤ i ≤ M}. At the same time one certainly needs to impose some conditions on potentials for the validity of the results. Indeed, if all the potentials have additional symmetry or effectively smaller period (which is in fact also an additional symmetry) then obviously the global minimizer is non-unique. However this seems to be the only necessary restriction. Unfortunately such a general result is quite difficult to prove.

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2. As we have already mentioned in the first section Ya. Sinai ([43]) has proved in the beginning of the 1990’s the existence and uniqueness of the stationary distribution µν for the viscous Burgers equation. It is natural to ask whether µν converges weakly to the stationary distribution µ for the inviscid equation. The answer to this question is affirmative. The corresponding result in the one-dimensional case was proven in ([15]) using rather complicated probabilistic estimates. However, it will be shown in ([25]) that the corresponding result holds in a very general situation and the proof is based on a simple conceptual argument. The approach uses the stochastic Cax formula and semiconcavity of solutions to the Hamilton-Jacobi equation. 3. It is also possible to show that the invariant measure κ b (dx, dv, dω) for the skewproduct extension Lˆs (or Lˆm in the “kicked” case) of the Lagrangian flow is in fact the unique action minimizing measure in the sense of Mather. To make this statement precise one should define the skew-product extension of the Lagrangian itself. This can be done in a natural way and the corresponding result will be proved elsewhere. 4. The most important problem which remains open in the theory is the problem of the hyperbolicity of the global minimizer. Since the invariant measure κ b (dx, dv, dω) is ergodic it is possible to define non-random Lyapunov exponents for the Jacobi cocycle: λ1 ≥ λ2 ≥ · · · ≥ λ2d . To prove hyperbolicity one has to show that λi = 0, 1 ≤ i ≤ 2d. Since the Lagrangian flow is symplectic the Lyapunov exponents appear in pairs of numbers having opposite sign. It follows that in the hyperbolic case there are d stable and d unstable Lyapunov exponents: λ1 ≥ λ2 ≥ · · · ≥ λd > 0 > λd+1 ≥ · · · ≥ λ2d . We strongly believe that the following hyperbolicity conjecture holds. Conjecture 1. Under assumptions of Theorems 4, 5 the invariant measure κ b (dx, dv, dω) is hyperbolic. Notice that in the one-dimensional case it is enough to show that λ1 > 0. This indeed was proven in ([15]). The proof is based on the exponential estimate for the diam(E(t)). Namely, it was shown that there exist constants α1 , α2 , K1 , K2 > 0 such that P (ω ∈  : diam(E ω,b (t)) ≥ K1 eα1 t ) ≤ K2 eα2 t .

(102)

Unfortunately the proof in ([15]) is purely one-dimensional. However if the estimate (102) holds in the multi-dimensional case it immediately gives the hyperbolicity conjecture. 5. The hyperbolicity conjecture implies that all “backward” minimizers are asymptotic to the global one as t → −∞. This brings us to an interesting concept of topological shocks. To explain this notion we have to describe the structure of minimizers on the universal cover Rd . Instead of one global minimizer on the torus Td one has a Zd lattice of them. Since every “backward” minimizer approaches the global one we get a random tiling of Rd onto domains of attraction to a particular global minimizer. The boundaries of these domains correspond to “global” shocks having pure topological nature. Other “local” shocks appear and later merge with the topological singularities. The structure of the topological shocks is quite rigid and crucially depends on the topology of the manifold. One can show that in the case of a two-dimensional torus the tiling is formed by a Z2 lattice of hexagonal domains. In the three-dimensional case the structure of the tiling can be more complicated. The notion of topological shock is discussed in more detail in ([6]). Notice that in the one-dimensional case the topological shock is formed

Burgers Turbulence and Random Lagrangian Systems

419

by just a single point. This point was called the main shock in [15]. It is an interesting question to study the structure of topological shocks for general Riemannian manifolds. 6. Another very interesting open problem is related to the random Hamilton-Jacobi and Burgers equation in a non-compact case. It is not known whether the stationary distribution exists even in the case of positive viscosity ν. The only result which we are aware of belongs to Yu. Kifer ([30]) who has constructed in the case d ≥ 3 the stationary distribution for the viscous Burgers equation driven by a force which is random in both x and t. We shall also mention recent papers [11, 22] where the non-compact HamiltonJacobi equation is studied in the autonomous case. 7. Finally, we shall say a few words about physically important quantitative predictions for correlation functions. We hope that in the multi-dimensional case one will be able to obtain non-trivial quantitative results, provided that the hyperbolicity of the global minimizer is established. In the case d = 1 such results were obtained in [16]). It was ω shown that the pdf g(z) for the velocity gradient z = du decays super-exponentially dx for large positive values of z, and algebraically for negative values, i.e. g(z) ∼ |z|Cα as z → −∞. The critical exponent α is universal. It means that α is independent of the precise formula for a driven force. In [7, 41] it was suggested that α is equal to 25 or 3. However, as it was shown in [16] the geometrical picture which follows from hyperbolicity suggests that the main contribution to g(z) for large negative values of z comes from small neighbourhoods of the preshock events. Then an easy calculation implies that α = 27 (see also [18, 19]). This prediction, although not rigorous, has been verified numerically ([4]) and accepted by the physical community. Appendix In this Appendix we prove Proposition 5. The proof of the proposition splits into several lemmas. We assume everywhere that ω ∈  is fixed. To simplify notation we do not indicate dependence on ω although all constants in the estimates below do depend on it. We start with two lemmas which provide estimates for the velocities of minimizers for the action Ac . Lemma 15. For P -almost all ω there exists a constant B > 0 such that for all  ∈ (0, 1] and for all c ∈ E, sup max |γ˙ (t)| ≤ B, γ ∈c,1 t∈[−1,1]

where c,1 is the set of backward minimizers for the action Ac starting at time t = 1. Proof. The proof is similar to the proof of Lemma 6. Since the large drift forces act only on a small time interval we can essentially reproduce the same estimates. For any v0 > 0 there exists a constant V (v0 ) > 0 such that if γ is a trajectory of the Lagrangian flow Lcs with c ∈ E,  ∈ (0, 1] and |γ˙ (t0 )| ≥ V (v0 ) for some t0 in [−1, 1] then |γ˙ (t) ≥ v0 for all t in [−1, 1]. Denote A¯ =

sup

max sup Ac−1,1 (δ)

√ c∈E ∈(0,1] ˙ δ(t):|δ(t)|≤ d

and A(v0 ) =

inf

min inf Ac−1,1 (δ).

˙ δ(t):|δ(t)|≥v 0 c∈E ∈(0,1]

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Clearly, A(v0 ) → ∞ as v0 → ∞. Since any√two points x, y ∈ Td can be connected by ˙ ≤ d for all t, it is enough to take B = V (v0 ), the curve δ(t), t ∈ [−1, 1] such that δ(t) ¯ where v0 is so large that A(v0 ) > A.   d d K Fix K > 0. Denote K  = {(x, y) ∈ T × T : ρ(x, y) ≤ K} and c,−, = {γ ∈ K c,−, : (γ (−), γ ()) ∈  }. Then the following lemma holds.

Lemma 16. For a fixed K > 0 and for P -almost all ω ∈  there exists a constant D(K) > 0 such that for all  ∈ (0, 1] and for all c ∈ E, sup K γ ∈c,−,

max |γ˙ (t)| ≤ D(K).

t∈[−,]

Proof. The proof follows the same lines. However one needs more estimates in order to prove uniformity in . As above for any v0 > 0 there exists a V (v0 ) > 0 independent of c and  such that if γ is a trajectory of the Lagrangian flow Lcs with c ∈ E,  ∈ (0, 1] and |γ˙ (t0 )| ≥ V (v0 ) for some t0 in [−, ] then |γ˙ (t)| ≥ v0 for all t in [−, ]. Denote  1 ¯ A(K) = sup max sup Ac−, (δ) c∈E  ˙ ∈(0,1] δ(t):|δ(t)|≤K/2 N  
− + k + (ck + ck + Wk () − Wk (−))F (δ()) k=1

and

 1 A1 (v0 ) = inf min inf Ac−, (δ) ˙ δ(t):|δ(t)|≥v 0 c∈E ∈(0,1]  +

N 




(ck− + ck+ + Wk () − Wk (−))F k (δ())

.

k=1

It follows from (16), (96) that for any K ≥ 0 there exists a constant C(K) such that   N 
 −   c + k (ck + ck + Wk () − Wk (−))F (δ())  ≤ C(K), A−, (δ) +   k=1

¯ ˙ provided |δ(t)| ≤ K/2 for all t ∈ [−, ]. Hence, A(K) is finite. We next show that A1 (v0 ) is also finite. Taking a curve δ0 (t) such that |δ˙0 (t)| = v0 for all t ∈ [−, ] we obtain

N
 −  1 + c k (ck + ck + Wk () − Wk (−))F (δ()) ≤ C0 A1 (v0 ) ≤ A−, (δ) +  k=1

for some constant C0 > 0. Denote   1 ˙ ) − b)2 dτ, I0 = (δ(τ − 2

 
N b2 k ˙ )(Wk (τ ) − Wk (−)) − I1 = ∇F (δ(τ )) · δ(τ dτ, 2 − k=1

I2 =

1 

N  
k=1 −

(F k (δ()) − F k (δ(τ )))(ck− χ− (τ ) + ck+ χ+ (τ )) dτ.

Burgers Turbulence and Random Lagrangian Systems

421

Then Ac−, (δ) +

 (ck− + ck+ + Wk () − Wk (−))F k (δ()) = I0 + I1 + I2 .

N 
k=1

It is easy to show that there exists a constant C1 > 0 such that  √ |I1 | , |I2 | ≤ C1 ( + I0 ). Hence,

 √ I0 + I1 + I2 ≥ I0 − 2C1 I0  − 2C1  ≥ −(C12 + 2C1 ).

(103)

This implies that A1 (v0 ) is finite. Moreover, it follows from (103) that A1 (v0 ) → ∞ as ¯ v0 → ∞. Take now v0 so large that A1 (v0 ) > A(K). It is easy to see the statement of the lemma holds for D(K) = V (v0 ).   The following lemma shows that two Lagrangian trajectories which start at the same point at time t = − do not intersectuntil time . Lemma 17. For a fixed D > 0 and for P -almost all ω ∈  there exists a constant 0 (D) > 0 such that for any two trajectories δ (1) (t), δ (2) (t) of the Lagrangian flow Lcs with c ∈ E,  ∈ (0, 0 (D)] we have 1 ρ(δ (1) (t), δ (2) (t)) ≥ √ (t + )|δ˙(1) (−) − δ˙(2) (−)| , t ∈ [−, ], 2 d

(104)

provided δ (1) (t), δ (2) (t) satisfy conditions δ (1) (−) = δ (2) (−), |δ˙(1) (−)|, |δ˙(2) (−)| ≤ D.

(105)

Proof. Let δ (1) (t), δ (2) (t) be the trajectories of the Lagrangian flow Lc which satisfy (105). Denote δ˙(1),(2) (t) = v (1),(2) (t), ρ(t) = ρ(δ (1) (t), δ (2) (t)), b(t) = |v (1) (t) − v (2) (t)|. Using (97) it is easy to show that there exists a constant D1 such that |v (1),(2) (t)| ≤ D1 for all t ∈ [−, ]. Applying this estimate and again (97) we have  t    τ 1 t b(t) ≤ b(−) + K1 b(τ ) dτ + b(s)ds dτ  − − − for some constant K1 > 0. It follows that there exists a constant K2 > 0 such that  t b(t) ≤ b(−) + K2 b(τ ) dτ. −

Hence, by the Gronwall inequality b(t) ≤ b(−)eK2 (t+) ,

(106)

ρ(t) ≤ K3 b(−), t ∈ [−, ],

(107)

which gives an estimate

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R. Iturriaga, K. Khanin

for  ∈ (0, 1] and a constant K3 > 0. Using once again (97) and the estimates (106), (107) we get |(v (1) (t) − v (2) (t)) − (v (1) (−) − v (2) (−))| ≤ K4 b(−), where K4 is a positive constant. To complete the proof choose a component of velocity such that 1 (1) (2) |vi (−) − vi (−))| ≥ √ b(−), 1 ≤ i ≤ d. d Then for all t ∈ [−, ],   1 (1) (2) |vi (t) − vi (t)| ≥ √ − K4  b(−). (108) d Take 0 (D) so small that 1 1 , K4 0 (D) ≤ √ . 2 2 d Then using (106) and (108) we finally obtain 4D0 (D)e2K2 0 (D) ≤

1 1 (1) (2) √ (t + )b(−) ≤ |δi (t) − δi (t)| ≤ , t ∈ [−, ], 2 2 d which implies ρ(t) ≥

1 √ (t 2 d

+ )b(−).

 

Remark 11. All estimates in Lemmas 15, 16 and 17 were proven for c ∈ E. In the same way one can prove all three lemmas for c ∈ Ea = {c ∈ R2N : |ci− |, |ci+ | ≤ a, 1 ≤ i ≤ N}. We shall denote corresponding constants B(a), D(K, a), 0 (D, a). Let B be the bound given by Lemma 15. Define K = 2B and  = {(x, y) ∈ Td × Td : ρ(x, y) ≤ K}. To formulate the next lemma we shall also need constants D = D(K + 1, 2) and 0 = 0 (D, 2) which are defined in Remark 11. Lemma 18. For all  ∈ (0, 0 ] and all c ∈ E the following statements hold. 1. If γ (τ ), τ ∈ (−∞, 1] belongs to c,1 then (γ (−), γ ()) ∈  . 2. For every (x, y) ∈  there exists a unique γ (τ ), τ ∈ [−, ] which minimizes the action Acs,t γ ) with fixed end-points x = γ (−), y = γ (). 3. The function V (x, y, c) is C ∞ ( × E). Proof. The proof of the first statement follows directly from Lemma 15 and the choice of the constant K. To prove the second statement notice that according to Lemma 16 all minimizers will have velocities with the norm bounded by D. Since all minimizers are Lagrangian trajectories it follows from Lemma 17 that for all  ∈ (0, 0 ] and all (x, y, c) in a neighbourhood of  × E there exists a unique minimizer. Finally, we shall show that the differentiability of V (x, y, c) is a consequence of the uniqueness of minimizers. Fix arbitary  ∈ (0, 0 ]. Let γx,v be a trajectory of the Lagrangian flow Lc with x = γ (−), v = γ˙ (−). Denote y = γ (), w = γ˙ (). Then the action Ac−, (γx,v ) and the Lagrangian flow Lc−, : (x, v) → (y, w) are C ∞ functions of the variables (x, v) ∈ Td × Rd and c ∈ R2N . Lemma 17 implies that for all (x, y, c) in a neighbourhood of  × E there exists a unique v = v(x, y, c) such that Lc−, : (x, v) → (y, ·) ∂y (x, v, c) is a nondegenerate matrix for all and |v| ≤ D. It follows from (104) that ∂v x ∈ Td , |v| ≤ D. Using the Implicit Function Theorem we conclude that v = v(x, y, c) and, hence, V (x, y, c) = Ac−, (γx,v ) are C ∞ functions on  × E.

Burgers Turbulence and Random Lagrangian Systems

423

In the next two lemmas we study the partial derivatives of the function V (x, y, c). Lemma 19. 1. The following asymptotic relations hold:    ∂ 2V  3    i lim max max  ∇F (x, y, c) + (x)  = 0,  →0 (x,y)∈ c∈E  ∂x∂c− 4 i    ∂ 2V  1    i lim max max  ∇F (x, y, c) + (x)  = 0,  →0 (x,y)∈ c∈E  ∂y∂c− 4 i    ∂ 2V  1    i lim max max  (x, y, c) + (x) ∇F  = 0,  →0 (x,y)∈ c∈E  ∂x∂c+ 4 i    ∂ 2V  3    lim max max  (x, y, c) + ∇F i (x) = 0. +  →0 (x,y)∈ c∈E  ∂y∂c 4 i 2. There exists a constant CV > 0 such that for all  ∈ (0, 1],    ∂ 3 V   max max  2 (x, y, c)  ≤ CV . (x,y)∈ c∈E ∂ (x, y)∂c

(109)

(110)

Proof. We shall prove only the first statement since the proof of the second statement follows the same path. For a fixed c ∈ E, (x0 , y0 ) ∈  denote γ (τ ; x0 , y0 , c) = γ (τ ), τ ∈ [−, ] a unique trajectory of the Lagrangian flow Lcs connecting points (x0 , −) and (y0 , ). Let v(τ ; x0 , y0 , c) = v(τ ) = γ˙ (τ ; x0 , y0 , c), τ ∈ [−, ] be the velocity of γ (τ ). Since V (x, y, c) is a smooth function on  × E it follows from Lemma 10 that ∂V (x0 , y0 , c) = v(; x0 , y0 , c) − b . ∂y Changing t to −t we also get ∂V (x0 , y0 , c) = −v(−; x0 , y0 , c) + b . ∂x Hence, ∂vx0 ,y0 ,c ∂vx0 ,y0 ,c ∂ 2 V ∂ 2 V (x0 , y0 , c) = (), (x0 , y0 , c) = − (−). ∂y∂c ∂c ∂x∂c ∂c Denote

∂γ ∂v (τ ; x0 , y0 , c), vc (τ ) = (τ ; x0 , y0 , c) = γ˙c (τ ). ∂c ∂c Differentiating with respect to c in Eq. (97) we get the following equation: γc (τ ) =

 vc (t) = vc (−) +  +

t

N


− k=1

t

N


− k=1

(vc (τ ) · ∇)∇F k (γ (τ ))(Wk (τ ) − Wk (s)) dτ

(v(τ ) · (γc (τ ) · ∇)∇)∇F k (γ (τ ))(Wk (τ ) − Wk (s)) dτ

(111)

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R. Iturriaga, K. Khanin

 −

−

 − −

k=1

1
∂ak ∇F k (γ (τ )) (τ, c) dτ  ∂c N

t

− N


1
(γc (τ ) · ∇)∇F k (γ (τ ))ak (τ, c) dτ  N

t

k=1

(γc (t) · ∇)∇F k (γ (t))(Wk (t) − Wk (s)),

(112)

k=1

where

 ak (τ, c) =

It follows from (113) that  ∂ak 1 − (τ, c) = 0 ∂ck

∂ak (τ, c) ∂ci−,+

ck− ck+

for τ ∈ [−, 0) for τ ∈ (0, ] .

(113)

= 0 for k = i and

 for τ ∈ [−, 0) ∂ak 0 (τ, c) = + for τ ∈ (0, ], ∂ck 1

for τ ∈ [−, 0) for τ ∈ (0, ] .

Using obvious boundary conditions γc (−) = 0, γc () = 0 we also have   |γc (t)| ≤ t max |vc (τ )|, vc (t)dt = 0. τ ∈[−,t]

−

(114)

(115)

Define v˜c (t) = vc (−) −

N
k=1

 ∇F (x0 ) k

t

−

1 ∂ak (τ, c) dτ, w(t) = max |vc (τ ) − v˜c (τ )|. τ ∈[−,t]  ∂c

Clearly there exists a constant K > 0 such that max |v˜c (t| ≤ K + |vc (−)|.

t∈[−,]

(116)

Using Lemma 16 and (112), (116) it is easy to show that there exist constants K1 , K2 > 0 such that for all  small enough  t w(τ )d(τ ). w(t) ≤ K1 (K + |vc (−)|) + K2 −

Hence, using Gronwall inequality we have w(t) ≤ K1 (K + |vc (−)|)eK2  , t ∈ [−, ]. Below we denote vc (−) by vc . Notice that  0      t + v˜c− (t)dt = 2vc− − ∇F i (x0 ) dt dt + i  − i − 0   3 = 2 vc− − ∇F i (x0 ) i 4

(117)

(118)

Burgers Turbulence and Random Lagrangian Systems

425

and 

 −

 v˜c+ (t)dt = 2vc+ − ∇F (x0 )



i

i

i

0

  t 1 i dt = 2 vc+ − ∇F (x0 ) . i  4

(119)

Using (115), (117)–(119) we obtain         v − − 3 ∇F i (x0 ) , v + − 1 ∇F i (x0 ) ≤ K1 (K + |vc |)eK2  , 1 ≤ i ≤ N. ci     ci 4 4 It follows that for  small enough max (|vc− |, |vc+ |) ≤ M = 1 + max max |∇F i (x0 )|,

1≤i≤N

i

1≤i≤N x0 ∈Td

i

which implies     v − − 3 ∇F i (x0 ) ,  ci  4

    v + − 1 ∇F i (x0 ) ≤ K3 , 1 ≤ i ≤ N  ci  4

(120)

(121)

for some constant K3 > 0. We next estimate vc (). Since v˜c− () = vc−,+ − ∇F i (x0 ) i

i

we finally get from (117), (120), (121)         v − () + 1 ∇F i (x0 ) , v + () + 3 ∇F i (x0 ) ≤ 2K3 , 1 ≤ i ≤ N. ci  ci    4 4

(122)

Since estimates (121), (122) are uniform in (x, y) ∈  and c ∈ E they together with (111) immediately imply (109).   Lemma 20.    ∂V     i lim max max  −,+ (x, y, c) − F (x) = 0.   →0 (x,y)∈ c∈E ∂c

(123)

i

Proof. We shall use the same notations as in the proof of Lemma 19. It follows from the proof of the previous lemma that there exists a constant K4 > 0 such that max |vc (t)| ≤ K4 , max |γc (t)| ≤ K4 .

t∈[−,]

t∈[−,]

(124)

Differentiating with respect to c in (96) and using (124) it is easy to show that there exists a constant K5 > 0 such that     ∂V    i  −,+ (x, y, c) − F (x) ≤ K5 ,   ∂c i which immediately implies the statement of lemma.

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Proof of Proposition 5. The first two statements were proven in Lemma 18. The third statement follows easily from Lemma 16 and (16), (96). We finish with the proof of the last statement. Since the mapping (F 1 (x), . . . , F N (x)) : Td → RN is an embedding, i the matrices ∂F ∂xj have rank d. Hence, there exists an absolute constant C1 > 0 such that for any vector r ∈ Rd , max |r · ∇F j (x)| ≥ C1 |r|.

1≤j ≤d

(125)

Let (x1 , y1 ), (x2 , y2 ) ∈  . Define rx = x2 − x1 , ry = y2 − y1 , A=

∂V ∂V (x2 , y2 ) − (x1 , y1 ) ∂c ∂c

and

   ∂V  ∂V    |A| = max max  − (x2 , y2 ) − − (x1 , y1 )  ∂ci  1≤i≤N ∂ci

   ∂V  ∂V    ,  + (x2 , y2 ) − + (x1 , y1 ) .  ∂ci  ∂ci

It follows from Lemma 19 that there exists a constant C2 > 0 such that A=

∂ 2 V (x1 , y1 )(rx , ry ) + Q, ∂(x, y)∂c

where |Q| ≤ C2 (|rx | + |ry |)2 . Lemma 19 also implies that for  small enough    ∂ 2V  3 1    i i (x , y )(r , r ) + · ∇F (x)) + · ∇F (x)) (r (r   1 1 x y x y  ∂(x, y)∂ci−  4 4 ≤

C1 max(|rx |, |ry |), 16

   ∂ 2V  3 1    i i (x , y )(r , r ) + · ∇F (x)) + · ∇F (x)) (r (r   1 1 x y x y  ∂(x, y)∂ci−  4 4 ≤ Denote

C1 max(|rx |, |ry |). 16   3  1 i i  Mi = max  (rx · ∇F (x)) + (ry · ∇F (x)) , 4 4   1  3  (rx · ∇F i (x)) + (ry · ∇F i (x)) . 4  4

It is easy to see that Mi ≥

1 max(|rx · ∇F i (x)| , |ry · ∇F i (x)|). 8

(126)

Burgers Turbulence and Random Lagrangian Systems

It follows from (125) that there exists 1 ≤ i ≤ N such that Mi ≥ Using the estimates above we have |A| ≥ max Mi − 1≤i≤N

≥

427 C1 8

max(|rx |, |ry |).

C1 max(|rx |, |ry |) − C2 (|rx | + |ry |)2 16

C1 max(|rx |, |ry |) − C2 (|rx | + |ry |)2 > 0, 16

C1 . Consider now the case when max(|rx |, |ry |) ≥ provided max(|rx |, |ry |) < m = 64C 2 m. Since (x1 , y1 ), (x2 , y2 ) ∈  it follows that |rx | ≥ m/2 if  is small enough. Using again the embedding property of the mapping (F 1 (x), . . . , F N (x)) : Td → RN we conclude that

F =

max

max |F i (x1 ) − F i (x2 )| > 0.

x1 ,x2 ∈Td :|x1 −x2 |≥m/2 1≤i≤N

This together with Lemma 20 imply that for  small enough and for all (x1 , y1 ), (x2 , y2 ) ∈  such that max(|rx |, |ry |) ≥ m we have ∂V ∂V (x1 , y1 ) = (x2 , y2 ). ∂c ∂c

(127)

 It follows from (127) that for  small enough the mapping ∂V ∂c (·, ·, c) is one-to-one on ω . The estimate (126) and Lemma 19 also imply that for  small enough and for all ∂ 2 V (x, y, c) have maximum rank 2d. Hence, the mapping (x, y) ∈  the matrices ∂(x,y)∂c

∂V ∂c (·, ·, c)

is an embedding.

 

Acknowledgement. We are grateful to J. Bec, A. Fahti, U. Frisch, G. Contreras, P. Plotnikov, V. Sedykh and Ya. Sinai for helpful discussions. The work was supported by EPSRC research grant GR/M45610, with further support for R. Iturriaga from Conacyt grant 28489-E.

References 1. Arnold, L.: Random Dynamical Systems. Berlin-Heidelberg-New York: Springer, 1998 2. Aubry, S.: The twist map, the extended Frenkel-Kontorova model and the devil’s staircase. Physica D, 7, 240–258 (1983) 3. Balkovsky, E., Falkovich, G., Kolokolov, I., Lebedev, V.: Intermittency of Burgers turbulence. Phys. Rev. Lett. 78, 1452–1455 (1997) 4. Bec, J.: Universality of velocity gradients in forced Burgers turbulence. Submitted to Phys. Rev. Lett. (2001) 5. Bec, J., Frisch, U., Khanin, K.: Kicked Burgers turbulence. J. Fluid Mech. 416, 239–267 (2000) 6. Bec, J., Iturriaga, R., Khanin, K.: Topological shocks in Burgers turbulence. Phys. Rev. Lett. 89, 024501-1–024501-4 (2002) 7. Boldyrev, S.: Velocity-difference probability density for Burgers turbulence. Phys. Rev. E 55, 6907– 6910 (1997) 8. Bouchaud, J.P., Mezard, M., Parisi, G.: Scaling and intermittency in Burgers turbulence. Phys. Rev. E 52, 3656–3674 (1995) 9. Bricmont, J., Kupiainen,A., Lefevere, R.: Ergodicity of the 2D Navier–Stokes equations with random forcing. Comm. Math. Phys. 224, 65–81 (2001) 10. Chekhlov, A., Yakhot, V.: Kolmogorov turbulence in a random force-driven Burgers equation. Phys. Rev. E 51, R2739–R2749 (1995) 11. Contreras, G.: Action potential and weak KAM solutions. Preprint (2000) 12. Contreras, G., Iturriaga, R.: Convex Hamiltonians without conjugate points. Ergod. Th. and Dynam. Sys. 19, 901–952 (1999)

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13. Contreras, G., Iturriaga, R.: Global Minimizers of Autonomous Lagrangians. 22 Colloqio Brasileiro de Matematica, 1999 14. Crauel, H.: Markov measures for random dynamical systems. Stochastics and Stochastics Reports 37, 153–173 (1991) 15. E, W., Khanin, K., Mazel, A., Sinai, Ya.: Invariant measures for Burgers equation with stochastic forcing. Annals of Math. 151, 877–960 (2000) 16. E, W., Khanin, K., Mazel, A., Sinai, Ya.: Probability distribution functions for the random forced Burgers equations. Phys. Rev. Lett. 78, 1904–1907 (1997) 17. E, W., Mattingly, J.C., Sinai, Ya.: Gibbsian dynamics and ergodicity for the stochastically forced Navier–Stokes equation. Comm. Math. Phys. 224, 83–106 (2001) 18. E, W., Vanden Eijnden, E.: Asymptotic theory for the probability density functions in Burgers turbulence. Phys. Rev. Lett. 83, 2572–2575 (1999) 19. E, W., Vanden Eijnden, E.: Statistical theory of the stochastic Burgers equation in the inviscid limit. Comm. Pure Appl. Math. 53, 852–901 (2000) 20. Elworthy, K.D.: Stochastic Differential Equations on Manifolds. LMS Lecture Notes. Cambridge: Cambridge University Press, 1982 21. Fathi, A.: Th´eor`eme KAM faible et Th´eorie de Mather sur les systems Lagrangiens. C.R. Acad. Sci. Paris, t. 324, S´erie I, 1043–1046 (1997) 22. Fathi, A., Maderna, E.: Weak KAM theorem on non-compact manifolds. Preprint (2000) 23. Flandoli, F., Maslowski, B.: Ergodicity of the 2-D Navier–Stokes equation under random perturbation. Commun. Math Physics 171, 119–141 (1995) 24. Frisch, U., Bec, J., Villone, B.: Singularities and the distribution of density in the Burgers/adhension model. Physica D (to appear, 2000) 25. Gomes, D., Iturriaga, R., Khanin, K., Padilla, P.: Convergence of stationary distributions for random forced Burgers equation in the viscosity limit, in preparation (2002) 26. Gotoh, T., Kraichnan, R.H.: Burgers turbulence with large scale forcing. Phys. Fluids A 10, 2859– 2866 (1998) 27. Gurarie, V., Migdal, A.: Instantons in Burgers equations. Phys. Rev. E 54, 4908–4914 (1996) 28. Hedlund, G.: Geodesics on a two dimensional Riemannian manifold with periodic coefficients. Annals Math. 33, 719–739 (1932) 29. Hopf, E.: The partial differential equation ut + uux = µuxx . Comm. Pure Appl. Math. 3, 201–230 (1950) 30. Kifer, Yu.: The Burgers equation with a random force and a general model for directed polymers in random environments. Prob. Theory Related Fields 108, 29–65 (1997) 31. Kuksin, S., Shirikyan, A.: Stochastic dissipative PDEs and Gibbs measures. Comm. Math. Phys. 213, 291–330 (2000) 32. Kunita, H.: Stochastic flows and stochastic differential equations. Cambridge Studies in Advanced Mathematics, vol. 24, Cambridge University Press, Cambridge, 1990 33. Lax, P.D.: Hyperbolic systems of conservation laws. Comm. Pure Appl. Math. 10, 537–566 (1957) 34. Lions, P.L.: Generalized solutions of Hamilton-Jacobi equations. Research Notes in Math. vol. 69, Pitman Advanced Publishing Program, Boston, 1982 35. Ma˜ne´ , R.: Generic properties and problems of minimizing measures of Lagrangian systems. Nonlinearity, 9, 273–310 (1996) 36. Ma˜ne´ , R.: Lagrangian flows: the dynamics of globally minimizing orbits. In: International Congress on Dynamical Systems in Montevideo (a tribute to Ricardo Ma˜ne´ ), F. Ledrappier, J. Lewowicz, S. Newhouse (eds), Pitman Research Notes in Math., vol. 362, Pitman Advanced Publishing Program, Boston, 1996, pp. 120–131. Reprinted in Bol. Soc. Bras. Mat. 28, 141–153 (1997) 37. Mather, J.: Existence of quasi-periodic orbits for twist homeomorphisms of the annulus. Topology 21, 457–467 (1982) 38. Mather, J.: Action minimizing measures for positive definite Lagrangian systems. Math. Zeitschrift 207, 169–207 (1991) 39. Morse, M.: Calculus of variations in the large. Amer. Math. Soc. Colloquium Publications, XVIII, 1934 40. Oleinik, O.A.: Discontinuous solutions of nonlinear differential equations. Uspekhi Mat. Nauk 12, 3–73 (1957) 41. Polyakov, A.: Turbulence without pressure. Phys. Rev. E 52, 6183–6188 (1995) 42. Revuz, D., Yor, M.: Continuous Martingales and Brownian Motion, Springer, 1991 43. Sinai, Ya.: Two results concerning asymptotic behavior of solutions of the Burgers equation with force. J. Stat. Phys. 64, 1–12 (1992) Communicated by A. Kupiainen

Commun. Math. Phys. 232, 429–455 (2003) Digital Object Identifier (DOI) 10.1007/s00220-002-0738-8

Communications in

Mathematical Physics

Couette Flows over a Rough Boundary and Drag Reduction Willi J¨ager1 , Andro Mikeli´c2 1 2

Institut f¨ur Angewandte Mathematik, Universit¨at Heidelberg, 69120 Heidelberg, Germany UFR Math´ematiques, Universit´e Claude Bernard Lyon 1, Bˆat. 101, 43, bd. du 11 novembre 1918, 69622 Villeurbanne Cedex, France

Received: 14 December 2001 / Accepted: 1 August 2002 Published online: 7 November 2002 – © Springer-Verlag 2002

Abstract: We consider the Couette flow between two plates. The lower plate is fixed and has periodically placed riblets of the characteristic size ε on it. In the limit ε → 0 we find the effective Couette-Navier flow as an O(ε 2 ) approximation for the effective mass flow and an O(ε2 ) L1 -approximation for the velocity. In the effective solution the effect of roughness enters through the Navier slip condition with the matrix coefficient in front of the effective shear stress, calculated using a boundary layer problem. Furthermore, an O(ε2 ) approximation for the tangential drag force is found. In all estimates explicit dependence on the kinematic viscosity ν, the velocity U of the upper plate and the distance between the plates L3 is kept. Also the uniqueness of the solution is expressed through a non-linear algebraic condition linking ε, ν, |U | and L3 . Then the result is applied to the viscous sub-layers around immersed bodies, strictly containing the surface riblets. It is found that for the riblets of the characteristic size ε, being of the order smaller or equal to ν 9/14 , the approximation obtained for the tangential drag could be applied. We compare ε and ν 9/14 for realistic data and our results lead to the conclusion that the riblets reduce significantly tangential drag, which may explain their presence on the skin of Nektons. 1. Introduction From a physical point of view, the no-slip condition v = 0, at an immobile solid boundary, is only justified where the molecular viscosity is concerned. Since the fluid cannot penetrate the solid, its normal velocity is equal to zero. This is the condition of non-penetration. To the contrary, the absence of slip is not very intuitive but corresponds to the observed situation at a smooth solid wall (see [22]). In many cases of practical significance the boundary is rough. An example are complex boundaries in the geophysical fluid dynamics. Compared with the characteristic size of a computational domain, such boundaries could be considered as rough. Other examples involve sea bottoms of random roughness and artificial bodies with periodic distribution of small bumps. A numerical simulation of the flow problems in the presence

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W. J¨ager, A. Mikeli´c

of a rough boundary is very difficult since it requires many mesh nodes and handling of many data. For computational purposes, an artificial smooth boundary, close to the original one, is taken and the equations are solved in the new domain. This way the rough boundary is avoided, but the boundary conditions at the artificial boundary aren’t given. It is clear that the non-penetration condition v · n = 0 should be kept, but there are no reasons to keep the no-slip. Usually it is supposed that the shear stress is a non-linear function F of the tangential velocity. F is determined empirically and its form varies for different problems. Such relations are called the wall laws and the classical Navier’s condition (see [21]) is an example. Another well-known example is modeling of the turbulent boundary layer close to the rough surface by a logarithmic velocity profile
vτ =

τw ρ



 
   1 y τw ln + C + ks+ , κ µ ρ

(1)

where vτ is the tangential velocity, y is the vertical coordinate and τw the shear stress. ρ denotes the density and µ the viscosity. κ ≈ 0.41 is the von K´arm´an’s constant and C + is a function of the ratio ks+ of the roughness height ks and the thin wall sublayer thickness δv = vµτ . For more details we refer to the book of Schlichting [23]. Justifying the logarithmic velocity profile in the overlap layer is mathematically out of reach for the moment. Nevertheless, after recent results we are able to justify the Navier’s condition for 2D laminar incompressible viscous flows over periodic rough boundaries. In this article we derive the Navier condition for a 3D laminar incompressible viscous flow over a rough plate. Then the result is used to find an accurate approximation for the skin friction. In the final section we apply these results to the drag reduction for a turbulent flow over a rough plate. Flow problems over rough surfaces were considered by O. Pironneau and collaborators in [20, 1, 2]. The paper [20] considers the flow over a rough surface and the flow over a wavy sea surface. It discusses a number of problems and announces a rigorous result for an approximation of the Stokes flow. Similarly, in the paper [1] numerical calculations are presented and rigorous results in [2] are announced. Finally, in the paper [2] the stationary incompressible flow at high Reynolds number Re ∼ 1ε over a periodic rough boundary, with the roughness period ε, is considered. An asymptotic expansion is constructed and, with the help of boundary layer correctors defined in a semi-infinite cell, effective wall laws are obtained. A numerical validation is presented, but there are no mathematically rigorous convergence results. The error estimate for the approximation, announced in [1], was not proved in [2]. The Couette flow over a rough plate and applications to the drag reduction were considered byY. Amirat, J. Simon and collaborators in [4–6]. Their approach has similarities with technique from [3, 16] and they concentrate on small Reynolds numbers. We are following the approach from [11, 13] for the auxiliary problems, used to calculate the effective coefficients in the Navier’s slip law, and the approach from [14] for the justification of the approximation. Our results are compared with the corresponding ones from [4–6]. In §2 we introduce the corresponding boundary layer problem, in §3 we obtain the Navier slip condition and in §4 the approximation for the skin friction is applied to the drag reduction.

Couette Flows over Rough Boundary and Drag Reduction

431

2. Navier’s Boundary Layer The effects of roughness occur in a thin layer surrounding the rough boundary. In this subsection we construct the 3D boundary layer, which will be used in taking into account the effects of roughness. They will enter through a matrix M. We note that M is independent of the flow regime and it is just a property of the flow domain. Corresponding 2D construction is in the articles [14, 19]. The detailed theory of such boundary layers is in [11]. We start by prescribing the geometry of the layer. Let bj , j = 1, 2, 3 be 3 positive constants. Let Z = (0, b1 ) × (0, b2 ) × (0, b3 ) and let ϒ be a Lipschitz surface y3 = ϒ(y1 , y2 ), taking values between 0 and b3 . We suppose that the rough surface ∪k∈Z2 ϒ + k is also a Lipschitz  surface. We introduce the canonical cell of roughness (the canonical hump) by Y = y ∈ Z | b3 > y3 > max{0, ϒ(y1 , y2 )} . Following the construction from [14], the crucial role is played by an auxiliary problem. It reads as follows: For a given constant vector λ ∈ R2 , find {β λ , ωλ } that solve in Z + ∪ (Y − b3 e3 ),

−y β λ + ∇y ωλ = 0

divy = 0 in Zbl ,  λ β S (·, 0) = 0 on S,  {∇y β λ − ωλ I }e3 S (·, 0) = λ on S, βλ

βλ

=0

{β λ , ωλ }

on (ϒ − b3 e3 ),

is

y

= (y1 , y2 )-periodic,

(2) (3) (4) (5) (6)

Z+

= (0, b1 ) × (0, b2 ) × (0, +∞), and Zbl = where S = (0, b1 ) × (0, b2 ) × {0}, Z + ∪ S ∪ (Y − b3 e3 ). Let V = {z ∈ L2loc (Zbl )3 : ∇y z ∈ L2 (Zbl )9 ; z = 0 on (ϒ − b3 e3 ); divy z = 0 in Zbl and z is y = (y1 , y2 )-periodic}. Then, by the Lax-Milgram lemma, there is a unique β λ ∈ V satisfying



∇β λ ∇ϕ dy = − ϕλ dy1 dy2 , ∀ϕ ∈ V . (7) Zbl

S

Using De Rham’s theorem we obtain a function ωλ ∈ L2loc (Zbl ), unique up to a constant and satisfying (2). By the elliptic theory, {β λ , ωλ } ∈ V ∩ C ∞ (Z + ∪ (Y − b3 e3 ))3 × C ∞ (Z + ∪ (Y − b3 e3 )) to (2)–(6).

Fig. 1. Boundary layer containing the canonical roughness

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W. J¨ager, A. Mikeli´c

In the neighborhood of S we have β λ − (λ1 , λ2 , 0)(y3 − y32 /2)e−y3 H (y3 ) ∈ W 2,q and ωλ ∈ W 1,q , ∀q ∈ [1, ∞). Then we have Lemma 1. ([11, 12, 19]). For any positive a, a1 and a2 , a1 > a2 , the solution {β λ , ωλ } satisfies  b1  b2 λ  0 0 β2 (y1 , y2 , a) dy1 dy2 = 0,       b1  b2 λ  b1  b2 λ     0 0 ω (y1 , y2 , a1 ) dy1 dy2 = 0 0 ω (y1 , y2 , a2 ) dy1 dy2 ,     b1  b2 λ b b (8) βj (y1 , y2 , a1 )dy1 dy2 = 0 1 0 2 βjλ (y1 , y2 , a2 )dy1 dy2 , j = 1, 2,  0 0       2      j,bl  Cλbl = Cλ λj = S β λ λ dy1 dy2 = − Zbl |∇β λ (y)|2 dy < 0.   j =1

 Lemma 2. Let λ ∈ R2 and let {β λ , ωλ } be the solution for (2)–(6) satisfying S ωλ dy1 2 2 j λ j j j dy2 = 0. Then β λ = j =1 β λj and ω = j =1 ω λj , where {β , ω } ∈ V ×  2 j Lloc (Zbl ), S ω dy1 dy2 = 0, is the solution for (2)–(6) for λ = ej , j = 1, 2. Lemma 3. Let a > 0 and let β a,λ be the solution for (2)–(6) with S replaced by Sa = (0, b1 ) × (0, b2 ) × {a} and Z + by Za+ = (0, b1 ) × (0, b2 ) × (a, +∞). Then we have

b1 b2 Cλa,bl = β a,λ (y1 , y2 , a)λ dy1 = Cλbl − a | λ |2 b1 b2 . (9) 0

0

Proof. It goes along the same lines as Lemma 2 from [14] and we omit it.



This simple result will imply the invariance of the obtained law on the position of the artificial boundary. Corollary 1. | Cλa,bl | is smallest for a = 0. Remark 1. If the boundary is flat, i.e. ϒ = cte., then the smallest constant | Cλa,bl | equals zero. We will see that this means that the no-slip condition remains. If b3 is the height of the biggest peak of ϒ, then we can’t extend Lemma 3 for a < 0. 1  j β dy1 dy2 be the b 1 b2 S i Navier’s matrix. Then the matrix M is symmetric negatively definite. Lemma 4. Let {β j , ωj } be as in Lemma 2 and let Mij =

Proof. We note that Mλλ =

1 1 Cλbl = − b1 b2 b1 b2

|∇β λ (y)|2 dy < 0,

 for λ = 0.

Zbl

Consequently, M is negatively definite. Since



∇β 1 ∇β 2 dy = β21 dy1 dy2 = b1 b2 M21 , b1 b2 M12 = β12 dy1 dy2 = − S

M is symmetric.



Zbl

S

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Lemma 5. Let Y have the mirror symmetry with respect to yj , where j is 1 or 2. Then the matrix M is diagonal. Proof. Let us suppose the mirror symmetry with respect to y1 , i.e. ϒ(y1 , y2 ) = ϒ(1 − y1 , y2 ). Then we introduce a new function ξ which coincides with β 1 (y) for y1 ∈ [0, b1 /2]. For b1 > y1 > b1 /2 we extend ξ1 evenly and the components ξ2 and ξ3 are extended as odd functions. We also introduce a function κ which coincides with ω1 for y1 ∈ [0, b1 /2]. For b1 > y1 > b1 /2 we extend it unevenly. The newly defined pair {ξ, κ} is also a variational solution for (2)–(6) and S κ(y1 , y2 , 0)dy1 dy2 = 0. Then by the uniqueness it coincides with {β 1 , ω1 }. Consequently,  



1 β21 dy1 dy2 = 0 = b1 b2 M21 , + b1 b2 M12 = β2 dy1 dy2 = S

S∩{y1 <1/2}

and the matrix M is diagonal.

S∩{y1 >1/2}



The next result gives the structure of the matrix M in an important particular case. Lemma 6. Let us suppose that the shape of the boundary doesn’t depend on y2 . Then for λ = e2 the system (2)–(6) has the solution β 2 = (0, β22 (y1 , y3 ), 0) and ω2 = 0, where β22 is determined by −

∂ 2 β22 ∂y12

−

∂ 2 β22 ∂y32

β22 = 0

=0

in (0, b1 ) × (0, +∞) ∪ (Y ∩ {y2 = 0} − b3 e3 ), (10)

 2 β2 (·, 0) = 0 on (0, b1 ) × {0},  ∂β 2  2 (·, 0) = 1 on (0, b1 ) × {0}, ∂y3 on (ϒ ∩ {y2 = 0} − b3 e3 ), β22 is y1 − periodic.

Furthermore, for λ = e1 , the system (2)–(6) has the solution (y1 , y3 )) and ω1 = ω(y1 , y3 ) satisfying −

∂βj1 ∂y12

−

∂βj1 ∂y32

+

β1

=

(β11 (y1 , y3 ),

Finally,

and | M11 | ≤ | M22 |.

(12) (13) 0, β31

∂ω = 0 in (0, b1 ) × (0, +∞) ∪ (Y ∩ {y2 = 0} − b3 e3 ), ∂yj j = 1 and j = 3, (14)

∂β11 ∂β 1 + 3 =0 in Zbl ∩ {y2 = 0}, ∂y1 ∂y3  1 βj (·, 0) = 0 on (0, b1 ) × {0}, j = 1 and j = 3,      ∂β11 ∂β31 ω = 0 and (·, 0) = 1, (·, 0) = 1 on (0, b1 ) × {0}, ∂y3 ∂y3 β11 = β31 = 0

(11)

on (ϒ ∩ {y2 = 0} − b3 e3 ), {β11 , β31 , ω} is y1 − periodic.   1 b1 1  M11 = b1 0 β1 (y1 , 0) dy1 M12 = M21 = 0  M = 1  b1 β 2 (y , 0) dy 22 1 2 1 b1 0

(15) (16) (17) (18)

(19)

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W. J¨ager, A. Mikeli´c

Proof. The solutions for (10)–(13) and for (14)–(18) satisfy the system (2)–(6) for λ = e2 and λ = e1 , respectively. Due to the uniqueness, they equal the solutions for (2)–(6). The inequality | M11 | ≤ | M22 | follows from the variational forms of (10)–(13) and (14)–(18).  In the next step we determine the decay in Z + by reductionto the Laplace operator. We choose the free constant in the pressure field in the way that S ωbl (y1 , y2 , 0) dy1 dy2 = 0. Lemma 7. Let {β j , ωj }, j = 1 and j = 3, be as in Lemma 2. Then we have   | D α curly β j (y) |≤ Ce−2πy3 min{1/b1 ,1/b2 } , y3 > 0, α ∈ N2 ∪ (0, 0)   | β j (y) − (M , M , 0) |≤ C(δ)e−δy3 , y > 0, ∀δ < 2π min{1/b , 1/b } 1j 2j 3 1 2 | D α β j (y) |≤ C(δ)e−δy3 , y3 > 0, α ∈ N2 , ∀δ < 2π min{1/b1 , 1/b2 }   | ωj (y) |≤ Ce−2πy3 min{1/b1 ,1/b2 } , y3 > 0.

