Commun. Math. Phys. 308, 1–21 (2011) Digital Object Identifier (DOI) 10.1007/s00220-011-1316-8
Communications in
Mathematical Physics
On the Correlation Function of the Characteristic Polynomials of the Hermitian Wigner Ensemble Tatyana Shcherbina Institute for Low Temperature Physics Ukr.Ac.Sci., 47 Lenin Ave., 61103 Kharkov, Ukraine. E-mail:
[email protected] Received: 7 July 2010 / Accepted: 19 March 2011 Published online: 27 August 2011 – © Springer-Verlag 2011
Abstract: We consider the asymptotic of the correlation functions of the characteristic polynomials of the hermitian Wigner matrices Hn = n −1/2 Wn . We show that for the correlation function of any even order the asymptotic coincides with this for the Gaussian Unitary Ensemble up to a factor, depending only on the fourth moment of the common probability law of entries W jk , W jk , i.e. that the higher moments do not contribute to the above limit. 1. Introduction Characteristic polynomials of random matrices have been actively studied in recent years. The interest was initially stimulated by the similarity between the asymptotic behavior of the moments of characteristic polynomials of a random matrix from the Circular Unitary Ensemble and the moments of the Riemann ζ -function along its critical line (see [8]). But with the emerging connections to the quantum chaos, integrable systems, combinatorics, representation theory and others, it has become apparent that the characteristic polynomials of random matrices are also of independent interest. This motivates the asymptotic studies of the moments of characteristic polynomials for other random matrix ensembles (see e.g. [3,10]). In this paper we consider the hermitian Wigner ensemble with symmetric entries distribution, i.e. hermitian n × n random matrices, Hn = n −1/2 Wn
(1)
with independent (modulo symmetry) and identically distributed entries W jk and W jk such that E{W jk } = E{(W jk )2 } = 0, E{2l+1 W jk } = E{2l+1 W jk } = 0,
E{|W jk |2 } = 1, j, k = 1, .., n, l, n ∈ N.
(2)
2
T. Shcherbina (n)
(n)
Denote by λ1 , . . . , λn the eigenvalues of Hn and define their Normalized Counting Measure (NCM) as Nn () = {λ(n) j ∈ , j = 1, .., n}/n,
Nn (R) = 1,
(3)
where is an arbitrary interval of the real axis. The global regime of the random matrix theory, centered around the weak convergence of the NCM of eigenvalues, is well-studied for many ensembles. It is shown that Nn converges weakly to a non-random limiting measure N known as the Integrated Density of States (IDS). The IDS is normalized to unity and is absolutely continuous in many cases N (R) = 1, N () = ρ(λ)d λ. (4)
The non-negative function ρ in (4) is called the limiting density of states of the ensemble. In the case of the hermitian Wigner ensemble ρ is given by the well-known semicircle law (see, e.g.,[11]): 1 ρsc (λ) = 4 − λ2 . (5) 2π The mixed moments (or the correlation functions) of characteristic polynomials are F2m () =
2m
Hn j=1
det(λ j − H )Pn (d Hn ),
(6)
where Hn is the space of hermitian n × n matrices, d Hn =
n
d Hjj
j=1
H j,k H j,k
(7)
1≤ j
is the standard Lebesgues measure on Hn , Pn (d Hn ) is a probability law of the n ×n random matrix Hn , and = {λ j }2m j=1 are real or complex parameters that may depend on n. We are interested in the asymptotic behavior of (6) for matrices (1) as n → ∞ and λ j = λ0 +
ξj , nρsc (λ0 )
j = 1, .., 2m,
where λ0 ∈ (−2, 2), ρsc is defined in (5) and ξ = {ξ j }2m j=1 are real parameters varying in a compact set K ⊂ R. In the case of the hermitian matrix model, i.e. the matrices with Pn (d Hn ) = Z n−1 e−n tr V (Hn ) d Hn , where V is bounded from below and grows sufficiently fast at infinity, the asymptotic behavior of (6) is known. Using the method of orthogonal polynomials, it was shown (see [2,13]) that 1 (nρ(λ0
2 ))m
F2m 0 + ξ /(nρ(λ0 ))
emnV (λ0 )+αV (λ0 ) = Cn
2m
j=1 ξ j
det
sin(π(ξi −ξm+ j )) m π(ξi −ξm+ j ) i, j=1
(ξ1 , .., ξm )(ξm+1 , .., ξ2m )
(1 + o(1)),
(8)
Characteristic Polynomials of the Wigner Ensemble
3
as n → ∞, where 0 = (λ0 , . . . , λ0 ) ∈ R2m , αV (λ) =
V (λ) , 2ρ(λ)
(9)
ρ is a density of (4), λ0 is such that ρ(λ0 ) > 0 and (x1 , . . . , xm ) is the Vandermonde determinants of x1 , . . . , xm . Unfortunately, the method of orthogonal polynomials can not be applied to the general case of the hermitian Wigner Ensembles (1)–(2). Thus, to find the asymptotic behavior of (6) other methods should be used. In [7] Gotze and Kosters use the exponential generating function to study the second moment, i.e. the case m = 1 in (6). It was shown that 1 F2 (λ0 + ξ1 /(nρsc (λ0 )), λ0 + ξ2 /(nρsc (λ0 ))) nρ(λ0 ) sin(π(ξ1 − ξ2 )) = 2π e−n exp{nλ20 /2 + α(λ0 )(ξ1 + ξ2 ) + κ4 } (1 + o(1)), π(ξ1 − ξ2 ) where α(λ) =
λ , κ4 = μ4 − 3/4, 2ρsc (λ)
(10)
and μ4 is the fourth moment of the common probability law of W jk , W jk . In this paper we consider the general case m ≥ 1 of (6) for the random matrices (1). Following [7], define 1 ξ ξ F2 λ0 + , λ0 + D (n) (ξ ) = nρ(λ0 ) nρsc (λ0 ) nρsc (λ0 )
2 nλ0 + 2α(λ0 )ξ + κ4 (1 + o(1)). = 2π e−n exp (11) 2 The main result of the paper is Theorem 1. Let the entries W jk , W jk of matrices (1) have a symmetric probability distribution with finite first 4m moments. Then we have for m ≥ 1, lim
n→∞
=
(nρsc (λ0 ))
m2
1 2m l=1
D (n) (ξl )
F2m 0 + ξ /(nρsc (λ0 ))
exp{m(m − 1)κ4 (λ20 − 2)2 /2} sin(π(ξi − ξm+ j )) m det , (ξ1 , . . ., ξm ) (ξm+1 , . . ., ξ2m ) π(ξi − ξm+ j ) i, j=1
(12)
where F2m and ρsc (λ) are defined in (6) and (5), 0 = (λ0 , . . . , λ0 ) ∈ R2m , λ0 ∈ (−2, 2), ξ = {ξ j }2m j=1 , and α(λ) and κ4 are defined in (10). The theorem shows that the above limits for the mixed moments of characteristic polynomials for random matrices (1) coincide with those for the Gaussian Unitary Ensemble corresponding to V = λ2 /2 and ρ = ρsc in (8) up to a factor, depending only on the fourth moment of the common probability law of entries W jk , W jk , i.e. that the higher moments of the law do not contribute to the above limit. This is a manifestation of the universality, that can be compared with the universality of the local bulk regime for Wigner matrices (see [5] and references therein).
4
T. Shcherbina
The paper is organized as follows. In Sect. 2 we obtain a convenient integral representation for F2m in the case of symmetric probability distribution of entries with 4m finite moments by using the integration over the Grassmann variables and Harish Chandra/Itzykson-Zuber formula for integrals over the unitary group. In Sect. 3 we prove Theorem 1 by applying the steepest descent method to the integral representation. We denote by C, C1 , etc. and c, c1 , etc. various n-independent quantities below, which can be different in different formulas. Integrals without limits denote the integrals over the whole real axis. 2. The Integral Representation In this section we obtain the integral representation for the mixed moments F2m (6) of the characteristic polynomials. To this end we use the integration over the Grassmann variables. The integration was introduced by Berezin and widely used in the physics literature (see [1] and [4]). For the reader’s convenience we give a brief outline of this technique here. 2.1. Grassmann integration. Let us consider the two sets of formal variables {ψ j }nj=1 , {ψ j }nj=1 , which satisfy the anticommutation conditions ψ j ψk + ψk ψ j = ψ j ψk + ψk ψ j = ψ j ψ k + ψ k ψ j = 0,
j, k = 1, .., n.
These two sets of variables {ψ j }nj=1 and {ψ j }nj=1 generate the Grassmann algebra A. Taking into account that ψ 2j = 0, we have that all elements of A are polynomials of {ψ j }nj=1 and {ψ j }nj=1 . We can also define functions of the Grassmann variables. Let χ be an element of A, which absolute term is a. For any analytical function f we mean by f (χ ) the element of A obtained by substituting χ − a in the Taylor series of f at the point a. Since χ is a polynomial of {ψ j }nj=1 , {ψ j }nj=1 , which absolute term is a, there exists such l that (χ − a)l = 0, and hence the series terminates after a finite number of terms and so f (χ ) ∈ A. Following Berezin [1], we define the operation of integration with respect to the anticommuting variables in a formal way: d ψ j = d ψ j = 0, ψ j d ψ j = ψ j d ψ j = 1. (13) This definition can be extended on the general element of A by the linearity. A multiple integral is defined to be a repeated integral. The “differentials” d ψ j and d ψ k anticommute with each other and with the variables ψ j and ψ k . Thus, if f (χ1 , . . . , χm ) = a0 +
m j1 =1
a j1 χ j1 +
a j1 j2 χ j1 χ j2 + · · · + a1,2,...,m χ1 . . . χm ,
j1 < j2
where {χ j }mj=1 are independent Grassmann variables, then f (χ1 , . . . , χm )d χm . . . d χ1 = a1,2,...,m . Let A be an ordinary hermitian matrix. The following Gaussian integral is wellknown:
Characteristic Polynomials of the Wigner Ensemble
5
n n
d z j d z j 1 = . exp − A j,k z j z k π det A j,k=1
(14)
j=1
One of the important formulas of the Grassmann variables theory is the analog of (14) for the Grassmann algebra (see [1]): n n
exp A j,k ψ j ψk d ψ j d ψ j = det A. (15) j,k=1
j=1
Besides, we have q n n
ψ l p ψs p exp A j,k ψ j ψk d ψ j d ψ j = det Al1 ,..,lq ;s1 ,..,sq , p=1
j,k=1
(16)
j=1
where Al1 ,..,lq ;s1 ,..,sq is a (n − q) × (n − q) minor of the matrix A without rows l1 , .., lq and columns s1 , .., sq .
2.2. Asymptotic integral representation for F2 . In this subsection we obtain the asymptotic integral representation of (6) for m = 1 by using the Grassmann integrals. This formula was obtained in [7] by using another method. We give here a detailed proof to show the basic ingredients of our technique that will be elaborated in the next subsection to obtain the asymptotic integral representation of (6) for m > 1. Denote 2 D2 = D (n) (ξl ), (17) l=1
where D (n) (ξ ) is defined in (11). In follows from (11) and the steepest descent method that 2 n 4 − λ 0 α(λ )ξ +κ /2 4 D (n) (ξ ) = e 0 2 n 2 (18) × (t − iλ0 /2)n e− 2 (t+iλ0 /2) d t (1 + o(1)) as n → ∞. Using (15), we obtain from (6), D2−1 F2 () = D2−1 E
=
D2−1 E
e
2
s=1 λs
e n
2 l=1
n
j,k=1 (λl −H ) j,k ψ jl ψkl
p=1 ψ ps ψ ps
d ψ qr d ψqr
r =1 q=1
exp
n 2
−
2 w j,k √ (ψ jl ψkl + ψ kl ψ jl ) n j
n 2 2 n iw j,k wjj − √ (ψ jl ψkl − ψ kl ψ jl ) − ψ jl ψ jl d ψ qr d ψqr , (19) √ n n j=1
l=1
r =1 q=1
6
T. Shcherbina
2 where {ψ jl }nj,l=1 are the Grassmann variables (n variables for each determinant in (6)). Denote
χ +j,k = χ +j, j =
2 2 (ψ jl ψkl + ψ kl ψ jl ), χ − = (ψ jl ψkl − ψ kl ψ jl ), j = k, j,k l=1 2
l=1
ψ jl ψ jl ,
j, k = 1, .., n.
(20)
l=1 2 s Using that (χ ± j,k ) = 0 for s > 4, j, k = 1, .., n (since ψ js = ψ js = 0 for any j = 1, .., n, s = 1, 2), we expand the second exponent under the integral in (19) into the series and integrate with respect to the measure (2). We get then n (χ +j,k )2 2 μ4 + 4 1+ + (χ ) D2−1 F2 () = D2−1 e s=1 λs p=1 ψ ps ψ ps 4n 4!n 2 j,k j
j
r =1 q=1
j=1
where μ4 is 4th moment of the common probability law of W jk , W jk of (2). Writing 1 ± 2 μ4 1 ± 2 κ4 ± 4 ± 4 1± (χ ) + (χ ) = exp ± (χ j,k ) + (χ ) , j = k, 4n j,k 4!n 2 j,k 4n 4!n 2 j,k 1 + 2 1 + 2 (χ j, j ) = exp (χ ) , j, k = 1, .., n, 1+ 2n 2n j, j where κ4 is defined in (10), and using (21), we get D2−1 F2 () = D2−1
2
e
s=1 λs
n
κ4 1 p=1 ψ ps ψ ps − 2n σ1 + n 2 σ2
2 n
d ψ qr d ψqr ,
(22)
ψ ks ψkl ,
(23)
r =1 q=1
where 1 + 2 2 (χ j,k ) − (χ − − ) (χ +j, j )2 j,k 2 n
σ1 = −
j
=
2 n l=1
j=1
j=1
ψ jl ψ jl
2
+2
n
1≤l<s≤2
j=1
ψ jl ψ js ·
n k=1
2 1 + 4 4 (χ j,k ) + (χ − = ψ j1 ψ j2 ψ j1 ψ j2 . j,k ) 4! n
σ2 =
j
j=1
Now we use the formulas π 2 exp{ab } = exp{−ax 2 − 2abx}d x, a π exp{abc} = exp{−auu − abu − acu}d ud u, a
(24)
Characteristic Polynomials of the Wigner Ensemble
7
where b, c are complex numbers or even Grassmann variables (i.e. sums of the products of even number of Grassmann variables), and a is a positive number. For the case of even Grassmann variables this formula can be obtained by expanding the exponent into the series and integrating each term by using (13). We obtain 1 n2 exp − σ1 = exp − 2n 2π 2 H2 n 2 exp iτ p ψ j p ψ j p + j=1
p=1
n 2 τq + 2 2 2
q=1
u ab u ab
1≤a
iu cd ψ jc ψ jd + iu cd ψ jd ψ jc d Q,
(25)
1≤c
where
Q=
τ1 u 12
u 12 τ2
,
(26)
H2 is the space of 2 × 2 hermitian matrices and d Q is given in (7) for n = 2. Write the formula (cf. (24)) n κ
|κ | 2 p ε(κ4 ) 2 4 4 exp 2 σ2 = exp ψ j1 ψ j2 ψ j1 ψ j2 d p (27) e−|κ4 | p n π n j=1
with σ2 of (23) and ε(x) =
x, x > 0, −i x, x < 0.
(28)
Substituting (25)–(27) in (22) and using (15)–(16) for integration over the Grassmann variables, we obtain D2−1 F2 () = Z2 dp
H2
n
d Qe− 2 tr Q
2 −|κ
4| p
2
2 p ε(κ4 ) n det(Q − i) + , n
(29)
where Q is defined in (26) and = diag {λ1 , λ2 },
Z2 =
(−1)n n 2 D2−1 . 2π 2 π |κ4 |−1
(30)
Recall that we are interested in = 0 + ξ /nρsc (λ0 ), where 0 = diag{λ0 , λ0 } and ξ = diag{ξ1 , ξ2 }. Change variables to τ j − iλ0 /2 − iξ j /nρsc (λ0 ) → τ j , j = 1, 2 and note that after the change we can move the integration with respect to τ j from line z = λ0 /2 + ξ j /nρsc (λ0 ) back to the real axis. Indeed, consider the contour C j R , which is the rectangle with vertices at (−R, 0), (−R, λ0 /2 + ξ j /n), (R, λ0 /2 + ξ j /n) and (R, 0). Since the integrand in (29) is analytic in {τ j }2j=1 , the integral with respect to τ j of this function over C j R is equal to 0. Besides, the integral over the segments of lines z = ±R tends to 0 as R → ∞, since the integrand in (29) is a polynomial of τ j multiplied by exp{−nτ 2j /2}. Thus, letting R → ∞, we obtain that the integral with
8
T. Shcherbina
respect to τ j over the line z = λ0 /2 + ξ j /n is equal to the integral over the real axis. We obtain i 2 i − n tr Q+ 2 0 − ρsc i(λ ) tr Q+ 2 0 ξ 0 e 2 D2−1 F2 () = Z 2 H2
i0 2 p ε(κ4 ) n det(Q − )+ dp d Q 2 n i0 i tr Q + ξ d Q exp − = Z 2 dp ρsc (λ0 ) 2 H2 n ξ2
1 2 p ε(κ4 ) −|κ4 | p 2 − tr μ (Q) 1 + , n 2n ρsc (λ0 )2 ndet(Q − i0 /2) ×e
1 −|κ4 | p 2 − 2n tr
ξ2 ρsc (λ0 )2
(31)
where Q is again a hermitian matrix (see (26)) and n
μn (Q) = det n (Q − i0 /2)e− 2 tr (Q+i0 /2) . 2
(32)
Let q1 , q2 be the eigenvalues of Q. Set n = {(Q, p) : a ≤ |ql − iλ0 /2| ≤ A, l = 1, 2, | p| ≤ log n}, nQ = {Q ∈ H2 : a ≤ |ql − iλ0 /2| ≤ A},
(33)
for sufficiently small a and sufficiently big A (note that if |λ0 | ≥ δ, then |ql − iλ0 /2| ≥ δ 2 /4 and we can omit the first inequality in (33)). The integral in (31) over the 2 domain maxl=1,2 |ql | ≥ A is O(e−n A /4 ), A → ∞ and the integral over the domain −1 minl=1,2 |ql | ≤ a is O(e−n log a ), a → 0. If a ≤ |ql − iλ0 /2| ≤ A and | p| ≥ log n, then according to (17), (18), and (26), the corresponding integral is bounded by 2 2 Z2 |μn (Q)|d Q (1 + C p/n)n e−|κ4 | p dp = O(e−C log n ), (34) nQ
| p|≥log n
and we can write D2−1 F2 () = Z 2 ×e
n
μn (Q)e
1 −|κ4 | p 2 − 2n tr
−i tr(Q+
i0 ξ 2 ) ρsc (λ0 )
(1 + f n (det(Q − i0 /2), p))
i ξ2 +2 p ε(κ4 ) det−1 (Q− 2 0 ) ρsc (λ0 )2
d p d Q + O(e−c log n ), (35) 2
where f n (det(Q − i0 /2), p) =e
−2 p ε(κ4 ) det−1 (Q−
i0 2 )
1+
2 p ε(κ4 ) n det(Q −
i0 2 )
n − 1.
(36)
If we write the term in the brackets above as (1 + T )n = exp{n log(1 + T )} and expand the exponent with respect to T , we obtain on n , | f n (det(Q − i0 /2), p)| ≤
logk n , n
(37)
Characteristic Polynomials of the Wigner Ensemble
9
where k is independent of n. Note also that f n is an analytic function in p and entries of Q. Besides, it is easy to check that i0 2 −1 e−|κ4 | p +2 p ε(κ4 ) det (Q− 2 ) dp I := | p|≤log n
=
π κ4 det−2 (Q−i0 /2) 2 e + O(e−c log n ), |κ4 |
n (see (33)). Thus, (35) yields hence |I | > C3 > 0 on n 2 D2−1 D2−1 F2 () = μn (Q) exp −itr(Q + i0 /2) ξ /ρsc (λ0 ) n 2 (−1) 2π nQ
+ κ4 det −2 (Q − i0 /2) 1 + f n(1) (det(Q − i0 /2)) d Q +O(e−c log n ), 2
(38)
where f n(1) (det (Q
×
− i0 /2)) = e
| p|≤log n
e−|κ4
| p 2 +2 p ε(κ
ξ2 ρsc (λ0 )2
1 − 2n tr
4
) det−1 (Q−i
−1+ I 0 /2)
1 −1 − 2n tr
e
ξ2 ρsc (λ0 )2
f n (det(Q − i0 /2), p)dp.
(39)
nQ and It follows from (37), that f n(1) (det(Q − i0 /2)) is analytic in elements of Q on | f n(1) (det(Q − i0 /2))| ≤ logk n/n,
(40)
where k is independent of n. Let us change variables to Q = U ∗ T U , where U is a unitary matrix and T = diag{t1 , t2 }. Then d Q of (7) for n = 2 becomes π (t1 − t2 )2 d t1 d t2 d μ(U ), 2 where μ(U ) is the normalized to unity Haar measure on the unitary group U (2) (see e.g. [9], Sect. 3.3). Since functions det(Q − i0 /2) and tr (Q + i0 /2)2 are unitary invariant, (38) implies D2−1 F2 () =
n 2 (−1)n 4π D2 ×e
− n2
2
U (2)
dμ(U )
iλ0 2 s=1 (ts + 2 ) −
tr
L aA ×L aA
d t1 d t2
2
(tl − iλ0 /2)n
l=1
i iλ ξ U ∗ (T + 2 0 )U ρsci(λ +κ4 r2=1 (tr − 20 )−2 0)
i0 2 )) + O(e−c log n ), ×(t1 − t2 )2 1 + f n(1) (det(T − 2
(41)
where L aA = {t ∈ R : a ≤ |t − iλ0 /2| ≤ A}.
(42)
The integral over the unitary group U (2) can be computed using the well-known Harish Chandra/Itsykson-Zuber formula (see e.g. [9], App. 5)
10
T. Shcherbina
n Proposition 1. Let A be the normal n × n matrix with distinct eigenvalues {ai }i=1 and B = diag{b1 , . . . , bn }. Then
1 exp − tr(A − U ∗ BU )2 2 (B) f (B)d U d B 2 U (n) n
(B) 1 f (b1 , . . . , bn )d B, j! exp − tr(a j − b j )2 (43) = 2 (A) j=1 where f (B) is any symmetric function of {b j }nj=1 , d B = nj=1 d b j and (A), (B) n , {b }n are the Vandermonde determinants for the eigenvalues {ai }i=1 i i=1 of A and B. We obtain finally from (41) 2 iλ0 2 n 2 iρsc (λ0 )n 2 D2−1 F2 () = (tl − iλ0 /2)n e− 2 l=1 (tl + 2 ) n 2π(−1) D2 L aA ×L aA −
l=1
2
iξl iλ0 −2 −2 l=1 ρsc (λ0 ) (tl + 2 )+κ4 (t1 −iλ0 /2) (t2 −iλ0 /2)
×e t 1 − t2 2 1 + f n(2) (T ) d t1 d t2 + O(e−c log n ), × ξ1 − ξ2
(44)
where L aA is defined in (42) and f n(2) (T ) = f n(1) (det(T −i0 /2)) is analytic and bounded by logk n/n if tl ∈ L aA , l = 1, 2. This asymptotic integral representation is used in Sect. 3 for the proof of Theorem 1 for m = 1. 2.3. Asymptotic integral representation for F2m . Denote 2m D2m = D (n) (ξl ),
(45)
l=1
where D (n) (ξ ) is defined in (11). Using (15), we obtain from (6) (cf. (19)) −1 −1 D2m F2m () = D2m E
−1 = D2m E
×e
−
2m n l=1
e
j,k=1 (λl −H ) j,k ψ jl ψkl
n 2m
d ψ qr d ψqr
r =1 q=1
2m
e
j
s=1 λs
w j,k 2m √ p=1 ψ ps ψ ps − j
n
n
iw j,k 2m w j j 2m n 2m √ √ l=1 (ψ jl ψkl −ψ kl ψ jl )− j=1 l=1 ψ jl ψ jl n n
n
d ψ qr d ψqr ,
r =1 q=1
(46) 2m {ψ jl }n, j,l=1
where are the Grassmann variables (n variables for each determinant). As in (20) we denote χ +j,k = χ +j, j
=
2m 2m (ψ jl ψkl + ψ kl ψ jl ), χ − = (ψ jl ψkl − ψ kl ψ jl ), j,k l=1 2m l=1
l=1
ψ jl ψ jl ,
j, k = 1, . . . , n.
j = k, (47)
Characteristic Polynomials of the Wigner Ensemble
11
2 s Using that (χ ± j,k ) = 0 for s > 4m, j, k = 1, .., 2m (since ψ jl = ψ jl = 0 for any j = 1, .., n, l = 1, .., 2m), we expand the exponent under the integral in (46) into the series and integrate with respect to the measure (2). We get then similarly to (22), n 2m n 2m κ2 p 2m 1 −1 −1 D2m F2m () = D2m e s=1 λs k=1 ψ ks ψks − 2n σ1 + p=2 n p σ p d ψ qr d ψqr , 2
r =1 q=1
(48) where κ2 p are cumulants of the entries w jk , w jk of (2), i.e. the coefficients in the expansion s κq (it)q + o(t s ), t → 0. l(t) := log E{eitw jk } = q! q=0
The function σ1 in (48) is the same as in (23) (but with χ ± j,k of (47) and the sums from 1 to 2m instead of from 1 to 2), n 1 + 4 2 + 4 4 σ2 = ((χ j,k ) + (χ − ) ) + (χ j, j ) j,k 4! 4! j
=2
+
j=1
n
ψ jl1 ψ jl2 ψ js1 ψ js2 ·
l1
1 4
n
ψkl1 ψkl2 ψks1 ψks2
k=1 n
ψ jl1 ψ js1 ψ jl2 ψ js2 ·
l1 =s1 ,l2 =s2 j=1
ψkl1 ψks1 ψ kl2 ψ ks2 ,
(49)
k=1
and we have for p ≥ 3, 1 + 2p 2 + 2p 2p σp = ((χ j,k ) + (−1) p (χ − (χ j, j ) j,k ) ) + (2 p)! (2 p)! j
2m
=
j
[ 2p ]
( p)
cs,l
n
l1 ,..,l2 p =1 s=0
×
n
ψ jl1 ..ψ jl p+2s ψ jl p+2s+1 ..ψ jl2 p
j=1
ψkl1 ..ψkl p+2s ψ kl p+2s+1 ..ψ kl2 p ,
k=1 ( p)
where cs,l are n-independent positive coefficients and l = (l1 , . . ., l2 p ). Using (24), we obtain −|κ | 2 w w + v v −2 4 l l s s l l s s l l s s l l s s l<s l1 =s1 ,l2 =s2 1 2 1 2 1 2 1 2 12 1 2 12 1 2 en κ4 σ2 = C2 e ×
n
e
ε(κ4 ) 2n
a1 =b1 ,a2 =b2
va1 a2 b1 b2 ψ ja1 ψ jb1 ψ ja2 ψ jb2 +v a1 a2 b1 b2 ψ ja1 ψ jb1 ψ ja2 ψ jb2
j=1
×
n
e
2ε(κ4 ) c
wc1 c2 d1 d2 ψ jc1 ψ jc2 ψ jd1 ψ jd2+w c1 c2 d1 d2 ψ jc1 ψ jc2 ψ jd1 ψ jd2
d W d V,
j=1
(50)
12
T. Shcherbina
where c < d and l < s mean that c1 < c2 < d1 < d2 , l1 < l2 < s1 < s2 and dW =
l<s
dV =
d wl1 l2 s1 s2 d wl1 l2 s1 s2 ,
d vl1 l2 s1 s2 d vl1 l2 s1 s2 ,
l1 =s1 ,l2 =s2
C2
=
π 2|κ4 |
−(2m ) 4
π |κ4 |
−(2m)2 (2m−1)2
(51)
,
and for p ≥ 3,
exp
×
κ
2p σp np
n
=
C p
exp
− |κ2 p |
2m
p
[2]
r l,s rl,s
l1 ,..,l2 p =1 s=0
exp
j=1
ε(κ2 p ) n p/2
2m
[ 2p ]
( p) (cl,q )1/2 (rl,q ψ jl1 ..ψ jl p+2q ψ jl p+2q+1 ..ψ jl2 p
l1 ,..,l2 p =1 q=0
+r l,q ψ jl1 ..ψ jl p+2q ψ jl p+2q+1 ..ψ jl2 p ) d R
(52)
with l = (l1 , . . . , l2 p ) and dR =
2m
p
[2]
C p
d rl,s d rl,s ,
=
l1 ,..,l2 p =1 s=0
π |κ2 p |
−[ p ](2m)2 p 2
,
p ≥ 3.
(53)
Substituting (50)–(52) and (25) (with sums from 1 to 2m) to (48), using (15)–(16), and integrating over the Grassmann variables in (48), we get −1 D2m F2m () = Zm
H2m
dQ
n
d V d R d W e− 2 tr Q νn (v, w, r )n (i Q + , v, w, r ), 2
(54)
where H2m is the space of 2m × 2m hermitian matrices, v w rp r
= {va1 a2 b1 b2 |a1 = b1 , a2 = b2 , a1 , a2 , b1 , b2 = 1, .., 2m}, = {wa1 a2 b1 b2 |a1 < a2 < b1 < b2 , a1 , a2 , b1 , b2 = 1, .., 2m}, = {rl,s |l1 , .., l2 p = 1, .., 2m, s = 0, .., [ p/2]}, = (r3 , . . . , r2m ),
(55)
and 2m
|κ2 p |r p r p . νn (v, w, r ) = exp − |κ4 |vv − 2|κ4 |ww − p=3
(56)
Characteristic Polynomials of the Wigner Ensemble
13
d Q, d V, d R, and d W are defined in (7) for n = 2m, (51), and (53), and ⎛
τ1 u 12 u 13 ..
u 12 τ2 u 23 ..
u 13 u 23 τ3 ..
⎜ ⎜ ⎜ Q=⎜ ⎜ ⎝u 1,2m−1 u 2,2m−1 u 3,2m−1 u 2m,1 u 2m,2 u 2m,3
⎞ .. u 1,2m−1 u 1,2m .. u 2,2m−1 u 2,2m ⎟ ⎟ .. u 3,2m−1 u 3,2m ⎟ ⎟ .. .. .. ⎟ .. τ2m−1 u 2m,2m−1 ⎠ .. u 2m−1,2m τ2m
(57)
is obviously hermitian. We denote also = diag{λ1 , λ2 , . . . , λ2m },
Zm =
2m 2m 2 −1 n D2m C p . 2 2m π 2m p=2
(58)
According to (15)–(16), (i Q+, v, w, r ) in (54) is a polynomial of the entries of i Q+ and of {vl1 l2 s1 s2 /n}, {wl1 l2 s1 s2 /n}, {rl1 ,..,l2 p ,s /n p/2 } with n-independent coefficients and degree at most 2m. Besides, possesses the properties: 1. The degree of each variable in (i Q + , v, w, r ) is at most one. 2. (i Q + , v, w, r ) does not contain terms C(i Q + )wl1 l2 s1 s2 /n or C(i Q +)wl1 l2 s1 s2 /n, since the terms ψ jl1 ψ jl2 ψ js1 ψ js2 or ψ jl1 ψ jl2 ψ js1 ψ js2 cannot 2m be completed to l=1 ψ jl ψ jl only by terms ψ jl ψ js . 3. (i Q + , v, w, r ) can be written as (i Q +, v, w, r ) = det(i Q + ) −
2ε(κ4 ) σ1 + f n (i Q + , v/n, w/n, r p /n p/2 ), n
(59)
where f n (i Q + , v/n, w/n, r p /n p/2 ) contains all the terms of (i Q + , v, w, r ) which are O(n −3/2 ) as n → ∞ but Q, v, w, r are fixed, and σ1 contains linear with respect to v terms. In view of (16),
σ1 =
(vl1 l2 s1 s2 ql1 ,s1 ,l2 ,s2 + vl1 l2 s1 s2 ql2 ,s2 ,l1 ,s1 ),
(60)
l1 =s1 ,l2 =s2
where qs,l, p,r is (2m − 2) × (2m − 2) minor of the matrix i Q + without rows with numbers s and l and columns with numbers p and r . ξ /nρsc (λ0 ), where 0 = diag{λ0 , .., λ0 } and Recall that we are interested in = 0 + ξ = diag{ξ1 , . . . , ξ2m }. Shift now τ j − iλ0 /2 − iξ j /nρsc (λ0 ) → τ j , j = 1, .., 2m to obtain in new variables (cf. (31)) −1 D2m F2m () = Z m
×e
H2m
dQ
− n2 tr (Q+
νn (v, w, r )n (i Q + 0 /2, v, w, r )
i0 2 2 ) −i
ξ2 sc (λ0 )2
tr (Q+ i20 )ξ /ρsc (λ0 )− 2n1 tr ρ
d V d R d W,
(61)
14
T. Shcherbina
where Q is the hermitian matrix of (57) and d Q, d V, d R and d W are defined in (7) for n = 2m, (51), and (53). We have in view of property (1), v 0 llss , v, w, r )| ≤ 1+C 12 1 2 (1 + C|(i Q + 0 /2)qs |) |(i Q + 2 n q,s l =s
×
a1
2m w aabb 1+C 1 2 1 2 n
2m
[ 2p ]
r l,s 1 + C p/2 n
(62)
p=3 l1 ,..,l2 p =1 s=0
with n-independent C and where l = s means l1 = s1 , l2 = s2 , and (59) yields v llss 1 + C(Q) 1 2 1 2 |(i Q + 0 , v, w, r )| ≤ |det(i Q + 0 /2)| n l1 =s1 ,l2 =s2
×
l1
w 2m llss 1 + C(Q) 1 2 1 2 n
2m
p
[2] r l,s 1 + C(Q) p/2 . n
(63)
p=3 l1 ,..,l2 p =1 s=0
2m are the Here C(Q) is bounded if a ≤ |ql − iλ0 /2| ≤ A, l = 1, .., 2m, and {ql }l=1 2 eigenvalues of Q. Note that if |λ0 | > δ > 0, then |ql − iλ0 /2| ≥ δ everywhere. Denote
n = {(Q, v, w, r ) : a ≤ |qs − iλ0 /2| ≤ A, |vl1 l2 s1 s2 | ≤ log n, |wl1 l2 s1 s2 | ≤ log n, |rl,s | ≤ log n}, nQ
(64)
= {Q ∈ H2m : a ≤ |qs − iλ0 /2| ≤ A, s = 1, .., 2m}.
According to (62), the integral over the domain maxl=1,..,2m |ql − iλ0 /2| ≥ A in (61) 2 is O(e−n A /4 ), A → ∞ and the integral over the domain minl=1,..,2m |ql − iλ0 /2| ≤ a −1 is O(e−n log a ), a → 0. Moreover, the bound (63) implies that this integral over the domain, where the absolute value of at least one of {vl1 l2 s1 s2 }, {wl1 l2 s1 s2 } or {rl,s } is 2 greater than log n but a ≤ |qs − iλ0 /2| ≤ A, s = 1, .., 2m, can be bounded by e−c log n (similarly to (34)). Hence, using (18), (45), and (58) for the estimate of the integral with |μn (Q)|, we can write 2 − 1 tr ξ −2ε(κ4 )det−1 (i Q+0 /2)σ1
−1 D2m F2m () = Z m νn (v, w, r )e 2n ρsc (λ0 )2 ×μn (Q)e
−tr
n i0 ξ (Q+ 2 ) ρsci(λ 0)
(1 + f n (Q, v, w, r )) d D + O(e−c log n ), 2
(65)
where d D = d Q d V d R d W, μn , σ1 and νn (v, w, r ) are defined in (32), (60), and (56) respectively, and n f n (i Q + 20 , v, w, r ) − 2ε(κ4 )σ1 /n 2ε(κ4 )det−1 (i Q+ 20 )σ1
f n (Q, v, w, r ) = e − 1. 1+ det(i Q + 0 /2) (66) If we write the term in the brackets above as (1 + T )n = exp{n log(1 + T )}, expand the exponent with respect to T and use property (3) and formula (63), we obtain on n , | f n (Q, v, w, r )| ≤ n −1/2 logk n,
(67)
Characteristic Polynomials of the Wigner Ensemble
15
where k is independent of n. Note also that f n is analytic in entries of Q. It is easy to check that I := νn (Q, v, w, r )d V d R d W n
=
2m
(C p )−1 eκ4 σ (i Q+0 /2)det
−2 (i Q+
0 /2)
+ O(e−c log n ), 2
(68)
p=2
where νn (Q, v, w, r ) = exp{−2ε(κ4 )det−1 (i Q + 0 /2)σ1 } νn (v, w, r ), ql1 ,s1 ,l2 ,s2 ql2 ,s2 ,l1 ,s1 σ (i Q + 0 /2) =
(69)
l1 =s1 ,l2 =s2
with ql1 ,s1 ,l2 ,s2 defined in (60) (but for the matrix i Q + 0 /2 instead of i Q + 0 ). Moreover, according to the Cauchy-Binet formula (see [6]), σ (i Q + 0 /2) is the sum S2m−2 (A) of the principal minors of order (2m − 2) × (2m − 2) for the matrix A = (i Q ∗ + 0 /2)(i Q + 0 /2) = U ∗ (i T0 + 0 /2)2 U, where U is a unitary 2m × 2m matrix diagonalizing Q and T0 = diag{q1 , .., q2m }, i.e. Q = U ∗ T0 U . Since S2m−2 (A) is the coefficient in front of λ2 in the characteristic polynomial det(A − λI ), S2m−2 (A) is unitary invariant, and therefore σ (i Q + 0 /2) is also unitary invariant. Thus, we have on n of (64), 0 1 0 ≤ C, )det −2 (i Q + ) = σ (i Q + λ0 2 λ0 2 2 2 (iq + ) (iq + ) s l 2 2 1≤s C > 0.
(70)
This, (65), and (68) yield −1 n 2m D2m 2
−1 D2m F2m () =
2m π
2m 2
e
Q
n
−tr (Q+
i0 0 0 2 i ξ 2 ) ρsc (λ0 ) +κ4 σ (i Q+ 2 )/det(i Q+ 2 )
2 ×μn (Q) 1 + f n(1) (Q) d Q + O(e−c log n ),
(71)
where μn is defined in (32) and f n(1) (Q) = e
1 − 2n tr
+I −1 e
ξ2 ρsc (λ0 )2 1 − 2n tr
−1
ξ2 ρsc (λ0 )2
n
νn (Q, v, w, r ) f n (Q, v, w, r )d V d R d W (72)
with I of (68) and νn of (69). According to (66) and (70), we have | f n(1) (Q)| ≤ logk n/n 1/2 . (1)
Besides, we will use below that f n (Q) is analytic in elements of Q.
(73)
16
T. Shcherbina
Let us change variables to Q = U ∗ T U , where U is a unitary 2m × 2m matrix and T = diag{t1 , . . . , t2m }. The differential d Q in (71) is π m(2m−1)
2m
j!
−1
2 (T )d T d μ(U )
j=1
2m d tl , (T ) is the Vandermonde determinant of in the new variables, where d T = l=1 2m {tl }l=1 , and μ(U ) is the normalized to unity Haar measure on the unitary group U (2m) (see e.g. [9], Sect. 3.3). Functions det(i Q + 20 ), tr (Q + i2 0 )2 , and σ (i Q + 20 ) are unitary invariant. Hence, (71) implies 2 2m (−1)mn n 2m iλ0 n −1 F2m () = dμ(U ) dT (tl − D2m ) 2m l=1 A 2m 2 D2m (2π )m j=1 j! U (2m) (L a ) 2m
tr U ∗ (T + i20 )U ρsci(λξ 0 ) +κ4 1≤l<s≤2m (tl − iλ20 )−2 (ts − iλ20 )−2 2 ×2 (T ) 1 + f n(1) (U ∗ T U ) + O(e−c log n ), (74) ×e
iλ0 2 l=1 (tl + 2 ) −
− n2
where L aA is defined in (42). Using Proposition 1, we have 2 iλ0 2 n 2m (−1)mn n 2m −1 D2m F2m () = e− 2 l=1 (tl + 2 ) m D2m (2π ) (L aA )2m ×e
κ4
× where
iλ0 −2 iλ0 −2 1≤i< j≤2m (ti − 2 ) (t j − 2 )
−i πme
2m
l=1 (tl +
(T ) 2
2m
(tl −
l=1 ξl iλ0 2 ) ρsc (λ0 )
(T ) (−i ξ /ρsc (λ0 ))
iλ0 n ) 2
2 + f n(2) (T ) d T + O(e−c log n ),
(75)
ξ (1) ∗ i0 )U f (U T U )d μ(U ). exp − itr U ∗ (T + 2 ρsc (λ0 ) n According to (73), we get that f n(2) (T ) =
| f n(2) (T )| ≤ n −1/2 logk n, tl ∈ L aA , l = 1, .., 2m.
(76)
We obtain finally (cf. (44)) −1 F2m () = D2m
(−1)mn n 2m D2m 2m π m ×e
− n2
2m
2
(L aA )2m
iλ0 2 l=1 (tl + 2 ) −i
2m
(tl −
l=1 2m
iλ0 n ) (iρsc (λ0 ))m(2m−1) 2
ξl iλ0 l=1 (tl + 2 ) ρsc (λ0 ) +κ4
iλ0 −2 iλ0 −2 l1
(T ) 2 1 + f n(2) (T ) d t j + O(e−c log n ), (ξ ) j=1 2m
× where
f n(2) (T ) = (T ) (−i ξ /ρsc (λ0 ))ei (2)
2m
l=1 tl ξl /ρsc (λ0 )
f n(2) (T ).
f n (T ) is an analytic function bounded by n −1/2 logk n if tl ∈ L aA , l = 1, .., 2m.
(77)
Characteristic Polynomials of the Wigner Ensemble
17
3. Asymptotic Analysis In this section we prove Theorem 1, passing to the limit n → ∞ in (77) for λ j = λ0 + ξ j /nρsc (λ0 ), where ρsc is defined in (5), λ0 ∈ (−2, 2) and ξ j ∈ [−M, M] ⊂ R, j = 1, .., 2m. To this end consider the function 4 − λ20 t 2 iλ0 + t − log(t − iλ0 /2) − . 2 2 8
V (t, λ0 ) =
(78)
Then (77) yields
−1 D2m
(nρsc (λ0
2 ))m
F2m () = Z m,n
Wn (t1 , . . . , t2m )d T + O(e−c log n ), 2
(L aA )2m
(79)
where D2m is defined in (45), 2m
2m
(T ) ( ξ) iλ0 −2 iλ0 −2 ×eκ4 1≤l<s≤2m (tl − 2 ) (ts − 2 ) 1 + f n(2) (T ) ,
Wn (t1 , . . . , t2m ) = e
−n
l=1
V (tl ,λ0 )−i
ξl l=1 ρsc (λ0 ) tl
(80)
and (−1)mn n m ρsc (λ0 )m(m−1) e−mκ4 = . (−i)m(2m−1) 22m π 2m 2
Z m,n
(81)
Now we need Lemma 1. The function V (t, λ0 ) for t ∈ R attains its minimum at 4 − λ20 t = x± := ± . 2
(82)
Moreover, if t ∈ Un (x± ) := (x± − n −1/2 log n, x± + n −1/2 log n), then we have for sufficiently big n, C log2 n . n
(83)
12 t − (4 − λ20 )/4 − log(t 2 + λ20 /4) , 2
(84)
V (t, λ) ≥ Proof. Note that for t ∈ R, V (t, λ0 ) = thus
d t V (t, λ0 ) = t − 2 , dt t + λ20 /4 d2 1 2t 2 V (t, λ ) = 1 − , + 0 d t2 t 2 + λ20 /4 (t 2 + λ20 /4)2
(85)
18
T. Shcherbina
and t = x± of (82) are the minimum points of V (t, λ0 ). Writing iλ0 4 − λ20 V+ := V (x+ , λ0 ) = − i arcsin(−λ0 /2), 4 iλ0 4 − λ20 V− := V (x− , λ0 ) = − + i arcsin(−λ0 /2) − iπ, 4 we conclude that
(86)
V (x± , λ0 ) = 0. Expanding V (t, λ0 ) into the Taylor series and using (85)–(86), we obtain for t ∈ Un (x± ): 4 − λ20 (t − x± )2 + O(n −3/2 log3 n), 4 where x± is defined in (82). This implies for t ∈ Un (x± ), V (t, λ0 ) =
V (t, λ) ≥
(87)
C log2 n . n
The lemma is proved. Taking into account that |t j − iλ0 /2| > a for t j ∈ L aA , j = 1, .., 2m, we have
−2 −2 exp κ4 (88) (tl − iλ0 /2) (ts − iλ0 /2) ≤ C. 1≤l<s≤2m
Since ξ1 , . . . , ξ2m are distinct, the inequality |(T )/( ξ )| ≤ C1 for |t j | ≤ A, j = 1, .., 2m, (83), and (88) yield 2 2 Z m,n .. Wn (t1 , . . . , t2m )d T ≤ C1 n m e−C2 log n , L aA \(U+ ∪U− )
L aA
L aA
where L aA , Wn and Z m,n are defined in (42), (80), and (81) respectively, and U± = {t ∈ R : |t − x± | ≤ n −1/2 log n}
(89)
with x± of (82). Note that we have for t ∈ U± in view of (78) and (86), (t − x± )2 1 + f ± (t − x± ), n → ∞, (90) V (t, λ0 ) = V± + 1 + (x± − iλ0 /2)2 2 where f ± (t − x± ) = O((t − x± )3 ). Shifting t j − x± → t j for t j ∈ U± and using (86) we obtain for the r.h.s. of (79), 2m 2m ncα j iξ t − t 2 − j j −n f (t ) Z m,n d tj e 2 j ρsc (λ0 ) α j j (Un )2m j=1 α j=1 κ4 1≤l<s≤2m (tl + pαl )−2 (ts + pαs )−2 − 2m j=1 (nVα j +i x α j ξ j /ρsc (λ0 ))
×e (t1 + xα1 , . . . , t2m + xα2m ) 2 (1 + f n(2) (T )) + O(e−c log n ), × (ξ1 , . . . , ξ2m )
(91)
Characteristic Polynomials of the Wigner Ensemble
19
where the sum is over the collection α = {α j }2m j=1 , α j = ±, j = 1, .., 2m and −2 , c± = 1 + p±
p± = x± − iλ0 /2, Un = (−2n −1/2 log n, 2n −1/2 log n). (92)
Define I :=
(Un )2m
2m 2m 2m
ncα j 2 iξ j t j exp − n f α j (t j ) tj − − 2 ρsc (λ0 ) j=1
j=1
×(t1 + xα1 , . . . , t2m + xα2m )
2m
j=1
d tj
j=1
t j + xα j −
k−1 iξ j nρsc (λ0 )cα j Un, j
2m ncα j 2 iξ j t − n fα j t j − × exp − , dt j 2 j nρsc (λ0 )cα j j,k=1
= det
(93)
where U j,n =
− 2n −1/2 log n +
iξ j iξ j . , 2n −1/2 log n + nρsc (λ0 )cα j nρsc (λ0 )cα j
Since f ± (t) = O(t 3 ), changing variables to
√
√ nt j → t j , expanding exp{−n f α j (t j / n−
iξ j /nρsc (λ0 )cα j )} in (93), and keeping the terms up to the order n −4m , we obtain 2
I =
(+)
k−1 iξ j 2π det xα j − ncα j nρsc (λ0 )cα j j=1 2m 1 (α ) + Pk,mj (ξ j /n) (1 + o(1)), n → ∞, n j,k=1 2m
(94)
(−)
where Pk,m and Pk,m are polynomials with n- and j-independent (but k-dependent) coefficients of degree at most 4m 2 . Introducing D(ξ/n, λ) = det
xα j −
k−1 iξ j (α ) + λPk,mj (ξ j /n) nρsc (λ0 )cα j
2m ,
(95)
j,k=1
taking into account that D(ξ/n, λ) is a polynomial in {ξ j /nρsc (λ0 )}2m j=1 and λ, and putting α1 = . . . = αs = +, αs+1 = . . . = α2m = −, we find that if ξ j = ξl for j, l = 1, .., s or j, l = s + 1, .., 2m, then D(ξ/n, λ) = 0. This implies D(ξ/n, λ) = (ξ1 /n, . . . , ξs /n) (ξs+1 /n, . . . , ξ2m /n)(C0 + λF(ξ/n, λ)), (96)
20
T. Shcherbina
where F(ξ/n, λ) is a polynomial with bounded coefficients. Substituting λ = 0 in (95) and computing the Vandermonde determinant, we obtain s(s−1) (2m−s)(2m−s−1) 2 2 −i −i ρsc (λ0 )c+ ρsc (λ0 )c− s 2m iξ j iξk x+ − x− − + nρsc (λ0 )c+ nρsc (λ0 )c−
C0 =
j=1 k=s+1
=
−i ρsc (λ0 )c+
s(s−1) 2
−i ρsc (λ0 )c−
(2m−s)(2m−s−1) 2
(x+ − x− )s(2m−s) (1 + o(1)).
Hence, for α1 = . . . = αs = +, αs+1 = . . . = α2m = −, we get from (96) 2
2
nm I 2m π m (−i/ρsc (λ0 ))m(m−1)+(m−s) = 2 2 ( ξ) (c+ )s /2 (c− )(2m−s) /2 2 n −(m−s) ( 4 − λ20 )s(2m−s) (1 + o(1)) × , n → ∞. s 2m j=1 l=s+1 (ξ j − ξl )
(97)
This expression is of order O(1), and for s = m it is of order o(1). Hence, only terms of (91), where exactly m of {α j }2m j=1 are pluses contribute in the limit (12). Consider one of such terms in (91), e.g. the term with α1 = .. = αm = 1, αm+1 = .. = α2m = −1. Using (81), (92), and (97) with s = m we can rewrite this term as i m(m+1) eiπ(ξm+1 +..+ξ2m −ξ1 −..−ξm ) m(m−1)κ4 (λ2 −2)2 /2 0 e . (2iπ )m i,m j=1 (ξi − ξm+ j ) In view of the identity
sin(π(ξ j −ξm+k )) m det π(ξ j −ξm+k )
j,k=1
det
iπ(ξ j −ξm+k )
iπ(ξm+k −ξ j )
−e ξ j −ξm+k
m j,k=1
, (2iπ )m (ξ1 , .., ξm ) (ξm+1 , .., ξ2m ) the determinant in the l.h.s. is the linear combination of exp{iπ 2m j=1 α j ξ j } over the 2m collection {α j } j=1 , in which exactly m elements are pluses. According to the identity (see [12], Prob. 7.3) # $m m(m−1) 1 k
=
e
(98)
the coefficient of exp{iπ(ξm+1 + .. + ξ2m − ξ1 − .. − ξm )} is
m m(m−1) det ξm+k1−ξ j (−1) 2 j,k=1 . = 2 (2iπ )m (ξ1 , .., ξm ) (ξm+1 , .., ξ2m ) (−1)m (2iπ )m i,m j=1 (ξi − ξm+ j ) Other coefficients can be computed analogously. Thus, restricting the sum in (91) to that over the collection {α j }2m j=1 , in which m elements are pluses, and m are minuses, and using (98), we obtain Theorem 1 after certain algebra.
Characteristic Polynomials of the Wigner Ensemble
21
Acknowledgements. The author is grateful to Prof.L.Pastur for many interesting discussions of the problem. This work was partially supported by joint project 17-01-10 of National Academy of Sciences of Ukraine and Russian Foundation for Basic Research.