(20)

Proof. As in [12] we take the curl of Eq. (2) and obtain the following problem for   j ξm = curl β j m , m = 1, 2, 3:  j in Z + ξm = 0 (21) j j ξm ∈ W 1−1/q,q (S), ∀q < +∞ ξm is periodic in y = (y1 , y2 ). j

Now Tartar’s lemma from [16] implies an exponential decay of ∇ξm to zero and of j j ξm . Since ξm ∈ L2 (Z + ), this constant equals zero. Furthermore, having established an exponential decay, we are in a situation to apply the separation of variables and explicit calculations, analogous to these in [15], and give the first estimate in (20). In the next step we use the following identity, holding for the divergence free fields: −β j = curl curl β j = curl ξ j , and the same arguing as above leads to the second and the third estimate. After taking the divergence of Eq. (2) we find out that the pressure is harmonic in Z + . Since the averages of the pressure over the sections {y3 = a} are zero, we obtain the fourth estimate in (21).  Corollary 2. The system (2)–(6) defines a boundary layer. 3. Justification of the Navier’s Slip Condition for the Laminar 3D Couette Flow A mathematically rigorous justification of the Navier’s slip condition for the 2D Poiseuille flow over a rough boundary is undertaken in [14]. Rough boundary was the periodic repetition of a basic cell of roughness, with characteristic heights and lengths of the impurities equal to a small parameter ε. Then the flow domain was decomposed on a rough layer and its complement. The no-slip condition was imposed on the rough boundary and there were inflow and outflow boundaries, not interacting with the humps. The flow was governed by a given constant pressure drop. The mathematical models were the stationary incompressible Navier-Stokes equations. In [14] the flow under moderate Reynolds numbers was considered and the following results were proved: a) A non-linear stability result with respect

Couette Flows over Rough Boundary and Drag Reduction

435

to small perturbations of the smooth boundary with a rough one; b) An approximation result of order ε3/2 ; c) Navier’s slip condition was justified. In this article we are going to establish analogous results for a 3D Couette flow. Since our goal is to find the contribution to the tangential drag, we skip details already given in [14] or other recent references. Our results will be applied to a flow in a tiny layer with a rough lower boundary and with thickness proportional to the square root of the kinematic viscosity. Consequently, we need estimates depending explicitly on ε and these 2 parameters. We consider a viscous incompressible fluid flow in a domain ε consisting of the parallelepiped P = (0, L1 ) × (0, L2 ) × (0, L3 ), the interface  = (0, L1) ×    (0, L2 ) × {0} and the layer of roughness R ε = ∩ {k∈Z2 } ε Y + (k1 , k2 , −b3 ) ((0, L1 ) × (0, L2 ) × (−εb3 , 0)). The canonical cell of roughness Y ⊂ (0, b1 )×(0, b2 )× (0, b3 ) is defined in §2. For simplicity we suppose that L1 /(εb1 ) and L2 /(εb2 ) are integers. Let I ={0 ≤ k1 ≤ L1 /b1 ; 0 ≤ k2 ≤ L2 /b2 ; k ∈ Z2 }. Then, our rough boundary B ε = {k∈I } ε ϒ + (k1 , k2 , −b3 ) is supposed to consist of a large number of periodically distributed humps of characteristic length and amplitude ε, small compared with a characteristic length of the macroscopic domain. Then, for a fixed ε > 0 and a given constant velocity U = (U1 , U2 , 0), the Couette flow is described by the following system: −νv ε + (v ε ∇)v ε + ∇p ε = 0 in ε , (22) ε ε in  , (23) div v = 0 ε ε v =0 on B , (24) v ε = U on 2 = (0, L1 ) × (0, L2 ) × {L3 }, (25) ε ε {v , p } is periodic in (x1 , x2 ) with period (L1 , L2 ), (26)  where ν > 0 is the kinematic viscosity and ε p ε dx = 0. Let us note that a similar problem was considered in [6], but in an infinite strip with a rough boundary. In [6] the authors were looking for solutions periodic in (x1 , x2 ), with the period ε(b1 , b2 ), i.e. with the Reynolds number proportional to |U |ε/ν. For high Reynolds numbers it is not clear that it is the only solution. Since we need not only existence for a given ε, but also the a priori estimates independent of ε, we give a non-linear stability result with respect to rough perturbations of the boundary, leading to uniform a priori estimates. Our proof follows the corresponding one from [14].

Fig. 2. Couette flow over a rough plate

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W. J¨ager, A. Mikeli´c

First, we observe that the Couette flow in P , satisfying the no-slip conditions at , is given by v0 =

U1 x 3 U 2 x3 x3 e1 + e2 = U , L3 L3 L3

p0 = 0.

(27)

 Let |U | = U12 + U22 . Then it is easy to see that v 0 is unique if |U |L3 < 2ν, i.e. if the Reynolds number is moderate. We extend the velocity to ε \ P by zero. The idea is to construct the solution to (22)–(26) as a small perturbation to the Couette flow (27). Before the existence result, we prove two auxiliary lemmas: Lemma 8. ([14]) Let ϕ ∈ H 1 (ε \ P ) be such that ϕ = 0 on B ε . Then we have   b3  ∂ϕ   , ϕL2 (ε \P ) ≤ ε √  2  ∂x3 L2 (ε \P )3    √   1/2  ∂ϕ  ϕL2 () ≤ b3 2ε  , ∂x3 L2 (ε \P )3     ∂ϕ   ϕL1 () ≤ L1 L2 b3 ε 1/2   ∂x  2 ε 3 , 3 L ( \P )   L3 + εb3  ∂ϕ    ϕL2 (P ) ≤ . √ 2  ∂x3 L2 (P )3

(28) (29) (30) (31)

Lemma 9. Let ϕ ∈ H 1 (ε )3 be (L1 , L2 )-periodic in (x1 , x2 ) and equal to zero on B ε ∪ 2 . Then for 2 ≤ q ≤ 6 we have  3/q−1/2 ϕLq (ε )3 ≤ C0,q L3 + εb3 ∇ϕL9 (ε )9 (32) with  C0,q = 3

13/2(1/2−1/q) 1/2−2/q

2



1 1 3/8 + 4 + 2 L21 L2



3/4−3/(2q) (L3 + εb3 )

2

. (33)

Proof. First we note that for any ϕ ∈ L6 (ε ) we have 3/q−1/2

3/2−3/q

ϕLq (ε ) ≤ ϕL2 (ε ) ϕL6 (ε ) , 2 ≤ q ≤ 6 leading to ϕLq (ε )3 ≤ 2

1/4−3/(2q)



L3 + εb3

3/q−1/2

3/2−3/q ϕL6 (ε )3

   ∂ϕ 3/q−1/2    ∂x  2 ε 3 . 3 L ( )

For the smooth functions u with zero trace we have u6L6 (ε )3 ≤ 48u6L2 (ε )9 and after applying this estimate to (1 + x1 /L1 )(2 − x1 /L1 )(1 + x2 /L2 )(2 − x2 /L2 )ϕ and inserting the result to the interpolation estimate for ϕLq (ε )3 , we obtain (32)–(33).  Now we are in position to prove the existence of a solution close to the Couette flow:

Couette Flows over Rough Boundary and Drag Reduction

437

Theorem 1. Let |U |L3 < 2ν and let R0 =

 |U | ν L1 L2 b3 . L3 ν − L3 |U |/2

(34)

L3 Then for ε < the problem (22)–(26) has a solution {v ε , pε } ∈ H 2 (ε )3 × H 1 (ε ),   b3 εb1 , εb2 -periodic in (x1 , x2 ) and satisfying √ ∇(v ε − v 0 )L2 (ε )9 ≤ R0 ε. (35) Moreover, √ v ε L2 (ε \P )3 ≤ √b3 R0 ε ε, 2 √ v ε L2 ()3 ≤ 21/4 b3 R0 ε.

(36)

Furthermore, it is C ∞ in P¯ . Proof. We follow the analogous proof from [14]. Let  Z ε = z ∈ H 1 (ε )3 : z = 0 on B ε ∪ 2 and z is periodic in (x1 , x2 ) .  Then wesearch v ε in the form v ε = v 0 + w ε , where w ε ∈ W ε = ϕ ∈ Z ε : div ϕ = 0 in ε is such that





ε ε ε 0 ε ν ∇w ∇ϕ + (w ∇)w ϕ + (v ∇)w · ϕ + (w ε ∇)v 0 ϕ ε ε P P

1  ε =ν U · ϕ dx1 dx2 , ∀ϕ ∈ W . (37)  L3 The proof of solvability for (37) consists of several steps:   a) Let H = ϕ ∈ H 1 (P )3 : ϕ = 0 on 2 and ϕ is (L1 , L2 ) – periodic in (x1 , x2 ) and let  (ψ, ϕ) be a bilinear form on H × H given by



 0     (ψ, ϕ) = ν v ∇ ψϕ dx + ψ∇ v 0 ϕ dx. ∇ψ∇ϕ dx + (38) P

P

P

Then for every ψ ∈ H we have



    2 0  (ψ, ψ) = |∇ψ|2 dx. ν|∇ψ| + (ψ∇)v ψ dx ≥ ν − L3 |U |/2 P

(39)

P

b) Now let wk ∈ W ε , w k H 1 (ε )2 ≤ R. We consider the problem



 (wk+1 , ϕ) + ν ∇w k+1 ∇ϕ dx + (w k ∇)w k+1 ϕ dx ε ε  \P 

1  ε =ν ∀ϕ ∈ W . U · ϕ dx1 dx2 ,  L3 Due to (39), for 2ν > L3 |U |, the problem (40) has a unique solution w k+1 ∈ W ε .

(40)

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W. J¨ager, A. Mikeli´c

c) Now we define a nonlinear mapping T by T (w k ) = wk+1 .

(41)

Let us check if T is a weakly continuous map, T : W ε → W ε . Let zj ∈ W ε , j = 1, 2 and let w j = T zj , j = 1, 2. Furthermore, let z = z1 − z2 and w = w 1 − w 2 . Then we have



  1 2 − (z∇)w w =  (w, w) + ν |∇w| ≥ ν − L3 |U |/2 |∇w|2 ε \P

ε

leading to

ε



 ν − L3 |U |/2 ∇wL2 (ε )9 ≤ zL4 (ε )3 w 1 L4 (ε )3 ,

which proves the weak continuity in W ε . d) It remains to check that T (BR ) ⊂ BR for  √ √ |U | ν R ≤ R0 ε = L1 L2 b3 ε, L3 ν − L3 |U |/2   as a map from L W ε , W ε . By Lemma 8, we have !

! ! !  √ U ν ! ! ϕ dx1 dx2 ! ≤ |U | L1 L2 b3 ε∇ϕL2 (ε \)9 , ! !  L3 ! L3 Consequently, (39) implies

√ ∇w k+1 L2 (ε )4 ≤ R0 ε

∀ϕ ∈ Z ε .

(42)

and T (BR ) ⊂ BR . Therefore, by the fixed point theorem of Schauder-Tychonov, T has a fixed point wε ∈ W ε , satisfying the estimate (35).  Remark 2. For simplicity and in accordance with the application in the last section we 2|U |  are going to suppose |U |L3 ≤ ν. Then R0 ≤ L1 L 2 b3 . L3
√ 2L3 , ε < L3 /b3 and |U |L3 ≤ ν. Proposition 1. Let 0 < ε ≤ ε1 = ν · max{bπ1 ,b2 } |U | Then for an arbitrary (εb1 , εb2 )-periodic in (x1 , x2 ) solution to (22)–(26), obtained in Theorem 1, we have the following equalities and a priori estimates:

L1 L2 v3ε (x1 , x2 , x3 ) dx1 dx2 = 0 ∀x3 ∈ [−εb3 , L3 ], (43) 0 0 ! L1 L2  !  ! ! 4|U | U j x3 ε ! v dx1 dx2 !! ≤ (x , x , x ) − b3 L1 L2 ε, ∀x3 ∈ [0, L3 ], (44) 1 2 3 j ! L3 L3 0 0 

L1 L2  U j x3 vjε (x1 , x2 , x3 ) − dx1 dx2 L3 0 0 



  x3 1 x3 ε = 1− vj dx1 dx2 + v3ε vjε dx 1− L3 ν L3  P



1 L3 L1 L2 ε ε − v3 vj dx, ∀x3 ∈ [0, L3 ], j = 1, 2, (45) ν x3 0 0

Couette Flows over Rough Boundary and Drag Reduction

and

439

! !

! ν ∂v ε νUj !! ν j ! ε dx1 dx2 + v dx1 dx2 − ! ! ! L1 L2  ∂x3 L1 L2 L 3  j L3 !    √ max{b1 , b2 } 2 ε 4b3 |U |2 2 ≤ + 2b3 ε , j = 1, 2. 2π L3 L23

(46)

Proof. Let wε = v ε − v 0 be a (εb1 , εb2 )-periodic solution in (x1 , x2 ), constructed in the proof of Theorem 1. Let z = νε w ε and y = xε . Then divy z = 0 in Zbl ∩{y3 < Lε3 } and after b b averaging it over an arbitrary horizontal section we get dyd 3 0 1 0 2 z3 (y1 , y2 , y3 ) dy1 dy2 b b = 0. Using the non-penetration condition for z at the boundary y3 = L3 /ε we get 0 1 0 2 z3 (y1 , y2 , y3 ) dy1 dy2 = 0, ∀y3 ∈ ] − b3 , L3 /ε[, implying (43). Next we have −y z + ∇y p +

2 ε 2  Uj y3 ∂z ε 2 U ε2 + z3 + 2 (z∇y )z = 0 ν L3 ∂yj ν L3 ν

(47)

j =1

  in ZL3 = Z + ∩ {y3 < L3 /ε} ∪ (Y − b3 e3 ). After testing (47) by z, we obtain





z3  |z|2 ε2 ε2 |∇y z| dy + z · ∇y U · (z1 , z2 ) dy + ν ZL3 L3 ν ZL3 2 ZL3  

  ν ν|U |   ∂z  = . U · (z1 , z2 ) dy1 dy2 ≤ b1 b2 b3  L3 S L3 ∂y3 L2 (ZL ) 2



3

Let  z dx1 dx2 be the average of " z over ]0, L1 [×]0, L2 [ for a given x3 ∈ [0, L3 ]. √ √ 2π L3 Then for ε ≤ ε1 = ν we have max{b1 , b2 } |U | ! !

! !!  !

! ε2 ! ! ε2 

! U z z j 3 3  ! ! U · (z1 , z2 ) dy ! = !! zj −  zj dy1 dy2 !! ! ! ν ZL L3 ! !ν ZL3 L3 ! 3 j  

  ε2 |U |  z3 L2 (ZL ) zj −  zj dy1 dy2  ≤  2 3 ν L3 L (ZL3 )   2  ε2 |U | max{b1 , b2 }  ∇y z2 2 ≤ L (ZL3 ) ν L3 2π 2 1 ≤ ∇y zL2 (Z ) giving L3 2   2ν|U |  ∇y z 2 ≤ b1 b2 b3 , (48) L (ZL3 ) L3 and leading to (35) with |U |L3 ≤ ν. The estimate (48) allows getting precise estimates on Lq -norms ofv ε . Letπ(y1 , y2 ) = y1 y1 y2 y2 b1 b2 1 81 = 64 . Let 4 (1 + b1 )(2 − b1 )(1 + b2 )(2 − b2 ). Then 0 ≤ π(y1 , y2 ) ≤ π 2 , 2

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W. J¨ager, A. Mikeli´c

˜ ∩ H 1 (Z), ˜ (b1 , b2 )−periodic, we Z˜ =] − 1, 2[2 × ] − b3 , L3 /ε[. Then ∀ϕ ∈ C ∞ (Z) have 1/6 ∇(π ϕ)L2 (Z) and π ϕL6 (Z) ˜ ≤ 48 ˜   81 1/6 3 1 ϕL6 (ZL ) ≤ 48 ∇ϕL2 (ZL )3 + ϕL2 (ZL ) . 3 3 3 16 4 min{b1 , b2 }

Let C¯ 2 =

max{b1 , b2 } . Since 2π  

  ϕ −  ϕ dy1 dy2   

L2 (ZL3 )

(49) reads  

  ϕ −  ϕ dy1 dy2   

L6 (ZL3 )

81 1/6 ≤ 48 16



≤ C¯ 2 ∇ϕL2 (ZL

3 3)

,

(49)

(50)

$  # 3 b1 b2 1 , + max ∇ϕL2 (ZL )3 3 2 b2 b 1 π

= C¯ 6 ∇ϕL2 (ZL

3 3)

(51)

.

 

   Moreover,  ϕ −  ϕ dy dy ≤ C¯ 3 ∇ϕL2 (ZL )3 1 2  3 L3 (ZL3 ) " $   # 81 1/6 3 b1 b2 1 = ∇ϕL2 (ZL )3 , 48 max{b1 , b2 } + max , (52) 3 64π 2 b2 b1 π and we have  

  2C¯ 2 |U |      ≤ b1 b2 b3 ν  z −  z dy1 dy2  2 L3 L (ZL3 )3  

.   2C¯ 3 |U |       z −  z dy dy ≤ b b b ν 1 2 1 2 3  L3 L3 (ZL )3

(53)

3

Consequently,  

 ε  2C¯ 2 |U |   ε   w −  w dx1 dx2  L1 L2 b3 ε 3/2   2 ε3≤ L 3 L ( )  

.   ε 2C¯ 3 |U |   ε 1/3 1/6 4/3     ≤ b3 (L1 L2 ) (b1 b2 ) ε w −  w dx1 dx2  L3 L3 (ε )3 Analogously, let Cˆ 2 =

b3 √ 2

81 1/6 and Cˆ 32 = b3 2−1/2 16 48

3 2

(54)

 + max{ bb21 , bb21 } π1 . Then

 2Cˆ 2 |U |    zL2 (Y −b3 e3 )3 ≤ ν b 1 b 2 b3 L3  2Cˆ 3 |U |   zL3 (Y −b e )3 ≤ ν b1 b2 b3 3 3 L3

(55)

Couette Flows over Rough Boundary and Drag Reduction

441

and  2Cˆ 2 |U |    w ε L2 (ε \P )3 ≤ L1 L2 b3 ε 3/2 L3 .  2Cˆ 3 |U |   w ε L3 (ε \P )3 ≤ b3 (L1 L2 )1/3 (b1 b2 )1/6 ε 4/3 L3

(56)

It remains to estimate the averages over the horizontal sections for x3 ≥ 0. We write Eq. (47) in the form  % Uj ε 2 y3 zj U ε 2 ε2 + zzj 2 + z3 = 0, in Z + ∩ {y3 < L3 /ε}. divy −∇y zj + pej + L3 ν ν L3 ν (57) After integrating (57) over an arbitrary horizontal section, and using (43), we obtain d dy3



b1



0

b2

#

b1



0

∂zj ε2 (y1 , y2 , y3 ) − z3 (y1 , y2 , y3 )zj (y1 , y2 , y3 ) 2 ∂y3 ν

$ dy1 dy2 = 0,

implying d dy3



b2

zj (y1 , y2 , y3 ) dy1 dy2 0



=

0 b1



b2

z3 (y1 , y2 , y3 )zj (y1 , y2 , y3 ) 0

0

ε2 ¯ dy1 dy2 + C, ν2

y3 ≥ 0,

Using the no-slip condition on y3 = L3 /ε, we obtain

b1

− 0



b2



L3 /ε

zj (y1 , y2 , y3 ) dy1 dy2 =



b1



b2

z3 (y1 , y2 , y3 )zj (y1 , y2 , y3 )

0

0

y3

×

0

ε2 ¯ 3 /ε − y3 ), dy1 dy2 + C(L ν2

y3 ≥ 0. (58)

¯ The equality (58) allows to calculate C: ε C¯ = − L3 Hence

0

b1



zj dS − S

ε3 ν 2 L3



L3 /ε



b1



b2

z3 (y)zj (y) dy. 0

0

(59)

0

b2

zj (y1 , y2 , y3 ) dy1 dy2  

  L3 /ε b1 b2 εy3 εy3 ε2 = 1− zj dS + 2 1− z3 (y)zj (y) dy L3 ν L3 S 0 0 0



ε2 L3 /ε b1 b2 + 2 z3 (y)zj (y) dy, ν y3 0 0 0

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W. J¨ager, A. Mikeli´c

implying, under the hypothesis ε ≤ ε1 , ! b1 b2 ! ! ! 4|U |ν ! zj (y1 , y2 , y3 ) dy1 dy2 !! ≤ b1 b2 b3 , ! L3 0 0 and

!

! ! ! ! wε (x1 , x2 , x3 ) dx1 dx2 ! ≤ 4|U | b3 ε, j ! ! L3

x3 ≥ 0,

y3 ≥ 0,

j = 1, 2.

(60)

Finally,





∂zj ε ε3 ε2 dy1 dy2 + zj dy1 dy2 = − 2 z3 zj dy + 2 z3 zj dS, L3 S ν L3 Z + ∩{y3
!

!

! ∂zj ! ε ! ! dy dy + z dy dy 1 2 j 1 2! ! ∂y L3 S 3 S     2 √ ε max{b1 , b2 } 2 2 |U | ≤ 4b1 b2 b3 ε . 2b3 + L3 2π L23

Consequently,

! !

! ν ∂w ε ! ν j ! ! dx1 dx2 + wjε dx1 dx2 ! ! ! L1 L2  ∂x3 ! L1 L2 L3     √ 4b3 |U |2 2 max{b1 , b2 } 2 ε ≤ + b3 2 ε , 2π L3 L23

and the proposition is proven.



Corollary 3. Under the hypothesis of the previous proposition, we have  

  ε w −  w ε dx1 dx2   q ε3  L ( )

2C¯ q |U |  ≤ b3 (L1 L2 )1/q (b1 b2 )1/2−1/q ε 1+1/q , 2 ≤ q ≤ 6, (61) L3 2Cˆ q |U |  wε Lq (ε \P )3 ≤ b3 (L1 L2 )1/q (b1 b2 )1/2−1/q ε 1+1/q , 2 ≤ q ≤ 6, (62) L3 

   |  wε (x1 , x2 , ·) dx1 dx2  ≤ 4|U (63) j   ∞ L3 b3 ε, j = 1, 2, L (0,L3 )

with C¯ q =



max{b1 , b2 } 2π

and Cˆ q =



b3 √ 2

3/q−1/2 

3/q−1/2 

81 1/6 48 16

81 1/6 48 16





$ 3/2−3/q # 3 b1 b2 1 , + max 2 b2 b1 π

$ 3/2−3/q # 3 b1 b2 1 , . + max 2 b2 b1 π

(64)

(65)

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1/2 Theorem 2. Let C0 (b1 , b2 , b3 ) = 2C0,6 max{C¯ 3 , Cˆ 3 }b3 (b1 b2 )1/6 , with C0,6 given by (33) and let us suppose the assumptions of Proposition 1. Furthermore let

ν>

|U | L3 |U | (L1 L2 )1/3 ε 4/3 . + 4|U |b3 ε + C0 (b1 , b2 , b3 ) 2 L3

(66)

Then v ε , constructed in Theorem 1, is a unique solution to (22)–(26). Proof. Let v ε be a (εb1 , εb2 )-periodic in (x1 , x2 ) solution constructed in Theorem 1 and let zε be another solution. We note that in general zε is only (L1 , L2 )-periodic in (x1 , x2 ). Then uε = v ε − zε satisfies the equation −νuε + (uε ∇)v 0 + ∇ π˜ ε + (uε ∇)w ε + (zε ∇)uε = 0 in ε , with w ε = v ε − v 0 ∈ W ε . π˜ ε is a corresponding pressure difference. After testing (67) with uε we get

(uε ∇)v 0 uε dx (ν − L3 |U |/2) ∇uε 2L2 (ε )9 ≤ ν∇uε 2L2 (ε )9 + ε

(uε ∇)w ε uε dx. =−

(67)

(68)

ε

 Let us estimate the term ε (uε ∇)w ε uε dx:



(uε ∇)w ε uε dx = − ∇uε : wε ⊗ uε ε ε   





=− ∇uε : wε ⊗ uε − ∇uε : wε −  w ε dx1 dx2 ⊗ uε ε \P P 



∂ ε  w dx1 dx2 − (uε uε3 ). ∂x 3 P By (61), (62) and (63) !

!  

! ! ! ∇uε : wε −  w ε dx1 dx2 ⊗ uε ! ! ! P

≤ ∇uε L2 (P )9 w ε −  w ε dx1 dx2 L3 (P )3 uε L6 (P )3

!

! ! !

ε \P

2|U | 1/2 ≤ C0,6 C¯ 3 ∇uε 2L2 (P )9 b (L1 L2 )1/3 (b1 b2 )1/6 ε 4/3 , L3 3 ! ! ∇uε : wε ⊗ uε !! ≤ ∇uε L2 (ε \P )9 uε L6 (ε \P )3 ∇w ε L3 (ε \P )3 ≤ C0,6 Cˆ 3 ∇uε 2L2 (P )9

and

2|U | 1/2 b (L1 L2 )1/3 (b1 b2 )1/6 ε 4/3 , (70) L3 3

! !

 

! ! ! ∇uε :  w ε dx1 dx2 ⊗ uε ! ≤ 4|U |b3 ε∇uε 2 2 ε 9 . ! ! L ( \P ) P

(69)

(71)

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After inserting (69)–(71) into (68) we get # L3 |U | 0≥ ν− − 4|U |b3 ε − C0,6 max{C¯ 3 , Cˆ 3 }· 2 $ 2|U | 1/2 b3 (L1 L2 )1/3 (b1 b2 )1/6 ε 4/3 ∇uε 2L2 (ε )9 L3 and the theorem is proven.



Remark 3. We note that in the previous theorem the (εb1 , εb2 )-periodic solution from the paper [6] is recovered. √ Remark√4. In our application to the flow in a viscous layer, we are going to have L3 ∼ ν, |U | ∼ ν. Then (66) means the non-linear stability, with respect to the perturbation of a smooth boundary by small rugosities of the characteristic size ε, for ε of the order smaller or equal to ν 3/4 . For bigger ε the contribution of the roughness should be taken into account even for establishing the stability. In the above considerations, we have obtained the uniform a priori estimates for {v ε , pε }. Moreover, we have found that Couette’s flow in P is an O(ε 3/2 ) L2 -approximation for v ε . At  it is an O(ε) L2 -approximation. Following the approach from [14], the Navier slip condition should correspond to taking into the account the next order corrections for the velocity. Then formally we get vε = v0 −

2  ε   j x  Uj β − (Mj 1 , Mj 2 , 0)H (x3 ) L3 ε j =1

ε − L3

2 



Uj

j =1

x3 1− L3

 (Mj 1 , Mj 2 , 0)H (x3 ) + O(ε 2 ),

where v 0 is the Couette velocity in P and the last term corresponds to the counterflow generated by the motion of . On the interface , ∂vjε ∂x3

=

i 2 Uj 1  ∂βj − Ui + O(ε) L3 L3 ∂y3

2 x  1 ε 1  Ui βji vj = − + O(ε). ε L3 ε

and

i=1

i=1

After averaging we obtain the familiar form of the Navier’s slip condition eff

uj

= −ε

2  i=1

eff

Mj i

∂ui ∂x3

on

,

(N F C)

where ueff is the average over the impurities and the matrix M is defined in Lemma 4. The higher order terms are neglected. Now let us make this formal asymptotic expansion rigorous. It is clear that in P the flow continues to be governed by the Navier-Stokes system. The presence of the irregularities would only contribute to the effective boundary conditions at the lateral boundary. The leading contribution for the estimate (35) were the

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445

 interface integral terms  ϕj . Following the approach from [14], we eliminate it by using the boundary layer-type functions β j,ε (x) = εβ j

x  ε

and ωj,ε (x) = ωj

x  ε

x ∈ ε , j = 1, 2,

,

(72)

where {β j , ωj } is defined in Lemma 2. We have, for all q ≥ 1 and j = 1, 2, 1 j,ε β − ε(M1j , M2j , 0)Lq (P )3 + ωj,ε Lq (P ) + ∇β j,ε Lq ()9 = Cε 1/q ε

(73)

and   −β j,ε + ∇ωj,ε = 0   div β j,ε = 0  j,ε  (·, 0) = 0 β    {∇β j,ε − ωj,ε I }e3  (·, 0) = ej

in ε \  in ε on  on .

(74)

  As in [14] stabilization of β j,ε towards a nonzero constant velocity ε M1j , M2j , 0 , at the upper boundary, generates a counterflow. It is given by the following Oseen’s systems in P , with i = 1,2:  2   ∂d i   i + ∇g i + x3   −νd U + d3i LU3 = 0 in P , j  L3 ∂xj j =1 (75)  i =0  div d in P ,    i d = ei on , d i is periodic in (x1 , x2 ) d i = 0 on 2 . Under the assumption |U |L3 < 2ν, the problem (75) has a unique solution in the form x3 )ei and g i = 0. of the 3D Couette flow d i = (1 − L3 Now, we would like to prove that the following quantities are o(ε) for the velocity and O(ε) for the pressure:  −1 %    2  ε ε  x U (x) = v − v + U I− M βj − (Mj 1 , Mj 2 , 0)H (x3 ) L3 L3 ε j =1 j   −1 %  2 ε  ε x3  + I− 1− (Mj 1 , Mj 2 , 0)H (x3 ) U M L3 L3 L3 j =1 j  −1 % −1 2  x  x+  ε ε ε  ε =v + I− U β j U (76) M M − 3 I− L3 L3 ε L3 L3 j =1 j  −1 % 2  ε ν P ε = pε + U ωj,ε . I− M (77) L3 L3 ε

ε

0

j =1

Then we have the following result

j

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Lemma 10. Let {U ε , P ε } be given by (76)–(77). Then  ε U ∈ H 1 (ε )3 , U ε = 0 on B ε , it is (L − periodic in (x1 , x2 ),   # 1 , L2 ) $  1 1 ε ε and ∀α ∈ N3 . , div U = 0 in  and ∀δ < 2π min  b b 1 2   α ε |D U | + |D α P ε | ≤ C(b1 , b2 , b3 ) exp{−δL3 /ε} on 2

(78)

Furthermore, ∀ϕ satisfying (78), U ε satisfies the variational equation  −1 %



2  ε  ε ε I− U ∇ U ∇U ∇ϕ dx − M ν L3 ε ε j =1 j  

2

+    εx Uj ϕ + ∂U ε ϕ j,ε ε  ϕ 3 × β − + U3 U Mj 1 , Mj 2 , 0 · + x3 ε L3 L3 L3 L3 ∂xj P j =1    

2

  ε  ε  εx3+  j,ε + U ∇ U ϕ dx − β − Mj 1 , Mj 2 , 0 ∇ ε L3 ε j =1    −1 % −1 %

2

 ε ϕ ε ε ε ×U I− M = U I−  ϕ dx +ν M U L3 L3 L3 ε 2 j =1

j

j

 ϕ  j,ε × ∇β − ωj,ε I e3 d, (79) L3 where   −1 % −1 % 2 2 x3+  ε ε 1  j,ε ε    = 2 I− I− U U χP + 2 U β M Uj M L3 L3 L3 j =1 3 L3 j,l=1 j l  %  %   2 −1 −1  ε ε ∂β l,ε I− I− × − M M U U ∂xj L3 L3 j,l=1 j l   +    l,ε εx3 1  j,ε × 2 β − εH (x3 ) Mj 1 , Mj 2 , 0 ∇ β − (Ml1 , Ml2 , 0) L3 L3    −1 %  −1 % 2  ε ε 1 x3 −ε I− I− 1 − M M U U L3 L3 L3 L23 j,k,l=1

× Mj k χP

j

l

∂β l,ε

. ∂xk Lemma 11. Let ε be given by (80). Then we have !

! ! ! ε 3/2 ! ! ¯ ! ε  ϕ dx ! ≤ ε |∇ϕ|L2 (ε )3 C(b1 , b2 , b3 )   % |U |2 |M| + b3 |M|b 3 × 1+ε , + ε2 L3 L3 L23 and the right-hand side in (79) is estimated by R1 ε 3/2

|U |2 ∇ϕL2 (ε )9 , where L3

(80)

(81)

Couette Flows over Rough Boundary and Drag Reduction

447



%

 ¯ 1 , b2 , b3 ) 1 + ε |M| + b3 + ε 2 |M|b3 + νε −3/2 L3 exp{−δL3 /ε} R1 = C(b 2 L3 L3

(82)

for any δ < 2π min{1/b1 , 1/b2 }. Lemma 12. Let a be the bilinear form given by



 2 

ε a(ψ, ϕ) = ν I− ∇ψ∇ϕ dx − M L3 ε ε j =1    εx3+  ϕ j,ε × β − Mj 1 , Mj 2 , 0 · L3 L3

−



2



β

j,ε

ε j =1 

+

P

−1

%



U

ψ∇



j

   −1 %  εx3+  ϕ ε − I− M Mj 1 , Mj 2 , 0 ∇ ψ U L3 L3 L3 j

ϕ ψ3 U + L3

2

 ε j =1 

Uj ϕ + ∂ψ x , L3 3 ∂xj

∀ψ, ϕ satisfying (78).

Then we have #

   |U |L3 ε|U |  a(ϕ, ϕ) + ϕ∇ ϕ · ϕ dx ≥ ν − − √ |M| + βL∞ (Zbl ) 2 2 ε # $$ 3 ε |U |3 3δL3 ˜ − C(b1 , b2 , b3 ) exp − ∇ϕ2L2 (ε )9 ε L33 (83) for all ϕ satisfying (78). Theorem 3. Let ν˜ = ν −

 |U |L3 ε|U |  − √ |M| + βL∞ (Zbl ) 2 2

(84)

and let us suppose that ν>

# $ 3 3  |U |L3 ε|U |  ˜ 1 , b2 , b3 ) ε |U | exp − 3δL3 . (85) + √ |M| + βL∞ (Zbl ) + C(b 2 ε L33 2

Then we have the following error estimate: |U |2 L3 ≤ # $. 3 |U |3 ε 3δL3 ˜ ν˜ − C(b1 , b2 , b3 ) exp − ε L33 R1 ε 3/2

∇U ε L2 (ε )9

Proof. It is a consequence of the preceding lemmas and the proof of Theorem 1.

(86)



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W. J¨ager, A. Mikeli´c

Remark 5. In the
applications in the last section, L3 /ε ≥ 10, δ ≈ 2π min{1/b1 , 1/b2 }, δL3 L3 3δL3 exp{− } < 1.63 · 10−27 and ε 3 exp{− } < 1.4·10−82 ε 3 . b1 , b2 ≤ 1, νε −1 ε ε ε Therefore, without losing generality we neglect the boundary layer effects on 2 . In the text which follows they are simply skipped. Proposition 2. Let us suppose (85). Then we have

U3ε (x1 , x2 , x3 ) dx1 dx2 = 0, ∀x3 ∈ (−εb3 , L3 ),

(87)



 

  ε U −  U ε dx1 dx2   

  Cq R1 |U |2 2+1/q L1 L2 1/q−1/2 ≤ ε , ν˜ L3 b1 b2 Lq (ε )3   C˜ q R1 |U |2 2+1/q L1 l2 1/q−1/2 ε ε , U Lq (ε \P )3 ≤ ν˜ L3 b1 b2 ! ! " !

! 2 2 ! ! ! U ε dx1 dx2 ! ≤ b3 b1 b2 R1 ε |U | , j = 1, 2. j ! ! L1 L2 ν˜ L3 ! !

(88) (89) (90)



Proof. Using the (εb1 , εb2 )-periodicity in (x1 , x2 ) of U ε we get  

 ε  U −  U ε dx1 dx2   

 ≤ Cq ε

Lq (ε )3

1/2+1/q

L 1 L2 b1 b2

1/q−1/2 ∇U ε L2 (ε )9 ,

(91)

and (88) follows from (91) and (86). Equations (89) and (90) are proved analogously.  Our next step is to calculate the tangential drag force or the skin friction ε Ft,j =



1 L1 L2

ν 

∂vjε ∂x3

(x1 , x2 , 0) dx1 dx2 , j = 1, 2.

Theorem 4. Let the assumptions of Theorem 2 and Proposition 2 be fulfilled. Then 

    U ε (x1 , x2 , ·) dx1 dx2  j   ≤ C˜ 1

ε 2 |U |2 ν˜ L3

# 1+

ν ν˜

L∞ (0,L3 )



1+

ε L3

 + ε2

|U | νL3

 1+

ε|U | ν˜

$ ,

(92)

j = 1, 2.

(93)

and ! !  ! −1 % !

∂v ε ! ν ! ε ν j ! I− dx1 dx2 − M U !! !L L L3 L3 ! 1 2  ∂x3 j!    $ #  ε ε 2 |U |2 ε|U | ˜ ≤ C1 2 1+ ε|U | , ν˜ + ν + 1 + ν˜ L3 L3 ν˜

Couette Flows over Rough Boundary and Drag Reduction

449

Proof. We write (79) as   −1 % 2   ε 1 ε ε ε ε div −ν∇Uk + P ek + U Uk − I− U M  L3 L3 j =1 j    $   εx3 εx3 j,ε + β j,ε − (Mj 1 , Mj 2 , 0) Ukε × U ε βk − M j k L3 L3 Uk ε + U = −εk in P , k = 1, 2. L3 3 After averaging (94) over arbitrary horizontal section we get  −1 %

# 2  ∂Ukε ε 1 d I−  −ν + U ε Ukε − M U dx3 ∂x3 L3 L3 j =1 j

 $ j,ε j,ε × U ε βk + β3 Ukε dx1 dx2 =  εk dx1 dx2 , k = 1, 2.

(94)

(95)

Let

   −1 % 

 2    ε 1 j,ε j,ε εk (x3 ) =  I− U ε Ukε − M U ε βk + β3 Ukε dx1 dx2 . U   L3 L3 j =1

j

(96) Then after integrating (95) between x3 and L3 we get  

   

x3 x3 1 L3 ε ε  Uk (L3 ) + 1− − H (η − x3 ) εk (x3 ) dη  Uk (x3 ) = L3 ν 0 L3    η



1 L3 x3 ε + 1− − H (η − x3 )  k dξ dη ν 0 L3 0  

x3 + 1−  Ukε , (97) L3 

and







∂Ukε ν ν ε  Uk (L3 ) − = ν  Ukε + εk (0) ∂x3 L3 L3 

1 − L3





L3 0

εk (η)

1 dη + L3



η



0

εk dξ dη.

(98)

Using the a priori estimates (87)−(88) and the properties of the boundary layer velocities we get ! ! L3  4 3  ! ! ε ! ≤ C˜ 1 (b1 , b2 , b3 , L1 , L2 ) ε |U | 1 + ε|U | , ! (99)  (η) dη k ! ! ν˜ L23 ν˜ 0   ε 3 |U |3 ε|U | ε ˜ |k (0)| ≤ C1 (b1 , b2 , b3 , L1 , L2 ) 2 . (100) 1+ ν˜ L3 ν˜

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W. J¨ager, A. Mikeli´c

Next we average εk over an arbitrary horizontal section and find out   −1 %  −1 % 

2  ε ε 1 ε  I− I− M M U U  k (x3 ) = − 2 L L L 3 3 j,l=1 3 j l  

d ε j,ε j,ε βk − ×  β3 Mlk . dx3 L3 Consequently, !

! ! !

L3

0



η



0

! ! ε 2 |U |2 εk dξ dη!! ≤ C˜ 1 (b1 , b2 , b3 , L1 , L2 ) , L3

(101)

and !

!    $ 2 2# ! ! ! U ε (x1 , x2 , ·) dx1 dx2 ! ≤ C˜ 1 ε |U | 1 + ν 1 + ε + ε 2 |U | 1 + ε|U | , j ! ! ν˜ L3 ν˜ L3 νL3 ν˜ proving (92). Next we get ! ! !

 $ #   ε !! ! ∂U ε 2 |U |2 ε|U | ε k! !ν  ˜ ≤ C1 2 ε|U | , ν˜ + ν + 1 + 1+ ! ∂x3 !! ν˜ L3 L3 ν˜ !

j = 1, 2.



(102) Since

 −1 %



∂vkε ∂Ukε ε 1  I− = + U , M ∂x3 ∂x3 L3 L3 

(102) implies (93).



k



Corollary 4. Let ε < L3 /b3 , let R1 be given by (82) and  #   |U |L3 ε|U |  ε 2 |U |2 ν ε ˜ ∞ ν> + √ |M| + βL (Zbl ) + C1 1+ 1+ 2 ν˜ ν˜ L3 2 $  1/6  |U | |U |2 ε|U | b1 b2 + ε2 + C0,6 max{C¯ 3 , Cˆ 3 } 1+ R1 ε 7/3 . (103) νL3 ν˜ L1 L2 L3 ν˜ Then v ε , constructed in Theorem 1, is a unique solution to (22)–(26). Proof. It is analogous to the proof of Theorem 2, but now Eq. (79) and the a priori estimates for U ε are used.  Remark 6. It is possible to add further correctors and then our problem would contain an exponentially decreasing forcing term. This is in accordance with [6] for the Navier-Stokes system and with [4] and [5] for the Stokes system. For the case of rough boundaries with different characteristic heights and lengths we refer to the doctoral dissertation of I. Cotoi [10]. The estimate (67) is of the same order in ε as the H 1 -estimate in [3], obtained for the Laplace operator. The advantage of our approach is that we are going to obtain the Navier’s slip condition with a negatively definite matricial coefficient.

Couette Flows over Rough Boundary and Drag Reduction

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The above estimates allow to justify Navier’s slip condition. First, we introduce the effective Couette-Navier flow through the following boundary value problem: Find a velocity field ueff and a pressure field p eff such that   −νueff + (ueff ∇)ueff + ∇p eff = 0 in P ,    eff = 0 eff = (U , U , 0) on  ,  div u in P , u  1 2 2  eff 2  ∂u (104) eff eff  uj = −ε Mj i i , j = 1, 2, and u3 = 0 on ,   ∂x3   i=1   eff eff {u , p } is periodic in (x1 , x2 ) with period (L1 , L2 ). If |U |L3 < 2ν, the problem (104) has a unique solution      −1  ε ueff = U + x3 − 1 U , 0 I− M L3 L3   eff p =0

for x ∈ P

.

(105)

for x ∈ P

The effective mass flow rate between the plates is then # −1 $ 

L3 1 ε eff eff  M U , = u˜ (x3 ) dx3 = L3 U − M I− 2 L3 0 where M is negatively definite, and  −1 ε ν eff Ft = I− M U . L3 L3

(106)

Let us estimate the error made when replacing {v ε , pε , Mε } by {ueff , peff , Meff }. We have Theorem 5. Under the assumptions of Theorem 4 we have #      |U |ε 1−1/q 1/q ν˜ ε  ε eff  1+1/q |U | 1+ 1+ L3 1+ C v − u  q 3 ≤ ε L (P ) L3 ν˜ ν L3  $ |U | ε|U | + ε2 1+ , (107) νL3 ν˜ #    |U | ν˜ |U | ε 1+ L3 1 + 1+ |Mε − Meff | ≤ ε2 C L3 ν˜ ν L3  $ |U | ε|U | + ε2 1+ . (108) νL3 ν˜ Proof. We have  −1 % 2  ε ε  ε eff ε x3 U eff v −u I− =U + −u − M U L3 L3 L3 j =1 j  x   j × β − (Mj 1 , Mj 2 , 0) ε    −1 %  2 ε  ε x3 − I− 1− (Mj 1 , Mj 2 , 0) (109) M U L3 L3 L3 j =1

j

in P . After a simple calculation we find out the identity

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W. J¨ager, A. Mikeli´c

 −1 %    2  x ε ε  βj U − (Mj 1 , Mj 2 , 0) I− v −u − M L3 L3 ε j =1 j   −1 %  2 ε x3 ε   I− 1− (Mj 1 , Mj 2 , 0) U M − L3 L3 L3 j =1 j  −1 %    2  ε  ε x =− I− βj U − (Mj 1 , Mj 2 , 0) . M (110) L3 L3 ε 0

eff

j =1

j

Now (107)–(108) follows from (92), (88) and (110).



Remark 7. We see that the presence of the periodic roughness diminishes the tangential drag. The contribution is linear in ε, and consequently rather small. It coincides with the conclusion from [5] that for laminar flows there are no palpable drag reductions. Nevertheless, we are going to see in the next subsection that the calculations from the laminar case could be useful for flows around immersed bodies. Remark 8. As in [14] we prove that a perturbation of the interface position of order O(ε) implies a perturbation in the solution of O(ε2 ). The result is a consequence of Lemma 3. In fact the matrix M would change but it is compensated by the change of the position of . Consequently, there is a freedom in fixing the position of . It influences the result only at the next order of the asymptotic expansion. 4. Drag Reduction and Homogenization Drag reduction for planes, ships and cars reduces significantly the spending of the energy, and consequently the cost for all types of land, sea and air transportation. Drag-reduction adaptations were important for the survival of Avians and Nektons, since their efficiency or speed, or both, have improved. Essentially, there are three forms of drag. The largest drag component is pressure or form drag. The two remaining drag components are skin-friction drag and drag due to lift. Skin-friction drag is the result of the no-slip condition on the surface. Those components are present for both laminar (low Reynolds number) or turbulent (high Reynolds number) flows. There are several drag-reduction methods and here we are interested in the use of dragreducing surfaces. For an overview of other techniques we refer to Bushnell, Moore [9]. The inspiration comes from morphological observations like the skin of fast sharks. It is covered with tiny scales having little longitudinal ribs on their surface (shark dermal denticles). These are tiny ridges, closely spaced (less than 100 µm apart and still less in height). We note that the considered sharks have a length of approximately 2 m and swim at Reynolds numbers Re ≈ 3 · 107 (see e.g. Vogel [24]). Such grooves are similar to ones used on the yacht “Stars and Strips” in America’s Cup finals and seem to reduce the skin-friction for O(10%) (see [9]). In the applications, the main interest is in the turbulent case. Mathematical modeling of the turbulent flows in the presence of solid walls is still out of reach. However the turbulent boundary layers on surfaces with fine roughness contain a viscous sub-layer. It was found that the viscous sub-layer exhibits a streaky structure. Those “low-speed streaks” are believed to be produced by slowly rotating longitudinal vortices. For a

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streaky structure, with a preferred lateral wavelength, a turbulent shear stress reduction was observed. The experimental facts were theoretically explained in the papers by Bechert and Bartenwerfer [7] and Luchini, Manzo and Pozzi [17] (see also [8] and references in mentioned articles). Bechert and Bertenwerfer argued that the velocity profile, which is an asymptotically linear Couette flow in the viscous sub-layer, appears as it was originated from a smooth solid wall placed at some distance below the riblet tips, but still above the bottom of roughness. They called that distance the protrusion height. In [7] the protrusion heights were calculated for the mean longitudinal flows for groove shapes. Similar calculations, but for the cross flow problems are in [17]. If h is the difference between the origins of the longitudinal and cross flows, v the friction velocity and if s is the lateral rib spacing, h then the quantity plays an important role. In [8] it is noted that, in particular for s h sv small Reynolds numbers s + = , the drag reduction is proportional to . ν s We suppose that our riblets remain all the time in the pure viscous sub-layer and try to apply the analysis from Sect. 3. √ Equations (22)–(26) are posed in a domain ε with L3 = δv = ν, the upper layer δv moves with the velocity (the friction velocity) U = e = (v1 , v2 , 0) at x3 = δv . Since ν δv |U | = ν < 2ν, our results from Sect. 3 are applicable and we get !    −1 !! ! ε ε|U | 2 ν ! ε ! I− M U ! ≤ CF . (111) !Ft − ! ! δv δv δv √ √ Furthermore, on the shark’s skin ε/δv = 0.1, L3 = δv = 10−3 = ν and |U | = ν = 10−3 . Obviously, the estimate (111) implies that the effects of roughness are very significant. Nevertheless, we can use the estimate (111) only if the constructed solution, which leads to it, was unique. A sufficient condition for uniqueness is (103). Supposing √ % √ √ 2 1 π 2 1 ε < ν max , , , 4 |M| + βL∞ b3 max{b1 , b2 } the uniqueness condition (103) reduces to ε ≤ Cν 9/14 , where C = C(b1 , b2 , b3 , L1 , L2 ). Since for shark’s skin ε ≈ 10−4 and ν 9/14 ≈ 1.389 · 10−4 we arrive at the conclusion that our laminar calculations describe the flow in the viscous sub-layer around the riblets on the shark’s skin. The formula (111) shows that the presence of the roughness has changed significantly the tangential drag. At first glance it is lower since (111) says that it is better for a Nekton to have riblets than a smooth skin at x3 = 0. Nevertheless, as pointed out in [18] and by S. Luckhaus in many discussions, this is not a correct comparison. Comparison should be made not with P but with a flow region of the same volume having a flat boundary in ε \ P . Answering the question as to which form of riblets gives the minimal tangential drag force is a shape design problem for a flow in the semi-infinite strip Zbl . We are going to address it in subsequent publications. The goal of this paper was to develop a rigorous mathematical theory which explains how the presence of riblets affects the tangential drag.