References 1. Berezin, F.A.: Introduction to the algebra and analysis of anticommuting variables. Moscow: Moscow State University Publ., 1983 (Russian) 2. Brezin, E., Hikami, S.: Characteristic polynomials of random matrices. Commun. Math. Phys. 214, 111– 135 (2000) 3. Brezin, E., Hikami, S.: Characteristic polynomials of real symmetric random matrices. Commun. Math. Phys. 223, 363–382 (2001) 4. Efetov, K.: Supersymmetry in disorder and chaos. New York: Cambridge University Press, 1997 5. Erdos, L.: Universality of Wigner random matrices: a survey of recent results. Uspekhi Mat. Nauk. 66, 67–198 (2011) 6. Gantmacher, F.R.: The Theory of Matrices. New York: Chelsea, 1959 7. Gotze, F., Kosters, H.: On the second-ordered correlation function of the characteristic polynomial of a hermitian Wigner matrix. Commun. Math. Phys. 285, 1183–1205 (2008) 8. Keating, J.P., Snaith, N.C.: Random matrix theory and ζ (1/2 + it). Commun. Math. Phys. 214, 57–89 (2000) 9. Mehta, M.L.: Random Matrices. New York: Academic Press, 1991 10. Mehta M.L. Normand J.-M.: Moments of the characteristic polynomial in the three ensembles of random matrices. J. Phys. A: Math. Gen. 34, 4627–4639 (2001) 11. Pastur, L.: The spectrum of random matrices. Teoret. Mat. Fiz. 10, 102–112 (1972) (Russian) 12. Polya, G., Szego, G.: Problems and theorems in analysis, Vol. II. Die Grundlehren der math. Wissenschaften. Berlin-Heidelberg-New York: Springer-Verlag, 1976 13. Strahov, E., Fyodorov, Y.V.: Universal Results for Correlations of Characteristic Polynomials: RiemannHilbert Approach. Commun. Math. Phys. 241, 343–382 (2003) Communicated by P. Forrester
Commun. Math. Phys. 308, 23–47 (2011) Digital Object Identifier (DOI) 10.1007/s00220-011-1324-8
Communications in
Mathematical Physics
Existence of Axially Symmetric Static Solutions of the Einstein-Vlasov System Håkan Andréasson1 , Markus Kunze2 , Gerhard Rein3 1 Mathematical Sciences, Chalmers University of Technology, Göteborg University, 41296 Göteborg,
Sweden. E-mail: [email protected]
2 Fakultät für Mathematik, Universität Duisburg-Essen, 45117 Essen, Germany.
E-mail: [email protected]
3 Fakultät für Mathematik, Physik und Informatik, Universität Bayreuth, 95440 Bayreuth, Germany.
E-mail: [email protected] Received: 24 August 2010 / Accepted: 1 April 2011 Published online: 4 September 2011 – © Springer-Verlag 2011
Abstract: We prove the existence of static, asymptotically flat non-vacuum spacetimes with axial symmetry where the matter is modeled as a collisionless gas. The axially symmetric solutions of the resulting Einstein-Vlasov system are obtained via the implicit function theorem by perturbing off a suitable spherically symmetric steady state of the Vlasov-Poisson system. 1. Introduction The aim of the present investigation is to prove the existence of static, asymptotically flat, and axially symmetric solutions of the Einstein-Vlasov system. This system describes, in the context of general relativity, the evolution of an ensemble of particles which interact only via gravity. Examples from astrophysics of such ensembles include galaxies or globular clusters where the stars play the role of the particles and where collisions among these particles are usually sufficiently rare to be neglected. The particle distribution is given by a density function f on the tangent bundle T M of the spacetime manifold M. We assume that all particles have the same rest mass which is normalized to unity. Hence the particle distribution function is supported on the mass shell P M = {gαβ p α p β = −c2 and p α is future pointing} ⊂ T M. Here gαβ denotes the Lorentz metric on the spacetime M and if x α are coordinates on M, then p α denote the corresponding canonical momentum coordinates; Greek indices always run from 0 to 3, and we have a specific reason for making the dependence on the speed of light c explicit. We assume that the coordinates are chosen such that ds 2 = c2 g00 dt 2 + gab d x a d x b , where Latin indices run from 1 to 3 and t = x 0 should be thought of as a timelike coordinate. On the mass shell p 0 can be expressed by the remaining coordinates, p 0 = −g 00 1 + c−2 gab pa p b ,
24
H. Andréasson, M. Kunze, G. Rein
and f = f (t, x a , p b ) ≥ 0. The Einstein-Vlasov system now consists of the Einstein field equations G αβ = 8π c−4 Tαβ
(1.1)
a p 0 ∂t f + pa ∂x a f − βγ p β p γ ∂ pa f = 0
(1.2)
coupled to the Vlasov equation
via the following definition of the energy momentum tensor: dp 1 dp 2 dp 3 1/2 . Tαβ = c|g| pα pβ f − p0
(1.3)
α are the ChrisHere |g| denotes the modulus of the determinant of the metric, and βγ toffel symbols induced by the metric. We note that the characteristic system of the Vlasov equation (1.2) are the geodesic equations written as a first order system on the mass shell P M which is invariant under the geodesic flow. For more background on the Einstein-Vlasov equation we refer to [3]. In [8,16,18,19] the existence of a broad variety of static, asymptotically flat solutions of this system has been established, all of which share the restriction that they are spherically symmetric. The purpose of the present investigation is to remove this restriction and prove the existence of static, asymptotically flat solutions to the EinsteinVlasov system which are axially symmetric but not spherically symmetric. From the applications point of view this symmetry assumption is more “realistic” than spherical symmetry, and from the mathematics point of view the complexity of the Einstein field equations increases drastically if one gives up spherical symmetry. We use usual axial coordinates t ∈ R, ρ ∈ [0, ∞[, z ∈ R, ϕ ∈ [0, 2π ] and write the metric in the form
ds 2 = −c2 e2ν/c dt 2 + e2μ dρ 2 + e2μ dz 2 + ρ 2 B 2 e−2ν/c dϕ 2 2
2
(1.4)
for functions ν, B, μ depending on ρ and z. The Killing vector fields ∂t and ∂φ correspond to the stationarity and axial symmetry of the spacetime. The additional assumption that the spacetime is static implies that the total angular momentum of the spacetime is zero, cf. [21, Ch. 11, Prob. 6]. Since the generalization to stationary spacetimes with non-zero angular momentum induces qualitatively new, additional difficulties it is postponed to a later investigation. The reason for writing ν/c2 instead of ν is so that below ν converges to the Newtonian potential U N in the limit c → ∞. The metric is to be asymptotically flat in the sense that the boundary values lim
|(ρ,z)|→∞
ν(ρ, z) =
lim
|(ρ,z)|→∞
μ(ρ, z) = 0,
lim
|(ρ,z)|→∞
B(ρ, z) = 1
(1.5)
are attained at spatial infinity with certain rates which are specified later. In addition we need to require the condition that the metric is locally flat at the axis of symmetry, i.e., ν(0, z)/c2 + μ(0, z) = ln B(0, z), z ∈ R.
(1.6)
We refer to [4] for more information on axially symmetric spacetimes and state our main result.
Existence of Axially Symmetric Static Solutions of Einstein-Vlasov System
25
Theorem 1.1. There exist static solutions of the Einstein-Vlasov system (1.1), (1.2), (1.3) with c = 1 such that the metric is of the form (1.4) and satisfies the boundary conditions (1.5), (1.6), and the spacetime is axially symmetric, but not spherically symmetric. The strategy of the proof of this result is as follows. Due to the symmetries of the metric the following quantities are constant along geodesics: E := −g(∂/∂t, p α ) = c2 e2ν/c p 0 2 2 = c2 eν/c 1 + c−2 e2μ ( p 1 )2 + e2μ ( p 2 )2 + ρ 2 B 2 e−2ν/c ( p 3 )2 , 2
L := g(∂/∂ϕ, p α ) = ρ 2 B 2 e
−2ν/c2
p3 ;
(1.7) (1.8)
E can be thought of as a local or particle energy and L is the angular momentum of a particle with respect to the axis of symmetry. Since up to regularity issues a distribution function f satisfies the Vlasov equation if and only if it is constant along geodesics, any distribution function f which depends only on E and L satisfies the Vlasov equation with a metric of the above form. Hence we make the ansatz f (x a , p b ) = φ(E, L),
(1.9)
and the Vlasov equation (1.2) holds. Upon insertion of this ansatz into the definition (1.3) of the energy momentum tensor, the latter becomes a functional Tαβ = Tαβ (ν, B, μ) of the yet unknown metric functions ν, B, μ, and we are left with the problem of solving the field Einstein equations (1.1) with this right-hand side. We obtain solutions by perturbing off spherically symmetric steady states of the Vlasov-Poisson system via the implicit function theorem; the latter system arises as the Newtonian limit of the Einstein-Vlasov system. Our main result specifies conditions on the ansatz function φ above such that a two parameter family of axially symmetric solutions of the Einstein-Vlasov system passes through the corresponding spherically symmetric, Newtonian steady state. The parameter γ = 1/c2 turns on general relativity and the second parameter λ turns on the dependence on L and hence axial symmetry; notice that L is not invariant under arbitrary rotations about the origin, so if f actually depends on L the solution is not spherically symmetric. The scaling symmetry of the Einstein-Vlasov system can then be used to obtain the desired solutions for the physically correct value of c. The detailed formulation of our result is stated in the next section together with the basic set up of its proof. The remaining sections of the paper are then devoted to establishing the various features of the basic set up which are needed to apply the implicit function theorem, and to prove various properties of the solutions we obtain. We conclude this introduction with some further references to the literature. The idea of using the implicit function theorem to obtain equilibrium configurations of self-gravitating matter distributions from already known solutions can be traced back to L. Lichtenstein who argued the existence of axially symmetric, stationary, selfgravitating fluid balls in this way [12,13]. His arguments were put into a rigorous and modern framework in [9]. The analogous approach was used in [17] to obtain axially symmetric steady states of the Vlasov-Poisson system, see also [20]. The approach has also been used to construct axially symmetric stationary solutions of the Einstein equations coupled to a matter model: In [10] matter was described as an ideal fluid whereas in [1,2] it was described as a static or a rotating elastic body respectively. Besides the different matter model our investigation differs from the latter two in that we employ the rather explicit form of the metric stated above and a reduced version of the Einstein field equations which closely follows [4].
26
H. Andréasson, M. Kunze, G. Rein
2. Set Up of the Proof In what follows we also use the Cartesian coordinates (x 1 , x 2 , x 3 ) = (ρ cos ϕ, ρ sin ϕ, z) ∈ R3 , which correspond to the axial coordinates ρ ∈ [0, ∞[, z ∈ R, ϕ ∈ [0, 2π ]; it should be noted that tensor indices always refer to the spacetime coordinates t, ρ, z, ϕ. By abuse of notation we write ν(ρ, z) = ν(x), etc. In Sect. 3 we collect the relevant information on the relation between regularity properties of axially symmetric functions expressed in the variables x ∈ R3 or ρ ∈ [0, ∞[, z ∈ R, respectively. We introduce two (small) parameters γ = 1/c2 ∈ [0, ∞[ and λ ∈ R. In order to obtain the correct Newtonian limit below we adjust the ansatz for f as follows. Let v 1 = eμ p 1 , v 2 = eμ p 2 , v 3 = ρ Be−γ ν p 3 , so that p =e 0
−γ ν
1 + γ |v|2 .
For the particle distribution function we make the ansatz f (x, v) = φ (E − 1/γ ) ψ(λL). The important point here is that eγ ν(x) 1 + γ |v|2 − 1 1 E − 1/γ = → |v|2 + ν(x) as γ → 0, γ 2
(2.1)
(2.2)
i.e., the limit is the non-relativistic energy of a particle with phase space coordinates (x, v) in case ν = U N is the Newtonian gravitational potential. For γ = 0 this limit is to replace the argument of φ in (2.1). We now specify the conditions on the functions φ and ψ. Conditions on φ and ψ. (φ1) φ ∈ C 2 (R) and there exists E 0 > 0 such that φ(η) = 0 for η ≥ E 0 and φ(η) > 0 for η < E 0 . (φ2) The ansatz f (x, v) = φ 21 |v|2 + U (x) leads to a compactly supported steady state of the Vlasov-Poisson system, i.e., there exists a solution U = U N ∈ C 2 (R3 ) of the semilinear Poisson equation 1 2 |v| + U dv, U (0) = 0, U = 4πρ N = 4π φ 2 U N (x) = U N (|x|) is spherically symmetric, and the support of ρ N ∈ C 2 (R3 ) is the closed ball B R N (0), where U N (R N ) = E 0 and U N (r ) < E 0 for 0 ≤ r < R N , U N (r ) > E 0 for r > R N . (φ3) 6 + 4πr 2 a N (r ) > 0, r ∈ [0, ∞[, where
1 2 |v| + U N (r ) dv. a N (r ) := φ 2 R3 (ψ) ψ ∈ C ∞ (R) is even with ψ(L) = 1 iff L = 0, and ψ ≥ 0.
Existence of Axially Symmetric Static Solutions of Einstein-Vlasov System
27
For such a steady state, lim U N (x) = U N (∞) > E 0 .
|x|→∞
The normalization condition U N (0) = 0 instead of U N (∞) = 0 is unconventional from the physics point of view, but it has technical advantages below. Examples for ansatz functions φ which satisfy (φ1) and (φ2) are found in [5,19], the most well-known ones being the polytropes φ(E) := (E 0 − E)k+
(2.3)
for 2 < k < 7/2; here E 0 > 0 and (·)+ denotes the positive part. In Sect. 7 we show that for this class of ansatz functions also (φ3) holds. Numerical checks indicate that (φ3) holds for general isotropic steady states of the Vlasov-Poisson system. We can now give a more detailed formulation of our result. Theorem 2.1. There exists δ > 0 and a two parameter family (νγ ,λ , Bγ ,λ , μγ ,λ )(γ ,λ)∈[0,δ[×]−δ,δ[ ⊂ C 2 (R3 )3 with the following properties: (i) (ν0,0 , B0,0 , μ0,0 ) = (U N , 1, 0), where U N is the potential of the Newtonian steady state specified in (φ2). (ii) If for γ > 0 a distribution function is defined by Eq. (2.1) and a Lorentz metric √ by (1.4) with c = 1/ γ then this defines a solution of the Einstein-Vlasov system (1.1), (1.2), (1.3) which satisfies the boundary condition (1.6) and is asymptotically flat. For λ = 0 this solution is not spherically symmetric. (iii) If for γ = 0 a distribution function is defined by Eq. (2.1), observing (2.2), this yields a steady state of the Vlasov-Poisson system with gravitational potential ν0,λ which is not spherically symmetric for λ = 0. (iv) In all cases the matter distribution is compactly supported both in phase space and in space. Remark. (a) The smallness restriction to γ = 1/c2 is undesired because c is, in a given set of units, a definite number. However, if ( f, ν, B, μ) is a static solution for some choice of c ∈ ]0, ∞[ then the rescaling f˜(ρ, z, p 1 , p 2 , p 3 ) = ν˜ (ρ, z) = ˜ B(ρ, z) = μ(ρ, ˜ z) =
c−3 f (cρ, cz, cp 1 , cp 2 , p 3 ), c−2 ν(cρ, cz), B(cρ, cz), μ(cρ, cz)
yields a solution of the Einstein-Vlasov system with c = 1. The factor c2 in the metric (1.4) is removed by a rescaling of time. (b) The smallness restriction to λ means that the solutions obtained are close to being spherically symmetric.
28
H. Andréasson, M. Kunze, G. Rein
(c) The metric does not satisfy the boundary conditions (1.5), but lim
|(ρ,z)|→∞
ν(ρ, z) = ν∞ ,
lim
|(ρ,z)|→∞
μ(ρ, z) = −ν∞ /c2 ,
lim
|(ρ,z)|→∞
B(ρ, z) = 1. (2.4)
However, if we by abuse of notation redefine ν = ν − ν∞ and μ = μ + ν∞ /c2 , then the original condition (1.5) is restored and the metric (1.4) takes the form
2 2 ds 2 = −c2 e2ν/c c12 dt 2 + c22 e2μ dρ 2 + e2μ dz 2 + ρ 2 B 2 e−2ν/c dϕ 2 (2.5) with constants c1 , c2 > 0, which simply amounts to a choice of different units of time and space. By general covariance of the Einstein-Vlasov system (1.1), (1.2), (1.3) the equations still hold. (d) In view of [17], part (iii) of the theorem does not give new information on steady states of the Vlasov-Poisson system and is stated mainly in order to understand the obtained two parameter family of states as a whole. However, we note that for the Newtonian set-up in [17] axially symmetric steady states were obtained as deformations of a spherically symmetric one. The present approach differs considerably from this and in principle is more direct. (e) In the course of the proof of the theorem additional regularity properties and specific rates at which the boundary values at infinity are approached will emerge. (f) An alternative approach to the one followed here would be to start with static, spherical symmetric solutions of the Einstein-Vlasov system and use the implicit function theorem to perturb off from these. The technical reason why we adopted the present approach is that we have much better information on the linearized equations in the Newtonian case than in the relativistic one, which is essential in the application of the implicit function theorem. The alternative of perturbing off spherically symmetric Einstein-Vlasov solutions is currently under investigation. In the rest of this section we transform the problem of finding the desired solutions into the problem of finding zeros of a suitably defined operator. The Newtonian steady state specified in (φ2) will be a zero of this operator for γ = λ = 0, and the implicit function theorem will yield our result. In order that the overall course of the argument becomes clear we will go through its various steps, postponing the corresponding detailed proofs to later sections. The Einstein field equations are overdetermined, and we need to identify a suitable subset of (combinations of) these equations which, on the one hand, suffice to determine ν, B, μ, and which are such that at the end of the day all the field equations hold once this reduced system is solved. We introduce the auxiliary metric function ξ = γ ν + μ. Let and ∇ denote the Cartesian Laplace and gradient operator respectively. Taking suitable combinations of the field equations one finds that ∇B 1 ν + (2.6) · ∇ν = 4π γ γ e(2ξ −4γ ν) T00 + T11 + T22 + 2 2 e2ξ T33 , B ρ B ∇ρ · ∇ B = 8π γ 2 B (T11 + T22 ), B + (2.7) ρ
Existence of Axially Symmetric Static Solutions of Einstein-Vlasov System
∂ρ B ∂z B 1+ρ ∂ρ ξ − ρ ∂z ξ B B
1 ρ = ∂ρ (ρ 2 ∂ρ B) − ∂zz B + γ 2 ρ (∂ρ ν)2 − (∂z ν)2 , 2ρ B 2B ∂ρ B ∂ρ (ρ∂z B) ∂z B 1+ρ ∂z ξ + ρ ∂ρ ξ = + 2γ 2 ρ ∂ρ ν∂z ν. B B B
29
(2.8) (2.9)
The last two equations arise from ρ (G 11 − G 22 ) = 0 and ρ G 12 = 0 respectively; note that due to (2.1), T11 = T22 and T12 = 0. Because of the asymptotic behavior of B and the structure of the left-hand side of (2.7) we write B = 1 + h/ρ. Next, we observe that by taking suitable combinations of (2.8) and (2.9) we obtain equations which contain only ∂ρ ξ or ∂z ξ respectively, and we chose the former. In the above equations the components of the energy momentum tensor are functions of the unknown quantities ν, h, ξ = γ ν + μ for which we therefore have obtained the following reduced system of equations: 1 ∇(h/ρ) · ∇ν, B ∂ρρ h + ∂zz h = 8π γ 2 ρ B11 (ν, B, ξ, ρ; γ , λ),
(1 + ∂ρ h)2 + (∂z h)2 ∂ρ ξ = ∂z h ∂zρ h + 2γ 2 (ρ + h)∂ρ ν∂z ν
1 2 2 2 (∂ρρ h − ∂zz h) + γ (ρ + h) (∂ρ ν) − (∂z ν) , + (1 + ∂ρ h) 2 ν = 4π (00 + γ 11 + γ 33 ) (ν, B, ξ, ρ; γ , λ) −
(2.10) (2.11)
(2.12)
where αβ are defined below in (2.14)–(2.16). We supplement this with the boundary condition (1.6) which in terms of the new unknowns and since necessarily h(0, z) = 0, reads ξ(0, z) = ln 1 + ∂ρ h(0, z) .
(2.13)
It remains to determine precisely the dependence of the functions αβ on the unknown quantities ν, h, ξ . Since the ansatz (2.1) is even in the momentum variables p 1 , p 2 , p 3 — the fact that ψ is even is needed here, all the off-diagonal elements of the energymomentum tensor vanish. The computation of its non-trivial components uses the new integration variables eγ ν 1 + γ |v|2 − 1 , s = v3, η= γ the abbreviation
m(η, B, ν, γ ) = Be−γ ν
e−2γ ν (1 + γ η)2 − 1 , γ
30
H. Andréasson, M. Kunze, G. Rein
and yields 00 (ν, B, ξ, ρ; γ , λ) = γ 2 e(2ξ −4γ ν) T00 m(η,B,ν,γ ) 4π (2ξ −4γ ν) ∞ e = φ(η)(1 + γ η)2 ψ(λρs) ds dη, (2.14) B (eγ ν −1)/γ 0 11 (ν, B, ξ, ρ; γ , λ) = T11 + T22 ∞ m(η,B,ν,γ ) 4π = 3 e2ξ φ(η) ψ(λρs)(m 2 (η, B, ν, γ ) − s 2 ) ds dη, (2.15) B (eγ ν −1)/γ 0 e2ξ T33 ρ2 B2 m(η,B,ν,γ ) φ(η) ψ(λρs) s 2 ds dη;
33 (ν, B, ξ, ρ; γ , λ) = 4π = 3 e2ξ B
∞
(eγ ν −1)/γ
(2.16)
0
we recall that T11 = T22 . The reason for keeping B as argument on the right-hand sides above is that the matter terms are differentiable in this variable, but taking a derivative with respect to h would yield an irritating factor 1/ρ. For elements of the function space chosen below h/ρ extends smoothly to the axis of symmetry ρ = 0. We now define the function spaces in which we will obtain the solutions of the system (2.10), (2.11), (2.12). As noted above we write, by abuse of notation, axially symmetric functions as functions of x ∈ R3 or of ρ ≥ 0, z ∈ R; regularity properties of axially symmetric functions are considered in Sect. 3. We fix 0 < α < 1/2 and 0 < β < 1, and consider the Banach spaces X1 := ν ∈ C 3,α (R3 ) | ν(x) = ν(ρ, z) = ν(ρ, −z) and ν X1 < ∞ , X2 := h ∈ C 4,α (R2 ) | h(ρ, z) = −h(−ρ, z) = h(ρ, −z) and h X2 < ∞ , X3 := ξ ∈ C 2,α (Z R ) | ξ(x) = ξ(ρ, z) = ξ(ρ, −z) and ξ X3 < ∞ , where Z R := {x ∈ R3 | ρ < R} is the cylinder of radius R > 0, the latter being defined in (2.17) below. The norms are defined by ν X1 := ν C 3,α (R3 ) + (1 + |x|)1+β ∇ν ∞ , h X2 := h C 4,α (R2 ) + (1 + |(ρ, z)|)2 ∇(h/ρ) ∞ , ξ X3 := ξ C 2,α (Z R ) , and X := X1 × X2 × X3 ,
(ν, h, ξ ) X := ν X1 + h X2 + ξ X3 .
L ∞ -norm, functions in C k,α (Rn ) have by definition continuous
Here · ∞ denotes the derivatives up to order k and all the highest order derivatives are Hölder continuous with exponent α, |D σ g(x) − D σ g(y)| g C k,α (Rn ) := D σ g ∞ + sup , |x − y|α x,y∈Rn ,x = y |σ |≤k
|σ |=k
Existence of Axially Symmetric Static Solutions of Einstein-Vlasov System
31
and D σ denotes the derivative corresponding to a multi-index σ ∈ Nn0 . We note that if h ∈ X2 , then B = 1+h/ρ ∈ C 3 (R3 ), cf. Lemma 3.2. Moreover, it will be straightforward to extend ξ to R3 once a solution is obtained in the above space. Now we recall the properties of the Newtonian steady state specified in (φ2). That condition implies that there exists R > R N > 0 such that U N (r ) > (E 0 + U N (∞))/2, r > R.
(2.17)
If ||ν − U N ||∞ < |E 0 − U N (∞)|/4 and 0 ≤ γ < γ0 , with γ0 > 0 sufficiently small, depending on E 0 and U N , then eγ ν(x) − 1 > E 0 for all |x| > R. γ This implies that there exists some δ > 0 such that for all (ν, h, ξ ; γ , λ) ∈ U, the matter terms resulting from (2.14)–(2.16) are compactly supported in B R (0), where U := {(ν, h, ξ ; γ , λ) ∈ X × [0, δ[×] − δ, δ[ | (ν, h, ξ ) − (U N , 0, 0) X < δ}. In addition we require that δ > 0 is sufficiently small so that for all elements in U it holds that B = 1 + h/ρ > 1/2, and the factor in front of ∂ρ ξ in (2.12) is larger than 1/2; since h vanishes on the axis of symmetry, h/ρ is controlled by ∇h. Now let an element (ν, h, ξ ; γ , λ) ∈ U be given and substitute it into the matter terms defined in (2.14)–(2.16). With the right hand sides obtained in this way the Eq. (2.10)–(2.12) can then be solved, observing the boundary condition (2.13) and the fact that we require h to vanish on the axis of symmetry. We define the corresponding solution operators by 1 1 G 1 (ν, h, ξ ; γ , λ)(x) := − − M1 (y) dy |y| R3 |x − y| 1 ∇(h/ρ)(y) · ∇ν(y) dy + , 4π R3 B(y) |x − y| G 2 (ν, h, ξ ; γ , λ)(x) := 4 ln |(ρ − ρ, ˜ z − z˜ )| ρ˜ M2 (ρ, ˜ z˜ ) d ρ˜ d z˜ , R2 ρ G 3 (ν, h, ξ ; γ , λ)(x) := ln 1 + ∂ρ h(0, z) + g(s, z) ds, 0 ≤ ρ < R. 0
Here M1 (x) := (00 + γ 11 + γ 33 )(ν(x), B(x), ξ(x), ρ, ; γ , λ), M2 (ρ, z) := γ 2 B(x) 11 (ν(x), B(x), ξ(x), ρ; γ , λ), M2 (ρ, z) = M2 (−ρ, z) for ρ < 0 and z ∈ R, and
−1
g := (1 + ∂ρ h)2 + (∂z h)2 ∂z h ∂zρ h + 2γ 2 (ρ + h)∂ρ ν∂z ν
1 2 2 2 (∂ρρ h − ∂zz h) + γ (ρ + h) (∂ρ ν) − (∂z ν) . + (1 + ∂ρ h) 2
(2.18)
32
H. Andréasson, M. Kunze, G. Rein
Finally we define the mapping to which we are going to apply the implicit function theorem as F : U → X , (ν, h, ξ ; γ , λ) → (ν, h, ξ ) − (G 1 , G 2 , G 3 )(ν, h, ξ ; γ , λ). The proof of Theorem 2.1 now proceeds in a number of steps. Step 1. As a first step we need to check that the mapping F is well defined, in particular it preserves the various regularity and decay assumptions. This is done in Sect. 4. Step 2. The next step is to see that F(U N , 0, 0; 0, 0) = 0. This is due to the fact that for γ = λ = 0 the choice h = ξ = 0 trivially satisfies (2.11), (2.12), while (2.10) reduces to ν = 4π 00 (ν, 1, 0; 0, 0) with
00 (ν, 1, 0; 0, 0) = 4π
∞ ν
φ(η) 2(η − ν) dη =
R3
φ
1 2 |v| + ν 2
dv;
notice that h = 0 implies that B = 1. By (φ2), ν = U N is a solution of this equation, and the fact that U N ∈ X1 is part of what was shown in the previous step. Step 3. Next we show that F is continuous, and continuously Fréchet differentiable with respect to (ν, h, ξ ). The fairly technical but straightforward details are covered in Sect. 5. Step 4. The crucial step is to see that the Fréchet derivative L := DF(U N , 0, 0; 0, 0) : X → X is one-to-one and onto. Indeed, L(δν, δh, δξ ) = (δν − L 1 (δν) − L 2 (δh, δξ ), δh, δξ − L 3 (δh)) , where
L 1 (δν)(x) := −
R3
1 1 − |x − y| |y|
a N (y)δν(y) dy,
1 dy ∇(δh/ρ)(y) · ∇U N (y) 4π R3 |x − y| 1 1 − δξ(y)ρ N (y) dy, +2 |y| R3 |x − y| ρ 1 (∂ρρ δh − ∂zz δh)(s, z) ds, 0 ≤ ρ < R, L 3 (δh)(x) := ∂ρ δh(0, z) + 2 0
L 2 (δh, δξ )(x) :=
with a N as defined in (φ3). To see that L is one-to-one let L(δν, δh, δξ ) = 0. Then the second component of this identity implies that δh = 0, and hence also δξ = 0 by the third component. It therefore remains to show that δν = 0 is the only solution of the equation δν = L 1 (δν), i.e., of the equation δν = 4πa N δν, δν(0) = 0
(2.19)
Existence of Axially Symmetric Static Solutions of Einstein-Vlasov System
33
in the space X1 . Under the assumption on a N stated in (φ3) this is correct and shown in Sect. 6. It is at this point that our unconventional normalization condition in (φ2) together with the shift in the solution operator G 1 become important; notice that L 1 (δν)(0) = 0. To see that L is onto let (g1 , g2 , g3 ) ∈ X be given. We need to show that there exists (δν, δh, δξ ) ∈ X such that L(δν, δh, δξ ) = (g1 , g2 , g3 ). The second component of this equation simply says that δh = g2 . Now δh ∈ X2 implies that L 3 (δh) ∈ X3 , cf. Lemma 3.1 (b). Hence we set δξ = g3 + L 3 (δh) to satisfy the third component of the onto equation, and it remains to show that the equation δν − L 1 (δν) = g1 + L 2 (δh, δξ )
(2.20)
has a solution δν ∈ X1 . Firstly, L 2 (δh, δξ ) ∈ X1 . The assertion therefore follows from the fact that L 1 : X1 → X1 is compact, as is shown in Lemma 6.2. We are now ready to apply the implicit function theorem, cf. [7, Thm. 15.1], to the mapping F : U → X ; strictly speaking we should suitably extend F to γ < 0, but this is not essential. We obtain the following result. Theorem 2.2. There exists δ1 , δ2 ∈]0, δ[ and a unique, continuous solution map S : [0, δ1 [×] − δ1 , δ1 [→ Bδ2 (U N , 0, 0) ⊂ X such that S(0, 0) = (U N , 0, 0) and F(S(γ , λ); γ , λ) = 0 for all(γ , λ) ∈ [0, δ1 [×] − δ1 , δ1 [. The definition of F implies that for any (γ , λ) the functions (ν, h, ξ ) = S(γ , λ) are a solution of Eqs. (2.10)–(2.12), and if f is defined by (2.1) then the equations (2.6), (2.7), (2.12) hold with the induced energy momentum tensor. We can extend ξ to the whole space using the solution operator G 3 for all x ∈ R3 . Also, the boundary condition (1.6) on the axis of symmetry is satisfied: ξ(0, z) = G 3 (ν, h, ξ )(0, z) = ln(1 + ∂ρ h(0, z)) = ln B(0, z); recall that ξ = γ ν + μ. For γ = 0 we conclude first that h = 0, cf. (2.11) or the G 2 -part of the solution operator respectively, then the G 3 -part implies that ξ = 0 so that the solution reduces to (ν, 0, 0), where ν solves ν = 4π 00 (ν, 1, 0, ρ; 0, λ). Since 00 (ν, 1, 0, ρ; 0, λ) = 4π
ν
∞
φ(η)
√
2(η−ν)
ψ(λρs) ds dη
0
coincides with the spatial density induced by the ansatz (2.1) for the Newtonian case, cf. [17, Lemma 2.1], part (iii) of Theorem 2.1 is established. If λ = 0 then condition (ψ) implies that f really depends on the angular momentum variable L which is not invariant under all rotations about the origin, but only invariant under rotations about the axis ρ = 0. Moreover, if the metric were spherically symmetric then the explicit dependence of the quantities j j on ρ would imply that the induced energy momentum tensor would not be spherically symmetric which is a contradiction. Hence the obtained solutions are not spherically symmetric if λ = 0. To complete the proof of Theorem 2.1 we must show that indeed all the field equations are satisfied by the obtained metric (1.4).
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H. Andréasson, M. Kunze, G. Rein
The corresponding argument relies on the Bianchi identity ∇α G αβ = 0 which holds for the Einstein tensor induced by any (sufficiently regular) metric, and on the identity ∇α T αβ = 0 which is a direct consequence of the Vlasov equation (1.2); ∇α denotes the covariant derivative corresponding to the metric (1.4). The details are carried out in Sect. 8. Finally we collect the additional information on the solution which we obtain in the course of the proof. Proposition 2.3. Let (ν, h, ξ ) = S(γ , λ) be any of the solutions obtained in Theorem 2.2 and define μ := ξ − ν/c2 and B = 1 + h/ρ. Then the limit ν∞ := lim|x|→∞ ν(x) exists, and for any σ ∈ N30 with |σ | ≤ 1 and x ∈ R3 the following estimates hold: |D σ (ν(x) − ν∞ )| ≤ C(1 + |x|)−(1+|σ |) , |D σ (B − 1)(x)| ≤ C(1 + |x|)−(2+|σ |) , |D σ ξ(x)| ≤ C(1 + |x|)−(2+|σ |) . In particular, the spacetime equipped with the metric (1.4) is asymptotically flat in the sense that (2.4) and, after a trivial change of coordinates, also (1.5) holds. 1 (y) Proof. By definition of G 1 , lim|x|→∞ ν(x) = M|y| dy. The first two estimates are established in Lemma 4.2. As to the third one we observe that by the boundary condition (2.13) and Lemma 4.2, |ξ(0, z)| ≤ C|∂ρ h(0, z)| ≤
C . (1 + |z|)2
By (2.12) and the known asymptotic behavior of the coefficients in that equation which are given in terms of ν and h and their derivatives, C , + |z|3
|∂ρ ξ(ρ, z)| ≤ cf. Lemma 4.2. Hence
ρ
|ξ(ρ, z)| ≤ |ξ(0, z)| + 0
≤
C +C 1 + |z|2
1 + ρ3
|∂ρ ξ(s, z)| ds ∞
0
ds C ≤ , 1 + s 3 + |z|3 1 + |z|2
which is the desired estimate for ξ(ρ, z), provided ρ < |z|. Since we already know that the metric under consideration satisfies the full set of the Einstein equations, we can now use (2.8) and (2.9) to see that also ∂z ξ is given in terms of ν and h and their derivatives and satisfies the same decay estimate as ∂ρ ξ . Starting from ρ |ξ(ρ, z)| ≤ |ξ(ρ, ρ)| + |∂z ξ(ρ, s)| ds, z
we can use the decay of ∂z ξ to obtain the decay estimate for ξ(ρ, z) for ρ ≥ z ≥ 0 (or ρ ≥ −z ≥ 0), and the proof is complete. Remark. The decay results obtained in Proposition 2.3 coincide with the ones derived in [4] for stationary, axisymmetric spacetimes.
Existence of Axially Symmetric Static Solutions of Einstein-Vlasov System
35
3. Regularity of Axially Symmetric Functions We call a function f : R3 → R axially symmetric if there exists a function f˜ : [0, ∞[×R → R such that f (x) = f˜(ρ, z), where ρ = x11 + x22 and z = x3 for x ∈ R3 . In this section we collect some results on the relation between the regularity properties of f and those of f˜. Lemma 3.1. Let f : R3 → R be axially symmetric and f (x) = f˜(ρ, z), where f˜ : [0, ∞[×R → R. Let k ∈ {1, 2, 3} and α ∈ ]0, 1[. (a) f ∈ C k (R3 ) iff f˜ ∈ C k ([0, ∞[×R) and all derivatives of f˜ of order up to k which are of odd order in ρ vanish for ρ = 0. (b) f is Hölder continuous with exponent α ∈ ]0, 1[ iff f˜ is. Proof. As to part (a) let f ∈ C k (R3 ) be axially symmetric. Then f is even in x1 and x2 and f˜(ρ, z) = f (ρ, 0, z). This proves the “only-if” part. For the “if” part one checks that the corresponding derivatives of f , which exist for ρ = 0, extend continuously to ρ = 0. As to part (b) one only needs to observe that x → ρ(x) = since |∇ρ(x)| = 1.
x12 + x22 is Lipschitz,
At several places in our analysis it is convenient to extend functions of (ρ, z) to negative values of ρ. Lemma 3.2. Let h = h(ρ, z) ∈ C 4 (R2 ) be odd in ρ and define h(ρ, z)/ρ, ρ = 0, b(ρ, z) := ∂ρ h(0, z), ρ = 0. Then b ∈ C 3 (R2 ) and all derivatives of b up to order 3 which are of odd order in ρ vanish for ρ = 0. By abuse of notation, b ∈ C 3 (R3 ). Proof. The regularity of b only needs to be checked at ρ = 0. Since h is odd in ρ it follows that h(0, z) = ∂ρρ h(0, z) = 0 for z ∈ R. Hence as ρ → 0, b(ρ, z) =
1 (h(ρ, z) − h(0, z)) → ∂ρ h(0, z), ρ
and by Taylor expansion, 1 1 ∂ρ h(ρ, z) − 2 h(ρ, z) ρ ρ 1 = ∂ρ h(0, z) + ∂ρρ h(τ, z)ρ ρ 1 1 2 − 2 h(0, z) + ∂ρ h(0, z)ρ + ∂ρρ h(σ, z)ρ ρ 2 1 = ∂ρρ h(τ, z) − ∂ρρ h(σ, z) 2 1 → ∂ρρ h(0, z) = 0, 2
∂ρ b(ρ, z) =
(3.1)
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H. Andréasson, M. Kunze, G. Rein
where σ, τ are between 0 and ρ. All other derivatives can be treated in a similar fashion, where one should observe that ∂z h(0, z) = 0. The regularity with respect to x then follows by Lemma 3.1. 4. F is Well Defined As a first step we investigate the regularity properties of the functions j j , j = 0, . . . , 3, and of the induced matter terms M1 , M2 . Lemma 4.1. Let φ and ψ satisfy the conditions (φ1) and (ψ) respectively. (a) The functions 00 and 33 have derivatives with respect to ν, ξ, ρ ∈ R and B ∈ ]1/2, 3/2[ up to order three and these are continuous in ν, ξ, B, ρ, γ , λ. The same is true for 11 for derivatives up to order four. (b) For (ν, ξ, h; γ , λ) ∈ U, M1 ∈ C 2 (R3 ) and M2 ∈ C 2,α (R2 ) are both compactly supported. Proof. As to part (a) we note that differentiability with respect to ξ and ρ is straightforward. Concerning differentiability with respect to ν and B we observe that for j = 0, . . . , 3 the expression j j is differentiable once with respect to the indicated variables, provided φ ∈ L ∞ loc , cf. the proof of [17, Lemma 2.1]. Under Assumption (φ1) we can first differentiate twice before the change to the integration variables η and s and obtain expressions which are essentially of the same form as j j , but with φ or φ instead of φ so that the resulting expression can be differentiated once more. The reason why 11 is one order more differentiable is that when differentiating this expression with respect to ν or B the integral with respect to s is preserved, its integrand is differentiated, and the resulting expression is qualitatively of the same type as 00 and can be differentiated three more times. Part (b) follows since the functions ν, B, ξ which are now substituted into j j are all at least in C 2,α (R3 ); the fact that ξ is defined only on the cylinder Z R does not matter here because the integrals in the definitions of j j yield functions with support in Z R . We now show that F is well defined, more precisely: Lemma 4.2. Let (ν, ξ, h; γ , λ) ∈ U. Then the following holds : (a) G 1 = G 1 (ν, ξ, h; γ , λ) ∈ C 3,α (R3 ) is axially symmetric, even z = x3 , and M1in (y) 2 (1 + |x|)(G 1 − a) ∞ , (1 + |x|) ∇G 1 ∞ < ∞, where a = |y| dy. (b) G 2 = G 2 (ν, ξ, h; γ , λ) ∈ C 4,α (R2 ) is odd in ρ, even in z, and (1 + |(ρ, z)|)G 2 ∞ , (1 + |(ρ, z)|)2 D 1 G 2 ∞ , (1 + |(ρ, z)|)3 D 2 G 2 ∞ < ∞. Moreover, (1 + |(ρ, z)|)2 (G 2 /ρ) ∞ , (1 + |(ρ, z)|)3 D 1 (G 2 /ρ) ∞ < ∞. Here D j stands for any derivative of order j with respect to (ρ, z) ∈ R2 . (c) G 3 = G 3 (ν, ξ, h; γ , λ) ∈ C 2,α (Z R ) is axially symmetric, even in z = x3 , and G 3 C 2,α (Z R ) < ∞. (d) F(ν, ξ, h; γ , λ) ∈ X .
Existence of Axially Symmetric Static Solutions of Einstein-Vlasov System
37
Proof. As to part (a) the potential induced by the matter term M1 , which is in C 1,α (R3 ) by Lemma 4.1 (b), has the desired regularity and decay properties due to standard regularity results in Hölder spaces, cf. [14, Thms. 10.2, 10.3], and the decay of 1/|x − y| and its derivatives together with the compact support of M1 . As to the source term g = ∇(h/ρ) · ∇ν of the second term in G 1 we notice that ν ∈ X1 and h ∈ X2 implies that g ∈ C 1,α (R3 ) with |g(x)| ≤ C(1 + |x|)−3−β , in particular g ∈ L 1 ∩ L ∞ (R3 ). This implies the regularity of the potential induced by g and also its decay:
|g(y)| dy ≤ |x − y|
|x−y|≤|x|/2
... +
|x−y|>|x|/2
2 dy + (1 + |y|)−3−β |g(y)| dy |x − y| |x| |x−y|≤|x|/2 C C dy + ≤ ≤ C(1 + |x|/2)−3−β |x − y| |x| |x| |x−y|≤|x|/2
≤C
for large |x| as desired; for the gradient of the potential induced by g we argue completely analogously. As to part (b) we first recall that M2 = M2 (ρ, z) is even in ρ, and the actual source term ρ M2 is odd, compactly supported, and by Lemma 4.1 (b) and Lemma 3.1 (b), M2 ∈ C 2,α (R2 ). Hence G 2 ∈ C 4,α (R2 ) is odd in ρ ∈ R. As to the decay of G 2 let supp M2 ⊂ B R (0) ⊂ R2 . Then for |(ρ, z)| ≥ 2R and (ρ, ˜ z˜ ) ∈ supp M2 the estimate |ln |(ρ − ρ, ˜ z − z˜ )| − ln |(ρ, z)|| ≤ holds, and since
2R |(ρ, z)|
ρ˜ M2 = 0 this implies that
C ; |G 2 (ρ, z)| = G 2 (ρ, z) − 4 ln |(ρ, z)| ρ˜ M2 (ρ, ˜ z˜ ) d z˜ d ρ˜ ≤ |(ρ, z)| the estimates for the derivatives of G 2 follow along the same lines. Finally, ∂ρ (G 2 /ρ) = −G 2 /ρ 2 + ∂ρ G 2 /ρ which implies that |∂ρ (G 2 /ρ)(ρ, z)| ≤
C C . + |(ρ, z)|ρ 2 |(ρ, z)|2 |ρ|
This yields the asserted decay when |ρ| becomes large. But we can also use (3.1) to see that |∂ρ (G 2 /ρ)(ρ, z)| ≤ C/|z|3 . Both estimates together yield the asserted decay for ∂ρ (G 2 /ρ), and the decay for G 2 /ρ and ∂z (G 2 /ρ) can be dealt with similarly. In order to prove part (c) we observe that (2.18) and the regularity of ν and h imply that g and hence G 3 ∈ C 2,α (Z R ). By construction, ∂ρ G 3 = g. Since h is odd in ρ we find that h(0, z) = ∂z h(0, z) = ∂zz h(0, z) = ∂ρρ h(0, z) = 0, which implies that g(0, z) = 0. Thus by Lemma 3.1, G 3 ∈ C 2,α (Z R ), and the proof is complete.
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H. Andréasson, M. Kunze, G. Rein
5. F is Continuous and Continuously Differentiable with Respect to ν, h, ξ In this section we give some details of the proof of the following result: Lemma 5.1. The mappings G i : U → Xi , i = 1, 2, 3 are continuous and continuously Fréchet differentiable with respect to ν, h, and ξ . Proof. We only show the differentiability assertion and focus on G 1 . Defining = 00 + γ 11 + γ 33 we consider the differentiability only with respect to ν, and neglecting the dependence on the remaining variables we look at the prototype mapping (ν(y)) G : V → X1 , G(ν)(x) := dy, R3 |x − y| where V ⊂ X1 is open, ∈ C 3 (R) and ◦ ν has support in a fixed ball for all ν ∈ V. Our first claim is that G has the Fréchet derivative (ν(y))δν(y) dy, ν ∈ V, δν ∈ X1 . [DG(ν)δν](x) = |x − y| R3 In order to prove this claim we need to show that for ν ∈ V there exists > 0 such that for δν ∈ B (0) ⊂ X1 , ||G(ν + δν) − G(ν) − DG(ν)δν||X1 = o(||δν||X1 ). The support property and the standard elliptic estimate imply that ||G(ν + δν) − G(ν) − DG(ν)δν||X1 ≤ C ||G(ν + δν) − G(ν) − DG(ν)δν||C 3,α (R3 ) ≤ C ||(ν + δν) − (ν) − (ν)δν||C 1,α (R3 ) ≤ C ||(ν + δν) − (ν) − (ν)δν||C 2 (R3 ) . b
Clearly, ||(ν + δν) − (ν) − (ν)δν||∞ = o(||δν||∞ ) ≤ o(||δν||X1 ). We need to establish analogous estimates for expressions where we take derivatives with respect to x up to second order of the left hand side. Let i, j ∈ {1, 2, 3}. Then ∂xi (ν + δν) − (ν) − (ν)δν = (ν + δν) − (ν) ∂xi δν + (ν + δν) − (ν) − (ν)δν ∂xi ν, where both terms on the right are o(||δν||X1 ). Similarly, ∂xi x j (ν + δν) − (ν) − (ν)δν = (ν + δν) − (ν) − (ν)δν ∂xi ν ∂x j ν + (ν + δν) − (ν) ∂xi ν ∂x j δν + ∂x j ν ∂xi δν + (ν + δν) − (ν) − (ν)δν ∂xi x j ν + (ν + δν) − (ν) ∂xi x j δν + (ν + δν) ∂xi δν ∂x j δν,
Existence of Axially Symmetric Static Solutions of Einstein-Vlasov System
39
and all the terms appearing on the right are o(||δν||X1 ). This proves the differentiability assertion for G. As to the continuity of this derivative, ( (ν) − (˜ν )) δν (y) dy ||DG(ν) − DG(˜ν )|| L(X1 ,X1 ) = sup 3 | · −y| R ||δν||X ≤1 1
≤C
sup
||δν||X1 ≤1
||( (ν) − (˜ν )) δν||C 1,α (R3 )
X1
≤ C || (ν) − (˜ν )||C 2 (R3 ) → 0 as ν˜ → ν in X1 . b
These arguments prove the continuous Fréchet differentiability of the first part of G 1 with respect to ν. The derivatives with respect to h or ξ can be dealt with in exactly the same manner. The source term in the potential which represents the second part of G 1 can be expanded explicitly in powers of δh and δν which together with the standard elliptic estimate proves the assertion for that term; note that both B and B + δh/ρ are bounded away from 0. The mapping G 2 is treated in the same way as our prototype G above, except that we have to estimate the source term including its third order derivatives, observing that 11 has derivatives up to order four with respect to ν, B, ξ . The mapping G 3 is easier since the term g defined in (2.18) can be expanded explicitly in powers of δν and δh, where again we observe that the denominator in that expression is bounded away from 0. 6. DF (U N , 0, 0; 0, 0) is One-to-One and Onto We recall from Sect. 2 and Eq. (2.19) that in order to prove that the map L is one-to-one it remains to show that g = 0 is the only solution of g = 4πa N g, g(0) = 0,
(6.1)
in the space X1 . Inspired by the method in [17] we expand g into spherical harmonics Ylm , l ∈ N0 , m = −l, . . . , l, where we use the notation of [11]; for a more mathematical reference on spherical harmonics see [15]. Denote by (r, θ, ϕ) and (s, τ, ψ) the spherical coordinates of a point x ∈ R3 and y ∈ R3 respectively. For l ∈ N0 and m = −l, . . . , l we define glm (r ) :=
1 r2
|x|=r
∗ Ylm (θ, ϕ) g(x) d Sx .
(6.2)
The symmetry assumptions in the function space X1 imply that g1−1 = g10 = g11 = 0, since up to multiplicative constants the spherical harmonics with l = 1 are given by sin θ e±iϕ and cos θ . To proceed, we use the following expansion, cf. [11], ∞ l l 4π r< 1 = Y ∗ (τ, ψ) Ylm (θ, ϕ), l+1 lm |x − y| 2l + 1 r> l=0 m=−l
40
H. Andréasson, M. Kunze, G. Rein
where r< := min (r, s) and r> := max (r, s). In view of (6.1), 1 1 Y ∗ (θ, ϕ) d Sx a N (s) g(y)dy glm (r ) = − 2 r R3 |x|=r |x − y| lm ∞ l r< 4π a N (s) l+1 Y ∗ (τ, ψ) g(y) d S y ds =− 2l + 1 0 r> |y|=s lm ∞ l 4π r< =− a N (s) l+1 s 2 glm (s)ds 2l + 1 0 r> r ∞ 4π s l+2 rl =− a N (s) l+1 glm (s) ds + a N (s) l−1 glm (s) ds . 2l + 1 0 r s r By a straightforward computation we find that glm satisfies the equation
r 2 glm = l(l + 1) + 4π r 2 a N (r ) glm ,
(6.3)
where prime denotes a derivative with respect to r . (s)| so We use this to show that g00 = 0 as follows. We define w(r ) := sup0≤s≤r |g00 that |g00 (r )| ≤ r w(r ); at this point it becomes essential that g(0) = g00 (0) = 0. Now (6.3) can be integrated to yield the Gronwall estimate r s|a N (s)| w(s) ds, r ≥ 0, w(r ) ≤ 4π 0
so that w = 0, and hence g00 = 0 as desired. It therefore remains to consider glm with l ≥ 2. For these we prove the following auxiliary result. Lemma 6.1. Let a ∈ Cc ([0, ∞[) and λ > 0 be such that λ + 4πr 2 a(r ) > 0 for r ∈ [0, ∞[. Let u ∈ C 2 ([0, ∞[) be a bounded solution to (r 2 u ) = (λ + 4πr 2 a(r ))u.
(6.4)
Then u = 0. Proof. We fix ra > 0 such that a(r ) = 0 for r ≥ ra . Multiplying (6.4) with u and integrating by parts we obtain for r > 0, r r 2 2 (λ + 4π s a(s)) u (s) ds = (s 2 u (s)) u(s) ds 0 0 r = r 2 u (r ) u(r ) − s 2 (u (s))2 ds. (6.5) 0
Now if there exists r0 > 0 so that u(r0 ) = 0 or u (r0 ) = 0 then (6.5) implies that u(r ) = u (r ) = 0 for r ∈ [0, r0 ]. The unique solvability of (6.4) for r ≥ r0 then shows that u = 0 as claimed. So we assume now that u(r ) = 0 and u (r ) = 0 for r > 0. Since (6.4) is invariant under u → −u, we may suppose that u(r ) > 0 and u (r ) > 0 for all r ∈]0, ∞[; note that (6.5) enforces uu > 0 on ]0, ∞[. For r ≥ ra > 0 (6.4) simplifies to (r 2 u ) = λu, which has the solution (l + 1) u(ra ) + ra u (ra ) r l l u(ra ) − ra u (ra ) ra l+1 u(r ) = + . (2l + 1) ra (2l + 1) r Therefore u is unbounded which is a contradiction.
Existence of Axially Symmetric Static Solutions of Einstein-Vlasov System
41
Since g ∈ X1 , Eq. (6.2) implies that glm is bounded. Due to (φ3) we can apply Lemma 6.1 to conclude that glm = 0 for all l ≥ 2, and thus g = 0 as desired. We now prove the compactness result which was needed to show that L is onto. Lemma 6.2. The mapping K : X1 → X1 , a N (y) w(y) dy (K w)(x) = |x − y| R3 is compact. We remark that the operator L 1 has the form L 1 (δν)(x) = −K (δν)(x) + K (δν)(0) and is compact if K is, since the mapping ν → ν(0) is continuous on X1 . Proof. First we observe that the mapping √ ∞ √ 1 2 u → |v| + u dv = 2 2π φ φ (E) E − u d E 2 R3 u is in C 2 (R), and since U N ∈ C 2 (R3 ) the function a N is in Cc2 (R3 ). Hence a N w ∈ C 1,1/2 (R3 ) for any w ∈ X1 , and since α < 1/2 the mapping K is well defined. We fix a function χ ∈ Cc∞ (R3 ) such that 0 ≤ χ ≤ 1, χ (x) = 1 for |x| ≤ 1, and χ (x) = 0 for |x| ≥ 2. Let χ R (x) = χ (x/R) for R > 0 and define (K R w)(x) = χ R (x)(K w)(x). We show that K R → K in the operator norm as R → ∞. To this end, let ζ R = 1 − χ R so that for w ∈ X1 and x ∈ R3 , (K w − K R w)(x) = ζ R (x)(K w)(x),
(6.6)
and the latter vanishes for |x| ≤ R. Now let w X1 ≤ 1. For σ ∈ N30 with |σ | ≤ 3, it follows that σ D (K w − K R w)(x) ≤ ζ R (x) D σ (K w)(x) cτ D τ ζ R (x)D σ −τ (K w)(x) + 0<τ ≤σ
≤ 1{|x|≥R}
1 C D σ −τ (K w)(x) ≤ C ; +C |x| R R 0<τ ≤σ
constants denoted by C do not depend on x or R. In order to estimate the Hölder norm of D σ (K w − K R w) for |σ | = 3 we take x, x˜ ∈ R3 with |x| ˜ ≥ |x| and again apply the product rule to the expression (6.6). Adding and subtracting terms we have to estimate expressions like C τ σ −τ (D ζ R (x) − D τ ζ R (x))D ˜ ˜ (K w)(x) ≤ |x − x| R and terms like the following: τ σ −τ −1 D ζ R (x) |y| |(a N w)(x − y) − (a N w)(x˜ − y)| dy; D R3
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H. Andréasson, M. Kunze, G. Rein
if |σ − τ | = 3 we throw one derivative onto a N w. The latter quantity together with its first order derivatives is Hölder continuous. The factor in front of the integral vanishes for |x| ≤ R, so we need only consider |x| ˜ ≥ |x| ≥ R. Since the domain of integration extends only over y with |y − x| ≤ R N or |y − x| ˜ ≤ R N we can on the domain of integration estimate |y| ≥ |x| − |x − y| ≥ R − R N ≥ R/2or analogously with x˜ instead of x, where we assume that R > 2R N . Since D σ −τ |y|−1 ≤ |y|− j with j ≥ 1 the term under consideration can be estimated by C R −1 |x − x| ˜ α and altogether we conclude that K w − K R w C 3,α (R3 ) ≤ C/R. Recalling the definition of the norm · X1 we see that the following chain of estimates finally shows that K R → K in the corresponding operator norm as desired: |a N (y)w(y)| dy |∇(K w)(x) − ∇(K R w)(x)| ≤ ζ R (x) 3 |x − y|2 R |a N (y)w(y)| dy + R −1 |∇χ (x/R)| 3 |x − y| R C C ≤ 1{|x|≥R} 2 + R −1 1{R≤|x|≤2R} |x| |x| ≤ C(1 + |x|)−β−1 R −(1−β) . To complete the proof we have to show that K R is compact for any R > 0 on the space X1 . First the fact that a N ∈ Cc2 (R3 ) implies that K R : C 3,α (R3 ) → C 3,1/2 (R3 ) is continuous, and the same is true for K R : C 3,α (R3 ) → C 3,1/2 (B 3R (0)), where we note that all the functions K R w with w ∈ C 3,α (R3 ) are supported in B3R (0). Since α < 1/2 the embedding C 3,1/2 (B 3R (0)) → C 3,α (B 3R (0)) is compact, and because of the support property we conclude that K R : X1 → X1 is compact; on ∇ K R w the weight (1 + |x|)1+β only amounts to multiplication with a bounded function. 7. Discussion of Condition (φ3) In this section we investigate Condition (φ3) for the case of the polytropic steady states (2.3). We first allow for the general range k ∈]−1/2, 7/2[ of polytropic exponent. Using the elementary integration formula, 1 2 k (k + 1) k+ 32 s − |v| dv = (2π )3/2 (7.1) s+ , s ∈ R, 2 (k + 25 ) R3 +
Existence of Axially Symmetric Static Solutions of Einstein-Vlasov System
43
the Poisson equation in (φ2) is found to be 1 2 k+ 23 3/2 (k + 1) (r U ) = 4π(2π ) − U ) (E 0 N + N r2 (k + 25 ) for U N = U N (r ). According to [19] there exists a solution U N such that U N (0) < E 0 , U N (0) = 0, U N (R N ) = E 0 , U N (r ) > E 0 for r > R N , and U N (r ) > 0 for r ∈]0, R N [. For z := E 0 − U N this means that −
1 2 (k + 1) 3 (r z ) = 4π cn z +n , where n := k + ∈ ]1, 5[, cn := (2π )3/2 , r2 2 (k + 25 )
and furthermore z(0) > 0, z (0) = 0, z(R N ) = 0, and z (r ) < 0 for r ∈ ]0, R N [. In terms of z the function a N from (φ3) reads a N (r ) = −(2π )3/2
k(k) (k +
3 2)
k+ 21
z(r )+
= −n cn z(r )n−1 + ,
where once more (7.1) was used. Thus condition (φ3) is equivalent to 4π n cn r 2 z(r )n−1 < 6. +
(7.2)
Now consider the function ζ (s) := z(αs) for α := (4π cn )−1/2 . It is found to satisfy the Emden-Fowler equation −
1 2 (s ζ ) = ζ+n s2
(7.3)
and ζ (0) > 0, ζ (0) = 0, ζ (s0 ) = 0 for s0 := R N /α, as well as ζ (s) < 0 for s ∈ ]0, s0 [. In terms of s = α −1r condition (7.2) becomes s 2 ζ (s)n−1 < +
6 . n
(7.4)
The left-hand side can be conveniently expressed by means of the dynamical systems representation of (7.3). For, let U (t) := −
sζ (s)n sζ (s) ≥ 0, V (t) := − ≥ 0, t := ln s, ζ (s) ζ (s)
where we consider t ∈] − ∞, ln s0 [. Then U˙ = U (3 − U − nV ), V˙ = V (U + V − 1),
(7.5)
and U (t)V (t) = s 2 ζ (s)n−1 + , which provides the relation to (7.4). Thus we have to verify that U (t)V (t) < 6/n. In the terminology of [6, p. 501], where m = 0, ζ is an E-solution to (7.3). Thus [6, Prop. 5.5] implies that (U (t), V (t)) lies in the unstable manifold of the fixed point P3 = (3, 0) of (7.5). In particular, we have limt→−∞ (U (t), V (t)) = (3, 0). Also note that P3 is of saddle type with eigenvalues −3 and 2; the corresponding eigenvectors are (1, 0) and (−3n/5, 1). Since the line V = n1 (3 − U ) separates the regions U˙ > 0 (below the line) and U˙ < 0 (above the line), a phase plane analysis reveals that
44
H. Andréasson, M. Kunze, G. Rein
we must always have U (t) ≤ 3, so that W (t) := U (t)V (t) ≤ 3V (t). In addition, it is calculated that V and W are solutions to the system V˙ = V (V − 1) + W, W˙ = W (2 − (n − 1)V ),
(7.6)
such that limt→−∞ (V (t), W (t)) = (0, 0). The origin is a fixed point of saddle type for (7.6), the eigenvalues are −1 and 2 with corresponding eigenvectors (1, 0) and (1, 3). 2 2 Note that W˙ > 0 for V < n−1 , W˙ < 0 for V > n−1 , V˙ > 0 above the curve V → V (1 − V ), and V˙ < 0 below this curve. Since the curve has unity slope at V = 0, it follows that (V (t), W (t)), lying in the unstable manifold of the origin, will be above the curve for t very negative. Then a phase plane analysis shows that this property persists for all times. In particular, we always have V˙ > 0, and W is increasing until 2 it reaches its maximal value for t0 such that V (t0 ) = n−1 . Thus our original problem of proving (φ3) is equivalent to showing that W (t0 ) = max W < 6/n. Thanks to the preceding observations, the parametrized curve t → (V (t), W (t)) for t ∈] − ∞, t0 ] can be rewritten as a curve W = W (V ) in the (V, W )-plane which solves dW W (2 − (n − 1)V ) = , dV V (V − 1) + W
(7.7)
2 and which is such that W (0) = 0 and W ( n−1 ) = max W .