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Nevertheless the fact that we obtain a reduction of the tangential drag doesn’t explain totally the presence of riblets since the physical flows are non-stationary and include non-linear effects as well. This analysis is essentially time independent and we found that in the considered physical situation the Couette flow in the viscous sub-layer could be approximated by the Couette-Navier flow (105). We have the following wall law: eff

uj

= −ε

2  i=1

eff

Mj i

∂ui eff , j = 1, 2, and u3 = 0 on . ∂x3

Hence we have the Navier’s slip condition as a wall law if ε is smaller or equal to Cν 9/14 . For bigger ε we could expect a non-linear time dependent version of the Navier’s slip condition. To conclude we point out that the geometry from Lemma 6 corresponds to the rough boundaries from [7] and [17]. For such geometries we have found out that M is ε diagonal and the origins of the cross and longitudinal flows are at y + = M11 and δv ε y + = M22 , respectively. The relative change in Ftε is then for small Reynolds numδv ε(M11 − M22 ) bers, giving a link to the above mentioned theory by Bechert and Barδv tenwerfer. Acknowledgement. This work was supported in part by the European project “HMS 2000: Homogenization and Multiple Scales”, contract: HPRN-CT-2000-00109, and the sonderforschungsbereich SFB 359.

References 1. Achdou, Y., Pironneau, O., Valentin, F.: Shape control versus boundary control. In: Equations aux d´eriv´ees partielles et applications. Articles d´edi´es a` J.L. Lions, eds. F. Murat et al., Paris: Elsevier, 1998, p. 1–18 2. Achdou,Y., Pironneau, O., Valentin, F.: Effective boundary conditions for laminar flows over periodic rough boundaries. J. Comp. Phys. 147, 187–218 (1998) 3. Allaire, G., Amar, M.: Boundary layer tails in periodic homogenization. ESAIM: Control, Optimisation and Calculus of Variations 4, 209–243 (1999) 4. Amirat, Y., Simon, J.: Influence de la rugosit´e en hydrodynamique laminaire. C. R. Acad. Sci. Paris, S´erie I, 323, 313–318 (1996) 5. Amirat, Y., Simon, J.: Riblet and Drag Minimization. In: Optimization Methods in PDEs, Contemp. Math. 209, ed. S. Cox, et al., Providence, RI: American Math. Soc., 1997, pp. 9–17 6. Amirat,Y., Bresch, D., Lemoine, J., Simon, J.: Effect of rugosity on a flow governed by Navier-Stokes equations. To appear in Quarterly of Appl. Maths. 2001 7. Bechert, D.W., Bartenwerfer, M.: The viscous flow on surfaces with longitudinal ribs. J. Fluid Mech. 206, 105–129 (1989) 8. Bechert, D.W., Bruse, M., Hage, W., van der Hoeven, J.G.T., Hoppe, G.: Experiments on drag reducing surfaces and their optimization with an adjustable geometry. Preprint, spring 1997 9. Bushnell, D.M., Moore, K.J.: Drag reduction in nature. Ann. Rev. Fluid Mech. 23, 65–79 (1991) 10. Cotoi, I.: Etude asymptotique de l’´ecoulement d’un fluide visqueux incompressible entre une plaque lisse et une paroi rugueuse. Doctoral dissertation, Universit´e Blaise Pascal, Clermont-Ferrand, January 2000 11. J¨ager, W., Mikeli´c, A.: On the Boundary Conditions at the Contact Interface between a Porous Medium and a Free Fluid. Annali della Scuola Normale Superiore di Pisa, Classe Fisiche e Matematiche – Serie IV 23, Fasc. 3, 403–465 (1996) 12. J¨ager, W., Mikeli´c, A.: On the effective equations for a viscous incompressible fluid flow through a filter of finite thickness. Commun. on Pure and App. Math. 51, 1073–1121 (1998)

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13. J¨ager, W., Mikeli´c, A.: On the interface boundary conditions by Beavers, Joseph and Saffman. SIAM J. Appl. Math. 60, 1111–1127 (2000) 14. J¨ager, W., Mikeli´c, A.: On the roughness-induced effective boundary conditions for a viscous flow. J. of Differ. Eqs. 170, 96–122 (2001) 15. J¨ager, W., Mikeli´c, A., Neuß, N.: Asymptotic analysis of the laminar viscous flow over a porous bed. SIAM J. on Scientific and Statistical Computing 22, 2006–2028 (2001) 16. Lions, J.L.: Some Methods in the Mathematical Analysis of Systems and Their Control. New York: Gordon and Breach, 1981 17. Luchini, P., Manzo, F., Pozzi, A.: Resistance of a grooved surface to parallel flow and cross-flow. J. Fluid Mech. 228, 87–109 (1991) 18. Luchini, P.: Asymptotic analysis of laminar boundary-layer flow over finely grooved surfaces. Eur. J. Mech. B/Fluids 14, 169–195 (1995) 19. Mikeli´c, A.: Homogenization Theory and Applications to Filtration Through Porous Media. In: Chapter in Filtration in Porous Media and Industrial Applications, Lecture Notes in Mathematics, 1734, eds. M. Espedal, A. Fasano, A. Mikeli´c, Berlin-Heidelberg, New York: Springer-Verlag, 2000, p. 127–214 20. Mohammadi, B., Pironneau, O., Valentin, F.: Rough boundaries and wall laws. Int. J. Numer. Meth. Fluids 27, 169–177 (1998) 21. Navier, C.L.M.H.: Sur les lois de l’´equilibre et du mouvement des corps e´ lastiques. Mem. Acad. R. Sci. Inst. France 369 (1827) 22. Panton, R.L.: Incompressible Flow. New York: John Wiley and Sons, 1984 23. Schlichting, H., Gersten, K.: Boundary-Layer Theory, 8th Revised and Enlarged Edition, Berlin: Springer-Verlag, 2000 24. Vogel, S.: Life in Moving Fluids. 2nd ed., Princeton, NJ: Princeton University Press, 1994 Communicated by A. Kupiainen

Commun. Math. Phys. 232, 457–500 (2003) Digital Object Identifier (DOI) 10.1007/s00220-002-0730-3

Communications in

Mathematical Physics

Ricci-Flat Metrics, Harmonic Forms and Brane Resolutions M. Cvetiˇc1,3,6 , G.W. Gibbons2 , H. Lu¨ 4 , C.N. Pope5,6 1 2 3 4 5 6

Department of Physics and Astronomy, University of Pennsylvania, Philadelphia, PA 19104, USA DAMTP, Centre for Mathematical Science, Cambridge University, Wilberforce Road, Cambridge CB3 OWA, UK Department of Physics and Astronomy, Rutgers University, Piscataway, NJ 08855, USA Department of Physics, University of Michigan, Ann Arbor, Mi 48109, USA Center for Theoretical Physics, Texas A&M University, College Station, TX 77843, USA Institute Henri Poincar´e, 11 rue Pierre et Marie Curie, 75231 Paris Cedex 05, France

Received: 22 February 2001 / Accepted: 16 August 2002 Published online: 7 November 2002 – © Springer-Verlag 2002

Abstract: We discuss the geometry and topology of the complete, non-compact, Ricci-flat Stenzel metric, on the tangent bundle of S n+1 . We obtain explicit results for all the metrics, and show how they can be obtained from first-order equations derivable from a superpotential. We then provide an explicit construction for the harmonic selfdual (p, q)-forms in the middle dimension p + q = (n + 1) for the Stenzel metrics in 2(n+1) dimensions. Only the (p, p)-forms are L2 -normalisable, while for (p, q)-forms the degree of divergence grows with |p − q|. We also construct a set of Ricci-flat metrics whose level surfaces are U (1) bundles over a product of N Einstein-K¨ahler manifolds, and we construct examples of harmonic forms there. As an application, we construct new examples of deformed supersymmetric non-singular M2-branes with such 8-dimensional transverse Ricci-flat spaces. We show explicitly that the fractional D3-branes on the 6-dimensional Stenzel metric found by Klebanov and Strassler is supported by a pure (2, 1)-form, and thus it is supersymmetric, while the example of Pando Zayas-Tseytlin is supported by a mixture of (1, 2) and (2, 1) forms. We comment on the implications for the corresponding dual field theories of our resolved brane solutions. Contents 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Stenzel Metrics . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1. Geometrical and topological considerations . . . . . . . . 2.2. Detailed calculations . . . . . . . . . . . . . . . . . . . . 2.3. Covariantly-constant spinors . . . . . . . . . . . . . . . . 2.4. K¨ahler form . . . . . . . . . . . . . . . . . . . . . . . . . 2.5. Explicit solutions for Ricci-flat Stenzel metrics . . . . . . 3. Harmonic Forms . . . . . . . . . . . . . . . . . . . . . . . . . 3.1. Harmonic (p, q)-forms in 2(p + q) dimensions . . . . . . 3.2. L2 -normalisable harmonic (p, p)-forms in 4p dimensions

. . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

458 461 461 463 466 467 467 469 469 470

458

4.

5.

6.

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M. Cvetiˇc, G.W. Gibbons, H. L¨u, C.N. Pope

3.3. Non-normalisable harmonic (p, q)-forms . . . . . . . . . . . . . . . 3.4. Canonical form, and special Lagrangian submanifold . . . . . . . . . Applications: Resolved M2-Branes and D3-Branes . . . . . . . . . . . . . 4.1. Fractional D3-brane using the 6-dimensional Stenzel metric . . . . . . 4.2. Fractional M2-brane using the 8-dimensional Stenzel metric . . . . . Ricci-Flat K¨ahler Metrics on Ck Bundles . . . . . . . . . . . . . . . . . . . 5.1. Curvature calculations, and superpotential . . . . . . . . . . . . . . . 5.2. Solving the first-order equations . . . . . . . . . . . . . . . . . . . . 5.3. General results for N Einstein-K¨ahler factors in the base space . . . . More Fractional D3-Branes and Deformed M2-Branes . . . . . . . . . . . 6.1. The resolved fractional D3-brane . . . . . . . . . . . . . . . . . . . . 6.1.1. Harmonic 3-form on the C2 bundle over CP1 . . . . . . . . . . 6.1.2. The issue of supersymmetry in the Pando Zayas-Tseytlin D3brane. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2. Harmonic 4-form for C2 /Z2 and C2 bundles over CP2 , and smooth M2-branes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3. Harmonic 4-form for C2 bundle over CP1 × CP1 , and smooth M2-brane 6.4. Deformed M2-brane on the complex line bundle over CP3 . . . . . . 6.5. Deformed M2-brane on an 8-manifold of Spin(7) holonomy . . . . . Conclusions and Comments on Dual Field Theories . . . . . . . . . . . . .

470 471 471 472 473 476 477 478 483 487 487 487 488 489 492 494 495 496

1. Introduction Fractional D3-branes have been extensively studied recently, since they can provide supergravity solutions that are dual to four-dimensional N = 1 super-Yang-Mills theories in the infra-red regime [1–8]. The idea is that by turning on fluxes for the R-R and NS-NS 3-form fields of the type IIB supergravity, in addition to the usual flux for the self-dual 5-form that supports the ordinary D3-brane, a deformed solution can be found that is free of the usual small-distance singular behaviour on the D3-brane horizon. This is achieved by first replacing the usual flat 6-metric transverse to the D3-brane by a non-compact Ricci-flat K¨ahler metric. It can then be shown that if there exists a suitable harmonic 3-form G(3) satisfying a complex self-duality condition, then the type IIB equations of motion are satisfied if the R-R and NS-NS fields are set equal to the real and imaginary parts of the harmonic 3-form, with the usual harmonic function H of the D3-brane 1 2 solution now satisfying the modified equation H = − 12 m |G(3) |2 in the transverse space. A key feature of the type IIB equations that allows such a solution to arise is that there is a Chern-Simons or “transgression” modification in the Bianchi identity for the self-dual 5-form, bilinear in the R-R and NS-NS 3-forms. The construction can be extended to encompass other examples of p-brane solutions, and in [6] a variety of such cases were analysed. These included heterotic 5-branes, dyonic strings, M2-branes, D2-branes, D4-branes and type IIA and type IIB strings. The case of M2-branes was also discussed in [9]. In all these cases, the ability to construct deformed solutions depends again upon the existence of certain Chern-Simons or transgression terms in Bianchi identities or equations of motion. The additional field strength contribution that modifies the standard p-brane configuration then comes from an appropriate harmonic form in the transverse space. One again replaces the usual flat transverse space by a more general complete non-compact Ricci-flat manifold. In order to get deformed solutions that are still supersymmetric, a necessary condition on this

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manifold is that it must have an appropriate special holonomy that admits the existence of covariantly-constant spinors. One can easily establish that if the harmonic form is L2 -normalisable, then it is possible to choose integration constants in such a way that the deformed solution is completely non-singular [6]. In particular, it can be arranged that the horizon is completely eliminated, with the metric instead smoothly approaching a regular “endpoint” at small radial distances. At large distances, the metric then has the same type of asymptotic structure as in the undeformed case, with a well-defined ADM mass per unit spatial world-volume. If, on the other hand, the harmonic form in the transverse manifold is not L2 -normalisable, then the deformed solution will suffer from some kind of pathology. Usually, one chooses a harmonic form that is at least square-integrable in the small-radius regime, and this can be sufficient to allow a solution which gives a useful infra-red description of the dual super-Yang-Mills theory. If the harmonic form fails to be square-integrable at large radius, then this will lead to some degree of pathology in the asymptotic structure of the deformed solution in that region. For example, the deformed KS D3-brane solution [2] is based on a non-normalisable harmonic 3-form in the six-dimensional Ricci-flat K¨ahler transverse space, for which the integral of |G(3) |2 diverges as the logarithm of the proper distance at large radius. This leads to a deformed D3-brane metric that is complete and everywhere non-singular, and for which the harmonic function H has the asymptotic structure H ∼ c0 +

Q + m2 log ρ ρ4

(1.1)

at large proper distance ρ. Although the metric is still asymptotic to dx µ dxµ + dsc2 , where dsc2 is the metric on the six-dimensional Ricci-flat conifold, the effect of the deformation involving the logarithm is that the associated ADM mass per unit 3-volume is no longer well-defined. This is because the effect of the log ρ term in H is to cause a slower fall-off at infinity than the normal ρ −4 dependence that picks up a finite and non-zero ADM contribution.1 This change in the asymptotic structure implies that the solution may not admit an AdS5 region, even when the constant c0 in (1.1) goes to zero in a decoupling limit. Of course this feature is itself of great interest, since it is associated with a breaking of conformal symmetry in the dual field theory picture. One might wonder whether there could be some other Ricci-flat K¨ahler 6-manifold for which an L2 -normalisable harmonic 3-form might exist. In fact rather general arguments establish that this is not possible, at least for the case where the 6-metric is asymptotically of the form of a cone, and the middle homology is one-dimensional.2 On the other hand, L2 -normalisable harmonic forms can exist in non-compact Ricci-flat manifolds in other dimensions, and indeed some examples of fully resolved p-brane solutions based on such harmonic forms were obtained in [6]. We shall obtain further examples in this paper, using Ricci-flat K¨ahler 8-manifolds to obtain smooth deformed M2-branes. Since the ADM mass is then well-defined, the asymptotic structure correspondingly may still allow an approach to AdS, if the constant term in the metric function H goes to zero, implying that the dual field theory will still be a conformal one (three-dimensional in the case of M2-branes). For practical purposes, the ADM mass measured relative to the fiducial metric dx µ dxµ + dsc2 is a certain constant times the limit of ρ 5 ∂H /∂ρ as ρ goes to infinity. 2 We are grateful to Nigel Hitchin for extensive discussions on this point. 1

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In this paper, we explore some of these questions in greater detail. To begin, in Sect. 2, we study the class of complete non-compact Ricci-flat K¨ahler manifolds whose metrics were constructed by Stenzel [10]. These are asymptotically conical, with level surfaces that are described by the coset space SO(n + 2)/SO(n), and they have real dimension d = 2n + 2. The n = 1 example is the Eguchi-Hanson instanton [11], and the n = 2 example is the six-dimensional “deformed conifold” found by Candelas and de la Ossa [13]. It is this example that is used in the fractional D3-brane KS solution in [2]. In Sect. 2.1 we describe the geometry and topology of the general Stenzel manifolds, and then in Sect. 2.2 we carry out detailed calculations of the curvature, and show how Ricciflat solutions can be obtained from a system of first-order equations derivable from a superpotential. In subsequent subsections we then obtain the explicit Ricci-flat Stenzel metrics and their K¨ahler forms, and then we derive integrability conditions for the covariantly-constant spinors. In Sect. 3 we obtain explicit results for harmonic forms in the middle dimension, that is to say, for harmonic (n + 1)-forms in the 2(n + 1)-dimensional Stenzel metrics.3 More precisely, we construct harmonic (p, q)-forms for all integers p and q satisfying p + q = n + 1, where p and q count the number of holomorphic and antiholomorphic indices. We show that these are L2 -normalisable if and only if p = q, which can, of course, occur only in dimensions d = 4p. In Sect. 4, we make use of some of these results in order to construct deformed p-brane solutions. Specifically, we first review the fractional D3-brane solution of [2]. Our results on harmonic forms allow us to give an explicit proof that their solution has a harmonic 3-form of type (2, 1), which therefore ensures supersymmetry. We then construct a smooth deformed M2-brane, using the L2 -normalisable (2, 2)-form in the 8-dimensional Stenzel metric. This is also supersymmetric. In Sect. 5 we construct another class of complete non-compact Ricci-flat K¨ahler manifolds. These are again of the form of resolved cones, but in this case the level surfaces are themselves U (1) bundles over the product of N Einstein-K¨ahler manifolds. Typical
mi examples would be to take the base space to be M = N i=1 CP , for an arbitrary set of integers mi . In fact the requirements of regularity of the metric mean that one of the factors in the base space M must be a complex projective space, but the others might be other Einstein-K¨ahler manifolds. Topologically, the total space is a Ck bundle over the remaining Einstein-K¨ahler factors.4 Having obtained general results for Ricci-flat K¨ahler metrics in all the cases, we present some more detailed explicit formulae for three 8-dimensional examples, corresponding to taking the base space to be S 2 × CP2 , CP2 × S 2 and S 2 × S 2 × S 2 . We also discuss some well-known examples corresponding to complex line bundles over CPm . In Sect. 6 we make use of our results for these Ricci-flat metrics, to obtain further examples of deformed p-brane solutions. We begin by considering the case where the base space is M = S 2 × S 2 (i.e. m1 = m2 = 1), meaning that the level surfaces are the 5-dimensional space known as T 1,1 or Q(1, 1), which is a U (1) bundle over S 2 × S 2 . Topologically, the 6-dimensional manifold is a C2 bundle over CP1 . Its Ricci-flat metric is present in [13], and it was discussed recently in [5], where it was used to provide an alternative resolution of the D3-brane. We construct the self-dual harmonic 3-form that was used in [5] in a complex basis, and by this means demonstrate that it contains 3 Nigel Hitchin has informed us that Daryl Noyce has independently constructed the unique harmonic form in the middle dimension in the 4N-dimensional Stenzel manifolds. 4 There are certain topological restrictions on the possible choices for the other Einstein-K¨ahler factors in the base space. For a detailed discussion, see [12].

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both (2, 1) and (1, 2) pieces. This implies that the resolved D3-brane solution of [5] is not supersymmetric [6]. We also construct L2 -normalisable harmonic 4-forms of type (2, 2) in the 8-dimensional examples based on S 2 × CP2 and S 2 × S 2 × S 2 , and then use these in order to construct additional deformed M2-branes, which are supersymmetric. A further smooth deformed M2-brane example, which is non-supersymmetric, results from taking the 8-dimensional transverse space to be the complex line bundle over CP3 . We also include a discussion of a fifth completely smooth deformed M2-brane, which was obtained previously in [6]. This solution uses an 8-manifold of exceptional Spin(7) holonomy rather than a Ricci-flat K¨ahler manifold. We give a simple proof of its supersymmetry. The paper ends with conclusions and discussions in Sect. 7. 2. Stenzel Metrics In this section we shall construct a sequence of complete non-singular Ricci-flat K¨ahler metrics, one for each even dimension, on the co-tangent bundle of the (n + 1) sphere T  S n+1 . Restricted to the base space S n+1 , the metric coincides with the standard round sphere metric. The sequence, which begins with the Eguchi-Hanson metric for n = 1, was first constructed in generality by Stenzel [10] following a method discussed in [14]. The case n = 2 was originally given, in rather different guise, by Candelas and de la Ossa [13] as a “deformation" of the conifold. The isometry group of these metrics is SO(n + 2), acting in the obvious way on T  S n+1 . The principal (i.e. generic) orbits are of co-dimension one, corresponding to the coset SO(n + 2)/SO(n). There is a degenerate orbit (i.e. a generalized “bolt”) corresponding to the zero section, i.e. to the base space S n+1 ≡ SO(n + 2)/SO(n + 1). It is therefore possible to obtain the ordinary differential equations satisfied by the metric functions using coset techniques, and this we shall do shortly. Before doing so, however, we wish to make some comments about the geometry and topology of the metrics, which are intended to illuminate the subsequent calculations.

2.1. Geometrical and topological considerations. Any K¨ahler metric is necessarily symplectic, and in the present case the symplectic structure coincides with the standard symplectic structure on T  S n+1 . The sphere S n+1 is thus automatically a Lagrangian sub-manifold. In other words the K¨ahler form restricted to the (n + 1)-sphere vanishes. The complex structure on T  S n+1 is however non-obvious, and arises from the fact that we may view T  S n+1 as a complex quadric in Cn+2 , za za = a 2 ,

(2.1)

where a = 1, 2, . . . , n + 2. Setting

    sinh pb pb pb pb x a + i pa , za = cosh √ p b pb

(2.2)

one obtains x b x b = a 2 and pb x b = 0. These are the equations defining a point x b lying on an (n + 1)-sphere of radius a in En+1 , and a cotangent vector pb . Note that as the radius a is sent to zero we obtain the conifold, which makes contact with the work of Candelas and de la Ossa [13].

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The strategy of Stenzel [10] is now to assume that the K¨ahler potential K depends only on the quantity √ τ = z¯ a za = cosh(2 pb pb ). (2.3) From this it is clear that the principal orbits of the isometry group correspond to the surfaces of constant energy H = 21 pb pb on the phase space T  S n+1 . The stabliser of each point on the orbit consists of rotations leaving fixed a point √ on S n+1 and a tangent vector pb . The transitivity of the action is equally obvious. Thus pb pb , or some function of it, it will serve as a radial variable. In fact the level sets H = constant can be viewed as circle bundles over the Grassmannian SO(n + 2)/(SO(n) × SO(2)). To see why, recall that the Hamiltonian H generates the geodesic flow on T  S n+1 . Each such geodesic is a great circle consisting of the intersection of a two-plane through the origin of En+2 with the (n + 1)-sphere. The circle factor in the denominator of the coset corresponds to the fact that geodesics or great circles are the orbits of a circle subgroup of the isometry group SO(n + 2) of the (n + 1)-sphere. Thus the circle fibre of the circle bundle is an orbit of the isometry group of the Ricci-flat K¨ahler metric. In terms of K¨ahler geometry, the quotient of T  S n+1 by the circle action corresponds to the Marsden-Weinstein or symplectic quotient, and gives at each radius a homogeneous K¨ahler metric of two less dimensions. At large distances the Stenzel metric tends to a Ricci-flat cone over the EinsteinSasaski manifold SO(n + 2)/SO(n). At small radius the orbits collapse to the zero-section of T  S n+1 . Thus it is clear that the (n + 1)-sphere  ∈ Hn+1 (T  S n+1 ) provides the only interesting homology cycle, and it is in the middle dimension. In the case that n is odd, its self-intersection number  ·  ∈ Z is, depending upon orientation convention, 2, while if n is even its self-intersection number vanishes. This is equivalent to the statement that the Euler characteristic of the even-dimensional spheres is 2, while for the odd-dimensional spheres it vanishes. To see this equivalence, recall that the topology of the co-tangent bundle is the same as that of the tangent bundle. Now the Euler characteristic of any closed orientable manifold is given by the number of intersections, suitably counted, of the zero section with any other section of its tangent bundle. In other words it is the number of zeros, suitably counted, of a vector field on the manifold. We shall see that these facts have consequences for the cohomology. In the case of a closed (2n + 2)-manifold M (i.e. compact, without boundary), one may use Poincar´e duality to see that if α and β are closed middle-dimensional (n + 1)-forms representing elements of H n+1 (M), then the cup product α ∪ β is an integer-valued bilinear form on H n+1 (M) given by  α∧β. (2.4) M

The cup product is symmetric or skew-symmetric depending upon whether n is odd or even respectively. Thus if n is even,  α ∧α = 0. (2.5) M

Moreover, the Hodge duality operator  acts on H n+1 (M), and  = (−1)n+1 .

(2.6)

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Thus if n is odd, H n+1 (M) decomposes into real self-dual or anti-self dual (n + 1) forms. Any such closed form must necessarily be harmonic, and its L2 norm will be proportional to the self-intersection number. The total number of linearly-independent harmonic middle-dimensional forms will depend only on the topology of the closed manifold M. If n is even, we can find a complex basis of self-dual harmonic forms in L2 , but there is no relation between their normalisability and the integral in (2.4). Our manifolds are non-compact, and the situation is therefore more complicated and we must proceed with caution. The usual one-to-one correspondence between harmonic forms and geometric cycles may break down. One generally expects at least as many L2 harmonic forms as topology requires, but there may be more (cf. [15]). It is still true that L2 harmonic forms must be closed and co-closed [16]. However, the notion of exactness must be modified since we are interested in whether closed forms in L2 are the exterior derivatives of forms of one lower degree which are also in L2 . For example, the Taub-NUT metric admits an exact harmonic two-form in L2 , but it is the exterior derivative of a Killing 1-form which is not in L2 . In the present case, if n is odd it seems reasonable to expect at least one harmonic form in the middle dimension, which is Poincar´e dual to the (n + 1)-sphere. Because the Stenzel metric behaves like a cone near infinity, all the Killing vectors are of linear growth. It follows [17] that any harmonic form must be invariant under the action of the isometry group. In the case of the Taub-NUT and Schwarzschild metrics, this observation permits the complete determination of the L2 cohomology [17, 18]. We shall obtain an L2 harmonic form in the middle dimension for all the Stenzel manifolds with odd n. We obtain a general explicit construction of harmonic (p, q)-forms in all the Stenzel manifolds, where p + q = n + 1. These middle-dimension harmonic forms include (p, p) forms when n is odd, and these are the L2 -normalisable examples mentioned above. All the others are non-normalisable, with a “degree of non-normalisability” that increases with |p − q| at fixed p + q. In particular, this accords with the expectation that if n is even we should not find any harmonic form in L2 .5

2.2. Detailed calculations. Let LAB be the left-invariant 1-forms on the group manifold SO(n + 2). These satisfy dLAB = LAC ∧ LCB .

(2.7)

We consider the SO(n) subgroup, by splitting the index as A = (1, 2, i). The Lij are the left-invariant 1-forms for the SO(n) subgroup. We make the following definitions: σi ≡ L1i ,

σ˜ i ≡ L2i ,

ν ≡ L12 .

(2.8)

These are the 1-forms in the coset SO(n + 2)/SO(n). We have dσi = ν ∧ σ˜ i + Lij ∧ σj , d σ˜ i = −ν ∧ σi + Lij ∧ σ˜ j , dLij = Lik ∧ Lkj − σi ∧ σj − σ˜ i ∧ σ˜ j .

dν = −σi ∧ σ˜ i , (2.9)

5 Nigel Hitchin and Tamas Hausel have both pointed out to us that results of Atiyah, Patodi and Singer on asyptotically cylindrical manifolds [19] and some propeties of K¨ahler manifolds used in [17] can be extended to asymptotically conical metrics, and they imply that the L2 cohomology is toplogical, i.e. isomorphic to the compactly-supported cohomolgy in ordinary cohomolgy. The results reported here are consistent with those theorems. We thank them for helpful communications.

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Note that the 1-forms Lij lie outside the coset, and so one finds that they do not appear eventually in the expressions for the curvature (see also [20]). We now consider the metric ds 2 = dt 2 + a 2 σi2 + b2 σ˜ i2 + c2 ν 2 ,

(2.10)

where a, b and c are functions of the radial coordinate t, and then we define the vielbeins e0 = dt ,

ei = a σi ,

˜

ei = b σ˜ i ,

˜

e0 = c ν .

(2.11)

Calculating the spin connection, we find a˙ ω0i = − ei , a ˜

i ω0i ˜ =Be ,

ωij = −Lij ,

b˙ ˜ ω0i˜ = − ei , b ω0˜ i˜ = −A ei ,

c˙ ˜ ω00˜ = − e0 , c ˜

ωi j˜ = C δij e0 ,

(2.12)

ωi˜j˜ = −Lij ,

where a dot means d/dt, and A=

(a 2 − b2 − c2 ) , 2a b c

B=

(b2 − c2 − a 2 ) , 2a b c

C=

(c2 − a 2 − b2 ) . 2a b c

(2.13)

From this, we obtain the curvature 2-forms   a¨ a˙ C b˙ B c˙ ˜ ˜ 0i = − e0 ∧ ei − + + e0 ∧ ei , a bc b c  ˙  b¨ 0 b C a˙ A c˙ ˜ i˜ 0i˜ = − e ∧ e + + + e0 ∧ ei , b ac a c   c¨ 0 c˙ B a˙ A b˙ ˜ 0˜ 00˜ = − e ∧ e + + + ei ∧ ei , c ab a b     1 a˙ 2 1 ˜ ˜ i j 2 ij = e ei ∧ ej , − ∧ e + − B (2.14) a2 a2 b2     1 b˙ 2 1 2 i˜ j˜ i˜j˜ = e ei ∧ ej , − ∧ e + − A b2 b2 a2   a˙ b˙ i Cc C c˙ ˜ j k i˜ j˜ k˜ ˙ i j˜ = A B e ∧ e − e ∧e − δij e ∧ e + C + δij e0 ∧ e0 , ab ab c     Bb a˙ c˙ B b˙ ˜ 0˜ i ˙ 0i + AC + e ∧e + B + e0 ∧ ei , ˜ =− ac ac b   ˙  Aa b c˙ A a˙ ˜ ˜ 0˜ i˜ = − +BC+ e0 ∧ ei − A˙ + e0 ∧ ei . bc bc a

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This implies that the Ricci tensor is diagonal, and that its vielbein components are given by n a¨ n b¨ c¨ − − , a b c   c¨ a˙ c˙ b˙ c˙ (a 2 − b2 )2 − c4 + R0˜ 0˜ = − − n , + c ac bc 2a 2 b2 c2 

 n a˙ b˙ a¨ a˙ 2 1 a˙ c˙ a 4 − (b2 − c2 )2 Rij = − + (n − 1) (2.15) − 2 − − + a a2 a ab ac 2a 2 b2 c2 × δij , ¨  

b 1 n a˙ b˙ b˙ 2 b˙ c˙ b4 − (a 2 − c2 )2 Ri˜j˜ = − + (n − 1) − δij , − − + b b2 b2 ab bc 2a 2 b2 c2 R00 = −

Defining a = eα , b = eβ , c = eγ , and introducing the new coordinate η by = dt, we find that the Ricci-flat equations can be derived from the Lagrangian L = T − V , where a n bn c dη

T = α  γ  + β  γ  + n α  β  + 21 (n − 1) α  + 21 (n − 1) β  , 2

2

V = 41 (a b)2n−2 (a 4 + b4 + c4 − 2a 2 b2 − 2n a 2 c2 − 2n b2 c2 ) ,

(2.16)

where a prime means d/dη, together with the constraint that the Hamiltonian vanishes, T + V = 0. (Note that the Hamiltonian comes from the G00 component of the Einstein tensor.) Writing the Lagrangian as L = 21 gij (dα i /dη) (dα j /dη) − V , where α i = (α, β, γ ), we find that the potential can be written in terms of a superpotential, as V = − 21 g ij

∂W ∂W ∂α i ∂α j

(2.17)

with W = 21 (a b)n−1 (a 2 + b2 + c2 ) .

(2.18)

It follows that the Lagrangian can be written, after dropping a total derivative, as  i  j  dα dα L = 21 gij (2.19) ± g ik ∂k W ± g j  ∂ W , dη dη where ∂i W ≡ ∂W/∂α i . This implies that the second-order equations for Ricci-flatness are satisfied if the first-order equations dα i /dη = ∓g ij ∂j W are satisfied. Thus we arrive at the first-order equations α˙ = 21 e−α−β−γ (e2β + e2γ − e2α ) , β˙ = 21 e−α−β−γ (e2α + e2γ − e2β ) , γ˙ =

−α−β−γ 1 2n e

(e

2α

+e

2β

(2.20)

− e ), 2γ

where the dot again denotes the radial derivative d/dt. Note that in terms of the quantities defined in (2.13), these equations take the simple form α˙ + A = 0 ,

β˙ + B = 0 ,

γ˙ + n C = 0 .

(2.21)

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If we now make use of the first-order Ricci-flat conditions (2.20) in the expressions (2.14) for the curvature 2-forms, we find that they can be simplified to   a¨  0 b¨  0 ˜ ˜ ˜ ˜ 0i˜ = − e ∧ ei − e0 ∧ ei , e ∧ ei + e0 ∧ ei , a b      c¨ 1 1 a˙ 2  i ˜ ˜ i˜ j˜ j ij = e =− − ∧ e + e ∧ e , e0 ∧ e0 + ei ∧ ei , c n a2 a2    1 b˙ 2  i i˜ j˜ j e , = − ∧ e + e ∧ e b2 b2   2 ˜ ˜ ˜ (2.22) = A B ei ∧ ej − ei ∧ ej + δij ek ∧ ek n    nC c 1 ˜ ˜ − 2A B + e0 ∧ e0 + ek ∧ ek δij , ab n    a˙ c˙ Bb ˜ ˜ =− + AC + e0 ∧ ei + e0 ∧ ei , ac ac ˙   b c˙ A a  0˜ ˜ =− +BC+ e ∧ ei − e0 ∧ ei . bc bc

0i = − 00˜ i˜j˜ i j˜

0i ˜ 0˜ i˜

2.3. Covariantly-constant spinors. Since the Stenzel metrics are K¨ahler, it follows that if they are Ricci flat then there should be two covariantly-constant spinors η. The integrability condition is Rabcd  cd η = 0 .

(2.23)

From the expressions for the curvature that we obtained in (2.22), we can then read off that the covariantly-constant spinors must satisfy (0i − 0˜ i˜ ) η = 0 ,

(2.24)

and it is easy to check that all the integrability conditions are satisfied if (2.24) is satisfied. It is useful to note that one can directly read off from (2.22) other consequent results (which can also be derived from (2.24)), such as ij η = −i˜j˜ η. The covariant-constancy condition D η ≡ d η + 41 ωab  ab η = 0 itself is now easily solved. From (2.12), and using the first-order equations (2.21) and the integrability relations (2.24), we find that η simply satisfies d η = 0. In other words, in the frame we are using the covariantly constant spinors have constant components, and satisfy the projection conditions (2.24). In fact we can reverse the logic, and derive the first-order equations (2.21) by requiring the existence of a covariantly-constant spinor η subject to the additional assumption that η has constant components. Equation (2.24) is then a further consequence. Moreover, the 2-form η¯ ab η

(2.25)

is covariantly constant and may be normalised so that it squares to −1; in other words, it gives us the K¨ahler form.

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2.4. K¨ahler form. From now on, we define a new radial coordinate r related to t by dt = h dr, where h can be chosen for convenience, and a prime will mean a derivative with respect to r. Thus the metric is now written as ds 2 = h2 dr 2 + a 2 σi2 + b2 σ˜ i2 + c2 ν 2 ,

(2.26)

and the vielbein is e0 = h dr ,

˜

ei = a σi ,

˜

ei = b σ˜ i ,

e0 = c ν .

(2.27)

It is easy to see that the K¨ahler form mentioned above is given by ˜

˜

J = −e0 ∧ e0 + ei ∧ ei = −h c dr ∧ ν + a b σi ∧ σ˜ i .

(2.28)

The closure of J follows from (a b) = h c, which can be seen from the first-order equations (2.20). Further checking, using the spin connection (2.12), shows that J is indeed covariantly constant. Again, the logic could be reversed, and by requiring the existence of a covariantly-constant 2-form that squares to −1, one could derive the first-order equations (2.21). From this, it follows that we can introduce a holomorphic tangent-space basis of complex 1-forms  α as follows: ˜

 0 ≡ −e0 + i e0 ,

˜

 i = ei + i ei .

(2.29)

In terms of this, we have that the K¨ahler form is J =

i 2

 α ∧ ¯ α¯ ,

(2.30)

and so it is manifestly of type (1, 1) (one barred, one unbarred, complex index). By looking at how other forms are expressed in terms of the complex holomorphic basis  α , we can see how they decompose into type (p, q) pieces, where p and q count the number of holomorphic and anti-holomorphic basis 1-forms in each term.

2.5. Explicit solutions for Ricci-flat Stenzel metrics. Here, we shall construct the explicit solutions to the first-order equations (2.20), for arbitrary n. This gives the class of Ricci-flat metrics on complete non-compact manifolds of dimension d = 2n + 2, as constructed by Stenzel. Starting from (2.20), and changing to the new radial coordinate r related to t by dt = h dr, we first make the coordinate gauge choice h = c. The first-order equations then give α  − β  = −2 sinh(α − β) ,

α  + β  = e−α−β+2γ ,

γ  + 21 n (α  + β  ) = n cosh(α − β) .

(2.31)

The first equation gives eα−β = coth r, the third gives eα+β = k e−2γ /n sinh 2r, where k is a constant, and then the second can be solved explicitly for γ . It is advantageous to introduce a function R(r), defined by  r R(r) ≡ (sinh 2u)n du . (2.32) 0

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Choosing k = (n + 1)−1/n without loss of generality, the solution is then given by a 2 ≡ e2α = R 1/(n+1) coth r , b2 ≡ e2β = R 1/(n+1) tanh r , 1 h2 = c2 ≡ e2γ = R −n/(n+1) (sinh 2r)n , n+1

(2.33)

with the Ricci-flat metric taking the form ds 2 = c2 dr 2 + c2 ν 2 + a 2 σi2 + b2 σ˜ i2 .

(2.34)

The integral (2.32) can be evaluated in general, in terms of a hypergeometric function: R=

 2n (sinh r)n+1 2 F1 21 (1 + n), 21 (1 − n), 21 (3 + n); − sinh2 r . n+1

(2.35)

For each n the result is expressible in relatively simple terms; for the first few values of n one has n = 1 : R = sinh2 r , n = 2 : R = 18 (sinh 4r − 4r) , n = 3 : R = 23 (2 + cosh 2r) sinh4 r , n=4: R= n=5: R= n=6: R=

(2.36)

1 64 (24r − 8 sinh 4r + sinh 8r) , 6 2 15 (19 + 18 cosh 2r + 3 cosh 4r) sinh r , 1 384 (−120r + 45 sinh 4r − 9 sinh 8r + sinh 12r) .

Note that when n is odd, one can always change to a new radial variable z = sinh r in terms of which the metric can be written using rational functions. It is evident from (2.35) that at small r we shall have R∼

2n n+1 , r n+1

(2.37)

and consequently, the metric near r = 0 takes the form  ds 2 ∼

2n n+1

1/(n+1) 

dr 2 + r 2 σ˜ i2 + σi2 + ν 2 .

(2.38)

Thus the radial coordinate runs from r = 0, where the metric approaches Rn+1 × S n+1 with an S n+1 “bolt,” to the asymptotic region at r = ∞. Note that the S n+1 bolt at r = 0 is a Lagrangian submanifold; in other words, the K¨ahler form (2.28) vanishes when restricted to it. When n = 1, the 4-dimensional metric is the Eguchi-Hanson instanton [11]. When n = 2, the 6-dimensional metric is the “deformed” conifold solution found by Candelas and de la Ossa [13]. For arbitrary n, the solutions were first obtained by Stenzel [10].

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3. Harmonic Forms 3.1. Harmonic (p, q)-forms in 2(p + q) dimensions. Here, we present a general construction of harmonic forms in the “middle dimension,” namely (n + 1)-forms in the 2(n + 1)-dimensional Stenzel manifolds. These can be further refined as (p, q) forms where p and q denote the numbers of holomorphic and antiholomorphic indices on the form, and p + q = n + 1. We begin by making the following ansatz for the (p, q) harmonic form: G(p,q) = f1 i1 ···iq−1 j1 ···jp ¯ 0 ∧ ¯ i1 ∧ · · · ∧ ¯ iq−1 ∧  j1 ∧ · · · ∧  jp +f2 i1 ···ip−1 j1 ···jq  0 ∧  i1 ∧ · · · ∧  ip−1 ∧ ¯ j1 ∧ · · · ∧ ¯ jq ,

(3.1)

where f1 and f2 are functions of r. It is easy to see that the epsilon tensors cause each term in each sum to be a product of complex vielbeins in distinct subspaces each of complex dimension one,6 and from this it follows that the Hodge dual is given by ∗G(p,q) = i p−q G(p,q) .

(3.2)

Since G(p,q) is an eigenstate under ∗, it follows that the condition for harmonicity reduces to dG(p,q) = 0. It is useful first to note that from the expressions for the vielbeins in the Stenzel metrics, we can rewrite (3.1), up to an irrelevant constant factor, as G(p,q) = f1 i1 ···iq−1 j1 ···jp (dr + i ν) ∧ h¯ i1 ∧ · · · ∧ h¯ iq−1 ∧ hj1 ∧ · · · ∧ hjp + f2 i1 ···ip−1 j1 ···jq (dr − iν) ∧ hi1 ∧ · · · ∧ hip−1 ∧ h¯ j1 ∧ · · · ∧ h¯ jq ,

(3.3)

hi ≡ σi cosh r + i σ˜ i sinh r .

(3.4)

where It is easy also to verify that dhi = 21 (tanh r + coth r) (dr − i ν) ∧ hi + 21 (tanh r − coth r) (dr − i ν) ∧ h¯ i . (3.5) Imposing dG(p,q) = 0, we now find that the functions f1 and f2 satisfy the equations f1 + f2 + 2(p f1 + q f2 ) coth r = 0 , f1 − f2 + 2(p f1 − q f2 ) tanh r = 0 .

(3.6)

These equations can be solved in terms of hypergeometric functions, to give  f1 = c1 q 2 F1 21 p, 21 (q + 1), 21 (p + q) + 1; −(sinh 2r)2  +c2 (sinh 2r)−p−q 2 F1 21 (1 − p), − 21 q, 1 − 21 (p + q); −(sinh 2r)2 , (3.7)  f2 = −c1 p 2 F1 21 q, 21 (p + 1), 21 (p + q) + 1; −(sinh 2r)2  +c2 (sinh 2r)−p−q 2 F1 21 (1 − q), − 21 p, 1 − 21 (p + q); −(sinh 2r)2 , where c1 and c2 are arbitrary constants. Note that for any specific choice of the integers p and q these expressions reduce to elementary functions of r, so the occurrence of hypergeometric functions here is just an artefact of writing formulae valid for all p and q. 6 There are no factors such as  1 ∧ ¯ 1 , for example. This also shows that these (p, q)-forms are entirely perpendicular to the K¨ahler form J = 2i  α ∧ ¯ α¯ .

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3.2. L2 -normalisable harmonic (p, p)-forms in 4p dimensions. In the special case where p = q, the above construction gives an harmonic (p, p)-form in the middle dimension of a Stenzel manifold of dimension 4p. In this case, we find that with c2 taken to be zero, the functions f1 and f2 in (3.7) become f1 = −f2 =

p c1 , (cosh r)2p

(3.8)

and so the harmonic (p, p)-form G(p,p) is given by G(p,p) =

1 i ···i j ···j ¯ 0 ∧ ¯ i1 ∧ · · · ∧ ¯ ip−1 ∧  j1 ∧ · · · ∧  jp (cosh r)2p 1 p−1 1 p 1 − i ···i j ···j  0 ∧  i1 ∧ · · · ∧  ip−1 ∧ ¯ j1 ∧ · · · ∧ ¯ jq , (cosh r)2p 1 p−1 1 p (3.9)

(after scaling out an irrelevant constant factor.) It therefore has magnitude given by constant . (3.10) (cosh r)4p √ 1 Since the 2(n+1)-dimensional Stenzel metric has g = n+1 (sinh 2r)n , and n = 2p−1 here, it follows that this harmonic form is L2 -normalisable (see footnote 2). One can also express this normalisable harmonic form in terms of the original real vielbein basis. Doing so, we find |G(p,p) |2 =

G(p,p) =

1 (cosh r)n+1 m 

m! ˜ ˜ ˜ × i1 ···i2s+1 j1 ···j2m−2s e0 ∧ ei1 ∧ · · · ∧ ei2s+1 ∧ ej1 ∧ · · · ∧ ej2m−2s s! (m−s)! s=0 ˜ ˜ +i1 ···i2m−2s j1 ···j2s+1 e0 ∧ ei1 ∧ · · · ∧ ei2m−2s ∧ ej1 ∧ c · · · ∧ ej2s+1 , (3.11)

where m is defined by n = 2m + 1 = 2p − 1. In fact another way to obtain the middledimension harmonic form when n is odd is to write down an ansatz of the form (3.11), with a different function of r for each term in the sum, and then impose closure. This leads to a recursive system of first-order differential equations for the functions, whose only solution giving an L2 harmonic form is (3.11).

3.3. Non-normalisable harmonic (p, q)-forms. We saw above that the special case of a harmonic (p, p)-form in a Stenzel manifold of dimension 4p yields the simple expression (3.9) for an L2 -normalisable form. It is not hard to see that for any case other than p = q, the construction in Sect. 3.1 always gives harmonic (p, q)-forms that are not L2 -normalisable. A divergence in the integral of |G(p,q) |2 at r = 0 is avoided if the constant c2 in (3.7) is chosen to be zero, but the integral diverges at large r unless p = q (which can only occur in dimensions that are a multiple of 4, since in general the dimension is 2(p + q)). In fact the degree of divergence becomes larger as |p − q| becomes larger.

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It follows from the above discussion that the “most nearly normalisable” harmonic (p, q)-form in a Stenzel manifold of dimension 4N + 2 will be for the case (p, q) = (N +1, N ) (or its complex conjugate). One then finds that with c2 = 0 the term involving f2 dominates at large r, and that |G(N +1,N ) |2 ∼

1 . (sinh 2r)2N

(3.12)

√ Since we have g = (sinh 2r)n /(n + 1), and n = 2N here, it follows that the harmonic (N + 1, N )-form is marginally not L2 normalisable, and the integral of |G(2N +1) |2 diverges as the logarithm of the proper distance, at large radius. Our findings for (p, q) middle-dimension harmonic forms, and especially, the fact that only in dimensions 4N can there exist L2 harmonic forms, are consistent with the general discussion in Sect. 2.1.

3.4. Canonical form, and special Lagrangian submanifold. If we take q = 0, implying that p = n + 1 in the 2(n + 1)-dimensional Stenzel manifold, then with c2 = 0 we see from (3.7) that f1 vanishes, while f2 becomes a constant. This gives the so-called canonical form, of type (n + 1, 0): G(n+1,0) =  0 ∧  1 ∧ · · · ∧  n .

(3.13)

It is easily verified that this is covariantly constant. From (3.3) and (3.4) we see that it restricts to −i ν ∧ σ1 ∧ · · · ∧ σn

(3.14)

on the S n+1 bolt at r = 0. Thus (G(4) ) restricted to the bolt vanishes. We have already seen that the K¨ahler form vanishes on the bolt, and so it follows that the bolt is a Special Lagrangian Submanifold. Hence it is a calibrated submanifold, and volume-minimising in its homology class; in other words, it is a supersymmetric cycle.