Lemma 7.1. If k < 7/2 is sufficiently close to 7/2, then (φ3) holds for φ given by (2.3). 2 Proof. If W (V ) < 1 < 6/n for all V ∈]0, n−1 ], then we are done. Hence we assume 2 that W (V0 ) = 1 for some V0 ∈]0, n−1 ]. Then 1 = W (V0 ) ≤ 3V0 yields V0 ≥ 1/3. Since W (V ) ≥ 1 for V ≥ V0 , it follows that V (V − 1) + W = (V − 1)2 + V + W − 1 ≥ V , so that by (7.7),
ln(max W ) =
max W W =1
dW ≤ W
2 n−1
V0
(2 − (n − 1)V˜ ) ˜ dV V˜
(2 − (n − 1)V˜ ) ˜ dV V˜ 1/3 6 1 = 2 ln − 2 + (n − 1). n−1 3 ≤
2 n−1
Therefore 36 1 (n − 1) − 2 . max W ≤ exp (n − 1)2 3
(7.8)
At n = 5 the relation 9 −2/3 6 < e 4 5 holds. Hence it follows from (7.8) that max W < 6/n is verified for n sufficiently close to n = 5.
Existence of Axially Symmetric Static Solutions of Einstein-Vlasov System
45
The method of proof for the preceding lemma can be refined as follows. Fix A < 6/n. 2 Then W (V ) < A for V ∈ [0, n−1 ] would be acceptable. Hence we can assume that 2 W (V0 ) = A for some V0 ∈]0, n−1 ]. Then A = W (V0 ) ≤ 3V0 shows that V0 ≥ A/3. From W (V ) ≥ A for V ≥ V0 we obtain ln(max W ) − ln A 2 max W ˜) n−1 (2 − (n − 1) V dW ≤ d V˜ = 2 W V˜ − V˜ + A V0 W (V0 ) 2 ˜) n−1 (2 − (n − 1) V d V˜ ≤ 2 V˜ − V˜ + A A/3 5−n 5−n 3 − 2A =√ arctan √ + arctan √ 4A − 1 4 A − 1(n − 1) 3 4A − 1 2 9[An − 2(A + 1)n + 6 + A] n−1 ln . − 2 A(A + 6)(n − 1)2 Therefore max W ≤ A (n), where n−1 2 A(A + 6)(n − 1)2 A (n) := A 9[An 2 − 2(A + 1)n + 6 + A] 5−n 3 − 2A 5−n arctan √ . + arctan √ × exp √ 4 A − 1(n − 1) 4A − 1 3 4A − 1
For different A it can be checked (e.g. using Maple) for which values n ∈ ]1, min{6/A, 5}[ the relation A (n) < 6/n is verified. Taking A = 1, we get at least n ∈ [2.6, 5[, for A = 6/5 we get at least n ∈ [2.35, 4.85], and for A = 2, we get at least n ∈ [2.1, 2.5]. In summary, the desired relation max W < 6/n can be obtained for at least n ∈ [2.1, 5[, which corresponds to at least k ∈ [0.6, 3.5[ in (2.3). Notice however that the regularity assumption on φ requires k > 2. 8. The Field Equations Hold For a metric of the form (1.4) the components 00, 11, 12, 22, and 33 of the field equations are nontrivial. We have so far obtained a solution ν, B, ξ of the reduced system (2.6), (2.7), (2.12), where the appearing components of the energy momentum tensor are induced by a phase space density f which satisfies the Vlasov equation (1.2). We define E αβ := G αβ − 8π c−4 Tαβ so that the Einstein field equations become E αβ = 0. By (2.7), E 11 + E 22 = 0.
(8.1)
Using this information (2.6) says that 2
ρ 2 B 2 E 00 + c2 e4ν/c E 33 = 0 or 2
c2 e4ν/c E 00 + ρ 2 B 2 E 33 = 0.
(8.2)
46
H. Andréasson, M. Kunze, G. Rein
The Vlasov equation implies that ∇α T αβ = 0, and ∇α G αβ = 0 due to the contracted Bianchi identity, where ∇α denotes the covariant derivative corresponding to the metric (1.4). We want to use these relations to show that the remaining components of E αβ vanish also, but there is a technical catch: The metric, more specifically ξ , is only C 2 . To overcome this complication we approximate ξ by C 3 functions ξn . The induced Einstein αβ tensor G n again satisfies the Bianchi identity. Taking β = 1 and letting n → ∞ we obtain the equation ∂z B E 12 − ρ Be−2ξ (B + ρ∂ρ B) E 33 = 0, ∂z E 12 + 4∂z μ + B
(8.3)
where (8.2) has been used to eliminate E 00 and we recall that ξ = ν/c2 + μ. Here ∂z E 12 is at first a distributional derivative, but since all other terms in the equation are continuous this derivative indeed exists in the classical sense. The same approximation maneuver can be performed for β = 2 to obtain the equation ∂ρ E
12
1 ∂ρ B E 12 − ρ 2 Be−2ξ ∂z B E 33 = 0, + 4∂ρ μ + + ρ B
(8.4)
which holds for ρ > 0. However, if we multiply this equation with ρ we obtain an equation which holds for ρ ≥ 0. This is because E 12 (0, z) = 0 which is nothing but the boundary condition (2.13) on the axis of symmetry which we have incorporated into our integration of (2.12). We eliminate E 33 from (8.3), (8.4) and write the resulting equation for E 12 in terms of X := ρe4μ B E 12 . The result is the equation ∂ρ X −
ρ∂z B ∂z X = 0, B + ρ∂ρ B
which again holds for ρ ≥ 0. Since X (0, z) = 0 and since any characteristic curve of this equation intersects the axis of symmetry ρ = 0, we conclude that X vanishes identically. By (8.3) the same is true for E 33 so that E 12 = E 33 = E 00 = 0. Finally we observe that by (2.12), ∂ρ B ∂z B 1+ρ E 12 = 0. (E 11 − E 22 ) + ρ B B Since E 12 = 0 this means that E 11 = E 22 , and with (8.1) we conclude that E 11 = E 22 = 0, and all the non-trivial field equations are satisfied. Acknowledgement. The authors would like to thank Marcus Ansorg for useful discussions.
Existence of Axially Symmetric Static Solutions of Einstein-Vlasov System
47
References 1. Andersson, L., Beig, R., Schmidt, B.: Static self-gravitating elastic bodies in Einstein gravity. Commun. Pure Appl. Math. 61, 988–1023 (2008) 2. Andersson, L., Beig, R., Schmidt, B.: Rotating elastic bodies in Einstein gravity. Commun. Pure Appl. Math. 63, 559–589 (2009) 3. Andréasson, H.: The Einstein-Vlasov System/Kinetic Theory. Living Rev. Relativity 8 (2005), available at http://relativity.livingreviews.org/Articles/lrr-2005-z, 2005 4. Bardeen, J.: Rapidly rotating stars, disks, and black holes. In: Black Holes / Les Astres Occlus, ed. by C. DeWitt, B. S. DeWitt, Les Houches, 1972, London-NewYork-Paris: Gordon and Breach, 1973 5. Batt, J., Faltenbacher, W., Horst, E.: Stationary spherically symmetric models in stellar dynamics. Arch. Rat. Mech. Anal. 93, 159–183 (1986) 6. Batt, J., Pfaffelmoser, K.: On the radius continuity of the models of polytropic gas spheres which correspond to positive solutions of the generalized Emden-Fowler equations. Math. Meth. Appl. Sci. 10, 499–516 (1988) 7. Deimling, K.: Nonlinear Functional Analysis, Berlin-New York: Springer, 1985 8. Fjällborg, M., Heinzle, M., Uggla, C.: Self-gravitating stationary spherically symmetric systems in relativistic galactic dynamics. Math. Proc. Cambridge Philos. Soc. 143, 731–752 (2007) 9. Heilig, U.: On Lichtenstein’s analysis of rotating Newtonian stars. Ann. de l’Inst. H. Poincaré, Physique Théorique 60, 457–487 (1994) 10. Heilig, U.: On the existence of rotating stars in general relativity. Commun. Math. Phys. 166, 457–493 (1995) 11. Jackson, D.: Classical Electrodynamics, New York: Wiley, 1975 12. Lichtenstein, L.: Untersuchung über die Gleichgewichtsfiguren rotierender Flüssigkeiten, deren Teilchen einander nach dem Newtonschen Gesetze anziehen. Erste Abhandlung. Homogene Flüssigkeiten. Allgemeine Existenzsätze. Math. Z. 1, 229–284 (1918) 13. Lichtenstein, L.: Gleichgewichtsfiguren rotierender Flüssigkeiten, Berlin: Springer, 1933 14. Lieb, E., Loss, M.: Analysis, Providence, RI: Amer. Math. Soc., 1997 15. Müller, C.: Spherical Harmonics. Lecture Notes in Mathematics 17, Berlin: Springer, 1966 16. Rein, G.: Static solutions of the spherically symmetric Vlasov-Einstein system. Math. Proc. Camb. Phil. Soc. 115, 559–570 (1994) 17. Rein, G.: Stationary and static stellar dynamic models with axial symmetry. Nonlinear Analysis; Theory, Methods & Applications 41, 313–344 (2000) 18. Rein, G., Rendall, A.: Smooth static solutions of the spherically symmetric Vlasov-Einstein system. Ann. de l’Inst. H. Poincaré, Physique Théorique 59, 383–397 (1993) 19. Rein, G., Rendall, A.: Compact support of spherically symmetric equilibria in non-relativistic and relativistic galactic dynamics. Math. Proc. Camb. Phil. Soc. 128, 363–380 (2000) 20. Schulze, A.: Existence of axially symmetric solutions to the Vlasov-Poisson system depending on Jacobi’s integral. Commun. Math. Sci. 6, 711–727 (2008) 21. Wald, R.: General Relativity, Chicago, IL: Chicago University Press, 1984 Communicated by P.T. Chru´sciel
Commun. Math. Phys. 308, 49–80 (2011) Digital Object Identifier (DOI) 10.1007/s00220-011-1343-5
Communications in
Mathematical Physics
Ergodic Properties of a Model for Turbulent Dispersion of Inertial Particles Krzysztof Gaw¸edzki1 , David P. Herzog2,3 , Jan Wehr2 1 Laboratoire de Physique, C.N.R.S., ENS-Lyon, Université de Lyon, 46 Allée d’Italie, 69364 Lyon, France 2 Department of Mathematics, The University of Arizona, 617 N. Santa Rita Ave., P. O. Box 210089, Tucson,
AZ 85721-0089, USA. E-mail: [email protected]
3 Department of Mathematics, Duke University, Box 90320, Durham, NC 27708-0320, USA
Received: 3 September 2010 / Accepted: 1 July 2011 Published online: 30 September 2011 – © Springer-Verlag 2011
Abstract: We study a simple stochastic differential equation that models the dispersion of close heavy particles moving in a turbulent flow. In one and two dimensions, the model is closely related to the one-dimensional stationary Schrödinger equation in a random δ-correlated potential. The ergodic properties of the dispersion process are investigated by proving that its generator is hypoelliptic and using control theory. 1. Introduction Transport by turbulent flows belongs to phenomena whose understanding is both important for practical applications and abounds in intellectual challenges. Unlike the reputedly difficult problem of turbulence per se, turbulent transport allows simple modeling that accounts, at least qualitatively, for many of its observable features. The simplest of such models study transport properties of synthetic random velocity fields with presupposed distributions that only vaguely render the statistics of realistic turbulent velocities. The advection by velocity fields of quantities like temperature or tracer density may be derived from the dynamics of the Lagrangian trajectories of fluid elements. In synthetic velocity ensembles, such dynamics is described by a random dynamical system. One of the best studied schemes of this type is the so called Kraichnan model based on a Gaussian ensemble of velocities decorrelated in time but with long-range spatial correlations [8,18]. In this case, the random dynamical system that describes the Lagrangian flow is given by stochastic differential equations (SDE’s). It was successfully studied with the standard tools of the theory of random dynamical systems, but it also led to non-trivial extensions of that theory [12,19,20]. The problem of turbulent transport of matter composed of small but heavy particles (like water droplets in turbulent atmosphere) may be also studied by modeling turbulent velocities by a random synthetic ensemble, but it requires a modification of the previous approach. The reason is that heavy particles do not follow Lagrangian trajectories due to their inertia. On the other hand, the assumptions of the time decorrelation of random
50
K. Gaw¸edzki, D. P. Herzog, J. Wehr
velocities may be more realistic for inertial particles on scales where the typical relaxation time of particle trajectories (called the Stokes time) is much longer than the typical correlation time of fluid velocities. There have been a number of papers that pursued the study of dynamics of inertial particles with various simplifying assumptions, see e.g. [1–7,9,17,23,24,30,34]. The primary focus of those studies, combining analytical and numerical approaches, was the phenomenon of intermittent clustering of inertial particles transported by turbulent flow. A good understanding of that phenomenon is of crucial importance for practical applications. The aim of the present article is to show that the simplest among the models of inertial particles dynamics are amenable to rigorous mathematical analysis. More concretely, we study the SDE’s that describe the pair dispersion of close inertial particles in shortly correlated moderately turbulent homogeneous and isotropic d-dimensional velocity fields (not necessarily compressible). Such models were discussed in some detail in [2,17,24,25,30,34]. In particular, it was noted in [34] that the d = 1 version of the model is closely related to the one-dimensional stationary Schrödinger equation with δ-correlated potential studied already in the sixties of the last century [14] as a model for Anderson localization. As was stressed in [17], the d = 2 model for the inertial particle dispersion is also related to the one-dimensional stationary Schrödinger equation, but this time with δ-correlated complex potential. The models for dispersion were used to extract information about the (top) Lyapunov exponent for the inertial particles which is a rough measure of the tendency of particles to separate or to cluster [2,34]. The numerical calculations of the Lyapunov exponents in two or more dimensional models of particle dispersion presumed certain ergodic properties that seemed consistent with the results of simulations but were not obvious. From the point of view of the first order SDE for the particle dispersion and its time derivative, the ergodic properties of the dispersion process are a cumulative effect of the noisy advection in the subspace of the time-derivatives of dispersion and of the deterministic drift acting in the transverse directions of the phase space. We shall establish such properties rigorously by showing the hypoellipticity of the generator of the Markov process solving the corresponding SDE and by proving the irreducibility of the process with the help of control theory. For a quick introduction to such, by now standard, methods, we refer the reader to [11,31]. More information about the ergodic theory of Markov processes may be found in the treatise [26]. The main trouble in our analysis comes from configurations where two inertial particles (almost) coincide in space but have different velocities, leading to phase-space caustics [35] and enhanced spatial concentrations of particles (the “sling effect” of [7]). The reason is that these are the configurations where the diffusive part of the generator of the dispersion process vanishes. We show that in more than one space dimension two close inertial particles avoid such situations since, almost surely, the pair dispersion does not vanish. Some quantitative measure of such avoidance is provided. The result does not preclude the presence of caustic-like configurations in the evolution of three or more close inertial particles [35]. The paper is organized as follows. In Sect. 2, we present the SDE modeling the inertial particle dispersion. In Sect. 3, we recall its relation to models of one-dimensional Anderson localization. Section 4 establishes the hypoelliptic properties of the generator of the dispersion process. In Sect. 6, we introduce the (real-)projective version of the dispersion process whose compact space of states may be identified with the (2d − 1)dimensional sphere S 2d−1 . Section 5 is devoted to proving that the dispersion process is controllable. Together with the hypoelliptic properties of the generator, this implies that the projectivized version of the process has a unique invariant probability measure
Ergodic Properties of a Model for Turbulent Dispersion of Inertial Particles
51
with a smooth strictly positive density. The analytic expression for such a measure may be written down explicitly in d = 1 but not in higher dimensions. The smoothness and strict positivity of its density provide, however, in conjunction with the isotropy assumption, valuable information about the equal-time statistics of the projectivized dispersion. The isotropy permits to project further the projectivized dispersion to the quotient space S 2d−1 /S O(d). For d = 2, this space may be identified with with the complex projective space PC1 = C ∪ ∞ and the projected process with the complex-projectivized dispersion. In fact, the point at infinity, corresponding to phase-space caustics, may be dropped from PC1 since the complex-projectivized dispersion process stays in C with probability one. For d ≥ 3, the quotient space S 2d−1 /S O(d) is not smooth but has an open dense subset that may be identified with the complex upper-half-plane that the projected process never leaves. These non-explosive behaviors are established in Sect. 8 by constructing a Lyapunov function with appropriate properties. Physically they mean that the relative motion of a pair of close inertial particles avoids the caustic configurations, and for d ≥ 3, also the direction along the line that joins the particles. Results about behavior of the invariant density of the dispersion process projected to S 2d−1 /S O(d), established in Sect. 7, provide some qualitative information about such avoidance. Finally, in Sect. 9 and Appendix B, we demonstrate how the ergodic properties of the projectivized dispersion process proven in the paper lead to the formulae for the top Lyapunov exponent for inertial particles that were used in the physical literature. Appendix A derives a formula, used in the main text, expressing the S O(2d)-invariant measure on S 2d−1 in terms of S O(d) invariants. 2. Basic Equations The motion in a turbulent flow of a small body of large density, called below an inertial particle, is well described by the equation [1,2,23,24,34] 1 r¨ = − τ r˙ − u(t, r) , (2.1) where r(t) is the position of the particle at time t and u(t, r) is the fluid velocity field. Relation (2.1) is the Newton equation with the particle acceleration determined by a viscous friction force proportional to the relative velocity of the particle with respect to the fluid. Constant τ is the Stokes time. Much of the characteristic features of the distribution of non-interacting inertial particles moving in the flow according to Eqs. (2.1) is determined by the dynamics of the separation δr(t) ≡ ρ(t), called particle dispersion, of very close trajectories. In a moderately turbulent flow, the particle dispersion evolves according to the linearized equation: 1 ρ¨ = − τ ρ˙ − (ρ · ∇)u(t, r(t)) (2.2) or, in the first-order form: 1
ρ˙ = τ χ ,
1
χ˙ = − τ χ + (ρ · ∇)u(t, r(t)).
(2.3)
For sufficiently heavy particles, the correlation time of (∇u)(t, r(t)) is short with respect to the Stokes time τ and one may set in good approximation [2] ∇ j u i (t, r(t)) dt = d S ij (t),
(2.4)
52
K. Gaw¸edzki, D. P. Herzog, J. Wehr
where d S(t) is a matrix-valued white noise with the isotropic covariance i ik i k i k D ik d S j (t) d Slk (t ) = D ik jl δ(t − t ) dt dt , jl = A δ δ jl + B(δ j δl + δl δ j ). (2.5) Positivity of the covariance requires that A ≥ |B|,
A + (d + 1)B ≥ 0.
(2.6)
Incompressibility implies that A + (d + 1)B = 0, but we shall not impose it, in general. We shall only assume that A + 2B > 0 for d = 1 and that A > 0 for d ≥ 2. After the substitution of (2.4), Eq. (2.2) becomes the linear SDE 1
ρ¨ = − τ ρ˙ +
1 d S(t) ρ τ dt
(2.7)
that may be written in the first order form in a more standard notation employing differentials as 1 0 ρ ρ τ dt d = . (2.8) χ χ d S(t) − τ1 dt We shall interpret the latter SDE using the Itô convention, but the Stratonovich convention would lead to the same process. The solution of Eq. (2.8) exists with probability 1 for all times and has the form
t 0 1 ρ(t) ρ(0) τ ds = T exp , (2.9) χ (t) χ (0) d S(s) − τ1 ds 0 where the time ordered exponential may be defined as the sum of its Wiener chaos decomposition
t 0 T exp d S(s) 0 ·
0 d S(sn )
0 0
1 τ ds − τ1 ds
(sn −sn−1 )
e
0 0
=
∞
0 0
e
1 τ − τ1
0<s1 <···<sn
n=0
1 τ − τ1
(t−sn )
···
0 d S(s1 )
0 0
s1
e
0 0
1 τ − τ1
(2.10)
that converges in the L 2 -norm for functionals of the white noise d S(t). The resulting stochastic process (ρ(t), χ (t)) ≡ p(t) is Markov and has generator 1 j l ik 1 ρ ρ D jl ∇χ i ∇χ k . (2.11) L = τ χ · ∇ρ − χ · ∇χ + 2 i, j,k,l
In other words, for smooth functions f , d f ( p(t)) = (L f )( p(t)) (2.12) dt for · · · denoting the expectation. For the process p(t) given by Eq. (2.9), p(t) = 0 for all t ≥ 0 if p(0) = 0. On the other hand, if p(0) = 0 then p(t) = 0 with probability 1 for all t ≥ 0 so that we may restrict the space of states of the Markov process p(t) to R2d \{0} ≡ R2d =0 .
Ergodic Properties of a Model for Turbulent Dispersion of Inertial Particles
53
3. Relation to One-Dimensional Localization In the lowest dimensions, there is a simple relation between the stochastic process p(t) = (ρ(t), χ (t)) constructed above and simple models of Anderson localization in one space dimension. Let us set t
ψ(t) = e 2τ ρ(t)
(3.1)
exponentially blowing up the long-time values of ρ(t). Equation (2.7) implies that − ψ¨ +
1 d S(t) τ dt
ψ =−
1 4τ 2
ψ,
(3.2)
or, in the first order form, 0 ψ d = ξ d S(t) +
1 4τ dt
ψ . ξ 0
1 τ dt
(3.3)
Similarly as before, the above SDE defines a Markov process. Clearly, t 1 (ψ(t), ξ (t)) = e 2τ ρ(t), χ (t) + 2 ρ(t) .
(3.4)
Viewing t as the one-dimensional spatial coordinate, Eq. (3.2) takes the form of the vector-like stationary Schrödinger equation −
d2 ψ dt 2
+ V (t) ψ = E ψ,
(3.5)
where V (t) = τ1 d S(t) dt plays the role of the random matrix-valued white-noise potential and E = − 4τ1 2 of the (negative) energy. In particular, in d = 1, ψ(t) is a real scalar function and so is the δ-correlated potential V (t) =
1√ dβ(t) , A + 2B τ dt
(3.6)
where β(t) is the Brownian motion. The scalar version of Eq. (3.5) was studied in [14] as a model of one-dimensional Anderson localization, see also [21]. In d = 2, interpreting ψ as a complex number ψ 1 + iψ 2 , one may replace the matrix valued δ-correlated potential V (t) in the SDE (3.5) with the complex valued one V (t) =
1 τ
√
A + 2B
dβ 1 (t) √ dβ 2 (t) , +i A dt dt
(3.7)
where β 1 (t), β 2 (t) are two independent Brownian motions (the two realizations of V (t) lead to the Markov processes with the same generator and, consequently, with the same law). Consequently, as stressed in [17], Eq. (3.5) in d = 2 may be viewed as a model of localization for a one-dimensional non-hermitian random Schrödinger operator of the type not studied before.
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K. Gaw¸edzki, D. P. Herzog, J. Wehr
4. Hypoelliptic Properties of the Generator The generator (2.11) of the particle dispersion process p(t) = (ρ(t), χ (t)) has certain non-degeneracy properties which imply smoothness of the transition probabilities. Let us start with the following fact about the covariance (2.5) of the matrix-valued white noise d S(t): Lemma 4.1. We have D ik jl =
d
Eδ ij δnm + Fδ im δ jn + Gδni δ mj Eδlk δnm + Fδ km δln + Gδnk δlm
(4.1)
m,n=1
for √ E = d −1 − A + B + A + (d + 1)B , √ 1 √ F = 2 ( A + B + A − B), √ 1 √ G = 2 ( A + B − A − B).
(4.2)
Proof. The right hand side of Eq. (4.1) is: (E 2 d + 2E F + 2E G)δ ij δlk + (F 2 + G 2 )δ ik δ jl + 2F Gδli δ kj .
(4.3)
Hence, in order to satisfy Eq. (4.1), we must have A = F 2 + G2,
B = E 2 d + 2E(F + G) = 2F G.
(4.4)
The assumed values of E, F, G solve these equations.
Define the vector fields 1 X 0 = τ χ · ∇ρ − χ · ∇χ ,
ρ j Eδ ij δnm + Fδ im δ jn + Gδni δ mj ∇χ i , X nm = i, j
Ynm
=−
ρ j Eδ ij δnm + Fδ im δ jn + Gδni δ mj ∇ρ i
i, j
+ (χ j + ρ j ) Eδ ij δnm + Fδ im δ jn + Gδni δ mj ∇χ i = τ [X 0 , X nm ], i, j
Z nm
=−
(2χ j + ρ j ) Eδ ij δnm + Fδ im δ jn + Gδni δ mj ∇ρ i
i, j
+ (χ j + ρ j ) Eδ ij δnm + Fδ im δ jn + Gδni δ mj ∇χ i = τ [X 0 , Ynm ].
(4.5)
i, j
Using Eq. (4.1), one infers that Eq. (2.11) giving the generator L may be rewritten in the form:
1 (X nm )2 . (4.6) L = X0 + 2 m,n
Ergodic Properties of a Model for Turbulent Dispersion of Inertial Particles
55
Remark 4.2. The process p(t) may be equivalently obtained from the SDE d p = X 0 ( p) dt +
d
X nm ( p) dβmn (t),
(4.7)
m,n=1
where βmn (t) are independent Brownian motions. Here, we adopt the Stratonovich convention and hence the generator corresponding to that equation has the form (4.6) so that the latter SDE leads to a process with the same law as p(t). The convention of the stochastic integral we choose, however, is insignificant as the process ρ(t) is of bounded variation. In order to establish hypoelliptic properties of L, we shall use the following nondegeneracy relation satisfied by the vector fields X nm , Ynm and Z nm : Proposition 4.3. Suppose that p = (ρ, χ ) = 0. Then the vectors X nm ( p), Ynm ( p), Z nm ( p)
with m, n = 1, . . . , d
(4.8)
Proof. First suppose that ρ = 0 so that χ = 0. We have
χ j Eδ ij δnm + Fδ im δ jn + Gδni δ mj ∇χ i . Ynm (0, χ ) =
(4.9)
span the 2d-dimensional space.
i, j
Let φ ∈ Rd . Then
(α χ n χ m +β χ n φ m ) Ynm (0, χ ) = [E(αχ 2 +βχ ·φ)+ Fα χ 2 +G(αχ 2 +βχ ·φ)]χ ·∇χ m,n
+ Fβ χ 2 φ · ∇χ .
(4.10)
Setting α=−
χ ·φ E +G , (E + F + G)F (χ 2 )2
β=
1 F χ2
(4.11)
(note that F > 0 and E + F + G > 0), we obtain
(α χ n χ m + β χ n φ m ) Ynm (0, χ ) = φ · ∇χ .
(4.12)
m,n
Hence the vector φ · ∇χ is in the span of (4.8) for arbitrary φ. We have still to show that an arbitrary vector σ · ∇ρ is in that span. To this aim note that
Z nm (0, χ ) = −2 χ j Eδ ij δnm + Fδ im δ jn + Gδni δ mj ∇ρ i + (...)i ∇χ i . (4.13) i, j
i
Proceeding as before, we show that an appropriate combination of Z nm (0, χ ) gives the vector
σ · ∇ρ + (...)i ∇χ i (4.14) i
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K. Gaw¸edzki, D. P. Herzog, J. Wehr
from which the term (...)i ∇χ i may be removed by subtracting an appropriate combination of Ynm (0, χ ). That ends the proof of the claim of Proposition 4.3 for ρ = 0. Suppose now that ρ = 0. Proceeding as before, we see that arbitrary vector φ · ∇χ may be obtained by taking an appropriate combination of the vectors X nm (ρ, χ ). Similarly, an arbitrary vector σ · ∇ρ may be obtained as an appropriate combination of the vectors Ynm (ρ, χ ) and X nm (ρ, χ ). This completes the proof of Proposition 4.3.
The representation (4.6) and Proposition 4.3 imply, by virtue of Hörmander’s theory [11,16,28,29], the following result: Corollary 4.4. The operators L , L † , ∂t − L , ∂t − L † and 2∂t − L ⊗ 1 − 1 ⊗ L † are 2d 2d 2d hypoelliptic1 on R2d =0 , R+ × R=0 and R+ × R=0 × R=0 , respectively. In particular, the hypoellipticity of 2∂t − L ⊗ 1 − 1 ⊗ L † implies that the transition probabilities of the dispersion process p(t), Pt ( p0 , d p) = Pt ( p0 , p) d p,
(4.15)
have densities (annihilated by 2∂t − L ⊗ 1 − 1 ⊗ L † ) that are smooth functions of (t, p0 , p) for t > 0 and away from the origin in R2d . 5. Control Theory and Irreducibility The additional important property of the process p(t) restricted to R2d =0 is its irreducibility assured by the strict positivity of the smooth transition probability densities Pt ( p0 , p) for all t > 0 and p0 = 0 = p. The latter property results, according to Stroock-Varadhan’s Support Theorem [33], see also [31], from the controllability of the process p(t) on R2d =0 that is established in the following Proposition 5.1. For every T > 0 and p0 = 0 = p1 there exists a piecewise smooth 2 curve [0, T ] t → (u nm (t)) ∈ Rd such that the solution of the ODE
u nm (t) X nm ( p) (5.1) p˙ = X 0 ( p) + m,n
with the initial condition p(0) = p0 satisfies p(T ) = p1 . Proof. First suppose that ρ0 = 0 = ρ1 . Let [0, T ] t → ρ(t) be any curve such that ρ(0) = ρ0 , ρ(T ) = ρ1 ,
˙ τ ρ(0) = χ0 , ˙ ) = χ1 , τ ρ(T
(5.2)
˙ and such that ρ(t) = 0 for all t ∈ [0, T ]. Set χ (t) = τ ρ(t). Let φ(t) = τ χ˙ (t) + χ (t). Then the formula u nm =
1 α ρn ρm τ
+ β ρn φm ,
(5.3)
(5.4)
1 A differential operator D on a domain is hypoelliptic if for all distributions f, g such that D f = g, smoothness of g on an open subset U ⊂ implies smoothness of f on U .
Ergodic Properties of a Model for Turbulent Dispersion of Inertial Particles
57
where now α=−
ρ·φ E+G , (E + F + G)F (ρ 2 )2
β=
1 , F ρ2
(5.5)
defines smooth control functions [0, T ] t → (u nm (t)) such that Eq. (5.1) holds. Now suppose that ρ0 = 0 = ρ1 . Choose 0 < < 21 T and for 0 ≤ t ≤ , set t ρ(t) = 1 − e− τ χ0
(5.6)
˙ and χ (t) = τ ρ(t). Then p(t) = (ρ(t), χ (t)) satisfies Eq. (5.1) with u nm (t) ≡ 0 for 0 ≤ t ≤ , with the correct initial condition at t = 0. Note that (5.7) (ρ( ), χ ( )) = (1 − e− τ )χ0 , e− τ χ0 . Since, by the assumptions, χ0 = 0, we infer that ρ( ) = 0 and the solution of Eq. (5.1) for ≤ t ≤ T may be constructed as in the previous point but taking (5.7) as the initial conditions at t = . Similarly, if ρ0 = 0 = ρ1 then set for T − ≤ t ≤ T, T −t ρ(t) = 1 − e τ χ1 , (5.8) ˙ and χ (t) = τ ρ(t). Then p(t) = (ρ(t), χ (t)) satisfies Eq. (5.1) with u nm (t) ≡ 0 for T − ≤ t ≤ T , with the correct final condition at t = T . One has (5.9) (ρ(T − ), χ (T − )) = (1 − e τ )χ1 , e τ χ1 . Since, by the assumptions, χ1 = 0 now, we infer that ρ(T − ) = 0 and the solution of Eq. (5.1) for 0 ≤ t ≤ T − with ρ(t) = 0 may be constructed as in the first point but taking (5.9) as the final condition at t = T − . Finally, if ρ0 = 0 = ρ1 , we combine the above solutions for 0 ≤ t ≤ and T − ≤ t ≤ T with vanishing u nm with the solution with ρ(t) = 0 and appropriate u nm (t) for ≤ t ≤ T − ∈.
Remark 5.2. Note that the solution p(t) of the ODE (5.1) satisfying p(0) = p0 = 0 and p(T ) = p1 = 0 is everywhere nonzero. 6. Projection of the Dispersion to S2 d−1 The generator L of the process commutes with the multiplicative action of R+ on R2d given by
σ
p −→ σ p
(6.1)
for σ > 0. It follows that if p(0) = 0 then the projection [ p(t)] ≡ π (t)
(6.2)
R2d =0 /R+
is also a Markov process whose genof the process p(t) on the quotient space erator may be identified with L acting on functions on R2d =0 that are homogeneous of 2d degree zero. The quotient space R=0 /R+ may be naturally identified with the sphere S 2d−1 = { (ρ, χ ) | ρ 2 + χ 2 = R 2 }
(6.3)
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K. Gaw¸edzki, D. P. Herzog, J. Wehr
for a fixed R and we shall often use this identification below. The transition probabilities Pt (π0 ; dπ ) of the process π (t) are obtained by projecting the original transition probm m m abilities from R2d =0 to the quotient space. Note that the vector fields X 0 , X n , Yn , Z n 2d also commute with the action R+ so may be identified with vector fields on R=0 /R+ and Eq. (4.6) still holds. Viewed as vector fields on S 2d−1 , X nm , Ynm and Z nm still span at each point the tangent space to S 2d−1 . It follows that the operators L , L † , ∂t − L , ∂t − L † , 2∂t − L ⊗ 1 − 1 ⊗ L † (with the adjoints defined now with respect to an arbitrary measure with smooth positive density on S 2d−1 , e.g. the normalized standard S O(2d)-invariant one μ0 (dπ )) are still hypoelliptic and the transition probabilities of the projected process have smooth densities Pt (π0 ; π ) with respect to μ0 (dπ ) for t > 0. Consequently, the process π (t) is strongly Feller: for bounded measurable functions f on S 2d−1 , the functions Pt (π0 ; dπ ) f (π ) = Pt (π0 ; π ) f (π ) μ0 (dπ ) (6.4) (Tt f )(π0 ) = S 2d−1
S 2d−1
are continuous (and even smooth) for t > 0. Besides, the projected process is still irreducible since Pt (π0 ; π ) > 0 for all t > 0 and π0 , π ∈ S 2d−1 . The latter property follows from the relation between Pt (π0 ; π ) and Pt ( p0 ; p) and from the strict positivity of the latter away from the origin of R2d . The gain from projecting the process p(t) to the compact space S 2d−1 is that the projected process π (t) has necessarily invariant probability measures μ(dπ ). In particular, each weak-topology accumulation point for T → ∞ of the Cesaro means T
−1
T
Pt (π0 ; dπ ) dt
(6.5)
0
provides such a measure.2 Since the (a priori distributional) density n(π ) of an invariant measure is annihilated by L † , the hypoellipticity of the latter operator assures that n(π ) is a smooth function. The invariance relation Pt (π0 , π ) n(π0 ) μ0 (dπ0 ) = n(π ) (6.6) S 2d−1
together with the strict positivity of Pt (π0 , π ) implies then the strict positivity of the density n(π ) of the invariant measure and, in turn, the uniqueness of the latter (different ergodic invariant measures have to have disjoint supports, so that there may be only one such measure), see e.g. [31] for more details. One obtains this way Theorem 6.1. The projected process π (t) has a unique invariant probability measure μ(dπ ) with a smooth strictly positive density n(π ). The smoothness of the densities Pt (π0 ; π ) implies by the Arzelà-Ascoli Theorem that the operators of the semigroup Tt on the space C(S 2d−1 ) of continuous function on S 2d−1 with the sup-norm, defined by Eq. (6.4), are compact for t > 0. The uniqueness of the invariant measure implies then that the spectrum of Tt is strictly inside the unit disk except for the geometrically simple eigenvalue 1 corresponding to the constant eigenfunctions, see [31]. It follows that the process π (t) is exponentially mixing: 2 Probability measures on a compact space form a compact set in weak topology.
Ergodic Properties of a Model for Turbulent Dispersion of Inertial Particles
59
Theorem 6.2.
f 1 (π (t1 )) f 2 (π (t2 ))
−→
f 1 (π ) μ(dπ )
t1 →∞ t2 −t1 →∞
f 2 (π ) μ(dπ ),
(6.7)
exponentially fast for continuous functions f 1 , f 2 .
7. Properties of the Invariant Measure Due to the isotropy of the covariance (2.5), the generator L of the process π (t) commutes with the action of the rotation group S O(d) induced on S 2d−1 by the mappings
O
(ρ, χ ) −→ (Oρ, Oχ )
(7.1)
for O ∈ S O(d). As a consequence, the process π (t) stays Markov when projected to the quotient space Pd = S 2d−1 /S O(d). The unique invariant measure μ(dπ ) of the process π (t) has to be also invariant under S O(d) and its projection to Pd provides the unique invariant probability measure of the projected process.3 The projected invariant measure may be expressed in terms of invariants of the S O(d)-action. Such invariants will be chosen as the following dimensionless combinations: • for d = 1 where P1 = S 1 , x=
χ , ρ
(7.2)
• for d = 2 where P2 = PC1 , x=
ρ·χ ρ2
and
y=
ρ1χ 2 − ρ2χ 1 ρ2
(7.3)
with z = x + i y providing the inhomogeneous complex coordinate of PC1 , • for d ≥ 3, ρ·χ x= 2 ρ
and
ρ 2 χ 2 − (ρ · χ )2 y= . ρ2
(7.4)
Note that |x| → ∞ for d = 1 and |(x, y)| → ∞ in d ≥ 2 correspond to phase-space caustics where the dispersion ρ tends to zero with χ = τ ρ˙ staying finite. The right-hand side of the d ≥ 3 expression for y would give in d = 2 the absolute value of y. The configurations with y = 0 correspond to collinear ρ and χ . The quotient spaces Pd are not smooth for d ≥ 3. 3 To see the uniqueness, note that averaging over the action of S O(d) maps C(S 2d−1 ) to C(P ) and that d dual map sends invariant measures for the projected process to invariant measures of π (t).
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K. Gaw¸edzki, D. P. Herzog, J. Wehr
7.1. d = 1 case. In one dimension, Eq. (2.8) implies that 1 d x = − τ x + x 2 dt + d S(t).
(7.5)
The invariant probability measure on S 1 is easily found [14,34] to have the form dμ = η(x)d x with 2 x 2 1 1 x 3 +x 2 −1 − τ (A+2B) 3 x 3 +x 2 3 τ (A+2B) η(x) = Z d x d x, e (7.6) e −∞
where Z is the normalization constant. Since the normalized rotationally invariant meadx sure on S 1 = {(ρ, χ ) | ρ 2 + χ 2 = R 2 } has the form dμ0 = π(1+x 2 ) , it follows from our
general result that the density n(x) = π(1 + x 2 ) η(x) of the invariant measure relative to dμ0 must be smooth and positive at x = ∞, i.e. at the origin when expressed in the variable x −1 . In particular, η(x) = O(|x|−2 )
|x| → ∞,
for
(7.7)
which may also be easily checked directly. In one dimension, the generator L given by Eq. (2.11) acts on a function f (x) according to the formula: 1
1
(L f )(x) = − τ (x 2 + x) ∂x f (x) + 2 (A + 2B) ∂x2 f (x).
(7.8)
It coincides with the generator of the process satisfying the SDE (7.5). The trajectories of the latter process with probability one explode to −∞ in finite time but, in the version of the process that describes the projectivized dispersion of the one-dimensional inertial particle, they re-enter immediately from +∞. 7.2. d = 2 case. In two dimensions, the invariant measure on S 3 has to have the form dμ =
1 2π
η(z, z¯ ) d 2 z d arg(ρ).
(7.9)
On the other hand, the S O(4)-invariant normalized measure on S 3 is dμ0 =
1 2π
η0 (z, z¯ ) d 2 z d arg(ρ)
(7.10)
1 . π(1 + |z|2 )2
(7.11)
with η0 (z, z¯ ) =
It follows from the general result obtained above that the density of dμ relative to dμ0 , n(z, z¯ ) =
η(z, z¯ ) , η0 (z, z¯ )
(7.12)
has to extend to a smooth positive function on PC1 , i.e. to be smooth and positive at zero when expressed in the variables (z −1 , z¯ −1 ). In particular, η(z, z¯ ) = O(|z|−4 )
for
|z| → ∞.
(7.13)
Ergodic Properties of a Model for Turbulent Dispersion of Inertial Particles
61
The unique invariant probability measure of the Markov process obtained by projecting 1 π (t) from S 3 to S 3 /S O(2) = PC1 has the form (7.9) with 2π d arg(ρ) on the right hand side dropped. Note that the relation (7.13) implies that ∞ η(x, y) dy = O(|x|−3 ) for |x| → ∞, (7.14) −∞
√ by changing variables y → 1 + x 2 y in the integral. Such behavior was heuristically argued for and numerically checked in [2]. In two dimensions, the generator L of Eq. (2.11) acts on S O(d) invariant functions f (x, y) according to the formula: 1
(L f )(x, y) = − τ (x 2 − y 2 + x) ∂x f (x, y) 1
− τ (2x y + y) ∂ y f (x, y) 1
1
+ 2 (A + 2B) ∂x2 f (x, y) + 2 A ∂ y2 f (x, y).
(7.15)
It coincides with the generator of the process z(t) = (x + i y)(t) in the complex plane given by the SDE [30] √ √ 1 (7.16) dz = − τ z + z 2 dt + A + 2B dβ 1 (t) + i A dβ 2 (t), where β 1 (t) and β 2 (t) are two independent Brownian motions. 7.3. d ≥ 3 case. Finally, in three or more dimensions, the invariant measure on S 2d−1 has to have the form dμ = η(x, y) d x d y d[O],
(7.17)
where O ∈ S O(d) is the rotation matrix such that O −1 ρ is along the first positive half-axis in Rd and O −1 χ lies in the half-plane spanned by the first axis and the second positive half-axis. Note that, generically, O is determined modulo rotations in (d − 2) remaining directions. d[O] stands for the normalized S O(d)-invariant measure on S O(d)/S O(d − 2). In the same notation, the S O(2d)-invariant normalized measure on S 2d−1 takes the form dμ0 = η0 (x, y) d x d y d[O].
(7.18)
for η0 (x, y) =
(d − 1)2d−1 y d−2 , π(1 + x 2 + y 2 )d
(7.19)
as is shown in Appendix A. As before, it follows from the general analysis that the function n(x, y) =
η(x, y) η0 (x, y)
(7.20)
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K. Gaw¸edzki, D. P. Herzog, J. Wehr
is smooth and positive on the sphere S 2d−1 = {(ρ, χ ) | ρ 2 + χ 2 = R 2 }. In particular, this implies that η(x, y) = O(y d−2 )
for
y 0,
(7.21)
i.e. for ρ and χ becoming parallel or χ 2 becoming small and η(x, y) = O(|x|−2d )
for
|x| → ∞
(7.22)
when ρ 2 → 0 but the angle between ρ and χ stays away from a multiple of π2 . The smoothness and positivity of n(x, y) on S 2d−1 imply (again by changing variables √ y → 1 + x 2 y in the integral) that now ∞ η(x, y) dy = O(|x|−d−1 ) for |x| → ∞. (7.23) 0
A straightforward calculation shows that, in three or more dimensions, the action on L on S O(d)-invariant functions f (x, y) is given by a generalization of Eq. (7.15): 1
(L f )(x, y) = − τ (x 2 − y 2 + x) ∂x f (x, y) 1
− τ (2x y + y −
τ A(d−2) ) ∂y 2y
1
1
f (x, y)
+ 2 (A + 2B) ∂x2 f (x, y) + 2 A ∂ y2 f (x, y).
(7.24)
It coincides with the generator of the process z(t) = (x + i y)(t) in the complex plane given by the SDE [2,17] √ √ 1 τ A(d−2) dz = − τ z + z 2 − i dt + A + 2B dβ 1 (t) + i A dβ 2 (t), (7.25) 2 Im(z) which upon setting d = 2 reduces to the SDE (7.16). 8. Absence of Explosion in the Complex (Half-)Plane Let us set
Qd =
R2 H+
if if
d = 2, d ≥ 3,
(8.1)
where H+ = {(x, y) | y > 0} is the upper-half plane. Note that Q d may be identified with an open dense subset of the quotient space Pd = S 2d−1 /S O(d) using the S O(d)-invariants (7.3) or (7.4) on S 2d−1 . We shall often use the complex combination x + i y as a coordinate on Q d . In the present section, we shall show that for d ≥ 2 the unique solution of the SDE (7.25) starting from z ∈ Q d remains in Q d for all times t ≥ 0 with probability one. This will also have to be the property of the projection of the process π (t) to the quotient space Pd = S 2d−1 /S O(d) when described in the complex coordinate z = x + i y. Indeed, the
Ergodic Properties of a Model for Turbulent Dispersion of Inertial Particles
63
coincidence of the generators of the two processes will assure that they have the same law. Let us start by generalizing and simplifying (7.25). Let w(t) = z(t) + 1/2, where z(t) solves (7.25) with z(0) = x + i y ∈ Q d . Clearly, w(t) satisfies an SDE of the form dw =
2κ1 2κ2 τ b(d−2) 1 −w 2 + α + i dt + dβ (t) + i dβ 2 (t), τ τ Im(w)
1 τ
(8.2)
where α = a1 + ia2 ∈ C, b > 0, κ1 ≥ 0, κ2 > 0, and β 1 (t) and β 2 (t) are two independent Brownian motions. When d = 2, the term proportional to b(d − 2) is absent from (8.2). When d ≥ 3, we suppose that τ b(d − 2) ≥ κ2 . Clearly in (7.25), all of these assumptions are met under the given substitution. Since w(t) is a horizontal shift of z(t), w(t) stays in Q d with probability one for all times if and only if z(t) does. Employing methods of refs. [15,22,27,31], we shall estimate the time at which the process w(t) leaves Q d . To this end, it is easy to see that there exists a sequence of precompact open subsets {On | n ∈ N} of Q d such that On ↑ Q d as n → ∞. Thus we may define stopping times: τn = inf{s > 0 | w(s) ∈ Onc },
(8.3)
for n ∈ N. Let τ∞ be the finite or infinite limit of τn as n → ∞. Definition 8.1. We say that the solution w(t) is non-explosive if P [τ∞ = ∞] = 1.
(8.4)
Naturally, in order to show that w(t) remains in Q d for t ≥ 0 with probability one, it is enough to prove that w(t) is non-explosive. Let M be the generator of the process w(t) = x(t) + i y(t). We see that for f ∈ C ∞ (Q d ): (Md f )(x, y) = − τ (x 2 − y 2 − a1 )∂x f (x, y)− τ (2x y − a2 −τ b(d − 2)y −1 )∂ y f (x, y) 1
+
κ1 2 ∂ τ x
1
f (x, y) +
κ2 2 ∂ τ y
f (x, y),
(8.5)
where the term τ b(d − 2)y −1 is absent for d = 2. Let us define ∂ Qd =
∞ {(x, y) ∈ R2 | y = 0} ∪ ∞
if d = 2, if d ≥ 3,
(8.6)
with ∞ denoting the point compactifying R2 . To ensure condition (8.4), it suffices to construct a (Lyapunov) function d ∈ C∞ (Q d ) that satisfies:
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K. Gaw¸edzki, D. P. Herzog, J. Wehr
(I) d (x, y) ≥ 0 for all (x, y) ∈ Q d , (II) d (x, y) → ∞ as (x, y) → ∂ Q d , (x, y) ∈ Q d , (III) Md d (x, y) ≤ Cd (x, y) for all (x, y) ∈ Q d , where C > 0 is a positive constant. See, for example, [27]. We will show: Theorem 8.2. If κ1 ≥ 0 and κ2 > 0 for d = 2 or τ b(d − 2) ≥ κ2 > 0 for d ≥ 3 then there exists d ∈ C ∞ (Q d ) that satisfies (I), (II), and Md d (x, y) → −∞ as (x, y) → ∂ Q d , (x, y) ∈ Q d .
(IV)
Given such d , clearly d + 1 will satisfy (I), (II) and (III). We will then have: Theorem 8.3. Under the assumptions of Theorem 8.2, the solution w(t) of the SDE (8.2) stays in Q d for all times t > 0 with probability one if w(0) = x + i y ∈ Q d . Corollary 8.4. This implies the same result about the solution z(t) of the SDE (7.25) with A > 0 and A + 2B ≥ 0. The existence of the Lyapunov function with the properties asserted in Theorem 8.2 has another consequence. It allows to show that T 1 lim lim inf T Pt (w, Onc ) dt = 0 (8.7) n→∞ T →∞
0
for the SDE (8.2) and On ↑ Q d as before, implying the existence of an invariant measure on Q d , see Theorems 4.1 and 5.1 in Chap. III of [15]. If the generator of the process is elliptic, then the same tools that we used for the projectivized dispersion (i.e. hypoellipticity and control theory [31]) show that the invariant measure must have a smooth strictly positive density and be unique. This gives: Theorem 8.5. Under the assumptions of Theorem 8.2, the system (8.2) on Q d has an invariant measure which is unique and has a smooth strictly positive density if κ1 > 0. Remark 8.6. Theorem 8.5 allows to reaffirm and strengthen what has already been proven earlier since it implies the existence of an invariant measure for the system (7.25) if A > 0 and A + 2B ≥ 0 and its uniqueness if A + 2B > 0. Given the non-explosivity result of Corollary 8.4, the approach taken earlier implied the existence and the uniqueness of an invariant measure for the system (7.25) under more stringent conditions: A > 0, A ≥ |B| and A + (d + 1)B ≥ 0. The construction of the Lyapunov function d is split up into two cases: d = 2 and d ≥ 3. The existence of d for d ≥ 3 will be easy, given 2 . Thus we shall first construct 2 . 8.1. d = 2 case. It is not easy to write down a globally defined function 2 that satisfies (I), (II), and (IV) in all of Q 2 = R2 . This is because the signs of the coefficients of the vector fields in M2 vary over different regions in R2 . We shall thus construct functions that satisfy these properties in different regions, the union of which is R2 . We shall then glue together these functions to form one single globally defined 2 . One should note that this idea is similar in spirit to that of M. Scheutzow in [32]. Let r = x 2 + y 2 . For the rest of Subsection 8.1, we will drop the use of the subscript 2 in M2 and 2 . We first need the following:
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65
Definition 8.7. Let X ⊂ R2 be unbounded. We say that a function f (x, y) → ±∞ as r → ∞ in X if f (x, y) → ±∞ as (x, y) → ∞, (x, y) ∈ X . Definition 8.8. Let X ⊂ R2 be unbounded and let ϕ ∈ C ∞ (X ) satisfy (i) ϕ ≥ 0 for all (x, y) ∈ X , (ii) ϕ → ∞ as r → ∞ in X , (iii) Mϕ → −∞ as r → ∞ in X . We call ϕ a Lyapunov function in X corresponding to M and denote N (α, κ1 , κ2 , X ) = { Lyapunov functions in X corresponding to M } . We shall abbreviate “Lyapunov function” by LF. Definition 8.9. Let X ⊂ R2 be unbounded and f, g : X → R. We shall say that f is asymptotically equivalent to g in X and write f X g if lim
r →∞
f (x, y) = 1, g(x, y)
where the limit is taken only over points (x, y) ∈ X . It is clearly sufficient to construct LFs in regions that cover R2 , except, possibly, a large ball about the origin. The constructions will be done in a series of propositions. The possibly daunting multitude of parameters is designed to make the gluing possible. There is a total of five LFs in five different regions and the details that follow are not difficult to verify. The crucial LF is the fifth one, ϕ5 , defined in a region where explosion occurs in a nonrandom equation, i.e., when α = κ1 = κ2 = 0 in (8.2). Proposition 8.10. Let X 1 = {x ≥ 1} ⊂ R2 , C1 > 0, and δ ∈ (0, 1/2). Define ϕ1 (x, y) = C1 (x 2 + y 2 )δ/4 .
(LF1)
We claim that ϕ1 ∈ N (α, κ1 , κ2 , X 1 ) for all α ∈ C, κ1 ≥ 0, κ2 > 0. Proof. ϕ1 is nonnegative everywhere in R2 , hence everywhere in X 1 . ϕ → ∞ as r → ∞ in all of R2 , hence in all of X 1 . It is easy to check that ∂x x ϕ1 and ∂ yy ϕ1 both go to zero as r → ∞. Thus dropping second order terms in the expression for Mϕ1 , we have C1 δ C 1 δ a1 x + a2 y x(x 2 + y 2 )δ/4 + 2 2 (x 2 + y 2 )1−δ/4 C1 δ x(x 2 + y 2 )δ/4 → −∞ − 2
τ Mϕ1 X 1 − X1
(8.8)
as r → ∞ in X 1 , since x ≥ 1 in X 1 .
We need a remark before we move onto the next region. Let R ⊂ R2 be the real axis. Remark 8.11. Let f (x, y) = u(x, |y|) be a twice differentiable function in X \R. Then (τ M f )(x, y) = κ1 u x x (x, |y|) + κ2 u |y||y| (x, |y|) + (y 2 − x 2 + a1 ) u x (x, |y|) + (−2x|y| + sgn(y)a2 ) u |y| (x, |y|). Proof. Apply the chain rule to the operator ∂|y| .
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In the following arguments, often the function will be of the form f (x, y) = u(x, |y|). The above remark will allow for simplifications in the argument for property (iii) in Definition 8.8. Proposition 8.12. Let C2 > 0, δ ∈ (0, 1/2) and ϕ2 = C2 (−x + |y|δ/2 ).
(LF2)
Then ϕ2 ∈ N (α, κ1 , κ2 , X 2 ) for all α ∈ C and all κ1 ≥ 0, κ2 > 0, where X 2 = {−2 ≤ x ≤ 2} ∩ {|y| ≥ 22/δ }. Proof. ϕ2 is indeed smooth in X 2 since X 2 is bounded away from R. Note that the region was chosen so that ϕ2 ≥ 0 in X 2 . Moreover, since x is bounded in this region, r → ∞ in X 2 if and only if |y| → ∞. Hence, ϕ2 → ∞ in X 2 . By Remark 8.11 and noting that ∂x x ϕ2 = 0 and that ∂|y||y| ϕ2 → 0 as |y| → ∞, we have δ δ τ Mϕ2 (x, y) X 2 C2 (x 2 − y 2 − a1 ) + C2 (−2x|y| + sgn(y)a2 ) |y| 2 −1 2 X 2 − C2 y 2 → −∞
as r → ∞ in X 2 .
Proposition 8.13. Let C3 > 0 and δ ∈ (0, 1/2). Define 2 δ x + y2 ϕ3 = C3 |y|3/2
(LF3)
on X 3 = {x ≤ −1} ∩ {|y| ≥ 1}. Then ϕ3 ∈ N (α, κ1 , κ2 , X 3 ) for all α ∈ C and all κ1 ≥ 0, κ2 > 0. Proof. Smoothness of ϕ3 is not a problem in this region as we are bounded away from R in X 3 . Clearly, ϕ3 ≥ 0 and note that ϕ3 → ∞ as r → ∞ in X 3 . After dropping the δ(δ − 1)-terms which are negative, we obtain: δ−1
2 15x 2 x3 2 x + y2 1 τ Mϕ3 (x, y) ≤ C3 δ + κ + κ − 1 2 |y|3/2 |y|3/2 4|y|7/2 4|y|3/2 |y|3/2 2 3x 2a1 x 1 + x|y|1/2 + 3/2 + sgn(y)a2 − + 5/2 |y| 2|y|1/2 2|y| 2 δ−1 x3 x + y2 X 3 C3 δ + x|y|1/2 |y|3/2 |y|3/2 (8.9) = δ x ϕ3 → −∞ as r → ∞ in X 3 since x ≤ −1 in X 3 .
Proposition 8.14. Let C4 > 0, η > 1 and δ ∈ (0, 1/2). Define ϕ4 (x, y) = C4 on X 4 = {x ≤ −1} ∩
|x|2δ + |y|2δ
(LF4)
3
|y| 2 δ
1 3 3 η κ2 2 ( 2 δ + 1) √
|x|
≤ |y| ≤ 2 .
Then ϕ4 ∈ N (α, κ1 , κ2 , X 4 ) for all α ∈ C and all κ1 ≥ 0, κ2 > 0.
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67
Proof. Note that ϕ4 is smooth in X 4 since this region excludes both x and y axes. Moreover, ϕ4 → ∞ as r → ∞ in X 4 since then x must approach ∞ as r → ∞, and y is bounded above. Dropping insignificant terms in the expression for Mϕ4 , we see that in X 4: |x|2δ |x|2δ+1 3 3 + κ2 2 δ 2 δ + 1 + δ|x||y|δ/2 τ Mϕ4 (x, y) ≤ C4 − δ 3 3 δ δ+2 |y| 2 |y| 2 δ |x|2δ−1 |x|2δ 1 3 −1 2 − a1 2δ + sgn(y)a δ|y| − sgn(y)a δ 22 22 3 3 |y| 2 δ |y| 2 δ+1 |x|2δ |x|2δ+1 3 3 X 4 C4 −δ + κ2 2 δ 2 δ + 1 3δ 3 |y| 2 δ+2 |y| 2 1 3 3 |x|2δ+1 −1 + κ2 δ+1 = C4 δ 3 2 2 |x||y|2 |y| 2 δ |x|2δ+1 → −∞ ≤ − C4 δ(1 − 1/η2 ) 3 |y| 2 δ in X 4 as r → ∞.
Proposition 8.15. Let C5 , β > 0 and E > 0 such that 2κ2 > Eβ, let ξ > 1, and let ϕ5 (x, y) = C5 (E|x|β − y 2 |x|β+1 )
(LF5)
be defined on 1 E X 5 = {x ≤ −1} ∩ |y| ≤ ξ |x| . Then ϕ5 ∈ N (α, κ1 , κ2 , X 5 ) for all α ∈ C and all κ1 ≥ 0. Proof. The fact that ϕ5 is smooth in X 5 is clear as x ≤ −1 in X 5 . Again by the choice of X 5 , ϕ5 ≥ 0 and ϕ5 → ∞ as r → ∞ in X 5 . Dropping irrelevant terms in the expression for Mϕ5 , we see that in X 5 : τ Mϕ5 (x, y) ≤ C5 κ1 Eβ(β − 1)|x|β−2 − 2κ2 |x|β+1 + Eβ|x|β+1 + (β + 1)y 4 |x|β + C5 a1 (−Eβ|x|β−1 + (β + 1)y 2 |x|β ) − 2a2 y|x|β+1 X 5 C5 (Eβ − 2κ2 ) |x|β+1 → −∞ as r → ∞ in X 5 , as |x| must approach ∞ when r → ∞ in X 5 .
We now have our desired LFs. It is not obvious, however, that the regions X 1 , X 2 , . . . , X 5 cover R2 except, possibly, a bounded region about the origin. To assure that one has to show that X 4 and X 5 overlap. In order to make this more tangible, we will choose some of the parameters given in the previous propositions. With the choices that follow, however, we first need a lemma that says that varying the diffusion coefficients (κ1 , κ2 ) is permitted. This lemma will also be of crucial use later when we glue the LFs to form a globally defined .
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Lemma 8.16. Fix κ2 > 0 and suppose that ∈ N (α, κ1 , κ2 , R2 ) for all α ∈ C and all κ1 ≥ 0. Then for every ι2 > 0, we can find ∈ N (α, ι1 , ι2 , R2 ) for all α ∈ C and all ι1 ≥ 0. α For the proof of Lemma 8.16, we temporarily use the notation M(κ for the gen1 ,κ2 ) erator M given by (8.5).
Proof. Let η > 0 be such that η3 ι2 = κ2 . Define (x, y) = (ηx, ηy). For the function , smoothness and properties (I) and (II) are immediate. Let s = ηx and t = ηy. Then by the chain rule,
as r =
τ M(ια1 ,ι2 ) (x, y) =
τ η
M(ι
=
τ η
M(ι
αη2 3 3 (s, t) 1 η ,ι2 η ) αη2 (s, t) 3 1 η ,κ2 )
→ −∞,
x 2 + y 2 → ∞.
By Lemma 8.16, it is enough to find a function ∈ N (α, κ1 , κ2 , R2 ) for some fixed κ2 > 0 for all α ∈ C and all κ1 ≥ 0. All of the ϕi satisfy these criteria. In fact, ϕi for i = 1, 2, 3, 4 work more generally. The reason that ϕ5 only works for 2κ2 > Eβ is due to the fact that when α = κ1 = κ2 = 0, the solution to (8.2) has an explosive trajectory along the negative real axis. √ Now we choose some parameters. Let E = 5 and ξ = 5/2, so as to make X 5 = {x ≤ −1 ∩ {|y||x|1/2 ≤ 2}. 1 55 1 Let β = 11 4 δ, δ = 7 κ2 , and κ2 ∈ (0, 1). Then Eβ = 28 κ2 < 2κ2 and δ ∈ (0, 2 ). Hence for all i = 1, 2, . . . , 5, ϕi is a LF in the region X i . Decrease κ2 > 0 so that
X 4 ⊃ {x ≤ −1} ∩ {1 ≤ |y||x|1/2 ≤ 2}. One can easily check that X 4 and X 5 overlap in such a way that we have covered all of R2 with X 1 , X 2 , . . . , X 5 except a bounded region about the origin. We fix κ2 > 0 sufficiently small (this will be made precise later). We will construct a function ∈ N (α, κ1 , κ2 , R2 ) for all α ∈ C and all κ1 ≥ 0. The idea is as follows. Note that ϕ1 and ϕ2 are LFs in the region X 1 ∩ X 2 . We shall define a nonnegative smooth auxiliary function ζ (x) ∈ C ∞ (R) such that ζ (x) = 0 for x ≥ 2 and ζ (x) = 1 for x ≤ 1, satisfying some additional properties. We will then show that the combination (1 − ζ )ϕ1 + ζ ϕ2 is a LF in the larger region X 1 ∪ X 2 . Proceeding inductively this way, we shall construct a LF in all of R2 . Let us first define some auxiliary functions needed to construct such a . Let ζ : R → R+ be a C ∞ function such that 1 for x ≤ 1, ζ (x) = 0 for x ≥ 2, and ζ (x) < 0 for all x ∈ (1, 2). We define the smooth function μ : R → R+ as the horizontal shift of ζ , three units to the left, i.e., μ(x) = ζ (x + 3) for x ∈ R.
Ergodic Properties of a Model for Turbulent Dispersion of Inertial Particles
Let
⎧ ⎪ ⎨ ζ (|y|) ν(x, y) = 0 ⎪ ⎩0
69
for x ≤ −2, for |y| ≥ 2, for x > −1,
and assume that ν is C ∞ outside of the ball B4 . Let q : (−∞, −1] × R → R be defined by ⎧ ⎪ if |x|1/2 |y| ≥ 2, ⎨1 q(x, y) = |x|1/2 |y| − 1 if 1 < |x|1/2 |y| < 2, ⎪ ⎩0 if |x|1/2 |y| ≤ 1, and
r (t) =
1 exp − 1−(2t−1) 2
if 0 < t < 1,
0
otherwise.
Let s(x) =
1 N
x
−∞
r (t) dt,
where N = R r (t)dt. Now define a function on R2 by ⎧ ⎪ ⎨ s(q(x, y)) if x ≤ −1, ρ(x, y) = 1 if |y| ≥ 3, ⎪ ⎩1 if x ≥ −1/2. 2 ) outside of B . Clearly, ρ is C ∞ (R 4 √ ˙ ϕ for x 2 + y 2 ≥ r 2 (δ) by Let r (δ) = max 4, 24/δ + 4 . Define
⎧ ϕ ⎪ ⎪ ⎪ 1 ⎪ ⎪ ζ ⎪ ⎪ ϕ2 + (1 − ζ )ϕ1 ⎪ ⎪ ϕ2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ μϕ3 + (1 − μ)ϕ2 ϕ(x, y) = ϕ3 ⎪ ⎪ ⎪ νϕ4 + (1 − ν)ϕ3 ⎪ ⎪ ⎪ ⎪ ϕ4 ⎪ ⎪ ⎪ ⎪ ⎪ ρϕ4 + (1 − ρ)ϕ5 ⎪ ⎪ ⎩ ϕ5 and now
(x, y) =
if if if if if if if if if
x ≥ 2, 1 < x < 2, − 1 ≤ x ≤ 1, − 2 < x < −1, x ≤ −2, |y| ≥ 2 x ≤ −2, 1 < |y| < 2, x ≤ −2, |y| ≤ 1, |x|1/2 |y| ≥ 2, x ≤ −2, 1 < |x|1/2 |y| < 2, x ≤ −2, |x|1/2 |y| ≤ 1,
ϕ(x, y) arbitrary positive and smooth
if x 2 + y 2 ≥ Br (δ) , if x 2 + y 2 < Br (δ) .
It is easy to see can be chosen to be nonnegative and C ∞ (R2 ). With the aid of Lemma 8.16, the following lemma implies Theorem 8.2 in the d = 2 case.
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Lemma 8.17. For κ2 sufficiently small, ∈ N (α, κ1 , κ2 , R2 ) for all α ∈ C and all κ1 ≥ 0. Proof. Clearly, is smooth and satisfies Properties I and II. Since Mϕi → −∞ as r → ∞ in X i for each i, all we must verify is that M → −∞ as r → ∞ in the overlapping regions. Let us recall the choices that have already been made: E = 5, ξ =
√
5 , 2
β=
11 δ, 4
δ=
κ2 , 7
and note that κ2 ∈ (0, 1) was chosen such that X 4 ⊃ {x ≤ −1} ∩ {1 ≤ |y||x|1/2 ≤ 2}. Pick C5 > C4 = C3 > C2 > C1 . Consider first ψ1 := ζ ϕ2 + (1 − ζ )ϕ1 defined in the region Y1 = {1 < x < 2} ∩ Brc(δ) . We have τ Mψ1 = ζ τ Mϕ2 + (1 − ζ )τ Mϕ1 + (y 2 − x 2 + a1 )ζ (ϕ2 − ϕ1 ) + κ1 (ζ (ϕ2 − ϕ1 ) + 2ζ (∂x ϕ2 − ∂x ϕ1 )) Y1 − ζ C2 y 2 − (1 − ζ )
C1 δ x(x 2 2
+ y 2 )δ/4 + (y 2 − x 2 + a1 )ζ (C2 − C1 )|y|δ/2
+ κ1 (ζ (C2 − C1 )|y|δ/2 − 2ζ C2 ).
(8.10)
Note that if x is bounded away from 1 and 2 in (1, 2), the dominant term above is y 2 ζ (x)(C2 − C1 )|y|δ/2 → −∞. Note also that as x → 1 or x → 2, ζ , ζ → 0. But, the first two terms in the expression above decay to −∞ at least as fast as −C|y|δ/2 , where C > 0 is a constant independent of ζ and x. Thus we may choose > 0 so that whenever x ∈ (1, 2)\(1 + , 2 − ), Mψ1 decays at least as fast as −D|y|δ/2 , where D is some positive constant. We now consider ψ2 := μϕ3 + (1 − μ)ϕ2 in the region Y2 = {−2 < x < −1} ∩ Brc(δ) . We have τ Mψ2 = μ τ Mϕ3 + (1 − μ)τ Mϕ2 + (y 2 − x 2 + a1 )μ (ϕ3 − ϕ2 ) + κ1 (μ (ϕ3 − ϕ2 ) + 2μ (∂x ϕ3 − ∂x ϕ2 ) Y2 μ δxϕ3 − (1 − μ)C2 y 2 + (y 2 − x 2 + a1 )μ (C3 − C2 )|y|δ/2 Y2
+ κ1 (μ (C3 − C2 )|y|δ/2 + 2μ C2 ) μ C3 x|y|δ/2 − (1 − μ)C2 y 2 + (y 2 − x 2 + a1 )μ (C3 − C2 )|y|δ/2 + κ1 (μ (C3 − C2 )|y|δ/2 + 2μ C2 ).
Note that, for the very same reasons as in the case of ψ1 , Mψ2 → −∞ as r → ∞ in Y2 .
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71
Let ψ3 := νϕ4 + (1 − ν)ϕ3 in the region Y3 = {x ≤ −2} ∩ {1 < |y| < 2} ∩ Brc(δ) . Thus τ Mψ3 = ν τ Mϕ4 + (1 − ν)τ Mϕ3 + (−2x|y| + sgn(y)a2 )(∂|y| ν)(ϕ4 − ϕ3 ) Y3
2 + κ2 (∂|y| ν)(ϕ4 − ϕ3 ) + 2κ2 (∂|y| ν)∂|y| (ϕ4 − ϕ3 ) ν τ Mϕ4 + (1 − ν)τ Mϕ3 → −∞.
This is true since we chose C3 = C4 . Hence the first order term (−2x|y| + sgn(y)a2 )(∂|y| ν)(ϕ4 − ϕ3 ) approaches infinity at worst as fast as C|x|, where C is a positive constant. The second 2 ν)(ϕ − ϕ ) is a bounded function in this region. Moreover the term order term κ2 (∂|y| 4 3 2κ2 (∂|y| ν)∂|y| (ϕ4 − ϕ3 ) at worst approaches infinity as fast as D|x|2δ , where D is some positive constant. But, both Mϕ3 and Mϕ4 approach negative infinity at least as fast as −D|x|2δ+1 , where D is another positive constant. This gives the desired result. Let ψ4 := ρϕ4 + (1 − ρ)ϕ5 , in the region Y4 = {1 < |x|1/2 |y| < 2} ∩ {x ≤ −2} ∩ Brc(δ) . Note that τ Mψ4 = s(q)τ Mϕ4 + (1 − s(q))τ Mϕ5 + ∂x (s(q))(ϕ4 − ϕ5 )(y 2 − x 2 + a1 ) + ∂|y| (s(q))(ϕ4 − ϕ5 )(−2x|y| + sgn(y)a2 ) + κ1 ∂x2 (s(q))(ϕ4 − ϕ5 ) 2 (s(q))(ϕ4 − ϕ5 ) + 2κ1 ∂x (s(q))∂x (ϕ4 − ϕ5 ) + κ2 ∂|y| + 2κ2 ∂|y| (s(q))∂|y| (ϕ4 − ϕ5 ).
Note that in the expression above, we may drop the κ1 ∂x2 (s(q))(ϕ4 − ϕ5 ) and 2κ1 ∂x (s(q))∂x (ϕ4 −ϕ5 ) terms, as they are asymptotically less than other terms. Dropping other obviously insignificant terms, we obtain: τ Mψ4 Y4 s(q)τ Mϕ4 + (1 − s(q))τ Mϕ5 − ∂x (s(q))(ϕ4 − ϕ5 )x 2 2 − ∂|y| (s(q))(ϕ4 − ϕ5 )2x|y| + κ2 ∂|y| (s(q))(ϕ4 − ϕ5 ) + 2κ2 ∂|y| (s(q))∂|y| (ϕ4 − ϕ5 ) ≤ F(x, y),
where F is a smooth function satisfying: C5 δ 11 δ+1 |x| 4 4 − ∂x (s(q))(ϕ4 − ϕ5 )x 2 − ∂|y| (s(q))(ϕ4 − ϕ5 )2x|y|
F Y4 − s(q)C4 (1 − 1/η2 )
δ
23δ/2
11
|x| 4 δ+1 − (1 − s(q))
2 (s(q))(ϕ4 − ϕ5 ) + 2κ2 ∂|y| (s(q))∂|y| (ϕ4 − ϕ5 ). + κ2 ∂|y|
(8.11)
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K. Gaw¸edzki, D. P. Herzog, J. Wehr
By the choice of C5 > C4 , it is easy to see that for large |x| in this region, there are constants C, D > 0 such that 11 |x| 4 δ+1 2 κ2 ∂|y| (s(q))(ϕ4 − ϕ5 ) + 2κ2 ∂|y| (s(q))∂|y| (ϕ4 − ϕ5 ) ≤ Cκ2 r (q) g(x, y)2
and 11
−∂x (s(q))(ϕ4 − ϕ5 )x 2 − ∂|y| (s(q))(ϕ4 − ϕ5 )2x|y| ≤ −Dr (q)|x| 4 δ+1 , where the constants C, D and the function r (q) are independent of κ2 . r (q) is a function that goes to zero faster than any power of the function g(x, y) := |(|x|1/2 |y| − 1)(|x|1/2 |y| − 2)| as |x|1/2 |y| → 1 or 2. But note that, for all > 0, we may choose κ2 so small so that D>
Cκ2 g(x, y)2
for 1 + ≤ |x|1/2 |y| ≤ 2 − . From the first two terms in (8.11), we obtain at least 11 −C δ x 4 δ+1 decay for some C > 0 independent of κ2 for all 1 ≤ x 1/2 |y| ≤ 2. But since every other term in the expression goes to zero faster than every power of g as |x|1/2 |y| → 1 or 2, we can choose an > 0 so small as above so that Mψ4 → −∞ for all 1 < |x|1/2 |y| < 2. This completes the proof.
8.2. d ≥ 3 case. Here we shall complete the proof of Theorem 8.2 for d ≥ 3. Recall that for d = 2 and > 0 sufficiently small, we defined 2 (x, y) := (x, y) ∈ C ∞ (Q 2 ) that satisfies (I), (II), and (IV) for all α ∈ C, κ1 ≥ 0, and κ2 ∈ (0, ). Lemma 8.16 implied then that for κ2 > 0 arbitrary, the function 2,η (x, y) := 2 (ηx, ηy) ∈ C ∞ (Q 2 ) satisfied (I), (II), and (IV) for all α, provided that η−3 = 2κ2 . Let us fix κ2 > 0. For d ≥ 3, we shall define d,η := 2,η + log(1 + log2 (ηy/2)).
(8.12)
Lemma 8.18. For fixed κ2 > 0, d,η is a smooth function on Q d that satisfies (I), (II), and (IV). Proof. By definition, d,η is smooth and nonnegative in Q d = H+ . Clearly, d,η → ∞ as (x, y) → ∂ Q d = {(x, y) ∈ R2 : y = 0} ∪ {∞}, (x, y) ∈ Q d . Thus we must verify property (IV). To this end, note that: τ Md d,η = τ Md 2,η + τ Md (log(1 + log2 (ηy/2))) b(d − 2) ∂ y 2,η = τ M2 2,η + τ y 2 log(ηy/2) + −2x y 2 + a2 y + τ b(d − 2) − κ2 2 y (1 + log2 (ηy/2)) 2κ2 (1 − log2 (ηy/2)) . + 2 y (1 + log2 (ηy/2))2
Ergodic Properties of a Model for Turbulent Dispersion of Inertial Particles
73
Case 1. Suppose first that y ≥ 2η−1 . It is easy to check that there exist positive constants K 1 , K 2 > 0 such that b(d − 2) ∂ y 2,η ≤ K 1 , y 2 log(ηy/2) 2κ2 (1 − log2 (ηy/2)) ≤ K2. + (a2 y + τ b(d − 2) − κ2 ) 2 y (1 + log2 (ηy/2)) y 2 (1 + log2 (ηy/2))2 τ
If also x > −2η−1 then − 2x
log(ηy/2) ≤ K3 1 + log2 (ηy/2)
(8.13)
for a positive constant K 3 , whereas for x ≤ −2η−1 , − 2x
log(ηy/2) ≤ −K 4 x. 1 + log2 (ηy/2)
(8.14)
Since M2 2,η → −∞ as (x, y) → ∂ Q d and y ≥ 2η−1 , and, besides, if x ≤ −2η−1 and y ≥ 2η−1 then 2,η (x, y) is equal to the rescaled function ϕ3 so that, by (8.9), M2 2,η ≤ K 5 x(x 2 + y 2 )δ/4 for some K 5 > 0, we infer that Md d,η → −∞ as (x, y) → ∂ Q d and y ≥ 2η−1 . Case 2. Suppose now that 0 < y < 2η−1 . If |x| < 2η−1 , then (x, y) → ∂ Q d if and only if y ↓ 0. Since 2,η is smooth on R2 , there exists a constant K 6 > 0 such that τ M2 2,η +
1 τ b(d − 2) ∂2,η ≤ K6 y ∂y y
for (x, y) ∈ (−2η−1 , 2η−1 ) × (0, 2η−1 ). Hence, recalling the assumption τ b(d − 2) ≥ κ2 , we have on this rectangle: 1 2 log(ηy/2) + −2x y 2 + a2 y + τ b(d − 2) − κ2 2 y y (1 + log2 (ηy/2)) 2 2κ2 (1 − log (ηy/2)) + 2 y (1 + log2 (ηy/2))2 1 2κ2 (1 − log2 (ηy/2)) ≤ K7 + 2 → −∞ y y (1 + log2 (ηy/2))2
τ Md d,η ≤ K 6
as y ↓ 0. If x ≥ 2η−1 , then 2,η is equal to the rescaled function ϕ1 . We see that by (8.8) there exist constants K 8 , K 9 , K 10 , K 11 > 0 such that τ Md d,η ≤ −K 8 x(x 2 + y 2 )δ/4 + K 9 (x 2 + y 2 )δ/4−1 2 log(ηy/2) + −2x y 2 + a2 y + τ b(d − 2) − κ2 2 y (1 + log2 (ηy/2))
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K. Gaw¸edzki, D. P. Herzog, J. Wehr
+
2κ2 (1 − log2 (ηy/2)) y 2 (1 + log2 (ηy/2))2
≤ −K 8 x(x 2 + y 2 )δ/4 + K 10 x + K 11
1 2κ2 (1 − log2 (ηy/2)) . + y y 2 (1 + log2 (ηy/2))2
Note that as (x, y) → ∂ Q d in this region then x → ∞ or y ↓ 0. It is thus easy to see that τ Md d,η → −∞. If x < −2η−1 , then it is easy to check that ∂ y 2,η is bounded above by the choice of C3 = C4 and C5 > C4 . Then, for some K 12 > 0, τ Md d,η ≤ τ M2 2,η + K 12
1 2κ2 (1 − log2 (ηy/2)) + y y 2 (1 + log2 (ηy/2))2
so that τ Md d,η → −∞ as (x, y) → ∂ Q d in this region.
9. Top Lyapunov Exponent The Lyapunov exponent λ for the dispersion process p(t) = (ρ(t), χ (t)) is the asymptotic rate of growth in time of the logarithm of the length ρ 2 + χ 2 . Suppose that the process starts at t = 0 from p0 = (ρ0 , χ0 ) = 0. Anticipating the existence of the limit below, we shall define: 1 T →∞ T
λ = lim
1 T →∞ T
T
= lim =
1 lim T →∞ T
=
1 lim T →∞ T
ln
0
0
T
T
ρ 2 (T ) + χ 2 (T ) − ln
ρ02 + χ02 )
d ln ρ 2 (t) + χ 2 (t) dt dt L ln ρ 2 (t) + χ 2 (t) dt
(L ln
ρ 2 + χ 2 ) Pt ( p0 , d p) dt,
(9.1)
0
whereL is the generator of the process p(t) given by Eq. (2.11). Note that the function L ln ρ 2 + χ 2 is smooth on R2d \{0} and homogeneous of degree zero. It may be viewed as a function f 0 (π ) on S 2d−1 that, besides, is S O(d)-invariant. We may then rewrite the definition of λ as T 1 f 0 (π ) Pt (π0 ; dπ ) dt. (9.2) λ = lim T T →∞
0
Now, the existence of the limit follows from the fact that the Cesaro means the transition probabilities Pt (π0 ; dπ ) tend in weak topology to the unique invariant probability measure μ(dπ ). Hence (9.3) λ= f 0 (π ) μ(dπ ) and is independent of p0 . The crucial input that allows to make the latter formula more explicit is the formula 1 ρ·χ x (9.4) L ln ρ 2 = = . 2 τ ρ τ
Ergodic Properties of a Model for Turbulent Dispersion of Inertial Particles
75
It implies that f 0 (π ) = L ln
1 χ2 x ρ 2 + χ 2 = + L ln 1 + 2 τ 2 ρ x 1 ln (1 + x 2 ) for d = 1 = + L 2 2 ln (1 + x + y ) for d ≥ 2 τ 2
(9.5)
in terms of the S O(d) invariants with L given by explicit formulae (7.8), (7.15) or (7.24). If the functions ln(1 + x 2 ) in d = 1 and ln (1 + x 2 + y 2 ) in d = 2 that are homogeneous of degree zero on R2d \{0} were smooth, then their contributions to the expectation with respect to the invariant measure on the right-hand side of (9.2) would drop out by the integration by parts. The problem is, however, the lack of smoothness of those functions at ρ = 0 and a more subtle argument is required. 9.1. d = 1 case. In one dimension, Eq. (9.3) reduces to the identity λ=
∞
x τ
−∞
+
1 L ln (1 + x 2 ) η(x) d x 2
(9.6)
with η(x) given by Eq. (7.6). Since the latter integral represents the integration of a smooth function against a smooth measure on S 1 , it converges absolutely. Consequently, the formula for λ may be rewritten in the form: λ = lim
n
n→∞ −n
x τ
+
1 L ln (1 + x 2 ) η(x) d x. 2
(9.7)
† Now the integration by parts and the formula L † η = 0 for the formal adjoint √ L defined with respect to the Lebesgue measure d x show that the term with L ln 1 + x 2 drops out (for the cancellation of the boundary terms it is crucial that the integral is over a symmetric finite interval [−n, n]). We obtain this way the identity
1 lim λ= τ n→∞
n
1 p.v. x η(x) d x ≡ τ −n
∞ −∞
x η(x) d x,
(9.8)
where “p.v.” stands for “principal value”. The result may be expressed [21] by the Airy functions [13]: λ=−
1 d 1 + ln Ai2 (c) + Bi2 (c) 1 2τ 4τ c 2 dc
for c =
1 2
(4τ (A + 2B)) 3
.
(9.9)
1 is the Lyapunov exponent for the one-dimensional Anderson problem The number λ+ 2τ (3.5), recall relation (3.1). It is always positive reflecting the permanent localization in one dimension. On the other hand, λ itself changes sign as a function of τ and A + 2B signaling a phase transition in the one-dimensional advection of inertial particles [34].
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9.2. d ≥ 2 case. In two or more dimensions, Eq. (9.3) becomes x 1 λ= + L ln (1 + x 2 + y 2 ) η(x, y) d x d y, τ 2
(9.10)
where η(x, y) is the density of the invariant measure from Eqs. (7.9) or (7.17). The asymptotic behavior of η(x, y) was established in Sect. 7.2 and Sect. 7.3. We show in Appendix B that it guarantees that the term with L ln (1 + x 2 + y 2 ) may, indeed, be dropped from the expectation on the right hand side of Eq. (9.2) so that 1 λ= x η(x, y) d x d y, (9.11) τ where the integral converges absolutely as follows from the estimates (7.14) and (7.23). In general, there is no closed analytic expressions for the right hand side, unlike in the one-dimensional case. The results of numerical simulations for λ, indicating its qualitative dependence on the parameters of the model, together with analytic arguments about its behavior when Aτ → ∞ with A/B = const. or when τ → 0 with A/τ = const. and B/τ = const. may be found in [2,4,17,24,25]. 10. Conclusions We have studied rigorously a simple stochastic differential equation (SDE) used to model the pair dispersion of close inertial particles moving in a moderately turbulent flow [2,17,24,25,30,34]. We have established the smoothness of the transition probabilities and the irreducibility of the dispersion process using Hörmander criteria for hypoellipticity and control theory. For the projectivized version of the dispersion process, these results implied the existence of the unique invariant probability measure with smooth positive density as well as exponential mixing. The latter properties permitted to substantiate the formulae for the top Lyapunov exponents for the inertial particles used in the physical literature. In two space dimensions, we also showed that the complex-projectivized version of the dispersion process is non-explosive when described in the inhomogeneous variable of the complex projective space, unlike the real-projective version of the dispersion in one space dimension. This shows that in d = 2 the particle dispersion, if non-zero initially, cannot vanish in finite time keeping a non-vanishing time-derivative, and hence avoids the (strict) caustic configurations [35]. A similar result was established in d ≥ 3 for the complex-valued process built from the S O(d) invariants of the projectivized dispersion that was shown to stay for all times in the upper halfplane, avoiding also the configurations when the particle dispersion becomes collinear with its time derivative. These non-explosive behaviors are the reason why the numerical simulation of the processes in the complex (half-)plane could lead to reliable numerical results [2]. There are other questions about the models studied here that may be amenable to rigorous analysis. Let us list some of them: What about dispersion processes for more than two particles and the expressions for the other 2d − 1 Lyapunov exponents (the 2d exponents have to sum to −d/τ [10])? Such multi-particle dispersion processes provide, among other things, more information about caustic-like configurations [35]. Can one establish the existence of the large deviations regime for the finite-time Lyapunov exponents (the corresponding rate function for the top exponent was numerically studied in the d = 2 model in [2,4]; it gives access to more subtle information about the clustering of inertial particles than the top Lyapunov exponent itself)? Is the SDE
Ergodic Properties of a Model for Turbulent Dispersion of Inertial Particles
77
modeling the inertial particle dispersion in fully developed turbulence, that was introduced and studied numerically in [3,4], amenable to rigorous analysis? All those open problems are left for a future study. Acknowledgements. We would like to thank J. Zabczyk for a reference to [32]. J.W. acknowledges the support from C.N.R.S. for an extended visit at ENS-Lyon during which this work was started. D.P.H. and J.W. acknowledge partial support under NSF Grant DMS 0623941.
Appendix A We establish here the expressions (7.18) and (7.19) for the S O(2d)-invariant normalized volume measure on S 2d−1 for d ≥ 3 (the same proof works also in d = 2, although there, the corresponding formulae are straightforward and well known). Note that for S 2d−1 identified with the set of (ρ, χ ) ∈ R2d such that ρ 2 + χ 2 = R 2 for fixed R, we may write dμ0 = const. δ(R − ρ 2 + χ 2 ) dρ dχ ,
(A.1)
provided that we identify functions on S 2d−1 with homogeneous function of degree zero on R2d \{0}. Let us parametrize: ρ = O(ρ, 0, . . . , 0),
χ = O(ρx, ρy, 0, . . . , 0),
(A.2)
where ρ = |ρ|, χ = |χ |, x and y are the S O(d)-invariants of Eq. (7.4), and O ∈ S O(d). Note that O −1 is the rotation that aligns ρ with first positive half-axis of Rd and brings χ into the half-plane spanned by the first axis and the second positive half-axis, as required in Sect. 7.3. O and O O , for O rotating in the subspace orthogonal to the first two axes, give the same (ρ, χ ). Let be a d × d antisymmetric matrix, i j = − ji . Setting O = e and differentiating Eqs. (A.2) at = 0, we obtain: (A.3) dρ = (dρ, ρ d21 , . . . , ρ dd1 ), dχ = (−ρy d21 , ρx d21 , ρx d31 + ρy d32 , . . . , ρx dd1 + ρy dd2 ) +(x dρ + ρ d x, y dρ + ρ dy, 0, . . . , 0). (A.4) Hence for the volume element, dρ dχ = ρ 2d−1 y d−2 dρ d x d y d21 · · · dd1 d32 · · · dd2 .
(A.5)
The product of di j gives, modulo normalization, the S O(d)-invariant volume element d[O] of the homogeneous space S O(d)/S O(d − 2) at point [1]. Using the S O(d)invariance, we infer that dρ dχ = const. ρ 2d−1 y d−2 dρ d x d y d[O].
(A.6)
Substituting the last expression to Eq. (A.1) and performing the integral δ(R − ρ 2 + χ 2 ) ρ 2d−1 dρ = δ(R − ρ 1 + x 2 + y 2 ) ρ 2d−1 dρ R 2d−1 (A.7) (1 + x 2 + y 2 )d that collects the entire ρ-dependence in the integration against homogeneous function of zero degree, we obtain Eq. (7.19), modulo a constant factor that is fixed by normalizing of the resulting measure. =
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Appendix B We show here that
(Lg)(x, y) η(x, y) d x d y = 0
(B.1)
in two or more dimensions, where g(x, y) = ln(1 + x 2 + y 2 ) and η(x, y) is the density of the invariant measure as defined by Eqs. (7.9) and (7.17). As mentioned in Sect. 9, the identity (B.1) does not follow immediately by integration by parts since the function g(x, y), is not smooth on S 2d−1 . We shall then replace g(x, y) by the functions g (x, y) = ln 1 +
ρ2 + χ 2 x 2 + y2 = ln 1 + (x 2 + y 2 ) ρ 2 + χ 2
(B.2)
that are smooth on S 2d−1 for > 0. The identity (B.1) will follow if we show that (Lg)(x, y) η(x, y) d x d y = lim (B.3) (Lg )(x, y) η(x, y) d x d y. 0
Note that 2x , + y 2 ))(1 + (x 2 + y 2 )) 2y , ∂ y g (x, y) = (1 + (1 + )(x 2 + y 2 ))(1 + (x 2 + y 2 )) 1 + (1 + 2 )(y 2 − x 2 ) + (1 + )(y 2 − 3x 2 ) ∂x2 g (x, y) = 2 , (1 + (1 + )(x 2 + y 2 ))2 (1 + (x 2 + y 2 ))2 1 + (1 + 2 )(x 2 − y 2 ) + (1 + )(x 2 − 3y 2 ) ∂ y2 g (x, y) = 2 (1 + (1 + )(x 2 + y 2 ))2 (1 + (x 2 + y 2 ))2 ∂x g (x, y) =
(1 + (1 + )(x 2
(B.4)
so that |∂x g (x, y)| ≤ |∂x2 g (x,
2|x| , 1 + x 2 + y2
y)| ≤ 10,
|∂ y g (x, y)| ≤ |∂ y2 g (x,
2|y| , 1 + x 2 + y2
(B.5)
y)| ≤ 10.
Using the explicit forms (7.15) and (7.24) of the generator L, we infer that |(Lg )(x, y)| ≤ C(1 + |x|) with an -independent constant C. Since the integral (1 + |x|) η(x, y) d x d y
(B.6)
(B.7)
converges due to the estimates (7.14) and (7.23), and point-wise lim (Lg )(x, y) = (Lg)(x, y),
0
relation (B.3) follows from the Dominated Convergence Theorem.
(B.8)
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References 1. Bec, J.: Multifractal concentrations of inertial particles in smooth random flows. J. Fluid Mech. 528, 255–277 (2005) 2. Bec, J., Cencini, M., Hillerbrand, R.: Heavy particles in incompressible flows: the large Stokes number asymptotics. Physica D 226, 11–22 (2007) 3. Bec, J., Cencini, M., Hillerbrand, R.: Clustering of heavy particles in random self-similar flow. Phys. Rev. E 75, 025301 (2007) 4. Bec, J., Cencini, M., Hillerbrand, R., Turitsyn, K.: Stochastic suspensions of heavy particles. Physica D 237, 2037–2050 (2008) 5. Duncan, K., Mehlig, B., Ostlund, S., Wilkinson, M.: Clustering in mixing flows. Phys. Rev. Lett. 95, 240602 (2005) 6. Elperin, T., Kleeorin, N., Rogachevskii, I.: Self-Excitation of fluctuations of inertial particle concentration in turbulent fluid flow. Phys. Rev. Lett. 77, 5373–5376 (1996) 7. Falkovich, G., Fouxon, A., Stepanov, M.G.: Acceleration of rain initiation by cloud turbulence. Nature 419, 151–154 (2002) 8. Falkovich, G., Gaw¸edzki, K., Vergassola, M.: Particles and fields in fluid turbulence. Rev. Mod. Phys. 73, 913–975 (2001) 9. Fouxon, I., Horvai, P.: Separation of heavy particles in turbulence. Phys. Rev. Lett. 100, 040601 (2008) 10. Fouxon, I., Horvai, P.: Fluctuation relation and pairing rule for Lyapunov exponents of inertial particles in turbulence. J. Stat. Mech.: Theor. & Experim. 08, L08002 (2007) 11. Friz, P.K.: An Introduction to Malliavin Calculus, lecture notes, http://www.math.nyu.edu/phd-students/ frizpete/malliavin/mall.pdf, 2002 12. Gaw¸edzki, K.: Soluble models of turbulent transport. In: Non-Equilibrium Statistical Mechanics and Turbulence, eds. S. Nazarenko, O. Zaboronski, Cambridge: Cambridge University Press 2008, pp. 44–107 13. Gradstein, I.S., Rhyzik, I.M.: Table of Integrals, Series, and Products, Vth edition. New York: Academic Press 1994 14. Halperin, B.I.: Green’s functions for a particle in a one-dimensional random potential. Phys. Rev. 139, A104–A117 (1965) 15. Has’minskii, R.Z.: Stochastic Stability of Differential Equations. alphen aanden Rija, Netherlands: Sijthoff and Noordhoff, 1980 16. Hörmander, L.: The Analysis of Linear Partial Differential Operators. Vol. III, Berlin: Springer, 1985 17. Horvai, P.: Lyapunov exponent for inertial particles in the 2D Kraichnan model as a problem of Anderson localization with complex valued potential, http://arxiv.org/abs/nlin/0511023v1 [nlin.co], 2005 18. Kraichnan, R.H.: Small scale structure of a scalar field convected by turbulence. Phys. Fluids 11, 945–953 (1968) 19. Le Jan, Y., Raimond, O.: Integration of Brownian vector fields. Ann. Probab. 30, 826–873 (2002) 20. Le Jan, Y., Raimond, O.: Flows, coalescence and noise. Ann. Probab. 32, 1247–1315 (2004) 21. Lifshitz, I.M., Gredeskul, S., Pastur, L.: Introduction to the Theory of Disordered Systems. New York: Wiley, 1988 22. Mattingly, J.C., Stuart, A.M., Higham, D.J.: Ergodicity for SDEs and approximations: locally Lipschitz vector fields and degenerate noise, Stochastic Process. Appl. 101, 185–232 (2002). doi:10.1016/S03044149(02)00150-3 http://dx.doi.org/10.1016/S0304-4149(02)00150-3 23. Maxey, M.R., Riley, J.J.: Equation of motion for a small rigid sphere in a nonuniform flow. Phys. Fluids 26, 883–889 (1983) 24. Mehlig, B., Wilkinson, M.: Coagulation by random velocity fields as a Kramers problem. Phys. Rev. Lett 92, 250602 (2004) 25. Mehlig, B., Wilkinson, M., Duncan, K., Weber, T., Ljunggren, M.: On the aggregation of inertial particles in random flows. Phys. Rev. E 72, 051104 (2005) 26. Meyn, S.P., Tweedie, R.L.: Markov Chains and Stochastic Stability. Communication and Control Engineering Series, London: Springer-Verlag, 1993 27. Meyn, S.P., Tweedie, R.L.: Stability of Markovian processes III: Foster-Lyapunov criteria for continuous-time processes. Adv. Appl. Prob. 25, 518–548 (1993) 28. Norriss, J.: Simplified Malliavin calculus. In: Séminaire de probabilité XX, Lectures Note in Math. 1204, Berlin: Springer, 1986, pp. 101–130 29. Nualart, D.: Malliavin Calculus and Related Topics, 2nd edition, Berlin-Heidelberg: Springer, 2006 30. Piterbarg, L.: The top Lyapunov exponent for a stochastic flow modeling the upper ocean turbulence. SIAM J. Appl. Math. 62, 777–800 (2001) 31. Rey-Bellet, L.: Ergodic properties of Markov processes. In: Open Quantum systems II. The Markovian approach, Lecture notes in Mathematics 1881, Berlin: Springer, 2006, pp. 1–78
80
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32. Scheutzow, M.: Stabilization and destabilization by noise in the plane. Stoch. Anal. Appl. 11, 97–113 (1993) 33. Stroock, D.W., Varadhan, S.R.S.: On the support of diffusion processes with applications to the strong maximum principle. In: Proc. 6-th Berkeley Symp. Math. Stat. Prob., Vol. III, Berkeley: Univ. California Press, 1972, pp. 361368 34. Wilkinson, M., Mehlig, B.: The path-coalescence transition and its applications. Phys. Rev. E 68, 040101 (2003) 35. Wilkinson, M., Mehlig, B.: Caustics in turbulent aerosols. Europhys. Lett. 71, 186–192 (2005) Communicated by M. Aizenman
Commun. Math. Phys. 308, 81–113 (2011) Digital Object Identifier (DOI) 10.1007/s00220-011-1325-7
Communications in
Mathematical Physics
On Quantization of Complex Symplectic Manifolds Andrea D’Agnolo1, , Masaki Kashiwara2 1 Dipartimento di Matematica Pura ed Applicata, Università di Padova, Via Trieste 63, 35121 Padova, Italy.
E-mail: [email protected]
2 Research Institute for Mathematical Sciences, Kyoto University, Kyoto 606-8502, Japan.
E-mail: [email protected] Received: 7 September 2010 / Accepted: 18 March 2011 Published online: 28 August 2011 – © Springer-Verlag 2011
Abstract: Let X be a complex symplectic manifold. By showing that any Lagrangian subvariety has a unique lift to a contactification, we associate to X a triangulated category of regular holonomic microdifferential modules. If X is compact, this is a Calabi-Yau category of complex dimension dim X + 1. We further show that regular holonomic microdifferential modules can be realized as modules over a quantization algebroid canonically associated to X . Contents 0. Introduction . . . . . . . . . . . . . . . . . . . 1. Gerbes and Algebroid Stacks . . . . . . . . . . 2. Contactification of Symplectic Manifolds . . . . 3. Holonomic Modules on Symplectic Manifolds . 4. Quantization Algebroid . . . . . . . . . . . . . 5. Quantization Modules . . . . . . . . . . . . . . Appendix A. Remarks on Deformation-Quantization References . . . . . . . . . . . . . . . . . . . . . . .
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0. Introduction Let X be a complex symplectic manifold. As shown in [16] (see also [13]), X is endowed with a canonical deformation quantization algebroid W X . Recall that an algebroid is to an algebra as a gerbe is to a group. The local model of W X is an algebra similar to the The first named author (A.D’A.) expresses his gratitude to the Research Institute for Mathematical Sciences of Kyoto University for hospitality during the preparation of this paper and acknowledges partial support from the Fondazione Cariparo through the project “Differential methods in Arithmetic, Geometry and Algebra”.
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one of microdifferential operators, with a central deformation parameter . The center of W X is a subfield k of formal Laurent series C[−1 , ]]. Deformation quantization modules have now been studied quite extensively (see [3,11,12] and also [14,19] for related results), and they turned out to be useful in other contexts as well (see e.g. [9]). Of particular interest are modules supported by Lagrangian subvarieties. It is conjectured in [11] that, if X is compact, the triangulated category of regular holonomic deformation-quantization modules is Calabi-Yau of dimension dim X over k. There are some cases (representation theory, homological mirror symmetry, quantization in the sense of [5]) where one would like to deal with categories whose center is C instead of k. In the first part of this paper, we associate to X a C-linear triangulated category of regular holonomic microdifferential modules. If X is compact, this category is Calabi-Yau of dimension dim X + 1 over C. Our construction goes as follows. For a possibly singular Lagrangian subvariety ⊂ X , we prove that there is a unique contactification ρ : Y − → X of a neighborhood of and a Lagrangian subvariety ⊂ Y such that ρ induces a homeomorphism between and . As shown in [6], the contact manifold Y is endowed with a canonical microdifferential algebroid EY . We define the triangulated category of regular holonomic microdifferential modules along as the bounded derived category of regular holonomic EY -modules along . We then take the direct limit over the inductive family of Lagrangian subvarieties ⊂ X . In the second part of this paper, we show that regular holonomic microdifferential modules can be realized as modules over a quantization algebroid E X canonically associated to X . More precisely, if ⊂ Y is a lift of ⊂ X as above, we prove that the category of coherent EY -modules supported on is fully faithfully embedded in the category of coherent E X -modules supported on . Our construction of E X is similar to the construction of W X in [16], which was in turn similar to the construction of EY in [6]. Here, we somewhat simplify matters by presenting an abstract way of obtaining an algebroid from the data of a gerbe endowed with an algebra valued functor. Let us briefly recall the constructions of EY , W X and present the construction of EX . Denote by P ∗ M the projective cotangent bundle to a complex manifold M and by E M the ring of microdifferential operators on P ∗ M as in [17]. Recall that, in a local system of coordinates, E M is endowed with the anti-involution given by the formal adjoint of total symbols. Let Y be a complex contact manifold. By Darboux theorem, the local model of Y is an open subset of P ∗ M. By definition, a microdifferential algebra E on an open subset V ⊂ Y is a C-algebra locally isomorphic to E M . Assume that E is endowed with an anti-involution ∗. Any two such pairs (E , ∗ ) and (E, ∗) are locally isomorphic. Such isomorphisms are not unique, and in general it is not possible to patch the algebras E together in order to get a globally defined microdifferential algebra on Y . However, the automorphisms of (E, ∗) are all inner and are in bijection with a subgroup of invertible elements of E. This is enough to prove the existence of a microdifferential algebroid EY , i.e. an algebroid locally represented by microdifferential algebras. Denote by T ∗ M the cotangent bundle to a complex manifold M, by (t; τ ) the symplectic coordinates on T ∗ C, and consider the projection ρ
→ T ∗ M, (x, t; ξ, τ ) → (x, ξ/τ ) P ∗ (M × C) −
On Quantization of Complex Symplectic Manifolds
83
defined for τ = 0. This is a principal C-bundle, with action given by translation in the t variable. Note that, for λ ∈ C, the outer isomorphism Ad(eλ∂t ) of ρ∗ E M×C acts by translation t → t + λ at the level of total symbols. Let X be a complex symplectic manifold. By Darboux theorem, the local model of X is an open subset of T ∗ M. Let ρ : V − → U be a contactification of an open subset U ⊂ X . By definition, this is a principal C-bundle whose local model is the projection {τ = 0} − → T ∗ M above. Consider a quadruple (ρ, E, ∗, ) of a contactification ρ: V − → U , a microdifferential algebra E on V , an anti-involution ∗ and an operator ∈ E locally corresponding to ∂t−1 . One could try to mimic the above construction of the microdifferential algebroid EY in order to get an algebroid from the algebras ρ∗ E. −1 This fails because the automorphisms of (ρ, E, ∗, ) given by Ad(eλ ) for λ ∈ C are not inner. There are two natural ways out. The first possibility, utilized in [16], is to replace the algebra ρ∗ E by its subalgebra W = C0 ρ∗ E of operators commuting with . Locally, this corresponds to the operators −1
of ρ∗ E M×C whose total symbol does not depend on t. Then the action of Ad(eλ ) is trivial on W, and these algebras patch together to give the deformation-quantization algebroid W X . −1 The second possibility, which we exploit here, is to make Ad(eλ ) an inner automorphism. This is obtained by replacing the algebra ρ∗ E by the algebra E =
−1 C∞ ρ∗ E eλ , λ∈C
where C∞ ρ∗ E = {a ∈ ρ∗ E; ad() N (a) = 0, ∃N ≥ 0} locally corresponds to operators in ρ∗ E M×C whose total symbol is polynomial in t. By patching these algebras we get the quantization algebroid E X . The deformation parameter is not central in E X . We −1 show that the centralizer of in E X is equivalent to the twist of W X ⊗C ( λ∈C Ceλ ) by the gerbe parameterizing the primitives of the symplectic 2-form. In an appendix at the end of the paper, we give an alternative construction of the deformation-quantization algebroid W X . Instead of using contactifications, we consider as objects deformation-quantization algebras endowed with compatible anti-involution and C-linear derivation. We thus show that W X itself is endowed with a canonical C-linear derivation. One could then easily prove along the lines of [15] that W X is the unique k-linear deformation-quantization algebroid which is trivial graded and is endowed with compatible anti-involution and C-linear derivation. Finally, we compare regular holonomic quantization modules with regular holonomic deformation-quantization modules. This paper is organized as follows. In Sect. 1, after recalling the definitions of gerbe and of algebroid on a topological space, we explain how to obtain an algebroid from the data of a gerbe endowed with an algebra valued functor. In Sect. 2, we review some notions from contact and symplectic geometry, discussing in particular the gerbe parameterizing the primitives of the symplectic 2-form. We further show how a Lagrangian subvariety lifts to a contactification. In Sect. 3, we first recall the construction of the microdifferential algebroid of [6] in terms of algebroid data. Then we show how to associate to a complex symplectic manifold a triangulated category of regular holonomic microdifferential modules.
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In Sect. 4, we start by giving a construction of the deformation-quantization algebroid of [16] in terms of algebroid data. Then, with the same algebroid data, we construct the algebroid EX . In Sect. 5, we prove coherency of quantization algebras and show how to realize regular holonomic microdifferential modules as modules over EX . In Appendix A, we give an alternative description of the deformation quantization algebroid using deformation-quantization algebras endowed with compatible anti-involution and C-linear derivation. We also compare regular holonomic deformation-quantization modules with regular holonomic quantization modules. The results of this paper were announced in [1], to which we refer. 1. Gerbes and Algebroid Stacks We review here some notions from the theory of stacks, in the sense of sheaves of categories, recalling in particular the definitions of gerbe and of algebroid (refer to [2,4,10,13]). We then explain how to obtain an algebroid from the data of a gerbe endowed with an algebra valued functor. 1.1. Review on stacks. Let X be a topological space. A prestack C on X is a lax analogue of a presheaf of categories, in the sense that for a chain of open subsets W ⊂ V ⊂ U the restriction functor C(U ) − → C(W ) coincides with the composition C(U ) − → C(V ) − → C(W ) only up to an invertible transformation (satisfying a natural cocycle condition for chains of four open subsets). The prestack C is called separated if for any U ⊂ X and any p, p ∈ C(U ) the presheaf U ⊃ V → Hom C(V ) ( p|V , p |V ) is a sheaf. We denote it by Hom C ( p, p ). A stack is a separated prestack satisfying a natural descent condition (see e.g. [10, Chap. 19]). If ρ : Y − → X is a continuous map, we denote by ρ −1 C the pull back on Y of a stack C on X . A groupoid is a category whose morphisms are all invertible. A gerbe on X is a stack of groupoids which is locally non empty and locally connected, i.e. any two objects are locally isomorphic. Let G be a sheaf of commutative groups. A G-gerbe is a gerbe P endowed with a group homomorphism G − → Aut(idP ). A G-gerbe P is called invertible if G|U − → AutP ( p) is an isomorphism of groups for any U ⊂ X and any p ∈ P(U ). We denote by P ×G Q the contracted product of two G-gerbes. This is the stack associated to the prestack whose objects are pairs ( p, q) of an object of P and an object of Q, with morphisms Hom
G
P×Q
G ( p, q), ( p , q ) = Hom P ( p, p ) × Hom Q (q, q ).