4. Applications: Resolved M2-Branes and D3-Branes The sequence of Stenzel metrics begins with n = 1, which is the 4-dimensional EguchiHanson metric. It admits a normalisable harmonic self-dual 2-form. It was shown in [6] that this can be used to smooth out the singularities in the heterotic 5-brane and in the dyonic string, including the singularity that is associated with the negative tension contribution in the dyonic string. The resolved solutions are smooth and supersymmetric, and have well-defined ADM masses. We refer the reader to [6] for details. In this section, we review the construction of the deformed fractional D3-brane of [2], which uses the 6-dimensional Stenzel metric. We also construct a new resolved M2-brane using the 8-dimensional Stenzel metric. Both solutions are smooth and supersymmetric. The D3-brane does not have a well-defined ADM mass, whilst the M2-brane does.

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4.1. Fractional D3-brane using the 6-dimensional Stenzel metric. The standard D3brane can be deformed when the six-dimensional transverse space admits a harmonic self-dual 3-form. In the notation we shall use here, the general solution is given by [6] 2 d sˆ10 = H −1/2 dx µ dx ν ηµν + H 1/2 ds62 ,

F(5) = d 4 x ∧ dH −1 + ∗ˆ dH

(4.1)

RR NS F(3) = F(3) + i F(3) = m G(3) ,

where ds62 is any six-dimensional Ricci-flat K¨ahler metric that admits a non-trivial complex harmonic self-dual 3-form ∗G(3) = i G(3) , and ∗ˆ and ∗ are Hodge duals with respect 2 and ds 2 respectively. The function H satisfies that to d sˆ10 6 1 2 H = − 12 m |G(3) |2 ,

(4.2)

where is the scalar Laplacian in the 6-dimensional transverse space. In [2], a particular fractional D3-brane was constructed where the six-dimensional Stenzel metric was used for the transverse ds62 , and we shall now review this solution. After making trivial redefinitions (including r −→ r/2) in order to adjust the conventions to those of [2], and taking n = 2, the solution found in Sect. 2.5 for the Stenzel metric becomes h2 = where

1 , 3K 2 K=

a 2 = 2K cosh2 (r/2) ,

b2 = 2K sinh2 (r/2) ,

c2 =

4 , 3K 2

(sinh 2r − 2r)1/3 , 21/3 sinh r

(4.3)

and the metric is then given by (2.26) with i running over 2 values. The Stenzel manifold is smooth, complete and non-compact, with r running from r = 0 to r = ∞. In these conventions, the general result (3.7) yields a harmonic (2, 1) form G(2,1) =

2(r coth r − 1) 0 (sinh 2r − 2r) 0 ¯ ∧  1 ∧  2 −  ∧ ( 1 ∧ ¯ 2 + ¯ 1 ∧  2 ) . 2 sinh r 2 sinh3 r (4.4)

This can be recognised as the self-dual harmonic 3-form constructed in [2], by noting that it can be expressed as G(3) = ω(3) − i ∗ω(3) ,

(4.5)

where

  ˜ ˜ ˜ ˜ ˜ ω(3) = g1 e0 ∧ e1 ∧ e2 + g2 e0 ∧ e1 ∧ e2 + g3 e0 ∧ e1 ∧ e2 − e2 ∧ e1 ,

(4.6)

and g1 =

sinh r − r , sinh r sinh2 (r/2)

g2 =

sinh r + r , sinh r cosh2 (r/2)

g3 =

2(r coth r − 1) . (4.7) sinh2 r

Calculating the norm of G(3) , one obtains the result |G(3) |2 = 12g12 + 12g22 + 24g32  96  2 2 2 4 = r) r − 3r sinh 2r + 3 sinh r + sinh r . (4.8) (3 + 2 sinh sinh6 r

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√ Since for the metric we have g = 23 sinh2 r, it follows that G(3) is not L2 normalisable; it does not fall off sufficiently rapidly at large r. It was argued in [2] that the self-dual harmonic 3-form was of type (2, 1), and then in [3, 4] arguments were presented that would show that the deformed D3-brane solution built using G(3) would be supersymmetric. Our explicit proof that G(3) is of type (2, 1) thus demonstrates the supersymmetry of the solution. Because the L2 norm of G(3) converges for small r, but diverges for large r, it follows that the function H is regular at small r, but does not fall off fast enough at large r to have a well-defined ADM mass. In fact, H has the large-r asymptotic behaviour given in (1.1). 4.2. Fractional M2-brane using the 8-dimensional Stenzel metric. As a consequence of the Chern-Simons modification to the equation of the motion of the 3-form potential in D = 11 supergravity, namely d ∗Fˆ(4) = 21 F(4) ∧ F(4) ,

(4.9)

it is possible to construct a deformed M2-brane, given by [9, 21, 6] 2 d sˆ11 = H −2/3 dx µ dx ν ηµν + H 1/3 ds82 ,

F(4) = d 3 x ∧ dH −1 + m G(4) ,

(4.10)

where G(4) is the harmonic self-dual 4-form in the Ricci-flat transverse space ds82 , and the function H satisfies 1 2 2 H = − 48 m G(4) .

(4.11)

Warped reductions of this type, were also discussed in [22–24]. In this section, we shall construct a deformed M2-brane using the 8-dimensional Stenzel metric for the transverse ds82 . In this case, the index i on σi and σ˜ i in the metric (2.26) runs over 3 values. The Ricci-flat solution coming from the first-order equations (2.20) is given by a 2 = 13 (2 + cosh 2r)1/4 cosh r ,

b2 = 13 (2 + cosh 2r)1/4 sinh r tanh r ,

h2 = c2 = (2 + cosh 2r)−3/4 cosh3 r ,

(4.12)

with the metric then given by (2.26). The radial coordinate runs from r = 0 to r = ∞, and the metric lives on a smooth complete non-compact manifold. In terms of the vielbein basis (2.27), we find from (3.11) that the following is an L2 -normalisable self-dual harmonic 4-form (of type (2, 2)):  3 1 2 3 0 0˜ 1˜ 2˜ 3˜ G(4) = e ∧ e ∧ e ∧ e + e ∧ e ∧ e ∧ e cosh4 r  1 k˜ j˜ k˜ 0˜ 0 i j i  ∧ e ∧ e ∧ e + e ∧ e ∧ e ∧ e e . (4.13) + ij k 2 cosh4 r We can easily see that |G(4) |2 =

360 . cosh8 r

(4.14)

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The 8-dimensional Stenzel manifold can be used as the transverse space to construct the deformed M2-brane. The solution is given by   2 = H −2/3 −dt 2 + dx12 + dx22 + H 1/3 ds82 , ds11 (4.15) F(4) = dt ∧ dx1 ∧ dx2 ∧ dH −1 + m G(4) . All the equations of motions are satisfied provided that 1 2 H = − 48 m |G(4) |2 ,

where √ g=

(4.16)

is the scalar Laplacian in the 8-dimensional transverse space. Since we have sinh3 (2r), assuming that H depends only on r, we have

1 216



h−2

√

g H



=−

5m2 (sinh 2r)3 . 144 cosh8 r

The first integration can be performed straightforwardly, giving   h2 7m2 cosh 2r  H =√ , β+ g 72 cosh4 r

(4.17)

(4.18)

where β is an arbitrary integration constant. In order for the solution regular at r = 0, we must have β=−

5m2 . 72

(4.19)

It is easier to perform the next integration by making a coordinate redefinition, 2 + cosh 2r = y 4 . In terms of y, with β given in (4.19), the function H is then given by  15m2 dy H =− √ 4 − 1)5/2 (y 2     5m2 (5y 5 − 7y) 25m2 1 = c0 − √ + √ F arcsin |−1 , y 4 2(y 4 − 1)3/2 4 2

(4.20)

(4.21)

where c0 is an integration constant, and F (φ|m) is the incomplete elliptic integral of the first kind,  φ F (φ|m) ≡ (1 − m sin2 θ)−1/2 dθ . (4.22) 0

It is easy to verify that the function H is regular for r running from 0 to infinity. For r = 0, H is just a constant. At large r, the function H behaves as H = c0 +

640m2 20480 21/3 m2 − + ··· , 6 2187ρ 28431 32/3 ρ 26/3

(4.23)

where ρ is the proper distance, defined by h dr = dρ. Thus the M2-brane has no singularity, and it has a well-defined ADM mass.

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It is worth commenting further on the choice (4.19) for the integration constant β. The solution to (4.16) has two integration constants β and c0 , which originate from the fact that one can add to H any solution H0 of the homogeneous equation H0 = 0 .

(4.24)

However, the solution for H0 has a singularity at small distance, and so it requires an external delta-function source at the singularity. For the usual M2-brane with flat transverse space, the divergence of H signals a breakdown of our coordinate system, and the delta function corresponds to a smooth horizon. However, if, as in the present case, the transverse space is a smooth non-flat manifold, the singularity is real, and not just a coordinate artefact. In the case when there is a |G(4) |2 source, it is possible to find a smooth everywhere-bounded positive function H . In this case there is no breakdown of our coordinate system, nor is there a naked singularity, and we get a complete non-singular solution without an horizon.7 The L2 normalisability then guarantees finiteness of the ADM mass, as may be easily seen by integrating (4.16). Thus our choice for the constant β in (4.19) ensures that our solution is not only smooth and non-singular, but it is also free of any horizon, and is a rigorous supergravity solution. Any other choice of the constant β would give a solution that was singular, requiring an external source at the spacetime singularity on the horizon. Let us now consider the supersymmetry of the deformed solution. From the D = 11 supersymmetry transformations, it follows that if any supersymmetry is to be preserved, the harmonic 4-form must satisfy: δψa =

1 288

(Gbcde abcde − 8Gabcd bcd ) η = 0 .

(4.25)

a ,

Multiplying by we deduce that the two terms separately must give zero, and in fact the supersymmetry condition can be reduced to [9, 23] Gabcd bcd η = 0 .

(4.26)

Now from (4.13), the vielbein components of the 4-form are given by G0ij ˜ k = 3u ij k ,

G0i˜j˜k˜ = 3u ij k ,

G0ij k˜ = u ij k ,

G0i ˜ j˜k˜ = u ij k ,

(4.27)

where u ≡ 1/ cosh4 r. Substituting into (4.26), we see that taking a = 0, i, i˜ and 0˜ respectively, we obtain the following conditions that must be satisfied if there is to be preserved supersymmetry:   a = 0 : ij k i˜j˜k˜ + ij k˜ η = 0 ,   a = i : ij k 30j + 2 +  ˜ k 0˜ j˜k˜ η = 0 , 0j k˜ (4.28)   a = i˜ : ij k 30j˜k˜ + 20˜ j˜k + 0j k η = 0 ,   a = 0˜ : ij k ij k + i j˜k˜ η = 0 . It is now a simple matter to show, using the integrability conditions (2.24) which we already established, that Eq. (4.28) are satisfied, for both of the covariantly-constant spinors on the Stenzel 8-manifold. In other words, turning on the deforming flux from the harmonic 4-form G(4) does not lead to any further breaking of supersymmetry, and so the resolved M2-brane preserves 41 of the original supersymmetry. 7 If the transverse space is a cone, and the harmonic function depends only on the radial coordinate, the singularity in H0 corresponds to AdS4 times the base of the cone.

476

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5. Ricci-Flat K¨ahler Metrics on Ck Bundles There are many possible ans¨atze that one can adopt for constructing classes of Ricciflat metrics. A classic procedure is to look for metrics of cohomogeneity one, in which there are level surfaces composed of homogeneous manifolds, with arbitrary functions of radius parameterising homogeneous deformations of these surfaces.8 The conditions for Ricci-flatness then reduce to ordinary second-order differential equations for these functions. If one is lucky, the equations are solvable and the solutions include ones that describe metrics on smooth complete manifolds. Indeed, the Stenzel construction that we studied in Sect. 2 is an example of this type. In cases where there are Ricci-flat solutions with special holonomy, such as hyper-K¨ahler, K¨ahler or the G2 and Spin(7) exceptional cases, we have always found that first-order equations, derivable from a superpotential, can be constructed. All solutions of these satisfy the second-order equations, but the converse is not necessarily true. In this section we study another general class of metrics of cohomogeneity one, where the level surfaces are taken to be U (1) bundles over a product of N Einstein-K¨ahler manifolds, which would typically themselves be homogeneous. We then introduce (N + 1) arbitrary functions of the radial coordinate r, parameterising the volumes of the N base-space factors, and the length of the U (1) fibres. Following the familiar pattern, we then calculate the curvature, derive the second-order equations for Ricci-flatness, and then look for a first-order system coming from a superpotential. Having done this, we are able to solve the equations and obtain complete non-compact Ricci-flat K¨ahler metrics. The Ricci-flat solutions that we obtain here are such that the metric coefficient for one of the factors in the base space goes to zero at r = 0, as does the coefficient in the U (1) fibre direction. This implies that this particular factor in the base space must be a complex projective space CPm , so that r = 0 can become the origin of spherical polar  coordinates on R2k , where k = m + 1. If we write the base space as M = CPm × M,  denotes the product of the remaining Einstein-K¨ahler manifolds in the base, where M  The manifold has a then the total manifold has the topology of a Ck bundle over M.  at r = 0. Global considerations impose constraints on the bolt with the topology M possible choices for the other Einstein-K¨ahler base space factors. See [12] for a detailed discussion. Our principal focus will be on the case where all the Einstein-K¨ahler factors in the base space are taken to be complex projective spaces CPmi . It is shown in [12] that if the metric coefficient for the CPm1 factor is the one that goes to zero at r = 0, then regularity at r = 0 implies that the other CPmi factors must be such that m1 + 1 = gcd(m1 + 1, m2 + 1, m3 + 1, . . . ) ,

(5.1)

where gcd denotes the greatest common divisor of its arguments. The special case of just two factors, with the first being the trivial zero-dimensional manifold CP0 , and  being CPm , gives a well-known sequence of Ricci-flat manifolds on the complex M line bundle over CPm . The m = 1 case is the Eguchi-Hanson instanton. We obtain an L2 -normalisable harmonic (m + 1)-form for all the Cm+1 bundles over CPm where m is odd. The special case of two factors CPm1 × CPm2 with m1 = m2 = 1, for which the base space is S 2 × S 2 and the topology of the total space is a C2 bundle over CP1 , is the 8

See [25, 26] for a general discussion of such metrics.

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6-dimensional “small resolution” of the conifold discussed in [13], and more recently in [5], as an alternative to the “deformation” of the conifold. We shall study this in some detail, and show that the non-normalisable harmonic 3-form used in [5] to construct a fractional D3-brane gives a non-supersymmetric solution. We shall also consider three other special cases in some detail, giving 8-dimensional examples where the base space is S 2 × CP2 or S 2 × S 2 × S 2 ,9 construct L2 -normalisable harmonic 4-forms in two of these manifolds, and use them to build further supersymmetric deformed M2-branes. 5.1. Curvature calculations, and superpotential. To begin with, since it illustrates most of the key features, we shall consider the case of a base space that is the product of just two factors, comprising Einstein-K¨ahler spaces of real dimensions n and n. ˜ In the next subsection, we shall present the general results for a product of N Einstein-K¨ahler spaces. We make the following ansatz for metrics of cohomogeneity one whose level surfaces are U (1) bundles over products of two Einstein-K¨ahler base spaces: d sˆ 2 = dt 2 + a 2 ds 2 + b2 d s˜ 2 + c2 σ 2 ,

(5.2)

where a, b and c are functions of the radial coordinate t, ds 2 and d s˜ 2 are Einstein-K¨ahler spaces of real dimensions n and n˜ respectively, and . σ = dψ + A + A

(5.3)

 living in ds 2 and d s˜ 2 respectively, have field strengths F = dA The potentials A and A,    = q J, where J and J are the K¨ahler forms on ds 2 and F = d A, given by F = p J , F 2 and d s˜ . Furthermore, we assume cosmological constants λ and λ˜ for the two spaces, so Rij = λ δij ,

ab = λ˜ δab , R

Fik Fj k = p2 δij ,

ac F bc = q 2 δab . F

(5.4)

Note that there is a considerable redundancy in the use of constants here, since λ and λ˜ could be absorbed into rescalings of the functions a and b. It is advantageous to keep ˜ p and q unfixed for now, since the choice of how to specify them all the constants λ, λ, most conveniently depends on what choice one makes for the Einstein-K¨ahler metrics in the base space. In the orthonormal basis eˆ0 = dt ,

˜

eˆ0 = c σ ,

eˆi = a ei ,

eˆa = b ea ,

(5.5)

we find that the non-vanishing components of the Ricci tensor are b¨ c¨ a¨ Rˆ 00 = −n − n˜ − , a b c b˙ c˙ c¨ n p 2 c2 a ˙ c ˙ n˜ q 2 c2 Rˆ 0˜ 0˜ = −n − n˜ − + + , ac bc c 4a 4 4b4   a¨ a˙ b˙ p 2 c2 a˙ c˙ a˙ 2 λ Rˆ ij = − δij , + + (n − 1) 2 + n˜ − 2+ a ac a ab a 2a 4   b˙ c˙ b˙ 2 λ˜ b¨ a˙ b˙ q 2 c2 ˆ Rab = − δab . + + (n˜ − 1) 2 + n − + b bc b a b b2 2b4

(5.6)

9 Note that the choice S 2 × CP2 for the base actually violates the condition (5.1) for regularity when one of the factors collapses at the origin. This means that the metric actually has orbifold-like singularities at r = 0. We shall discuss a way to avoid this difficulty later.

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M. Cvetiˇc, G.W. Gibbons, H. L¨u, C.N. Pope

From this, after introducing the new radial variable η defined by dt = c a n bn˜ dη, we find that the conditions for Ricci-flatness can be derived from the Lagrangian L = T − V , where T = n α  γ  + n˜ β  γ  + n n˜ α  β  + 21 n(n − 1) α  + 21 n( ˜ n˜ − 1) β  , 2

2

˜ β+4γ V = 18 n p2 e(2n−4) α+2n˜ β+4γ + 81 n˜ q 2 e2n α+(2n−4)

(5.7)

˜ β+2γ , − 21 n λ e(2n−2) α+2n˜ β+2γ − 21 n˜ λ˜ e2n α+(2n−2)

together with the requirement that T + V vanishes. Here, a prime means a derivative with respect to η. Defining α i = (α, β, γ ) as usual, we find that the Lagrangian can be written as L = 21 gij (dα i /dη) (dα j /dη) + 21 g ij ∂W/∂α i ∂W/∂α j , where the superpotential is given by ˜ β+2γ + k en α+n˜ β W = 41 n p e(n−2) α+n˜ β+2γ + 41 n˜ q en α+(n−2)

(5.8)

and the various constants must be chosen so that k=

λ λ˜ = . p q

(5.9)

This leads to the first-order equations α  = 21 p e(n−2) α +n˜ β+2γ ,

˜ β+2γ β  = 21 q en α+(n−2) ,

˜ β+2γ γ  = − 41 n p e(n−2) α+n˜ β+2γ − 41 n˜ q en α+(n−2) + k en α+n˜ β .

(5.10)

5.2. Solving the first-order equations. We proceed here by introducing a new radial variable r, defined by10 dr = e(n−1) α+n˜ β+2γ dη .

(5.11)

The first-order equations (5.10) now become dα = 21 p e−α , dr

dβ = 21 q eα−2β , dr

dγ = − 41 n p e−α − 41 n˜ q eα−2β + k eα−2γ . dr (5.12)

The first can be solved at sight; the second can then be solved, and then using these results the third can be solved. After making an appropriate choice of integration constants, the result is e2α = 41 p 2 r 2 , e2β = 41 p q (r 2 + 2 ) , −n/2

 ˜ r2 k p r2 r2 1 1 e2γ = 1+ 2 ˜ 2 + 21 n, − 2 , 2 F1 1 + 2 n, − 2 n, n+2  

(5.13)

10 Note that another choice is to take dr = en α+(n−1) ˜ β+2γ dη; this will reverse the rˆ oles of the two metrics ds 2 and d s˜ 2 , with consequences that will become clear later.

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where  is a constant. The Ricci-flat metric is given by d sˆ 2 = e2α−2γ dr 2 + e2γ σ 2 + e2α ds 2 + e2β d s˜ 2 .

(5.14)

(Note that once one plugs in specific integer values for n and n, ˜ the hypergeometric function in the expression for e2γ becomes purely algebraic.) At small r, we have e2α = 41 p 2 r 2 ,

e2β ∼ 41 p q 2 ,

e2γ ∼

kp 2 r . n+2

(5.15)

Bearing in mind that k = λ/p, we therefore find that near r = 0, the metric approaches d sˆ 2 ∼

(n + 2) p 2 dS 2 + 41 p q 2 d s˜ 2 , 4λ

where

 dS = dr + r 2

2

2

4λ2 λ σ2 + ds 2 2 2 p (n + 2) n+2

(5.16)  .

(5.17)

Regularity at r = 0 therefore requires that the quantity enclosed in the parentheses be the unit (n + 1)-sphere metric. This means in particular that ds 2 should be the standard Fubini-Study metric on CPm , where n = 2m. The canonical choice for the cosmological constant that gives a “unit” CPm is in fact λ = n+2,

(5.18)

2 , where and the Fubini-Study metric is then ds 2 = dm 2 dm = F −1 d z¯ a dza − F −2 z¯ a zb dza d z¯ b ,

(5.19)

and F = 1 + z¯ a za . After setting λ = n + 2, we therefore find that d2 ≡

4 2 σ + ds 2 , p2

(5.20)

 term in σ is irrelevant here, and so should be the unit (2m + 1)-sphere metric. The A regularity demands that  2 2 (5.21) d2 = dψ + A + ds 2 p must be the unit (n + 1)-sphere. Recalling that we originally required that dA = p J , where J is the K¨ahler form on ds 2 , we see that this means that regularity requires that the potential B in d2 = (dψ + B)2 + ds 2 should give dB = 2J . This is precisely what one finds in the description of S 2m+1 as the Hopf fibration over CPm . More detailed discussions of the regularity conditions at r = 0 are discussed in [12]: In order for the fibre coordinate to have the correct periodicity, it must be that the other Einstein-K¨ahler factor must impose no further restriction above those implied by the CPm itself. For example, if the other factor is CPm˜ , then it must be that [12] m + 1 = gcd(m + 1, m ˜ + 1) .

(5.22)

480

M. Cvetiˇc, G.W. Gibbons, H. L¨u, C.N. Pope

We can summarise the above results as follows. We have found that the Ricci-flat metric given by (5.13) and (5.14) can be regular at r = 0, if the n-dimensional EinsteinK¨ahler metric ds 2 is taken to be the Fubini-Study metric on CPm , with n = 2m. Furthermore the Einstein-K¨ahler manifold for the metric d s˜ 2 must satisfy certain topological conditions [12], which reduce to (5.22) if it is CPm˜ . Then r = 0 is a regular region in the manifold, corresponding to a bolt whose topology is that of the Einstein-K¨ahler manifold with metric d s˜ 2 . For r > 0, we have level surfaces that are U (1) bundles over the product of the two Einstein-K¨ahler spaces whose metrics are ds 2 and d s˜ 2 . Of course the constants p and q must be chosen appropriately, to be commensurate with the periodicity of the fibre coordinate ψ. For example, if one takes the base space to be the product CPm × CPm˜ , and chooses the canonical values λ = 2(m + 1) and λ˜ = 2(m ˜ + 1) for the cosmological constants so as to give unit Fubini-Study metrics, then, after taking into account the relation (5.9), we may without loss of generality take p = m + 1, q = m ˜ + 1. The fibre coordinate z must then have period 2π , implying that the U (1) bundle over CPm × CPm˜ is simply-connected, or 2π/s, where s is any integer, in which case the bundle space is not simply connected.11 Thus when we consider CPm × CPm˜ base spaces, we shall typically make the choices λ = 2(m + 1) ,

λ˜ = 2(m ˜ + 1) ,

p = m + 1,

q=m ˜ + 1.

(5.23)

We can, of course, consider instead the situation where the rˆoles of the two metrics ds 2 and d s˜ 2 are interchanged, as mentioned in the footnote above. Everything goes through, mutatis mutandis, in exactly the same way as described above. It will now be the metric d s˜ 2 that is required to be the Fubini-Study metric on CPm˜ , with n˜ = 2m. ˜ Substituting the first-order equations (5.10) back into the expressions for the curvature 2-forms, we can read off the integrability conditions Rˆ ABCD CD η = 0 for the existence of covariantly-constant spinors. These conditions give (0i + Jij  ˜ ) η = 0 , (0a + Jab  ˜ ) η = 0 . (5.24) 0j

0b

The spinors that satisfy these conditions are the expected complex pair of covariantlyconstant spinors in the Ricci-flat K¨ahler metrics. It is straightforward to establish that the K¨ahler form is given by ˜ Jˆ = eˆ0 ∧ eˆ0 + e2α J + e2β J,

(5.25)

or, in other words, the vielbein components JˆAB are given by Jˆ00˜ = 1, Jˆij = Jij , Jˆab = Jab . (As in our discussion of the Stenzel metrics, we could again instead derive the first-order equations (5.10) by requiring that (5.25) be covariantly constant.) It can be useful to obtain complex holomorphic coordinates zµ for the K¨ahler metrics k spaces. This can be done by solving the holomorphicity conditions onNthe C bundle  δM + i JM N ∂M zµ = 0. It is most convenient to do this in the orthonormal frame, for which we therefore need the basis of vector fields dual to the vielbein (5.5). Using the r coordinate of (5.14), it is given by ∂ ∂ Eˆ 0 = eγ −α , Eˆ 0˜ = e−γ , ∂r ∂ψ     ∂ ∂ −α −β ˆ ˆ   Ei = e Ei − A i Ea − A a , Ea = e , ∂ψ ∂ψ

(5.26)

11 See, for example, [27] for a detailed discussion. It is also shown in [27] that these specific U (1) bundles over CPm × CPm˜ admit Killing spinors when the scalings are chosen so that the metric is Einstein.

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a are the vielbeins inverse to ei and e˜a on ds 2 and d s˜ 2 . From this we where Ei and E obtain the following holomorphicity conditions: 

∂ ∂ +i ∂ρ ∂ψ

 zµ = 0 ,

∂zµ = 0, ∂ψ µ a + i Jab E b ) zµ − (A b ) ∂z = 0 , a + i Jab A (E ∂ψ (Ei + i Jij Ej ) zµ − (Ai + i Jij Aj )

(5.27)

where we have defined the new radial coordinate ρ by dρ = eα−2γ dr .

(5.28)

It is evident therefore that the holomorphic coordinates for the K¨ahler metrics ds 2 and d s˜ 2 themselves can be used as holomorphic coordinates in the total manifold. It therefore remains to find one more complex coordinate z0 , constructed from the additional real coordinates ρ and ψ. The first equation in (5.27) shows that z0 should be a function of a can be written in terms of the K¨ahler ρ + i ψ. Noting that the vector potentials Ai and A  on ds 2 and d s˜ 2 as functions K and K Ai = p Ji j ∂j K ,

a = q Ja b ∂b K , A

(5.29)

it therefore follows that the extra complex coordinate can be taken to be 

z0 = eρ+i ψ+p K+q K .

(5.30)

We conclude this subsection with a number of explicit examples. Our first two make use of an S 2 × CP2 base space. These will actually still have orbifold-like singularities at r = 0, as we discussed above, since (5.22) is not satisfied. However, they are still of interest for the purposes of constructing deformed M2-brane solutions, since the remaining singularities are rather mild ones. We shall discuss this, and a complete resolution of the singularities, later.

C2 /Z2 bundle over CP2 . A particular class of examples would be to take the base space to be S 2 × CP2 , in which case we get 8-dimensional Ricci-flat K¨ahler metrics. Note that there are two distinct types of solution; one of them has a CP2 bolt at r = 0, whilst the other has instead an S 2 bolt. Consider first the case with the CP2 bolt; with our form of the solution where the untilded metric is singled out as the one whose coefficient goes to zero at r = 0, we therefore take ds 2 to be the S 2 metric, and d s˜ 2 to be the CP2 metric. From our general results, after making the conventional choices (5.23), i.e. λ = 4, λ˜ = 6, p = 2, q = 3 here, the Ricci-flat K¨ahler 8-metric is then given by (5.14), e2α = r 2 ,

e2β = 23 (r 2 + 2 ) ,

e2γ =

r 2 (3r 4 + 82 r 2 + 64 ) . 6(r 2 + 2 )2

(5.31)

482

M. Cvetiˇc, G.W. Gibbons, H. L¨u, C.N. Pope

(Note that the unit CP1 is actually a 2-sphere of radius 21 .) Thus the Ricci-flat K¨ahler metric is d sˆ82 = U −1 dr 2 + r 2 U σ 2 + 41 r 2 (dθ 2 + sin2 θ dφ 2 ) + 23 (r 2 + 2 ) d22 ,

(5.32)

where σ = dψ −

1 2

, cos θ dφ + A

U=

3r 4 + 82 r 2 + 64 . 6(r 2 + 2 )2

(5.33)

Here the maximum allowed periodicity for ψ is (ψ)max = π (see, for example, [28]),  = 3J, where J is the K¨ahler form on the unit CP2 metric d 2 , given in (5.19). and d A 2 If ψ had the period 2π , then U (1) fibres over S 2 would describe S 3 , and the metric would approach R4 × CP2 locally near r = 0. Instead, we get the lens space S 3 /Z2 , and the metric therefore approaches (R4 /Z2 ) × CP2 ; the 8-manifold is a C2 /Z2 bundle over CP2 . We could, of course, replace CP2 by the standard Einstein-K¨ahler metric on S 2 × S 2 in this metric, in which case the fibre coordinate would be allowed to have the period 2π needed for complete regularity at r = 0. In fact this would give a special case of a more general class of Ricci-flat K¨ahler metrics on C2 bundles over S 2 × S 2 , which we shall construct in Sect. 5.3. C3 /Z3 bundle over CP1 . The other possibility using S 2 × CP2 in the base space is to interchange the rˆoles of the S 2 and CP2 factors, so that now ds 2 is the CP2 metric, and d s˜ 2 is the S 2 metric. It is convenient to refer to this therefore as a CP2 × S 2 base, with the understanding that it is always the first factor whose metric coefficient goes to zero at r = 0. For this example, it is therefore convenient to choose the constants so that λ = 6, λ˜ = 4, p = 3 and q = 2. The resulting Ricci-flat K¨ahler 8-metric is then d sˆ82 = U −1 dr 2 + 49 r 2 r 2 U σ 2 + 49 r 2 d22 + 23 (r 2 + 2 ) d12 ,

(5.34)

where in this case we have , σ = dψ + A + A

U=

3r 2 + 42 , 9(r 2 + 2 )

(5.35)

 = 2J. The metrics d 2 and d 2 are the unit metrics on CP2 and and dA = 3J , d A 2 1 1 CP respectively, given by (5.19), and J and J are their respective K¨ahler forms. (Note that d12 = 41 d22 = 41 (dθ 2 + sin2 θ dφ 2 ).) The maximum allowed periodicity for ψ is again (ψ)max = π , while the period that would be need for the U (1) bundle over CP2 to describe S 5 would be ψ = 3π. This means that we instead get the lens space S 5 /Z3 , and so near r = 0 the metric approaches R6 /Z3 × S 2 ; the 8-manifold is a C3 /Z3 bundle over S 2 (or CP1 ). Complex line bundle over CPm . Another possibility is to take one factor in the product base manifold to be trivial, and the other to be CPm (or any other Einstein-K¨ahler manifold). The case where m = 1 is Eguchi-Hanson; for general m the corresponding Ricci-flat K¨ahler metrics were constructed in [29], and also in [30]. Since we shall make use of one of these examples later, we shall summarise the general results here. By taking

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p = n = λ = 0, q = 1, λ˜ = n+2 ˜ = 2m+2, and setting α = 0, the first-order equations (5.10) can be solved to give the 2(m + 1)-dimensional Ricci-flat K¨ahler metric 2 d sˆ 2 = U −1 dr 2 + 4r 2 U σ 2 + r 2 dm ,

(5.36)

where r here is related to the r variable in (5.10) by r −→ r 2 , the function U is given here by  r 2m+2 0 U =1− , (5.37) r 2 is the metric on the unit Fubini-Study metric on CPm , with r0 being a constant, and dm  here, where d A  = J , the K¨ahler form on the given in (5.19). Note that σ = dψ + A CPm . The radial coordinate r runs from r = r0 , where the metric approaches R2 × CPm , to infinity. Topologically, the manifold is a C1 bundle over CPm . For future reference we note that it is very easy to solve for an L2 -normalisable (anti)-self-dual harmonic form in the middle dimension, when m is odd. It is given by

1 2 ˜ (5.38) r m+1 J (m+1)/2 . G(m+1) = 2m+2 r m−1 eˆ0 ∧ eˆ0 ∧ J (m−1)/2 − r m+1

Note that the factors of r within the square brackets just convert each power of the K¨ahler form J on CPm into a 2-form of unit magnitude in the metric d sˆ 2 , i.e. r 2 J = 21 Jab eˆa ∧eˆb . Thus each term within the square brackets is just a constant times a wedge product of hatted vielbeins. The magnitude of G(m+1) is therefore given by constant , r 4m+4

|G(m+1) |2 =

(5.39)

and so the L2 -normalisability is manifest.

5.3. General results for N Einstein-K¨ahler factors in the base space. As we indicated above, the construction of the previous subsection has a straightforward generalisation to the case where we have N Einstein-K¨ahler factors in the base space, M = M1 × M2 × · · · × MN ,

(5.40)

with real dimensions ni and metrics dsi2 . Thus we write d sˆ 2 = dt 2 +

N

ai2 dsi2 + c2 σ 2 ,

(5.41)

i=1

where σ = dψ +

Ai ,

(5.42)

i

where dAi = pi J i , and J i is the K¨ahler form on the factor Mi in the base manifold. By comparing with the previous subsection, our notation here and in what follows should be self-evident.

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We find that the Ricci tensor for d sˆ 2 has components Rˆ 00 = −

ni

c¨ a¨ i − , a c

ni

a˙ i c˙ c¨ ni pi2 c2 , − + ai c c 4ai4

i

Rˆ 0˜ 0˜ = −

i

 2 c2

p a˙ j a¨ i a˙ i c˙ a˙ i λi = − + + nj − 2 + i 4  δai bi . − ai ai c ai2 ai aj a 2ai i j 

Rˆ ai bi

(5.43)

i

a˙ i2

Defining ai = eαi , c = eγ , the conditions for Ricci-flatness can be derived from the Lagrangian

2 L = 21 ni nj αi αj − 21 ni αi + ni αi γ  − V , (5.44) i,j

i

where V =

1 8

ni pi2 e2µi −

i

1 2

i

ni λi e2µi +2αi −2γ ,

(5.45)

i

with µi ≡ 2γ − 2αi +

(5.46)

nj αj .

j

The primes denote derivatives with respect to η, defined by dt = e

 i

ni αi +γ

(5.47)

dη .

Defining α0 = γ , and indices a = (0, i), the Lagrangian (5.44) can be written as L = 21 gab (dα a /dη) (dα b /dη) − V , with gij = ni nj − ni δij , g0i = ni , g00 = 0. This has the inverse g ij =

1 , D

g 0i =

1 , D

g 00 =

1 − 1, D

(5.48)

 where D = i ni is the total dimension of the base space. It is then straightforward to show that the potential V can be written in terms of a superpotential W , as V = − 21 g ab (∂W/∂α a ) (∂W/∂α b ), where 

W = 41 ni pi eµi + k e i ni αi , (5.49) i

provided that the constants pi and λi satisfy λi = k pi .

(5.50)

It follows that the following first-order equations imply Ricci-flatness: 

αi = 21 pi eµi , γ  = k e i ni αi − 41 ni pi eµi . i

(5.51)

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We can solve these by defining a new radial coordinate12 r: dr = eµ1 +α1 dη ,

(5.52)

which leads to dαi = 21 pi eα1 −2αi , dr

dγ = k eα1 −2γ − dr

1 4

ni pi eα1 −2αi .

(5.53)

i

The equation for α1 can be solved immediately, and then those for the remaining αi can be integrated. We find   e2αi = 41 p1 pi r 2 + 2i ,

(5.54)

where 1 = 0 and the other i are constants of integration. Defining γ˜ ≡ γ + 21 in an intermediate step, and x ≡ r 2 , the equation for γ can be solved to give e2γ = 21 p1 k



x + 2i

−ni /2 

x

dy 0

i



y + 2j

nj /2

.

 i

n i αi

(5.55)

j

The integration is elementary, giving an expression for e2γ as a rational function of x for any given choice of the integers ni , but the general expression for arbitrary dimensions ni requires the use of hypergeometric functions. In terms of the r coordinate, the metric is given by d sˆ 2 = e2α1 −2γ dr 2 +

e2αi dsi2 + e2γ σ 2 .

(5.56)

i

The analysis of the structure of the Ricci-flat metrics proceeds in a fashion that is analogous to that of the previous section. The radial coordinate runs from r = 0, where the metric functions e2α1 and e2γ vanish, to r = ∞. Regularity at r = 0 requires that the Einstein-K¨ahler metric ds12 on the factor M1 in the base space (5.40) be the Fubini-Study metric on CPm1 , where n1 = 2m1 , so that r = 0 becomes the origin of spherical polar coordinates on Rn1 +2 . Even though the other metric functions e2αi for i ≥ 2 are non-zero for the entire range 0 ≤ r ≤ ∞, there are again topological restrictions on the choice of Einstein-K¨ahler manifolds for these factors, stemming from the requirement that the U (1) fibre coordinate should have the periodicity needed for the U (1) bundle over CPm1 to be the (2m1 + 1)-sphere and not a lens space. (See [12] for a detailed discussion: If, for example, all the other factors are complex projective spaces CPmi , then they must satisfy (5.1).) Topologically, the manifold on which the metric d sˆ 2 is then defined is a Ck bundle over the product of the remaining base-space factors M2 × M3 × · · · × MN , where k = 21 n1 + 1.

12 We single out the i = 1 factor in the base space purely as a matter of convention; there is no loss of generality, since we have not yet specified the choices for these factors.

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Arguments analogous to those of the previous subsection show that the K¨ahler form for the metric d sˆ 2 is given by

˜ Jˆ = eˆ0 ∧ eˆ0 + ai2 J i , (5.57) i

where J i denotes the K¨ahler form on the i th factor in the product of Einstein-K¨ahler manifolds (5.40) in the base space. The two covariantly-constant spinors will satisfy the integrability conditions (0ai + Jai bi 0b ˜ i)η = 0,

(5.58)

where Jai bi are the vielbein components of the K¨ahler form J i . It is straightforward to see, generalising the discussion of Sect. 5.2, that for holomorphic complex coordinates on the total space we can use the holomorphic coordinates of the various factors Mi in the base space M, together with the additional complex coordinate z0 given by 

z0 = eρ+i ψ+

i

pi Ki

,

(5.59)

where Ki denotes the K¨ahler function on the factor Mi , and ρ is defined by dρ = eα1 −2γ dr. Let us present one explicit example of the more general Ricci-flat K¨ahler solutions: C2 bundle over CP1 × CP1 . Consider the case where we take the base space to be S 2 × S 2 × S 2 , so n1 = n2 = n3 = 2. Then we find     e2α1 = 41 p12 r 2 , e2α2 = 41 p1 p2 r 2 + 22 , e2α3 = 41 p1 p3 r 2 + 23 ,     p1 k r 2 22 23 + 23 22 + 23 r 2 + 21 r 4 2γ    e = , (5.60) 4 r 2 + 22 r 2 + 23 and after making convenient choices pi = 1, λi = 1 for the constants, the metric is given by     d sˆ82 = U −1 dr 2 + 41 r 2 U σ 2 + 41 r 2 d21 + 41 r 2 + 22 d22 + 41 r 2 + 23 d23 , (5.61) where

  3r 4 + 4 22 + 23 r 2 + 622 23    U= , 6 r 2 + 22 r 2 + 23

(5.62)

d21 , d22 and d23 are metrics on three unit 2-spheres, and in an obvious notation we have σ = dψ + cos θ1 dφ1 + cos θ2 dφ2 + cos θ3 dφ3 ,

(5.63)

where ψ has period 4π. The metric approaches R4 × S 2 × S 2 at r = 0, with an S 2 × S 2 bolt; topologically, the manifold is a C2 bundle over S 2 × S 2 (or CP1 × CP1 ).

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6. More Fractional D3-Branes and Deformed M2-Branes 6.1. The resolved fractional D3-brane. 6.1.1. Harmonic 3-form on the C2 bundle over CP1 . This is a special case of the construction Sect. 5, in which the base space is taken to be just S 2 × S 2 . It gives a complete non-compact manifold that provides a “small resolution” of the singular conifold [13]. The metric can be written in the form [5]     r 2 + 62 2 1 r 2 + 92 22 , ds62 = 2 r 2 σ 2 + 16 r 2 d22 + 16 r 2 + 62 d  dr + 9 2 2 2 r + 9 r + 6 (6.1) where 22 = d θ˜ 2 + sin2 θ˜ d φ˜ 2 , d

d22 = dθ 2 + sin2 θ dφ 2 ,

(6.2)

σ = dψ + cos θ dφ + cos θ˜ d φ˜ ,

and  is a constant. The radial coordinate runs from r = 0 to r = ∞. Near r = 0, the metric smoothly approaches flat R 4 times a 2-sphere of radius , while at large r the metric describes the cone with level surfaces that are the U (1) bundle over S 2 × S 2 . (We are us˜ ing the notation of [5] here;  it corresponds in our notation to taking p = q = λ = λ = 1, and then sending r −→ 23 r and  −→ 2.) From (5.25) we see that a holomorphic basis of 1-forms is ˜

 0 = −e0 + i e0 ,

 1 = e1 + i e2 ,

 2 = e3 + i e4 ,

(6.3)

where e0 = h dr ,

e5 = c σ ,

e1 = a dθ ,

e3 = b d θ˜ ,

e4 = b sin θ˜ d φ˜ ,

e2 = a sin θ dφ ,

and a, b and c and h are the metric coefficients in (6.1), given by  2  r + 92 r2 , b2 = 16 (r 2 + 2 ) , c2 = 19 a 2 = 16 r 2 , r 2 + 62

h2 =

(6.4)

r 2 + 62 . r 2 + 92 (6.5)

It can be useful also to obtain complex coordinates zµ compatible with the complex  b  b µ structure. Solving the conditions δa + i Ja ∂b z = 0, we are led to the following choice of complex coordinates: z1 = tan 21 θ ei φ ,

˜ z2 = tan 21 θ˜ ei φ ,

z3 = sin θ sin θ˜ eρ−i ψ ,

(6.6)

where ρ is related to r by 3h2 dr = r dρ, i.e. eρ = r 2 (r 2 + 92 )1/2 . The complex vielbein basis given in (6.3) then takes the form   dz1 dz2 dz3 dz1 dz2 0 ˜ ,  1 = a sin θ + cos θ − ,  2 = b sin θ˜ ,  = c cos θ z1 z2 z3 z1 z2 (6.7)

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which shows that it is indeed holomorphic. Note that the complex coordinate z3 is related to the z0 coordinate of the general discussion in Sect. 5.2 by simple coordinate transformations. There is a complex self-dual 3-form, satisfying ∗G(3) = i G(3) , given by G(3) =

1 1 (e5 ∧ e1 ∧ e2 − i e0 ∧ e3 ∧ e4 ) − 2 (e5 ∧ e3 ∧ e4 − i e0 ∧ e1 ∧ e2 ) . 2 ca cb (6.8)

From this, it follows that G(3) is given by G(3) = −f1 ¯ 0 ∧ ( 1 ∧ ¯ 1 +  2 ∧ ¯ 2 ) + f2  0 ∧ ( 1 ∧ ¯ 1 −  2 ∧ ¯ 2 ) ,

(6.9)

where f1 ≡

1 1 − , 4c a 2 4c b2

f2 ≡

1 1 + . 4c a 2 4c b2

(6.10)

Thus we see that G(3) in general has (2, 1) and (1, 2) pieces. It would become pure (2, 1) if f1 vanished. This would happen only if the scale parameter  were set to zero, since then a and b become equal. In this limit, the metric reverts to the original unresolved conifold. The (1, 2) piece does, of course, go to zero faster than the (2, 1) piece as r tends to infinity in the resolved metric. Thus the harmonic 3-form G(3) becomes “asymptotically pure” at large distances. This 3-form was used to construct a fractional D3-brane in [5]. Owing to the (marginal) non-normalisability of the 3-form at large distance, it follows that the solution has a logarithmic correction to the D3-brane metric function H at large proper distance, as in (1.1). The solution also has a repulsion type of singularity owing to the non-normalisability of G(3) at small distance. In the next subsection, we shall address the issue of supersymmetry. 6.1.2. The issue of supersymmetry in the Pando Zayas-Tseytlin D3-brane. In the general discussions of supersymmetry for fractional D3-branes in [3, 4], it is argued that the deformed solution will only be supersymmetric if the complex self-dual harmonic 3-form is purely of type (2, 1). In fact, it was argued in [3, 4] that the self-duality of the 3-form already implied that it could contain only (2, 1) and (0, 3) pieces, and in [4] it was proved that the presence of a (0, 3) term would imply that there would be no supersymmetry. Since we have found that the self-dual harmonic 3-form in the resolved D3-brane solution of [5] has both (2, 1) and (1, 2) pieces, it is appropriate first to discuss why the (1, 2) piece can in fact be present. After that, we shall discuss its implications for supersymmetry. The general statement about the duality of (p, q)-forms in six-dimensional K¨ahler spaces is as follows. One must distinguish between (2, 1) or (1, 2)-forms that are perpendicular to the K¨ahler form, Gabc J ab = 0, and those that are parallel, Gabc = K[a Jbc] . Denoting these by (p, q)⊥ and (p, q) , we then have, in an obvious notation, ∗(2, 1)⊥ = i (2, 1)⊥ , ∗(2, 1) = −i (2, 1) , ∗(1, 2)⊥ = −i (1, 2)⊥ , ∗(1, 2) = i (1, 2) , ∗(0, 3) = i (0, 3) , ∗(3, 0) = −i (3, 0) .

(6.11)

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We can indeed verify by inspection of (6.9) that the first term is of type (1, 2) , and the second term is of type (2, 1)⊥ . This is therefore compatible with the fact that G(3) is self-dual, ∗G(3) = i G(3) .13 Now let us turn to the question of supersymmetry. It is shown in [3, 4] that in the Majorana basis of [31], the criterion for unbroken supersymmetry for fractional D3-branes is that in addition to the usual requirements of the standard D3-brane, the harmonic self-dual 3-form should satisfy Gabc  abc η = 0 ,

Gabc  abc η∗ = 0 ,

(6.12)

where η is covariantly-constant in the six-dimensional Ricci-flat K¨ahler metric. The Majorana basis implies that the ten-dimensional Dirac matrices ˆ A with spatial indices are symmetric and real, while the Dirac matrix with the timelike index is antisymmetric and real. (These are the conventions of [31], modified to our notation where the metric signature is mostly positive.) In terms of a 4 + 6 decomposition, we shall have ˆ µ = γµ ⊗ 1l ,

ˆ m = γ5 ⊗ m ,

(6.13)

where γ5 = 4!i  µνρσ γµνρσ is antisymmetric and imaginary, and the Dirac matrices m in the six-dimensional space are also antisymmetric and imaginary. We also have that the chirality operator 7 = 6!i  a1 ···a6 a1 ···a6 is imaginary and antisymmetric, while ˆ 11 is symmetric and real. Note that because 7 is imaginary in the Majorana basis, this means that η∗ has the opposite chirality to η. We can now see that if the harmonic self-dual 3-form is written as Im G(3) = GRe (3) + i G(3) ,

(6.14)

Im where GRe (3) and G(3) are both real, then the criterion for supersymmetry is equivalent to

GRe abc abc η = 0 ,

GIm abc abc η = 0 .

(6.15)

Expressing the conditions in this form has the advantage that it is now independent of the choice of basis for the Dirac matrices. In particular, substituting (6.9) into (6.15), and making use of the conditions 12 η = 1˜ 2˜ η = −00˜ η satisfied by the covariantlyconstant spinor η (see [6]), we arrive at the conclusion that the resolved D3-brane solution of [5], using the Ricci-flat metric on the C2 bundle over CP1 is not supersymmetric, since f1 is non-zero. This is consistent with the fact that the (1, 2) piece in G(3) is non-vanishing. One can also demonstrate the breaking of supersymmetry by a direct substitution of G(3) into (6.12) in the Majorana basis. 6.2. Harmonic 4-form for C2 /Z2 and C2 bundles over CP2 , and smooth M2-branes. Let us now consider examples where the 8-dimensional Ricci-flat solution is obtained by taking the level surfaces to be the U (1) bundle over S 2 × CP2 . First, we shall choose the case where the bolt at r = 0 is CP2 , so the metric is given by (5.32); by our general 13 Note that there would be no such harmonic form of type (1, 2) in a compact Calabi-Yau 3-fold  since it would require the existence of a harmonic (0, 1)-form Kα¯ , which is excluded by the fact that the first cohomology group H 1 (Z) vanishes. However, in a non-compact manifold, where furthermore the harmonic forms are not being required to be L2 -normalisable, such arguments break down.