Let R be a commutative sheaf of rings. For an R-algebra A denote by Mod(A) the stack of left A-modules. An R-linear stack is a stack A such that for any U ⊂ X and any p, p ∈ A(U ) the sheaves Hom A ( p , p) have an R|U -module structure compatible with composition and restriction. The stack of left A-modules Mod(A) = FctR (A, Mod(R)) has R-linear functors as objects and transformations of functors as morphisms. Let L be a commutative R-algebra and A an R-linear stack. An action of L on A is the data of R|U -algebra morphisms L|U − → EndA ( p) for any U ⊂ X and any p ∈ A(U ), compatible with restriction. Then L acts as a Lie algebra on Hom A ( p , p) by [l, f ] = l p f − f l p , where l p denotes the image of l ∈ L(U ) in EndA ( p). This gives a filtration of A by the centralizer series
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0 CL Hom A ( p , p) = { f ; [l, f ] = 0, ∀l ∈ L},
i−1 i Hom A ( p , p) = { f ; [l, f ] ∈ CL , ∀l ∈ L} for any i > 0. CL 0 A and C ∞ A the substacks of A with the same objects as A and morphisms Denote by CL i L 0 0 A is an L-linear stack and C ∞ A Hom A , respectively. Note that CL CL Hom A and i CL L is an R-linear stack. An R-algebroid A is an R-linear stack which is locally non empty and locally connected by isomorphisms. Thus, an algebroid is to a sheaf of algebras as a gerbe is to a sheaf of groups. For p ∈ A(U ), set A p = EndA ( p). Then A|U is equivaop lent to the full substack of Mod(A p ) whose objects are locally free modules of rank op one. (Here A p denotes the opposite ring of A p .) Moreover, there is an equivalence Mod(A|U ) Mod(A p ). One says that A is represented by an R-algebra A if A A p for some p ∈ A(X ). The R-algebroid A is called invertible if A p R|U for any U ⊂ X and any p ∈ A(U ). The pull-back and tensor product of algebroids are still algebroids. The following lemma is obvious. 0 A is locally Lemma 1.1.1. Let A be an R-algebroid endowed with an action of L. If CL 0 ∞ connected by isomorphisms, then CL A and CL A are algebroids.
1.2. Algebroid data. Let R-Alg be the stack on X with R-algebras as objects and R-algebra homomorphisms as morphisms. Definition 1.2.1. An R-algebroid data is a triple (P, , ) with P a gerbe, : P − → R-Alg a functor of stacks and a collection of liftings of group homomorphisms
( p)× 7 o o p ooo Ad ooo o o oo
/ AutR-Alg ( ( p)) EndP ( p)
∀U ⊂ X, ∀ p ∈ P(U ),
(1.2.1)
compatible with restrictions and such that for any g ∈ Hom P ( p , p) and any φ ∈ EndP ( p ) one has p (gφ g −1 ) = (g)( p (φ )).
(1.2.2)
Remark 1.2.2. Denote by Grp the stack on X with sheaves of groups as objects and group homomorphisms as morphisms. The R-algebroid data (P, , ) induces three natural functors E, A, F : P − → Grp defined by E( p) = EndP ( p), A( p) = AutR-Alg ( ( p)), F( p) = ( p)× for p ∈ P and by E(g) = Ad(g), A(g) = Ad(g), F(g) = (g) for g : p − → p. Then condition (1.2.2) states that : E − → F is a transformation of functors and the commutative diagram (1.2.1) corresponds to a commutative diagram of transformations of functors ;F vv
vvv
E
v vv vv
Ad
/ A.
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Remark 1.2.3. There is a natural interpretation of R-algebroid data in terms of 2categories (refer to [18, §9], where 2-categories are called bicategories). Denote by R-Alg the 2-prestack on X obtained by enriching R-Alg with a set of 2-arrows f ⇒ f given by {b ∈ A; b f (a ) = f (a )b, ∀a ∈ A }, → A. In particular, f f if and only if for two R-algebra morphisms f, f : A − f = Ad(b) f for some b ∈ A× . The R-algebroid data (P, , ) is equivalent to the data of the lax functor of 2-prestacks : P − → R-Alg, where P has trivial 2-arrows and is obtained by enriching at the level of 2-arrows −1 → g in P( p). by (id g − →g ) = p (g g ) for a morphism g − We will prove in the next proposition that the following description associates an R-prestack A0 to the data (P, , ). (i) For an open subset U ⊂ X , objects of A0 (U ) are the same as those of P(U ). (ii) For p, p ∈ A0 (U ), the sheaf of morphisms is defined by Hom A0 ( p , p) = ( p)
E ndP ( p)
×
Hom P ( p , p).
→ p in A0 are equivalence classes [a, g] of pairs This means that morphisms p − (a, g) with a ∈ ( p) and g : p − → p in P, for the relation (a, φg) ∼ (a p (φ), g),
∀φ ∈ EndP ( p).
→ p and [a , g ] : p − → p is given by (iii) Composition of [a, g] : p − [a, g] ◦ [a , g ] = [ag(a ), gg ]. Here we set for short g(a ) = (g)(a ). (iv) For two morphisms [a, g], [a , g ] : p − → p and r ∈ R, the R-linear structure of A0 is given by r [a, g] = [ra, g], [a, g] + [a , g ] = [a + a p (g g −1 ), g]. (v) The restriction functors are the natural ones. Proposition 1.2.4. Let (P, , ) be an R-algebroid data. The description (i)–(v) above defines a separated R-prestack A0 on X . The associated stack A is an R-algebroid endowed with a functor J : P − → A such that EndA (J ( p)) ( p) for any p ∈ P. Proof. (a) Let us show that the composition is well defined. Consider two composable → p and [a , g ] : p − → p . At the level of representatives, morphisms [a, g] : p − set (a, g) ◦ (a , g ) = (ag(a ), gg ).
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(a-i) Let us show that for φ ∈ EndP ( p) we have (a, φg) ◦ (a , g ) ∼ (a p (φ), g) ◦ (a , g ). For this, we have to check that (aφ(g(a )), φgg ) ∼ (a p (φ)g(a ), gg ). This follows from p (φ)g(a ) = φ(g(a )) p (φ), which is a consequence of the commutativity of (1.2.1). (a-ii) Similarly, for φ ∈ EndP ( p ) we have to prove that (a, g) ◦ (a , φ g ) ∼ (a, g) ◦ (a p (φ ), g ). In other words, we have to check that (ag(a ), gφ g ) ∼ (ag(a p (φ )), gg ). This follows from gφ g = (gφ g −1 )gg and g(a p (φ )) = g(a )g( p (φ )) = g(a ) p (gφ g −1 ), where the last equality is due to (1.2.2). (a-iii) Associativity is easily checked. (b) The R-linear structure is well defined by an argument similar to that in part (a) above. (c) The functor J : P − → A is induced by the functor J0 : P − → A0 defined by p → p on objects and g → [1, g] on morphisms. The morphism ( p) − → EndA (J ( p)), a → [a, id] has an inverse given by [a, g] → a p (g). Note that the functor J : P − → A is neither faithful nor full, in general. Remark 1.2.5. For an R-algebroid A, denote by A× the gerbe with the same objects as A and isomorphisms as morphisms. Then A is the R-algebroid associated with the data (A× , A , ), where A ( p) = EndA ( p) and p is the identity. Example 1.2.6. Let X be a complex manifold and O X its structure sheaf. To an invertible O X -module L one associates an invertible Z/2Z-gerbe PL⊗1/2 defined as follows: ∼
→ (i) Objects on U are pairs (F, f ), where F is an invertible OU -module and f : F ⊗2 − L is an OU -linear isomorphism. (ii) If (F , f ) is another object, a morphism (F , f ) − → (F, f ) is an OU -linear ∼ ⊗2 → F, such that f = f ϕ . isomorphism ϕ : F − Note that any ψ ∈ EndPL⊗1/2 (F, f ) is a locally constant Z/2Z-valued function. Denote by CL⊗1/2 the invertible C-algebroid associated with the data (PL⊗1/2 , , ), where (F, f ) = CU , (ϕ) = id, (F , f ) (ψ) = ψ.
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2. Contactification of Symplectic Manifolds We first review here some notions from contact and symplectic geometry. In particular, we discuss the gerbe parameterizing the primitives of the symplectic 2-form. Then, we show how any Lagrangian subvariety of a complex symplectic manifold can be uniquely lifted to a local contactification. 2.1. The gerbe of primitives. Let X be a complex manifold and O X its structure sheaf. Denote by T X and T ∗ X the tangent and cotangent bundle, respectively, and by X and 1X their sheaves of sections. For k ∈ Z, denote by kX the sheaf of holomorphic k-forms k → k−1 the and by d : kX − → k+1 X X the differential. For v ∈ X denote by i v : X − k k inner derivative and by L v : X − → X the Lie derivative. Let ω ∈ Γ (X ; 2X ) be a 2-form which is closed, i.e. dω = 0. Definition 2.1.1. The gerbe Cω on X is the stack associated with the separated prestack defined as follows. (1) Objects on U ⊂ X are primitives of ω|U , i.e. 1-forms θ ∈ Γ (U ; 1X ) such that dθ = ω|U . (2) If θ is another object, a morphism θ − → θ is a function ϕ ∈ Γ (U ; O X ) such that dϕ = θ − θ . Composition with ϕ : θ − → θ is given by ϕ ◦ ϕ = ϕ + ϕ . The following result is clear. Lemma 2.1.2. (i) The stack Cω is an invertible C-gerbe. 2 (ii) If ω ∈ X (X ) is another closed 2-form, there is an equivalence C
∼
→ Cω+ω . Cω × Cω − For a principal C-bundle ρ : Y − → X , denote by →Y Tλ : Y − the action of λ ∈ C and by va =
d dλ Tλ λ=0
∈ Y
the infinitesimal generator of the C-action. Definition 2.1.3. The gerbe Cω on X is defined as follows: ρ
→ U, α) of a principal C-bundle ρ and a (1) Objects on U ⊂ X are pairs ρ = (V − 1-form α ∈ Γ (V ; 1V ) such that i va α = 1 and ρ ∗ ω = dα. In particular, L va α = 0. ρ
→ U, α ), morphisms χ : ρ − → ρ are morphisms of (2) For another object ρ = (V − principal C-bundles such that χ ∗ α = α . Denote by p1 : X ×C − → X the trivial principal C-bundle given by the first projection. Let t be the coordinate of C. For a primitive θ of ω, an object of Cω is given by ( p1 , p1∗ θ + dt). By the next lemma, any object ρ of Cω is locally of this form and any automorphism of ρ is locally of the form Tλ , for λ ∈ C. (See [16, Remark 9.3] for similar observations.)
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Lemma 2.1.4. There is a natural equivalence Cω − → Cω . In particular, Cω is an invertible C-gerbe. Proof. As above, denote by p1 : U × C − → U the first projection and by t the coordinate of C. Consider the functor B : Cω − → Cω given by θ → ( p1 , p1∗ θ + dt) on objects and ϕ → (x, t) → (x, t + ϕ(x)) on morphisms. As B is clearly faithful, we are left to prove that it is locally full and locally essentially ρ surjective. For the latter, let ρ = (V − → U, α) be an object of Cω (U ). Up to shrinking U , we may assume that the bundle ρ is trivial. Choose an isomorphism of principal C-bundles ξ : U × C − → V . As i va (ξ ∗ α − dt) = L va (ξ ∗ α − dt) = 0, there exists a 1 unique 1-form θ ∈ X (U ) such that ξ ∗ α − dt = p1∗ θ . Then ω|U = dθ and ρ B(θ ). It remains to show that any morphism χ : ρ − → ρ of Cω (U ) is in the image of B. Up to shrinking U , we may assume that ρ = ( p1 , p1∗ θ + dt) and ρ = ( p1 , p1∗ θ + dt). Then χ : U × C − → U × C is given by (x, t) → (x, t + ϕ(x)) for some ϕ ∈ O X (U ). Since χ ∗ ( p1∗ θ + dt) = p1∗ θ + dt, it follows that dϕ = θ − θ . Hence χ = B(ϕ). Definition 2.1.5. Let R be a commutative ring endowed with a group homomorphism : C − → R × . The stack Rω is the invertible R-algebroid associated with the data (Cω , R , ), where
R (ρ) = RU , R (χ ) = id RU , ρ (Tλ ) = (λ), ρ
for ρ = (V − → U, α), χ : ρ − → ρ and λ ∈ C. Note that by Lemma 2.1.2 there is an R-linear equivalence ∼
→ Rω+ω . Rω ⊗R X Rω − Remark 2.1.6. Equivalence classes of invertible C-gerbes and of invertible R-algebroids are classified by H 2 (X ; C) and H 2 (X ; R × ), respectively. The class of Cω coincides with the de Rham class [ω] of the closed 2-form ω, and the class of Rω is the image of [ω] by : H 2 (X ; C) − → H 2 (X ; R × ). 2.2. Symplectic manifolds. A complex symplectic manifold X = (X, ω) is a complex manifold X of even dimension endowed with a holomorphic closed 2-form ω ∈ Γ (X ; 2X ) which is non-degenerate, i.e. the n-fold exterior product ω ∧ · · · ∧ ω never vanishes for n = 21 dim X . ∼
→ X be the Hamiltonian isomorphism induced by the symplectic Let H : 1X − form ω. The Lie bracket of ϕ, ϕ ∈ O X is given by {ϕ, ϕ } = Hϕ (ϕ ), where Hϕ = H (dϕ) is the Hamiltonian vector field of ϕ. Example 2.2.1. Let M be a complex manifold. Its cotangent bundle T ∗ M has a natural symplectic structure (T ∗ M, dθ ), where θ denotes the canonical 1-form. Let (x) = (x1 , . . . , xn ) be a system of local coordinates on M. The associated system (x; u) of local symplectic coordinates on T ∗ M is given by p = i u i ( p)d xi . Then the canonical 1-form is written θ = i u i d xi and the Hamiltonian vector field of ϕ ∈ O M is written Hϕ = i ϕu i ∂xi − ϕxi ∂u i .
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An analytic subset ⊂ X is called involutive if for any f, g ∈ O X with f | = g| = 0 one has { f, g}| = 0. The analytic subset is called Lagrangian if it is involutive and dim X = 2 dim . Let X = (X , ω ) be another symplectic manifold. A symplectic transformation ψ : X − → X is a holomorphic isomorphism such that ψ ∗ ω = ω . By Darboux theorem, for any complex symplectic manifold X there locally exist symplectic transformations ψ
X ⊃U − → U M ⊂ T ∗ M, for a complex manifold M with dim M =
1 2
(2.2.1)
dim X .
2.3. Contact manifolds. Let γ : Z − → Y be a principal C× -bundle over a complex manifold Y . Denote by vm the infinitesimal generator of the C× -action on Z . For k ∈ Z, let O Z (k) be the sheaf of k-homogeneous functions, i.e. solutions ϕ ∈ O Z of vm ϕ = kϕ. Let OY (k) = γ∗ O Z (k) be the corresponding invertible OY -module, so that OY (−1) is × the sheaf of sections of the line bundle C ×C Z . γ → Y, θ ) is a complex manifold Y endowed with A complex contact manifold Y = (Z − a principal C× -bundle γ and a holomorphic 1-form θ ∈ Γ (Z ; 1Z ) such that (Z , dθ ) is a complex symplectic manifold, i vm θ = 0 and L vm θ = θ , i.e. θ is 1-homogeneous. Example 2.3.1. Let M be a complex manifold and θ the canonical 1-form on T ∗ M as in Example 2.2.1. The projective cotangent bundle P ∗ M has a natural contact structure (γ , θ ) with γ : T ∗ M\M − → P ∗ M the projection. Here T ∗ M \ M denotes the cotangent bundle with the zero-section removed. Note that the 1-form θ on Z may be considered as a global section of 1Y ⊗O OY (1). In particular, there is an embedding ι : OY (−1) − → 1Y , ϕ → ϕθ.
(2.3.1)
Note also that the symplectic manifold Z is homogeneous with respect to the C× -action, i.e. θ = i vm (dθ ). Moreover, there exists a unique C× -equivariant embedding Z → T ∗ Y such that θ is the pull-back of the canonical 1-form on T ∗ Y . Since dθ is 1-homogeneous, the Hamiltonian vector field Hϕ of ϕ ∈ O Z (k) is (k −1)homogeneous, i.e. [vm , Hϕ ] = (k − 1) Hϕ . An analytic subset of Y is called involutive (resp. Lagrangian) if γ −1 is involutive (resp. Lagrangian) in Z . γ
→ Y , θ ) be another contact manifold. A contact transformation Let Y = (Z − χ: Y − → Y is an isomorphism of principal C× -bundles Z
χ
γ
Y such that χ ∗ θ = θ .
χ
/Z /Y
γ
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By the Darboux theorem, for any complex contact manifold Y there locally exist contact transformations χ
Y ⊃V − → VM ⊂ P ∗ M,
(2.3.2)
for a complex manifold M with dim M = 21 (dim Y + 1). 2.4. Contactifications. Let X = (X, ω) be a complex symplectic manifold. A contactification of X is a global object of the stack Cω described in Definition 2.1.3. Morphisms of contactifications are morphisms in Cω . ρ For a contactification ρ = (Y − → X, α) of X , the total space Y of ρ has a natural q1 complex contact structure given by (Y × C× − → Y, τ q1∗ α), where q1 is the first pro× jection and τ is the coordinate of C ⊂ C. Note that, in terms of contact structures, a morphism ρ − → ρ of contactifications is a contact transformation χ : Y − → Y over X . Example 2.4.1. Let M be a complex manifold and denote by (t; τ ) the symplectic coordinates of T ∗ C. Consider the principal C-bundle, ρ
P ∗ (M × C) ⊃ {τ = 0} − → T ∗ M, (x, t; ξ, τ ) → (x; ξ/τ ), with the C-action given by translation in the t variable. Note that the bundle ρ is trivialized by ∼
→ (T ∗ M) × C, (x, t; ξ, τ ) → ((x; ξ/τ ), t). χ : {τ = 0} − Consider the projection p1 : (T ∗ M) × C − → T ∗ M. As in Example 2.2.1, denote by θ the canonical 1-form of T ∗ M. Then a contactification of (T ∗ M, dθ ) is given by (ρ, α), with ρ as above and α = χ ∗ ( p1∗ θ +dt). In a system (x; u) of local symplectic coordinates on T ∗ M, one has θ = u d x and α = (ξ/τ )d x +dt. As the canonical 1-form of T ∗ (M × C) is τ α = ξ d x + τ dt, the map (2.3.1) is given by ι : O P ∗ (M×C) (−1)|{τ =0} − → 1P ∗ (M×C) |{τ =0} , ϕ → ϕ τ α. 2.5. Contactification of Lagrangian subvarieties. In this section we show how any Lagrangian subvariety of a complex symplectic manifold lifts to a contactification (see e.g. [3, Lemma 8.4] for the case of Lagrangian submanifolds). Let us begin with a preliminary lemma. Lemma 2.5.1. Let M be a complex manifold, S ⊂ M a closed analytic subset and θ ∈ 1M a 1-form such that dθ | Sreg = 0. Then there locally exists a continuous function f on S such that f is holomorphic on the non-singular locus Sreg , and d f | Sreg = θ | Sreg . → S be a resolution of singularities and let p : S − → M be the composite Proof. Let S − S − → S → M. Thus S is a complex manifold, p is proper and p −1 (Sreg ) − → Sreg is an isomorphism. Consider the global section θ = p ∗ θ of 1S . As dθ | Sreg = 0 and p −1 (Sreg ) is dense in S , we have dθ = 0. Fix a point s0 ∈ S and set S0 = p −1 (s0 ). Since θ |(S0 )reg = 0, there exists a unique holomorphic function f defined on a neighborhood of S0 such that d f = θ and
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f | S0 = 0. As p is proper, replacing M by a neighborhood of s0 we may assume that f is globally defined on S . Set S = S × S S and S0 = S0 × S S0 . We may assume that S0 intersects each connected component of S . Consider the diagram Sreg
q
/ S
p1 p2
/
/ S
p
/ M,
where p1 and p2 are the projections S × S S − → S . To conclude, it is enough to prove that g = p1∗ f − p2∗ f vanishes, for then we can set f (w) = f (w ) with p(w ) = w. Since pp1 = pp2 , one has dq ∗ g = d( pp1 q)∗ θ − d( pp2 q)∗ θ = 0 so that g is locally . Hence g is locally constant by Sublemma 2.5.2 below with T = S constant on Sreg and U = Sreg . Since g vanishes on S0 , it vanishes everywhere. Sublemma 2.5.2. Let T be a Hausdorff topological space and U ⊂ T a dense open subset. Assume there exists a basis B of open subsets of T such that any B ∈ B is connected and B ∩ U has finitely many connected components. If a continuous function on T is locally constant on U , then it is locally constant on T . Let now X = (X, ω) be a complex symplectic manifold. Proposition 2.5.3. Let be a Lagrangian subvariety of X . Then there exist a neighborhood U of in X and a pair (ρ, ) with ρ : V − → U a contactification and a Lagrangian subvariety of V such that ρ| is a homeomorphism over and a holomorphic isomorphism over reg . Proof. Let {Ui }i∈I be an open cover of in X such that for each i ∈ I there is a primitive θi ∈ 1X (Ui ) of ω|Ui . Set i = ∩ Ui . Using Lemma 2.5.1, up to shrinking the cover we may assume that there is a continuous function f i on i such that f i |i,reg is a primitive of θi |i,reg . Set Ui j = Ui ∩ U j and similarly for i j . Up to further shrinking the cover we may assume that i j intersects each connected component of Ui j and that there is a function ϕi j ∈ O X (Ui j ) such that dϕi j = θi − θ j |Ui j and ϕi j |i j,reg = f i − f j |i j,reg . Set Ui jk = Ui ∩ U j ∩ Uk and similarly for i jk . Note that d(ϕi j + ϕ jk + ϕki ) = 0, so that ϕi j + ϕ jk + ϕki is locally constant on Ui jk . Since it vanishes on i jk , it vanishes everywhere. p1
Set ρi = (Vi −→ Ui , αi ), where Vi = Ui × C and αi = p1∗ θi + dt. Let (ρi , i ) be the pair with i = {(x, t) ∈ Vi ; x ∈ i , t + f i (x) = 0}. Then the pair (ρ, ) is obtained by patching the (ρi , i )’s via the maps (x, t) → (x, t + ϕi j (x)). Let us give an example that shows how, in general, and are not isomorphic as complex spaces. Example 2.5.4. Let X = (T ∗ C, dθ ) with symplectic coordinates (x; u), and Y = (X × C, α) with extra coordinate t. Then θ = u d x and α = u d x + dt. Take as ⊂ X a parametric curve = {(x(s), u(s)); s ∈ C}, with x(0) = u(0) = 0. Then = {(x, u, t); x = x(s), u = u(s), t + f (s) = 0},
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where f satisfies the equations f (s) = u(s)x (s) and f (0) = 0. For x(s) = s 3 , u(s) = s 7 + s 8 ,
f (s) =
3 10 10 s
+
3 11 11 s ,
we have an example where f cannot be written as an analytic function of (x, u). In fact, 11 ∈ s 11 = 11x(s)u(s) − 110 / C[[s 3 , s 7 + s 8 ]]. 3 f (s) and s 3. Holonomic Modules on Symplectic Manifolds We start by giving here a construction of the microdifferential algebroid of [6] in terms of algebroid data and by recalling some results on regular holonomic microdifferential modules. Then, using the results from the previous section, we show how it is possible to associate to a complex symplectic manifold a natural C-linear category of holonomic modules.
3.1. Microdifferential algebras. Let us review some notions from the theory of microdifferential operators (refer to [7,17]). Let M be a complex manifold. Denote by E M the sheaf on P ∗ M of microdifferential operators, and by Fk E M its subsheaf of operators of order at most k ∈ Z. Then E M is a sheaf of C-algebras on P ∗ M, filtered over Z by the Fk E M ’s. Take a local symplectic coordinate system (x; ξ ) on T ∗ M. For an open subset U ⊂ T ∗ M, a section a ∈ Γ (U ; Fk E M ) is represented by its total symbol, which is a formal series
a(x, ξ ) = a j (x, ξ ), a j ∈ Γ (U ; O P ∗ M ( j)) j≤k
satisfying suitable growth conditions. In terms of total symbols, the product in E M is given by Leibniz rule. More precisely, for a ∈ E M with total symbol a (x, ξ ), the product aa has total symbol
1 ∂ξJ a(x, ξ )∂xJ a (x, ξ ). J ! n
J ∈N
For a ∈ Fk E M , the top degree component ak ∈ O P ∗ M (k) of its total symbol does not depend on the choice of coordinates. The map σk : Fk E M − → O P ∗ M (k), a → ak induced by the isomorphism Fk E M /Fk−1 E M O P ∗ M (k) is called the symbol map. Recall that an operator a ∈ Fk E M \Fk−1 E M is invertible at p ∈ P ∗ M if and only if σk (a)( p) = 0. For a ∈ Fk E M and a ∈ Fk E M , one has {σk (a), σk (a )} = σk+k −1 ([a, a ]). op
An anti-involution of E M is an isomorphism of C-algebras ∗ : E M − → E M such that ∗∗ = id.
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Remark 3.1.1. In a local system of symplectic coordinates, an example of anti-involution ∗ of E M is given by the formal adjoint. This is described at the level of total symbols by
1 ∂ξJ ∂xJ a(x, −ξ ) . a ∗ (x, ξ ) = J ! n J ∈N
The formal adjoint depends on the choice of the top-degree form d x1 ∧ · · · ∧ d xn . Consider a contact transformation χ
P∗ M ⊃ V − → V ⊂ P ∗ M, where M, M are complex manifolds with the same dimension. It is a fundamental result of [17] that contact transformations can be locally quantized. Theorem 3.1.2. With the above notations: ∼ (i) Any C-algebra isomorphism f : χ∗ E M |V − → E M |V is a filtered isomorphism, and σk ( f (a )) = χ∗ σk (a ) for any a ∈ Fk E M . (ii) For any p ∈ V there exists a neighborhood U of p in V and a C-algebra isomor∼ phism f : χ∗ E M |U − → E M |U . (iii) Let ∗ and ∗ be anti-involutions of E M |V and E M |V , respectively. For any p ∈ V there exists a neighborhood U of p in V and a C-algebra isomorphism f as in (ii) such that f ∗ = ∗ f . An isomorphism f as in (ii) is called a quantized contact transformation over χ . Quantized contact transformations over χ are not unique. It was noticed in [6] that one can reduce the ambiguity to an inner automorphism by considering anti-involutions as in (iii) (see Lemma 3.2.4 below). The C-algebra E M is left and right Noetherian. It is another fundamental result of [17] that the support of a coherent E M -module is a closed involutive subvariety of P ∗ M. A coherent E M -module supported by a Lagrangian subvariety is called holonomic. We refer e.g. to [7] for the notion of regular holonomic E M -module. 3.2. Microdifferential algebroid. Let Y be a complex contact manifold. Definition 3.2.1. A microdifferential algebra E on Y is a sheaf of C-algebras such that, locally on Y , there is a C-algebra isomorphism E|V χ −1 E M in a Darboux chart (2.3.2). By Theorem 3.1.2, any C-algebra automorphism of E M is filtered and symbol preserving. It follows that a microdifferential algebra E on Y is filtered and has symbol maps σk : Fk E − → OY (k). Example 3.2.2. Let Y = P ∗ M be the projective cotangent bundle to a complex M the invertible O -module of top-degree manifold M and denote by M = dim M M forms. Consider the algebra of twisted microdifferential operators ⊗1/2
E⊗1/2 = M M
⊗−1/2
⊗O M E M ⊗O M M
.
Then E⊗1/2 is a microdifferential algebra on P ∗ M, and the formal adjoint ∗ of M Remark 3.1.1 gives a canonical anti-involution of E⊗1/2 . M
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Definition 3.2.3. The gerbe PY on Y is defined as follows: (1) For an open subset V ⊂ Y , objects of PY (V ) are pairs p = (E, ∗) of a microdifferential algebra E on V and an anti-involution ∗ of E. (2) If p = (E , ∗ ) is another object, Hom PY ( p , p) = { f ∈ Isom C-Alg (E , E); f ∗ = ∗ f }. (The fact that the stack of groupoids PY is a gerbe follows from Theorem 3.1.2.) Lemma 3.2.4 ([6, Lemma 1]). For any p = (E, ∗) ∈ PY there is an isomorphism of sheaves of groups ∼
ψ : {b ∈ E × ; b∗ b = 1, σ0 (b) = 1} − → EndPY ( p), b → Ad(b). By this lemma, we have a natural C-algebroid data on Y , and hence a C-algebroid, defined as follows. Definition 3.2.5. The microdifferential algebroid on Y is the C-algebroid EY associated to (PY , E , ), where
E ( p) = E, E ( f ) = f, p (g) = b, → p and g = ψ(b). for p = (E, ∗), f : p − By the construction in § 1.2, this means that objects of EY are microdifferential algebras (E, ∗) endowed with an anti-involution. Morphisms (E , ∗ ) − → (E, ∗) in EY are ∼ → E such that f ∗ = ∗ f . equivalence classes of pairs (a, f ) with a ∈ E and f : E − The equivalence relation is given by (a, Ad(b) f ) ∼ (ab, f ) for b ∈ E × with b∗ b = 1 and σ0 (b) = 1. Note that EY is locally represented by microdifferential algebras. In fact, the sheaf of endomorphisms of (E, ∗) in EY is isomorphic to E. Remark 3.2.6. Let Y = P ∗ M be the projective cotangent bundle to a complex manifold M. With notations as in Example 3.2.2, a global object of E P ∗ M is given by (E⊗1/2 , ∗). This implies that the algebroid E P ∗ M is represented by the microdifferential M algebra E⊗1/2 . M
γ
3.3. Holonomic modules on contact manifolds. Let Y = (Z − → Y, θ ) be a complex contact manifold. Consider the stack Mod(EY ) of modules over the microdifferential algebroid EY . For a subset S ⊂ Y , denote by Mod S (EY ) the full substack of Mod(EY ) of objects supported on S. By construction, EY is locally represented by microdifferential algebras. As the notions of coherent and regular holonomic microdifferential modules are local and invariant by quantized contact transformations, they make sense also for objects of Mod(EY ). Denote by Modcoh (EY ) and Modrh (EY ) the full substacks of Mod(EY ) whose objects are coherent and regular holonomic, respectively. Let R be an invertible C-algebroid. Then Mod(R) is locally equivalent to Mod(CY ). Hence the notion of local system makes sense for objects of Mod(R). Denote by LocSys(R) the full substack of Mod(R) whose objects are local systems. Consider the invertible C-algebroid C⊗1/2 on Y as in Example 1.2.6. Y By [6, Prop. 4] (see also [3, Cor. 6.4]), one has
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Proposition 3.3.1. For a smooth Lagrangian submanifold ⊂ Y there is an equivalence Mod,rh (EY ) p∗ LocSys( p −1 C⊗1/2 ),
where p :
γ −1
− → is the restriction of γ : Z − → Y.
Recall that a C-linear triangulated category T is called Calabi-Yau of dimension d if for each M, N ∈ T the vector spaces Hom T (M, N ) are finite-dimensional and there are isomorphisms Hom T (M, N )∨ Hom T (N , M[d]), functorial in M and N . Here H ∨ denotes the dual of a vector space H . Denote by Dbrh (EY ) the full triangulated subcategory of the bounded derived category of EY -modules whose objects have regular holonomic cohomologies. The following theorem is obtained in [11]1 as a corollary of results from [8]. Theorem 3.3.2. If Y is compact, then Dbrh (EY ) is a C-linear Calabi-Yau triangulated category of the same dimension as Y . 3.4. Holonomic modules on symplectic manifolds. Let X = (X, ω) be a complex symplectic manifold and ⊂ X a closed Lagrangian subvariety. By Proposition 2.5.3 there exists a neighborhood U ⊃ , a contactification ρ : V − → U and a closed Lagrangian subvariety ⊂ V such that ρ induces an isomorphism − → . Let us still denote by ρ the composition V − →U − → X . We set RH X, = ρ∗ Mod,rh (EV ), DRH (X ) = Db,rh (EV ). By unicity of the pair (ρ, ), the stack RH X, and the triangulated category DRH (X ) only depend on . For ⊂ , there are natural fully faithful, exact functors → RH X, , DRH (X ) − → DRH (X ). RH X, − The family of closed Lagrangian subvarieties of X , ordered by inclusion, is filtrant. Definition 3.4.1. (i) The stack of regular holonomic microdifferential modules on X is the C-linear abelian stack defined by RH X = lim RH X, . − →
(ii) The triangulated category of complexes of regular holonomic microdifferential modules on X is the C-linear triangulated category defined by DRH(X ) = lim DRH (X ). − →
1 The statement in [11, Theorem 9.2 (ii)] is not correct. It should be read as Theorem 3.3.2 in the present paper.
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As a corollary of Proposition 3.3.1, we get Theorem 3.4.2. For a closed smooth Lagrangian submanifold ⊂ X , there is an equivalence RH X, p1 ∗ LocSys( p1−1 C⊗1/2 ),
where p1 :
× C×
− → is the projection.
Remark 3.4.3. When X is reduced to a point, the category of regular holonomic microdifferential modules on X is equivalent to the category of local systems on C× . As a corollary of Theorem 3.3.2, we get Theorem 3.4.4. If X is compact, then DRH(X ) is a C-linear Calabi-Yau triangulated category of dimension dim X + 1. 4. Quantization Algebroid In this section, we first recall the construction of the deformation-quantization algebroid of [16] in terms of algebroid data. Then, with the same data, we construct a new C-algebroid where the deformation parameter is no longer central. Its centralizer is related to the deformation-quantization algebroid through a twist by the gerbe parameterizing the primitives of the symplectic 2-form. ρ
4.1. Quantization data. Let X be a complex symplectic manifold. Let ρ = (Y − → X, α) be a contactification of X and E a microdifferential algebra on Y . γ Recall that the contact structure on Y is given by Y = (Z − → Y, θ ), where Z = Y × C× , γ = q1 is the first projection and θ = τ q1∗ α for τ the coordinate of C× ⊂ C. Definition 4.1.1. A deformation parameter is an invertible section ∈ F−1 E such that ι(σ−1 ()) = α, under the embedding (2.3.1). Example 4.1.2. Let (t; τ ) be the symplectic coordinates on T ∗ C. Recall from Example 2.4.1 the contactification of the conormal bundle T ∗ M to a complex manifold M given by ρ
P ∗ (M × C) ⊃ {τ = 0} − → T ∗ M. In this case the condition ι(σ−1 ()) = α reads σ−1 () = τ −1 . Denote by ∂t ∈ F1 EC the operator with total symbol τ . It induces a deformation parameter = ∂t−1 in E M×C |{τ =0} . → Y (for λ ∈ C) denotes the C-action on Y and va denotes its Recall that Tλ : Y − infinitesimal generator. Note that d −1 Ad(eλ )|λ=0 dλ is a C-linear derivation of E inducing va on symbols. This derivation is integrable, and induces the isomorphism ad(−1 ) =
−1 )
eλ Ad(
−1
∼
= Ad(eλ ) : (T−λ )∗ E − → E.
This is a quantized contact transformation over T−λ .
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Definition 4.1.3. The gerbe P X on X is defined as follows: (1) Objects on U ⊂ X are quadruples q = (ρ, E, ∗, ) of a contactification ρ = ρ → U, α), a microdifferential algebra E on V , an anti-involution ∗ of E and a (V − deformation parameter ∈ F−1 E such that ∗ = −. (2) If q = (ρ , E , ∗ , ) is another object, Hom P X (q , q) = {(χ , f ); χ ∈ Hom Cω (ρ , ρ), f ∈ Isom C-Alg (χ∗ E , E),
f ∗ = ∗ f, f ( ) = },
with composition given by (χ , f ) ◦ (χ , f ) = χ χ , f (χ∗ f ) . −1
Note that Ad(eλ ) commutes with ∗ for λ ∈ C, since ∗ = −. Remark 4.1.4. Let M be a complex manifold. With notations as in Example 4.1.2, the operator ∂t ∈ F1 EC induces a deformation parameter = ∂t−1 in the algebra E⊗1/2 of M×C
twisted microdifferential operators. Hence PT ∗ M has a global object given by (ρ, E⊗1/2 {τ =0} , ∗, ∂t−1 ), M×C
with ∗ the anti-involution given by the formal adjoint. Lemma 4.1.5 ([16, Lemma 5.4]). For any q = (ρ, E, ∗, ) ∈ P X (U ), there is an isomorphism of sheaves of groups ∼
ψ : CU × {b ∈ ρ∗ F0 E × ; [, b] = 0, b∗ b = 1, σ0 (b) = 1} − → EndP X (q) −1 given by ψ(μ, b) = Tμ , Ad(beμ ) . One could now try to mimic the construction of the microdifferential algebroid EY in order to get an algebroid from the algebras ρ∗ E. This fails because the automorphisms −1 of (ρ, E, ∗, ) are not all inner, an outer automorphism being given by Ad(eλ ) for λ ∈ C. −1 There are two natural ways out: consider subalgebras where Ad(eλ ) acts as the −1 identity, or consider bigger algebras where Ad(eλ ) becomes inner. The first solution, utilized in [16] to construct the deformation-quantization algebroid, is recalled in Sect. 4.2. The second solution is presented in Sect. 4.3, and will allow us to construct the quantization algebroid. 4.2. Deformation-quantization algebroid. Let X be a complex symplectic manifold. We can now describe the deformation-quantization algebroid of [16] in terms of algebroid data. ρ Let ρ = (Y − → X, α) be a contactification of X . Let E be a microdifferential algebra on Y and ∈ F−1 E a deformation parameter. The deformation-quantization algebra associated with the data (ρ, E, ) is the algebra W = C0 ρ∗ E. This is the subalgebra of ρ∗ E of operators commuting with . Denote by k ⊂ C[−1 , ]] the center of W, a subfield of the field of formal Laurent series.
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Example 4.2.1. As in Example 4.1.2, consider the contactification of the conormal bundle T ∗ M to a complex manifold M given by ρ
P ∗ (M × C) ⊃ {τ = 0} − → T ∗ M. Then = ∂t−1 is a deformation parameter in E M×C . Set W M = C∂0t ρ∗ E M×C |{τ =0} . Take a local symplectic coordinate system (x; ξ ) on T ∗ M. Since an element a ∈ Fk W M commutes with ∂t , its total symbol is a formal series independent of t,
a j (x, ξ, τ ), a j ∈ O P ∗ (M×C) ( j),
j≤k
a− j (x, u, 1) and recalling that satisfying suitable growth conditions. Setting a j (x, u) = = ∂t−1 , the total symbol of a can be written as a(x, u, ) =
a j (x, u) j , a j ∈ OT ∗ M .
j≥−k
To make the link with usual deformation-quantization, consider two operators a, a ∈ F0 W M of degree zero with total symbol a(x, u), a (x, u) ∈ OT ∗ M . Then the product aa has a total symbol given by the Leibniz star-product a(x, u) a (x, u) =
|J | ∂uJ a0 (x, u)∂xJ a0 (x, u). J ! n
J ∈N
Note that there is a natural identification k = W{pt} . Recall the gerbe P X from Definition 4.1.3 and the isomorphism ψ of Lemma 4.1.5. Definition 4.2.2. The deformation-quantization algebroid on X is the k-algebroid W X associated to the data (P X , W , ), where
W (q) = W, W (χ , f ) = ρ∗ f, q (ψ(μ, b)) = b, for q = (ρ, E, ∗, ), W = C0 ρ∗ E, (χ , f ) : q − → q, and for (μ, b) as in Lemma 4.1.5. −1
Note that is indeed a lifting of since the action of Ad(eμ ) is trivial on W. Recall that objects of W X are objects q = (ρ, E, ∗, ) of P X . Morphisms q − → q in W X are equivalence classes [a, (χ , f )] with a ∈ W and (χ , f ) : q − → q a morphism in P X . M be a complex manifold and X = T ∗ M. With notations as the algebroid WT ∗ M is represented by the algebra W⊗1/2 = M . {τ =0}
Remark 4.2.3. Let in Remark 4.1.4, C 0 ρ∗ E ⊗1/2
M×C
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4.3. Quantization algebras. Let ρ = (Y − → X, α) be a contactification of the complex symplectic manifold X = (X, ω). Let E be a microdifferential algebra on Y and ∈ F−1 E a deformation parameter. Let us set E[ρ] = C∞ ρ∗ E, where C∞ E = {a ∈ E; ad() N (a) = 0, locally for some N > 0}. In local coordinates (x, t; ξ, τ ), sections of C∞ E are sections of E whose total symbol is polynomial in t. Definition 4.3.1. The quantization algebra associated with (ρ, E, ) is the C-algebra −1 E = E[ρ] eλ , λ∈C
whose product is given by −1
−1
eλ eλ
−1
−1
−1
−1
= e(λ+λ ) , eλ a = Ad(eλ )(a) eλ ,
for λ, λ ∈ C and a ∈ E[ρ] . Denote by R the group ring of the additive group C with coefficients in C, so that −1 R C eλ . λ∈C
Then one has an algebra isomorphism C0 E W ⊗C R, where W = ρ∗ C0 E is the deformation-quantization algebra associated with (ρ, E, ). In particular, C 0 E is a k ⊗ R-algebra.
C
4.4. Quantization algebroid. Let X = (X, ω) be a complex symplectic manifold. Recall the gerbe P X on X from Definition 4.1.3 and the isomorphism ψ of Lemma 4.1.5. Definition 4.4.1. The quantization algebroid on X is the C-algebroid E X associated to the data (P X , , ), where E μ−1
, E (q) = E, E (χ , f ) = ρ∗ f, q (ψ(μ, b)) = be
→ q, and for (μ, b) as in Lemma 4.1.5. for q = (ρ, E, ∗, ), (χ , f ) : q − Recall that objects of E X are objects q = (ρ, E, ∗, ) of P X . Morphisms q − →q in E X are equivalence classes [a, ˜ (χ , f )] with a˜ ∈ E and (χ , f ) : q − → q a morphism in EX . Note that there is a natural action of C[] on E X . With the notations of §1.1, we set for short 0 C0 E X = CC [] E X .
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Remark 4.4.2. Let M be a complex manifold and X = T ∗ M. With notations as in Remark 4.1.4, the algebroid ET ∗ M is represented by the algebra E⊗1/2 {τ =0} . M×C
Recall the notation R by Definition 2.1.5 for
λ∈C C e
λ−1 .
Let Rω be the invertible R-algebroid given −1
: C − → R × , λ → eλ . Note that Rω R X if X admits a contactification. The following proposition can be compared with [16, Remark 9.3]. Proposition 4.4.3. There is an equivalence of k ⊗C R-algebroids W X ⊗C X Rω C0 EX . → W X ⊗C X Rω defined by Proof. Consider the functor ψ : C0 EX − −1 −1 (ρ, E, ∗, ) → (ρ, E, ∗, ), ρ , [aeλ , (χ , f )] → [a, (χ , f )] ⊗ [eλ , χ ] −1
on objects and morphisms, respectively. Since a ∈ C0 E commutes with eλ , ψ is indeed compatible with composition of morphisms. The fact that ψ is an equivalence is a local problem, and thus follows from the isomorphism of the representative algebras C0 E W ⊗C R. In particular, W X is equivalent to the homogeneous component of degree zero in −1 C0 C eλ . E X ⊗R X R−ω W X ⊗C λ∈C
5. Quantization Modules Here, after establishing some algebraic properties of quantization algebras, we show how the category RH X of regular holonomic microdifferential modules can be embedded in the category of quantization modules.
5.1. A coherence criterion. Let us state a non-commutative version of Hilbert’s basis theorem. For a sheaf of rings A on a topological space, consider the sheaf of rings AS A ⊗Z Z[S] of polynomials in a variable S which is not central but satisfies the rule Sa = ϕ(a)S + ψ(a), ∀a ∈ A, where ϕ is an automorphism of A and ψ is a ϕ-twisted derivation, i.e. a linear map such that ψ(ab) = ψ(a)b + ϕ(a)ψ(b). The following result can be proved along the same lines as [7, Theorem A.26]. Theorem 5.1.1. If A is Noetherian, then AS is Noetherian.
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5.2. Algebraic properties of quantization algebras. As the results in the rest of this section are of a local nature, we will consider the geometrical situation of Example 2.4.1. In particular, for (t; τ ) the symplectic coordinates of T ∗ C, we consider the projection ρ
→ T ∗ M = X. P ∗ (M × C) ⊃ Y = {τ = 0} − For = ∂t−1 , we set E = E M×C |τ =0 , E[ρ] = C∞ ρ∗ E, W = C0 ρ∗ E, E =
−1
E[ρ] eλ .
λ∈C
Theorem 5.2.1. The ring E[ρ] is Noetherian. ∼
Proof. Note that there is an isomorphism WS − → E[ρ] given by S → t. Using the results of [7, Appendix], one proves that W is Noetherian. Then E[ρ] is also Noetherian by Theorem 5.1.1. Theorem 5.2.2. The sheaves of rings E and C0 E are coherent. as the arguments for C 0 E are similar. Proof. We shall only consider E, −1 For a finitely generated Z-submodule of C, set E = λ∈ E[ρ] eλ . By induction on the minimal number of generators of one proves that E is Noetherian. In fact, ∼ → E let = 0 + Zλ and assume that E0 is Noetherian. If 0 ⊕ Zλ, then E0 S − −1 by S → eλ . Hence E0 is Noetherian by Theorem 5.1.1. Otherwise, let N be the −1 smallest integer such that nλ ∈ 0 . Then E E0 S/S − enλ is again Noetherian. → E are As E is Noetherian, it is in particular coherent. Since the morphisms E − flat for ⊂ , coherence is preserved at the limit E lim E . − →
For M ∈ Mod(E[ρ] ), let us set for short ρE∗ M = E ⊗ρ −1 E[ρ] ρ −1 M, Supp(M) = supp(ρE∗ M) ⊂ Y. Let us denote by Modρ-f,coh (E[ρ] ) the full abelian substack of Modcoh (E[ρ] ) whose objects M are such that ρ is finite on Supp(M). Let us denote by Modρ-f,coh (E) the full abelian substack of Modcoh (E) whose objects N are such that ρ is finite on supp(N ). Proposition 5.2.3. (i) The ring E is flat over ρ −1 E[ρ] . (ii) There is an equivalence of categories Modρ-f,coh (E[ρ] ) o
ρE∗ ρ∗
/ ρ Mod ∗ ρ-f,coh (E),
meaning that the functors ρE∗ and ρ∗ are quasi-inverse to each other. Let us set for short Ak = ρ −1 Fk E[ρ] , Note that A−k =
k A
0
= A0
k ,
B−k =
k B0
Bk = Fk E. = B0
k
(5.2.1)
and
A0 /A−1 ρ −1 O X [t], B0 /B−1 OY . The above proposition is a non commutative analogue of the following classical result
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Proposition 5.2.4. (i) The ring OY is flat over ρ −1 O X [t]. (ii) There is an equivalence of categories ρ∗
Modρ-f,coh (O X [t]) o
ρ∗
/ ρ Mod ∗ ρ-f,coh (OY ).
Proof of Proposition 5.2.3 (i). With notations (5.2.1), it is enough to show that B0 is flat over A0 . Thus, for a coherent A0 -module M, we have to prove that L H −1 (B0 ⊗A M) = 0.
(5.2.2)
0
One says that u ∈ M is an element of -torsion if N u = 0 for some N ≥ 0, i.e. if A−N u = 0. Denote by Mtor ⊂ M the coherent submodule of -torsion elements. One says that M is an -torsion module if Mtor = M and that M has no -torsion if Mtor = 0. Considering the exact sequence 0− → Mtor − →M− → M/Mtor − → 0, it is enough to prove (5.2.2) in the case where M is either an -torsion module or has no -torsion. (a) Assume that M has no -torsion. Then the multiplication map A−1 ⊗A0 M − →M is injective. Setting M−1 = A−1 M = M, this implies the isomorphism (A0 /A−1 ) ⊗A0 M M/M−1 . By Proposition 5.2.4 (i), we have L L H −1 ((B0 /B−1 ) ⊗BL B0 ⊗A M) H −1 ((B0 /B−1 ) ⊗A (M/M−1 )) = 0. 0 0 0 /A−1
From the exact sequence 0 − → B−1 − → B0 − → B0 /B−1 − → 0 we thus obtain the exact sequence L L B−1 ⊗B0 H −1 (B0 ⊗A M) − → H −1 (B0 ⊗A M) − → 0. 0
0
L M) = 0. By Nakayama’s lemma, we get H −1 (B0 ⊗A 0 (b) Let M be an -torsion module. As M is coherent, there locally exists N > 0 such that N M = 0. Considering the exact sequence
0− → M−1 − →M− → M/M−1 − → 0, by induction on N one reduces to the case N = 1. Then M = M/M−1 has a structure of A0 /A−1 -module. Hence L L L L B0 ⊗A M B0 ⊗A A0 /A−1 ⊗A M B0 /B−1 ⊗A M, 0 0 0 /A−1 0 /A−1
and (5.2.2) follows from Proposition 5.2.4 (i).
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We shall consider an operator a ∈ F0 E[ρ] monic in the t variable, i.e. an operator of the form a = tm +
m−1
bi t i ,
m ∈ N>0 , bi ∈ F0 W.
(5.2.3)
i=0
Lemma 5.2.5. Let a be of the form (5.2.3). Then there are isomorphisms ρE∗ (E[ρ] /E[ρ] a) E/Ea, ρ∗ (E/Ea) E[ρ] /E[ρ] a. Proof. The first isomorphism is clear. For the second, note that ρ∗ (E/Ea) ρ∗ E/ρ∗ Ea since ρ is finite on supp(E/Ea). Note also that, by division, any c ∈ ρ∗ E can be written as ∼ → E[ρ] /E[ρ] a c = da + b with d ∈ ρ∗ E and b ∈ E[ρ] . Then the isomorphism ρ∗ E/ρ∗ Ea − is given by c → b. Proof of Proposition 5.2.3 (ii). (a) Let N0 be a coherent F0 E-module such that ρ is finite on supp N0 . We will show that N0 is F0 W-coherent. As this is a local problem on Y , we can assume that (x0 , t; ξ0 , 1) ∈ supp N0 only for t = 0. Thus supp N0 ⊂ {t p + ϕ(x, t, ξ/τ ) = 0} with ϕ ∈ O X [t] vanishing for t = 0 and of degree less than p in the t variable. Choose a system u 1 , . . . , u N of generators for N0 . By division, for each i there exists ai of the form (5.2.3) such that ai u i = 0. One thus gets an exact sequence 0− → N0 − →
N
F0 E/F0 Eai − → N0 − → 0.
i=1
As F0 E/F0 Eai is F0 W-coherent, N0 is a finitely generated F0 W-module. Since also N0 is finitely generated over F0 W, it follows that N0 is F0 W-coherent. In particular, this shows that any N ∈ ρ∗ Modρ-f,coh (E) is a coherent E[ρ] -module. (b) Let N ∈ ρ∗ Modρ-f,coh (E) and choose a system u 1 , . . . , u N ∈ N of generators. By (a), ρ∗ F0 Eu i is F0 W-coherent. Hence, {t j F0 Wu i } j>0 is stationary in ρ∗ F0 Eu i , so that there exist m i > 0 and bi j ∈ F0 W such that t m i u i = j<m i bi j t j u i . In other words, for each i there exists ai = t m i − j bi j t j of the form (5.2.3) such that ai u i = 0. One thus gets an exact sequence
0− →N − →
N
E/Eai − →N − → 0.
i=1
Applying the same argument to N one gets a presentation
N
E/Eai − →
i=1
N
E/Eai − →N − → 0.
i=1
Since ρ∗ = ρ! is exact on this sequence, by Lemma 5.2.5 the module ρ∗ N has the presentation
N i=1
E[ρ] /E[ρ] ai − →
N
E[ρ] /E[ρ] ai − → ρ∗ N − → 0.
i=1 ∼
Applying the exact functor ρE∗ and using again Lemma 5.2.5, we get that ρE∗ ρ∗ N − → N.
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(c) For M ∈ Modρ-f,coh (E[ρ] ), let us show that the map M − → ρ∗ ρE∗ M is injective. Let M0 be a lattice of M, that is a coherent sub-F0 E[ρ] -module such that E[ρ] M0 = M. Since ρ F∗0 E M0 is a lattice for ρE∗ M, it is enough to prove the injectivity of the map → ρ∗ ρ F∗0 E M0 . Assume that u ∈ M0 is sent to 0. By Proposition 5.2.4 there are M0 − isomorphisms ∼
M0 /F−1 EM0 − → ρ∗ ρ ∗ (M0 /F−1 EM0 ) ρ∗ ρ F∗0 E M0 /F−1 Eρ∗ ρ F∗0 E M0 . It follows that u ∈ F−1 EM0 . By induction we then get u ∈ k>0 F−k EM0 , so that u = 0. ∼ (d) We finally have to prove the isomorphism M − → ρ∗ ρE∗ M. Let u 1 , . . . , u N be a system of generators of M. By the same arguments as in (b), for each i there exists ai of the form (5.2.3) such that ai u i = 0 in ρE∗ M. By (c) this implies ai u i = 0 in M. As in (b) we thus get a resolution
N
E[ρ] /E[ρ] ai − →
i=1
N
E[ρ] /E[ρ] ai − →M− → 0,
i=1 ∼
→ ρ∗ ρE∗ M by Lemma 5.2.5. giving the isomorphism M −
For S ⊂ Y , let us denote by Mod S,coh (E[ρ] ) the full abelian substack of Modcoh (E[ρ] ) whose objects M are such that Supp(M) ⊂ S. For T ⊂ X , let us denote by ModT,coh (E) the full abelian substack of Modcoh (E) whose objects M are such that supp(M) ⊂ T . We set for short = E ⊗ M. EM E[ρ] Proposition 5.2.6. (i) The ring E is faithfully flat over E[ρ] . (ii) Let S ⊂ Y be an analytic subset such that ρ| S is proper and injective. Then the functor : Mod S,coh (E[ρ] ) − E(·) → Modρ(S),coh (EX ) is fully faithful. Proof. (i) is straightforward. (ii) For a coherent E[ρ] -module M, there is an isomorphism of E[ρ] -modules EM
−1
eλ M.
λ∈C −1
Here, the E[ρ] -module structure of eλ M is given by −1
a(eλ
−1
· b) = eλ
−1
· Ad(e−λ )(a)b, −1
for a ∈ E[ρ] and b ∈ M. Note that Supp(eλ M) = Tλ Supp(M).
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For M, M ∈ Mod S,coh (E[ρ] ), one has , EM) Hom E[ρ] (M , Hom E(EM
λ∈C
λ∈C
−1
eλ M)
λ∈C −1
Hom E[ρ] (M , eλ M) −1
Hom E (ρE∗ M , ρE∗ (eλ M))
Hom E (ρE∗ M , ρE∗ M) Hom E[ρ] (M , M), where the second to last isomorphism is due to the fact that Supp(M ) ∩ −1 Supp(eλ M) = ∅ for λ = 0. 5.3. Induced modules. Assume that the symplectic manifold X admits a contactificaρ tion ρ = (Y − → X, α). In this section we show how the constructions from the previous section can be globalized. Definition 5.3.1. For a contactification ρ of X , the gerbe Pρ on X is defined as follows: (1) Objects on U ⊂ X are triples p = (E, ∗, ) of a microdifferential algebra E on ρ −1 (U ), an anti-involution ∗ of E and a deformation parameter such that ∗ = −. (2) If p = (E , ∗ , ) is another object, Hom Pρ ( p , p) = { f ∈ Isom R-Alg (E , E); f ∗ = ∗ f, f ( ) = }. As a corollary of Lemma 3.2.4, one has Lemma 5.3.2. For any p = (E, ∗, ) ∈ Pρ there is an isomorphism of sheaves of groups ∼
→ EndPρ ( p) ψρ : {b ∈ E × ; [, b] = 0, b∗ b = 1, σ0 (b) = 1} − given by ψρ (b) = Ad(b). Definition 5.3.3. For a contactification ρ of X , the stack E[ρ] is the C-algebroid on X associated to the data (Pρ , E[ρ] , ) where
E[ρ] ( p) = E[ρ] , E[ρ] ( f ) = ρ∗ f, p (g) = b, for p = (E, ∗, ), f : p − → p and g = ψρ (b). Note that Proposition 4.4.3 implies W X C0 E[ρ] . As in the local case, for M ∈ Mod(E[ρ] ) we set for short Supp(M) = supp(ρE∗ M) ⊂ Y.
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Consider the faithful C-linear functors → EY , (E, ∗, ) → (E, ∗), ρ −1 E[ρ] − [a, f ] → [a, f ], E[ρ] − → E X , (E, ∗, ) → (ρ, E, ∗, ), −1 [a, f ] → [ae0 , idρ , f ],
on objects, on morphisms, on objects, on morphisms.
For S ⊂ Y they induce the functors → ρ∗ Modρ-f,coh (EY ), ρE∗ : Modρ-f,coh (E[ρ] ) − E(·) : Mod S,coh (E[ρ] ) − → Modρ(S),coh ( E X ). By Propositions 5.2.3 and 5.2.6 we have Proposition 5.3.4. (i) The functor ρE∗ is an equivalence. (ii) Let S ⊂ Y be an analytic subset such that ρ| S is proper and injective. Then E(·) is fully faithful. We can thus embed regular holonomic microdifferential modules in the stack of coherent E X -modules. Thus, with notations as in Definition 3.4.1, we have Corollary 5.3.5. There is a fully faithful embedding E X ). RH X ⊂ Modcoh ( Remark 5.3.6. We do not know if the above result extends to give an embedding DRH(X ) ⊂ Dbcoh ( E X ) at the level of derived categories. Appendix A. Remarks on Deformation-Quantization We give in this appendix an alternative description of the deformation quantization algebroid using triples (W, ∗, v) of a deformation-quantization algebra W endowed with an anti-involution ∗ and an order preserving C-linear derivation v. We also compare regular holonomic deformation-quantization modules with regular holonomic quantization modules. A.1. Deformation-quantization and derivations. Let X = (X, ω) be a complex symplectic manifold and W a deformation-quantization algebra on X . Lemma A.1.1. Let w be an order preserving k-linear derivation of W. Then w is locally of the form ad(−1 d) for some d ∈ F0 W. Proof. Let (x; u) be a local system of quantized symplectic coordinates (see [9, §2.2.3]). For i = 1, . . . , n, set ei = w(xi ) ∈ F−1 W. From w([xi , x j ]) = 0 we get [ei , x j ] = [e j , xi ] for any i, j = 1, . . . , n. Hence there locally exists e ∈ F0 W with ei = [xi , e]. Replacing w by w − ad(−1 e) we may assume w(xi ) = 0. Set di = w(u i ) ∈ F−1 W. From w([xi , u j ]) = 0 we get [xi , d j ] = 0, so that di = di (x) does not depend on u. From w([u i , u j ]) = 0 we get [di , u j ] = [d j , u i ]. Hence there locally exists d = d(x) ∈ F0 W with di = [u i , d]. Replacing w by w − ad(−1 d) we have w(xi ) = w(u j ) = 0, and hence w = 0.
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Definition A.1.2. Let PX be the stack on X associated with the separated prestack PX,0 defined as follows. (1) Objects on U ⊂ X are triples q = (W, ∗, v) of a deformation quantization algebra W on U , an anti-involution ∗ and an order preserving C-linear derivation v of W such that v() = and v∗ = ∗v. (2) If q = (W , ∗ , v ) is another object, Hom PX,0 (q , q) = {(g, d); g ∈ Isom R-Alg (W , W), d ∈ F0 W, g∗ = ∗g, d = d ∗ , v − gv g −1 = ad(−1 d)}, with composition given by (g, d) ◦ (g , d ) = (gg , d + g(d )). Using Lemma A.1.1 one gets Lemma A.1.3. The stack PX is a gerbe. Remark A.1.4. Let M be a complex manifold and X = T ∗ M. With notations as in Remark 4.1.4, where = ∂t−1 , a global object of PX is given by (W⊗1/2 , ∗, ad(t∂t )). M
Lemma A.1.5. For any q = (W, ∗, v) ∈
PX (U )
there is a group isomorphism ∼
ψω : CU × {b ∈ F0 W × ; b∗ b = 1, σ0 (b) = 1} − → EndPX (q) given by ψω (μ, b) = (Ad(b), μ + v(b)b−1 ). (i) Let us prove injectivity. Assume that Ad(b) = id and μ + v(b)b−1 = 0. Then b ∈ k(0), μ = 0 and v(b) = 0. As v(b) = ∂∂ b, we get b ∈ C. Since σ0 (b) = 1, this finally gives b = 1. (ii) Let us prove surjectivity. Take (g, d) ∈ EndPX (q). Since any k-algebra automorphisms of W is inner, we can locally write g = Ad(b) for some b ∈ F0 W × . As g commutes with the anti-involutions, we have Ad(b)(a ∗ ) = (Ad(b)(a))∗ = Ad(b∗−1 )(a ∗ ) for any a ∈ W. This implies Ad(b∗ b) = id, so that b∗ b ∈ k(0). Take k ∈ k(0) with k ∗ k = b∗ b. Up to replacing b with bk −1 we may thus assume that b∗ b = 1. This implies σ0 (b) = ±1 and we may further assume that σ (b) = 1. Replacing (g, d) by (g, d) · ψω (b−1 , 0) we may thus assume g = id. Since ad(−1 d) = 0, we have d ∈ k(0). As d ∗ = d and ∗ = −, the coefficients of the odd powers of in d vanish, and we may write d = μ + 2 d for μ ∈ C and d ∈ k(0). Take d ∈ k(0) such that ∂∂ d = d , and set b = exp(d ). Since v(b)b−1 = d , we have d = μ + v(b)b−1 . Hence ψω (μ, b) = (id, d).
Proof.
Definition A.1.6. The algebroid WX is the k-algebroid on X associated to the data (PX , W , ), where
W (q) = W, W (g, d) = g, q (h, e) = b, for q = (W, ∗, v), (g, d) : q − → q and (h, e) = ψω (μ, b). Proposition A.1.7. There is a k-linear equivalence WX W X .
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This follows from the following proposition. Proposition A.1.8. There is an equivalence of gerbes PX P X . Proof. Let us consider the gerbe PX whose objects on U ⊂ X are quintuples q = (ρ, E, ∗, , t) such that π(q) = (ρ, E, ∗, ) is an object of P X and t ∈ F0 E is an operator with [−1 , t] = 1. (The local model in a Darboux chart is obtained by Example 4.1.2 with −1 = ∂t and t = t.) We set Hom PX (q , q) = Hom P X (π(q ), π(q)). There is a natural equivalence ∼
→ P X , q → π(q). PX − Consider the functor ψ : PX − → PX given by q → (C0 ρ∗ E, ∗, ad(t−1 )), (χ , f ) → ρ∗ f, t − f (t ) ,
for q = (ρ, E, ∗, , t), for (χ , f ) : q − → q.
This is well defined since ad(t−1 ) − f ad(t −1 ) f −1 = ad((t − f (t ))−1 ). It follows from Lemmas A.1.5 and 4.1.5 that ψ is fully faithful. As PX and PX are gerbes, ψ is an equivalence. Recall that if q = (W, ∗, v) is an object of PX on an open subset U ⊂ X , then W X |U is represented by W. As shown in [15], the filtration and the anti-involution of W extend to W X . As we will now explain, also the derivation of W extends to W X . Let ε be a formal variable with ε2 = 0. Consider the natural morphisms i
π
W− → W[ε] − → W. Let us extend the anti-involution ∗ to W[ε] by setting ε∗ = −ε. Lemma A.1.9. Let ϕ : W − → W[ε] be an order preserving C-algebra morphism such that π ϕ = idW , ϕ() = + ε2 and ϕ∗ = ∗ϕ. Then ϕ = i + εv for an order preserving C-linear derivation v of W such that v∗ = ∗v. Remark A.1.10. There is an isomorphism of W ⊗C W op -modules (W[ε])ϕ C1 ρ∗ E such that the multiplication by ε corresponds to ad(−1 ). In local coordinates where −1 = ∂t and v = ad(t∂t ), this isomorphism is given by a + εb → at + b. The above lemma motivates the following definition. Definition A.1.11. A derivation of a C-linear stack A is the data of a pair ϕ = (C, ϕ), where C is an invertible C[ε]-algebroid such that C/ε is represented by C X and ϕ : A − → A ⊗C C is a C-linear functor such that π ϕ idA . Here π : A ⊗C C − → A is the functor induced by C − → C/ε.
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Consider the following algebroid. Definition A.1.12. The algebroid WεX is the k[ε]-algebroid associated to the data (PX , εW , ), where
εW (q) = W[ε], εW (g, d) = (1 + ε ad(d))g, q (h, e) = (1 + εμ)b, for q = (W, ∗, v), (g, d) : q − → q and (h, e) = ψω (μ, b). There is a natural morphism ϕ : WX − → WεX satisfying ϕ() = + ε2 and ϕ∗ = ∗ϕ. Similarly to Proposition 4.4.3, one proves that there is an equivalence of k[ε]-algebroids WεX W X ⊗C C[ε]ω , where C[ε]ω is the invertible C[ε]-algebroid given by Definition 2.1.5 for : C − → C[ε]× , λ → (1 + ελ). Thus W X is endowed with the derivation ϕ = (C[ε]ω , ϕ). Summarizing, W X is a filtered k-stack endowed with an anti-involution ∗ and with a C-linear derivation ϕ such that F0 W X /F−1 W X is represented by O X , ϕ() = and ϕ∗ = ∗ϕ. One can prove along the lines of [15] that W X is unique among the stacks which satisfy these properties and which are locally represented by deformation quantization algebras. A.2. Comparison of regular holonomic modules. We shall compare here regular holonomic quantization-modules with regular holonomic deformation-quantization modules. Let us start by recalling the definition of regular holonomic deformation-quantization modules from [11]. Let X be a complex symplectic manifold and a closed Lagrangian subvariety of X . Let W be a deformation-quantization algebra on X . Definition A.2.1. (i) One says that a coherent F0 W-module M0 is regular holonomic along if supp(M0 ) ⊂ and M0 /M0 is a coherent O -module. (ii) One says that a coherent W-module M is regular holonomic along if supp(M) ⊂ and there exists locally a coherent F0 W-submodule M0 of M such that M0 generates M over W and M0 is regular holonomic along . Recall that W X denotes the deformation-quantization algebroid on X . As the above definition is local, there is a natural notion of regular holonomic W X -module along . Let us denote by Mod,rh (W X ) the full substack of Modcoh (W X ) whose objects are regular holonomic along . Up to shrinking X , we may assume that there exist a contactification ρ : Y − → X and a Lagrangian subvariety of Y such that ρ induces an isomorphism − → . By definition, regular holonomic E X -modules along are equivalent to regular holonomic EY -modules along . In order to compare quantization and deformation-quantization modules, let us thus consider the forgetful functor for : ρ∗ Mod,rh (EY ) − → Mod,rh (W X ) induced by the equivalence W X C0 E[ρ] and the functor ρ −1 E[ρ] − → EY from §5.3.
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Proposition A.2.2. (i) The functor for is faithful but not locally full in general. (ii) If is a smooth submanifold, the functor for is locally essentially surjective but not essentially surjective in general. (iii) The functor for is not locally essentially surjective in general. Proof. (i) holds more generally for the forgetful functor ρ∗ Mod(EY ) − → Mod(W X ). (ii) Let be a smooth submanifold. Consider the commutative diagram / Mod,rh (W X ) O
for
ρ∗ Mod,rh (EY ) O ∼
ρ∗ p1 ∗ LocSys( p1−1 C⊗1/2 )
∼
/ LocSys(k⊗1/2 ),
where p1 : × C× − → is the projection. The vertical equivalences are due to Proposition 3.3.1 and [3, Cor. 9.2], respectively. The bottom arrow is given by L → k ⊗C L|s=1 , where s is the coordinate of C× . This shows that the forgetful functor is locally essentially surjective. To prove that it is not surjective in general, take X = T ∗ (C× ) and the zero section of T ∗ (C× ). Then the local system with monodromy 1 + around the origin is not in the essential image of the forgetful functor. (iii) follows from Proposition A.2.3 below. Before stating Proposition A.2.3 let us introduce some notations. Let M = C. Denote by (x, t; ξ, τ ) the symplectic coordinates of P ∗ (M × C) and by (x; u) those of T ∗ M. Let W = W M , and recall that = ∂t−1 . We will identify elements a ∈ W with their total symbol a(x, u, τ ), and write for example ax for the operator with total symbol ∂∂x a(x, u, τ ). Denote by OM = W/W∂x the canonical regular holonomic module along the zero section 1 = {(x, u); u = 0}. The quotient map W − → OM , b → [b] induces an isomorphism of vector spaces ∼ − C x0 W with the subring of operators whose total symbol does not depend on ∂x . OM ← For m ∈ Z>0 , consider the Lagrangian subvariety = 1 ∪ 2 , with 2 = {(x, u); u = x m }. For a ∈ C x0 W, let Ma be the regular holonomic module along with generators v1 , v2 and relations ∂x v1 = 0, (∂x − x m ∂t )v2 = av1 . Note that Ma C x0 W v1 ⊕ C x0 W v2 . ∼
→ Ma . Let a ∈ C x0 W be another operator. If [a − a ] ∈ (∂x − x m ∂t )OM , then Ma − ∼ 0 m → Ma is In fact, if e ∈ C x W satisfies a − a = ex − x e∂t , an isomorphism Ma −
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given by v1 → v1 , v2 → v2 + ev1 . Since OM /(∂x − x m ∂t )OM thus assume that
m−1 i=0
kx i , we may
a = a0 + a1 x + · · · + am−1 x m−1 with ai ∈ k. The following counterexample was developed by the second author (M.K.) while working with Pierre Schapira at [11]. Proposition A.2.3. If Ma for(N ) for some EY -module N , then a is homogeneous, i.e. a = ai0 x i0 for some i 0 ∈ {0, . . . , m − 1}. Proof. The existence of such an N is equivalent to the existence of an endomorphism t of Ma such that [t, x] = [t, ∂x ] = 0 and [t, ∂t ] = −1. (i) Let tv1 = bv1 + cv2 for b, c ∈ C x0 W. Then 0 = t∂x v1 = ∂x tv1 = ∂x (bv1 + cv2 ) = bx v1 + cx v2 + c(x m ∂t v2 + av1 ). Hence bx + ac = 0, x m c∂t + cx = 0. It follows from the second equation that c = 0. Thus the first equation implies that b ∈ k. Up to replacing t by t − b, we may assume that tv1 = 0. (ii) Let tv2 = bv1 + cv2 for b, c ∈ C x0 W. Then 0 = t (∂x − x m ∂t )v2 − av1 = (∂x − x m ∂t )tv2 + x m v2 − [t, a]v1 = (∂x − x m ∂t )(bv1 + cv2 ) + x m v2 − [t, a]v1 = bx v1 + cx v2 + c(x m ∂t v2 + av1 ) − x m b∂t v1 − x m c∂t v2 + x m v2 − [t, a]v1 . Hence ac + bx − x m b∂t − [t, a] = 0, cx + x m = 0.
(A.2.1)
m+1
The second equation gives c = − xm+1 + d for d ∈ k. Then, the first equation in (A.2.1) can be rewritten (ad(∂x ) − x m ∂t )(xa + (m + 1)b∂t ) − (xax − ea + (m + 1)∂t [t, a]) = 0, for e = (m + 1)d∂t − 1 ∈ k. Considering the degree in x, we have xa + (m + m−1 i 1)b∂t = xax − ea + (m + 1)∂t [t, a] = 0. Since a = i=0 ai x , it implies m−1 that i=0 ((e − i)ai − (m + 1)∂t [t, ai ])x i = 0. Hence we have (e − i)ai − (m + t [t,ai ] +i. Since 1)∂t [t, ai ] = 0 for every i. Thus we have either ai = 0 or e = (m+1)∂ ai (m+1)∂t [t,ai ] ai
∈ (m + 1)Z + F−1 k, this implies a = ai0 x i0 for some 0 ≤ i 0 ≤ m − 1.
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References 1. D’Agnolo, A., Kashiwara, M.: A note on quantization of complex symplectic manifolds. http://arxiv.org/ abs/1006.0306v2 [math.AG], 2010 2. D’Agnolo, A., Polesello, P.: Deformation quantization of complex involutive submanifolds. In: Noncommutative geometry and physics (Yokohama, 2004), Rivers Edge, NJ: World Scientific, 2005, pp. 127–137 3. D’Agnolo, A., Schapira, P.: Quantization of complex Lagrangian submanifolds. Adv. Math. 2131, 358– 379 (2007) 4. Giraud, J.: Cohomologie non abelienne. Grundlehren der Math. Wiss. 179, Berlin: Springer, 1971 5. Gukov, S., Witten, E.: Branes and quantization. Adv. Theor. Math. Phys. 13(5), 1445–1518 (2009) 6. Kashiwara, M.: Quantization of contact manifolds. Publ. Res. Inst. Math. Sci. 32(1), 1–7 (1996) 7. Kashiwara, M.: D-modules and Microlocal Calculus. Translations of Mathematical Monographs 217, Providence, RI: Amer. Math. Soc. 2003 8. Kashiwara, M., Kawai, T.: On holonomic systems of microdifferential equations III. Publ. RIMS Kyoto Univ. 17, 813–979 (1981) 9. Kashiwara, M., Rouquier, R.: Microlocalization of rational Cherednik algebras. Duke Math. J. 144(3), 525–573 (2008) 10. Kashiwara, M., Schapira, P.: Categories and sheaves. Grundlehren der Math. Wiss. 332, Berlin-Heidelberg-New York: Springer, 2006 11. Kashiwara, M., Schapira, P.: Constructibility and duality for simple holonomic modules on complex symplectic manifolds. Amer. J. Math. 130(1), 207–237 (2008) 12. Kashiwara, M., Schapira, P.: Deformation quantization modules, http://arxiv.org/abs/1003.3304v2. [math.AG], 2010, to appear in Astérisque 13. Kontsevich, M.: Deformation quantization of algebraic varieties. In: EuroConférence Moshé Flato, Part III (Dijon, 2000), Lett. Math. Phys. 56(3), 271–294 (2001) 14. Nest, R., Tsygan, B.: Remarks on modules over deformation quantization algebras. Mosc. Math. J. 4(4), 911–940 (2004) 15. Polesello, P.: Classification of deformation quantization algebroids on complex symplectic manifolds. Publ. Res. Inst. Math. Sci. 44(3), 725–748 (2008) 16. Polesello, P., Schapira, P.: Stacks of quantization-deformation modules on complex symplectic manifolds. Int. Math. Res. Notices 2004:49, 2637–2664 (2004) 17. Sato, M., Kawai, T., Kashiwara, M.: Microfunctions and pseudo-differential equations. In: Hyperfunctions and pseudo-differential equations (Katata 1971), Lecture Notes in Math. 287, Berlin-HeidelbergNew York: Springer, 1973, pp. 265–529 18. Street, R.: Categorical structures. In: Handbook of algebra, Vol. 1, Amsterdam: North-Holland, 1996, pp. 529–577 19. Tsygan, B.: Oscillatory modules. Lett. Math. Phys. 88(1–3), 343–369 (2009) Communicated by N.A. Nekrasov
Commun. Math. Phys. 308, 115–132 (2011) Digital Object Identifier (DOI) 10.1007/s00220-011-1323-9
Communications in
Mathematical Physics
Exactness of the Fock Space Representation of the q -Commutation Relations Matthew Kennedy, Alexandru Nica Pure Mathematics Department, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada. E-mail: [email protected]; [email protected] Received: 29 September 2010 / Accepted: 21 March 2011 Published online: 9 September 2011 – © Springer-Verlag 2011
Abstract: We show that for all q in the interval (−1, 1), the Fock representation of the q-commutation relations can be unitarily embedded into the Fock representation of the extended Cuntz algebra. In particular, this implies that the C ∗ -algebra generated by the Fock representation of the q-commutation relations is exact. An immediate consequence is that the q-Gaussian von Neumann algebra is weakly exact for all q in the interval (−1, 1). 1. Introduction The q-commutation relations provide a q-analogue of the bosonic (q = 1) and the fermionic (q = −1) commutation relations from quantum mechanics. These relations have a natural representation on a deformed Fock space which was introduced by Bozejko and Speicher in [1], and was subsequently studied by a number of authors (see e.g. [2,5–7,9,10]). For the entirety of this paper, we fix an integer d ≥ 2. Consider the usual full Fock space F over Cd , F = ⊕∞ n=0 Fn
(orthogonal direct sum),
(1.1)
where F0 = C and Fn = (Cd )⊗n for n ≥ 1. Corresponding to the vectors in the standard orthonormal basis of Cd , one has left creation operators L 1 , . . . , L d ∈ B(F). Define the C∗ -algebra C by C := C ∗ (L 1 , . . . , L d ) ⊆ B(F).
(1.2)
It is well known that C is isomorphic to the extended Cuntz algebra. (Although it is customary to denote the extended Cuntz algebra by E, we use C here to emphasize that we are working with a concrete C ∗ -algebra of operators.) Research supported by a CGS Scholarship from NSERC Canada (M. Kennedy) and by a Discovery Grant from NSERC Canada (A. Nica).
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Now let q ∈ (−1, 1) be a deformation parameter. We consider the q-deformation F (q) of F as defined in [1]. Thus (q)
F (q) = ⊕∞ n=0 Fn
(orthogonal direct sum),
(1.3)
(q)
where every Fn is obtained by placing a certain deformed inner product on (Cd )⊗n . (The precise definition will be reviewed in Subsect. 2.1 below.) For q = 0, one obtains the usual non-deformed Fock space F from above. (q) (q) In this deformed setting, one also has natural left creation operators L 1 , . . . , L d ∈ (q) B(F ), which satisfy the q-commutation relations (q)
(q)
(q)
(q)
(L i )∗ L j = δi j I + q L j (L i )∗ , 1 ≤ i, j ≤ d. Define the C∗ -algebra C (q) by (q)
(q)
C (q) := C ∗ (L 1 , . . . , L d ) ⊆ B(F (q) ).
(1.4)
For q = 0, this construction yields the extended Cuntz algebra C from above. It is widely believed that the algebra C and the deformed algebra C (q) are actually unitarily equivalent. In fact, this is known for sufficiently small q. In [5], a unitary U : F (q) → F was constructed which embeds C into C (q) for all q ∈ (−1, 1), i.e. C ⊆ U C (q) U ∗ , and it was shown that for |q| < 0.44 this embedding is actually surjective, i.e. C = U C (q) U ∗ . The main purpose of the present paper is to show that it is possible to unitarily embed C (q) into C for all q ∈ (−1, 1). Specifically, we construct a unitary operator ∗ ⊆ C. The unitary U Uopp : F (q) → F such that Uopp C (q) Uopp opp is closely related to the unitary U from [5], as we will now see. Definition 1.1. Let J : F → F be the unitary conjugation operator which reverses the order of the components in a tensor in (Cd )⊗n , i.e. J (η1 ⊗ · · · ⊗ ηn ) = ηn ⊗ · · · ⊗ η1 , ∀η1 , . . . , ηn ∈ Cd .
(1.5)
Note that for n = 0, Eq. (1.5) says that J () = . Let J (q) : F (q) → F (q) be the operator which acts as in Eq. (1.5), where the tensor (q) is now viewed as an element of the space Fn . It is known that J (q) is also unitary operator (see the review in Subsect. 2.1). Definition 1.2. Let q ∈ (−1, 1) be a deformation parameter and let U : F (q) → F be the unitary defined in [5]. Define a new unitary Uopp : F (q) → F by Uopp = J U J (q) . The following theorem is the main result of this paper. Theorem 1.3. For every q ∈ (−1, 1) the unitary Uopp from Definition 1.2 satisfies ∗ ⊆ C. Uopp C (q) Uopp
The following corollary follows immediately from Theorem 1.3. Corollary 1.4. For every q ∈ (−1, 1) the C ∗ -algebra C (q) is exact.
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To prove Theorem 1.3, we first consider the more general question of how to verify that an operator T ∈ B(F) belongs to the algebra C. It is well known that a necessary condition for T to be in C is that it commutes modulo the compact operators with the C ∗ -algebra generated by right creation operators on F. Unfortunately, this condition isn’t sufficient (and wouldn’t be sufficient even if we were to set d equal to 1, cf. [4]). Nonetheless, by restricting our attention to a ∗-subalgebra of “band-limited operators” on F and considering commutators modulo a suitable ideal of compact operators in this algebra, we do obtain a sufficient condition for T to belong to C. This bicommutant-type result is strong enough to help in the proof of Theorem 1.3. In addition to this Introduction, the paper has four other sections. In Sect. 2, we provide a brief review of the requisite background material. In Sect. 3, we prove the above-mentioned bicommutant-type result, Theorem 3.8. In Sect. 4, we establish the main results, Theorem 1.3 and Corollary 1.4. In Sect. 5, we apply these results to the family of q-Gaussian von Neumann algebras, showing in Theorem 5.1 that these algebras are weakly exact for every q ∈ (−1, 1). 2. Review of Background 2.1. Basic facts about the q-deformed Fock space. As explained in the Introduction, there is a fairly large body of research devoted to the q-deformed Fock framework and its generalizations. Here we provide only a brief review of the terminology and facts which will be needed in Sect. 4. 2.1.1. The q-deformed inner product As mentioned above, the integer d ≥ 2 will remain fixed throughout this paper. Also fixed throughout this paper will be an orthonormal basis ξ1 , . . . , ξd for Cd . For every n ≥ 1 this gives us a preferred basis for (Cd )⊗n , namely {ξi1 ⊗ · · · ⊗ ξin | 1 ≤ i 1 , . . . , i n ≤ d}.
(2.1) (Cd )⊗n
(obtained This basis is orthonormal with respect to the usual inner product on by tensoring n copies of the standard inner product on Cd ). As in the Introduction, we will use Fn to denote the Hilbert space (Cd )⊗n endowed with this inner product. The full Fock space over Cd is then the Hilbert space F from Eq. (1.1), with the convention that F0 = C for a distinguished unit vector , referred to as the “vacuum vector”. Now let q ∈ (−1, 1) be a deformation parameter. It was shown in [1] that there exists a positive definite inner product ·, · q on (Cd )⊗n , uniquely determined by the requirement that for vectors in the natural basis (2.1), one has the formula ξi1 ⊗ · · · ⊗ ξin , ξ j1 ⊗ · · · ⊗ ξ jn q = q inv(σ ) δi1 ,σ ( j1 ) · · · δin ,σ ( jn ) . (2.2) σ
The sum on the right-hand side of Eq. (2.2) is taken over all permutations σ of {1, . . . , n}, and inv(σ ) denotes the number of inversions of σ , i.e. inv(σ ) := |{(i, j) | 1 ≤ i < j ≤ n, σ (i) > σ ( j)}| . Note that under this new inner product, the natural basis (2.1) will no longer be orthogonal if q = 0. (q) We will use Fn to denote the Hilbert space (Cd )⊗n endowed with this deformed (q) inner product. In addition, we will use the convention that F0 is the same as F0 , i.e. it is spanned by the same vacuum vector . The q-deformed Fock space over Cd is then the Hilbert space F (q) from Eq. (1.3). For q = 0, the construction of F (q) yields the usual non-deformed Fock space F from Eq. (1.1).
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2.1.2. The deformed creation and annihilation operators For every 1 ≤ j ≤ d, one has (q) deformed left creation operators L j ∈ B(F (q) ) and deformed right creation operators (q)
Rj
(q)
(q)
(q)
∈ B(F (q) ), which act on the natural basis of Fn by L j () = R j () = ξ j and
(q)
L j (ξi1 ⊗ · · · ⊗ ξin ) = ξ j ⊗ ξi1 ⊗ · · · ⊗ ξin ,
(2.3)
(q)
R j (ξi1 ⊗ · · · ⊗ ξin ) = ξi1 ⊗ · · · ⊗ ξin ⊗ ξ j . (q)
Their adjoints are the deformed left annihilation operators (L j )∗ and the deformed (q)
(q)
right annihilation operators (R j )∗ , which act on the natural basis of Fn
by
⎧ (q) ⎪ (L j )∗ (ξi1 ⊗ · · · ⊗ ξin ) ⎪ ⎪ ⎪ ⎪ n ⎪ ⎪ ⎪ ⎪ = q m−1 δ j,im ξi1 ⊗ · · · ⊗ ξ ⎪ i m ⊗ · · · ⊗ ξi n , ⎪ ⎨ m=1
(2.4)
(q) ⎪ ⎪ (R j )∗ (ξi1 ⊗ · · · ⊗ ξin ) ⎪ ⎪ ⎪ ⎪ n ⎪ ⎪ ⎪ ⎪ = q n−m δim , j ξi1 ⊗ · · · ⊗ ξ ⎪ i m ⊗ · · · ⊗ ξi n , ⎩ m=1
where the “hat” symbol over the component ξim means that it is deleted from the tensor (e.g. ξi1 ⊗ ξ i2 ⊗ ξi3 = ξi1 ⊗ ξi3 ). It’s clear from these formulas that the left creation (left annihilation) operators commute with the right creation (right annihilation) operators. For the commutator of a left annihilation operator and a right creation operator, a direct calculation (see also Lemma 3.1 from [10]) gives the formula (q)
(q)
[(L i )∗ , R j ] | F (q) = δi j q n IF (q) , ∀n ≥ 1. n n
(2.5)
Taking adjoints gives the formula for the commutator of a left creation operator and a right annihilation operator. When we are working on the non-deformed Fock space F corresponding to the case when q = 0, it will be convenient to suppress the superscripts and write L j and R j for the left and right creation operators respectively. Note that in this case, Eq. (2.3) and Eq. (2.4) imply that d j=1
L j L ∗j =
d
R j R ∗j = 1 − P0 ,
(2.6)
j=1
where P0 is the orthogonal projection onto F0 . (q)
(q)
(q)
2.1.3. The unitary conjugation operator. For every n ≥ 1, let Jn : Fn → Fn be (q) the operator which reverses the order of the components in a tensor in (Cd )⊗n , i.e, Jn acts by the formula in Eq. (1.5) of the Introduction. A consequence of Eq. (2.2), which (q) (q) defines the inner product ·, · q , is that Jn is a unitary operator in B(Fn ). Indeed,
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this is easily seen to follow from Eq. (2.2) and the following basic fact about inversions of permutations: if θ denotes the special permutation which reverses the order on {1, . . . , n}, then one has inv(θ τ θ ) = inv(τ ) for every permutation τ of {1, . . . , n}. Therefore, we can speak of the unitary operator J (q) ∈ B(F (q) ) from Definition 1.1, (q) (q) is an involution, i.e. (J (q) )2 = which is obtained as J (q) := ⊕∞ n=0 Jn . Note that J IF (q) , and that it intertwines the left and right creation operators, i.e. (q)
Rj
(q)
= J (q) L j J (q) , 1 ≤ j ≤ d.
(2.7)
2.2. The unitary U. In this subsection, we review the construction of the unitary U : F (q) → F from [5], which appears in Definition 1.2. An important role in the construction of this unitary is played by the positive operator M (q) :=
d
(q)
(q)
L j (L j )∗ ∈ B(F (q) ).
j=1 (q)
(q)
Clearly M (q) can be written as a direct sum M (q) = ⊕∞ n=0 Mn , where Mn is a positive (q) (q) operator on Fn , for every n ≥ 0. Using Eq. (2.3) and Eq. (2.4), one can show that Mn (q) acts on the natural basis of Fn by (q)
Mn (ξi1 ⊗ · · · ⊗ ξin ) =
n
q m−1 ξim ⊗ ξi1 ⊗ · · · ⊗ ξ i m ⊗ · · · ⊗ ξi n .
(2.8)
m=1
(Recall that the “hat” symbol over the component ξim means that it is deleted from the tensor.) (q) (q) With the exception of M0 (which is zero), the operators Mn are invertible. This is implied by Lemma 4.1 of [5], which also gives the estimate (q)
(Mn )−1 ≤ (1 − |q|)
∞
1 + |q|k < ∞, ∀n ≥ 1. 1 − |q|k
(2.9)
k=1
An important thing to note about Eq. (2.9) is that the upper bound on the right-hand side is independent of n. The unitary operator U is defined as a direct sum, U := ⊕∞ n=0 Un , where the unitaries (q) Un : Fn → Fn are defined recursively as follows: we first define U0 by U0 () = , and for every n ≥ 1 we define Un by (q)
Un := (I ⊗ Un−1 )(Mn )1/2 .
(2.10)
In Proposition 3.2 of [5] it was shown that Un as defined in Eq. (2.10) is actually a unitary operator, and hence that U is a unitary operator. Moreover, in Sect. 4 of [5] it was shown that C ⊆ U C (q) U ∗ for every q ∈ (−1, 1).
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2.3. Summable band-limited operators. Throughout this subsection, we fix a Hilbert space H, and in addition we fix an orthogonal direct sum decomposition of H as H = ⊕∞ n=0 Hn .
(2.11)
We will study certain properties an operator T ∈ B(H) can have with respect to this decomposition of H. We would like to emphasize that the concepts considered here depend not only on H, but also on the orthogonal decomposition for H in Equation (2.11). Definition 2.1. Let T be an operator in B(H). If there exists a non-negative integer b such that T (Hn ) ⊆ Hm , ∀n ≥ 0, (2.12) m≥0 |m−n|≤b
then we will say that T is band-limited. A number b as in Eq. (2.12) will be called a band limit for T . The set of all band-limited operators in B(H) will be denoted by B. Definition 2.2. Let T be an operator in B. We will say that T is summable when it has the property that ∞
T | Hn < ∞,
n=0
where we have used T | Hn ∈ B(Hn , H) to denote the restriction of T to Hn . The set of all summable band-limited operators in B(H) will be denoted by S. Proposition 2.3. With respect to the preceding definitions, (1) B is a unital ∗-subalgebra of B(H) and, (2) S is a two-sided ideal of B which is closed under taking adjoints. Proof. The proof of (1) is left as an easy exercise for the reader. To verify (2), we first show that S is closed under taking adjoints. Suppose T ∈ S, and let b be a band limit for T . By examining the matrix representations of T and of T ∗ with respect to the orthogonal decomposition (2.11), it is easily verified that T ∗ |Hn ≤ T |Hm , ∀n ≥ 0. m≥0 |m−n|≤b
This implies that ∞
T ∗ |Hn ≤ (2b + 1)
n=0
∞
T |Hm < ∞,
m=0
which gives T ∗ ∈ S. Next, we show that S is a two-sided ideal of B. Since S was proved to be self-adjoint, it will suffice to show that it is a left ideal. It is clear that S is closed under linear combinations. The fact that S is a left ideal now follows from the simple observation that for T ∈ B and S ∈ S we have ∞ ∞ T S | Hn ≤ T S | Hn < ∞, n=0
which implies T S ∈ S.
n=0
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121
In the following definition, we identify some special types of band-limited operators. Definition 2.4. Let T be an operator in B. (1) If T satisfies T (Hn ) ⊆ Hn for all n ≥ 0, then we will say that T is block-diagonal. (2) If there is k ≥ 0 such that T satisfies T (Hn ) ⊆ Hn+k for n ≥ 0, then we will say that T is k-raising. (3) If there is k ≥ 0 such that T satisfies T (Hn ) ⊆ Hn−k for n ≥ k and T (Hn ) = {0} for n < k, then we will say that T is k-lowering. Note that a block-diagonal operator is both 0-raising and 0-lowering. The following proposition gives a Fourier-type decomposition for band-limited operators. As pointed out to us by the referee, it may be useful for intuition to think of this decomposition in terms of the group of block-diagonal unitaries {Dz | z ∈ T} on H, where for every z ∈ T (i.e. z ∈ C with |z| = 1) the unitary Dz sends ξ → z n ξ for all n ≥ 0 and ξ ∈ Hn . It is easily seen that for a decomposition like the one discussed in Eq. (2.13) below, one then has Dz∗ T Dz =
b
z −k X k +
k=0
b
z k Yk , ∀ z ∈ T.
k=1
This the components X k , Yk to be retrieved via integrals on T, e.g. X k = k allows ∗ T D dz. (This observation could have been used for an alternative proof of z D z z T Proposition 2.5. But it is also very easy, as shown below, to prove this proposition directly from the definitions.) Proposition 2.5. Let T be an operator in B with a band-limit b ≥ 0, as in Definition 2.1. Then we can decompose T as T =
b k=0
Xk +
b
Yk ,
(2.13)
k=1
where each X k is a k-raising operator for 0 ≤ k ≤ b, and each Yk is a k-lowering operator for 1 ≤ k ≤ b. This decomposition is unique. Moreover, if T is summable in the sense of Definition 2.2, then each of the X k and Yk are summable. Proof. First, fix an integer k satisfying 0 ≤ k ≤ b. For each n ≥ 0, consider the linear operator Pn+k T |Hn ∈ B(Hn , Hn+k ) which results from composing the orthogonal projection Pn+k onto Hn+k with the restriction T |Hn . Clearly Pn+k T |Hn ≤ T . This allows us to define an operator X k ∈ B(H) which acts on Hn by X k ξ = Pn+k T ξ, ∀ξ ∈ Hn .
(2.14)
It follows from this definition that X k is a k-raising operator. Similarly, for an integer k satisfying 1 ≤ k ≤ b, we can define a k-lowering operator Yk ∈ B(H) which acts on ξ ∈Hn by Pn−k T ξ if k ≤ n, Yk ξ = (2.15) 0 if k > n. It’s clear that Eq. (2.13) holds with each X k and Yk defined as above. Conversely, if Eq. (2.13) holds, then it’s clear that each X k and Yk is completely determined
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as in Eq. (2.14) and Eq. (2.15) respectively. This implies the uniqueness of this decomposition. Finally, suppose T is summable. The fact that each X k and Yk is summable then follows from the observation that Eq. (2.14) and Eq. (2.15) imply X k |Hn ≤ T |Hn and Yk |Hn ≤ T |Hn for every n ≥ 0. The following result about commutators will be needed in Sect. 4. Proposition 2.6. Let T ∈ B be a positive block-diagonal operator, and let V ∈ B be a 1-raising operator. Suppose that the commutator [T, V ] satisfies ∞
[T, V ] | Hn 1/2 < ∞.
(2.16)
n=0
Then the commutator [T 1/2 , V ] is a summable 1-raising operator. Proof. For every n ≥ 0, let Tn = T |Hn ∈ B(Hn ) and let Vn = V |Hn ∈ B(Hn , Hn+1 ). Since T is block-diagonal and V is 1-raising, it’s clear that [T, V ] and [T 1/2 , V ] are 1-raising operators which satisfy [T, V ] | Hn = Tn+1 Vn − Vn Tn , ∀n ≥ 0, and 1/2
1/2
[T 1/2 , V ] | Hn = Tn+1 Vn − Vn Tn , ∀n ≥ 0. It follows that the hypothesis (2.16) can be rewritten as ∞
Tn+1 Vn − Vn Tn 1/2 < ∞,
n=0
while the required conclusion that [T 1/2 , V ] ∈ S is equivalent to ∞
1/2
1/2
Tn+1 Vn − Vn Tn < ∞.
n=0
We will prove that this holds by showing that for every n ≥ 0, 1/2
1/2
Tn+1 Vn − Vn Tn ≤
5 V 1/2 Tn+1 Vn − Vn Tn 1/2 . 4
(2.17)
For the rest of the proof, fix n ≥ 0. Consider the operators A, B ∈ B(Hn ⊕ Hn+1 ) which, written as 2 × 2 matrices, are given by
0 Vn∗ 0 Tn . , B := A := 0 Tn+1 Vn 0 Since T is positive, it follows that A is positive, with 1/2 Tn 0 1/2 A = 1/2 . 0 Tn+1
Exactness of the q-Commutation Relations
123
A well-known commutator inequality (see e.g. [8]) gives [A1/2 , B] ≤
5 B1/2 [A, B]1/2 . 4
(2.18)
From the definitions of A and B, we compute
0 (Tn+1 Vn − Vn Tn )∗ , [A, B] = Tn+1 Vn − Vn Tn 0 1/2
and this implies [A, B] = Tn+1 Vn − Vn Tn . Similarly, [A1/2 , B] = Tn+1 Vn − 1/2 Vn Tn , and it’s clear that B = Vn . By substituting these equalities into (2.18) we obtain 1/2
1/2
Tn+1 Vn − Vn Tn ≤
5 Vn 1/2 Tn+1 Vn − Vn Tn 1/2 . 4
Since Vn ≤ V , this clearly implies that (2.17) holds.
3. An Inclusion Criterion In this section, we work exclusively in the framework of the (non-deformed) extended Cuntz algebra C. We will use the terminology of Subsect. 2.3 with respect to the natural decomposition F = ⊕∞ n=0 Fn . In particular, we will refer to the unital ∗-subalgebra B ⊆ B(F) which consists of band-limited operators as in Definition 2.1, and to the ideal S of B which consists of summable band-limited operators as in Definition 2.2. The main result of this section is Theorem 3.8. This is an analogue in the C ∗ -framework of the bicommutant theorem from von Neumann algebra theory, where we restrict our attention to the ∗-algebra B and consider commutators modulo the ideal S. In this framework, the role of “commutant” is played by the C ∗ -algebra generated by right creation operators on F. For clarity, we will first consider the special case of a block-diagonal operator. Definition 3.1. Let T ∈ B be a block-diagonal operator. The sequence of C-approximants for T is the sequence (An )∞ n=0 of block-diagonal elements of C defined recursively as follows: we first define A0 by A0 = T (), IF , and for every n ≥ 0 we define An+1 by ∗ ci1 ,...,in+1 ; j1 ,..., jn+1 L i1 · · · L in+1 L j1 · · · L jn+1 , (3.1) An+1 := An + 1≤i 1 ,...,i n+1 ≤d 1≤ j1 ,..., jn+1 ≤d
where the coefficients ci1 ,...,in+1 ; j1 ,..., jn+1 are defined by ci1 ,...,in+1 ; j1 ,..., jn+1 := T (ξ j1 ⊗ · · · ⊗ ξ jn+1 ), ξi1 ⊗ · · · ⊗ ξin+1 −δin+1 , jn+1 · T (ξ j1 ⊗ · · · ⊗ ξ jn ), ξi1 ⊗ · · · ⊗ ξin .
(3.2)
The main property of the approximant An is that it agrees with the operator T on each subspace Fm for m ≤ n. More precisely, we have the following lemma.
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Lemma 3.2. Let T ∈ B be a block-diagonal operator, and let (An )∞ n=0 be the sequence of C-approximants for T , as in Definition 3.1. Then for every m ≥ 0, if m ≤ n, T | Fm A n | Fm = (3.3) (T | Fn ) ⊗ Im−n if m > n. Proof. We will show that for every fixed n ≥ 0, Eq. (3.3) holds for all m ≥ 0. The proof of this statment will proceed by induction on n. The base case n = 0 is left as an easy exercise for the reader. The remainder of the proof is devoted to the induction step. Fix n ≥ 0 and assume that Eq. (3.3) holds for this n and for all m ≥ 0. We will prove the analogous statement for n + 1. From Eq. (3.1), it is immediate that An+1 |Fm = An |Fm = T |Fm , ∀m ≤ n. Thus it remains to fix m ≥ n + 1 and verify that An+1 | Fm = (T | Fn+1 ) ⊗ Im−n−1 ∈ B(Fm ). In light of how (T | Fn+1 ) ⊗ Im−n−1 acts on the canonical basis of Fm , this amounts to showing that for every 1 ≤ k1 , . . . , km , 1 , . . . , m ≤ d, one has An+1 (ξ 1 ⊗ · · · ⊗ ξ m ), ξk1 ⊗ · · · ⊗ ξkm = δkn+2 , n+2 · · · δkm , m T (ξ 1 ⊗ · · · ⊗ ξ n+1 ), ξk1 ⊗ · · · ⊗ ξkn+1 .
(3.4)
On the left-hand side of Eq. (3.4) we substitute for An+1 using the recursive definition given by Eq. (3.1). This gives An+1 (ξ 1 ⊗ · · · ⊗ ξ m ), ξk1 ⊗ · · · ⊗ ξkm = An (ξ 1 ⊗ · · · ⊗ ξ m ), ξk1 ⊗ · · · ⊗ ξkm + ci1 ,...,in+1 ; j1 ,..., jn+1 α(i 1 , . . . , i n+1 ; j1 , . . . , jn+1 ),
(3.5)
i 1 ,...,i n+1 j1 ,..., jn+1
where for every 1 ≤ i 1 , . . . i n+1 , j1 , . . . , jn+1 ≤ d, we have written α(i 1 , . . . , i n+1 ; j1 , . . . , jn+1 ) ∗ = L i1 · · · L in+1 L j1 · · · L jn+1 (ξ 1 ⊗ · · · ⊗ ξ m ), (ξk1 ⊗ · · · ⊗ ξkm ) . It is clear that an inner product like the one just written simplifies as follows: ∗ L i1 · · · L in+1 L j1 · · · L jn+1 (ξ 1 ⊗ · · · ⊗ ξ m ), (ξk1 ⊗ · · · ⊗ ξkm ) ∗ = L j1 · · · L jn+1 (ξ 1 ⊗ · · · ⊗ ξ m ), (L i1 · · · L in+1 )∗ (ξk1 ⊗ · · · ⊗ ξkm ) = δi1 ,k1 · · · δin+1 ,kn+1 δ j1 , 1 · · · δ jn+1 , n+1 ξ n+2 ⊗ · · · ⊗ ξ m , ξkn+2 ⊗ · · · ⊗ ξkm = δi1 ,k1 · · · δin+1 ,kn+1 δ j1 , 1 · · · δ jn+1 , n+1 δ n+2 ,kn+2 · · · δ m ,km . Thus in the sum on the right-hand side of Eq. (3.5), the only term that survives is the one corresponding to i 1 = k1 , . . . , i n+1 = kn+1 and j1 = 1 , . . . , jn+1 = n+1 , and we obtain that An+1 (ξ 1 ⊗ · · · ⊗ ξ m ), ξk1 ⊗ · · · ⊗ ξkm = An (ξ 1 ⊗ · · · ⊗ ξ m ), ξk1 ⊗ · · · ⊗ ξkm + δ n+2 ,kn+2 · · · δ m ,km ck1 ,...,kn+1 ; 1 ,..., n+1 .
(3.6)
Exactness of the q-Commutation Relations
125
Finally, we remember our induction hypothesis, which gives An (ξ 1 ⊗ · · · ⊗ ξ m ), ξk1 ⊗ · · · ⊗ ξkm = δkn+1 , n+1 · · · δkm , m T (ξ 1 ⊗ · · · ⊗ ξ n ), ξk1 ⊗ · · · ⊗ ξkn .
(3.7)
A straightforward calculation shows that if we substitute Eq. (3.7) into Eq. (3.6) and use Formula (3.2) which defines the coefficient ck1 ,...,kn+1 ; 1 ,..., n+1 , then we arrive at the right-hand side of Eq. (3.4). This completes the induction argument. Lemma 3.3. Let T ∈ B be a block-diagonal operator, and let (An )∞ n=1 be the sequence of C-approximants for T , as in Definition 3.1. Then for every n ≥ 1 one has the equation An+1 − An = T | Fn+1 − (T | Fn ) ⊗ I
(3.8)
(where the norm on the right-hand side is calculated in B(Fn+1 ), and the operator (T | Fn ) ⊗ I sends ξi1 ⊗· · ·⊗ξin ⊗ξin+1 to T (ξi1 ⊗· · ·⊗ξin )⊗ξin+1 for 1 ≤ i 1 , . . . , i n+1 ≤ d). Proof. Note that since An+1 − An is block-diagonal, An+1 − An = sup An+1 |Fm −An |Fm . m≥0
To compute this supremum, there are three cases to consider. In each case we apply Lemma 3.2. First, for m ≤ n, An+1 |Fm −An |Fm = 0. Next, for m = n + 1, An+1 |Fn+1 −An |Fn+1 = T |Fn+1 −(T |Fn ) ⊗ I . Finally, for m > n + 1, An+1 |Fm −An |Fm = (T |Fn+1 ) ⊗ Im−n−1 − (T |Fn ) ⊗ Im−n = (T |Fn+1 −(T |Fn ) ⊗ I ) ⊗ Im−n−1 = T |Fn+1 −(T |Fn ) ⊗ I . This makes it clear that the supremum over all m ≥ 0 is equal to the right-hand side of Eq. (3.8), as required. Lemma 3.4. Let T be a block-diagonal operator. If T satisfies ∞
(T |Fn+1 ) − (T |Fn ) ⊗ I < ∞,
n=1
then T ∈ C. Proof. Let (An )∞ n=1 be the sequence of C-approximants for T , as in Definition 3.1. In view of Lemma 3.3, the hypothesis of the present lemma implies that the sum ∞ A − A is finite. This in turn implies that the sequence (An )∞ n+1 n n=1 n=1 converges in norm to an operator A. Since each An belongs to C, it follows that A belongs to C. But we must have A = T , as Lemma 3.2 implies that A |Fm = lim An |Fm = T |Fm , ∀ m ≥ 0. n→∞
Hence T ∈ C, as required.
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Proposition 3.5. Let T be a block-diagonal operator. If the block-diagonal operator d Ri T Ri∗ belongs to the ideal S, then T ∈ C. T − i=1 Proof. The hypothesis is equivalent to ∞
(T −
d
i=1 Ri T
Ri∗ ) |Fn < ∞.
(3.9)
n=1
It’s easy to verify that for n ≥ 1, d Ri T Ri∗ |Fn = (T |Fn−1 ) ⊗ I, i=1
which gives (T −
d
Ri T Ri∗ ) |Fn = T |Fn −(T |Fn−1 ) ⊗ I .
i=1
Therefore, (3.9) implies that the hypothesis of Lemma 3.4 holds, and the result follows by applying said lemma. Corollary 3.6. Let T ∈ B be a block-diagonal operator such that [T, R ∗j ] ∈ S for 1 ≤ j ≤ d. Then T ∈ C. d Proof. By Proposition 3.5, it suffices to show that T − i=1 Ri T Ri∗ ∈ S. We can write T−
d i=1
Ri T Ri∗ = (P0 +
d
Ri Ri∗ )T −
i=1
= P0 T −
d
d
Ri T Ri∗
i=1
Ri [T, Ri∗ ],
i=1
where P0 is the orthogonal projection onto F0 , and where we have used Eq. (2.6). Since P0 and [T, Ri∗ ] belong to S, and since T and Ri belong to B, the result follows from the fact that S is a two-sided ideal of B. We now apply the above results on block-diagonal operators in order to bootstrap the case of general band-limited operators. It is convenient to first consider the case of k-raising/lowering operators, which were introduced in Definition 2.4. Proposition 3.7. Let T ∈ B be a k-raising or k-lowering operator for some k ≥ 0. If T satisfies [T, R ∗j ] ∈ S for 1 ≤ j ≤ d, then T ∈ C. Proof. First, suppose that T is k-raising. Let T = T (L ∗1 )k , so that T is block diagonal. The fact that the left and right annihilation operators commute implies that [T , R ∗j ] = [T (L ∗1 )k , R ∗j ] = [T, R ∗j ](L ∗1 )k , 1 ≤ j ≤ d. Since [T, R ∗j ] ∈ S by hypothesis, and since S is a two-sided ideal of B, it follows that [T , R ∗j ] ∈ S. Thus Corollary 3.6 gives T ∈ C, so T = T (L 1 )k ∈ C. Now suppose that T is k-lowering with k ≥ 1 (the case k = 0 is covered by the preceding paragraph). Let T = L k1 T , so that T is block diagonal. An immediate calculation shows that for every 1 ≤ j ≤ d we have
Exactness of the q-Commutation Relations
127
[T , R ∗j ] = L k1 [T, R ∗j ] + [L k1 , R ∗j ]T = L k1 [T, R ∗j ] − δ1, j L k−1 1 P0 T, where P0 is the orthogonal projection onto F0 . Since [T, R ∗j ] and P0 belong to S while L 1 , T ∈ B, we can invoke once again the fact that S is a two-sided ideal of B, and infer that [T , R ∗j ] ∈ S. Thus Corollary 3.6 gives T ∈ C, and we conclude that T = (L ∗1 )k T ∈ C. Theorem 3.8. Let T ∈ B be an operator such that either [T, R ∗j ] ∈ S for 1 ≤ j ≤ d, or [T, R j ] ∈ S for 1 ≤ j ≤ d. Then T ∈ C. Proof. First, suppose that T satisfies [T, R ∗j ] ∈ S for every 1 ≤ j ≤ d. Let b ≥ 0 be a band-limit for T . By Proposition 2.5, we can decompose T as T =
b
Xk +
k=0
b
Yk ,
k=1
where each X k is a k-raising operator, and each Yk is a k-lowering operator. We will prove that each X k ∈ C and each Yk ∈ C. Fix for the moment 1 ≤ j ≤ d. We have [T, R ∗j ] =
b b [X k , R ∗j ] + [Yk , R ∗j ] k=0
=
b+1 k=0
where X k =
and Yk
X k +
Yk ,
(3.10)
k=0
[X k+1 , R ∗j ] if 0 ≤ k ≤ b − 1, 0 if k = b or k = b + 1,
=
k=1 b+1
if k = 1, [X 0 , R ∗j ] [Yk−1 , R ∗j ] if 2 ≤ k ≤ b + 1.