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arguments in Sect. 2.1, we can expect that a harmonic 4-form should exist for this manifold. Of course in this case there is still an orbifold-type singularity in the metric near the origin, as we discussed earlier. Afterwards, we shall present a completely regular generalisation of the solution. Making a natural ansatz for a self-dual harmonic 4-form that is invariant under the isometry group, we obtain equations that admit the simple solution G(4) =

(r 2

 2 ˜ ˜ e2β eˆ0 ∧ eˆ0 ∧ J − 2e2α eˆ0 ∧ eˆ0 ∧ J + e2α+2β J ∧ J − e4β J ∧ J , 2 3 + ) (6.16)

where J is the K¨ahler form (i.e. volume form) on S 2 , and J is the K¨ahler form on CP2 . Note that 21 J ∧ J is the volume form on CP2 . We therefore find G2(4) =

2884 , (r 2 + 2 )6

(6.17)

from which it follows that the harmonic 4-form G(4) is L2 normalisable. By making a canonical choice for the vielbeins and K¨ahler structures on S 2 and CP2 , ˜ ˜ ˜ ˜ we may write J = e1 ∧ e2 , J = e˜1 ∧ e˜2 + e˜3 ∧ e˜4 . It then follows from (5.25) that a holomorphic vielbein basis for the 8-dimensional metric is ˜

 0 = eˆ0 + i eˆ0 ,

 1 = eˆ1 + i eˆ2 ,

˜

˜

 2 = eˆ1 + i eˆ2 ,

˜

˜

 3 = eˆ3 + i eˆ4 ,

(6.18)

and the K¨ahler form is given by Jˆ =

i 2

( 0 ∧ ¯ 0 +  1 ∧ ¯ 1 +  2 ∧ ¯ 2 +  3 ∧ ¯ 3 ) .

(6.19)

The harmonic 4-form (6.16) can then be rewritten as  4  0 ∧ ¯ 0 ∧  2 ∧ ¯ 2 +  0 ∧ ¯ 0 ∧  3 ∧ ¯ 3 − 2 0 ∧ ¯ 0 ∧  1 ∧ ¯ 1 4(r 2 + 2 )3 − 1 ∧ ¯ 1 ∧  2 ∧ ¯ 2 + 2 2 ∧ ¯ 2 ∧  3 ∧ ¯ 3 +  1 ∧ ¯ 1 ∧  3 ∧ ¯ 3 , (6.20)

G(4) = −

which shows that it is a (2, 2)-form. Furthermore, it satisfies Gabcd Jˆab = 0, and so it is perpendicular to the K¨ahler form. In the notation we used earlier, it is therefore a 4-form of type (2, 2)⊥ . Solving Eq. (4.16) for the function H in the deformed M2-brane (4.16), we first find that r 3 (3r 4 + 82 r 2 + 64 ) H  = β +

3 m2 4 (3r 2 + 2 ) . (r 2 + 2 )3

(6.21)

If the constant of integration β is chosen to be β = −3 m2 , then the solution for H is non-singular at r = 0. Explicitly, we find   √ √ 3m2 (3r 2 + 22 ) 27 2 m2 2 2 H =1− . (6.22) + arctan 24 (r 2 + 2 )2 46 3r 2 + 42

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This tends to a constant at small r, and at large r it has the asymptotic form H ∼1+

m2 m 2 2 19m2 4 − + + ··· . 6r 6 3r 8 90r 10

(6.23)

The asymptotic behaviour is best analysed using the proper distance ρ, defined by eα−γ dr = dρ, as the radial coordinate. The coordinates ρ and r are related by   22 86 1 r∼ √ ρ− + ··· . (6.24) + 2 3ρ 45 ρ 5 Thus in terms of ρ, the function H behaves as follows in the asymptotic region: H ∼1+

4m2 416m2 4 − + ··· . 3ρ 6 45ρ 10

(6.25)

As discussed in Sect. 4.2, the condition for supersymmetry of the deformed M2-brane is that the harmonic 4-form should satisfy GABCD BCD η = 0 ,

(6.26)

where η is covariantly constant in the 8-dimensional transverse metric. From the integrability conditions (5.24) for η, and the form of the harmonic 4-form (6.16), it is straightforward to show that (6.26) is satisfied, and so this deformed M2-brane solution is supersymmetric. As we mentioned earlier, the maximum periodicity (ψ)max = π of the ψ coordinate on the U (1) fibres, implied by the requirement of regularity of the principal orbits, means that near r = 0 the U (1) bundle over S 2 gives S 3 /Z2 rather than S 3 , and so there is an orbifold singularity at r = 0. This can be avoided if we replace CP2 by S 2 × S 2 , since then ψ can have period 2π instead. Since this is just a special case of more general C2 bundles over S 2 × S 2 that we discuss in the next section, we shall not consider this further here. It is worth noting, however, that there does exist a generalisation of the metrics on complex bundles over S 2 × CP2 that avoids the orbifold singularity. The solution is obtained, along with wide classes of more general related examples, in [12]. The investigation of these generalisations was motivated by a construction of a six-dimensional example in [32]. Here, we shall just quote the result for the new eight-dimensional metric over S 2 × CP2 , referring to [12] for explicit details. It is given by     (6.27) d sˆ 2 = e−2γ dr 2 + e2γ σ 2 + 2 r + 21 d12 + 3 r + 22 d22 , 2 denotes the Fubini-Study metric on the unit CPm . The function γ where as usual dm is given by     421 r r 2 + 322 r + 342 + r 2 3r 2 + 822 r + 622 2γ e = . (6.28)  2  3 r + 21 r + 22

When 1 = 0 this reduces, after simple coordinate transformations, to the metric given in (5.32). However, when 1 is non-zero the entire S 2 × CP2 remains uncollapsed at r = 0. The regularity conditions at r = 0 now imply that ψ should have period π , which is the same as the value dictated by regularity of the principal orbits.

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It is easy to repeat the analysis of the harmonic 4-form for this metric. One finds that there is again a normalisable self-dual form, given by  22 0˜ 2β 0  − 2e2α eˆ0 ∧ eˆ0˜ ∧ J + e2α+2β J ∧ J − e4β J ∧ J , G(4) =  e e ˆ ∧ e ˆ ∧ J  3 r 2 + 22 (6.29) where J is the K¨ahler form on CP1 , and Jis the K¨ahler form on CP2 . It is again of type (2, 2)⊥ , and so it will give a supersymmetric deformed M2-brane solution. The solution for H is now given by   12m2 3r + 21 + 222 H = c0 +   2 22 22 − 411 r + 22

221 log(r − ri ) + ri log(r − ri ) 108m2     ,  22 22 − 421 i=1 9ri2 + 8 21 + 222 ri + 622 22 + 221 3

−

(6.30)

where ri are the three roots of the cubic expression in e2γ . The function H approaches a positive constant at r = 0, and at large r it becomes      4m2 221 + 22 4m2 221 + 22 21 + 222 H = c0 + − 322 r 3 322 r 4   4m2 3261 + 7241 22 + 12021 42 + 1962 + . (6.31) 4522 r 5 In terms of proper distance ρ, defined by e−γ dr = dρ, we have 106496m2 22 256m2 − . 3ρ 6 45ρ 10 Note that when 1 = 1/22 , the solution becomes rather simpler:   m2 20r 2 + 5522 r + 2342 H = c0 + . 5  10 r + 22 H = c0 +

(6.32)

(6.33)

6.3. Harmonic 4-form for C2 bundle over CP1 × CP1 , and smooth M2-brane. We can also construct a harmonic self-dual 4-form for the 8-dimensional metric with the S 2 × S 2 × S 2 base space, which we obtained in (5.61). The natural self-dual ansatz is ˜

G(4) = eˆ0 ∧ eˆ0 ∧ [e2α1 f1 1 + e2α2 f2 2 + e2α3 f3 3 ] −e2α1 +2α2 f3 1 ∧ 2 − e2α1 +2α3 f2 1 ∧ 3 − e2α2 +2α3 f1 2 ∧ 3 . (6.34) If we let x ≡ r 2 , then the equations that follow from dG(4) = 0 are       x f1 + x + 22 f2 = x x + 22 f3 ,       x f1 + x + 23 f3 = x x + 23 f2 ,         x + 22 f2 + x + 23 f3 = x + 22 x + 23 f1 , where a prime means d/dx here.

(6.35)

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It is straightforward to solve these equations. By choosing the integration constants appropriately we can obtain a solution which gives a self-dual harmonic 4-form that is normalisable, namely   222 23 + 22 + 23 r 2 23 f1 =  , f2 = −       2 , 2 2 r 2 + 22 r 2 + 23 r 2 + 22 r 2 + 23 (6.36) 22 f3 = −   2 . r 2 + 23 r 2 + 22 One can see that in the special case where 3 = 2 , the solution reduces to the one found in (6.20). This is not surprising, since then the final S 2 ×S 2 factors in S 2 ×S 2 ×S 2 become an Einstein-K¨ahler 4-manifold, and the equations arising from solving for the harmonic 4-form reduce to those that we had to solve previously for the S 2 × CP2 base space. Using the harmonic 4-form given by (6.34) and (6.36), we can construct another completely regular deformed M2-brane. It is easily seen from (6.34) that the magnitude of G(4) will be given by   (6.37) |G(4) |2 = 48 f12 + f22 + f32 . 1 From the expression (5.61) for the metric, we find that H = − 48 m2 |G(4) |2 becomes  √ √ 1 g U H  = − 48 m2 g |G(4) |2 , (6.38)     √ where g = r 3 r 2 + 22 r 2 + 23 /128. The first integration gives rise to    63 62 1 m2     . (6.39) β+ H =√ 2 2 −  2 gU 256 22 − 23 r + 3 r 2 + 2 2

The singularity at r = 0 is avoided by choosing β = −m2 /256. Then we find that the function H is given by   3m2 22 + 23 + 3r 2     H = c0 −  2 (6.40) 2 23 − 22 222 − 23 r 2 + 22 r 2 + 23     √ 2 2 223 − 22 222 − 23 27m 2 .   +  3/2  2 3/2 arctan  3r 2 + 2 22 + 23 4 22 − 2 2 − 2 3

2

2

3

The coordinate r runs from 0 to infinity, and the function H is finite and positive definite. For small r, H approaches a constant, and for large r, it behaves as     m2 22 + 23 m2 742 + 522 23 + 743 m2 + + ··· . (6.41) H ∼ c0 + 6 − 6r 6r 8 90r 10 As usual, it is helpful to express the asymptotic behaviour in terms of proper distance ρ, √ defined by dr/ U = dρ. The r and ρ coordinates are related, at large r, by    2 2 2 − 23 22 + 23 1 + ··· . (6.42) + r∼ √ ρ− 3ρ 6ρ 3 2

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In terms of ρ, H has the following large-distance behaviour:   32m2 442 + 522 23 + 443 4m2 H ∼ c0 + − + ··· . 3ρ 6 45ρ 10

(6.43)

This deformed M2-brane is therefore completely regular, and it has a well-defined ADM mass. It is again supersymmetric. Note that this solution for H reduces to the solution (6.22) if the parameters 2 and 3 are set equal, as would be expected in the light of our earlier discussion. It is interesting also to note that the solution (6.40) becomes especially simple if the parameters satisfy 22 = 223 or 23 = 222 . Choosing the first of these two equivalent cases, we then find that the solution can be written as   m2 r 2 + 423 H = c0 +  (6.44)  3 . 6 r 2 + 23 r 2 + 223 6.4. Deformed M2-brane on the complex line bundle over CP3 . At the end of Sect. 5.3 we described the 2(m + 1)-dimensional Ricci-flat K¨ahler metrics on the complex line bundles over CPm , and we obtained an L2 -normalisable (anti)-self-dual harmonic (m+1)-form for each case when m is odd. In particular, we can take m = 3, and consider the 8-dimensional complex line bundle over CP3 . The metric is given in (5.36), and the harmonic 4-form can be read off from (5.38). Equation (4.16) for the M2-brane metric function H can be straightforwardly solved in this case, giving H = c0 +

m2 r06 . 6 r6

(6.45)

(We have made an appropriate choice for the normalisation of the harmonic 4-form.) Since the radial coordinate r runs from r0 to infinity, it follows that again we have a completely non-singular deformed M2-brane. In terms of the proper radial distance ρ defined by U −1/2 dr = dρ for this metric, the asymptotic large-distance behaviour of the function H in the corresponding resolved M2-brane is easily seen to be H =1+

m2 r06 m2 r014 − + ··· . 6 ρ 14ρ 14

(6.46)

It should be noted that this solution is not supersymmetric. This can be shown by substituting G(4) directly into the supersymmetry condition GABCD BCD η = 0, and making use of the integrability conditions (5.24), which reduce here to just (0a + Jab 0b ˜ ) η = 0. One finds that the only solution to all these conditions is η = 0. Alternatively, we may  observe that although  the harmonic (m + 1)-form constructed in (5.38) is of type 21 (m + 1), 21 (m + 1) , it is not perpendicular to the K¨ahler form ˜ Jˆ = eˆ0 ∧ eˆ0 + r 2 J , when the odd integer m is greater than 1. In particular, the harmonic 4-form in the complex line bundle over CP3 is of type (2, 2) but does not satisfy GABCD JˆCD = 0, and, as shown in [9], the vanishing of this quantity is another way of expressing the criterion for supersymmetry.

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6.5. Deformed M2-brane on an 8-manifold of Spin(7) holonomy. Recently a resolved M2-brane was constructed using a Ricci-flat 8-manifold of Spin(7) holonomy [6]. We shall summarise the key features of that solution here, in order to allow a comparison with the resolved M2-branes using Ricci-flat K¨ahler 8-manifolds (which have SU (4) holonomy) that we have obtained in this paper. The metric for the Spin(7) manifold, which is an R4 bundle over S 4 , is given by −1    a 10/3 a 10/3 9 2 9 2 ds82 = 1 − 10/3 1 − 10/3 (σi − Ai )2 + 20 dr 2 + + 100 r r d24 , r r (6.47) where σi are left-invariant 1-forms on SU (2), d24 is the metric on the unit 4-sphere, and i Ai is the SU (2) Yang-Mills instanton on S 4 [33, 34]. The Yang-Mills field strengths Fαβ j

k i F satisfy the algebra of the imaginary unit quaternions, Fαγ γβ = −δij δαβ + ij k Fαβ . A normalisable anti-self-dual harmonic 4-form was found in [6], with orthonormal components given by

G0ij k = 6f ij k

k i Gαβγ δ = −6f αβγ δ , Gij αβ = f ij k Fαβ , G0iαβ = −f Fαβ , (6.48)

where f = r −14/3 . The deformed M2-brane is given by (4.16), with [6]    2  a 10/3 3 1 − ar 2 40000m2  9− + H = c0 −  10/3  729a 16/3 r 2/3 r 1 − ar     √ √  a 10/3 √  5+1+4 r  32000 2 5 m2  ( 5 − 1) arctan  &  +      √ √ 243a 6 2 5 5−1   √  a 10/3 √  5−1+4 r   +( 5 + 1) arctan    . (6.49)  & √ √ 2 5 5+1 At large r, H has the asymptotic form 2 105 m2 28 104 a 4/3 m2 − + ··· . (6.50) 37 r 6 2673 r 22/3 In terms of the proper distance ρ, the asymptotic behaviour of the first two terms in H is the same as in the r coordinate. The supersymmetry of the solution was not discussed in [6], but has since been demonstrated in [24]. Here, we note that another simple proof of supersymmetry can be given by making use of the results in [34] on the integrability conditions for the covariantly-constant spinor in the Spin(7) manifold. These are all encapsulated in the equations H = c0 +

i 40i η + Fαβ αβ η = 0 .

(6.51)

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It useful also to note that these imply other equations, including 0i η = 21 ij k j k η ,

i Fαβ 0iβ η = 3α η ,

ij k 0ij k = 6η .

(6.52)

Using these equations, and the expressions given in (6.48) for the components of the harmonic 4-form, it is now elementary to verify that Gabcd bcd η = 0, and hence that the single supersymmetry allowed by the Spin(7) holonomy is preserved in the deformed solution. 7. Conclusions and Comments on Dual Field Theories The purposes of this paper were manifold. Our first motivation was purely formal. We have provided an explicit construction of self-dual harmonic forms for a class of complete non-compact Ricci-flat K¨ahler manifolds in 2(n + 1) real dimensions. Specifically, we focused on the Stenzel metrics [10]. These spaces have SO(n + 2) isometry, with level surfaces corresponding to SO(n + 2)/SO(n) coset spaces. The degenerate orbit (“bolt”) corresponds to the base space S n+1 ≡ SO(n+2)/SO(n+1). (The n = 1 case is the Eguchi-Hanson instanton, and the n = 2 case was first constructed by Candelas and de la Ossa [13] as the deformed conifold.) For these manifolds we provided an explicit construction of all the the harmonic, self-dual, middle dimension forms. Specifically, the solution for the harmonic (p, q)-forms in p + q = 2(n + 1) dimensions reduces to finding the solution to two coupled first-order differential equations, which we solved explicitly. Interestingly, the (p, p)-form (which implies n is odd) is proportional to (cosh r)−2p and thus turns out to be L2 -normalisable. On the other hand all the other (p, q)-forms (for n odd or even) are not L2 normalisable, with the degree of divergence increasing with the value |p − q|. We also gave a construction of another general set of complete Ricci-flat metrics, whose homogeneous level surfaces are U (1) bundles over a product of N EinsteinK¨ahler base spaces. The regularity of the solution implies that one of the base spaces has to be CPm with its Fubini-Study metric, while the other Einstein-K¨ahler factors are restricted by topological considerations. For example, if they are complex projective spaces CPmi , then they must satisfy (5.1) The total space is topologically a Cm+1 bundle over the remaining base-space factors. (The 6-dimensional example where there are just two S 2 factors appeared in [13] and was further discussed in [5]; the metric has level surfaces that are the 5-manifold known as T 1,1 , which is a U (1) bundle over S 2 × S 2 .) We discussed explicit examples, and constructed normalisable harmonic 4-forms for two 8-dimensional cases, where the base spaces are S 2 × CP2 and S 2 × S 2 × S 2 , and harmonic (m + 1)-forms for all the cases with CPm as base space, for all odd m. These formal constructions of self-dual harmonic forms turn out to have intriguing applications in the study of deformed p-brane configurations whose transverse spaces are non-compact Ricci-flat manifolds. In particular, the fractional D3-brane found in [2] provides the non-singular gravity dual of N = 1 super Yang-Mills theory in four dimensions. A generalisation to a number of deformed p-brane configurations with odd or even dimensional Ricci-flat transverse spaces was recently given in [6]. The systematic construction of the middle-dimension harmonic forms for the Stenzel spaces, as well as the generalisations given in Sect. 5 allowed us to provide another set of regular gravity solutions corresponding in particular to deformed M2-branes with 8-dimensional transverse Ricci-flat spaces. We constructed two examples using Ricci-flat K¨ahler 8manifolds, and in each case the deformed M2-branes are supported by (2, 2)-harmonic

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forms that are normalisable, and so the M2-branes are regular everywhere. In both cases, as well as for the case of the M2-brane on the Spin(7) manifold that was constructed in [6], the solutions are supersymmetric. This should be contrasted with the 6-dimensional Ricci-flat K¨ahler metric on the C2 bundle over CP1 , which has a harmonic form with both (1, 2) and (2, 1) contributions. Consequently, we show that the fractional D3-brane using this metric is not supersymmetric. The deformed M2-branes that we constructed in this paper, and the previously-known fractional D3-branes, provide supergravity duals to field theories with less than maximal supersymmetry. In fact, the lower-dimensional conformal symmetry associated with the AdS/CFT correspondence can be broken by the extra contributions to the “harmonic” function H of these resolved branes. Indeed, in all the known fractional D3-branes the function H has a universal asymptotic logarithmic modification, given by (1.1), owing to the (marginal) non-normalisability of the complex harmonic self-dual 3-forms in six-dimensions. This implies that the geometry no longer has an AdS5 background, and consequently the dual four-dimensional Yang-Mills field theory has no conformal symmetry. General mathematical arguments imply that for any six-dimensional Ricci-flat K¨ahler metric with an asymptotically conical structure, complex harmonic 3-forms will necessarily be non-normalisable. By contrast, deformed M2-branes have a richer structure, with a larger range of possibilities for the asymptotic behaviour. At large distance the modification to H takes the form   c Q H = c0 + 6 1 + γ + · · · . (7.1) ρ ρ For our Ricci-flat K¨ahler examples constructed in this paper γ takes the values 83 , 4 for supersymmetric M2-branes, and 8 for the non-supersymmetric solution, whilst for the Spin(7) example in [6], which is supersymmetric, we have γ = 43 . (The constant c is negative in all cases.) Thus in all these examples we have γ > 0, implying that the breaking of the conformal symmetry of the 3-dimensional field theories is much milder. In fact after dropping the constant 1 in the function H , the solutions are all asymptotically AdS4 × M7 at large r.14 The resolved M2-brane and dyonic string solutions can reduce on the compact level surfaces of the transverse spaces to give rise to domain walls that are asymptotically AdS. The asymptotically AdS geometry is supported, from the viewpoint of the dimensionally-reduced theory, by a non-trivial (and possibly massive) scalar potential that has a fixed point. Thus these geometries describe the renormalisation group flows of the corresponding dual field theories. However, they are very different from those associated with continuous distributed brane configurations [35–41]. Notably, there are fewer supersymmetries in our resolved brane solutions than there are in the distributed brane solutions, which do not break further supersymmetry. Furthermore, the solutions we obtained in this paper are completely free of singularities, while the distributed branes in general have singularities, including naked ones. Finally, while the distributed brane configurations are naturally dual to the Coulomb branch of the corresponding dual field theory, the resolved M2-branes we obtained here, which are coincident rather than distributed, are related to the Higgs branch. 14 Similarly, the resolved dyonic string using the Eguchi-Hanson metric, which was constructed in [6], has H ∼ c0 + Q ρ −2 − c ρ −6 + · · · , in terms of large proper distance ρ. As a consequence, the solution with c0 = 0 is also asymptotically AdS3 [6].

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In [6], a second deformed M2-brane with Spin(7) holonomy supported by a harmonic 4-form of the opposite duality was also explicitly constructed. In this case the 4-form is non-normalisable at large r, and as a consequence, the modification to the function H in (7.1) has a negative value of γ , namely γ = − 43 . Thus unlike the deformed M2-branes we discussed above, this solution will not approach AdS4 spacetime, and the corresponding three-dimensional field theory dual would have no conformal symmetry. An analogous solution with marginally non-normalisable large-distance behaviour appears to be absent for the dyonic string with an Eguchi-Hanson transverse space, which is perhaps consistent with the more central rˆole of conformal symmetry in two dimensional field theories. In general a deformed p-brane solution has a reduced number of supersymmetries, or none at all.15 In order for the solution to be free of (naked) singularities, the relevant harmonic form has to be normalisable at small proper distance. If the harmonic form is also normalisable at large proper distance, the solution may become asymptotically AdS in the decoupling limit, describing the renormalisation group flow of the Higgs branch of the corresponding less-supersymmetric dual conformal field theory. If, on the other hand, the harmonic form is non-normalisable at large distance, then the correction terms to the function H will break the AdS structure completely, and the dual field theory will have no conformal symmetry. There are clearly open avenues to be investigated along the formal directions, by constructing harmonic forms not only in the middle dimension, and for other types of Ricci-flat even-dimensional manifolds, such as hyper-K¨ahler ones, as well as odddimensional ones. In particular, the construction of harmonic forms in other than the middle-dimension may prove to be useful in the study of a larger class of deformed branes, thus providing gravity dual candidates for a larger class of models. Another intriguing question relates to the many exact Ricci-flat metrics that we obtained in this paper. It would be interesting to see how these are related to the general investigation of the integrability of the Einstein equations for cohomogeneity one metrics contained in [26]. Acknowledgements. We are grateful to Michael Atiyah and Nigel Hitchin for useful discussions on L2 harmonic forms, and to Joe Polchinski for discussions about the resolved D3-brane of [5]. We also thank Igor Klebanov for helpful discussions, which, together with enlightening material in [42], have clarified the precise interpretation of fractional and deformed branes. Accordingly, we have adjusted the terminology somewhat from an earlier version of this paper. M.C. is grateful to the Rutgers High Energy Theory Group for support and hospitality. M.C. is supported in part by DOE grant DE-FG02-95ER40893 and NATO grant 976951; H.L. is supported in full by DOE grant DE-FG02-95ER40899; C.N.P. is supported in part by DOE DE-FG03-95ER40917. The work of M.C., G.W.G. and C.N.P. was supported in part by ´ the programme Supergravity, Superstrings and M-theory of the Centre Emile Borel of the Institut Henri Poincar´e, Paris (UMS 839-CNRS/UPMC).

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4. Gubser, S.S.: Supersymmetry and F -theory realization of the deformed conifold with three-form flux. hep-th/0010010 5. Pando Zayas, L.A., Tseytlin, A.A.: 3-branes on a resolved conifold. JHEP 0011, 028 (2000) [hep-th/0010088] 6. Cvetiˇc, M. L¨u, H., Pope, C.N.: Brane resolution through transgression. Nucl. Phys. B600, 103–132 (2001) [hep-th/0011023] 7. Bigazzi, F., Giradello, L., Zaffaroni, A.: A note on regular type 0 solutions and confining gauge theories. Nucl. Phys. B598, 530–542 (2001) [hep-th/0011041] 8. Bertolini, M., Di Vecchia, P., Frau, M., Lerda, A. Marotta, R. Pesando, I.: Fractional D-branes and their gauge duals. JHEP 0102, 014 (2001) [hep-th/0011077] 9. Hawking, S.W., Taylor-Robinson, M.M.: Bulk charges in eleven dimensions. Phys. Rev. D58, 025006 (1998) [hep-th/9711042] 10. Stenzel, M.B.: Ricci-flat metrics on the complexification of a compact rank one symmetric space. Manuscripta Mathematica 80, 151 (1993) 11. Eguchi, T., Hanson, A.J.: Asymptotically flat self-dual solutions to Euclidean gravity. Phys. Lett. B74, 249 (1978) 12. Cvetiˇc, M., Gibbons, G.W., L¨u, H. Pope, C.N.: Supersymmetric non-singular fractional D2-branes and NS-NS 2-branes. Nucl. Phys. B606, 18–44 (2001) [hep-th/0101096] 13. Candelas, P., de la Ossa, X.C.: Comments on conifolds. Nucl. Phys. B342, 246 (1990) 14. Gibbons, G.W., Pope, C.N.: The positive action conjecture and asymptotically Euclidean metrics in quantum gravity. Commun. Math. Phys. 66, 267 (1979) 15. Segal, G.B., Selby, A.: The cohomology of the space of magnetic monopoles. Commun. Math. Phys. 177, 775 (1996) 16. de Rham, G.: Differential Manifolds. Springer-Verlag, Berlin, Heidelberg, New York (1988) 17. Hitchin, N.J.: L2 cohomology of hyperk¨ahler quotients. math.DG/9909002 18. Etesi, G., Hausel, T.: Geometric interpretation of Schwarzschild instantons. J. Geom. Phys. 37, 126–136 (2001) [hep-th/0003239] 19. Atiyah, M.F., Patodi, V.K. Singer, I.M.: Spectral Assymetry and Riemannnian Geometry I. Math. Proc. Camb. Phil. Soc. 77, 43 (1975) 20. Dancer, A.S., Strachan, I.A.B.: Einstein metrics of cohomogeneity one. Unpublished preprint 21. Duff, M.J., Evans, J.M., Khuri, R.R., Lu, J.X., Minasian, R.: The octonionic membrane. Phys. Lett. B412, 281 (1997) [hep-th/9706124] 22. Greene, B.R., Schalm, K., Shiu, G.: Warped compactifications in M and F theory. Nucl. Phys. B584, 480 (2000) [hep-th/0004103] 23. Becker, K., Becker, M.: Compactifying M-theory to four dimensions. JHEP 0011, 029 (2000) [hepth/0010282] 24. Becker, K.: A note on compactifications on Spin(7)-holonomy manifolds. JHEP 0105, 003 (2001) [hep-th/0011114] 25. Dancer, A.S., Wang, M.Y.: K¨ahler-Einstein metrics of cohomogeneity one. Math. Ann. 312, 503 (1998) 26. Dancer, A.S., Wang, M.Y.: The comhogeneity one Einstein equations from the Hamiltonian viewpoint. J. Reine Angew. Math. 524, 97 (2000) 27. Hoxha, P., Martinez-Acosta, R.R., Pope, C.N.: Kaluza-Klein consistency, Killing vectors and K¨ahler spaces. Class. Quant. Grav. 17, 4207 (2000) [hep-th/0005172] 28. Page, D.N., Pope, C.N.: Stability analysis of compactifications of D = 11 supergravity with SU (3)× SU (2) × U (1) symmetry. Phys. Lett. B145, 337 (1984) 29. Berard-Bergery, L.: Quelques exemples de varietes riemanniennes completes non compactes a courbure de Ricci positive. C.R. Acad. Sci., Paris, Ser. I 302, 159 (1986). 30. Page, D.N., Pope, C.N.: Inhomogeneous Einstein metrics on complex line bundles. Class. Quantum Grav. 4, 213 (1987) 31. Schwarz, J.H.: Covariant Field Equations of Chiral N = 2 D = 10 Supergravity. Nucl. Phys. B226, 269 (1983) 32. Pando Zayas, L.A., Tseytlin, A.A.: 3-branes on spaces with R × S 2 × S 3 topology. [hep-th/0101043] 33. Bryant, R.L., Salamon, S.: On the construction of some complete metrics with exceptional holonomy. Duke Math. J. 58, 829 (1989) 34. Gibbons, G.W., Page, D.N., Pope, C.N.: Einstein metrics on S 3 , R3 and R4 bundles. Commun. Math. Phys. 127, 529 (1990) 35. Kraus, P., Larsen, F., Trivedi, S.P.: The Coulomb branch of gauge theory from rotating branes. JHEP 9903, 003 (1999) [hep-th/9811120] 36. Freedman, D.Z., Gubser, S.S., Pilch, K., Warner, N.P.: Continuous distributions of D3-branes and gauged supergravity. JHEP 0007, 038 (2000) [hep-th/9906194]

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Commun. Math. Phys. 232, 501–534 (2003) Digital Object Identifier (DOI) 10.1007/s00220-002-0741-0

Communications in

Mathematical Physics

Dimension and Dynamical Entropy for Metrized C ∗ -Algebras David Kerr Graduate School of Mathematical Sciences, University of Tokyo, 3-8-1 Komaba, Meguro-ku, Tokyo 153-8914, Japan. E-mail:[email protected] Received: 13 February 2002 / Accepted: 20 August 2002 Published online: 22 November 2002 – © Springer-Verlag 2002

Abstract: We introduce notions of dimension and dynamical entropy for unital C ∗ -algebras “metrized” by means of c Lip-norms, which are complex-scalar versions of the Lip-norms constitutive of Rieffel’s compact quantum metric spaces. Our examples involve the UHF algebras Mp∞ and noncommutative tori. In particular we show that the entropy of a noncommutative toral automorphism with respect to the canonical c Lip-norm coincides with the topological entropy of its commutative analogue. 1. Introduction The idea of a noncommutative metric space was introduced by Connes [5–7] who showed in a noncommutative-geometric context that a Dirac operator gives rise to a metric on the state space of the associated C ∗ -algebra. The question of when the topology thus obtained agrees with the weak∗ topology was pursued by Rieffel [25, 26], whose line of investigation led to the notion of a quantum metric space defined by specifying a Lip-norm on an order-unit space [27]. This definition includes Lipschitz seminorms on functions over compact metric spaces and more generally applies to unital C ∗ -algebras, the subspaces of self-adjoint elements of which form important examples of order-unit spaces. We would like to investigate here the structures which arise by essentially specializing and complexifying Lip-norms to obtain what we call “c Lip-norms” on unital C ∗ -algebras. We thereby propose a notion of dimension for c Lip-normed unital C ∗ -algebras, along with two dynamical entropies (the second a measure-theoretic version of the first) which operate within the restricted domain of c Lip-norms satisfying the Leibniz rule (our version of noncommutative metrics). Means for defining dimension appeared within Rieffel’s work on quantum GromovHausdorff distance in [27], where it is pointed out that Definition 13.4 therein gives rise to possible “quantum” versions of Kolmogorov ε-entropy. We take here a different approach which has its origin in Rieffel’s prior study of Lip-norms in [25, 26], where the total boundedness of the set of elements of Lip-norm and order-unit norm no greater

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than 1 was shown to be a fundamental property. This total boundedness leads us to a definition of dimension using approximation by linear subspaces (Sect. 3). This also makes sense for order-unit spaces, but we will concentrate on C ∗ -algebras, the particular geometry of which will play a fundamental role in our examples, which involve the UHF algebras Mp∞ and noncommutative tori (Sect. 4). We will also show that for usual Lipschitz seminorms we recover the Kolmogorov dimension (Proposition 3.9). We also use approximation by linear subspaces to define our two dynamical “product” entropies (Sect. 5). The definitions formally echo that of Voiculescu-Brown approximation entropy [33, 4], but here the algebraic structure enters in a very different way. One drawback of the Voiculescu-Brown entropy as a noncommutative invariant is the difficulty of obtaining nonzero lower bounds (however frequently the entropy is in fact positive) in systems in which the dynamical growth is not ultimately registered in algebraically or statistically commutative structures. We have in mind our main examples, the noncommutative toral automorphisms. For these we only have partial information about the Voiculescu-Brown entropy in the nonrational case (see [33, Sect. 5] and the discussion in the following paragraph), and even deciding when the entropy is positive is a problem (in the rational case, i.e., when the rotation angles with respect to pairs of canonical unitaries are all rational, we obtain the corresponding classical value, as follows from the upper bound established in [33, Sect. 5] along with the fact that the corresponding commutative toral automorphism sits as a subsystem, so that we can apply monotonicity). We show that for general noncommutative toral automorphisms the product entropy relative to the canonical “metric” coincides with the topological entropy of the corresponding toral homeomorphism (Sect. 7). In analogy with the relation between the discrete Abelian group entropy of a discrete Abelian group automorphism and the topological entropy of its dual [23], product entropy (which is an analytic version of discrete Abelian group entropy) may roughly be thought of as a “dual” counterpart of Voiculescu-Brown entropy, as illustrated by the key role played by unitaries in obtaining lower bounds for the former. When passing from the commutative to the noncommutative in an example like the torus, the “dual” unitary description persists (ensuring a metric rigidity that facilitates computations) while the underlying space and the transparency of the complete order structure vanish. The shift on Mp∞ , on the other hand, is equally amenable to analysis from the canonical unitary and complete order viewpoints due to the tensor product structure, and its value can be precisely calculated for both product entropy (Sect. 6) and Voiculescu-Brown entropy [33, Prop. 4.7] (see also [9, 33, 32, 13, 1] for computations with respect to other entropies which we will discuss below). Since we are not dealing with discrete entities as in the discrete Abelian group entropy setting, product entropies will not be C ∗ -algebraic conjugacy invariants, but rather bi-Lipschitz C ∗ -algebraic conjugacy invariants (see Definition 2.8). In particular, if we consider c Lip-norms arising via the ergodic action of a compact group G equipped with a length function (see Example 2.13), the entropies will be “G-C ∗ -algebraic” invariants, that is, they will be invariant under C ∗ -algebraic conjugacies respecting the given group actions. To put this in context, we first point out that there have been two basic approaches to developing C ∗ -algebraic and von Neumann algebraic dynamical invariants which extend classical entropies. While the definitions of Voiculescu [33] and Brown [4] are based on local approximation, the measure-theoretic Connes-Narnhofer-Thirring (CNT) entropy [8] (a generalization of Connes-Størmer entropy [9]) and Sauvageot-Thouvenot entropy [29] take a physical observable viewpoint and are defined via the notions of an Abelian model and a stationary coupling with an Abelian system, respectively (see [31] for a survey). Because of the role played by Abelian systems in their respective definitions, the CNT and Sauvageot-Thouvenot entropies (which are known to

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coincide on nuclear C ∗ -algebras [29, Prop. 4.1]) function most usefully as invariants in the asymptotically Abelian situation. For instance, their common value for noncommutative 2-toral automorphisms with respect to the canonical tracial state is zero for a set of rotation parameters of full Lebesgue measure [20], while for rational parameters the corresponding classical value is obtained [16] and for the countable set of irrational rotation parameters for which the system is asymptotically Abelian (at least when restricted to a nontrivial invariant C ∗ -subalgebra generated by a pair of products of powers of the canonical unitaries) the value is positive when the associated matrix is hyperbolic (see [31, Chap. 9]) (in this case the Voiculescu-Brown entropy is thus also positive by [33, Prop. 4.6]). Other entropies which are not C ∗ -algebraic or von Neumann algebraic dynamical invariants have been introduced in [14, 32, 1]. The definitions of [14, 32] take a noncommutative open cover approach and hence are difficult to compute for examples like the noncommutative toral automorphisms (see the discussion in the last section of [14]). In [2] the Alicki-Fannes entropy [1] for general noncommutative 2-toral automorphisms was shown to coincide with the corresponding classical value if the dense algebra generated by the canonical unitaries is taken as the special set required by the definition. What is particular about the product entropies is that, from the perspective of noncommutative geometry as exemplified in noncommutative tori [24], they provide computable quantities which reflect the metric rigidity but require no additional structure to function (i.e., they are “metric” dynamical invariants). The organization of the paper is as follows. In Sect. 2 we recall Rieffel’s definition of a compact quantum metric space, and with this motivation then introduce c Lip-norms and the relevant maps for c Lip-normed unital C ∗ -algebras, after which we examine some examples. In Sect. 3 we introduce metric dimension and establish some properties, including its coincidence with Kolmogorov dimension for usual Lipschitz seminorms. Section 4 is subdivided into two subsections in which we compute the metric dimension for examples arising from compact group actions on the UHF algebras Mp∞ and noncommutative tori, respectively. The two subsections of Sect. 5 are devoted to introducing the two respective product entropies and recording some properties, and in Sects. 6 and 7 we carry out computations for the shift on Mp∞ and automorphisms of noncommutative tori, respectively. In this paper we will be working exclusively with unital (i.e., “compact”) C ∗ -algebras as generally indicated. For a unital C ∗ -algebra A we denote by 1 its unit, by S(A) its state space, and by Asa the real vector space of self-adjoint elements of A. Other general notation is introduced in Notation 2.2, 3.1, and 5.1. 2. c Lip-Norms on Unital C ∗ -Algebras The context for our definitions of dimension and dynamical entropy will essentially be a specialization of Rieffel’s notion of a compact quantum metric space to the complexscalared domain of C ∗ -algebras. A compact quantum metric space is defined by specifying a Lip-norm on an order-unit space (see below), and this has a natural self-adjoint complex-scalared interpretation on a unital C ∗ -algebra in what will call a “c Lip-norm” (Definition 2.3). In fact c Lip-norms will make sense in more general complex-scalared situations (e.g., operator systems), as will our definition of dimension (Definition 3.3), but we will stick to C ∗ -algebras as these will constitute our examples of interest and multiplication will ultimately enter the picture when we come to dynamical entropy, for which the Leibniz rule will play an important role.

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We begin by recalling from [27] the definition of a compact quantum metric space. Recall that an order-unit space is a pair (A, e) consisting of a real partially ordered vector space A with a distinguished element e, called the order unit, such that, for each a ∈ A, (1) there exists an r ∈ R with a ≤ re, and (2) if a ≤ re for all r ∈ R>0 then a ≤ 0. An order-unit space is a normed vector space under the norm a = inf{r ∈ R : −re ≤ a ≤ re}, from which we can recover the order via the fact that 0 ≤ a ≤ e if and only if a ≤ 1 and e − a ≤ 1. A state on an order-unit space (A, e) is a norm-bounded linear functional on A whose dual norm and value on e are both 1 (which automatically implies positivity). The state space of A is denoted by S(A). An important and motivating example of an order-unit space is provided by the space of self-adjoint elements of a unital C ∗ -algebra. In fact every order-unit space is isomorphic to some order-unit space of self-adjoint operators on a Hilbert space (see [27, Appendix 2]). Via Kadison’s function representation we also see that order-unit spaces are precisely, up to isomorphism, the dense unital subspaces of spaces of affine functions over compact convex subsets of topological vector spaces (see [26, Sect. 1]). Definition 2.1 ([27, Defs. 2.1 and 2.2]). Let A be an order-unit space. A Lip-norm on A is a seminorm L on A such that (1) for all a ∈ A we have L(a) = 0 if and only if a ∈ Re, and (2) the metric ρL defined on the state space S(A) by ρL (µ, ν) = sup{|µ(a) − ν(a)| : a ∈ A and L(a) ≤ 1} induces the weak∗ topology. A compact quantum metric space is a pair (A, L) consisting of an order-unit space A with a Lip-norm L. As mentioned above, the subspace Asa of self-adjoint elements in a unital C ∗ -algebra A forms an order-unit space, and so we can specialize Rieffel’s definition in a more or less straightforward way to the C ∗ -algebraic context. We would like, however, our “Lip-norm” to be meaningfully defined on the C ∗ -algebra A as a vector space over the complex numbers. Such a “Lip-norm” should be invariant under taking adjoints, and thus, after introducing some notation, we make the following definition, which seems reasonable in view of Proposition 2.4. Notation 2.2. Let L be a seminorm on the unital C ∗ -algebra A which is permitted to take the value +∞. We denote the sets {a ∈ A : L(a) < ∞} and {a ∈ A : L(a) ≤ r} (for a given r > 0) by L and Lr (or in some cases for clarity by LA and LA r ), respectively. For r > 0 we denote by Ar the norm ball {a ∈ A : a ≤ r}. We write ρL to refer to the semi-metric defined on the state space S(A) by ρL (σ, ω) = sup |σ (a) − ω(a)| a∈L1

for all σ, ω ∈ S(A). We write diam(S(A)) to mean the diameter of S(A) with respect to the metric ρL . We say that L separates S(A) if for every pair σ, ω of distinct states on A there is an a ∈ L such that σ (a) = ω(a), which is equivalent to ρL being a metric.

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Definition 2.3. By a c Lip-norm on a unital C ∗ -algebra A we mean a seminorm L on A, possibly taking the value +∞, such that (i) L(a ∗ ) = L(a) for all a ∈ A (adjoint invariance), (ii) for all a ∈ A we have L(a) = 0 if and only if a ∈ C1 (ergodicity), (iii) L separates S(A) and the metric ρL induces the weak∗ topology on S(A). Proposition 2.4. Let L be a c Lip-norm on a unital C ∗ -algebra A. Then the restriction L of L to the order-unit space L ∩ Asa is a Lip-norm, and the restriction map from S(A) to S(L ∩ Asa ) is a weak∗ homeomorphism which is isometric relative to the respective metrics ρL and ρL . Also, if L is any adjoint-invariant seminorm on A, possibly taking the value +∞, such that the restriction L to L ∩ Asa is a Lip-norm which separates S(Asa ) ∼ = S(A), then L is a c Lip-norm, and the restriction from S(A) to S(L ∩ Asa ) is a weak∗ homeomorphism which is isometric relative to the respective metrics ρL and ρL . Proof. The proposition is a consequence of the fact that if L is an adjoint-invariant seminorm then for any σ, ω ∈ S(A) the suprema of |σ (a) − ω(a)| over the respective sets L1 and L1 ∩ Asa are the same, as shown in the discussion prior to Definition 2.1 in [27]. The second statement of the proposition also requires the fact that the ergodicity of L (condition (1) of Definition 2.1) implies the ergodicity of L, which can be seen by noting that if a ∈ A and L(a) < ∞ then setting Re(a) = (a + a ∗ )/2 and Im(a) = (a − a ∗ )/2i (the real and imaginary parts of a) we have L (Re(a)) = 0 and L (Im(a)) = 0 by adjoint invariance, so that Re(a), Im(a) ∈ R1 by condition (1) of Definition 2.1, and hence a = Re(a) + iIm(a) ∈ C1.

 The following proposition follows immediately from Theorem 1.8 of [25] (note that the remark following Condition 1.5 therein shows that this condition holds in our case). Condition (4) in the proposition statement will provide the basis for our definitions of dimension and dynamical entropy. Proposition 2.5. A seminorm L on a unital C ∗ -algebra A, possibly taking the value +∞, is a c Lip-norm if and only if it separates S(A) and satisfies (1) (2) (3) (4)

L(a ∗ ) = L(a) for all a ∈ A, for all a ∈ A we have L(a) = 0 if and only if a ∈ C1, sup{|σ (a) − ω(a)| : σ, ω ∈ S(A) and a ∈ L1 } < ∞, and the set L1 ∩ A1 is totally bounded in A for  · .

When we come to dynamical entropy, c Lip-norms satisfying the Leibniz rule will be of central importance, and so we also make the following definition, which we may think of as describing one possible noncommutative analogue of a compact metric space (cf. Example 2.12). Definition 2.6. We say that a c Lip-norm L on a unital C ∗ -algebra A is a Leibniz c Lipnorm if it satisfies the Leibniz rule L(ab) ≤ L(a)b + aL(b) for all a, b ∈ L.

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Although we do not make lower semicontinuity a general assumption for c Lip-norms, it will typically hold in our examples, and has the advantage that we can recover the restriction of L to Asa in a straightforward manner from ρL , as shown by the following proposition, which is a consequence of [26, Thm. 4.1] and Proposition 2.4. Proposition 2.7. Let L be a lower semicontinuous c Lip-norm on a unital C ∗ -algebra A. Then for all a ∈ Asa we have L(a) = sup {|σ (a) − ω(a)|/ρL (σ, ω) : σ, ω ∈ S(A) and σ = ω} . As for metric spaces, the essential maps in our c Lip-norm context are ones satisfying a Lipschitz condition, which puts a uniform bound on the amount of “stretching” as formalized in the following definition, for which we will adopt the conventional metric space terminology (see [37, Def. 1.2.1]). Definition 2.8. Let A and B be unital C ∗ -algebras with c Lip-norms LA and LB , respectively. A positive unital (linear) map φ : A → B is said to be Lipschitz if there exists a λ ≥ 0 such that LB (φ(a)) ≤ λLA (a) for all a ∈ LA . The least such λ is called the Lipschitz number of φ. When φ is invertible and both φ and φ −1 are Lipschitz positive we say that φ is bi-Lipschitz. If LB (φ(a)) = LA (a) for all a ∈ A then we say that φ is isometric. The collection of bi-Lipschitz ∗ -automorphisms of A will be denoted by AutL (A). The category of interest for dimension will be that of c Lip-normed unital C ∗ -algebras and Lipschitz positive unital maps, with the bi-Lipschitz positive unital maps forming the categorical isomorphisms. For entropy we will incorporate the algebraic structure in the definitions so that we will want our positive unital maps to be in fact ∗ homomorphisms. We remark that, as for usual metric spaces, the isometric maps are too rigid to be usefully considered as the categorical isomorphisms, and that our dimension and dynamical entropies will indeed be invariant under general bi-Lipschitz positive unital maps and bi-Lipschitz ∗ -isomorphisms, respectively. We also remark that positive unital maps are C ∗ -norm contractive [28, Cor. 1], and hence any bi-Lipschitz positive unital map is C ∗ -norm isometric. The following pair of propositions capture facts pertaining to Lipschitz maps. The first one is clear. Proposition 2.9. Let A, B, and C be unital C ∗ -algebras with respective c Lip-norms LA , LB , and LC . If φ : A → B and ψ : B → C are Lipschitz positive unital maps with Lipschitz numbers λ and ζ , respectively, then ψ ◦ φ is Lipschitz with Lipschitz number bounded by the product λζ . Lemma 2.10. If L is a c Lip-norm on a unital C ∗ -algebra A and a ∈ L ∩ Asa then denoting by s(a) the infimum of the spectrum of a we have a − s(a)1 ≤ L(a)diam(S(A)), and hence for any σ, ω ∈ S(A) we have ρL (σ, ω) = sup{|σ (a) − ω(a)| : a ∈ Asa , L(a) ≤ 1, and a ≤ diam(S(A))}

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Proof. Let a be an element of L ∩ Asa and s(a) the infimum of its spectrum. Then there are σ, ω ∈ S(A) such that σ (a − s(a)1) = a − s(a)1 and ω(a) = s(a). We then have a − s(a)1 = |σ (a − s(a)1) − ω(a − s(a)1)| = |σ (a) − ω(a)| ≤ L(a)diam(S(A)). The second statement of the lemma follows by noting that, for any σ, ω ∈ S(A), ρL (σ, ω) = sup {|σ (a) − ω(a)| : a ∈ Asa and L(a) ≤ 1} (see the first sentence in the proof of Proposition 2.5), while if L(a) ≤ 1 then a − s(a)1 ≤ diam(S(A)) from above, L(a − s(a)1) = L(a) ≤ 1 by the ergodicity of L, and |σ (a − s(a)1) − ω(a − s(a)1)| = |σ (a) − ω(a)|. 

Proposition 2.11. If L is a lower semicontinuous Leibniz c Lip-norm on a unital C ∗ algebra A and u ∈ L is a unitary then Adu is bi-Lipschitz, and the Lipschitz numbers of Adu and its inverse are bounded by 2(1 + 2L(u)diam(S(A))). Proof. By the Leibniz rule and the adjoint-invariance of L, for any a ∈ L we have

 L uau∗ ≤ L(u)a + L(a) + aL u∗ = L(a) + 2aL(u). For any σ, ω ∈ S(A) we therefore have, using Lemma 2.10 for the first equality, ρL (σ ◦ Adu, ω ◦ Adu) 

= sup{|σ uau∗ − ω uau∗ | : a ∈ Asa , L(a) ≤ 1 and a ≤ diam(S(A))} ≤ sup{|σ (a) − ω(a)| : a ∈ Asa and L(a) ≤ 1 + 2L(u)diam(S(A))} ≤ (1 + 2L(u)diam(S(A))) sup{|σ (a) − ω(a)| : a ∈ Asa and L(a) ≤ 1} = (1 + 2L(u)diam(S(A)))ρL (σ, ω). Since L is lower semicontinuous we can thus appeal to Proposition 2.7 to obtain, for any a ∈ L ∩ Asa ,
 L uau∗ =

|(σ ◦ Adu)(a) − (ω ◦ Adu)(a)| ρL (σ, ω) σ,ω∈S(A)

≤

|(σ ◦ Adu)(a) − (ω ◦ Adu)(a)| ρL (σ ◦ Adu, ω ◦ Adu) σ,ω∈S(A)

sup

sup

×

ρL (σ ◦ Adu, ω ◦ Adu) ρL (σ, ω) σ,ω∈S(A) sup

= L(a)(1 + 2L(u)diam(S(A))).

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Thus, for any a ∈ L, setting Re(a) = (a + a ∗ )/2 and Im(a) = (a − a ∗ )/2i we have L(uau∗ ) ≤ L(uRe(a)u∗ ) + L(uIm(a)u∗ ) ≤ (L(Re(a)) + L(Im(a)))(1 + 2L(u)diam(S(A))) ≤ 2L(a)(1 + 2L(u)diam(S(A))) using adjoint invariance. The same argument applies to (Adu)−1 = Adu∗ , and so we obtain the result.

 We conclude this section with some examples of c Lip-norms. Example 2.12 (commutative C ∗ -algebras). For a compact metric space (X, d) we define the Lipschitz seminorm Ld on C(X) by Ld (f ) = sup{|f (x) − f (y)|/d(x, y) : x, y ∈ X and x = y}, from which we can recover d via the formula d(x, y) = sup {|f (x) − f (y)| : f ∈ C(X) and Ld (f ) ≤ 1} . The seminorm Ld is an example of a Leibniz c Lip-norm. For a reference on Lipschitz seminorms and the associated Lipschitz algebras see [37]. Example 2.13 (ergodic compact group actions). For us the most important examples of compact noncommutative metric spaces will be those which arise from ergodic actions of compact groups, as studied by Rieffel in [25]. Suppose γ is an ergodic action of a compact group G on a unital C ∗ -algebra A. Let e denote the identity element of G. We assume that G is equipped with a length function , that is, a continuous function  : G → R≥0 such that, for all g, h ∈ G, (1) (gh) ≤  (g) + (h),
(2)  g −1 = (g), and (3) (g) = 0 if and only if g = e. The length function  and the group action γ combine to produce the seminorm L on A defined by   γg (a) − a  , L(a) = sup (g) g∈G\{e} which is evidently adjoint-invariant. It is easily verified that L(a) = 0 if and only if a ∈ C1. Also, by [25, Thm. 2.3] the metric ρL induces the weak∗ topology on S(A), and the Leibniz rule is easily checked, so that L is Leibniz c Lip-norm. Example 2.14 (quotients). Let A and B be unital C ∗ -algebras and let φ : A → B be a surjective unital positive linear map. For instance, φ may be a surjective unital ∗ -homomorphism or a conditional expectation, as will be the case in our applications. Let L be a c Lip-norm on A. Then L induces a c Lip-norm LB on B via the prescription LB (b) = inf{L(a) : a ∈ A and φ(a) = b} for all b ∈ B. This is the analogue of restricting a metric to a subspace. To see that LB is indeed a c Lip-norm we observe that the restriction of φ to LA ∩ Asa yields a

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surjective morphism LA ∩
Asa → LB∩ Bsa of order-unit spaces (for surjectivity note that if φ(a) ∈ Bsa then φ 21 (a + a ∗ ) = φ(a)) so that we may appeal to [27, Prop. 3.1] to conclude that the restriction of LB to L ∩ Bsa is a Lip-norm, so that LB is a c Lip-norm
by  Proposition 2.4 (note that the restriction of LB to L ∩ Bsa separates S(B), since φ LA = LB and the restriction of LA to L ∩ Asa separates S(A) by the first part of Proposition 2.4). 3. Dimension for c Lip-Normed Unital C ∗ -Algebras Let A be a unital C ∗ -algebra with c Lip-norm L. Recall from Notation 2.2 our convention that L and Lr refer to the sets {a ∈ A : L(a) < ∞} and {a ∈ A : L(a) ≤ r}, respectively. The following notation will be extensively used for the remainder of the article. Notation 3.1. For a normed linear space (X,  · ) (which in our case will either be a C ∗ -algebra or a Hilbert space) we will denote by F(X) the collection of its finite-dimensional subspaces, and if Y and Z are subsets of X and δ > 0 we will write Y ⊂δ Z, and say that Z approximately contains Y to within δ, if for every y ∈ Y there is an x ∈ Z such that y − x < δ. Using dim X to denote the vector space dimension of a subspace X, for any subset Z ⊂ A and δ > 0 we set D(Z, δ) = inf {dim X : X ∈ F(A) and Z ⊂δ X} (or D(Z, δ) = ∞ if the set on the right is empty) and if σ is a state on A then we set Dσ (Z, δ) = inf {dim X : X ∈ F(Hσ ) and πσ (Z)ξσ ⊂δ X} (or Dσ (Z, δ) = ∞ if the set on the right is empty), with πσ : A → B(Hσ ) referring to GNS representation associated to σ , with canonical cyclic vector ξσ . Proposition 3.2. D(L1 , δ) is finite for every δ > 0. Proof. Let a ∈ L, and set Re(a) = (a + a ∗ )/2 and Im(a) = (a − a ∗ )/2i (the real and imaginary parts of a). Let s(Re(a)) and s(Im(a)) be the infima of the spectra of Re(a) and Im(a), respectively. Using Lemma 2.10 and the adjoint invariance of L we have a − (s(Re(a)) + is(Im(a)))1 ≤ Re(a) − s(Re(a))1 + Im(a) − s(Im(a))1 ≤ L(Re(a))diam(S(A)) + L(Im(a))diam(S(A)) ≤ 2L(a)diam(S(A)). Set r = 2 diam(S(A)). Since L1 ∩A1 is totally bounded by Proposition 2.5, so is Lr ∩Ar by a scaling argument. Let δ > 0. Then there is an X ⊂ F(A) which approximately contains Lr ∩ Ar to within δ, and if a ∈ L1 then from above we have a − (s(Re(a)) + is(Im(a)))1 ∈ Lr ∩ Ar , so that there exists an x ∈ X with a − (s(Re(a)) + is(Im(a)))1 − x < δ. But (s(Re(a)) + is(Im(a)))1 − x ∈ span(X ∪ {1}), and so we conclude that L1 ⊂δ span(X ∪ {1}). Hence D(L1 , δ) is finite.



In view of Proposition 3.2 we make the following definition.

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Definition 3.3. We define the metric dimension of A with respect to L by MdimL (A) = lim sup δ→0+

log D (L1 , δ) . log δ −1

We may think of D(L1 , δ) as the δ-entropy of A with respect to L in analogy with Kolmogorov ε-entropy [17], and indeed when L is a Lipschitz seminorm on a compact metric space (X, d) we will recover from MdimL (C(X)) the Kolmogorov dimension (Proposition 3.9). We emphasize that in using D(·, ·) in Definition 3.3 (and also in the definition of entropy in Sect. 5) we are not making any extra geometric assumptions in our finitedimensional approximations by linear subspaces. For example, we are not requiring that these subspaces be images of positive or completely positive maps which are close to the identity on the set in question. In computing lower bounds we are thus left to rely on the Hilbert space geometry implicit in the C ∗ -algebraic structure, making repeated use of Lemma 3.8 below. Proposition 3.4. Let A and B be unital C ∗ -algebras with c Lip-norms LA and LB , respectively. Suppose φ : A → B is a bi-Lipschitz positive unital map. Then MdimLA (A) = MdimLB (B). B Proof. Let λ > 0 be the Lipschitz number of φ. Then φ(LA 1 ) ⊂ Lλ , so that if X ∈ F(B) B and Lλ ⊂δ X then −1 LA 1 ⊂δ φ (X),

since φ is isometric for the C ∗ -norm (see the remark after Definition 2.8). As a consequence     B , δ ≤ D L , δ D LA λ 1 and so MdimLA (A) = lim sup

log D(LA 1 , δ) −1 log δ

≤ lim sup

log D(LB λ , δ) log δ −1

= lim sup

−1 log D(LB log λ−1 δ −1 1 , λ δ) · lim −1 −1 log λ δ δ→0+ log δ −1

= lim sup

−1 log D(LB 1 , λ δ) log λ−1 δ −1

δ→0+

δ→0+

δ→0+

δ→0+

= MdimLB (B). The reverse inequality follows by a symmetric argument. The following is immediate from Definition 3.3.



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Proposition 3.5. Let L and L be c Lip-norms on a unital C ∗ -algebra such that L ≤ L , that is, L(a) ≤ L (a) for all a ∈ A. Then MdimL (A) ≥ MdimL (A). Proposition 3.6. Let A and B be unital C ∗ -algebras, LA a c Lip-norm on A, φ : A → B a surjective positive unital map, and LB the c Lip-norm on B induced from LA by φ. Then MdimLB (B) ≤ MdimLA (A). Proof. Since LB is induced from LA (Example 2.14) for any b ∈ LB 1 there is an a ∈ A with φ(a) = b and L(a) ≤ 2. Thus if X is a linear subspace of A with LA 2 ⊂δ X it follows that LB ⊂ φ(X). Hence δ 1 MdimLB (B) = lim sup

log D(LB 1 , δ) log δ −1

≤ lim sup

log D(LA 2 , δ) −1 log δ

= lim sup

−1 log D(LA log 2δ −1 1 , 2 δ) · lim log 2δ −1 δ→0+ log δ −1

δ→0+

δ→0+

δ→0+

= MdimLA (A). 

Proposition 3.7. Let A and B be unital C ∗ -algebras with c Lip-norms LA and LB , respectively. Let L be a c Lip-norm on A ⊕ B which induces LA and LB via the quotients onto A and B, respectively (see Example 2.14). Then MdimL (A ⊕ B) = max (MdimL (A), MdimL (B)) . Proof. The inequality MdimL (A ⊕ B) ≥ max(MdimL (A), MdimL (B)) follows from Proposition 3.6. To establish the reverse inequality, let δ > 0, and let X ∈ F(A) and A⊕B B then L(a) and L(b) Y ∈ F(B) be such that LA 1 ⊂δ X and L1 ⊂δ Y . If (a, b) ∈ L1 are no greater than 1, and hence there exist x ∈ X and y ∈ Y such that x − a < δ and y − b < δ, so that (x, y) − (a, b) < δ. Thus LA⊕B ⊂δ span ({(x, 0) : x ∈ X} ∪ {(0, y) : y ∈ Y }), 1 and so we infer that

      B D LA⊕B , δ ≤ D LA 1 , δ + D L1 , δ . 1

For each δ > 0 the sum on the right in the above display is bounded by twice the maximum of its two summands, and so

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MdimL (A ⊕ B) = lim sup δ→0+

  log D LA⊕B ,δ 1 log δ −1



 log 2D LA log 2D LB 1 ,δ 1 ,δ ≤ max lim sup , lim sup log δ −1 log δ −1 δ→0+ δ→0+ 

= max(MdimL (A), MdimL (B)). 

As we show in Proposition 3.9 below, if L is a Lipschitz seminorm on a compact metric space (X, d) then MdimL (C(X)) coincides with the Kolmogorov dimension [17, 18], whose definition we recall. Let (X, d) be a compact metric space. A set E ⊂ X is said to be δ-separated if for any distinct x, y ∈ E we have d(x, y) > δ, while a set F ∈ X is said to be δ-spanning if for any x ∈ X there is a y ∈ F such that d(x, y) ≤ δ. We denote by sep(δ, d) the largest cardinality of an δ-separated set and by spn(δ, d) the smallest cardinality of a δ-spanning set. We furthermore denote by N (δ, d) the minimal cardinality of a cover of X by δ-balls. The Kolmogorov dimension of (X, d), which we will denote by Kdimd (X), is the common value of the three expressions lim sup δ→0+

log sep(δ, d) , log δ −1

lim sup δ→0+

log spn(δ, d) , log δ −1

lim sup δ→0+

log N (δ, d) . log δ −1

This also goes by other names in the literature, such as box dimension and limit capacity (see [22, Chap. 2]). We will need the following lemma from [33], which will also be of use later on. Lemma 3.8 ([33, Lemma 7.8]). If B is an orthonormal set of vectors in a Hilbert space H and δ > 0 then inf {dim X : X ∈ F(H) and X ⊂δ B} ≥ (1 − δ 2 )card(B). Proposition 3.9. Let (X, d) be a compact metric space, and let L be the associated Lipschitz seminorm on C(X), that is, L(f ) = sup {|f (x) − f (y)|/d(x, y) : x, y ∈ X and x = y} for all f ∈ C(X). Then MdimL (C(X)) = Kdimd (X). Proof. Let δ > 0 and let U = {B(x1 , δ), . . . , B(xr , δ)} be a cover of X by δ-balls. Let  = {f1 , . . . , fr } be a partition of unity subordinate to U. If f ∈ L1 and x and y are points of X contained in the same member of U, then |f (x) − f (y)| < 2δ. Thus for any x ∈ X we have        f (x) − ≤ f (x )f (x) |f (x) − f (xi )| fi (x) i i     1≤i≤r 1≤i≤r ≤ |f (x) − f (xi )| fi (x) {i:x∈B(xi ,δ)}

< 2δ.

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Thus L1 ⊂2δ span(), and since dim(span()) = card(U) we conclude that MdimL (C(X)) = lim sup δ→0+

log D (L1 , 2δ) ≤ Kdimd (X). log δ −1

To establish the reverse inequality, let δ > 0 and let E = {x1 , . . . , xr } be a δseparated set of maximal cardinality. The idea will be to consider the probability measure µ uniformly supported on E and to construct unitaries in C(X) with sufficiently small Lipschitz seminorm which, when viewed as elements of L2 (X, µ), form an orthonormal basis, so that we can appeal to Lemma 3.8. For each j = 1, . . . , r define the function fj by   fj (x) = max 0, 1 − δ −1 d(x, xj ) for all x ∈ X, and observe that L(fj ) = δ −1 . For each k = 1, . . . , r define the function gk by gk =

n



 j kr −1 fj ,

j =1 −1 where [ · ] means take the fractional part. We then have k ) ≤ δ , as can be seen by al L(g −1 fj (note that the supports of ternatively expressing gk as the join of the functions j kr the fj ’s are pairwise disjoint) and applying the inequality L(f ∨g) ≤ max(L(f ), L(g)) relating L to the lattice structure of real-valued functions on X. For each k = 1, . . . , r set

uk = e2πigk . Repeated application of the Leibniz rule yields, for each n ≥ 1,   n n n j (2πig ) (2π)j (2π )j k j ≤ L L(gk ) ≤ j L(gk ) j! j! j! j =0 j =0 j =0   n−1 j (2π )  L(gk ) = 2π  j! j =0

≤ 2π e L(gk ), 2π



 j

(2πigk ) n and thus, since the sequence converges uniformly to uk , we can j =0 j! n∈N appeal to the lower semicontinuity of L to obtain the estimate

L (uk ) ≤ 2πe2π L(gk ) ≤ 2π e2π δ −1 . Setting U (δ) = {uk : k = 1, . . . , r} and C = 2π e2π , we thus have that the set {C −1 u : u ∈ U (δ)}, which we will simply denote by C −1 U (δ), lies in L1 if δ ≤ C. Next, let µ be the probability measure uniformly supported on E and let πµ : C(X) → B(L2 (X, µ)) be the associated GNS representation, with canonical cyclic vector ξµ . Then, for each k = 1, . . . , r, πµ (uk )ξµ is the unit vector       −1 −1 2 −1 r−1 1, e2πikr , e2πikr , . . . , e2πikr

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under the obvious identification of L2 (X, µ) with Cr which respects the order of the indexing of the points x1 , . . . , xr . Hence we see that the set {πµ (u)ξµ : u ∈ U (δ)} forms an orthonormal basis for L2 (X, µ), and so by Lemma 3.8 we have Dµ (U (δ), 2−1 ) ≥ (1 − 2−2 ) card(U (δ)) = 43 card(E) = 43 sep(δ, d), (for the meaning of Dµ (·, ·) see Notation 3.1). Carrying out the above construction for each δ > 0, we then have MdimL (C(X)) = lim sup

log D(L1 , 2−1 C −1 δ) log 2Cδ −1

= lim sup

log D(L1 , 2−1 C −1 δ) log δ −1

≥ lim sup

log D(C −1 δU (δ), 2−1 C −1 δ) log δ −1

= lim sup

log D(U (δ), 2−1 ) log δ −1

≥ lim sup

log Dµ (U (δ), 2−1 ) log δ −1

≥ lim sup

log 43 sep(δ, d) log δ −1

δ→0+

δ→0+

δ→0+

δ→0+

δ→0+

δ→0+

= Kdimd (X). 

4. Group Actions and Dimension Here we compute the dimension for some examples in which the c Lip-norm is defined by means of an ergodic compact group action. 4.1. The UHF algebra Mp∞ . We consider here the infinite tensor product Mp⊗Z (usually denoted Mp∞ ) of p × p matrix algebras Mp over C with the infinite product of Weyl actions. As shown in [21] there is a unique ergodic action of G = Zp × Zp on a simple C ∗ -algebra up to conjugacy, namely the Weyl action on Mp , defined as follows. Let ρ be the p th root of unity e2πi/p , and consider the unitary u = diag(1, ρ, ρ 2 , . . . , ρ p−1 ) along with the unitary v which has 1’s on the superdiagonal and in the bottom left-hand entry and 0’s elsewhere. Then we have vu = ρuv, and u and v generate Mp C ∗ -algebraically. The Weyl action γ : G → Aut(Mp ) is given by the following specification on the generators u and v: γ(r,s) (u) = ρ r u, γ(r,s) (v) = ρ s v. We may then consider the infinite product action γ ⊗Z of the product group GZ on Mp⊗Z .

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515

Consider the metric on G obtained by viewing G as a subgroup of R2 /Z2 with the metric induced from the Euclidean metric on R2 , and let  be the length function on G defined by taking the distance to 0. Given 0 < λ < 1 we define the length function λ on GZ by 
 |j |
λ (gj , hj )j ∈Z = λ  (gj , hj ) . j ∈Z

We could also define length functions on GZ by using suitable choices of weightings of  on the factors other than the above geometric ones (and in many cases compute the metric dimension as in Proposition 4.1 below), but for simplicity we will restrict our attention to length functions of the form λ . Let L be the c Lip-norm on Mp⊗Z arising from the action γ ⊗Z and the length function λ . Proposition 4.1. We have   4 log p MdimL Mp⊗Z = . log λ−1 Proof. For each n consider the conditional expectation En of Mp⊗Z onto the subalgebra Mp⊗[−n,n] given by

 En (a) =

GZ\[−n,n]

γg⊗Z (a) dg,

where dg is normalized Haar measure on GZ and GZ\[−n,n] is the subgroup of GZ of elements which are the identity at the coordinates in the interval [−n, n]. Then for each a ∈ L we have       ⊗Z En (a) − a =  γ (a) − a dg   Z\[−n,n] g   G    ⊗Z  ≤ γg (a) − a  dg GZ\[−n,n]  ≤ L(a)λ (g) dg GZ\[−n,n] 2λn+1

≤ L(a)

. 1−λ Let δ > 0. If δ is sufficiently small there is an n ∈ N such that 2λn+1 (1 − λ)−1 ≤ δ ≤ 2λn (1 − λ)−1 . Then, in view of the above estimate on En (a) − a when a ∈ L1 , we have that L1 is approximately contained in Mp⊗[−n,n] to within δ. Since Mp⊗[−n,n] has linear dimension p2(2n+1) it follows that log D(L1 , δ) log D(L1 , 2λn+1 (1 − λ)−1 ) ≤ log δ −1 log(2(1 − λ)λ−n ) (4n + 2) log p ≤ log(2(1 − λ)λ−n ) and so

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  log D(L1 , δ) MdimL Mp⊗Z = lim sup log δ −1 n→∞ (4n + 2) log p ≤ lim n→∞ log(2(1 − λ)λ−n ) 4 log p = . log λ−1 To prove the reverse inequality, consider for each n ∈ N the subset  Un = ui−n v j−n ⊗ ui−n+1 v j−n+1 ⊗ · · · ⊗ uin v jn :

 0 ≤ ik , jk ≤ p − 1 for k = −n, . . . , n

of Mp⊗[−n,n] (i.e., all elementary tensors in Mp⊗[−n,n] whose components are Weyl unitaries in the respective copies
of Mp ). It is easily  checked that the c Lip-norm of any  element in Un is bounded by 2 1 + 2 nk=1 λk ≤ (4n + 2)λn . Now the product of any two distinct products of powers of Weyl generators in Mp is zero under evaluation at the unique tracial state τ on Mp⊗Z , as can be seen from the commutation relation between

u and v. Thus, since τ is a tensor product of traces in its restriction to Mp⊗[−n,n] , the product of any two distinct elements of n is zero under evaluation by τ . This implies that πτ (Un )ξτ is an orthonormal set in the GNS representation Hilbert space associated to τ with canonical cyclic vector ξτ , and so by Lemma 3.8 we have     Dτ Un , 2−1 ≥ 1 − 2−1 card (πτ (Un )ξτ ) = 43 p 2(2n+1) . Thus setting

  Wn = (4n + 2)−1 λn w : w ∈ Un

(which is contained in L1 ) we have D(Wn , (4n + 2)−1 λ−n 2−1 ) ≥ Dτ (Wn , (4n + 2)−1 λ−n 2−1 ) ≥ Dτ (Un , 2−1 ) ≥ 43 p 2(2n+1) and so

  log(D(Wn , (4n + 2)−1 λ−n 2−1 ) MdimL Mp⊗Z ≥ lim sup log((4n + 2)−1 λ−n 2−1 ) n→∞ ≥ lim sup n→∞

log 43 + (4n + 2) log p log((4n + 2)−1 2−1 ) + n log λ−1

4 log p = , log λ−1 completing the proof.



Because we have used the canonical unitary description of Mp⊗Z in an essential way, we cannot expect to be able to carry out a computation for much more general types of tensor products by extending the arguments of this subsection, although such a computation would be possible, for example, for tensor products of noncommutative tori, in which case we could incorporate the methods of the next subsection.

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517

4.2. Noncommutative tori. Let ρ : Zp × Zp → T be an antisymmetric bicharacter and for 1 ≤ i, j ≤ k set
 ρij = ρ ei , ej , where {e1 , . . . , ep } is the standard basis for Zp . The universal C ∗ -algebra Aρ generated by unitaries u1 , . . . , up satisfying uj ui = ρij ui uj is referred to as a noncommutative p-torus. Slawny showed in [30] that Aρ is simple if and only if ρ is nondegenerate (meaning that ρ(g, h) = 1 for all h ∈ Zp implies that g = 0), and these two conditions are furthermore equivalent to the existence of a unique tracial state on Aρ (see [11]). Let Aρ be a noncommutative p-torus with generators u1 , . . . , up . There is an ergodic action γ : Tp ∼ = (R/Z)p → Aut(Aρ ) determined by γ(t1 ,... ,tp ) (uj ) = e2πitj uj (see [21]). We will consider the c Lip-norm L arising from the action γ as in Example 2.13, with the length function given by taking the distance to 0 with respect to the metric induced from the Euclidean metric on Rp scaled by 2π (scaling will not affect the value of MdimL (Aρ ) but our choice of length function ensures for convenience that L(uj ) = 1 for each j = 1, . . . , p). We denote by τ the tracial state defined by  
γ(t1 ,... ,tp ) (a) d t1 , . . . , tp τ (a) = Tp

for all a ∈ Aρ , where d(t1 , . . . , tp ) is normalized Haar measure on Tp ∼ = (R/Z)p . p For (n1 , . . . , np ) ∈ N let R(n1 , . . . , np ) denote the set of points (k1 , . . . , kp ) in Zp such that |ki | ≤ ni for i = 1, . . . , p. For each a ∈ Aρ , we define for each (n1 , . . . , np ) ∈ Np the partial Fourier sum   −k k 1 τ aup p · · · u−k uk11 · · · upp s(n1 ,... ,np ) (a) = 1 (k1 ,... ,kp )∈R(n1 ,... ,np )

and for each n ∈ N the Ces`aro mean  σn (a) = 

 s(n1 ,... ,np ) (a)

(n + 1)p .

(n1 ,... ,np )∈R(n,n,... ,n)

Weaver showed in [35, Thm. 22] for the case p = 2 that σn (a) → a in norm for all a ∈ L. To compute MdimL (Aρ ) we will need a handle on the rate of this convergence, and so we have in Lemma 4.3 below an extension to the noncommutative case of a standard result in classical Fourier analysis (see for example [15]). To make the required estimate we will use the expression for σn (a) − a given by the following lemma, which can be proved in the same way as its specialization to the case p = 2, which appears in a more general form in [36] as Lemma 3.1 and is established in the course of the proof of [35, Thm. 22]. Recall the classical Fej´er kernel Kn defined by    n  |k| sin((n + 1)t/2) 2 1 2πikt 1− e = . Kn (t) = n+1 n+1 sin(t/2) k=−n

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Lemma 4.2. If a ∈ Aρ then for all n ∈ N we have 
 γ(t1 ,... ,tp ) (a)Kn (t1 ) · · · Kn (tp ) d t1 , . . . , tp σn (a) = Tp

and a − σn (a) =

p  k−1 k=1 T

 γ(t1 ,... ,tk−1 ,0,... ,0)

T

 (a − γrk (tk ) (a))Kn (tk )dtk

×Kn (t1 ) · · · Kn (tk−1 ) d(t1 , . . . , tk−1 ), with the integrals taken in the Riemann sense and rk (t) denoting the p-tuple which is t at the k th coordinate and 0 elsewhere. Notice that the right-hand expression in the second display of the statement of Lemma 4.2 is a telescoping sum, so that the second display is an immediate consequence of the first display in view of the fact that the integral of the Fej´er kernel over T is 1. Note also that the first display shows that σn (a) ≤ a for all n ∈ N and a ∈ Aρ , a fact which will be of use in the proof of Proposition 7.4. Lemma 4.3. If a ∈ LAρ then there is a C > 0 such that a − σn (a) < L(a)C

log n n

for all n ∈ N. Proof. It suffices to show that each of the summands on the right-hand side of the second display of Lemma 4.2 is bounded by Mn−1 log n for some M > 0 and all n ∈ N. We thus observe that if 1 ≤ k ≤ p then, with rk (t) denoting the p-tuple which is t at the k th coordinate and 0 elsewhere,    



Tk−1

γ(t1 ,... ,tk−1 ,0,... ,0)

T

 (a − γrk (tk ) (a))Kn (tk )dtk

  × Kn (t1 ) · · · Kn (tk−1 ) d(t1 , . . . , tk−1 )        (a − γr (t ) (a))Kn (tk )dtk  Kn (t1 ) · · · Kn (tk−1 ) d(t1 , . . . , tk−1 ) ≤ k k   k−1 T T ≤ a − γrk (tk ) (a)Kn (tk ) dtk T  ≤ L(a) |t|Kn (t)dt. T

 Estimating the integral T |t|Kn (t) dt is a standard exercise from classical Fourier analysis (see [15, Exercise 3.1]): using the fact that | sin(π t)| > 2|t| and hence   1 Kn (t) ≤ min n + 1, 4(n + 1)t 2 for 0 < |t| < 21 , we readily obtain, for the integral of |t|Kn (t) over each of the intervals 1 1 1 1 [− 21 , − 2(n+1) ], [− 2(n+1) , 2(n+1) ], and [ 2(n+1) , 21 ], an upper bound of n−1 log n times some constant independent of n, yielding the result.



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Proposition 4.4. We have MdimL (Aρ ) = p. Proof. Let δ > 0, and assume δ is sufficiently small so that there is an n ∈ N such that C(n + 1)−1 log(n + 1) ≤ δ ≤ Cn−1 log n. Lemma 4.3 then yields log D(L1 , δ) log D(L1 , Cn−1 log n) ≤ log δ −1 log(C(n + 1)−1 log(n + 1))−1 p log(2n + 1) ≤ log(Cn−1 log n)−1 so that MdimL (Aρ ) ≤ lim sup n→∞

p log(2n + 1) = p. log(Cn−1 log n)−1

To prove the reverse inequality, for each n ∈ N consider the set   k Un = uk11 uk22 · · · upp : |ki | ≤ n for i = 1, . . . , p of unitaries in Aρ . By repeated application of the Leibniz inequality and using the fact that L(ui ) = 1 for each i = 1, . . . , p we have the following estimate for the c Lip-norm of an arbitrary element of Un :   k L uk11 uk22 · · · upp ≤ k1 L(u1 ) + k2 L(u2 ) + · · · + kp L(up ) ≤ pn. Thus the set Wn = {(pn)−1 u : u ∈ Un } is contained in L1 . Now products of distinct elements of the (self-adjoint) set Un evaluate to zero under the tracial state τ , so that, in the GNS representation Hilbert space associated to τ with canonical cyclic vector ξτ , πτ (Un )ξτ forms an orthonormal set of vectors. Thus, given δ > 0 we can apply Proposition 3.8 to obtain, for each n ≥ 1,   D Wn , (pn)−1 δ ≥ D(Un , δ) ≥ Dτ (Un , δ) ≥ (1 − δ 2 )(2n + 1)p so that, assuming δ < 1, log D(Wn , (pn)−1 δ) log(pnδ −1 ) n→∞ log(1 − δ 2 ) + p log(2n + 1) ≥ lim sup log(pnδ −1 ) n→∞ = p,

MdimL (Aρ ) ≥ lim sup

as desired.



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5. Product Entropies We now study dynamics within the framework of unital C ∗ -algebras with Leibniz c Lip-norms, concentrating on iterative growth as captured in the “product” entropy of Subsect. 5.1 and its measure-theoretic version in Subsect. 5.2. That the Leibniz rule is important here can be seen by examining the proofs of Propositions 5.4 and 5.6 (although the latter only requires that L be closed under multiplication). 5.1. Product entropy. We begin by introducing some notation. Notation 5.1. For any set X we will denote by Pf (X) the collection of finite subsets of X. If X1 , X2 , . . . , Xn are subsets of the C ∗ -algebra A we will use the notation X1 · X2 · · · · · Xn or nj=1 Xj to refer to the set {a1 a2 · · · an : ai ∈ Xi for each i = 1, . . . , n} . Recall from Notation 2.2 that, for a C ∗ -algebra A and r > 0, Ar refers to the set {a ∈ A : a ≤ r}. For the meaning of D(· , ·) see Notation 3.1. Definition 5.2. Let A be a unital C ∗ -algebra with Leibniz c Lip-norm L, and let α ∈ Aut L (A). For  ∈ Pf (L ∩ A1 ) and δ > 0 we define   1 EntpL (α, , δ) = lim sup log D  · α() · α 2 () · · · · · α n−1 (), δ , n→∞ n EntpL (α, ) = sup EntpL (α, , δ), δ>0

EntpL (α) =

sup

∈Pf (L∩A1 )

EntpL (α, ).

We will call EntpL (α) the product entropy of α. We record in the following proposition the evident fact that EntpL (A) is invariant under bi-Lipschitz ∗ -isomorphisms. Proposition 5.3. Let A and B be unital C ∗ -algebras with Leibniz c Lip-norms LA and LB , repectively. Let α ∈ Aut LA (A) and β ∈ AutLB (B). Suppose  : A → B is a bi-Lipschitz ∗ -isomorphism which intertwines α with β (i.e.,  ◦ α = β ◦ ). Then EntpL (α) = EntpL (β). The entropy Entp(α) is related to the metric dimension of A by the following inequality, which formally parallels a familiar fact about topological entropy (see [10, Prop. 14.20]). We remark that we don’t know whether the Lipschitz number of a bi-Lipschitz automorphism α can be strictly less than 1, although it is evident that in general at least one of α and α −1 must have Lipschitz number at least 1. Proposition 5.4. If α ∈ Aut L (A) and MdimL (A) is finite then EntpL (α) ≤ MdimL (A) · log max(λ, 1), where λ is the Lipschitz number of α.

Dimension and Dynamical Entropy for Metrized C ∗ -Algebras

521

Proof. Let  ∈ Pf (L ∩ A1 , δ) and δ > 0. Set M = maxa∈ L(a). Then by repeated application of the Leibniz inequality we see that elements of the set n =  · α() · α 2 () · · · · · α n−1 () have c Lip-norm at most M(1 + λ + λ2 + · · · + λn−1 ), which is bounded above by Mnλn . Hence L1 contains the set {(Mnλn )−1 a : a ∈ n }, which we will denote simply by (Mnλn )−1 n . It follows that 1 log D(n , δ) n n→∞ 1 = lim sup log D((Mnλn )−1 n , (Mnλn )−1 δ) n→∞ n 1 ≤ lim sup log D(L1 , (Mnλn )−1 δ). n→∞ n

EntpL (α, , δ) = lim sup

If λ < 1 then this last limit supremum is clearly zero. If on the other hand λ ≥ 1 then 1 log D(L1 , (Mnλn )−1 δ) n→∞ n 1 log D(L1 , (Mnλn )−1 δ) ≤ lim sup log(Mnλn δ −1 ) log(Mnλn δ −1 ) n→∞ n log D(L1 , (Mnλn )−1 δ) 1 = lim sup · lim log(Mnλn δ −1 ) n δ −1 ) n→∞ log(Mnλ n n→∞ = MdimL (A) · log λ.

lim sup

We thus obtain the result by taking the supremum over all  and δ.



Corollary 5.5. If MdimL (A) is finite and α ∈ AutL (A) is Lipschitz isometric then EntpL (α) = 0. In particular EntpL (idA ) = 0. Corollary 5.5 shows that the appropriate domain for our notion of entropy as a measure of dynamical growth is the class of c Lip-normed unital C ∗ -algebras A for which MdimL (A) is finite, in analogy to the situation of topological approximation entropies [4, 33] which function under conditions of “finiteness” like nuclearity or exactness. Proposition 5.6. If α ∈ AutL (A) and k ∈ Z then EntpL (α k ) = |k| EntpL (α). Proof. Suppose first that k ≥ 0. Let  ∈ Pf (L ∩ A1 ) and δ > 0, and suppose 1 ∈ . Then n−1  j =0

so that

α () ⊂ jk

(n−1)k  j =0

α j ()

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D. Kerr

  n−1  1 EntpL (α k , , δ) = lim sup log D  α j k (), δ  n→∞ n j =0   (n−1)k  1 ≤ k lim sup α j (), δ  log D  n→∞ kn j =0

= k EntpL (α, , δ).  j On the other hand setting k = k−1 j =0 α () (which is contained in Pf (L ∩ A1 ) in view of the Leibniz rule) we have k  n

α j (k ) ⊂

j =0

so that

n−1 

α j ()

j =0

 n  k  k EntpL (α k , k , δ) = lim sup log D  α j (k ), δ  n→∞ n j =0   n−1  1 ≤ k lim sup log D  α j (), δ  n→∞ n j =0

= k EntpL (α, , δ), and hence

  EntpL α k , k , δ ≤ k EntpL (α, , δ) .

Taking the supremum over all  ∈ Pf (L ∩ A1 ) and δ > 0 yields EntpL (α k ) = k EntpL (α). To prove the assertion for k < 0 it suffices, in view of the first part, to show that EntpL (α −1 ) = EntpL (α). Since   n−1 n−1   α −n+1  α j () = α −j () j =0

we have

 D

n−1 

j =0

and hence

j =0





α j (), δ  = D 

n−1 

 α −j (), δ 

j =0

  EntpL (α, , δ) = EntpL α −1 , , δ ,

from which we reach the conclusion by taking the supremum over all  ∈ Pf (L ∩ A1 ) and δ > 0.

 The following proposition is clear from Definition 5.2.

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523

Proposition 5.7. Let A be a unital C ∗ -algebra with c Lip-norm LA and B ⊂ A a unital C ∗ -subalgebra with c Lip-norm LB such that LB is the restriction of LA to B. Suppose that there is a C ∗ -norm contractive idempotent linear map of A onto B. If α ∈ Aut L (A) leaves B invariant then EntpLB (α|B ) ≤ EntpLA (α). Proposition 5.8. Let A and B be unital C ∗ -algebras, LA a Leibniz c Lip-norm on A, φ : A → B a surjective unital ∗ -homomorphism, and LB the Leibniz c Lip-norm induced on B via φ. Suppose there exists a positive C ∗ -norm contractive (not necessarily unital) Lipschitz map ψ : B → A such that φ ◦ ψ = idB . Let α ∈ Aut LA (A) and β ∈ AutLB (B) and suppose φ ◦ α = β ◦ φ. Then EntpLB (β) ≤ EntpLA (α). Proof. Let  ∈ Pf (LB ∩ B1 ) and δ > 0. Since ψ is norm-decreasing we have ψ() ∈ Pf (LA ∩ A1 ). Now if X ∈ F(A) is such that ψ() · α(ψ()) · · · · · α n−1 (ψ()) ⊂δ X, then  · β() · · · · · β n−1 () = (φ ◦ ψ)() · β((φ ◦ ψ)()) · · · · · β n−1 ((φ ◦ ψ)()) = φ(ψ()) · φ(α(ψ())) · · · · · φ(α n−1 (ψ())) = φ(ψ() · α(ψ()) · · · · · α n−1 (ψ())) ⊂δ φ(X) and so D( · β() · · · · · β n−1 (), δ) ≤ D(ψ() · α(ψ()) · · · · · α n−1 (ψ()), δ), from which the proposition follows.



5.2. Product entropy with respect to an invariant state. We define now a version of MdimL (A) relative to a dynamically invariant state σ . As in Subsect. 5.1 we are assuming that L is a Leibniz c Lip-norm. For the meaning of Dσ (· , ·) see Notation 3.1. Definition 5.9. Let α ∈ AutL (A) and let σ be a state of A which is α-invariant, i.e., σ ◦ α = σ . For  ∈ Pf (L ∩ A1 ) and δ > 0 we define   1 EntpL,σ (α, , δ) = lim sup log Dσ  · α() · α 2 () · · · · · α n−1 (), δ , n→∞ n EntpL,σ (α, ) = sup EntpL,σ (α, , δ), δ>0

EntpL,σ (α) =

sup

∈Pf (L∩A1 )

EntpL,σ (α, ).

We will call EntpL,σ (α) the product entropy of α with respect to σ . The following two propositions follow immediately from the definition.

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D. Kerr

Proposition 5.10. Let A and B be unital C ∗ -algebras with respective Leibniz c Lipnorms LA and LB . Let α ∈ AutLA (A) and β ∈ AutLB (B), and let σ and ω be α- and β-invariant states on A and B, respectively. Suppose  : A → B is a bi-Lipschitz ∗ -isomorphism such that  ◦ α = β ◦  and ω ◦  = σ . Then EntpL,σ (α) = EntpL,ω (β). Proposition 5.11. Let A be a unital C ∗ -algebra with Leibniz c Lip-norm LA and B ⊂ A a unital C ∗ -subalgebra with Leibniz c Lip-norm LB such that LB is the restriction of LA to B. Let σ be a state on A with σ ◦ α = σ , and suppose that there is a idempotent linear map of A onto B which is contractive for the Hilbert space norm under the GNS construction associated to σ . If α ∈ Aut L (A) leaves B invariant then EntpLB ,σ (α|B ) ≤ EntpLA ,σ (α). The next proposition can be established in the same way as its counterpart Proposition 5.6 in Subsect. 5.1. Proposition 5.12. If α ∈ Aut L (A), σ is an α-invariant state on A, and k ∈ Z, then EntpL,σ (α k ) = |k| EntpL,σ (α). Proposition 5.13. If α ∈ Aut L (A) and σ is an α-invariant state on A then EntpL,σ (α) ≤ EntpL (α). Proof. It suffices to show that, for a given  ∈ Pf (L ∩ A1 ) and δ > 0, Dσ (, δ) ≤ D(, δ), and for this inequality we need only observe that if X is a finite-dimensional subspace of A such that  ⊂δ X, then whenever a ∈  and x ∈ X satisfy a − x < δ we have π(a)ξσ − π(x)ξσ σ = π(a − x)ξσ σ ≤ π(a − x) ≤ a − x < δ, so that π(X)ξσ is a subspace of Hσ with π()ξσ ⊂δ π(X)ξσ and dim π()ξσ ≤ dim X.

 Corollary 5.14. If MdimL (A) is finite and α ∈ AutL (A) is Lipschitz isometric then EntpL,σ (α) = 0. In particular EntpL,σ (idA ) = 0. Proof. This follows by combining Proposition 5.13 with Corollary 5.5.



6. Tensor Product Shifts The fundamental prototypical system for topological entropy is the shift on the infinite product {1, . . . , p}Z , with entropy log p. Here we consider the noncommutative analogue of this map, the (right) shift on the infinite tensor product Mp⊗Z of p × p matrix algebras Mp over C,here with  the Leibniz c Lip-norm L furnished by the infinite

product γ ⊗Z : GZ → Aut Mp⊗Z of Weyl actions and length function λ (for a given 0 < λ < 1) as described in Subsect. 4.1. Before computing the entropy of the shift we will show that it is a bi-Lipschitz ∗ -automorphism.

Dimension and Dynamical Entropy for Metrized C ∗ -Algebras

525

Proposition 6.1. The shift α on Mp⊗Z is a bi-Lipschitz ∗ -automorphism, and α and its inverse have Lipschitz numbers bounded by λ. Proof. Let T : GZ → GZ be the right shift homeomorphism. Then it is readily seen that if a is an elementary tensor in Mp[m,n] ⊂ Mp⊗Z for some m, n ∈ Z then γg⊗Z (α(a)) =   α γT⊗gZ (a) for all g ∈ GZ , and since such a generate Mp⊗Z we have γg⊗Z ◦α = α ◦γT⊗gZ for all g ∈ GZ . Thus, for any a ∈ Mp⊗Z , L(α(a)) =

= ≤

sup g∈GZ \{e}

sup g∈GZ \{e}

sup g∈GZ \{e}

 ⊗Z  γ (α(a)) − α(a) g

λ (g)   ⊗Z   α γ (a) − α(a) Tg

λ (g)  ⊗Z  γ (a) − a  Tg

λ (T g)

·

λ (T g) g∈GZ \{e} λ (g) sup

≤ L(a) · L(T ), where L(T ) is the Lipschitz number of the homeomorphism T with respect to the metric defining λ (see Subsect. 4.1), and it is straightforward to verify that L(T ) = λ. We can argue similarly for α −1 to reach the desired conclusion.

 Z Proposition 6.2. Let α be the shift on Mp⊗Z and τ = tr⊗ p the unique (and hence αinvariant) tracial state on Mp⊗Z . Then

EntpL,τ (α) ≥ 2 log p. Proof. Let u, v ∈ Mp⊗Z be the Weyl generators for the zeroth copy of Mp (identified as a subalgebra of Mp⊗Z ) and let  be the finite subset {ui v j : 0 ≤ i, j ≤ k − 1} of   L ∩ Mp⊗Z . Then the set n =  · α() · α 2 () · · · · · α n−1 () is precisely the subset 1

  ui0 v j0 ⊗ ui1 v j1 ⊗ · · · ⊗ uin−1 v jn−1 : 0 ≤ ik , jk ≤ p − 1 for k = 0, . . . , n − 1 of Mp⊗[0,n] as considered sitting in Mp⊗Z . Thus πτ (n )ξτ is an orthonormal set in the GNS representation Hilbert space associated to τ with canonical cyclic vector ξτ (see the second half of the proof of Proposition 4.1), and so by Lemma 3.8 for any δ > 0 we have Dτ (n , δ) ≥ (1 − δ 2 )card(πτ (n )ξτ ) = (1 − δ 2 )p 2n . Thus if δ < 1 we obtain EntpL,σ (α, , δ) = lim sup n→∞

which yields the proposition.



1 log Dσ (n , δ) ≥ 2 log p, n

526

D. Kerr

Note that by Propositions 5.4, 4.1, and 6.1 the shift α satisfies EntpL (α) ≤ 4 log p. The following proposition yields the sharp upper bound of 2 log p. Proposition 6.3. With α the shift we have EntpL (α) ≤ 2 log p. 



Proof. Let  ∈ Pf L ∩ Mp⊗Z

  1

and δ > 0. Set C = maxa∈ Lλ (a). For each n

consider the conditional expectation En : Mp⊗Z → Mp⊗[−n,n] given by  En (a) =

GZ\[−n,n]

γg⊗Z (a) dg,

where dg is normalized Haar measure on GZ . We then have       ⊗Z  En (a) − a =  γg (a) − a dg   Z\[−n,n]  G  ⊗Z  γ (a) − a  dg ≤ g Z\[−n,n] G ≤ Cλ (g) dg GZ\[−n,n]

2Cλn+1 ≤ . 1−λ If a1 , . . . , an ∈  then, estimating the norm of differences of products in the usual way and using the fact that the conditional expectations are norm-decreasing, we have  E



n−1 √ √ (E√n (an )) − a1 α(a2 ) · · · α n−1 (an )  n (a1 )α(E n (a2 )) · · · α n   k−1 α (E √ (ak )) − α k−1 (ak ) ≤  n k=1 n   E √ (ak ) − ak  =  n k=1 √ 2Cnλ n+1

≤

1−λ

,

which is smaller than δ for all n greater than some n0 ∈ N (here · denotes the ceiling function). Next we observe that the product     E√n (a1 )α E√n (a2 ) · · · α n−1 E√n (an )

Dimension and Dynamical Entropy for Metrized C ∗ -Algebras √ √ ⊗[− n, n+n]

is contained in the subalgebra Mp √ p2(2 n+n) .

527

of Mp⊗Z , and this subalgebra has lin-

ear dimension In view√of the first paragraph, for all n ≥ n0 , the set n is √ ⊗[− n, n+n] approximately contained in Mp to within δ, and so we have EntpL (α, , 2δ) ≤ lim sup n→∞

√ 1 log p2(2 n+n) = 2 log p. n

The proposition now follows by taking the supremum over all  and δ.