It is clear that each X k is a k-raising operator, and that each Yk is a k-lowering operator. Hence Eq. (3.10) provides the (unique) Fourier-type decomposition for [T, R ∗j ], as in Proposition 2.5. Since it is given that [T, R ∗j ] ∈ S, Proposition 2.5 implies that each X k ∈ S and each Yk ∈ S. This in turn implies that [X k , R ∗j ] ∈ S for every 0 ≤ k ≤ b, and that [Yk , R ∗j ] ∈ S for every 1 ≤ k ≤ b. Now let us unfix the index j from the preceding paragraph. For every 0 ≤ k ≤ b, we have proved that [X k , R ∗j ] ∈ S for all 1 ≤ j ≤ d, hence Proposition 3.7 implies that X k ∈ C. The fact that Yk ∈ C for every 1 ≤ k ≤ b is obtained in the same way. This concludes the proof in the case when the hypothesis on T is that [T, R ∗j ] ∈ S for all 1 ≤ j ≤ d. If T satisfies [T, R j ] ∈ S for all 1 ≤ j ≤ d, then since the ideal S is closed under taking adjoints, it follows that [T ∗ , R ∗j ] ∈ S for all 1 ≤ j ≤ d. The above arguments therefore apply to T ∗ , and lead to the conclusion that T ∗ ∈ C, which gives T ∈ C.
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4. Construction of the Embedding In this section we fix a deformation parameter q ∈ (−1, 1) and consider the C ∗ -algebra (q) (q) C (q) = C ∗ (L 1 , . . . , L d ) ⊆ B(F (q) ) from Eq. (1.4). The main result of this section (and also this paper), Theorem 1.3, shows that it is possible to unitarily embed C (q) into the C ∗ -algebra C = C∗ (L 1 , . . . , L d ) ⊆ B(F) from Eq. (1.2). We will once again utilize the terminology of Subsect. 2.3 with respect to the natural decomposition F = ⊕∞ n=0 Fn . In particular, we will refer to the unital ∗-algebra B ⊆ B(F) consisting of band-limited operators, and to the ideal S of B consisting of summable band-limited operators. (q) The deformed Fock space F (q) also has a natural decomposition F (q) = ⊕∞ n=0 Fn , and we will also need to utilize the terminology of Subsect. 2.3 with respect to this decomposition. We will let B (q) ⊆ B(F (q) ) denote the unital ∗-algebra consisting of band-limited operators, and we will let S (q) denote the ideal of B (q) which consists of summable band-limited operators. (q)
Remark 4.1. Recall the positive block-diagonal operator M (q) = ⊕∞ ∈ B (q) , n=0 Mn (q) which was reviewed in Subsect. 2.2. It was recorded there that for n ≥ 1, Mn is an (q) invertible operator on Fn . Moreover, for every n ≥ 1, one has the upper bound (2.9) (q) −1 for the norm (Mn ) , and this upper bound is independent of n. Therefore, the only obstruction to the operator M (q) being invertible on F (q) is the (q) fact that M0 = 0. We can overcome this obstruction by working instead with the (q) defined by operator M (q) := P (q) + M (q) , M 0
(4.1)
(q)
(q)
where P0 ∈ B(F (q) ) is the orthogonal projection onto the subspace F0 . It’s clear (q) )−1 . (q) is invertible, and that the bound from (2.9) applies to ( M that M (q) satisfies [( M (q) )−1/2 , R (q) ] ∈ S (q) for all 1 ≤ j ≤ d. Lemma 4.2. The operator M j (q) and R (q) satisfy the hypotheses of Proposition 2.6. Proof. First, we will show that M (q) It’s clear that M is block-diagonal and that R (q) is 1-raising, but it will require a bit of work to check that ∞ n=0
(q)
(q) , R ] | (q) 1/2 < ∞, ∀1 ≤ j ≤ d. [ M j Fn
(4.2)
(q) , In order to show that (4.2) holds, fix 1 ≤ j ≤ d. Using Eq. (4.1), which defines M we can write (q)
(q)
(q) , R ] = [P , R (q) ] + [M 0 j
d (q) (q) (q) [L i (L i )∗ , R j ] i=1
(q)
= [P0 , R (q) ] +
d i=1
(q)
(q)
(q)
L i [(L i )∗ , R j ],
Exactness of the q-Commutation Relations
129 (q)
(q)
where the last equality follows from the fact that L i and R j commute. The sum in this equation has only a single non-zero term. Indeed, as a consequence of Eq. (2.5), we (q) (q) have [(L i )∗ , R j ] = 0 whenever i = j. Thus we arrive at the following formula: (q)
(q)
(q)
(q)
(q)
(q) , R ] = [P , R (q) ] + L [(L )∗ , R ]. [M 0 j j j j
(4.3)
(q)
We next restrict the operators on both sides of (4.3) to a subspace Fn , for n ≥ 1. Noting (q) (q) (q) (q) (q) that [P0 , R j ] = −R j P0 vanishes on Fn , we obtain that (q)
(q)
(q)
(q)
(q) , R ] | (q) = L [(L )∗ , R ] | (q) , ∀n ≥ 1. [M j j j j Fn Fn
(4.4)
Finally, we take norms in Eq. (4.4) and invoke Eq. (2.5) once more to obtain that (q)
(q)
(q) , R ] | (q) ≤ |q|n L , ∀n ≥ 1. [ M j j Fn n/2 < ∞. The conclusion that (4.2) holds follows from here, since ∞ n=1 |q| (q) (q) and R , and conclude that Therefore, we can apply Proposition 2.6 to M j (q) (q) 1/2 (q) (q) −1/2 is bounded and block[( M ) , R ] ∈ S . Note that the operator ( M ) j
diagonal, meaning in particular that it belongs to the ∗-algebra B (q) . The desired result now follows from the obvious identity (q)
(q)
(q) )−1/2 [( M (q) )1/2 , R ]( M (q) )−1/2 , (q) )−1/2 , R ] = −( M [( M j j and the fact that S (q) is a two-sided ideal of B (q) .
Lemma 4.3. For 1 ≤ j ≤ d, the unitary U = ⊕∞ n=0 Un from Subsect. 2.2 satisfies (q)
(q)
∗ Un−1 L ∗j Un = (L j )∗ (Mn )−1/2 , ∀n ≥ 1.
(4.5)
(Note that on the left-hand side of Eq. (4.5), we view L ∗j as an operator in B(Fn , Fn−1 ). (q)
(q)
(q)
On the right-hand side of Eq. (4.5), we view (L j )∗ as an operator in B(Fn , Fn−1 ).) (q)
Proof. Consider the operator A j (q)
Fn
(q)
: Fn
(q)
→ Fn−1 which acts on the natural basis of
by (q)
A j (ξi1 ⊗ · · · ⊗ ξin ) = δ j,i1 ξi2 ⊗ · · · ⊗ ξin , ∀1 ≤ i 1 , . . . , i n ≤ d. (q)
We claim that A j satisfies (q)
(q)
(q)
A j = (L j )∗ (Mn )−1 . To see this, note that for 1 ≤ i 1 , . . . , i n ≤ d, (q)
(q)
(q)
A j Mn (ξi1 ⊗ · · · ⊗ ξin ) = A j
n
q m−1 ξim ⊗ ξi1 ⊗ · · · ⊗ ξ i m ⊗ · · · ⊗ ξi n
m=1
= =
n−1
q m−1 δ j,im ξi1 ⊗ · · · ⊗ ξ i m ⊗ · · · ⊗ ξi n
m=1 (q) (L j )∗ (ξi1
⊗ · · · ⊗ ξin ),
(4.6)
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M. Kennedy, A. Nica
where the first and last equalities follow from Eq. (2.8) and Eq. (2.4) respectively. Hence (q) (q) (q) (q) A j Mn = (L j )∗ |F (q) , so multiplying on the right by (Mn )−1 establishes the claim. n Now, from Eq. (2.10), which defines Un , we see that (q)
∗ ∗ Un−1 L ∗j Un = Un−1 L ∗j (I ⊗ Un−1 )(Mn )1/2 , (q)
and from the definition of A j it’s immediate that (q)
L ∗j (I ⊗ Un−1 ) = Un−1 A j . Together, this allows us to write (q)
(q)
∗ ∗ Un−1 L ∗j Un = Un−1 Un−1 A j (Mn )1/2 (q)
(q)
= A j (Mn )1/2 . Applying Eq. (4.6) now gives Eq. (4.5), as required.
Proposition 4.4. For 1 ≤ i, j ≤ d, the unitary U from Subsect. 2.2 satisfies (q) [U ∗ L ∗j U, Ri ] ∈ S (q) . (q)
Proof. Fix i and j and denote C := [U ∗ L ∗j U, Ri ]. It’s clear that C is a block-diagonal operator on F (q) . In order to show that C ∈ S (q) , we will need to estimate the norm of its diagonal blocks. For n ≥ 1, Lemma 4.3 gives (q)
C |F (q) = Un∗ L ∗j Un+1 Ri n (q)
(q)
∗ − Ri Un−1 L ∗j Un
(q)
(q)
= (L j )∗ (Mn+1 )−1/2 Ri (q)
(q)
(q)
= (L j )∗ ((Mn+1 )−1/2 Ri (q)
(q)
+((L j )∗ Ri
(q)
(q)
(q)
(q)
− Ri (L j )∗ (Mn )−1/2 (q)
(q)
− Ri (Mn )−1/2 ) (q)
(q)
− Ri (L j )∗ )(Mn )−1/2 .
Since C is block-diagonal, this gives (q) (q) )−1/2 , R (q) ] + [(L (q) )∗ , R (q) ]( M (q) )−1/2 . C = (L j )∗ [( M i j i (q)
(q) )−1/2 , R ] ∈ S (q) by Lemma 4.2. By Eq. (2.5), Now, [( M i (q)
(q)
[(L j )∗ , Ri ] |F (q) = δi j q n IF (q) , n n (q)
(q)
and since the operator [(L j )∗ , Ri ] is block-diagonal, this implies that it also belongs (q) (q) )−1/2 both belong to B (q) , and since S (q) is a two-sided to S (q) . Since (L )∗ and ( M j
ideal of B (q) , it follows that C ∈ S (q) .
We are now able to complete the proof of the embedding theorem.
Exactness of the q-Commutation Relations
131 (q)
∗ ∈ C, for 1 ≤ i ≤ d. Since Proof of Theorem 1.3. It suffices to show that Uopp L i Uopp (q)
∗ belongs to the algebra B of all band-limited operators, by Theorem 3.8 it Uopp L i Uopp will actually be sufficient to verify that (q)
∗ , R ∗j ] ∈ S, ∀1 ≤ i, j ≤ d. [Uopp L i Uopp
By Definition 1.1, we can write (q)
(q)
∗ = J U J (q) L i J (q) U ∗ J Uopp L i Uopp (q)
= J U Ri U ∗ J, where the last equality follows from Eq. (2.7). This gives (q)
(q)
∗ , R ∗j ] = [J U Ri U ∗ J, R ∗j ] [Uopp L i Uopp (q)
= J U [Ri , U ∗ J R ∗j J U ]U ∗ J (q)
= J U [Ri , U ∗ L ∗j U ](J U )∗ , (q)
and we know from Proposition 4.4 that [Ri , U ∗ L ∗j U ] ∈ S (q) . It is clear that conjugation by the unitary J U takes S (q) onto S, so this gives the desired result. The proof that C (q) is exact now follows from some simple observations about nuclear and exact C∗ -algebras (see e.g. [3]). Proof of Corollary 1.4. The extended Cuntz algebra C is (isomorphic to) an extension of the Cuntz algebra. Since the Cuntz algebra is nuclear, this implies that C is nuclear, and in particular that C is exact. Since exactness is inherited by subalgebras (see e.g. ∗ is exact, and hence that Chap. 2 of [3]), it follows from Theorem 1.3 that Uopp C (q) Uopp (q) C is exact. Remark 4.5. Since Theorem 1.3 holds for all q ∈ (−1, 1), a natural thought is that the methods used above could also be applied to establish the inclusion U C (q) U ∗ ⊆ C for all q ∈ (−1, 1), and hence (since the opposite inclusion was shown in [5]) that U C (q) U ∗ = C. To do this, it would be necessary to establish that [U L (q) U ∗ , R ∗j ] ∈ S, ∀1 ≤ i, j ≤ d.
(4.7)
This condition looks superficially similar to the condition from Proposition 4.4, but this is deceptive. We believe that establishing (4.7) will require a deeper understanding of the combinatorics which underlie the q-commutation relations. 5. An Application to the q-Gaussian von Neumann Algebras The q-Gaussian von Neumann algebra M(q) is the von Neumann algebra generated by (q) (q) {L i + (L i )∗ | 1 ≤ i ≤ d}. This algebra can be considered as a type of deformation of L(Fd ), the von Neumann algebra of the free group on d generators. Indeed, for q = 0, a basic result in free probability states that M(q) is precisely the realization of L(Fd ) as the von Neumann algebra generated by a free semicircular family (see e.g. Sect. 2.6 of [13] for the details). For general q ∈ (−1, 1) it is known that M(q) is a von Neumann algebra in standard form, with being a cyclic and separating trace-vector. The commutant of M(q) is the (q) (q) von Neumann algebra generated by {Ri + (Ri )∗ | 1 ≤ i ≤ d} (see Sect. 2 of [2]).
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Not much is known about the isomorphism class of the algebras M(q) for q = 0. The major open problem is to determine the extent to which they behave like L(Fd ). The best results to date show that M(q) does share certain properties with L(Fd ). Nou showed in [7] that M(q) is non-injective, Sniady showed in [12] that it is non-Gamma, and Ricard showed in [9] that it is a I I1 -factor. Shlyakhtenko showed in [10] that if we assume |q| < 0.44, then the results in [6] and [5] can be used to obtain that M(q) is solid in the sense of Ozawa. In addition, Shlyakhtenko showed in [11] that for small q, M(q) has no Cartan subalgebra. Based on the results in Sect. 4, we show here that M(q) is weakly exact. For more details on weak exactness, we refer the reader to Chapter 14 of [3]. Theorem 5.1. For every q in the interval (−1, 1), the q-Gaussian von Neumann algebra M(q) is weakly exact. Proof. It is known that a von Neumann algebra is weakly exact if it contains a weakly dense C ∗ -algebra which is exact (see e.g. Theorem 14.1.2 of [3]). Consider the unital (q) (q) C ∗ -algebra A(q) generated by {L i +(L i )∗ | 1 ≤ i ≤ d}. It is clear that A(q) is weakly (q) dense in M , while on the other hand, we have A(q) ⊆ C (q) . Therefore, the exactness of A(q) follows from Corollary 1.4, combined with the fact that exactness is inherited by subalgebras. Acknowledgement. The authors are grateful to the referee, whose suggestions led to an improved exposition of this work.
References 1. Bozejko, M., Speicher, R.: An example of a generalized Brownian motion. Commun. Math. Phys. 137(3), 519–531 (1991) 2. Bozejko, M., Kummerer, B., Speicher, R.: q-Gaussian processes: Non-commutative and classical aspects. Commun. Math. Phys. 185(1), 129–154 (1997) 3. Brown N., Ozawa N.: C∗ -algebras and finite dimensional approximations. Graduate Studies in Mathematics, Vol. 88. Providence, RI: Amer. Math. Soc., 2008 4. Davidson, K.: On operators commuting with Toeplitz operators modulo the compact operators. J. Funct. Anal. 24(3), 291–302 (1977) 5. Dykema, K., Nica, A.: On the Fock representation of the q-commutation relations. J. Fur Reine Und Ang. Math. 440, 201–212 (1993) 6. Jorgensen, P.E.T., Schmitt, L.M., Werner, R.F.: q-canonical commutation relations and stability of the Cuntz algebra. Pacific J. Math. 165(1), 131–151 (1994) 7. Nou, A.: Non-injectivity of the q-deformed von Neumann algebras. Math. Ann. 330(1), 17–38 (2004) 8. Pedersen, G.K.: A commutator inequality. In: Operator Algebras, Mathematical Physics and LowDimensional Topology (Istanbul 1991), Research Notes in Mathematics 5, Wellesley, MA: AK Peters, 1993, pp. 233–235 9. Ricard, E.: Factoriality of q-gaussian von Neumann algebras. Commun. Math. Phys. 257(3), 659–665 (2005) 10. Shlyakhtenko, D.: Some estimates for non-microstates free dimension, with applications to q-semicircular families. Int. Math. Res. Not. 51, 2757–2772 (2004) 11. Shlyakhtenko, D.: Lower estimates on microstates free entropy dimension. Anal. PDE 2(2), 119–146 (2009) 12. Sniady, P.: Factoriality of Bozejko-Speicher von Neumann algebras. Commun. Math. Phys. 246(3), 561– 567 (2004) 13. Voiculescu D., Dykema K., Nica A.: Free random variables. CRM Monograph Series, Vol. 1. Providence, RI: Amer. Math. Soc. 1992 Communicated by Y. Kawahigashi
Commun. Math. Phys. 308, 133–146 (2011) Digital Object Identifier (DOI) 10.1007/s00220-011-1321-y
Communications in
Mathematical Physics
Invariant Algebraic Surfaces for Generalized Raychaudhuri Equations Claudia Valls Departamento de Matemática, Instituto Superior Técnico, Universidade Técnica de Lisboa, Av. Rovisco Pais, 1049-001, Lisboa, Portugal. E-mail: [email protected] Received: 8 October 2010 / Accepted: 5 April 2011 Published online: 25 August 2011 – © Springer-Verlag 2011
Abstract: We consider a generalized Raychaudhuri equation, 1 x˙ = − x 2 − αx − 2(y 2 + z 2 − w 2 ) − 2β, 2 y˙ = −(α + x)y − γ , z˙ = −(α + x)z − δ, w˙ = −(α + x)w, where α, β, γ , δ are real parameters. This model has appeared in modern string cosmology. We study the algebraic invariants of this model for all values of the parameters α, β, γ , δ ∈ R. We prove that when γ = δ = 0 the system is integrable and for any other values of the parameters γ , δ, α, β we characterize all the invariant surfaces of this system. In particular we characterize all the polynomial and proper rational first integrals. 1. Introduction In this work we consider the Raychaudhuri equation for a two dimensional curved surface of constant curvature in the following form (see [2,3] and the references therein): 1 x˙ = − x 2 − αx − 2(y 2 + z 2 − w 2 ) − 2β, 2 y˙ = −(α + x)y − γ , z˙ = −(α + x)z − δ, w˙ = −(α + x)w,
(1)
where α, β, γ , δ are real parameters. In general relativity, the Raychaudhuri equation is a fundamental result describing the motion of nearby bits of matter. It is quite relevant
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since it is used as a fundamental lemma for the Penrose-Hawking singularity theorems (see [4] for details) and for the study of exact solutions in general relativity. However, it has an independent interest since it offers a simple and general validation of our intuitive expectation that gravitation should be a universal attractive force between any two bits of mass-energy in general relativity, as it is in Newton’s theory of gravitation. Here we further contribute to the understanding of the complexity, or more precisely of the topological structure of the dynamics of system (1) by studying its integrability. For a four dimensional system of differential equations the existence of one or two first integrals reduces the complexity of its dynamics and the existence of three first integrals that are functionally independent solves completely the problem (at least theoretically) of determining its phase portraits. In general for a given differential system it is a difficult problem to determine the existence or non–existence of first integrals. During recent years the interest in the study of integrability of differential equations has attracted much attention from the mathematical community. Darboux theory of integrability plays a central role in the integrability of the polynomial differential systems since it gives a sufficient condition for the itegrability inside the family of rational functions (for definition see Sect. 2). We highlight that it works for real or complex polynomial ordinary differential equations and that the study of complex invariant algebraic curves is necessary for obtaining all the real first integrals of a real polynomial differential equation, for more details see [5]. We introduce the new variable X = x + α. In this new variable sytem (1) becomes 1 ¯ X˙ = − X 2 − 2(y 2 + z 2 − w 2 ) − β, 2 y˙ = −X y − γ , z˙ = −X z − δ, w˙ = −X w, where β¯ = 2β −
α2 2 .
(2)
The associated vector field to (2) is
1 ∂ ∂ ∂ ∂ χ = − X 2 + 2(y 2 + z 2 − w 2 ) + β¯ − (X y + γ ) − (X z + δ) − X w . 2 ∂X ∂y ∂z ∂w (3) We study the existence of polynomial and rational first integrals of system (2). For proving our main results we shall use the information about invariant algebraic surfaces of this system. We will work with system (2) instead of system (1). We first consider the case in which γ = δ = 0. We start with the following result. Theorem 1. System (2) with γ = δ = 0 is integrable with the following rational first integrals: H1 =
y z 2β¯ − X 2 + 4(y 2 + z 2 − w 2 ) , H2 = and H3 = . w w w
It is straightforward to verify that H1 , H2 and H3 in the statement of the theorem are first integrals of system (2) when γ = δ = 0. Therefore the proof of Theorem 1 will be omitted and from now on we study the integrability of system (2) when γ 2 + δ 2 > 0.
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If γ ∈ R\{0} and δ = 0 then we introduce the rescaling Y = y/γ and system (2) becomes 1 ¯ X˙ = − X 2 − 2(γ 2 Y 2 + z 2 − w 2 ) − β, 2 Y˙ = −X Y − 1, (4) z˙ = −X z, w˙ = −X w. Theorem 2. The following statements hold system (4): (a) It does not admit any polynomial first integral; (b) The unique irreducible Darboux polynomials with nonzero cofactor are w, z with cofactor K = −X and 2n j 2n− j with a ∈ R, a a = 0 with cofactor K = −2n X for any n ≥ 1; j 0 2n j=0 a j w z (c) The unique rational first integrals are rational functions in the variable z/w. The proof of Theorem 2 is given in Sect. 3. If γ = 0 and δ ∈ R\{0} then we introduce the rescaling Z = z/δ and system (2) becomes 1 ¯ X˙ = − X 2 − 2(y 2 + δ 2 Z 2 − w 2 ) − β, 2 y˙ = −X y, Z˙ = −X Z − 1,
(5)
w˙ = −X w. System (5) is the same as system (4) interchanging the roles of Y with Z , z with y and γ with δ. Then we have the following theorem. Theorem 3. The following statements hold system (5): (a) It does not admit any polynomial first integral; (b) The unique irreducible Darboux polynomials with nonzero cofactor are w, y with j 2n− j with a ∈ R, a a cofactor K = −X and 2n j 0 2n = 0 with cofactor j=0 a j w y K = −2n X for any n ≥ 1; (c) The unique rational first integrals are rational functions in the variable y/w. The proof of Theorem 3 is exactly the same as the proof of Theorem 2 interchanging the roles of Y with Z , z with y and γ with δ. Therefore the proof is omitted. Now we assume γ , δ ∈ R\{0}. Introducing the rescaling Y = y/γ and Z = z/δ we have that system (2) becomes 1 ¯ X˙ = − X 2 − 2(γ 2 Y 2 + δ 2 Z 2 − w 2 ) − β, 2 Y˙ = −X Y − 1, Z˙ = −X Z − 1,
(6)
w˙ = −X w. Now we introduce the change of variables Y = (2u + v)/2 and Z = (2u − v)/2. In these new variables (u, v) system (6) becomes
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1 1 ¯ X˙ = − X 2 − 2 (γ 2 + δ 2 )u 2 + (γ 2 + δ 2 )v 2 − w 2 + (γ 2 − δ 2 )uv − β, 2 4 u˙ = −X u − 1, v˙ = −X v, w˙ = −X w.
(7)
Theorem 4. The following statements hold system (7): (a) It does not admit any polynomial first integral; (b) The unique irreducible Darboux polynomials with nonzero cofactor are v, w with j 2n− j with a ∈ R, a a cofactor K = −X and 2n j 0 2n = 0 with cofactor j=0 a j w v K = −2n X for any n ≥ 1; (c) The unique rational first integrals are rational functions in the variable v/w. The proof of Theorem 4 is given in Sect. 4. 2. Preliminary Results In this section we introduce some basic definitions and results related to the Darboux theory of integrability that we shall need in order to prove Theorem 2. Let U ⊂ R3 be an open subset. We say that the non–constant function H : U → R is a first integral of the polynomial vector field (3) on U associated to system (2), if H (X (t), y(t), z(t), w(t)) = constant for all values of t for which the solution (X (t), y(t), z(t), w(t)) of χ is defined on U . Clearly H is a first integral of χ on U if and only if χ H = 0 on U . When H is a polynomial we say that H is a polynomial first integral and when H is a proper rational function we say that H is a rational first integral. Let h = h(X, y, z, w) ∈ C[X, y, z, w] be a non–constant polynomial. We say that h = 0 is an invariant algebraic surface of the vector field χ in (3) if it satisfies χ h = K h, for some polynomial K = K (X, y, z, w) ∈ C[X, y, z, w], called the cofactor of h. Note that K has degree at most 1. The polynomial h is called a Darboux polynomial, and we also say that K is the cofactor of the Darboux polynomial h. We note that a Darboux polynomial with zero cofactor is a polynomial first integral. We recall the following well-known proposition (see [1] for a proof). Proposition 5. Assume that f (X, y, z, w) is a polynomial function in the real polynomial ring R[X, y, z, w]. Let f = f 1n 1 · · · f mn m be the factorization of f in irreducible factors over R[X, y, z, w]. Then f is a Darboux polynomial of system (1) with cofactor K f if and only if each f i is a Darboux polynomial with cofactor K fi for i = 1, . . . , m. m Moreover K f = i=1 n i K fi . Proposition 5 implies that in order to classify all the Darboux polynomials with non-zero cofactor of a polynomial differential system it is enough to classify all the irreducible ones. The following proposition characterizes the rational first integrals of a polynomial vector field. It can be easily proved from the definitions (for a proof see [6]). Proposition 6. Let X be a polynomial vector field in Rn . Then f /g with f and g being relatively prime polynomial functions is a rational first integral of system (2) if and only if both f and g are Darboux polynomials with the same cofactor.
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3. Proof of Theorem 2 Setting z = I w, system (4) becomes 1 ¯ X˙ = − X 2 − 2(γ 2 Y 2 + (I 2 − 1)w 2 ) − β, 2 Y˙ = −X Y − 1, I˙ = 0,
(8)
w˙ = −X w. Furthermore, if we introduce the change of variables X 1 = X − 2γ Y, Y1 = X + 2γ Y,
(9)
with inverse change X=
X 1 + Y1 Y1 − X 1 , Y = , 2 4γ
then system (8) becomes X 12 + 2γ − β¯ − 2(I 2 − 1)w 2 , 2 Y2 Y˙1 = − 1 − 2γ − β¯ − 2(I 2 − 1)w 2 , 2 ˙I = 0, X 1 + Y1 w. w˙ = − 2
X˙ 1 = −
(10)
We separate the proof of Theorem 2 into different propositions. Proposition 7. The unique first integrals of system (10) which are polynomials in the variables X 1 , Y1 , I do not depend on X 1 , Y1 , w and are polynomials in the variable I . Proof. Let g = g(X 1 , Y1 , w, I ) be a first integral of system (10) which is a polynomial in the variables X 1 , Y1 , I . We write it in the form g = nj=0 g j (X 1 , Y1 , I, w), where each g j is a homogeneous polynomial in the variables X 1 , Y1 . We have that either g = 0 or gn = 0. In the first case it is clear that g is a polynomial in the variable I . Now we assume that g = 0. Since g is a first integral of system (10) it satisfies
∂g ∂g Y2 X 12 + 2γ − β¯ − 2(I 2 − 1)w 2 − 1 + 2γ + β¯ + 2(I 2 − 1)w 2 2 ∂ X1 2 ∂Y1 X 1 + Y1 ∂g w = 0. (11) − 2 ∂w
−
Computing the terms of degree n + 1 in the variables X 1 , Y1 we have −
X 12 ∂gn Y 2 ∂gn X 1 + Y1 ∂gn − 1 − w = 0. 2 ∂ X1 2 ∂Y1 2 ∂w
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Solving this equation we get that gn = K n
X 1 − Y1 w I, , , X 1 Y1 X 1 Y1
where K n is an arbitrary function. Since gn must be an homogeneous polynomial of degree n in the variables X 1 , Y1 we must have that X 1 − Y1 n gn = cn (I ) , w where cn (I ) is a polynomial in the variable I . Now computing the terms of degree n − 1 in the variables X 1 , Y1 in Eq. (10) we get X 12 ∂gn−2 Y 2 ∂gn−2 X 1 + Y1 ∂gn−2 w − 1 − 2 ∂ X1 2 ∂Y1 2 ∂w ∂gn ∂gn 2 2 − (2γ + β¯ + 2(I 2 − 1)w 2 ) = 0. + (2γ − β¯ − 2(I − 1)w ) ∂ X1 ∂Y1
−
Solving this equation and using that gn−2 is a homogeneous polynomial of degree n − 2 in the variables X 1 , Y1 , we get gn−2
−4cn (I )γ n X 1 − Y1 n−2 n−1 = (X 1 − Y1 ) (3X 1 − Y1 ) + cn−2 (I ) , w 3X 12 w n
where cn−2 is a polynomial in the variable I . Since gn−2 must be a homogeneous polynomial of degree n − 2 in the variables X 1 , Y1 and gn = 0 (and hence cn (I ) = 0) we must have that n=0 (note that γ = 0). Therefore, g = g0 (w, I ). Imposing that g satisfies (11) we obtain −
X 1 + Y1 ∂g w = 0, that is g = g(I ). 2 ∂w
This concludes the proof of the proposition. Proposition 8. System (4) has no polynomial first integrals. Proof. We proceed by contradiction. Let h = h(X, Y, z, w) be a polynomial first integral of system (4). Without loss of generality we can assume that it has no constant term. Setting z = I w, introducing the change of variables in (9) and taking the notation g = g(X 1 , Y1 , I, w) = h(X 1 , Y1 , z, w) we get that g is a polynomial in the variables X 1 , Y1 , I, w that satisfies (11). In view of Proposition 8 we get that g is a polynomial in the variable I , that is g = nj=1 c j I j with c j ∈ R. This implies that h=
n j=1
cj
z j w
.
Since h is a polynomial we get that c j = 0 for j = 1, . . . , n and then h = 0 a contradiction with the fact that h is a first integral. This completes the proof of the proposition.
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Proposition 9. If f (X, y, z, w) is an irreducible Darboux polynomial of system (4) with nonzero cofactor and of degree n ≥ 2, then the cofactor is K = α0 − m X , where m is an integer. Proof. Let h = h(X, Y, z, w) be an irreducible Darboux polyomial with non-zero cofactor and degree n ≥ 2 and with cofactor K = α0 + α1 X + α2 Y + α3 w + α4 z, that is, it satisfies 1 ∂h ∂h ∂h ∂h − X 2 + 2(γ 2 Y 2 + z 2 − w 2 ) + β¯ − (X Y + 1) − Xz − Xw 2 ∂X ∂Y ∂z ∂w = (α0 + α1 X + α2 Y + α3 w + α4 z)h. We set z = I w and system (4) becomes (8). Now if we denote by g the Darboux polynomial h in these variables, we have that g = g(X, Y, w, I ) and satisfies 1 ∂g ∂g ∂g X 2 + 2(γ 2 Y 2 + (I 2 − 1)w 2 ) + β¯ − (X Y + 1) − Xw − 2 ∂X ∂Y ∂w = (α0 + α1 X + α2 Y + α3 w + α4 I w)g. Let g˜ be the restriction of g to I = 0. Then g˜ satisfies ∂ g˜ 1 ∂ g˜ ∂ g˜ − (X Y + 1) − Xw − X 2 + 2(γ 2 Y 2 − w 2 ) + β¯ 2 ∂X ∂Y ∂w = (α0 + α1 X + α2 Y + α3 w)g. ˜ (12) n We write g˜ = i=0 g˜i (X, Y, w), where each g˜i is a homogeneous polynomial of degree i in the variables (X, Y, w). Now computing the terms of degree n + 1 in (12) we get 1 ∂ g˜ ∂ g˜ n ∂ g˜ n n − X 2 + 2(γ 2 Y 2 − w 2 ) − XY − Xw = (α1 X + α2 Y + α3 w)g˜ n . 2 ∂X ∂Y ∂w Solving this equation we obtain that 2α1 + √
α2 Y
+ √
α3 w
2 γ 2 Y 2 −w2 2 γ 2 Y 2 −w2 Y −α1 g˜ n = α1α1 2 α w 4γ 2 X Y 2 − 4w 2 X + γ 2 Y 2 − w 2 (−X 2 + 4w 2 − 4γ 2 Y 2 ) √ α2 Y + √ 3 2 γ 2 Y 2 −w2 2 γ 2 Y 2 −w2 × Y w 4γ 2 y 2 − X 2 − 4w 2 , × Kn Y Y or
g˜ n = α1α1 32−α1 (w 2 − γ 2 Y 2 )−α1 α w 8γ 2 X Y 2 −8w 2 X +2γ 2 Y 2 − w 2 (X 2 −4w 2 +4γ 2 Y 2 ) − √ α2 Y − √ 3 2 γ 2 Y 2 −w2 2 γ 2 Y 2 −w2 × Y w 4γ 2 y 2 − X 2 − 4w 2 , × Kn Y Y 4γ y −X −4w where K n is a function in the variables w . Y and Y In both cases, since g˜ n must be a homogeneous polynomial we get that α2 = α3 = 0 and α1 = −m with m an integer. Hence K = α0 − m X + α4 z. 2 2
2
2
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Now we set w = J z. Then system (4) becomes 1 ¯ X˙ = − X 2 − 2(γ 2 Y 2 + (1 − J 2 )z 2 ) − β, 2 Y˙ = −X Y − 1, J˙ = 0, z˙ = −X z. Now if we denote by f the Darboux polynomial h in these variables, we have that f = f (X, Y, z, J ) and satisfies −
1
¯ X 2 + 2(γ 2 Y 2 + (1 − J 2 )z 2 − β)
2 = (α0 − m X + α4 z) f.
∂f ∂f ∂f − (X Y + 1) − Xz ∂X ∂Y ∂z
Now proceeding for f as we did with g, we readily get that α4 = 0. This concludes the proof of the proposition. Proposition 10. If f (X, y, z, w) is an irreducible Darboux polynomial of system (4) with nonzero cofactor and of degree n ≥ 2, then the cofactor is K = −2m X where m is a positive integer. Proof. We assume that h = h(X, Y, z, w) is an irreducible Darboux polyomial with nonzero cofactor and degree n ≥ 2. By Proposition 9 the cofactor has the form K = α0 −m X , where m is a non-negative integer. We set z = I w and system (4) becomes (8). Now we introduce the change of variables in (9) and system (8) becomes (10). Now if we denote by f the Darboux polynomial h in these variables, and by fˆ the restriction of f to I = w = 0 it satisfies
−
∂ fˆ ∂ fˆ Y2 X 12 m + 2γ − β¯ − 1 + 2γ + β¯ = c − (X 1 + Y1 ) fˆ. 2 ∂ X1 2 ∂Y1 2
(13)
We consider three different cases. Case 1. β¯ ∈ {−2γ , 2γ }. In this case solving (13) we get √
2
m/2 − √ ¯ 2c arctan √ √X 1¯ 2 2 β−2γ β−2γ ˆ ¯ ¯ f = (X 1 + 2β − 4γ )(Y1 + 2β + 4γ ) e
X 1 arctan √ √ arctan √ √Y1 ¯ ¯ 2 β+2γ 2 β−2γ + , ×C − β¯ − 2γ β¯ + 2γ
where C is an arbitrary function. Since fˆ must be a polynomial we must have c = 0 and C a constant and m must be an even integer, that is, m = 2 j with j an integer. This concludes the proof of the proposition in this case. Case 2. β¯ = 2γ . In this case solving (13) we get fˆ = e
2c X1
1
arctan √ Y√ 1 2 γ m 2 m/2 X 1 (8γ + Y1 ) C + , √ √ X1 2 2 γ
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where C is an arbitrary function. Since fˆ must be a polynomial we must have c = 0 and C a constant and m must be an even integer, that is, m = 2 j with j an integer. This concludes the proof of the proposition in this case. Case 3. β¯ = −2γ . In this case solving (13) we get X
c arctanh √ 1√ arctanh √ X√1 2 2 γ √ √ 1 2 γ 2 γ Y1m (−8γ + X 12 )m/2 C − + fˆ = e , √ √ Y1 2 2 γ where C is an arbitrary function. Since fˆ must be a polynomial we must have c = 0 and C a constant and m must be an even integer, that is, m = 2 j with j an integer. Thus, the proof of the proposition is completed. Proposition 11. The unique irreducible Darboux polynomials of system (4) with non j 2n− j , with a ∈ R, zero cofactor are z, w with cofactor K = −X and 2n j j=0 a j w z a0 a2n = 0 with cofactor −2n X for any n ≥ 1. Proof. It follows by direct computations that the unique irreducible Darboux polynomials of system (4) with nonzero cofactor and of degree one are z and w that have cofactor K = −X . Now we assume that h = h(X, Y, z, w) is an irreducible Darboux polyomial with non-zero cofactor and degree n ≥ 2. In view of Propositions 9 and 10 its cofactor is of the form K = −2m X with m ∈ N. Now we set z = I w and introduce the change of variables in (9). Then if we denote by g = g(X 1 , Y1 , I, w) the irreducible Darboux polynomial h in these variables we have that g satisfies X2 ∂g ∂g Y2 − 1 + 2γ − β¯ − 2(I 2 − 1)w 2 − 1 + 2γ + β¯ + 2(I 2 − 1)w 2 2 ∂ X1 2 ∂Y1 X 1 + Y1 X 1 + Y1 ∂g w = −2m g. (14) − 2 ∂w 2 We write g = w 2m h, where h = h(X 1 , Y1 , I, w) is a polynomial in the variables X 1 , Y1 , I and a rational function in the variable w. Introducing g in (14) we obtain that, after simplifying by w 2m , h satisfies ∂h ∂h X2 Y2 − 1 + 2γ − β¯ − 2(I 2 − 1)w 2 − 1 + 2γ + β¯ + 2(I 2 − 1)w 2 2 ∂ X1 2 ∂Y1 X 1 + Y1 ∂h w = 0, − 2 ∂w which is (11). Then either h = 0 or h is a first integral of system (10) which is a polynomial in the variables X 1 , Y1 , I . In view of Proposition 7 we have that h = h(I ) = l j 2m j j=0 c j I , with c j ∈ R, and hence g = w j=0 c j I . This implies that h=w
2m
j=0
cj
z j w
z 2m+1 z + · · · + −2m . w w Since h is a polynomial we must have c2m+1 = · · · = c = 0. Furthermore since it is irreducible we also have c0 = 0 and c2m = 0 and finally since the degree of h is n ≥ 2 we must have that n = 2m with m ≥ 1. This concludes the proof of the proposition. = c0 w 2m + c1 zw 2m−1 + · · · + c2m z 2m + c2m+1
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3.1. Proof of Theorem 2. Statements (a) and (b) in the theorem follow directly from Propositions 8 and 11. In what follows we prove statement (c) by contradiction. Assume that G is a rational first integral. Then taking into account Propositions 6 and 5, G must be of the form G = z n1 wn2
2n
a j z j w 2n− j
n 3
, n 1 , n 2 , n 3 ∈ Z.
j=0
Since G is a first integral it must satisfy χ G = 0, that is, 1 ∂G ∂G ∂G ∂G − (X Y + 1) − Xz − Xw χ G = − X 2 + 2(γ 2 Y 2 + z 2 − w 2 ) + β¯ 2 ∂X ∂Y ∂z ∂w = −(n 1 + n 2 + 2nn 3 )X G = 0. Hence, n 1 + n 2 + 2nn 3 = 0, that is n 2 = −n 1 − 2nn 3 . Then G = (z/w)n 1
2n
a j z j w− j
j=0
n 3
= (z/w)n 1
2n
a j (z/w) j
n 3
,
j=0
with n 1 , n 3 ∈ Z. This concludes the proof of the theorem. 4. Proof of Theorem 4 Setting v = I w, system (7) becomes 1 1 ¯ X˙ = − X 2 − 2(γ 2 + δ 2 )u 2 − 2 (γ 2 + δ 2 )I 2 − 1 w 2 − 2(γ 2 − δ 2 )I vw − β, 2 4 u˙ = −X u − 1, (15) I˙ = 0, w˙ = −X w. Furthermore, if we introduce the change of variables X 1 = X − 2 γ 2 + δ 2 u, Y1 = X + 2 γ 2 + δ 2 u,
(16)
with inverse change X=
Y1 − X 1 X 1 + Y1 , u= , 2 4 γ 2 + δ2
then system (15) becomes X˙ 1 = −
X 12 γ 2 + δ2 2 2 γ 2 − δ2 + 2 γ 2 + δ 2 − β¯ + 2 − I w − I w(Y1 − X 1 ), 2 2 2 γ 2 + δ2
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Y˙1 = − I˙ = 0, w˙ = −
143
Y12 γ 2 + δ2 2 2 γ 2 − δ2 I w(Y1 − X 1 ), − 2 γ 2 + δ 2 − β¯ + 2 − I w − 2 2 2 γ 2 + δ2 (17) X 1 + Y1 w. 2
We separate the proof of Theorem 2 into different propositions. Proposition 12. The unique first integrals of system (17) which are polynomials in the variables X 1 , Y1 , I do not depend on X 1 , Y1 , w and are polynomials in the variable I . Proof. Let g = g(X 1 , Y1 , w, I ) be a first integral of system (17) which is a polynomial in the variables X 1 , Y1 , I . We write it in the form g = nj=0 g j (X 1 , Y1 , I, w), where each g j is a homogeneous polynomial in the variables X 1 , Y1 . We have that either g = 0 or gn = 0. In the first case it is clear that g is a polynomial in the variable I . Now we assume that g = 0. Since g is a first integral of system (17) it satisfies ∂g γ 2 + δ2 2 2 γ 2 − δ2 − 2 γ 2 + δ 2 + β¯ − 2 − I w + I w(Y1 − X 1 ) 2 2 ∂ X1 2 γ 2 + δ2 Y2 ∂g γ 2 + δ2 2 2 γ 2 − δ2 I w + − 1 + 2 γ 2 + δ 2 + β¯ − 2 − I w(Y1 − X 1 ) 2 2 ∂Y1 2 γ 2 + δ2
−
X2
−
1
X 1 + Y1 ∂g w = 0. 2 ∂w
Computing the terms of degree n + 1 in the variables X 1 , Y1 we have −
X 12 ∂gn Y 2 ∂gn X 1 + Y1 ∂gn w = 0, − 1 − 2 ∂ X1 2 ∂Y1 2 ∂w
and solving it we get X 1 − Y1 w , gn = K n I, , X 1 Y1 X 1 Y1 where K n is an arbitrary function. Since gn must be an homogeneous polynomial of degree n in the variables X 1 , Y1 we must have that X 1 − Y1 n , gn = cn (I ) w where cn (I ) is a polynomial in the variable I . Computing the terms of degree n in the variables X 1 , Y1 in Eq. (17) we get X 12 ∂gn−1 Y 2 ∂gn−1 X 1 + Y1 ∂gn−1 w − 1 − 2 ∂ X1 2 ∂Y1 2 ∂w 2 2 2 γ −δ ∂gn γ − δ2 ∂gn − I w(Y1 − X 1 ) − I w(Y1 − X 1 ) = 0. 2 2 2 2 ∂ X ∂Y 1 1 2 γ +δ 2 γ +δ
−
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Solving this equation and using that gn−1 is a homogeneous polynomial of degree n − 1 in the variables X 1 , Y1 we get X 1 − Y1 n−1 , gn−1 = cn−1 (I ) w where cn−1 (I ) is a polynomial in the variable I . Now computing the terms of degree n − 1 in the variables X 1 , Y1 in Eq. (17) we get Y 2 ∂gn−2 X 12 ∂gn−2 X 1 + Y1 ∂gn−2 − 1 − w 2 ∂ X1 2 ∂Y1 2 ∂w γ 2 − δ2 ∂gn−1 γ 2 − δ2 ∂gn−1 − I w(Y1 − X 1 ) − I w(Y1 − X 1 ) 2 2 2 2 ∂ X1 ∂Y1 2 γ +δ 2 γ +δ 2 2 γ + δ 2 2 ∂gn I w + 2 γ 2 + δ 2 − β¯ + 2 − 2 ∂ X1 ∂g 2 + δ2 γ n I 2 w2 + (−2 γ 2 + δ 2 − β¯ + 2 − = 0. 2 ∂ X2
−
Solving this equation and using that gn−2 is a homogeneous polynomial of degree n − 2 in the variables X 1 , Y1 we get X 1 − Y1 n−2 −2cn (I ) γ 2 + δ 2 n n−1 n (X 1 − Y1 ) (3X 1 − Y1 ) + cn−2 (I ) , gn−2 = w 3X 12 w n where cn−2 is a polynomial in the variable I . Since gn−2 must be a homogeneous polynomial of degree n − 2 in the variables X 1 , Y1 and gn = 0 (and hence cn (I ) = 0) we must have that n=0 (note that γ 2 + δ 2 = 0). Therefore, g = g0 (w, I ). Imposing that g satisfies (11) we obtain −
X 1 + Y1 ∂g w = 0, that is g = g(I ). 2 ∂w
This concludes the proof of the proposition. Proposition 13. System (7) has no polynomial first integrals. Proof. The proof is completely analogous to the proof of Proposition 8 changing z by v. Proposition 14. If f (X, u, v, w) is an irreducible Darboux polynomial of system (7) with nonzero cofactor and of degree n ≥ 2, then the cofactor is K = α0 − m X , where m is an integer. Proof. Let h = h(X, u, v, w) be an irreducible Darboux polyomial with non-zero cofactor and degree n ≥ 2 and with cofactor K = α0 + α1 X + α2 u + α3 w + α4 v, that is, it satisfies 1 ∂h 1 − X 2 + 2 (γ 2 + δ 2 )u 2 + (γ 2 + δ 2 )v 2 − w 2 + (γ 2 − δ 2 )uv) + β¯ 2 4 ∂X ∂h ∂h ∂h − Xv − Xw = (α0 + α1 X + α2 u + α3 w + α4 v)h. − (X u + 1) ∂u ∂v ∂w
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We set v = I w and system (7) becomes (15). Now if we denote by g the Darboux polynomial h in these variables, and by g¯ the restriction of g to I = 0, then we have that g¯ satisfies (12) with Y replaced by u and γ 2 replaced by γ 2 + δ 2 . Then proceeding as in the proof of Proposition 9 we get that α2 = α3 = 0. Now setting w = J v and proceeding as in the proof of Proposition 9 we also get that α4 = 0. This completes the proof of the proposition. Proposition 15. If f (X, u, v, w) is an irreducible Darboux polynomial of system (15) with nonzero cofactor and of degree n ≥ 2, then the cofactor K = −2m X , where m is a positive integer. Proof. We assume that h = h(X, u, v, w) is an irreducible Darboux polyomial with non-zero cofactor and degree n ≥ 2. By Proposition 14 the cofactor has the form K = α0 −m X , where m is a non-negative integer. We set v = I w and system (7) becomes (15). Now we introduce the change of variables in (16) and system (15) becomes (17). If we denote by f the Darboux polynomial h in these variables, and by fˆ the restriction of f to I = w = 0, we get that fˆ satisfies (10) with γ replaced by γ 2 + δ 2 . Now the proof of the proposition follows in an analogous way to the proof of Proposition 10. Proposition 16. The unique irreducible Darboux polynomials of system (7) with non j 2n− j , with a ∈ R, zero cofactor are v, w with cofactor K = −X and 2n j j=0 a j w v a0 a2n = 0 with cofactor −2n X for any n ≥ 1. Proof. It follows by direct computations that the unique irreducible Darboux polynomials of system (7) with nonzero cofactor and of degree one are v and w that have cofactor K = −X . Now we assume that h = h(X, u, v, w) is an irreducible Darboux polyomial with non-zero cofactor and degree n ≥ 2. In view of Propositions 14 and 15 its cofactor is of the form K = −2m X with m ∈ N. Now we set v = I w and introduce the change of variables in (16). Then if we denote by g = g(X 1 , Y1 , I, w) the irreducible Darboux polynomial h in these variables we have that g satisfies X2 γ 2 + δ2 ∂g γ 2 − δ2 1 − 2 γ 2 + δ 2 + β¯ + 2 I 2 − 1 w2 + − I w(Y1 − X 1 ) 2 4 ∂ X1 2 γ 2 + δ2 Y2 γ 2 + δ2 2 2 γ −δ ∂g I 2 − 1 w2 + − 1 + 2 γ 2 + δ 2 + β¯ + 2 I w(Y1 − X 1 ) 2 2 2 4 ∂Y1 2 γ +δ −
X 1 + Y1 X 1 + Y1 ∂g w = −2m g. 2 ∂w 2
We write g = w 2m h, where h = h(X 1 , Y1 , I, w) is a polynomial in the variables X 1 , Y1 , I and a rational function in the variable w. After simplifying by w2n we get that h is a first integral of system (17) which is a polynomial in the variables X 1 , Y1 , I . In view of Proposition 12 we have that h = h(I ) = lj=0 c j I j , with c j ∈ R, and hence g = w2m j=0 c j I j . This implies that h = w 2m
j=0
cj
v j w
= c0 w 2m + c1 vw 2m−1 + · · · + c2m v 2m + c2m+1
v v 2m+1 + · · · + −2m . w w
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Since h is a polynomial we must have c2m+1 = · · · = c = 0. Furthermore since it is irreducible, we also have c0 = 0 and c2m = 0 and finally since the degree of h is n ≥ 2 we must have that n = 2m with m ≥ 1. This concludes the proof of the proposition. 4.1. Proof of Theorem 4. Statements (a) and (b) in the theorem follow directly from Propositions 13 and 16. In what follows we prove the statement (c) by contradiction. Assume that G is a rational first integral. Then taking into account Propositions 6 and 5, G must be of the form G = vn1 wn2
2n
a j v j w 2n− j
n 3
, n 1 , n 2 , n 3 ∈ Z.
j=0
Since G is a first integral it must satisfy χ G = 0. Since χ G = −(n 1 + n 2 + 2nn 3 )X G we deduce that n 1 + n 2 + 2nn 3 = 0, that is n 2 = −n 1 − 2nn 3 . Then G = (v/w)n 1
2n j=0
a j v j w− j
n 3
= (v/w)n 1
2n
a j (v/w) j
n 3
,
j=0
with n 1 , n 3 ∈ Z. This concludes the proof of the theorem. Acknowledgements. Partially supported by FCT through CAMGDS, Lisbon.
References 1. Christopher, C., Llibre, J.: Integrability via invariantalgebraic curves for planar polynomial differential systems. Ann. Diff. Eq. 14, 5–19 (2000) 2. Dasgupta, A., Nandan, H., Kar, S.: Kinematics of flows on curved, deformable media. Int. J. Geom. Meth. Mod. Phys. 6, 645–666 (2009) 3. Ghose, A., Guha, P., Khanra, B.: Determination of elementaryfirst integrals of a generalized Raychaudhuri equation by the Darboux integrability method. J. Math. Phys. 50, 102502 (2009) 4. Kar, S., Sengupta, S.: The Raychaudhuri equations: A briefreview. Pramana 69, 49–76 (2009) 5. Llibre, J.: Integrability of polynomial differential systems. In: Handbook of Differential Equations, Ordinary Differential Equations, Eds. A. Cañada, P. Drabek , A. Fonda, Amsterdam: Elsevier/North Holland, 2009, pp. 437–532 6. Llibre, J., Valls, C.: Integrability of the Bianchi IX system. J. Math. Phys. 46, 072901 (2005) Communicated by G. Gallavotti
Commun. Math. Phys. 308, 147–200 (2011) Digital Object Identifier (DOI) 10.1007/s00220-011-1318-6
Communications in
Mathematical Physics
Collisions of Particles in Locally AdS Spacetimes I. Local Description and Global Examples Thierry Barbot1, , Francesco Bonsante2, , Jean-Marc Schlenker3, 1 Laboratoire D’analyse Non Linéaire et Géométrie, Université d’Avignon et Des Pays de Vaucluse, 33,
Rue Louis Pasteur, 84 018 Avignon, France. E-mail: [email protected]
2 Dipartimento di Matematica dell’Università Degli Studi di Pavia, Via Ferrata 1, 27100 Pavia, Italy.
E-mail: [email protected]
3 Institut de Mathématiques de Toulouse, UMR CNRS 5219, Université Paul Sabatier,
31062 Toulouse Cedex 9, France. E-mail: [email protected] Received: 17 October 2010 / Accepted: 26 April 2011 Published online: 10 September 2011 – © Springer-Verlag 2011
Abstract: We investigate 3-dimensional globally hyperbolic AdS manifolds (or more generally constant curvature Lorentz manifolds) containing “particles”, i.e., cone singularities along a graph . We impose physically relevant conditions on the cone singularities, e.g. positivity of mass (angle less than 2π on time-like singular segments). We construct examples of such manifolds, describe the cone singularities that can arise and the way they can interact (the local geometry near the vertices of ). We then adapt to this setting some notions like global hyperbolicity which are natural for Lorentz manifolds, and construct some examples of globally hyperbolic AdS manifolds with interacting particles. Contents 1.
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Introduction . . . . . . . . . . . . . . . . . . . . . 1.1 Three-dimensional cone-manifolds . . . . . . . 1.2 AdS manifolds . . . . . . . . . . . . . . . . . 1.3 A classification of cone singularities along lines 1.4 Interactions and convex polyhedra . . . . . . . 1.5 A classification of HS-structures . . . . . . . . 1.6 Global hyperbolicity . . . . . . . . . . . . . . 1.7 Construction of global examples . . . . . . . . 1.8 Further extension . . . . . . . . . . . . . . . . Preliminaries . . . . . . . . . . . . . . . . . . . . . 2.1 (G, X )-structures . . . . . . . . . . . . . . . . 2.2 Background on the AdS space . . . . . . . . . Singularities in Singular AdS-Spacetimes . . . . . .
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T. B. and F. B. were partially supported by CNRS, ANR GEODYCOS.
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J.-M. S. was partially supported by the A.N.R. programs RepSurf, ANR-06-BLAN-0311,
GeomEinstein, 06-BLAN-0154, and ETTT, ANR-09-BLAN-0116-01, 2009-2013.