 As a consequence of Propositions 6.2, 6.3, and 5.13 we obtain the following. Proposition 6.4. With α the shift and τ the unique tracial state on Mp⊗Z we have EntpL (α) = EntpL,τ (α) = 2 log p. 7. Noncommutative Toral Automorphisms Let Aρ be a noncommutative p-torus with generators u1 , . . . , up , canonical ergodic action γ : Tp → Aut(Aρ ), and associated Leibniz c Lip-norm L and γ -invariant tracial state τ , as defined in Subsect. 4.2. We let πτ : Aρ → B(Hτ ) be the GNS representation associated to τ , with canonical cyclic vector ξτ . Let T = (sij ) be a p × p integral matrix with det T = ±1, and suppose that T defines an automorphism αT of Aρ via the specifications s

s

αT (uj ) = u11j · · · uppj on the generators (this will always be the case if det T = 1 owing to the universal property of noncommutative tori). These noncommutative versions of toral automorphisms were introduced in the case p = 2 in [34] and [3]. Since τ is zero on products of powers of generators which are not equal to the unit, we see that it is invariant under the automorphism αT and the action γ . Fix a t = (t1 , . . . , tp ) ∈ Tp ∼ = (R/Z)p and consider the automorphism γt coming from the action γ . We will compute the entropies EntpL (αT ◦ γt ) and EntpL,τ (αT ◦ γt ) and furthermore show that their common value bounds above the entropies EntpL (Adu ◦ αT ◦ γt ) and EntpL,τ (Adu ◦ αT ◦ γt ) for any unitary u ∈ L. We remark that in the case p = 2, when Aρ is a rotation C ∗ -algebra Aθ , Elliott showed in [12] that if the angle θ satisfies a generic Diophantine property then all automorphisms preserving the dense ∗ -subalgebra of smooth elements (i.e., all “diffeomorphisms”) are of the form Adu ◦ αT ◦ γt , where u is a smooth unitary (and hence of finite c Lip-norm). Proposition 7.1. The ∗ -automorphism α = Adu ◦ αT ◦ γt is bi-Lipschitz, and α and its inverse have Lipschitz numbers bounded by 2T (1 + 2L(u)diam(S(A))) and

    2 T −1  (1 + 2L(u)diam(S(A))),

respectively, where T  and T −1  are the respective norms of T and T −1 as operators on the real inner product space Rp .

528

D. Kerr

Proof. If we consider T as acting on Tp then γg ◦ α = α ◦ γT g for all g ∈ Tp , as can be seen by checking this equation on the generators u1 , . . . , up . As in the proof of Proposition 6.1 we thus have, for any a ∈ L, the bound L(α(a)) ≤ L(a) · L(T ), where L(T ) is the Lipschitz number of the homeomorphism T . If we consider T as an operator on Rp , then its Lipschitz number is T  by definition of the operator norm, and so by linearity the Lipschitz number L(T ) of T on the quotient Tp ∼ = Rp /Zp must again be T . Next note that γt is isometric, for if a ∈ L then L(γt (a)) =

γs+t (a) − γt (a) γs (a) − a = sup = L(a). (s) (s) s∈Tp \{0} s∈Tp \{0} sup

Also, since L is readily checked to be lower semicontinuous, by Proposition 2.11 the Lipschitz number of Adu is bounded by 2(1 + 2L(u)diam(S(A))). Thus by Proposition 2.9 we get the desired bound on the Lipschitz number of Adu ◦ αT ◦ γt . A similar argument can be applied to (Adu ◦ αT ◦ γt )−1 = γ−t ◦ αT −1 ◦ Adu∗ .

 Proposition 7.2. We have EntpL,τ (αT ◦ γt ) ≥



log |λi |,

|λi |≥1

where λ1 , . . . , λp are the eigenvalues of T counted with spectral multiplicity. Proof. Let K be a finite subset of Zp and set   k UK = uk11 uk22 · · · upp : (k1 , . . . , kp ) ∈ K . The elements of UK , being products of powers of generators, all have finite c Lip-norm. k Observe that αT ◦ γt takes a product of the form ηuk11 uk22 · · · upp , with η a complex number of modulus one, to a product of the same form, with the exponents on the ui ’s respecting the action of the group automorphism ζT of Zp defined via the action of T . Thus if K is a finite subset of Zp then the set UK · (αT ◦ γt )(UK ) · · · · · (αT ◦ γt )n−1 (UK ) contains a subset UK,n of the form   k η(k1 ,... ,kp ) uk11 uk22 · · · upp : (k1 , . . . , kp ) ∈ K + ζT K + · · · + ζTn−1 K , where each η(k1 ,... ,kp ) is a complex number of modulus one. Note that πτ (UK,n )ξτ is an orthonormal set of vectors in the GNS representation Hilbert space associated to τ with canonical cyclic vector ξτ , since the product of any two distinct vectors in this set is a k scalar multiple of a product of the form uk11 uk22 · · · upp with the ki ’s not all zero, in which case evaluation under τ yields zero. It thus follows from Lemma 3.8 that if δ > 0 then Dτ (UK,n , δ) ≥ (1 − δ 2 )card(π(UK,n )ξτ ) = (1 − δ 2 )card(K + ζT K + · · · + ζTn−1 K), so that whenever δ < 1 we get

Dimension and Dynamical Entropy for Metrized C ∗ -Algebras

529

1 log Dτ (UK · α(UK ) · · · · · α n−1 (UK ), δ), n→∞ n 1 ≥ lim sup log Dτ (UK,n , δ) n→∞ n 1 ≥ lim sup log card(K + ζT K + · · · + ζTn−1 K). n n→∞

EntpL,σ (αt ◦ γt , UK , δ) = lim sup

We thus reach the desired conclusion by recalling from the computation of the discrete Abelian group entropy of ζT [23] that lim lim sup K

n→∞

1 log |λi |, log card(K + ζT K + · · · + ζTn−1 K) = n |λi |≥1

where the limit is taken with respect to the net of finite subsets K of Zp .



To compute upper bounds we need a couple of lemmas. Lemma 7.3. Let ζT be the group automorphism of Zp defined via the action of an p × p integral matrix T with det(T ) = ±1. Let λ1 , . . . , λp be the eigenvalues of T counted with spectral multiplicity. For each m ∈ N let Km be the cube   (k1 , . . . , kp ) ∈ Zp : |ki | ≤ m for each i = 1, . . . , p and define recursively for n ≥ 0 the sets Lm,n ∈ Zp by Lm,0 = Km and Lm,n+1 = ζT (Lm,n ) + Km . Then for every δ > 0 there is a Q > 0 such that, for all m, n ∈ N, 
 card Lm,0 + Lm,1 + · · · + Lm,n−1 ≤ Q(mn2 )p (1 + δ)n |λi |n . |λi |≥1

Proof. For any subset K of Zp we will denote its convex hull as a subset of Rp by ˜ With ζT also referring to the linear map on Rp defined by T , we consider the conK. vex set L˜ m,0 + L˜ m,1 + · · · + L˜ m,n−1 . By amplifying this set by a linear factor of 2p we can ensure that it contains every cube of unit side length centred at some point in Lm,0 + Lm,1 + · · · + Lm,n−1 , so that  
 card Lm,0 + Lm,1 + · · · + Lm,n−1 ≤ 2p vol L˜ m,0 + L˜ m,1 + · · · + L˜ m,n−1 . To estimate this volume on the right we assemble a basis B of Rp by picking a basis for the spectral subspace associated to each real eigenvalue and each pair of conjugate complex eigenvalues. Working from this point on with respect to the basis B, we note that the sets K˜ m are now parallelipipeds, and they can be contained in cubes Bm centred at 0 of side length rm for some r > 0 independent of m by the linearity of our basis change. If we define the sets Mm,n recursively by Mm,0 = Bm and Mm,n+1 = ζT (Mm,n ) + Bm ,

530

D. Kerr

then the set Mm,0 + Mm,1 + · · · + Mm,n−1 is a p-dimensional rectangular box which is centred at the origin with each face perpendicular to some coordinate axis, and this box contains L˜ m,0 + L˜ m,1 + · · · + L˜ m,n−1 , so that it suffices to show that
 vol Mm,0 + Mm,1 + · · · + Mm,n−1 is bounded by the last expression in the lemma statement for some C > 0. Let v be a vector in B associated to a real eigenvalue λ or a complex conjugate pair ¯ We can then find a Q > 0 such that for all n ∈ N the length of the vector T n (v) {λ, λ}. is bounded by Q(1 + δ)n |λ|n , where the factor (1 + δ)n is required to handle additional polynomial growth in the presence of a possible non-trivial generalized eigenspace. In view of the recursion defining Mm,n we then see that any scalar multiple of v which lies in Mm,n must be bounded in length by Qrm(1 + δ)n−1 |λ|n−1 + Qrm(1 + δ)n−2 |λ|n−2 + · · · + Qrm, which in turn is bounded by Qrmn(1 + δ)n max(|λ|n , 1). It follows that any scalar multiple of v contained in Mm,0 + Mm,1 + · · · + Mm,n−1 is bounded in length by Qrm

n−1

j (1 + δ)j max(|λ|j , 1),

j =0

and this expression is less than Qrmn2 (1 + δ)n max(|λ|n , 1). Since the set Mm,0 + Mm,1 + · · · + Mm,n−1 is a rectangular box squarely positioned with respect to the basis B and centred at the origin (as described above), we combine these length estimates to conclude that  vol(Mm,0 + Mm,1 + · · · + Mm,n−1 ) ≤ (Qrmn2 )p (1 + δ)n |λi |n , |λi |≥1

which yields the result.



Proposition 7.4. Suppose u ∈ Aρ is a unitary with L(u) < ∞. Then EntpL (Adu ◦ αT ◦ γt ) ≤



log |λi |,

|λi |≥1

where λ1 , . . . , λp are the eigenvalues of T counted with spectral multiplicity.

Dimension and Dynamical Entropy for Metrized C ∗ -Algebras

531

Proof. Set α = Adu ◦ αT ◦ γt for notational brevity. Let  ∈ Pf (L ∪ (Aρ )1 ) and δ > 0. By Lemma 4.3 we can find an C > 0 such that a − σn (a) ≤ C

log n n

for all n ∈ N and a ∈  ∪ {u}, where σn (a) is the nth Ces`aro mean, as defined in the paragraph preceding the statement of Lemma 4.2. Since σn (u∗ ) = σn (u)∗ , we also then have  ∗  u − σn (u∗ ) ≤ C log n n for all n ∈ N. Furthermore

  log n  j  α (a) − α j (σn (a)) ≤ C n

for all j, n ∈ N. By applying the triangle inequality n times in the usual way to estimate differences of products and using the fact that the operation of taking a Ces`aro is norm-decreasing (as can be seen from the first display in the statement of Lemma 4.2), we then have, for any a1 , . . . , an ∈ ,   log n2   , a1 α(a2 ) · · · α n−1 (an ) − σn2 (a1 )α(σn2 (a2 )) · · · α n−1 (σn2 (an )) ≤ C n and this last quantity is less than δ for all n greater than or equal to some n0 ∈ N. With the notation of the statement of Lemma 7.3 we next note that for any a ∈ A and n ∈ N we have by definition   k σn2 (a) ∈ span uk11 uk22 · · · upp : (k1 , . . . , kp ) ∈ Kn2 , while if

  k a ∈ span uk11 uk22 · · · upp : (k1 , . . . , kp ) ∈ K

for some finite K ⊂ Zp then

  k (Adu)(σn2 (a)) ∈ span uk11 uk22 · · · upp : (k1 , . . . , kp ) ∈ K + K2n2

for all n ∈ N (the factor of 2 in the subscript of K2n2 is required to handle multiplication of a by both u and u∗ ). Thus, since γt commutes with the operation of taking a Ces`aro sum of a given order, the set of all products σn2 (a1 )α(σn2 (a2 )) · · · α n−1 (σn2 (an )) with ai ∈  for i = 1, . . . , n is contained in the subspace   k Xn = span uk11 uk22 · · · upp : (k1 , . . . , kp ) ∈ L2n2 ,0 + L2n2 ,1 + · · · + L2n2 ,n−1 , again using the notation in the statement of Lemma 7.3 (taking m = 2n2 here). In view of the first paragraph Xn approximately contains  · α() · · · · · α n−1 () to within 2δ for all n ≥ n0 , and by Lemma 7.3 there exists a Q > 0 such that  dim(Xn ) ≤ (2Qn3 )p (1 + δ)n |λi |n |λi |≥1

for all n ∈ N. Therefore

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D. Kerr

   1 EntpL (α, , 2δ) ≤ lim sup log (2Qn3 )p (1 + δ)n |λi |n  n→∞ n |λi |≥1 = log(1 + δ) + log |λi |. |λi |≥1

Taking the supremum over all δ > 0 then yields EntpL (α, ) ≤



log |λi |,

|λi |≥1

from which the proposition follows.



Theorem 7.5. We have

EntpL (αT ◦ γt ) = EntpL,τ (αT ◦ γt ) =

log |λi |,

|λi |≥1

where λ1 , . . . , λp are the eigenvalues of T counted with spectral multiplicity. In particular, EntpL (αT ) = EntpL,τ (αT ) =



log |λi |.

|λi |≥1



Proof. This follows by combining Propositions 7.2, 7.4, and 5.13.

We also have the following, which is a consequence of Propositions 5.13 and 7.4. Proposition 7.6. If u ∈ A is a unitary with L(u) < ∞, then EntpL (Adu) = EntpL,τ (Adu) = 0. k

It is readily seen that if u is a unitary of the form ηuk11 uk22 · · · upp for some integers k1 , . . . , kp and complex number η of unit modulus, then the automorphism Adu◦αT ◦γt can be alternatively expressed as αT ◦ γt  for some t  ∈ Tp , in which case Theorem 7.5 applies. We leave open the problem of computing the product entropies of Adu ◦ αT ◦ γt when u ∈ L is a unitary not of this form and the eigenvalues of T do not all lie on the unit circle. We expect however that the entropies are positive when αT is asymptotically Abelian (see [19] for a description of when this occurs in the case p = 2) and the partial Fourier sums or Ces`aro means of u converge sufficiently fast to u, for we could then aim to apply the argument of the proof of Proposition 7.2 up to a degree of approximation. Acknowledgements. This work was supported by the Natural Sciences and Engineering Research Council of Canada. I thankYasuyuki Kawahigashi and the operator algebra group at the University of Tokyo for their hospitality and for the invigorating research environment they have provided. I also thank Hanfeng Li for pointing out an oversight in an initial draft and the referee for making suggestions that resulted in significant improvements to the paper.

Dimension and Dynamical Entropy for Metrized C ∗ -Algebras

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References 1. Alicki, R., Fannes, M.: Defining quantum dynamical entropy. Lett. Math. Phys. 32, 75–82 (1994) 2. Andries, J., Fannes, M., Tuyls, P., Alicki, R.: The dynamical entropy of the quantum Arnold cat map. Lett. Math. Phys. 35, 375–383 (1995) 3. Brenken, B.: Representations and automorphisms of the irrational rotation algebra. Pacific J. Math. 111, 257–282 (1984) 4. Brown, N.P.: Topological entropy in exact C ∗ -algebras. Math. Ann. 314, 347–367 (1999) 5. Connes, A.: Compact metric spaces, Fredholm modules and hyperfiniteness. Ergod. Th. Dynam. Sys. 9, 207–220 (1989) 6. Connes, A.: Noncommutative Geometry. San Diego: Academic Press, 1994 7. Connes, A.: Gravity coupled with matter and the foundation of non-commutative geometry. Commun. Math. Phys. 182, 155–176 (1996) 8. Connes, A., Narnhofer, H., Thirring, W.: Dynamical entropy of C ∗ -algebras and von Neumann algebras. Commun. Math. Phys. 112, 691–719 (1987) 9. Connes, A., Størmer, E.: Entropy of automorphisms of II1 -von Neumann algebras. Acta. Math. 134, 289–306 (1975) 10. Denker, M., Grillenberger, C., Sigmund, K.: Ergodic theory on compact spaces. Lecture Notes in Math, Vol. 527. Berlin: Springer-Verlag, 1976 11. Elliott, G.A.: On the K-theory of the C ∗ -algebra generated by a projective representation of a torsion-free discrete abelian group. In: Operator Algebras and Group Representations, Vol. I. Boston: Pitman, 1984, pp. 159–164 12. Elliott, G.A.: The diffeomorphism group of the irrational rotation C ∗ -algebra. C. R. Math. Rep. Acad. Sci. Canada 8, 329–334 (1986) 13. Hudetz, T.: Quantum topological entropy: First steps of a “pedestrian” approach. In: Quantum Probability & Related Topics, River Edge, NJ: World Scientific, 1993, pp. 237–261 14. Hudetz, T.: Topological entropy for appropriately approximated C ∗ -algebras. J. Math. Phys. 35, 4303–4333 (1994) 15. Katznelson, Y.: An Introduction to Harmonic Analysis, Second Edition. New York: Dover Publications, 1976 16. Klimek, S., Le´sniewski, A.: Quantized chaotic dynamics and non-commutative KS entropy. Ann. Physics 248, 173–198 (1996) 17. Kolmogorov, A.N., Tihomirov, V.M.: ε-entropy and ε-capacity of sets in functional analysis. Amer. Math. Soc. Trans. (2) 17, 277–364 (1961) 18. Makarov, B.M., Goluzina, M.G., Lodkin, A.A., Podkorytov, A.N.: Selected problems in real analysis. Translations of Mathematical Monographs, Vol. 107. Providence: AMS, 1992 19. Narnhofer, H.: Ergodic properties of automorphisms on the rotation algebra. Rep. Math. Phys. 39, 387–406 (1997) 20. Narnhofer, H., Thirring, W.: C ∗ -dynamical systems that are asymptotically highly anticommutative. Lett. Math. Phys. 35, 145–154 (1995) 21. Olesen, D., Pedersen, G.K., Takesaki, M.: Ergodic actions of compact Abelian groups. J. Operator Theory 3, 237–269 (1980) 22. Pesin, Ya. B.: Dimension Theory in Dynamical Systems: Contemporary Views and Applications. Chicago: The University of Chicago Press, 1997 23. Peters, J.: Entropy on discrete abelian groups. Adv. Math. 33, 1–13 (1979) 24. Rieffel, M.A.: Noncommutative tori – A case study of noncommutative differentiable manifolds. Contemporary Math. 105, 191–211 (1990) 25. Rieffel, M.A.: Metrics on states from actions of compact groups. Doc. Math. 3, 215–229 (1998) 26. Rieffel, M.A.: Metrics on state spaces. Doc. Math. 4, 559–600 (1999) 27. Rieffel, M.A.: Gromov-Hausdorff distance for quantum metric spaces. arXiv:math.OA/0011063 v2 (2001) 28. Russo, B., Dye, H.A.: A note on unitary operators in C ∗ -algebras. Duke Math. J. 33, 413–416 (1966) 29. Sauvageot, J.-L., Thouvenot, P.: Une nouvelle d´efinition de l’entropie dynamique des syst`emes non-commutatifs. Commun. Math. Phys. 145, 411–423 (1992) 30. Slawny, J.: On factor representations and the C ∗ -algebra of canonical commutation relations. Commun. Math. Phys. 24, 151–170 (1972) 31. Størmer, E.: A survey of noncommutative dynamical entropy. In: Classification of Nuclear C ∗ -algebras. Entropy in Operator Algebras. Berlin: Springer, 2002, pp. 147–198 32. Thomsen, K.: Topological entropy for endomorphisms of local C ∗ -algebras. Commun. Math. Phys. 164, 181–193 (1994) 33. Voiculescu, D.V.: Dynamical approximation entropies and topological entropy in operator algebras. Commun. Math. Phys. 170, 249–281 (1995)

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34. Watatani, Y.: Toral automorphisms on irrational rotation algebras. Math. Japon. 26, 479–484 (1981) 35. Weaver, N.: Lipschitz algebras and derivations of von Neumann algebras. J. Funct. Anal. 139, 261– 300 (1996) 36. Weaver, N.: α-Lipschitz algebras on the noncommutative torus. J. Operator Theory 39, 123–138 (1998) 37. Weaver, N.: Lipschitz Algebras. River Edge, NJ: World Scientific, 1999 Communicated by A. Connes

Commun. Math. Phys. 232, 535–563 (2003) Digital Object Identifier (DOI) 10.1007/s00220-002-0743-y

Communications in

Mathematical Physics

Connections on Naturally Reductive Spaces, Their Dirac Operator and Homogeneous Models in String Theory∗ Ilka Agricola Institut f¨ur Reine Mathematik, Humboldt-Universit¨at zu Berlin, Sitz: WBC Adlershof, 10099 Berlin, Germany. E-mail: [email protected] Received: 19 February 2002 / Accepted: 26 August 2002 Published online: 22 November 2002 – © Springer-Verlag 2002

Abstract: Given a reductive homogeneous space M = G/H endowed with a naturally reductive metric, we study the one-parameter family of connections ∇ t joining the canonical and the Levi-Civita connection (t = 0, 1/2). We show that the Dirac operator D t corresponding to t = 1/3 is the so-called “cubic” Dirac operator recently introduced by B. Kostant, and derive the formula for its square for any t, thus generalizing the classical Parthasarathy formula on symmetric spaces. Applications include the existence of a new G-invariant first order differential operator D on spinors and an eigenvalue estimate for the first eigenvalue of D 1/3 . This geometric situation can be used for constructing Riemannian manifolds which are Ricci flat and admit a parallel spinor with respect to some metric connection ∇ whose torsion T = 0 is a 3-form, the geometric model for the common sector of string theories. We present some results about solutions to the string equations and a detailed discussion of a 5-dimensional example. 1. Introduction This paper proposes a differential geometric approach to some recent results of B. Kostant on an algebraic object called “cubic Dirac operator” ([Kos99]). The key observation is that one can introduce a metric connection on certain homogeneous spaces whose torsion (viewed as a (0, 3)-tensor) is 3-form such that the associated Dirac operator has Kostant’s algebraic object as its symbol. At the same time, there has recently been growing interest in connections with totally skew symmetric torsion for constructing models in string theory and supergravity. We show that the mentioned class of homogeneous spaces yields interesting candidates for such solutions and use Dirac operator techniques to prove some vanishing theorems. In the first part of this paper, we consider a reductive homogeneous space M = G/H endowed with a Riemannian metric that induces a naturally reductive metric ,  on m, ∗ This work was supported by the SFB 288 “Differential geometry and quantum physics” of the Deutsche Forschungsgemeinschaft and the Max-Planck Society.

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where we set g = h ⊕ m. The one-parameter family of G-invariant connections defined by t 0 ∇X Y = ∇X Y + t [X, Y ]m joins the canonical (t = 0) and the Levi-Civita (t = 1/2) connection. Its torsion T (X, Y, Z) = (2t − 1) · [X, Y ]m , Z is a 3-form. For an orthonormal basis Z1 , . . . , Zn of m, it induces the third degree element H :=

3
[Zi , Zj ]m , Zk Zi · Zj · Zk 2 i<j
inside the Clifford algebra C(m) of m. The fact that the Dirac operator associated with the connection ∇ t may then be written as
Dt ψ = Zi · Zi (ψ) + t · H · ψ i

led B. Kostant to suggest the name “cubic Dirac operator” for it. We will show that one main achievement in [Kos99] was to realize that, for the parameter value t = 1/3, the square of D t may be expressed in a very simple way in terms of Casimir operators and scalars only ([Kos99, Thm 2.13], [Ste99, 10.18]). It is a remarkable generalization of the well-known Parthasarathy formula for D 2 on symmetric spaces (Theorem 3.1 in this article, see [Par72]). In fact, S. Slebarski had already noticed independently that the parameter value t = 1/3 has distinguished properties (see Theorem 1 and the introduction in [Sle87a]). He used it to prove a “vanishing theorem” for the kernel of the twisted Dirac operator, which can be easily recovered from Kostant’s formula (see [Lan00, Thm 4]). His articles [Sle87a] and [Sle87b] contain several formulas of Weitzenb¨ock type for D 2 , but none of them is of Parthasarathy type. We shall compute the general expression for (D t )2 in Theorem 3.2 and show how it can be simplified for this particular parameter value in Theorem 3.3. We emphasize one difference between our work and [Kos99]. While Kostant studies the algebraic action of D 1/3 as an element of U(g) ⊗ C(m) on L2 -functions G → m (the spinor representation), we are mainly interested in spinors, i. e. L2 -sections of the spinor bundle S = G ×Ad m . In particular, this implies that one of his terms in the formula for (D t )2 (the “diagonally” embedded Casimir operator of h) vanishes independently of t (but see Remark 3.4 for how to derive the diagonal term from our computations). An immediate consequence of Theorem 3.2 is the existence of a new G-invariant first order differential operator
Dψ := [Zi , Zj ]m , Zk Zi · Zj · Zk (ψ) i,j,k

on spinors (Remark 3.6) that has no analogue on symmetric spaces. Furthermore, under some additional hypotheses (the lifted Casimir operator g has to be non-negative), Theorem 3.3 yields an eigenvalue estimate which is discussed in Corollary 3.1. In the second part of this paper, we use the preceding approach for studying the string equations on naturally reductive spaces. Stated in a differential geometric way, one wants to construct a Riemannian manifold (M, g) with a metric connection ∇ such that its torsion T = 0 is a 3-form and such that there exists at least one spinor field ψ satisfying the coupled system Ric∇ = 0,

δ(T ) = 0,

∇ = 0,

T ·  = 0.

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The number of preserved supersymmetries depends essentially on the number of ∇-parallel spinors. For a general background on these equations, we refer to the article by A. Strominger [Str86], where they appeared for the first time. Thus, if one looks for homogeneous solutions, the family of connections ∇ t yields canonical candidates for the desired connection ∇, and the results on the associated Dirac operator can be used to discuss the solution space to these equations. We discuss the significance of constant spinors (which do not always exist) in Theorem 4.2 and show that the last two string equations cannot have any solutions at all if the lifted Casimir operator g is non-negative (Theorem 4.3). In order to discuss the first equation, we present a representation theoretical expression for the Ricci tensor of the connection ∇ t , which generalizes previous results by Wang and Ziller (Theorem 4.4). The article ends with a thorough discussion of an example, namely, the naturally reductive metrics on the 5-dimensional Stiefel manifold. Although we rarely refer to it, this paper is in spirit very close (and in some sense complementary) to a recent article by Friedrich and Ivanov ([FI01]). There, the authors study metric connections with totally skew symmetric torsion preserving a given geometry. 2. A Family of Connections on Naturally Reductive Spaces Consider a Riemannian homogeneous space M = G/H . We suppose that M is reductive, i. e. the Lie algebra g of G may be decomposed into a vector space direct sum of the Lie algebra h of H and an Ad (H )-invariant subspace m such that g = h ⊕ m and Ad (H )m ⊂ m. We identify m with T0 M by the map X → X0∗ , where X ∗ is the Killing vector field on M generated by the one parameter group exp(tX) acting on M. We pull back the Riemannian metric  , 0 on T0 M to an inner product  ,  on m. Let Ad : H → SO(m) be the isotropy representation of M. By a theorem of Wang ([KN96, Ch. X, Thm 2.1]), there is a one-to-one correspondence between the set of G-invariant metric affine connections and the set of linear mappings m : m → so(m) such that   m hXh−1 = Ad (h)m (X)Ad (h)−1 for X ∈ m and h ∈ H. Its torsion and curvature are then given for X, Y ∈ m by ([KN96, Ch. X, Prop. 2.3]) T (X, Y ) = m (X)Y − m (Y )X − [X, Y ]m , R(X, Y ) = [m (X), m (Y )] − m ([X, Y ]m ) − Ad ([X, Y ]h ), where the Lie bracket is split into its m and h part, [X, Y ] = [X, Y ]m + [X, Y ]h . Lemma 2.1. The (0, 3)-tensor corresponding to the torsion (X, Y, Z ∈ m), T (X, Y, Z) := T (X, Y ), Z is totally skew symmetric if and only if the map m satisfies for all X, Y, Z ∈ m the invariance condition m (X)Y, Z + m (Z)Y, X = [X, Y ]m , Z + [Z, Y ]m , X. For a general map m , this is all one can say. We are interested in the one parameter family of connections defined by tm (X)Y := t · [X, Y ]m .

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It is well known that t = 0 corresponds to the canonical connection ∇ 0 , which, by the Ambrose-Singer theorem, is the unique metric connection on M such that its torsion and curvature are parallel, ∇ 0 T 0 = ∇ 0 R 0 = 0. By Lemma 2.1, the torsion of ∇ 0 is a 3-form if and only if M is naturally reductive. Definition 2.1. A homogeneous Riemannian metric on M is said to be naturally reductive (with respect to G) if the map [X, −]m : m → m is skew symmetric, [X, Y ]m , Z + Y, [X, Z]m  = 0

for all X, Y, Z ∈ m.

Note that if G1 ⊂ G2 are two transitive groups of isometries of M, then the properties of being naturally reductive with respect to G1 and G2 are independent of each other. Remark 2.1. Under the assumption that M is naturally reductive, the right-hand side in the criterion of Lemma 2.1 vanishes, and the remaining condition may be restated – using the skew symmetry of m (X) and m (Z) – as Y, m (X)Z + m (Z)X = 0. Since this equation has to hold for all X, Y and Z in m, we obtain that the torsion is a 3-form if and only if m (X)X = 0 for all X ∈ m. If M is naturally reductive, the torsion of the family ∇ t of connections is given by the simple expression T t (X, Y ) = (2t − 1) [X, Y ]m . One sees that the Levi-Civita connection is attained for t = 1/2. The general formula for the connection ∇ t is t 0 ∇X Y = ∇X Y + t [X, Y ]m . (1) Notice that for a symmetric space, [m, m] ⊂ h, so all connections of this one-parameter family coincide and are equal to the Levi-Civita connection. Assumption 2.1. We will assume that M = G/H is naturally reductive with respect to G. We begin by computing a few characteristic entities for this family of connections, which will be needed in the subsequent sections. We start by recalling a theorem of B. Kostant. Theorem 2.1 ([Kos56]). Suppose G acts effectively on M = G/H . If the inner product  ,  is naturally reductive with respect to G, then g˜ := m + [m, m] is an ideal in g ˜ ⊂ G is transitive on M, and there exists a unique whose corresponding subgroup G ˜ Ad (G)-invariant, symmetric, non-degenerate, bilinear form Q on g˜ (not necessarily positive definite) such that Q(h ∩ g˜ , m) = 0

and

Q|m =  , ,

where h ∩ g˜ will be the isotropy algebra in g˜ . Conversely, if G is connected, then, for any Ad (G)-invariant, symmetric, non-degenerate, bilinear form Q on g, which is nondegenerate on h and positive definite on m := h⊥ , the metric on M defined by Q|m is naturally reductive. In this case, g = g˜ .   Assumption 2.2. We shall assume from now on that g = g˜ and use the Ad (G)-invariant extension Q of the inner product  ,  as well as its restriction Q|h =: Qh to h where needed without further comment.

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Lemma 2.2. The curvature of the connection ∇ t is given by R t (X, Y )Z = t 2 [X, [Y, Z]m ]m +t 2 [Y, [Z, X]m ]m +t [Z, [X, Y ]m ]m +[Z, [X, Y ]h ]. If Zi , . . . , Zn is an orthonormal basis of m, the Ricci tensor and the scalar curvature are
Rict (X, Y ) = (t − t 2 )[X, Zi ]m , [Y, Zi ]m  + Qh ([X, Zi ], [Y, Zi ]), i

Scal = t




  (t − t 2 )[Zi , Zj ]m , [Zi , Zj ]m  + Qh [Zi , Zj ], [Zi , Zj ] .

i,j

At a later stage, we will give a further expression for the Ricci tensor due to Wang and Ziller ([WZ85]). For the time being, we observe that the connection with t = 1 has also special properties, for example, it has the same Ricci tensor as the canonical connection. This is why we propose to call it the anticanonical connection. We compute the covariant derivative of the torsion tensor. Lemma 2.3. As a map m × m → m, the covariant derivative of T is   (∇Zt T t )(X, Y ) = t (2t − 1) [X, [Y, Z]m ]m + [Y, [Z, X]m ]m + [Z, [X, Y ]m ]m . For the first time we encounter here an expression that will play an important role in different places. Let us define Jacm(X, Y, Z) := [X, [Y, Z]m ]m + [Y, [Z, X]m ]m + [Z, [X, Y ]m ]m , Jach(X, Y, Z) := [X, [Y, Z]h ] + [Y, [Z, X]h ] + [Z, [X, Y ]h ]. Note that the summands of Jach(X, Y, Z) automatically lie in m by the assumption that M is reductive. The Jacobi identity for g implies Jacm(X, Y, Z)+Jach(X, Y, Z), m = 0. As the connection ∇ t is metric, the covariant derivatives of T viewed as a (0, 3)- resp. (1, 2)-tensor are related by (∇Zt T t )(X, Y, V ) = (∇Zt T t )(X, Y ), V  = t (2t − 1)Jacm(X, Y, Z), V .

(2)

For completeness, we recall the formula for the exterior derivative of a differential form in terms of a connection with torsion. Lemma 2.4. If ω is an r-form, then (dω)(X0 , . . . , Xr ) =

r
i=0

−

(−1)i (∇Xi ω)(X0 , . . . , Xˆ i , . . . , Xr )


(−1)i+j ω(T (Xi , Xj ), X0 , . . . , Xˆ i , . . . , Xˆ j , . . . , Xr ).

0≤i<j ≤r

Lemma 2.5. The codifferential of the 3-form T t vanishes, δT t = 0, while its exterior derivative is given by dT t (X, Y, Z, V ) = 2(2t − 1) · Jacm(X, Y, Z), V . t T t = 0. Then it follows Proof. For the first claim, one deduces from Eq. (2) that X ∇X for the orthonormal basis Zi , . . . , Zn of m that

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δt T t =

n


Zi

∇Zt i T t = 0.

i=1

In particular, the divergence of T with respect to ∇ t coincides with its Riemannian divergence (a more general fact, see [FI01]), δ t T t = δ 1/2 T t = 0. Hence we shall drop the superscript, as we did in the statement of the lemma. The second claim follows from Lemma 2.4 by a simple algebraic computation.   Remark 2.2. We finish this section with a remark about the connection between the torsion and the Lie algebra structure. If some torsion 3-form T is given as a fundamental datum and is to be the torsion of the canonical connection of some space with naturally reductive metric, then the m-part of the commutators [m, m] may be reconstructed by
[X, Y ]m = − T (X, Y, Zi )Zi . i

This formula is fundamental for the point of view taken in the article [Kos99] (formula 1.23). The full Lie algebra structure of g can now be viewed as consisting of the torsion 3-form, the isotropy representation and the subalgebra structure of h, with some compatibility condition resulting from the Jacobi identity. This point of view will be useful in the last section, where we will study examples. 3. The Dirac Operator of the Family of Connections ∇t 3.1. General remarks and formal self adjointness. Assume that there exists a homoge : H → Spin(m) of the isotropy representation neous spin structure on M, i. e. a lift Ad such that the diagram Spin(m)  Ad H

 λ

? AdSO(m)

commutes, where λ denotes the spin covering. Moreover, we denote by  ad the corresponding lift into spin(m) of the differential ad : h → so(m) of Ad . Let κ : Spin(m) → GL(m ) be the spin representation, and identify sections of the spinor bundle S = G ×Ad m with functions ψ : G → m satisfying  (h−1 ))ψ(g) ψ(gh) = κ(Ad for all h ∈ H . For any G-invariant connection defined by a map m : m → so(m), we ˜ m : m → spin(m), which is given by  ˜ m := dλ−1 ◦ m . Then the consider its lift  covariant derivative on spinors may be expressed as ([Ike75, Lemma 2]) ˜ m (Z)ψ ∇Z ψ = Z(ψ) +  and thus the Dirac operator associated with this connection has the form
˜ m (Zi )ψ, Zi · Zi (ψ) + Zi ·  Dψ = i

(3)

(4)

Connections on Naturally Reductive Spaces

541

where Z1 , . . . , Zn denotes any orthonormal basis of m. In the same article, Ikeda states a criterion for the formal self adjointness of this operator. We restate the result here, since there is some confusion about the assumptions on the scalar product in the original version. Proposition 3.1. Let M = G/H be a homogeneous reductive manifold with a homogeneous spin structure,  ,  the scalar product on m induced by the Riemannian metric on M, and ∇ the G-invariant metric connection defined by some map m : m → so(m). Then the Dirac operator D associated with the connection ∇ is formally self adjoint if and only if for any vector X ∈ m and any orthonormal basis Z1 , . . . , Zn of m, one has

m (Zi )X, Zi  = [Zi , X]m , Zi . (∗) i

i

In particular, this condition is always satisfied if the torsion T (X, Y, Z) is totally skew symmetric. If the metric  ,  is in addition naturally reductive, condition (∗) is equivalent  to m (Zi )Zi = 0. Proof. By a result of Friedrich and Sulanke ([FS79]), the Dirac operator D ∇ associated with any metric connection ∇ is formally self adjoint if and only if the ∇-divergence of any vector X coincides with its Riemannian divergence,

div∇ (X) := Zi , ∇Zi X = Zi , ∇ZLC X =: div(X), i i

i

where ∇ LC denotes the Levi-Civita connection. But for any vector X, the covariant derivatives are related by 1 ∇Zi X = ∇ZLC X + T (Zi , X), i 2  thus equality of divergences holds if and only if i T (Zi , X), Zi  = 0. Inserting the general formula for the torsion and using the fact that m (X)Zi , Zi  = 0, one checks that this is equivalent to condition (∗). Since T (Zi , X), Zi  = T (Zi , X, Zi ), condition (∗) is always fulfilled if the (0, 3)-tensor T is totally skew symmetric. Alternatively, one easily deduces equation (∗) from the antisymmetry condition in Lemma 2.1 by a contraction. Finally, if the metric is naturally reductive, the right-hand side of (∗) vanishes, and by the antisymmetry of m (Zi ) one obtains X, m (Zi )Zi  = 0. This finishes the proof.   Returning to the family ∇ t , our aim is to rewrite the connection term of the Dirac operator in Eq. (4) as an element of the Clifford algebra C(m). Basically this amounts to the identification of spin(m) with the elements of second degree in C(m). We implement the Clifford relations via Zi · Zj + Zj · Zi = −2δij , in contrast to [Kos99] (see [Fri00] for notational details). The following lemma due to Parthasarathy expresses the lift of the isotropy representation as an element of the Clifford algebra. Lemma 3.1 ([Par72, 2.1]). For any element Y in h, one has n 1
 ad (Y ) = [Y, Zi ], Zj Zi · Zj . 4 i,j =1

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I. Agricola

Similarly, any skew symmetric map m (X) : m → m may be expanded in the standard basis Eij of so(m) as m (X) =


m (X)Zi , Zj Eij . i<j

Since Eij lifts to Zi · Zj /2 in the Clifford algebra, we obtain in complete analogy to the Parthasarathy Lemma: Lemma 3.2. For any map m : m → so(m), one has ˜ m (X) = 

1
1
m (X)Zi , Zj Zi · Zj = m (X)Zi , Zj Zi · Zj . 2 4 i<j

i,j

In particular, the image of 1m (Zi ) = [Zi , −]m in C(m) may be written ˜ 1m (Zi ) = 

1
[Zi , Zj ]m , Zk Zj · Zk . 4 j,k

Thus, by defining the element H :=

n
i=1

˜ 1m (Zi ) = Zi · 

1
[Zi , Zj ]m , Zk Zi · Zj · Zk 4 i,j,k

3
= [Zi , Zj ]m , Zk Zi · Zj · Zk , 2 i<j
we can rewrite the Dirac operator corresponding to the connection ∇ t from Eq. (4) as
Dt ψ = Zi · Zi (ψ) + t · H · ψ. (5) i

Remark 3.1. We identify differential forms with elements of the Clifford algebra by

ω1...r Zi1 ∧ . . . ∧ Zir −→ ω1...r Zi1 · · · · · Zir . i1 <...
i1 <...
Thus, the torsion form T t (X, Y, Z) = (2t − 1)[X, Y ]m , Z induces the element
T t = (2t − 1) [Zi , Zj ]m , Zk Zi · Zj · Zk i<j
of the Clifford algebra, which differs from H only by a numerical factor, Tt =

2(2t − 1) H. 3

The simplicity of Eq. (5) is the main reason why we prefer to work with the element H instead of T t .

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3.2. The cubic element H , its square and the Casimir operator. It is the cubic element H inside the Clifford algebra C(m) which suggested the name “cubic Dirac operator” to B. Kostant. We see that the fact that H is of degree 3 inside C(m) does not depend on the particular choice for m . The square of H will play an eminent role in our considerations, both for a Kostant-Parthasarathy type formula and for general vanishing theorems. Notice that the square of any element of degree 3 inside C(m) has only terms of degree zero and 4. Proposition 3.2. The terms of degree zero and 4 of H 2 are given by (H 2 )0 =

3
[Zi , Zj ]m , [Zi , Zj ]m , 8 i,j

(H 2 )4 = −

9 2




Zi , Jacm(Zj , Zk , Zl )Zi · Zj · Zk · Zl .

i<j
The first formula is valid for all n ≥ 3, while the second holds only for n ≥ 5. For n = 3, 4, one has (H 2 )4 = 0. Proof. The contributions of degree zero in H 2 are exactly the squares of the summands of H . Because of (Zi · Zj · Zk )2 = 1, we have (H 2 )0 =

9
9
[Zi , Zj ]m , Zk 2 = [Zi , Zj ]m , Zk [Zi , Zj ]m , Zk . 4 24 i<j
i,j,k

 For fixed i, j , the sum over k is the coordinate expansion of the scalar product [Zi , Zj ]m , [Zi , Zj ]m , thus 3
[Zi , Zj ]m , [Zi , Zj ]m , (H 2 )0 = 8 i,j

as claimed. Contributions of degree 4 occur if Zi · Zj · Zk is multiplied by Zi  · Zj  · Zk  with exactly one common index. Since this requires at least 5 different indices, it follows that there are no terms of fourth degree for n ≤ 4. For the moment, put aside the overall factor 9/4 of H 2 . We explain the occurrence of the term proportional to Z1234 := Z1 · Z2 · Z3 · Z4 in detail, the others are obtained in a similar way. Since H consists of ordered tuples proportional to Zij k := Zi · Zj · Zk , i < j < k, the only way to obtain a term in Z1234 is to multiply Z12k by Z34k , Z13k by Z24k and Z14k by Z23k for any index k ≥ 5. First we notice that the order of multiplication is irrelevant, since Z12k ·Z34k = Z34k ·Z12k ,

Z13k ·Z24k = Z24k ·Z13k ,

and

Z14k ·Z23k = Z23k ·Z14k .

Every term will thus have multiplicity two. In the next step, these products have to be rearranged in order to be proportional to Z1234 : Z12k · Z34k = −Z1234 ,

Z13k · Z24k = +Z1234 ,

Z14k · Z23k = −Z1234 .

The total contribution coming from the products Z12k by Z34k is thus
[Z1 , Z2 ]m , Zk [Z3 , Z4 ]m , Zk . (∗) := −2 Z1234 k≥5

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This is equal to the sum over all k, since the additional terms are zero. However, it shows that the sum is precisely the expansion of the scalar product [Z1 , Z2 ]m , [Z3 , Z4 ]m : (∗) = −2 Z1234

n


[Z1 , Z2 ]m , Zk [Z3 , Z4 ]m , Zk 

k=1

= −2 Z1234 [Z1 , Z2 ]m , [Z3 , Z4 ]m . After a similar simplification of the other two contributions, the fourth degree term in H 2 proportional to Z1234 is finally equal to (∗∗) := 2 [−[Z1 , Z2 ]m , [Z3 , Z4 ]m  + [Z1 , Z3 ]m , [Z2 , Z4 ]m  −[Z1 , Z4 ]m , [Z2 , Z3 ]m ] · Z1234 . This, in turn, may be rewritten as (∗∗) = −2Z1 , Jacm(Z2 , Z3 , Z4 ) · Z1234 . Reinserting the factor 9/4, we obtain the factor −9/2 as stated in the lemma.

 

For later reference, we compute the anticommutator of H with an element Zl for arbitrary l. 3
Lemma 3.3. For any l, one has H · Zl + Zl · H = − Zl , [Zi , Zj ]m Zi · Zj . 2 i,j

Proof. By definition, H · Zl + Zl · H =

 1
[Zi , Zj ]m , Zk  Zi · Zj · Zk · Zl + Zl · Zi · Zj · Zk ). 4 i,j,k

If all four indices i, j, k, l are pairwise different, Zi · Zj · Zk · Zl = −Zl · Zi · Zj · Zk , and the corresponding summand vanishes. Thus, the sum may be split into those parts where l is one of the indices i, j and k, respectively:  1
H · Zl + Zl · H = [Zl , Zj ]m , Zk  Zl · Zj · Zk · Zl + Zl · Zl · Zj · Zk ) 4 j,k

 1
+ [Zi , Zl ]m , Zk  Zi · Zl · Zk · Zl + Zl · Zi · Zl · Zk ) 4 i,k

 1
[Zi , Zj ]m , Zl  Zi · Zj · Zl · Zl + Zl · Zi · Zj · Zl ). + 4 i,j

We simplify the mixed products to get 1
1
H · Zl + Zl · H = − [Zl , Zj ]m , Zk Zj · Zk + [Zi , Zl ]m , Zk Zi · Zk 2 2 j,k

1
− [Zi , Zj ]m , Zl Zi · Zj . 2

i,k

i,j

Using the invariance property of the scalar product and renaming the summation indices, this is easily seen to be the desired expression.  

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545

Finally, we compute the image of the quadratic Casimir operator of h inside the Clifford algebra. Since the Ad (G)-invariant extension Q of  ,  is not necessarily positive definite when restricted to h, it is more appropriate to work with dual rather than with orthonormal bases. So pick bases Xi , Yi of h which are dual with respect to Qh , i. e. Qh (Xi , Yj ) = δij . The lift of the Casimir operator of h is defined as
h = − C  ad (Xi ) ◦  ad (Yi ). i

By the Parthasarathy Lemma (Lemma 3.1), this is equal to

h = − 1 C [Xi , Zj ], Zk [Yi , Zl ], Zp Zj · Zk · Zl · Zp . 16 i

j,k,l,p

We may get rid of the sum over i immediately. Since m is orthogonal to h, we can rewrite h as C

h = − 1 C Q(Xi , [Zj , Zk ])Q(Yi , [Zl , Zp ])Zj · Zk · Zl · Zp . 16 i

j,k,l,p

For fixed j, k, l and p, the sum over i is again the expansion of the h part of Q([Zj , Zk ], [Zl , Zp ]), yielding
h = − 1 C Qh ([Zj , Zk ], [Zl , Zp ])Zj · Zk · Zl · Zp . 16

(6)

j,k,l,p

This expression has the advantage that it does not contain the dual bases Xi , Yi any more. h has no second degree term, for such a term would occur if the two It turns out that C index pairs (j, k) and (l, p) had exactly one common index, for example, j = l. But such a term would appear twice, namely, as Zj · Zk · Zj · Zp and as Zj · Zp · Zj · Zk , and these cancel out each other. h are given for all n ≥ 3 by Proposition 3.3. The terms of degree zero and 4 of C h )0 = (C

1
Qh ([Zi , Zj ], [Zi , Zj ]), 8 i,j

h )4 = − 1 (C 2




Zi , Jach(Zj , Zk , Zl )Zi · Zj · Zk · Zl .

i<j
h )4 vanishes identically for n ≤ 3, but not for n = 4. In particular, (C Proof. As the form of the result suggests, the proof is similar to the computation of H 2 (Proposition 3.2). This is why we shall be brief. For the zero degree term, (j, k) = (l, p), and each term of this kind appears twice, thus h )0 = − (C

1
Qh ([Zi , Zj ], [Zi , Zj ])Zi · Zj · Zi · Zj . 8 i,j

Since Zi · Zj · Zi · Zj = −1, we obtain the first part of the proposition. For the fourth degree contribution, rewrite the Casimir operator as

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h = − 1 C 4




Qh([Zj , Zk ], [Zl , Zp ])Zj · Zk · Zl · Zp .

(7)

j
Then the index pairs (j, k) and (l, p) have to be completely disjoint. Again, we look only at the term that is proportional to Z1234 := Z1 · Z2 · Z3 · Z4 . It may be obtained by multiplying Z12 by Z34 , Z13 by Z24 and Z14 by Z23 . Again, these elements commute, so we only need to consider each product in the order of multiplication just given and count it twice. Restoring the order of indices in these products, one sees that the term in h)4 proportional to Z1234 looks like (C 2

Qh([Z1 , Z2 ], [Z3 , Z4 ]) − Qh([Z1 , Z3 ], [Z2 , Z4 ]) 4  +Qh ([Z1 , Z4 ], [Z2 , Z3 ]) · Z1234 .

(∗) := −

By the properties of Q, the first scalar product may be formulated differently: Qh([Z1 , Z2 ], [Z3 , Z4 ]) = Q([Z1 , Z2 ], [Z3 , Z4 ]h ) = Q(Z1 , [Z2 , [Z3 , Z4 ]h ]). Rewriting the other two products in a similar way, we see that 1 (∗) = − Q(Z1 , Jach(Z2 , Z3 , Z4 )) · Z1234 . 2   3.3. A Kostant-Parthasarathy type formula for (D t )2 . If M = G/H is a symmetric space, it is well known that besides the general Schr¨odinger-Lichnerowicz formula for D 2 , which is valid on any Riemannian manifold, there exists a formula expressing D 2 in terms of Casimir operators due to Parthasarathy (see also [Kos99, Remark 1.63]). The Dirac operator D is defined relative to the Levi-Civita connection, which coincides with our one-parameter family ∇ t , and  ,  denotes an Ad (G)-invariant scalar product on g whose restriction to m is positive definite. Let Scal be the scalar curvature of the symmetric space M and G the Casimir operator of G, viewed as a second order differential operator. Theorem 3.1 ([Par72, Prop.3.1], [Fri00, Ch. 3]). On a symmetric space M = G/H , one has 1 D 2 = G + Scal, 8   and the scalar curvature may be rewritten as Scal = 8 ·  g , g  −  h , h  . This formula is the starting point for vanishing theorems, the realization of discrete series representations in the kernel of D, and it allows the computation of the full spectrum of D on M. If we now go back to the situation studied in this article, i. e. a reductive homogeneous space G/H endowed with a naturally reductive metric  ,  on m, then, a priori, the steps in the proof of Theorem 3.1 cannot be performed any longer. To prove a Kostant-Parthasarathy type formula in this situation, we recall the general expression for the Dirac operator associated with the connection ∇ t from Eq. (5) and split it into the terms coming from the canonical connection and the 3-form H , respectively:

Connections on Naturally Reductive Spaces

Dt ψ =




547

t ˜ tm (Zi )ψ =: D 0 ψ + DH Zi · Zi (ψ) + Zi ·  ψ.

(8)

i

First, notice that the equivariance property of spinors implies that the action on spinors of vector fields coming from m is by “true” differential operators, while the action of vector fields in h is in fact purely algebraic. Lemma 3.4 ([Fri00, p. 85]). Let ψ be a spinor, i. e. a section in S = G ×Ad m and X an element of h, identified with the left invariant vector field it induces. Then X(ψ) = − ad (X) · ψ, where  ad (X) · ψ denotes the Clifford product of the spinor ψ with the element  ad (X) ⊂ spin(m) ⊂ C(m). Remark 3.2. In [Kos99, Sect. 2] and [Ste99, Chap. 10.5], the map assigning to X ∈ h the sum ζ (X)(−) := X(−) +  ad (X) · − is called the “diagonal” map from h to U(g) ⊗ C(m). The assumption that the action is on spinors thus implies that this diagonal map is equal to zero. In Remark 3.4, we show how the omission of this assumption leads to the diagonally embedded Casimir operator appearing in Kostant’s formula. Proposition 3.4. The square of D 0 , the Dirac operator corresponding to the canonical connection, is given by (D 0 )2 ψ = −




h · ψ + Zi2 (ψ) + 2 C

i

1
[Zi , Zj ]m , Zk Zi · Zj · Zk (ψ). 2 i,j,k

Before proceeding to the proof, let us make a short remark on how this formula is to be understood. In the first term, one has to take the derivative of ψ along all vector fields h, we mean the image Zi twice, thus yielding a second order differential operator. By C of the Casimir operator of h inside C(m) as described in Sect. 3.2. Finally, Zi · Zj · denotes the Clifford product of Zi and Zj , whereas Zk acts again as a derivative. Thus the last term is a first order differential operator. Notice that Clifford multiplication by any constant element in C(m) commutes with derivation along m. Proof. We compute (D 0 )2 as follows:


   (D 0 )2 ψ = Zi · Z i Zj · Zj (ψ) = Zi · Zj · Zi Zj (ψ) . i

j

i,j

We divide the sum into the diagonal (i = j ) and off-diagonal (i = j ) terms and see that this separates the second and the first order differential operator contribution, (D 0 )2 ψ = −


i

Zi2 (ψ) +

1
Zi · Zj · [Zi , Zj ](ψ). 2 i,j

We concentrate our attention on the second term. Split the commutator into its m and h part, then write the m part again in the orthonormal basis Z1 , . . . , Zn to obtain

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1
1
Zi · Zj · [Zi , Zj ](ψ) = Zi · Zj · ([Zi , Zj ]m (ψ) + [Zi , Zj ]h (ψ)) 2 2 i,j

i,j

1
= Zk , [Zi , Zj ]m Zi · Zj · Zk (ψ) 2 i,j,k

+

1
Zi · Zj · [Zi , Zj ]h (ψ). 2 i,j

This takes care of the last term in the formula of Proposition 3.4. Thus it remains to show that 1
h · ψ. Zi · Zj · [Zi , Zj ]h (ψ) = 2 C 2 i,j

The action of the commutators [Zi , Zj ]h on the spinor ψ is first transformed into Clifford multiplication by the adjoint representation as explained in Lemma 3.4, then rewritten in terms of an orthonormal basis according to the Parthasarathy Lemma (Lemma 3.1), 1
1
Zi · Zj · [Zi , Zj ]h(ψ) = − Zi · Z j ·  ad ([Zi , Zj ]h ) · ψ 2 2 i,j

i,j


1
Zi · Z j [[Zi , Zj ]h, Zp ], Zq Zp · Zq · ψ. =− 8 p,q i,j

h by Eq. (6). But since [[Zi , Zj ]h, Zp ], Zq  = Qh ([Zi , Zj ], [Zp , Zq ]), this is 2 C

 

With the preparations of Sect. 3.2, the other two terms in the expression for (D t )2 are relatively easy to compute. We denote the Casimir operator of the full Lie algebra g by g ,
h · ψ. g (ψ) = − Zi2 (ψ) + C i

We decided to use a symbol different from C in order to emphasize that g is a real h, which is a constant element of the second order differential operator, as opposed to C Clifford algebra. In particular, the result of Proposition 3.4 may be restated as
h · ψ + 1 (D 0 )2 ψ = g (ψ) + C [Zi , Zj ]m , Zk Zi · Zj · Zk (ψ). (9) 2 i,j,k

First we state the formula in its most general form. Theorem 3.2 (General Kostant-Parthasarathy formula). For n ≥ 5, the square of D t is given by
1 (D t )2 ψ = g (ψ) + (1 − 3t) [Zi , Zj ]m , Zk Zi · Zj · Zk (ψ) 2 1 − 2




i,j,k

Zi , Jach(Zj , Zk , Zl ) + 9t 2 Jacm(Zj , Zk , Zl )

i<j
·Zi · Zj · Zk · Zl · ψ 1
3
+ Qh ([Zi , Zj ], [Zi , Zj ])ψ + t 2 Qm ([Zi , Zj ], [Zi , Zj ])ψ. 8 8 i,j

i,j

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549

For n ≤ 4, one has
1 (D t )2 ψ = g (ψ) + (1 − 3t) [Zi , Zj ]m , Zk Zi · Zj · Zk (ψ) 2 1 − 2 +




i,j,k

Zi , Jach(Zj , Zk , Zl ) · Zi · Zj · Zk · Zl · ψ

i<j
1
3
Qh ([Zi , Zj ], [Zi , Zj ])ψ + t 2 Qm ([Zi , Zj ], [Zi , Zj ])ψ. 8 8 i,j

i,j

Proof. The mixed term is the first order differential operator t t + DH D 0 )ψ = t (D 0 DH




 Zp · Zp (H · ψ) + H · Zp · Zp (ψ)

p

=t




 Zp · H + H · Zp · Zp (ψ).

p

In Lemma 3.3, we computed the anticommutator of H with the vector Zp , which leads us to

 3

t t D 0 DH + DH D0 ψ = − t Zp , [Zi , Zj ]m Zi · Zj Zp (ψ). 2 p



i,j

By Lemma 3.2, we have 9 t 2 (DH ) ψ = − t2 2




Zi , Jacm(Zj , Zk , Zl )Zi · Zj · Zk · Zl · ψ

i<j
3
+ t2 [Zi , Zj ]m , [Zi , Zj ]m ψ 8 i,j

for n ≥ 5 and t 2 (DH ) ψ =

3 2
[Zi , Zj ]m , [Zi , Zj ]m ψ t 8 i,j

h from Proposition 3.3, one otherwise. Together with Eq. (9) and the formula for C obtains the desired formulas.   Now it becomes clear that the particular choice t = 1/3 leads to substantial simplifications in case of n = 3 or n ≥ 5. In fact, the second part of the first line vanishes identically, the second line is zero by the Jacobi identity in g (n ≥ 5) or for dimensional reason (n = 3), and the scalar contributions in the last line appear in a very precise ratio, which will allow some further simplification. It is a strange effect that no simplification is possible for n = 4.

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Theorem 3.3 (The Kostant-Parthasarathy formula for t = 1/3). For n = 3 or n ≥ 5 and t = 1/3, the general formula for (D t )2 reduces to 1
1/3 2 (D ) ψ = g (ψ) + Qh ([Zi , Zj ], [Zi , Zj ]) 8 i,j

  1
+ Qm [Zi , Zj ], [Zi , Zj ] ψ 3 i,j

  1 1
= g (ψ) + Qm [Zi , Zj ], [Zi , Zj ] ψ. Scal1/3 + 8 9 i,j

Remark 3.3. In particular, one immediately recovers the classical Parthasarathy formula for a symmetric space (Theorem 3.1), since then all scalar curvatures coincide and [Zi , Zj ] ∈ h. Remark 3.4. If the action is not on sections of G × ad m , but on arbitrary maps ψ : G → m , Lemma 3.4 does not hold anymore. By definition of the diagonal map ζ (Remark 3.2), one uses the alternate substitution [Zi , Zj ]h (ψ) = ζ ([Zi , Zj ]h )(ψ) −  ad ([Zi , Zj ]h ) · ψ in the proof of Proposition 3.4, thus yielding the modified result

h · ψ + 1 (D 0 )2 ψ = − Zi2 (ψ) + 2 C [Zi , Zj ]m , Zk Zi · Zj · Zk (ψ) 2 i

+

1
2

i,j,k

Zi · Zj ζ ([Zi , Zj ]h )(ψ).

i,j

This is the only place where Lemma 3.4 was explicitly used in the proof of the general Kostant-Parthasarathy formula. However, there is subtle point in the definition of the h : if Lemma 3.4 holds, Casimir operator C

−  ad (Xi ) ·  ad (Yi ) · ψ = − Xi (Yi (ψ)), i

i

and hence it was safe to define the Casimir operator of h as either of these expressions. h for this element. Without Lemma 3.4, the former equality We shall keep the notation C becomes wrong and the Casimir operator should correctly be defined as


h · ψ − Ch (ψ) := − Xi (Yi (ψ)) = C ζ (Xi )(ζ (Yi )ψ) + 2  ad (Xi ) · ζ (Yi )ψ. i

i

The second term shall be denoted henceforth ζ (Ch )ψ and is the “diagonally” embedded Casimir operator appearing in [Kos99]. The full Casimir operator of g becomes  g (ψ) = − i Zi2 (ψ) + Ch (ψ). With this modification, one checks that the general Kostant-Parthasarathy formula (Theorem 3.2) reads as before with the following additional term not depending on the parameter t,
1
−ζ (Ch )ψ − 2  ad (Xi ) · ζ (Yi )ψ + Zi · Zj ζ ([Zi , Zj ]h )(ψ). 2 i

i,j

But it is a routine computation to show that the last two terms cancel each other, hence only the diagonal term remains. This finishes the proof that Theorem 3.2 is, from the

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551

computational point of view, equivalent to [Kos99, Thm 2.13]. Since we are interested in spinors, these considerations will not play any further role in this paper. As in the classical Parthasarathy formula, the scalar term as well as the eigenvalues of g (ψ) may be expressed in representation theoretical terms if G (and hence M) is compact. Consider the unique Ad (G)-invariant extension Q of the scalar product  ,  on m to the full Lie algebra g, which exists by Kostant’s Theorem. Thus, Q is a multiple of the Killing form on any simple factor of g; however, Q is not necessarily positive definite, hence the scaling factors may be of different sign. If they are such that Q is positive definite, the metric  ,  is said to be normal homogeneous. We begin with a more careful analysis of the Casimir operator g (ψ). By the same arguments as in the symmetric space case, g (ψ) is a G-invariant differential operator, and this property does not depend on the signs of Q. We sketch the argument for completeness: On every simple summand gi of g, Qi := Q|gi is either a positive or a negative multiple of the Killing form, and Ad (g) maps gi into itself. Hence, in either case, the adjoint action of G transforms an orthonormal base Z˜ 1 , . . . , Z˜ m of gi into an orthonormal base, and dual bases X˜ 1 , Y˜1 , . . . , X˜ m , Y˜m of gi are mapped to dual bases: Qi (Ad (g)Z˜ k , Ad (g)Z˜ l ) = Qi (Z˜ k , Z˜ l ),

Qi (Ad (g)X˜ k , Ad (g)Y˜l ) = Qi (X˜ k , Y˜l ).

Now consider the Frobenius decomposition of the square integrable spinors into irreducible finite-dimensional representations Vλ of G,
L2 (S) = Mλ ⊗ V λ , ˆ λ∈G

where Mλ denotes the multiplicity space of Vλ . Let λ : G → GL(Vλ ) be the representation with highest weight λ, and d λ its differential. Then gi acts on Vλ by d λ (gi ) = −

m
k=1

d λ (Z˜ k )2 or d λ (gi ) = −

m


d λ (X˜ k )d λ (Y˜k ).

k=1

However, for any element X ∈ gi , one checks immediately  

λ (g)d λ (X) λ g −1 = d λ (Ad (g)X), hence gi commutes with the action of g ∈ G on Vλ , as claimed. Furthermore, it acts by multiplication by the well-known eigenvalue Qi (λ + i , λ + i ) − Qi ( i , i ), whose sign, however, depends on whether Qi is a positive or a negative multiple of the Killing form on gi . Here, i denotes the half sum of positive roots of gi , as usual. Since the center of G does not contribute to the total eigenvalue of g , we conclude: Lemma 3.5. For a compact group G, the operator g is non-negative if the metric  ,  is normal homogeneous or if the negative definite contribution to Q comes from an abelian summand in g.   In a forthcoming paper, we will discuss examples where Q has also a simple summand on which Q is negative definite and show that g has negative eigenvalues. We use these remarks to express the scalar term in Theorem 3.3 in a different way.

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Lemma 3.6. Let G be compact, n = 3 or n ≥ 5, and denote by g and h the half sum of the positive roots of g and h, respectively. Then the Kostant-Parthasarathy formula for (D 1/3 )2 may be restated as

 (D 1/3 )2 ψ = g (ψ) + Q( g , g ) − Q( h , h ) ψ = g (ψ) +  g − h , g − h ψ. In particular, the scalar term is positive independently of the properties of Q. Proof. Consider the eightfold multiple of the term under consideration and regroup it as 8((D 1/3 )2 − g ) =




Qh ([Zi , Zj ], [Zi , Zj ]) +

i,j

  1
Qm [Zi , Zj ], [Zi , Zj ] 3 i,j




   1
 = Q [Zi , Zj ], [Zi , Zj ] + 2 Qh [Zi , Zj ], [Zi , Zj ] . 3 i,j

i,j

The first summand can easily be seen to be a trace over m,

Q([Zi , Zj ], [Zi , Zj ]) = − Q([Zi , [Zi , Zj ]], Zj ) i,j

i,j

=−







Q (ad Zi )2 , Zj

j

 = −tr m

i




(ad Zi )2 .

i

For the second term, we first notice that it may be rewritten by expanding and contracting in two different ways as

  Qh ([Zi , Zj ], [Zi , Zj ]) = Q Xk , [Zi , Zj ] Q(Yk , [Zi , Zj ]) i,j

i,j,k

=




Q([Xk , Zi ], Zj )Q([Yk , Zi ], Zj )

i,j,k

=




Q ([Xk , Zi ], [Yk , Zi ]) .

i,k

This, in turn, can be identified with two different kinds of traces: On the one hand, this is

− Q([Zi , [Zi , Xk ]], Yk ) = −tr h (ad Zi )2 , i,k

i

on the other hand, this reads

− Q([Xk , [Yk , Zi ]], Zi ) = −tr m (ad Xk )(ad Yk ) = tr m Ch , i,k

k

where Ch denotes the “unlifted” Casimir operator of h, i. e. its usual action on g via the adjoint representation. Now, since the sum we have just treated appears twice, we use each way of writing it once to obtain

Connections on Naturally Reductive Spaces

8((D

553

 

1 2 2 ) − g ) = (ad Zi ) − tr h (ad Zi ) + tr m Ch −tr m 3 i i  
1 2 = −tr g (ad Zi ) + tr g Ch − tr h Ch 3

1/3 2

i

 1

= tr g Cg − tr h Ch . 3 Again, Cg is not to be confused with the action of the Casimir operator of g on spinors. By looking separately at every simple summand where Q is just a multiple of the Killing form, one easily sees that these traces are the rescaled lengths of the half sum of positive roots, tr g Cg = 24 Q( g , g ), and similarly for h (Proposition 1.84 in [Kos99]). This proves the formula. To see that the scalar is positive even for non-normal homogeneous metrics, decompose g = h + R, where R ∈ m. Since m and h are orthogonal with respect to Q, one obtains Q( g , g )−Q( h , h ) = Q( h +R, h +R)−Q( h , h ) = Q(R, R) = R, R > 0, since by dimensional reasons R = 0 and the scalar product on m is positive definite.   We can formulate our first conclusion from Theorem 3.3: Corollary 3.1. Let G be compact. If the operator g is non-negative, the first eigenvalue 1/3 λ1 of the Dirac operator D 1/3 satisfies the inequality  1/3 2 λ1 ≥ Q( g , g ) − Q( h , h). Equality occurs if and only if there exists an algebraic spinor in m which is fixed under  H ) of the isotropy representation. the lift κ(Ad Proof. By our assumption on g , its eigenvalue on a spinor ψ can be zero if and only if the Casimir eigenvalue of every simple summand gi of g vanishes, hence ψ has to lie in the trivial G-representation and is thus constant.   Remark 3.5. This eigenvalue estimate is remarkable for several reasons. Firstly, for homogeneous non-symmetric spaces, it is sharper than the classical Parthasarathy formula. For a symmetric space, one classically obtains λ21 ≥ Scal/8. But since the Schr¨odingerLichnerowicz formula yields immediately λ21 ≥ Scal/4, the lower bound in the classical Parthasarathy formula is never attained, and hence of little interest. In contrast, there exist many examples of homogeneous non-symmetric spaces with constant spinors. Secondly, it uses a lower bound which is always strictly positive; for naturally reductive metrics where the non-positive definite part of the metric comes from an abelian factor, the scalar curvature can become negative and hence a pure curvature bound would again be of small interest. Finally, of the known formulas of Weitzenb¨ock type which generalize the Schr¨odinger-Lichnerowicz formula, one does not yield an eigenvalue estimate ([FI01, Thm. 3.1] for any metric connection with skew symmetric torsion), another one yields an eigenvalue estimate which is not sharp and applies only to the normal homogeneous case (see [Goe99, Lemma 1.17]). The example of Sect. 5 illustrates the situation described here.

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Remark 3.6. Since D t is a G-invariant differential operator on M by construction, Theorem 3.2 implies that the linear combination of the first order differential operator and the multiplication by the element of degree four in the Clifford algebra appearing in the formula for (D t )2 is again G-invariant for all t. Hence, the first order differential operator
Dψ := [Zi , Zj ]m , Zk Zi · Zj · Zk (ψ) i,j,k

has to be a G-invariant differential operator, a fact that cannot be seen directly by any simple arguments. It has no analogue on symmetric spaces and certainly deserves further separate investigations. 4. The Equations of Type II String Theory on Naturally Reductive Spaces 4.1. The field equations. The common sector of type II string theories may be geometrically described as a tuple (M n ,  , , H, , ) consisting of a manifold M n with a Riemannian metric  , , a 3-form H, a so-called dilaton function and a spinor field  satisfying the coupled system of field equations 1 LC −2 H ) = 0, RicLC ij − Himn Hj mn + 2∇i ∂j = 0, δ(e 4 1 1 LC (∇X + X H ) = 0, (d − H ) = 0. 4 2 The first equation generalizes the Einstein equation, the second is a conservation law, while the first of the spinorial field equations suggests that the 3-form H should be the torsion of some metric connection ∇ with totally skew-symmetric torsion tensor T = H . Then the equations may be rewritten in terms of ∇: 1 1 Ric∇ + δ(T )+2∇ LC d = 0, δ(T ) = 2 · d # T , ∇ = 0, (d − T )· = 0. 2 2 If the dilaton is constant, the equations may be simplified even further, Ric∇ = 0,

δ(T ) = 0,

∇ = 0,

T ·  = 0.

In particular, the last equation becomes a purely algebraic condition. The number of preserved supersymmetries depends essentially on the number of ∇-parallel spinors. For a general background on these equations, we refer to the article by A. Strominger where they appeared first [Str86]. A routine calculation shows that δ(T ) is the skew symmetric part of the Ricci curvature, hence the first equation implies the second (see [FI01]). In any event, for the family of connections ∇ t , the second equation is always satisfied by Lemma 2.5. Before proceeding further, we add a general observation which follows easily from the formulas in [FI01] and which resulted from discussions with Bogdan Alexandrov (Humboldt-Universit¨at zu Berlin). Theorem 4.1. Let M n be a compact Riemannian manifold with metric  ,  and a metric connection ∇ with totally skew symmetric torsion T . Suppose that there exists a spinor field ψ such that all the equations Ric∇ = 0,

∇ = 0,

T · = 0

hold. Then T = 0 and ∇ is the Levi-Civita connection.

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Proof. If ψ is ∇-parallel, the Riemannian Dirac operator D LC acts on ψ by D LC ψ = −3T · ψ/4. The last equation thus implies D LC ψ = 0. By the classical Schr¨odingerLichnerowicz formula,   1 0 = ||∇ LC ψ||2 dM n + ScalLC ||ψ||2 dM n . 4 Mn Mn On the other hand, the two Ricci tensors are related by the equation n 1
 1 RicLC (X, Y ) = Ric∇ (X, Y ) + (δT )(X, Y ) + T (X, ei ), T (Y, ei ) , 2 4 i=1

where e1 , . . . , en denotes an orthonormal basis. If Ric∇ = 0, then δT = 0 (see above), and this implies that the Riemannian scalar curvature is non-negative and given by 4 ScalLC =

n


T (ei , ej ), T (ei , ej ).

i,j =1

Consequently, the scalar curvature ScalLC has to vanish identically, and the torsion form T is zero, too.   Hence, compact solutions to all equations have to be Calabi-Yau manifolds in dimensions 4 and 6, Joyce manifolds in dimensions 7 and 8, etc.

4.2. Some particular spinor fields. Consider the situation that the lift of the isotropy  H ) contains the trivial representation, i. e. an algebraic spinor ψ representation κ(Ad that is fixed under the action of H . Any such spinor induces a section of the spinor bundle S = G ×κ(Ad ) m if viewed as a constant map G → m and is thus of particular interest. Theorem 4.2. (1) Any constant spinor field ψ satisfies the equation ∇Zt ψ =

t (Z 3

H )ψ.

In particular, it is parallel with respect to the canonical connection (t = 0). Conversely, any spinor field ψ satisfying ∇ 0 ψ = 0 is necessarily constant. (2) Any constant spinor field ψ is an eigenspinor of the square of the Dirac operator (D t )2 , and its eigenvalue does not depend on the special choice of ψ:

 (D t )2 ψ = 9t 2 Q( g , g ) − Q( h , h ) ψ. In particular, H · ψ = 0 and hence the last string equation can never hold for a constant spinor. Proof. For a constant spinor field, the formula for the covariant derivative of a spinor ˜ tm (Z)ψ. By Lemma 3.2,  ˜ t (Z) may be expressed field (Eq. 3) reduces to ∇Zt ψ = 0 +  in terms of an orthonormal basis as

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∇Zt ψ =

t
[Z, Zj ]m , Zk Zj · Zk · ψ. 2 j
By the definition of H , this is easily seen to be t (Z H )/3. Conversely, assume that ψ is parallel with respect to the canonical connection, i. e. Zi (ψ) = 0 for all i. Then [Zi , Zj ](ψ) = 0, and the commutator [Zi , Zj ] may be split into its m and h part. But the m part acts again trivially on ψ, hence we obtain [Zi , Zj ]h (ψ) = 0. By Assumption 2.2, [m, m] spans all of h, hence h also acts trivially on ψ, which finishes the argument. For the second part of the theorem, we use that the Dirac operator on a constant spinor is given by D t ψ = t H · ψ for any t. Since any constant spinor lies in the trivial G-representation in the Frobenius decomposition of (S), the eigenvalue of g on ψ is zero. For t = 1/3, the Kostant-Parthasarathy formula (Theorem 3.3) thus yields

 1 (D 1/3 )2 ψ = Q( g , g ) − Q( h , h ) ψ = H 2 ψ. 9 This may be understood as a formula for H 2 ψ, from which we immediately derive the general formula through (D t )2 ψ = t 2 H 2 ψ. In particular, H · ψ cannot vanish.   Remark 4.1. Easy examples show that ψ might not be an eigenspinor of D t itself, since not all constant spinors are eigenspinors of H . For the canonical connection, ∇ 0 T 0 = 0 implies that the space of parallel spinors is invariant under T 0 , hence there exists a basis of the space of parallel spinors consisting of eigenspinors. 4.3. Vanishing theorems. This section is devoted to non-existence theorems for solutions in certain geometric configurations. It allows us to draw quite a precise picture of what a promising naturally reductive metric should look like. First, the Kostant-Parthasarathy formula yields that we should be interested in precisely those metrics where g is not non-negative. Theorem 4.3. If the operator g is non-negative and ∇ t is not the Levi-Civita connection, there do not exist any non-trivial solutions to the system of equations ∇ t ψ = 0,

T t · ψ = 0.

Proof. If the spinor ψ is ∇ t -parallel, then it lies in the kernel of D t = D 0 +tH . Since ∇ t is assumed not to be the Levi-Civita connection, T t does not vanish and hence T t ·ψ = 0 implies H · ψ = 0. Thus ψ is also in the kernel of D 0 . For the Dirac operator to the parameter t = 1/3, we obtain 1 D 1/3 ψ = D 0 ψ + H · ψ = 0, 3 which contradicts Corollary 3.1.

 

For the Levi-Civita connection, it is well known that the existence of a parallel spinor implies vanishing Ricci curvature. By repetition of the same argument, one sees that this conclusion does no longer hold for a metric connection with torsion. Rather, we get restrictions on the algebraic type of the derivatives of the torsion.

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Proposition 4.1. If the canonical connection ∇ 0 is Ricci flat and admits a parallel spinor, then the exterior derivative of its torsion T 0 satisfies (X dT 0 ) · ψ = 0 for all vectors X in m. Proof. In [FI01, Cor. 3.2], Friedrich and Ivanov showed that a spin manifold with some connection ∇ whose torsion T is totally skew symmetric and a ∇-parallel spinor ψ satisfies

1 X dT + ∇X T · ψ = Ric∇ (X) · ψ. 2 Since the canonical connection satisfies ∇ 0 T 0 = 0, the claim follows.

 

These conditions are independent of the equation T 0 · ψ = 0. If dT 0 = 0 and the dimension is sufficiently small, it can happen that the intersection of all kernels of X dT 0 is already empty, thus showing the non-existence of solutions. Models with dT 0 = 0 are of particular interest and are called closed in string theory. For further investigations of the Ricci tensor
   Rict (X, Y ) = (t − t 2 ) [X, Zi ]m , [Y, Zi ]m + Qh [X, Zi ], [Y, Zi ] , i

it is useful to describe it from a more representation theoretical point of view. Wang and Ziller derived the general formula we shall present for t = 1/2 in [WZ85]. Their proof may easily be generalized to the case of arbitrary t, hence we omit it here. The main idea is to use a more elaborate version of the core computation in the proof of Lemma 3.6. Recall that Ch denotes the (unlifted) Casimir operator of h, i. e. Ch = −




ad Xi ad Yi .

i

It defines a symmetric endomorphism A : m × m → m by A(X, Y ) := Ch X, Y . Similarly, we denote by β(X, Y ) = −tr g ad Xad Y the Killing form of the full Lie algebra g. We make no notational difference between β itself and its restriction to m.   Theorem 4.4. The endomorphisms A and β satisfy the identities

   A(X, Y ) = Qh [X, Zi ], [Y, Zi ] , β(X, Y ) = [X, Zi ]m , [Y, Zi ]m +2A(X, Y ). i

i

Thus, the Ricci tensor is given by Rict (X, Y ) = (t − t 2 )β(X, Y ) + (2t 2 − 2t + 1)A(X, Y ).   Remark 4.2. We observe that the coefficient of β vanishes for t = 0 and t = 1, and is positive between these parameter values, whereas the coefficient of A is always positive and attains its minimum for the Levi-Civita connection (t = 1/2).

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5

6

7

8

SU(2) SU(3) G2 Spin(7)

The endomorphism A has block diagonal structure, with every block corresponding to an irreducible summand of the isotropy representation. In particular, the block of the trivial representation vanishes, since its Casimir eigenvalue is zero. Since β is positive definite for G compact, we can deduce: Proposition 4.2. Assume that G is compact. If the isotropy representation Ad : H → SO(m) has fixed vectors, only the connections t = 0 and t = 1 can be Ricci flat.   Typically, the eigenvalues of Ch are linear functions of some deformation parameters, hence, they can vanish for some particular parameter choices without belonging to a trivial h-summand of m. This makes it difficult to make more precise predictions for the vanishing of the Ricci tensor. Proposition 4.3. If the canonical connection has vanishing scalar curvature, H cannot be simple and the metric cannot be normal homogeneous.  Proof. The scalar curvature for the canonical connection is i,j Qh ([Zi , Zj ], [Zi , Zj ]). By Assumption 2.2, not all vectors [Zi , Zj ]h can be zero. Since Qh is non-degenerate, we conclude that Qh can be neither positive nor negative definite. However, on every simple factor of h, Qh has to be a multiple of the Killing form; hence h cannot be simple.   This fact, as elementary as its proof might be, has far-reaching consequences for the geometry of homogeneous models of string theory. The existence of a parallel spinor severely restricts the holonomy group of ∇. In fact, it needs to be a subgroup of the isotropy subgroup of a spinor inside SO(n), and these subgroups are well-known. By a theorem of Wang ([KN96, Ch.X, Cor. 4.2]), the Lie algebra of the holonomy group is spanned by m0 + [m (m), m0 ] + [m (m), [m (m), m0 ]] + . . . , where the subspace m0 is defined as m0 = {[m (X), m (Y )] − m ([X, Y ]m ) − ad ([X, Y ]h ) : X, Y ∈ m }. For the canonical connection and using our assumption that [m, m]h spans all of h, we conclude that its holonomy Lie algebra is precisely h. For t = 0, the holonomy can only increase, hence we obtain Table 1 for the maximally possible subgroups Hmax . If we restrict our attention to the canonical connection, Proposition 4.3 implies that H cannot be equal to Hmax itself, but rather has to be a non-simple subgroup of it. This excludes many homogeneous spaces that would naturally come to mind. Of course, they might yield models for other connections than the canonical one, but such an analysis can only be performed on a case by case basis.

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5. Examples 5.1. The Jensen metric on V4,2 . The 5-dimensional Stiefel manifold V4,2 = SO(4)/SO(2) carries a one-parameter family of metrics constructed by G. Jensen [Jen75] with many remarkable properties. Embed H = SO(2) into G = SO(4) as the lower diagonal 2 × 2 block. Then the Lie algebra so(4) splits into so(2) ⊕ m, where m is given by    0 −a     −X t a 0  m =  =: (a, X) : a ∈ R, X ∈ M2,2 (R) .  00     X 00 Denote by β(X, Y ) := tr(Xt Y ) the Killing form of so(4). Then the Jensen metric on m to the parameter s ∈ R is defined by  1 1 (a, X), (b, Y ) = β(X, Y ) + sβ(a, b) = β(X, Y ) + 2s · ab. 2 2 For s = 2/3, G. Jensen proved that this metric is Einstein, and Th. Friedrich showed that it admits two Riemannian Killing spinors [Fri80] and thus realizes the equality case in his estimate for the first eigenvalue of the Dirac operator. It will become clear in the discussion that this metric is only naturally reductive with respect to G = SO(4) for s = 1/2. 5.2. General remarks. Denote by Eij the standard basis of so(4). Then the elements 1 Z1 := E13 , Z2 := E14 , Z3 = E23 , Z4 = E24 , Z5 = √ E12 2s form an orthonormal base of m. To start with, we compute all nonvanishing commutators in m. These are √ √ 1 [Z1 , Z3 ]m = 2s Z5 , [Z1 , Z5 ]m = − √ Z3 , [Z2 , Z4 ]m = 2s Z5 , 2s (*) 1 1 1 [Z2 , Z5 ]m = − √ Z4 , [Z3 , Z5 ]m = √ Z1 , [Z4 , Z5 ]m = √ Z2 . 2s 2s 2s Note that all these commutators have no h-contribution. We identify m with R 5 via the cos θ − sin θ chosen basis, and denote elements in H = SO(2) by g(θ) = ∈H = sin θ cos θ SO(2). As in [Fri80], we use a suitable basis ψ1 , . . . , ψ4 for the 4-dimensional spinor representation κ : Spin(R5 ) → GL(5 ). Then one derives for the isotropy representation and its lift the expressions     cos θ − sin θ 0 0 0 eiθ 0 0 0 0 0 0  sin θ cos θ −iθ 0 0       .  g(θ) =  0 e 0 cos θ − sin θ 0  , κ Ad Ad g(θ ) =  0   0 0 1 0  0  0 sin θ cos θ 0 0 0 0 1 0 0 0 0 1 Thus, the elements ψ3 and ψ4 define sections of the spinor bundle S = G ×κ(Ad ) 5 if viewed as constant maps G → 5 . In fact, for s = 2/3, ψ ± := ψ3 ∓ iψ4 are

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exactly the Riemannian Killing spinors from [Fri80] as we will see below. The sections induced by ψ1 and ψ2 are not constant and thus more difficult to treat. We will not consider them in our discussion. In [Jen75, Prop. 3], the author computed the map ∼ 5 LC m : m = R → so(5) (see Wang’s Theorem in Sect. 2) defining the Levi-Civita connection: 1 [Zα , Zβ ], LC m (Z5 )Zα = (1 − s)[Z5 , Zα ], 2 LC m (Zα )Z5 = s[Zα , Z5 ] for α, β = 1, . . . , 4.

LC m (Zα )Zβ =

Thus, one sees that for s = 1/2, m (X)Y is not globally proportional to the commutator [X, Y ]m , and both m (X)Y, Z and −[X, Y ]m , Z (the torsion of the canonical connection) fail to define a 3-form: The first is not skew symmetric in X and Y , the second is not skew symmetric in X and Z. In any case, by using the commutator relations (∗), the Levi-Civita connection can be identified with an endomorphism of R5 as follows: LC m (Z1 ) =

s E35 , 2

LC m (Z4 ) = −

LC m (Z2 ) = s E25 , 2

s E45 , 2

LC m (Z3 ) = −

s E15 , 2

1−s LC (E13 + E24 ). m (Z5 ) = √ 2s

The lift into the spin representation yields a global factor 1/2 and replaces Eij by Zi ∧Zj . By setting T˜ := (Z1 ∧ Z3 + Z2 ∧ Z4 ) ∧ Z5 , the Levi-Civita connection may be rewritten in a unified way as ˜ LC  m (Z5 ) =

1 2(1 − s)  Z5 √ 4 2s

 T˜ ,

˜ LC  m (Zα ) =

1√  2s Zα 4

T˜



for α = 1, . . . , 4.

(10) A careful analysis shows that V4,2 carries three different contact structures, one of which is Sasakian, one quasi-Sasakian but not Sasakian, and the third one has no special name, although special properties. Because the Nijenhuis tensor vanishes for all these metric almost contact structures, V4,2 admits a unique √ almost contact connection ∇ by [FI01, Thm. 8.2], whose torsion is given by T = − 2s T˜ . Theorem 5.1. (1) The constant spinors are parallel with respect to the contact connection ∇ if and only if s = 1/2; (2) The constant spinors ψ ± are Riemannian Killing spinors if and only if s = 2/3. Proof. In Eq. (10), we gave the general formula for the Levi-Civita connection in direction Zi as the inner product of Zi and the 3-form T˜ . If a constant spinor ψ is to be parallel with respect to ∇, ! " 1 LC ˜ 0 = ∇X ψ = m (X) + X T ψ, 4 ˜ LC then√the coefficients of  m as in Eq. (10) have to be equal for all Zi , hence, 2(1 − √ ˜ LC (X) + s)/ 2s = 2s, which means that s = 1/2. For this value, the combination 

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1 4X

T vanishes, so both constant spinors are parallel indeed. For the discussion of Riemannian Killing spinors, one uses a standard realization of the spin representation (see [Fri00, Prop., p. 13]) to check (Z5

T˜ ) · ψ ± = ±2Z5 · ψ ± ,

(Zα

T˜ ) · ψ ± = ±Zα · ψ ± for α = 1, . . . , 4.

LC ψ = µ X · ψ implies that the Looking at Z5 , we conclude that the Killing equation ∇X √ √ coefficients in Eq. (10) have to satisfy 2(1 − s)/ 2s = 2s/2. The solution is now s = 2/3, and one checks that ψ ± are Killing spinors indeed.  

In [FI01a], Friedrich and Ivanov studied the 5-dimensional contact case in more detail. 5.3. The naturally reductive space approach. We would like to interpret the metric  ,  ¯ and the connection as a naturally reductive metric with respect to some other group G, with the torsion √ T = − 2s (Z1 ∧ Z3 + Z2 ∧ Z4 ) ∧ Z5 ¯ H¯ with the Lie algebra decomposition as its canonical connection. So write M = G/ ¯ and assume that the original isotropy representation is a subrepresentag¯ = h¯ ⊕ m, ¯ remains tion of the new isotropy representation, i. e. the action of h ⊂ h¯ on m ∼ = m unchanged. This point of view necessarily enlarges the holonomy group H already for dimensional reasons. In fact, we can deduce a lot of information about the new isotropy representation from the formula for T . In Remark 2.2, we explained the relation between m-commutators and torsion. For example, the formula above implies √ √ [Z1 , Z3 ]m 2sZ5 , [Z4 , Z5 ]m 2sZ2 , [Z1 , Z4 ]m ¯ = ¯ = ¯ = [Z3 , Z4 ]m ¯ = 0. Then we can compute Jacm ¯ (Z1 , Z3 , Z4 ) = 2s Z2 . On the other hand, !

Jach¯ (Z1 , Z3 , Z4 ) = −Z2 +[Z4 , [Z1 , Z3 ]h¯ ]+[Z3 , [Z4 , Z1 ]h¯ ] = −Jacm ¯ (Z1 , Z3 , Z4 ). ¯ not both Thus, there must be two elements H1 := [Z1 , Z3 ]h¯ and H2 := [Z4 , Z1 ]h¯ in h, zero, such that [H1 , Z4 ] + [H2 , Z3 ] = (2s − 1)Z2 . By some more careful analysis, one obtains H2 = 0, H1 = [Z2 , Z4 ]h¯ and the action of H1 on the other vectors Zi . The systematic description of  ,  as a naturally reductive metric can be given using a deformation construction due to Chavel and Ziller ([Cha70], [Zil77]). It is based on the remark that for s = 1/2, m splits into an orthogonal direct sum of m1 := {(0, X)} and m2 := {(a, 0)} such that [h, m2 ] = 0 and [m2 , m2 ] ⊂ m2 . ¯ = G × M2 , Let M2 ⊂ G be the subgroup of G with Lie algebra m2 , and set G ¯ acts on M = G/H by (k, m)gH = kgH m−1 , H¯ = H × M2 . An element (k, m) of G and then H¯ can indeed be identified with the isotropy group of this action. We endow g¯ = g ⊕ m2 with the direct sum Lie algebra structure. The trick is now to choose a ¯ that depends on the deformation parameter s of the metric. Writing all realization of m

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elements of g¯ as 4-tuples (H, U, X, Y ) with H ∈ h, U ∈ m1 and X, Y ∈ m2 , we can realize the Lie algebra of H¯ as h¯ = {(H, 0, X, X) ⊂ g¯ : H ∈ h, X ∈ m2 } and choose

¯ = {(0, X, 2s Y, (2s − 1)Y ) : X ∈ m1 , Y ∈ m2 } m

as an orthogonal complement. Here, (0, 0, 2s Y, (2s − 1)Y ) will be identified with Y ∈ m2 . Since m2 is abelian in this example, the Lie algebra structure of g¯ is particularly simple. h¯ is a Lie algebra with commutator [(H, 0, X, X), (H  , 0, X , X )] = ([H, H  ], 0, 0, 0), the full isotropy representation is [(H, 0, X, X), (0, U, 2sY, (2s − 1)Y )] = (0, [H + X, U ], 0, 0) ¯ splits into its h¯ and m ¯ part as follows: and the commutator of two elements in m [(0, U, 2s X, (2s − 1)X), (0, V , 2s Y, (2s − 1)Y )]   = [U, V ]h , 0, −(2s − 1)[U, V ]m2 , −(2s − 1)[U, V ]m2   + 0, [U, V ]m1 + 2s([U, Y ] + [X, V ]), 2s[U, V ]m2 , (2s − 1)[U, V ]m2 . ¯ ¯ the metric  ,  is naturally reductive with respect to G, With these choices for h¯ and m, the torsion of its canonical connection is precisely T and the Ricci tensor is given by Ric0 = 2(1 − s)diag(1, 1, 1, 1, 0). For s = 1, the canonical connection is thus Ricci flat, and by Proposition 4.2, we know that no other connection can have this property. However, the holonomy H¯ ∼ = SO(2) × SO(2) is too large to admit parallel spinors. For s = 1/2, we have two parallel spinors for the canonical connection as seen in the preceding section, but the Ricci curvature does not vanish. In this case, one can ask the question whether some other connection of the family ∇ t admits parallel spinors. But using Wang’s Theorem ([KN96, Ch.X, Cor. 4.2]) for computing the holonomy, one sees that ∇ t has full holonomy SO(m) for t = 0, excluding again the existence of parallel spinors. We close this section with a look at the eigenvalue estimate for (D 1/3 )2 . Since the extension of H is by the abelian group SO(2), the Casimir operator g is non-negative by Lemma 3.5, and Corollary 3.1 can be applied. We compute the scalar in the general Kostant-Parthasarathy formula (Theorem 3.2) 1
3
Qh ([Zi , Zj ], [Zi , Zj ]) + t 2 Qm ([Zi , Zj ], [Zi , Zj ]) 8 8 i,j

i,j

1 3 = · 8(1 − s) + t 2 · 24s = 1 + (9t 2 − 1)s 8 8 and see that it is independent of the deformation parameter s precisely for the Kostant connection t = 1/3. If s = 1/2, there exist no constant spinors and hence Corollary 3.1 is a strict inequality, (λ1/3 )2 > 1.

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For s = 1/2, there exists a constant spinor ψ and it satisfies by Theorem 4.2, (D t )2 ψ = 9t 2 · 1 · ψ = 9t 2 ψ. Unfortunately, we have been unable to relate this bound with the infimum of the spectrum of (D t )2 for other values of t. In particular, it seems to be difficult to deduce from Corollary 3.1 any information about the Riemannian Dirac spectrum. Acknowledgements. I am grateful to Thomas Friedrich (Humboldt-Universit¨at zu Berlin) for many valuable discussions on the topic of this paper. My thanks are also due to the Erwin-Schr¨odinger Institute in Vienna and the Max-Planck Institute for Mathematics in the Natural Sciences in Leipzig for their hospitality.

References [Cha70] Chavel, I.: A class of Riemannian homogeneous spaces. J. Differ. Geom. 4, 13–20 (1970) [FI01a] Friedrich, Th., Ivanov, S.: Almost contact manifolds, connections with torsion, and parallel spinors, to appear in Journ. Reine Angew. Math. [FI01] Friedrich, Th., Ivanov, S.: Parallel spinors and connections with skew-symmetric torsion in string theory. Asian Journ. Math. 6, 303–336 (2002) [Fri80] Friedrich, Th.: Der erste Eigenwert des Dirac-Operators einer kompakten, Riemannschen Mannigfaltigkeit nichtnegativer Skalarkr¨ummung. Math. Nachr. 97, 117–146 (1980) [Fri00] Friedrich, Th.: Dirac operators in Riemannian geometry. Graduate Studies in Mathematics, Vol. 25, Providence, RI: American Mathematical Society, 2000 [FS79] Friedrich, Th., Sulanke, S.: Ein Kriterium f¨ur die formale Selbstadjungiertheit des Dirac-Operators. Coll. Math. XL, 239–247 (1979) [Goe99] Goette, S.: Equivariant η-invariants on homogeneous spaces. Math. Z. 232, 1–42 (1999) [Ike75] Ikeda, A.: Formally self adjointness for the Dirac operator on homogeneous spaces. Osaka J. Math. 12, 173–185 (1975) [Jen75] Jensen, G.: Imbeddings of Stiefel manifolds into Grassmannians. Duke Math. J. 42 (3), 397– 407 (1975) [KN91] Kobayashi, S., Nomizu, K.: Foundations of Differential Geometry I. Wiley Classics Library, Princeton, NJ: Wiley Inc., 1963, 1991 [KN96] Kobayashi, S., Nomizu, K.: Foundations of Differential Geometry II. Wiley Classics Library, Princeton, NJ: Wiley Inc., 1969, 1996 [Kos56] Kostant, B.: On differential geometry and homogeneous spaces II. Proc. N. A. S. 42, 354–357 (1956) [Kos99] Kostant, B.: A cubic Dirac operator and the emergence of Euler number multiplets of representations for equal rank subgroups. Duke Math. J. 100 (3), 447–501 (1999) [Lan00] Landweber, G.: Harmonic spinors on homogeneous spaces. Repr. Theory 4, 466–473 (2000) [Par72] Parthasarathy, R.: Dirac operator and the discrete series. Ann. Math. 96 (1), 1–30 (1972) [Sle87a] Slebarski, S.: The Dirac operator on homogeneous spaces and representations of reductive Lie groups I. Amer. J. Math. 109, 283–301 (1987) [Sle87b] Slebarski, S.: The Dirac operator on homogeneous spaces and representations of reductive Lie groups II. Amer. J. Math. 109, 499–520 (1987) [Ste99] Sternberg, S.: Lie Algebras, Lecture notes, 1999 [Str86] Strominger, A.: Superstrings with torsion. Nucl. Phys. B 274, 253–284 (1986) ´ Norm. [WZ85] Wang, M.Y., Ziller, W.: On normal homogeneous Einstein manifolds. Ann. Sci. Ec. Sup., 4e s´erie 18, 563–633 (1985) [Zil77] Ziller, W.: The Jacobi equation on naturally reductive compact Riemannian homogeneous spaces. Comment. Math. Helv. 52, 573–590 (1977) Communicated by H. Nicolai