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3.1 HS geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Suspension of regular HS-surfaces . . . . . . . . . . . . . . . . 3.3 Singularities in singular HS-surfaces . . . . . . . . . . . . . . . 3.4 Singular HS-surfaces . . . . . . . . . . . . . . . . . . . . . . . 3.5 Classification of singular lines . . . . . . . . . . . . . . . . . . 3.6 Local future and past of singular points . . . . . . . . . . . . . . 3.7 Geometric description of HS-singularities and AdS singular lines 3.8 Positive HS-surfaces . . . . . . . . . . . . . . . . . . . . . . . 4. Particle Interactions and Convex Polyhedra . . . . . . . . . . . . . . 4.1 The space HS3 . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Convex polyhedra in HS3 . . . . . . . . . . . . . . . . . . . . . 4.3 Induced HS-structures on the boundary of a polyhedron . . . . . 4.4 From a convex polyhedron to a particle interaction . . . . . . . 4.5 From a particle interaction to a convex polyhedron . . . . . . . 5. Classification of Positive Causal HS-Surfaces . . . . . . . . . . . . . 5.1 Photons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Hyperbolic regions . . . . . . . . . . . . . . . . . . . . . . . . 5.3 De Sitter regions . . . . . . . . . . . . . . . . . . . . . . . . . . 6. Global Hyperbolicity . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Local coordinates near a singular line . . . . . . . . . . . . . . 6.2 Achronal surfaces . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Time functions . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 Cauchy surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5 Maximal globally hyperbolic extensions . . . . . . . . . . . . . 7. Global Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 An explicit example . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Surgery . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3 Spacetimes containing BTZ-type singularities . . . . . . . . . . 7.4 Surgery on spacetimes containing BTZ-type singularities . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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1. Introduction 1.1. Three-dimensional cone-manifolds. The 3-dimensional hyperbolic space can be defined as a quadric in the 4-dimensional Minkowski space: H3 = {x ∈ R3,1 | x, x = −1 & x0 > 0} . Hyperbolic manifolds, which are manifolds with a Riemannian metric locally isometric to the metric on H3 , have been a major focus of attention for modern geometry. More recently attention has turned to hyperbolic cone-manifolds, which are the types of singular hyperbolic manifolds that one can obtain by gluing isometrically the faces of hyperbolic polyhedra. Three-dimensional hyperbolic cone-manifolds are singular along lines, and at “vertices” where three or more singular segments intersect. The local geometry at a singular vertex is determined by its link, which is a spherical surface with cone singularities. Among key recent results on hyperbolic cone-manifolds are rigidity results [HK98,MM,Wei] as well as many applications to three-dimensional geometry (see e.g. [Bro04,BBES03]).
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1.2. AdS manifolds. The three-dimensional anti-de Sitter (AdS) space can be defined, similarly as H 3 , as a quadric in the 4-dimensional flat space of signature (2, 2): AdS3 = {x ∈ R2,2 | x, x = −1} . It is a complete Lorentz space of constant curvature −1, with fundamental group Z. AdS geometry provides in certain ways a Lorentz analog of hyperbolic geometry, a fact mostly discovered by Mess (see [Mes07,ABB+ 07]). In particular, the so-called globally hyperbolic AdS 3-manifolds are in key ways analogs of quasifuchsian hyperbolic 3-manifolds. Among the striking similarities one can note an analog of the Bers double uniformization theorem for globally hyperbolic AdS manifolds, or a similar description of the convex core and of its boundary. Three-dimensional AdS geometry, like 3-dimensional hyperbolic geometry, has some deep relationships with Teichmüller theory (see e.g. [Mes07,ABB+ 07,BS09a,BKS06,KS07,BS09b,BS10]). Lorentz manifolds have often been studied for reasons related to physics and in particular gravitation. In three dimensions, Einstein metrics are the same as constant curvature metrics, so the constant curvature 3-dimensional Lorentz manifolds – and in particular AdS manifolds – are the 3-dimensional models of gravity. From this point of view, cone singularities have been extensively used to model point particles, see e.g. [tH96,tH93]. The goal pursued here is to start a geometric study of 3-dimensional AdS manifolds with cone singularities. We will in particular • describe the possible “particles”, or cone singularities along a singular line, • describe the singular vertices – the way those “particles” can “interact”, • show that classical notions like global hyperbolicity can be extended to AdS conemanifolds, • give examples of globally hyperbolic AdS particles with “interesting” particles and particle interactions. We focus here on the presentation of AdS manifolds for simplicity, but most of the local study near singular points extends to constant curvature-Lorentz 3-dimensional manifolds. More specifically, the first three points above extend from AdS manifolds with particles to Minkowski or de Sitter manifolds. The fourth point is mostly limited to the AdS case, although some parts of what we do here can be extended to the Minkowski or de Sitter case. We outline in more details those main contributions below. 1.3. A classification of cone singularities along lines. We start in Sect. 3 an analysis of the possible local geometry near a singular point. For the hyperbolic cone-manifold this local geometry is described by the link of the point, which is a spherical surface with cone singularities. In the AdS (as well as the Minkowski or de Sitter) setting there is an analog notion of link, which is now what we call a singular HS-surface, that is, a surface with a geometric structure locally modelled on the space of rays starting from a point in R2,1 (see Sect. 3.4). We then describe the possible geometry in the neighborhood of a point on a singular segment (Proposition 3.1). For hyperbolic cone-manifolds, this local description is quite simple: there is only one possible local model, depending on only one parameter, the angle. For AdS cone-manifolds – or more generally cone manifolds with a constant curvature Lorentz metric – the situation is more complicated, and cone singularities along segments can be of different types. For instance it is clear that the fact that the singular segment is space-like, time-like or light-like should play a role.
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There are two physically natural restrictions which appear in this section. The first is the degree of a cone singularity along a segment c: the number of connected components of time-like vectors in the normal bundle of c (Sect. 3.3). In the “usual” situation where each point has a past and a future, this degree is equal to 2. We restrict our study to the case where the degree is at most equal to 2. There are many interesting situations where this degree can be strictly less than 2, see below. The second condition (see Sect. 3.6) is that each point should have a neighborhood containing no closed causal curve – also physically relevant since closed causal curves induce causality violations. AdS manifolds with cone singularities satisfying those two conditions are called causal here. We classify and describe all cone singularities along segments in causal AdS manifolds with cone singularities, and provide a short description of each kind. They are called here: massive particles, tachyons, Misner singularities, BTZ-like singularities, and light-like and extreme BTZ-like singularities. We also define a notion of positivity for those cone singularities along lines. Heuristically, positivity means that those geodesics tend to “converge” along those cone singularitites; for instance, for a “massive particle” – a cone singularity along a time-like singularity – positivity means that the angle should be less than 2π , and it corresponds physically to the positivity of mass. Remark 1.1. All this analysis is local, even infinitesimal. It applies in a much wider setting than the one we restricted ourselves to here, and leads to a general description of all possible singularities in a 3-dimensional Lorentzian spacetime. Our first concern here is the case of singular AdS-spacetimes, hence we will not develop here further the other cases. 1.4. Interactions and convex polyhedra. In Sect. 4 we turn our attention to the vertices of the singular locus of AdS manifolds with cone singularities, in other terms the “interaction points” where several “particles” – cone singularities along lines – meet and “interact”. The construction of the link as an HS-surface, in Sect. 3, means that we need to understand the geometry of singular HS-surfaces. The singular lines arriving at an interaction point p correspond to the singular points of the link of p. An important point is that the positivity of the singular lines arriving at p, and the absence of closed causal curves near p, can be read directly on the link; this leads to a natural notion of causal singular HS-surface, those causal singular HS-surfaces are precisely those occurring as links of interaction points in causal singular AdS manifolds. The first point of Sect. 4 is the construction of many examples of positive causal singular HS-surfaces from convex polyhedra in HS3 , the natural analog of HS2 in one dimension higher. Given a convex polyhedron in HS3 one can consider the induced geometric structure on its boundary, and it is often an HS-structure and without closed causal curve. Moreover the positivity condition is always satisfied. This makes it easy to visualize many examples of causal HS-structures, and should therefore help in following the arguments used in Sect. 5 to classify causal HS-surfaces. However the relation between causal HS-surfaces and convex polyhedra is perhaps deeper than just a convenient way to construct examples. This is indicated in Theorem 4.3, which shows that all HS-surfaces having some topological properties (those which are “causally regular”) are actually obtained as induced on a unique convex polyhedron in HS3 . 1.5. A classification of HS-structures. Section 5 contains a classification of causal HS-structures, or, in other terms, of interaction points in causal singular AdS manifolds
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(or, more generally, in any singular spacetime). The main result is Theorem 5.6, which describes what types of interactions can, or cannot, occur. The striking point is that there are geometric restrictions on what kind of singularities along segments can interact at one point. 1.6. Global hyperbolicity. In Sect. 6 we consider singular AdS manifolds globally. We first extend to this setting the notion of global hyperbolicity which plays an important role in Lorentz geometry. A key result for non-singular AdS manifolds is the existence, for any globally hyperbolic manifold M, of a unique maximal globally hyperbolic extension. We prove a similar result in the singular context (see Proposition 6.22 and Proposition 6.24). However this maximal extension is unique only under the condition that the extension does not contain more interactions than M. Once more, this analysis could have been performed in a wider context. It applies in particular in the case of singular spacetimes locally modeled on the Minkowski spacetime, or the de Sitter spacetime. 1.7. Construction of global examples. Finally Sect. 7 is intended to convince the reader that the general considerations on globally hyperbolic AdS manifolds with interacting particles are not empty: it contains several examples, constructed using basically two methods. The first relies again on 3-dimensional polyhedra, but not used in the same way as in Sect. 4: here we glue their faces isometrically so as to obtain cone singularities along the edges, and interactions points at the vertices. The second method is based on surgery: we show that, in many situations, it is possible to excise a tube in an AdS manifold with non-interacting particles (like those arising in [BS09a]) and replace it by a more interesting tube containing an interaction point.
1.8. Further extension. We wish to continue in [BBS10] the investigation of globally hyperbolic AdS metrics with interacting particles, and to prove that the moduli space of those metrics is locally parameterized by 2-dimensional data (a sequence of pairs of hyperbolic metrics with cone singularities on a surface). 2. Preliminaries 2.1. (G, X )-structures. Let G be a Lie group, and X an analytic space on which G acts analytically and faithfully. In this paper, we are essentially concerned with the case where X = AdS3 and G its isometry group, but we will also consider other pairs (G, X ). A (G, X )-structure on a manifold M is a covering of M by open sets with homeomorphisms into X , such that the transition maps on the overlap of any two sets are (locally) in G. A (G, X )-manifold is a manifold equipped with a (G, X )-structure. Observe that if X˜ denotes the universal covering of X , and G˜ the universal covering of G, any (G, X )-struc˜ X˜ )-structure, and, conversely, any (G, ˜ X˜ )-structure defines a ture defines a unique (G, unique (G, X )-structure. An isomorphism between two (G, X )-manifolds is a homeomorphism whose local expressions in charts of the (G, X )-structures are restrictions of elements of G.
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→ X (where M A (G, X )-manifold is characterized by its developing map D : M denotes the universal covering of M) and the holonomy representation ρ : π1 (M) → G. Moreover, the developing map is a local homeomorphism, and it is π1 (M)-equivariant is the action by deck transformations). (where the action of π1 (M) on M For more details, we refer to the recent expository paper [Gol10], or to the book [Car03] oriented towards a physics audience. 2.2. Background on the AdS space. Let R2,2 denote the vector space R4 equipped with a quadratic form q2,2 of signature (2, 2). The Anti-de Sitter AdS3 space is defined as the −1 level set of q2,2 in R2,2 , endowed with the Lorentz metric induced by q2,2 . On the Lie algebra gl(2, R) of 2 × 2 matrices with real coefficients, the determinant defines a quadratic form of signature (2, 2). Hence we can consider the anti-de Sitter space AdS3 as the group SL(2, R) equipped with its Killing metric, which is bi-invariant. There is therefore an isometric action of SL(2, R) × SL(2, R) on AdS3 , where the two factors act by left and right multiplication, respectively. It is well known (see [Mes07]) that this yields an isomorphism between the identity component Isom0 (AdS3 ) of the isometry group of AdS3 and SL(2, R) × SL(2, R)/ ± (I, I ). It follows directly that the identity component of the isometry group of AdS3,+ (the quotient of AdS3 by the antipodal map) is PSL(2, R) × PSL(2, R). In all of this paper, we denote by Isom0,+ the identity component of the isometry group of AdS3,+ , so that Isom0,+ is isomorphic to PSL(2, R) × PSL(2, R). Another way to identify the identity component of the isometry group of AdS3 is by considering the projective model of AdS3,+ , as the interior (one connected component of the complement) of a quadric Q ⊂ RP 3 . This quadric is ruled by two families of lines, which we call the “left” and “right” families and denote by Ll , Lr . Those two families of lines have a natural projective structure (given for instance by the intersection of the lines of Ll with a fixed line of Lr ). Given an isometry u ∈ Isom0,+ , it acts projectively on both Ll and Lr , defining two elements ρl , ρr of PSL(2, R). This provides an identification of Isom0,+ with PSL(2, R) × PSL(2, R). The projective space RP 3 referred to above is of course the projectivization of R2,2 , and the elements of the quadric Q are the projections of q2,2 -isotropic vectors. The geodesics of AdS3,+ are the intersections between projective lines of RP 3 and the interior of Q. Such a projective line is the projection of a 2-plane P in R2,2 . If the signature of the restriction of q2,2 to P is (1, 1), then the geodesic is said to be space-like, if it is (0, 2) the geodesic is time-like, and if the restriction of q2,2 to P is degenerate then the geodesic is light-like. Similarly, totally geodesic planes are projections of 3-planes in R2,2 . They can be space-like, light-like or time-like. Observe that space-like planes in AdS3,+ , with the induced metric, are isometric to the hyperbolic disk. Actually, their images in the projective model of AdS3,+ are Klein models of the hyperbolic disk. Time-like planes in AdS3,+ are isometric to the anti-de Sitter space of dimension two. Consider an affine chart of RP 3 , complement of the projection of a space-like hyperplane of R2,2 . The quadric in such an affine chart is a one-sheeted hyperboloid. The interior of this hyperboloid is an affine chart of AdS3 . The intersection of a geodesic of AdS3,+ with an affine chart is a component of the intersection of the affine chart with an affine line . The geodesic is space-like if intersects1 twice the hyperboloid, light-like if is tangent to the hyperboloid, and time-like if avoids the hyperboloid. 1 Of course, such an intersection may happen at the projective plane at infinity.
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For any p in AdS3,+ , the q2,2 -orthogonal p ⊥ is a space-like hyperplane. Its complement is therefore an affine chart, that we denote by A( p). It is the affine chart centered at p. Observe that A( p) contains p, any non-time-like geodesic containing p is contained in A( p). Unfortunately, affine charts always miss some region of AdS3,+ , and we will consider regions of AdS3,+ which do not fit entirely in such an affine chart. In this situation, one can consider the conformal model: there is a conformal map from AdS3 to D2 × S1 , equipped with the metric ds02 − dt 2 , where ds02 is the spherical metric on the disk D2 , i.e. where (D2 , ds02 ) is a hemisphere (see [HE73, pp. 131–133]). One needs also to consider the universal covering AdS3 . It is conformally isometric to D2 × R equipped with the metric ds02 − dt 2 . But it is also advisable to consider it as the union of an infinite sequence (An )(n∈Z) of closures of affine charts. This sequence is totally ordered, the interior An of every term lying in the future of the previous one and in the past of the next one. The interiors An are separated one from the other by a space-like plane, i.e. a totally geodesic plane isometric to the hyperbolic disk. Observe that each space-like or light-like geodesic of AdS3 is contained in such an affine chart; whereas each time-like geodesic intersects every copy An of the affine chart. If two time-like geodesics meet at some point p, then they meet infinitely many times. More precisely, there is a point q in AdS3 such that if a time-like geodesic contains p, then it contains q also. Such a point is said to be conjugate to p. The existence of conjugate points corresponds to the fact that for any p in AdS3 ⊂ R2,2 , every 2-plane containing p contains also − p. If we consider AdS3 as the union of infinitely many copies An (n ∈ Z) of the closure of the affine chart A( p) centered at p, with A0 = A( p), then the points conjugate to p are precisely the centers of the An , all representing the same element in the interior of the hyperboloid. The center of A1 is the first conjugate point p + of p in the future. It has the property that any other point in the future of p and conjugate to p lies in the future of p + . Inverting the time, one defines similarly the first conjugate point p − of p in the past as the center of A−1 . Finally, the future in A0 of p is the interior of a convex cone based at p (more precisely, the interior of the convex hull in RP 3 of the union of p with the space-like 2-plane between A0 and A1 ). The future of p in AdS3 is the union of this cone with all the An with n > 0. In particular, one can give the following description of the domain E( p), intersection between the future of p − and the past of p + : it is the union of A0 , the past of p + in A1 and the future of p − in A−1 . We will need a similar description of 2-planes in AdS3 (i.e. of totally geodesic hypersurfaces) containing a given space-like geodesic. Let c be such a space-like geodesic, consider an affine chart A0 centered at a point in c (therefore, c is the segment joining two points in the hyperboloid). The set composed of the first conjugate points in the future of points in c is a space-like geodesic c+ , contained in the chart A1 . Every time-like 2-plane containing c contains also c+ , and vice versa. The intersection between the future of c and the past of c+ is the union of: • a wedge between two light-like half-planes both containing c in their boundary, • a wedge between two light-like half-planes both containing c+ in their boundary, • the space-like 2-plane between A0 and A1 .
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3. Singularities in Singular AdS-Spacetimes In this paper, we require spacetimes to be oriented and time oriented. Therefore, by (regular) AdS-spacetime we mean an (Isom0 (AdS3 ), AdS3 )-manifold. In this section, we classify singular lines and singular points in singular AdS-spacetimes. Actually, our first concern is the AdS background, but all this analysis can be easily extended to a more general situation, leading in a straightforward way to the notion of singular dS-spacetimes; or singular flat spacetimes (with regular part locally modelled on the Minkowski space). In order to understand the notion of singularities, let us consider first the similar situation in the classical case of Riemannian geometric structures, for example, of (singular) Euclidean manifolds (see p. 523-524 of [Thu98]). Locally, a singular point p in a singular Euclidean space is the intersection of various singular rays, the complement of these rays being locally isometric to R3 . The singular rays look as if they were geodesic rays. Since the singular space is assumed to have a manifold topology, the space of rays, singular or not, starting from p is a topological 2-sphere L( p): the link of p. Outside the singular rays, L( p) is locally modeled on the space of rays starting from a point in the regular model, i.e. the 2-sphere S2 equipped with its usual round metric. But this metric degenerates on the singular points of L( p), i.e. the singular rays. The way it may degenerate is described similarly: let r be a singular point in L( p) (a singular ray), and let ( p) be the space of rays in L( p) starting from r . It is a topological circle, locally modeled on the space 0 of geodesic rays at a point in the metric sphere S2 . The space 0 is naturally identified with the 1-sphere S1 of perimeter 2π , and locally S1 -structures on topological circles ( p) are easily classified: they are determined by a positive real number, the cone angle, and ( p) is isomorphic to 0 if and only if this cone angle is 2π . Therefore, the link L( p) is naturally equipped with a spherical metric with coneangle singularities, and one easily recovers the geometry around p by a fairly intuitive construction, the suspension of L( p). We refer to [Thu98] for further details. Our approach in the AdS case is similar. The neighborhood of a singular point p is the suspension of its link L( p), this link being a topological 2-sphere equipped with a structure whose regular part is locally modeled on the link HS2 of a regular point in AdS3 , and whose singularities are suspensions of their links (r ), which are circles locally modeled on the link of a point in HS2 . However, the situation in the AdS case is much more intricate than in the Euclidean case, since there is a bigger variety of singularity types in L( p): a singularity in L( p), i.e. a singular ray through p can be time-like, space-like or light-like. Moreover, nontime-like lines may differ through the causal behavior near them (for the definition of the future and past of a singular line, see Sect. 3.6). Proposition 3.1. The various types of singular lines in AdS spacetimes are: • • • •
Time-like lines: they correspond to massive particles (see Sect. 3.7.1). Light-like lines of degree 2: they correspond to photons (see Remark 3.24). Space-like lines of degree 2: they correspond to tachyons (see Sect. 3.7.2). Future BTZ-like singular lines: These singularities are characterized by the property that it is space-like, but has no future. • Past BTZ-like singular lines: These singularities are characterized by the property that it is space-like, but has no past. • (Past or future) extreme BTZ-like singular lines: they look like past/future BTZ-like singular lines, except that they are light-like.
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• Misner lines: they are space-like, but have no future and no past. Moreover, any neighborhood of the singular lines contains closed time-like curves. • Light-like or space-like lines of degree k ≥ 4: they can be described as k/2-branched cover over light-like or space-like lines of degree 2 (in particular, the degree k is even). They have the “unphysical” property of admitting a non-connected future. The several types of singular lines, as a not-so-big surprise, reproduce the several types of particles considered in physics. Some of these singularities appear in the physics litterature, but, as far as we know, not all of them (for example, the terminology tachyons, that we feel is adapted, does not seem to appear anywhere). In Sect. 3.1 we briefly present the space HS2 of rays through a point in AdS3 . In Sect. 3.2, we give the precise definition of regular HS-surfaces and their suspensions. In Sect. 3.3 we classify the circles locally modeled on links of points in HS2 , i.e. of singularities of singular HS-surfaces which can then be defined in the following Sect. 3.4. In this Sect. 3.4, we can state the definition of singular AdS spacetimes. In Sect. 3.5, we classify singular lines. In Sect. 3.6 we define and study the causality notion in singular AdS spacetimes. In particular we define the notion of causal HS-surface, i.e. singular points admitting a neighborhood containing no closed causal curve. It is in this section that we establish the description of the causality relation near the singular lines as stated in Proposition 3.1. Finally, in Sect. 3.7, we provide a geometric description of each singular line; in particular, we justify the “massive particle”, “photon” and “tachyon” terminology. Remark 3.2. More generally, HS2 is the model of links of points in arbitrary Lorentzian manifolds. Analogs of Proposition 3.1 still hold in the context of flat or locally de Sitter manifolds. 3.1. HS geometry. Given a point p in AdS3 , let L( p) be the link of p, i.e. the set of (non-parametrized) oriented geodesic rays based at p. Since these rays are determined by their tangent vector at p up to rescaling, L( p) is naturally identified with the set of AdS3 . Geometrically, T p AdS3 is a copy of Minkowski space R1,2 . Denote by rays in T p 2 HS the set of geodesic rays issued from 0 in R1,2 . It admits a natural decomposition in five subsets: • the domains H2+ and H2− composed respectively of future oriented and past oriented time-like rays, • the domain dS2 composed of space-like rays, • the two circles ∂H2+ and ∂H2− , boundaries of H2± in HS2 . The domains H2± are the Klein models of the hyperbolic plane, and dS2 is the Klein model of de Sitter space of dimension 2. The group SO0 (1, 2), i.e. the group of timeorientation preserving and orientation preserving isometries of R1,2 , acts naturally (and projectively) on HS2 , preserving this decomposition. The classification of elements of SO0 (1, 2) ≈ PSL(2, R) is presumably well-known by most of the readers, but we stress here that it is related to the HS2 -geometry: let g be a non-trivial element of SO0 (1, 2). • g is elliptic if and only if it admits exactly two fixed points, one in H2+ , and the other (the opposite) in H2− , • g is parabolic if and only if it admits exactly two fixed points, one in ∂H2+ , and the other (the opposite) in ∂H2− ,
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• g is hyperbolic if and only if it admits exactly 6 fixed points: two pairs of opposite points in ∂H2± , and one pair of opposite points in dS2 . In particular, g is elliptic (respectively hyperbolic) if and only if it admits a fixed in H2± (respectively in dS2 ). 3.2. Suspension of regular HS-surfaces. Definition 3.3. A regular HS-surface is a topological surface endowed with a (SO0 (1, 2), HS2 )-structure. The SO0 (1, 2)-invariant orientation on HS2 induces an orientation on every regular HS-surface. Similarly, the dS2 regions admit a canonical time orientation. Hence any regular HS-surface is oriented, and its de Sitter regions are time oriented. Given a regular HS-surface , and once a point p is fixed in AdS3 , we can construct a locally AdS manifold e(), called the suspension of , defined as follows: • for any v in HS2 ≈ L( p), let r (v) be the geodesic ray issued from p tangent to v. If v lies in the closure of dS2 , it defines e(v) := r (v); if v lies in H2± , let e(v) be the portion of r (v) between p and the first conjugate point p ± . • for any open subset U in HS2 , let e(U ) be the union of all e(v) for v in U . Observe that e(U )\{ p} is an open domain in AdS3 , and that e(HS2 ) is the intersection E( p) between the future of the first conjugate point in the past and the past of the first conjugate point in the future (cf. the end of Sect. 2.2). The regular HS-surface can be understood as the disjoint union of open domains Ui in HS2 , glued one to the other by coordinate change maps gi j given by restrictions of elements of SO0 (1, 2): gi j : Ui j ⊂ U j → U ji ⊂ Ui . But SO0 (1, 2) can be considered as the group of isometries of AdS3 fixing p. Hence every gi j induces an identification between e(Ui j ) and e(U ji ). Define e() as the disjoint union of the e(Ui ), quotiented by the relation identifying q in e(Ui j ) with gi j (q) in e(U ji ). This quotient space contains a special point p, ¯ represented in every e(Ui ) by p, and called the vertex (we will sometimes abusively denote p¯ by p). The fact that is a surface implies that e()\ p¯ is a three-dimensional manifold, homeomorphic to × R. The topological space e() itself is homeomorphic to the cone over . Therefore e() is a (topological) manifold only when is homeomorphic to the 2-sphere. But it is easy to see that every HS-structure on the 2-sphere is isomorphic to HS2 itself; and the suspension e(HS2 ) is simply the regular AdS-manifold E( p). Hence in order to obtain singular AdS-manifolds that are not merely regular AdSmanifolds, we need to consider (and define!) singular HS-surfaces. Remark 3.4. A similar construction holds for locally flat or locally de Sitter spacetimes, leading, mutatis mutandis to the notion of flat or de Sitter suspensions of HS-surfaces. 3.3. Singularities in singular HS-surfaces. The classification of singularities in singular HS-surfaces essentially reduces (but not totally) to the classification of RP1 -structures on the circle.
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3.3.1. Real projective structures on the circle. Let RP1 be the real projective line, and 1 and the real 1 be its universal covering. We fix a homeomorphism between RP let RP 1 . Let G be the group PSL(2, R) line: this defines an orientation and an order < on RP 1 of projective transformations of RP , and let G˜ be its universal covering: it is the group of projective transformations of RP1 . We have an exact sequence: 0 → Z → G˜ → G → 0. 1 the inequality Let δ be the generator of the center Z such that for every x in RP 1 by Z is projectively isomorphic to RP1 . δx > x holds. The quotient of RP The elliptic-parabolic-hyperbolic classification of elements of G induces a similar ˜ according to the nature of their projection in G. Observe classification for elements in G, 1 as translations, i.e. freely and properly disthat non-trivial elliptic elements act on RP continuously. Hence the quotient space of their action is naturally a real projective structure on the circle. We call these quotient spaces elliptic circles. Observe that it includes the usual real projective structure on RP1 . Parabolic and hyperbolic elements can all be decomposed as a product g˜ = δ k g, where g has the same nature (parabolic or hyperbolic) as g, ˜ but admits fixed points in 1 RP . The integer k ∈ Z is uniquely defined. Observe that if k = 0, the action of g˜ on 1 is free and properly discontinuous. Hence the associated quotient space, which is RP naturally equipped with a real projective structure, is homeomorphic to the circle. We call it a parabolic or hyperbolic circle, according to the nature of g, of degree k. Inverting g˜ if necessary, we can always assume, up to a real projective isomorphism, that k ≥ 1. 1. Finally, let g be a parabolic or hyperbolic element of G˜ fixing a point x0 in RP Let x1 be the unique fixed point of g such that x1 > x0 and such that g admits no fixed point between x0 and x1 : if g is parabolic, x1 = δx0 ; and if g is hyperbolic, x1 is the unique g-fixed point in ]x0 , δx0 [. Then the action of g on ]x0 , x1 [ is free and properly discontinuous, the quotient space is a parabolic or hyperbolic circle of degree 0. These examples exhaust the list of real projective structures on the circle up to a real 1 projective isomorphism. We briefly recall the proof: the developing map d : R → RP of a real projective structure on R/Z is a local homeomorphism from the real line into the real line, hence a homeomorphism onto its image I . Let ρ : Z → G˜ be the holonomy morphism: being a homeomorphism, d induces a real projective isomorphism between the initial projective circle and I /ρ(Z). In particular, ρ(1) is non-trivial, preserves I , and acts freely and properly discontinuously on I . An easy case-by-case study leads to a proof of our claim. It follows that every cyclic subgroup of G˜ is the holonomy group of a real projective circle, and that two such real projective circles are projectively isomorphic if and only if their holonomy groups are conjugate one to the other. But some subtlety appears if one takes into consideration the orientations: usually, by real projective structure we mean a (PGL(2, R), RP1 )-structure, i.e. coordinate changes might reverse the orientation. In particular, two such structures are isomorphic if there is a real projective transformation conjugating the holonomy groups, even if this transformation reverses the orientation. But here, by RP1 -circle we mean a (G, RP1 )-structure on the circle, with G = PSL(2, R). In particular, it admits a canonical orientation, preserved by the holonomy group: the one whose lifting to R is such that the developing map is orientation preserving. To be a RP1 -isomorphism, a real projective conjugacy needs to preserve this orientation.
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Let L be a RP1 -circle. Let γ0 be the generator of π1 (L) such that, for the canonical orientation defined above, and for every x in the image of the developing map: ρ(γ0 )x > x.
(1)
1. Let ρ(γ0 ) = δ k g be the decomposition such that g admits fixed points in RP According to the inequality (1), the degree k is non-negative. Moreover: The elliptic case. Elliptic RP1 -circles (i.e. with elliptic holonomy) are uniquely parametrized by a positive real number (the angle). The case k ≥ 1. Non-elliptic RP1 -circles of degree k ≥ 1 are uniquely parametrized by the pair (k, [g]), where [g] is a conjugacy class in G. Hyperbolic conjugacy classes are uniquely parametrized by a positive real number: the modulus of their trace. There are exactly two parabolic conjugacy classes: the positive parabolic class, composed of 1 , and the negative parathe parabolic elements g such that gx ≥ x for every x in RP 1 (this bolic class, made of the parabolic elements g such that gx ≤ x for every x in RP terminology is justified in Sect. 3.7.5, and Remark 3.18). The case k = 0. In this case, L is isomorphic to the quotient by g of a segment ]x0 , x1 [ admitting as extremities two successive fixed points of g. Since we must have gx > x for every x in this segment, g cannot belong to the negative parabolic class: Every parabolic RP1 -circle of degree 0 is positive. Concerning the hyperbolic RP1 -circles, the conclusion is the same as in the case k ≥ 1: they are uniquely parametrized by ˜ any RP1 -circle of a positive real number. Indeed, given a hyperbolic element g in G, degree 0 with holonomy g is a quotient of a segment ]x0 , x1 [, where the left extremity x0 is a repelling fixed point of g, and the right extremity an attractive fixed point. 3.3.2. HS-singularities. For every p in HS2 , let ( p) the link of p, i.e. the space of rays in T p HS2 . Such a ray v defines an oriented projective line cv starting from p. Let p be the stabilizer in SO0 (1, 2) ≈ PSL(2, R) of p. Definition 3.5. A ( p , ( p))-circle is the data of a point p in H S 2 and a ( p , ( p))structure on the circle. Since HS2 is oriented, ( p) admits a natural RP1 -structure, and thus every ( p , ( p))circle admits a natural underlying RP1 -structure. Given a ( p , ( p))-circle L, we construct a singular HS-surface e(L): for every element v in the link of p, define e(v) as the closed segment [− p, p] contained in the projective ray defined by v, where − p is the antipodal point of p in HS2 , and then operate as we did for defining the AdS space e() associated to a regular HS-surface. The resulting space e(L) is topologically a sphere, locally modeled on HS2 in the complement of two singular points corresponding to p and − p. These singular points will be typical singularities in singular HS-surfaces. Here, the singularity corresponding to p as a preferred status, as representation a ( p , ( p))-singularity. There are several types of singularity, mutually non isomorphic: • time-like singularities: they correspond to the case where p lies in H2± . Then, p is a 1-parameter elliptic subgroup of G, and L is an elliptic RP1 -circle. When p lies in H2+ (respectively H2− ), then the singularity is a future (respectively past) time-like singularity. • space-like singularities: when p lies in dS2 , p is a one-parameter subgroup consisting of hyperbolic elements of SO0 (1, 2), and L is a hyperbolic RP1 -circle.
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• light-like singularities: it is the case where p lies in ∂H2± . The stabilizer p is a one-parameter subgroup consisting of parabolic elements of SO0 (1, 2), and the link L is a parabolic RP1 -circle. We still have to distinguish between past and future light-like singularities. It is easy to classify time-like singularities up to (local) HS-isomorphisms: they are locally characterized by their underlying structure of the elliptic RP1 -circle. In other words, time-like singularities are nothing but the usual cone singularities of hyperbolic surfaces, since they admit neighborhoods locally modeled on the Klein model of the hyperbolic disk. But there are several types of space-like singularities, according to the causal structure around them. More precisely: recall that every element v of ( p) is a ray in T p HS2 , tangent to a parametrized curve cv starting at p and contained in a projective line of HS2 = P(R1,2 ). Taking into account that dS2 is the Klein model of the 2-dimensional de Sitter space, it follows that v, as a direction in a Lorentzian spacetime, can be a timelike, light-like or space-like direction. Moreover, in the two first cases, it can be future oriented or past oriented. Definition 3.6. If p lies in dS2 , we denote by i + (( p)) (respectively i − (( p))) the set of future oriented (resp. past oriented) directions. Observe that i + (( p)) and i − (( p)) are connected, and that their complement in ( p) has two connected components. This notion can be extended to light-like singularities: Definition 3.7. If p lies in ∂H2+ , the domain i + (( p)) (respectively i − (( p))) is the set of directions v such that cv (s) lies in H2+ (respectively dS2 ) for s sufficiently small. Similarly, if p lies in ∂H2− , the domain i − (( p)) (respectively i + (( p))) is the set of directions v such that cv (s) lies in H2− (respectively dS2 ) for s sufficiently small. In this situation, i + (( p)) and i − (( p)) are the connected components of the complement of the two points in ( p) which are directions tangent to ∂H2± . For time-like singularities, we simply define i + (( p)) = i − (( p)) = ∅. Finally, observe that the extremities of the arcs i ± (( p)) are precisely the fixed points of p . Definition 3.8. Let L be a ( p , ( p))-circle. Let d : L˜ → ( p) the developing map. ˜ preserved by The preimages d −1 (i + (( p))) and d −1 (i − (( p))) are open domain in L, the deck transformations. Their projections in L are denoted respectively by i + (L) and i − (L). We invite the reader to convince himself that the RP1 -structure and the additional data of i ± (L) determine the ( p , ( p))-structure on the link, hence the HS-singular point up to HS-isomorphism. In the sequel, we present all the possible types of singularities, according to the position in HS2 of the reference point p, and according to the degree of the underlying RP1 -circle. Some of them are called BTZ-like or Misner singularities; the reason for this terminology will be explained later in Sects. 3.7.4, 3.7.3, respectively. (1) time-like singularities: We have already observed that they are easily classified: they can be considered as H2 -singularities. They are characterized by their cone angle, and by their future/past quality.
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(2) space-like singularities of degree 0: Let L be a space-like singularity of degree 0, i.e. a ( p , ( p))-circle such that the underlying hyperbolic RP1 -circle has degree 0. Then the holonomy of L is generated by a hyperbolic element g, and L is isomorphic to the quotient of an interval I of ( p) by the group g generated by g. The extremities of I are fixed points of g, therefore we have three possibilities: • If I = i + (( p)), then L = i + (L) and i − (L) = ∅. The singularity is then called a BTZ-like past singularity. • If I = i − (( p)), then L = i − (L) and i + (L) = ∅. The singularity is then called a BTZ-like future singularity. • If I is a component of ( p) \ (i + (( p)) ∪ i − (( p))), then i + (L) = i − (L) = ∅. The singularity is a Misner singularity. (3) light-like singularities of degree 0: When p lies in ∂H2+ , and when the underlying parabolic RP1 -circle has degree 0, then L is the quotient of i + (( p)) or i − (( p)) by a parabolic element. • If I = i + (( p)), then L = i + (L) and i − (L) = ∅. The singularity is then called a future cuspidal singularity. Indeed, in that case, a neighborhood of the singular point in e(L) with the singular point removed is an annulus locally modelled on the quotient of H2+ by a parabolic isometry, i.e., a hyperbolic cusp. • If I = i − (( p)), then L = i − (L) and i + (L) = ∅. The singularity is then called a extreme BTZ-like future singularity. The case where p lies in ∂H2− and L of degree 0 is similar; we get the notion of past cuspidal singularity and extreme BTZ-like past singularity. (4) space-like singularities of degree k ≥ 1: when the singularity is space-like of degree k ≥ 1, i.e. when L is a hyperbolic ( p , ( p))-circle of degree ≥ 1, the situation is slightly more complicated. In that situation, L is the quotient of the universal 1 by a group generated by an element of the form δ k g, where δ covering L˜ p ≈ RP is in the center of G˜ and g admits fixed points in L˜ p . Let I ± be the preimage in L˜ p of i ± (( p)) by the developing map. Let x0 be a fixed point of g in L˜ p which is a left extremity of a component of I + (recall that we have prescribed an orientation, i.e. an order, on the universal covering of any RP1 -circle: the one for which the developing map is increasing). Then, this component is an interval ]x0 , x1 [, where x1 is another g-fixed point. All the other g-fixed points are the iterates x2i = δ i x0 and x2i+1 = δ i x1 . The components of I + are the intervals δ 2i ]x0 , x1 [ and the components of I − are δ 2i+1 ]x0 , x1 [. It follows that the degree k is an even integer. We have a dichotomy: • If, for every integer i, the point x2i (i.e. the left extremities of the components of I + ) is a repelling fixed point of g, then the singularity is a positive space-like singularity of degree k. • In the other case, i.e. if the left extremities of the components of I + are attracting fixed points of g, then the singularity is a negative space-like singularity of degree k. In other words, the singularity is positive if and only if for every x in I + we have gx ≥ x. (5) light-like singularities of degree k ≥ 1: Similarly, parabolic ( p , ( p))-circles have even degree, and the dichotomy past/future among parabolic ( p , ( p))-circles of degree ≥ 2 splits into two subcases: the positive case for which the parabolic element g satisfies gx ≥ x on L˜ p , and the negative case satisfying the reverse
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Fig. 1. A cuspidal singularity appears by taking the quotient of a half-sphere in HS2 containing H2+ and tangent to ∂H2+ at a point p. The opposite point − p then corresponds to a past extreme BTZ-like singularity
1 -cirinequality (this positive/negative dichotomy is inherent of the structure of RP cle data, cf. the end of Sect. 3.3.1). Remark 3.9. In the previous section we observed that there is only one RP1 hyperbolic circle of holonomy g up to RP1 -isomorphism, but this remark does not extend to hyperbolic ( p , ( p))-circles since a real projective conjugacy between g and g −1 , if preserving the orientation, must permute time-like and space-like components. Hence positive hyperbolic ( p , ( p))-circles and negative hyperbolic ( p , ( p))-circles are not isomorphic. Remark 3.10. Let L be a ( p , ( p))-circle. The suspension e(L) admits two singular points p, ¯ − p, ¯ corresponding to p and − p. Observe that when p is space-like, p¯ and − p, ¯ as HS-singularities, are always isomorphic. When p is time-like, one of the singularities is future time-like and the other is past time-like. If p¯ is a future light-like singularity of degree k ≥ 1, then − p¯ is a past light-like singularity of degree k, and vice versa. Finally, let p¯ be a future cuspidal singularity. The ( p , ( p))-circle L is the quotient by a cyclic group of the set of rays in T p HS2 tangent to projective rays contained in H2+ . It follows that the suspension e(L) is a cyclic quotient of the domain in HS2 delimited by the projective line tangent to ∂H2+ at p and containing H2+ . This half-space does not contain H2− . It follows that − p¯ is not a past cuspidal singularity, but rather a past extreme BTZ-like singularity (see Fig. 1).
3.4. Singular HS-surfaces. Once we know all possible HS-singularities, we can define singular HS-surfaces: Definition 3.11. A singular HS-surface is an oriented surface containing a discrete subset S such that \ S is a regular HS-surface, and such that every p in S admits a neighborhood HS-isomorphic to an open subset of the suspension e(L) of a ( p , ( p))circle L. The construction of AdS-manifolds e() extends to singular HS-surfaces: Definition 3.12. A singular AdS spacetime is a 3-manifold M containing a closed subset L (the singular set) such that M \ L is a regular AdS-spacetime, and such that every x in L admits a neighborhood AdS-isomorphic to the suspension e() of a singular HS-surface.
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Since we require M to be a manifold, each cone e() must be a 3-ball, i.e. each surface must be actually homeomorphic to the 2-sphere. There are two types of points in the singular set of a singular AdS spacetime: Definition 3.13. Let M be a singular AdS spacetime. A singular line in M is a connected subset of the singular set composed of the points x such that every neighborhood of x is AdS-isomorphic to the suspension e(x ), where x is a singular HS-surface e(L x ), where L x is a ( p , ( p))-circle. An interaction (or collision) in M is a point x in the singular set which is not on a singular line. Consider point x in a singular line. Then, by definition, a neighborhood U of x is isomorphic to the suspension e(x ), where the HS-sphere x is the suspension of a ( p , ( p))-circle L. The suspension e(L) contains precisely two opposite points p¯ and − p. ¯ Each of them defines a ray in U , and every point x in these rays are singular points, whose links are also described by the same singular HS-sphere e(L). Therefore, we can define the type of the singular line: it is the type of the ( p , ( p))circle describing the singularity type of each of its elements. Therefore, a singular line is time-like, space-like or light-like, and it has a degree. On the other hand, when x is an interaction, then the HS-sphere x is not the suspension of a ( p , ( p))-circle. Let p¯ be a singularity of x . It defines in e(x ) a ray, and for every y in this ray, the link of y is isomorphic to the suspension e(L) of the ( p , ( p))-circle defining the singular point p. ¯ It follows that the interactions form a discrete closed subset. In the neighborhood of an interaction, with the interaction removed, the singular set is an union of singular lines, along which the singularity-type is constant (however see Remark 3.10).
3.5. Classification of singular lines. The classification of singular lines, i.e. of ( p , ( p))-circles, follows from the classification of singularities of singular HS-surfaces: • • • • •
time-like lines, space-like or light-like line of degree 2, BTZ-like singular lines, extreme or not, past or future, Misner lines, space-like or light-like line of degree k ≥ 4. Recall that the degree is necessarily even.
Indeed, according to Remark 3.10, what could have been called a cuspidal singular line, is actually an extreme BTZ-like singular line. 3.6. Local future and past of singular points. In the previous section, we almost completed the proof of Proposition 3.1, except that we still have to describe, as stated in this proposition, what is the future and the past of the singular line (in particular, that the future and the past of non-time-like lines of degree k ≥ 2 has k/2 connected components), and to see that Misner lines are surrounded by closed causal curves. Let M be a singular AdS-manifold M. Outside the singular set, M is isometric to an AdS manifold. Therefore one can define as usual the notion of time-like or causal curve, at least outside singular points. If x is a singular point, then a neighborhood U of x is isomorphic to the suspension of a singular HS-surface x . Every point in x , singular or not, is the direction of a
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line in U starting from x. When x is singular, is a singular line, in the meaning of Definition 3.13; if not, , with x removed, is a geodesic segment. Hence, we can extend the notion of causal curves, allowing them to cross an interaction or a space-like singular line, or to go for a while along a time-like or a light-like singular line. Once this notion is introduced, one can define the future I + (x) of a point x as the set of final extremities of future oriented time-like curves starting from x. Similarly, one defines the past I − (x), and the causal past/future J ± (x). Let H+x (resp. H− x ) be the set of future (resp. past) time-like elements of the HS-surface x . It is easy to see that the local future of x in e(x ), which is locally isometric to M, is the open domain e(H+x ) ⊂ e(x ). Similarly, the past of x in e(x ) is e(H− x ). It follows that the causality relation in the neighborhood of a point in a time-like singular line has the same feature as the causality relation near a regular point: the local past and the local future are non-empty connected open subsets, bounded by lightlike geodesics. The same is true for a light-like or space-like singular line of degree exactly 2. On the other hand, points in a future BTZ-like singularity, extreme or not, have no future, and only one past component. This past component is moreover isometric to the quotient of the past of a point in AdS3 by a hyperbolic (parabolic in the extreme case) isometry fixing the point. Hence, it is homeomorphic to the product of an annulus by the real line. If L has degree k ≥ 4, then the local future of a singular point in e(e(L)) admits k/2 components, hence at least 2, and the local past as well. This situation is quite unusual, and in our further study we exclude it: from now on, we always assume that light-like or space-like singular lines have degree 0 or 2. Points in Misner singularities have no future, and no past. Besides, any neighborhood of such a point contains closed time-like curves (CTC in short). Indeed, in that case, e(L) is obtained by glueing the two space-like sides of a bigon entirely contained in the de Sitter region dS2 by some isometry g, and for every point x in the past side of this bigon, the image gx lies in the future of x: any time-like curve joining x to gx induces a CTC in e(L). But: Lemma 3.14. Let be a singular HS-surface. Then the singular AdS-manifold e() contains closed causal curves (CCC in short) if and only if the de Sitter region of contains CCC. Moreover, if it is the case, every neighborhood of the vertex of e() contains a CCC of arbitrarily small length. Proof. Let p¯ be the vertex of e(). Let H± p¯ denote the future and past hyperbolic part of , and let dS p¯ be the de Sitter region in . As we have already observed, the future of p¯ is the suspension e(H+p¯ ). Its boundary is ruled by future oriented lightlike lines, singular or not. It follows, as in the regular case, that any future oriented time-like line entering in the future of p¯ remains trapped therein and cannot escape anymore: such a curve cannot be part of a CCC. Furthermore, the future e(H+p¯ ) is isometric to the product (−π/2, π/2) × H+p¯ equipped with the singular Lorentz metric −dt 2 + cos2 (t)ghyp , where ghyp is the singular hyperbolic metric with cone singularities on H+p¯ induced by the HS-structure. The coordinate t induces a time function, strictly increasing along causal curves. Therefore, e(H+p¯ ) contains no CCC. It follows that CCC in e() avoid the future of p. ¯ Similarly, they avoid the past of p: ¯ all CCC are entirely contained in the suspension e(dS2p¯ ) of the de Sitter region of .
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For any real number , let f : dS2p¯ → e(dS2p¯ ) be the map associating to v in the de Sitter region the point at distance to p¯ on the space-like geodesic r (v). Then the image of f is a singular Lorentzian submanifold locally isometric to the de Sitter space rescaled by a factor λ( ). Moreover, f is a conformal isometry: its differential multiply by λ( ) the norms of tangent vectors. Since λ( ) tends to 0 with , it follows that if has a CCC, then e() has a CCC of arbitrarily short length. Conversely, if e() has a CCC, it can be projected along the radial directions on a surface corresponding to a fixed value of , keeping it causal, as can be seen from the explicit form of the metric on e() above. It follows that, when e() has a CCC, also has one. This finishes the proof of the lemma. The proof of Proposition 3.1 is now complete. Remark 3.15. All this construction can be adapted, with minor changes, to the flat or de Sitter situation, leading to a definition of singular flat or de Sitter spacetimes, locally modeled on suspensions of singular HS-surfaces. For examples, in the proof of Lemma 3.14, one has just to change the metric −dt 2 + cos2 (t)ghyp by −dt 2 + y 2 ghyp in the flat case, and by −dt 2 + cosh2 (t)ghyp in the de Sitter case. From now on, we will restrict our attention to HS-surfaces without CCC and corresponding to singular points where the future and the past, if non-empty, are connected: Definition 3.16. A singular HS-surface is causal if it admits no singularity of degree ≥ 4 and no CCC. A singular line is causal if the suspension e(L) of the associated ( p , ( p))-circle L is causal. In other words, a singular HS-surface is causal if the following singularity types are excluded: • space-like or light-like singularities of degree ≥ 4, • Misner singularities. 3.7. Geometric description of HS-singularities and AdS singular lines. The approach of singular lines we have given so far has the advantage to be systematic, but is quite abstract. In this section, we give cut-and-paste constructions of singular AdS-spacetimes which provide a better insight on the geometry of AdS singularities. 3.7.1. Massive particles. Let D be a domain in AdS3 bounded by two time-like totally geodesic half-planes P1 , P2 sharing as common boundary a time-like geodesic c. The angle θ of D is the angle between the two geodesic rays H ∩ P1 , H ∩ P2 issued from c ∩ H , where H is a totally geodesic hyperbolic plane orthogonal to c. Glue P1 to P2 by the elliptic isometry of AdS3 fixing c pointwise. The resulting space, up to isometry, only depends on θ , and not on the choices of c and of D with angle θ . The complement of c is locally modeled on AdS3 , while c corresponds to a cone singularity with some cone angle θ . We can also consider a domain D, still bounded by two time-like planes, but not embedded in AdS3 , wrapping around c, maybe several times, by an angle θ > 2π . Glueing as above, we obtain a singular spacetime with angle θ > 2π . In these examples, the singular line is a time-like singular line, and all time-like singular lines are clearly produced in this way.
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Remark 3.17. There is an important literature in physics involving such singularities, in the AdS background like here or in the Minkowski space background, where they are called wordlines, or cosmic strings, describing a massive particle in motion, with mass m := 1 − θ/2π . Hence θ > 2π corresponds to particles with negative mass - but they are usually not considered in physics. See for example [Car03, p. 41-42]. Let us mention in particular a famous example by R. Gott in [Got91], followed by several papers (for example, [Gra93,CFGO94,Ste94]) where it is shown that a (flat) spacetime containing two such singular lines may present some causal pathology at large scale. 3.7.2. Tachyons. Consider a space-like geodesic c in AdS3 , and two time-like totally geodesic planes Q 1 , Q 2 containing c. We will also consider the two light-like totally geodesic subspaces L 1 and L 2 of AdS3 containing c, and, more generally, the space P of totally geodesic subspaces containing c. Observe that the future of c, near c, is bounded by L 1 and L 2 . We choose an orientation of c: the orientation of AdS3 then induces a (counterclockwise) orientation on P, hence on every loop turning around c. We choose the indexation of the various planes Q 1 , Q 2 , L 1 and L 2 such that every loop turning counterclockwise around x, enters in the future of c through L 1 , then crosses successively Q 1 , Q 2 , and finally exits from the future of c through L 2 . Observe that if we had considered the past of c instead of the future, we would have obtained the same indexation. The planes Q 1 and Q 2 intersect each other along infinitely many space-like geodesics, always under the same angle. In each of these planes, there is an open domain Pi bounded by c and another component c+ of Q 1 ∩ Q 2 in the future of c and which does not intersect another component of Q 1 ∩ Q 2 . The component c+ is a space-like geodesic, which can also be defined as the set of first conjugate points in the future of points in c (cf. the end of Sect. 2.2). AdS3 . One of these components, denoted The union c ∪ c+ ∪ P1 ∪ P2 disconnects W , is contained in the future of c and the past of c+ . Let D be the other component, containing the future of c+ and the past of c. Consider the closure of D, and glue P1 to P2 by a hyperbolic isometry of AdS3 fixing every point in c and c+ . The resulting spacetime contains two space-like singular lines, still denoted by c, c+ , and is locally modeled on AdS3 on the complement of these lines (see Fig. 2). Clearly, these singular lines are space-like singularities, isometric to the singularities associated to a space-like ( p , ( p))-circle L of degree two. We claim furthermore that c is positive. Indeed, the ( p , ( p))-circle L is naturally identified with P. Our choice of indexation implies that the left extremity of i + (L) is L 1 . Since the holonomy sends Q 1 onto Q 2 , the left extremity L 1 is a repelling fixed point of the holonomy. Therefore, the singular line corresponding to c is positive according to our terminology. On the other hand, a similar reasoning shows that the space-like singular line c+ is negative. Indeed, the totally geodesic plane L 1 does not correspond anymore to the left extremities of the time-like components in the ( p , ( p))-circle associated to c+ , but to the right extremities. Remark 3.18. Consider a time-like geodesic in AdS3 , hitting the boundary of the future of c at a point in P1 . This geodesic corresponds to a time-like geodesic in the singular spacetime defined by our cut-and-paste surgery which coincides with before crossing P1 , and, after the crossing, with the image of by the holonomy. The direction of is closer to L 2 than was .
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Fig. 2. By removing the domain W and glueing P1 to P2 one gets a spacetime with two tachyons. If we keep W and glue P1 to P2 , we obtain a spacetime with one future BTZ singular line and one past BTZ singular line
In other words, the situation is as if the singular line c were attracting the lightrays, i.e. had positive mass. This is the reason why we call c a positive singular line (Sect. 3.8). There is an alternative description of these singularities: start again from a space-like geodesic c in AdS3 , but now consider two space-like half-planes S1 , S2 with common boundary c, such that S2 lies above S1 , i.e. in the future of S1 , and such that every timelike geodesic intersecting S1 intersects S2 (see Fig. 3). Then remove the intersection V between the past of S2 and the future of S1 , and glue S1 to S2 by a hyperbolic isometry fixing every point in c. The resulting singular spacetime contains a singular space-like line. It should be clear to the reader that this singular line is space-like of degree 2 and negative. If instead of removing a wedge V we insert it in the spacetime obtained by cutting AdS3 along a space-like half-plane S, we obtain a spacetime with a positive space-like singularity of degree 2. Last but not least, there is another way to construct space-like singularities of degree 2. Given the space-like geodesic c, let L +1 be the future component of L 1 \ c. Cut along L +1 , and glue back by a hyperbolic isometry γ fixing every point in c. More precisely, we consider the singular spacetime such that for every future oriented time-like curve in AdS3 \ L +1 terminating at L +1 , a point x can be continued in the singular spacetime by a future oriented time-like curve starting from γ x. Once more, we obtain a singular AdS-spacetime containing a space-like singular line of degree 2. We leave to the reader the proof of the following fact: the singular line is positive mass if and only if for every x in L +1 the light-like segment [x, γ x] is past-oriented, i.e. γ sends every point in L +1 in its own causal past. Remark 3.19. As a corollary we get the following description space-like HS-singularities of degree 2: consider a small disk U in dS2 and a point x in U . Let r be one light-like geodesic ray contained in U issued from x, cut along it and glue back by a hyperbolic dS2 -isometry γ like described in Fig. 4 (be careful that in this figure, the isometry, glueing the future copy of r in the boundary of U \ r into the past copy of r ; hence γ is the inverse of the holonomy). Observe that one cannot match one side on the other, but the resulting space is still homeomorphic to the disk. The resulting HS-singularity is
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Fig. 3. The cylinder represents the boundary of the conformal model of AdS. If we remove the domain V and glue S1 to S2 we get a spacetime with one tachyon. If we keep V and glue S1 to S2 , we obtain a spacetime with one Misner singular line
Fig. 4. Construction of a positive space-like singular line of degree 2
space-like, of degree 2. If r is future oriented, the singularity is positive if and only if for every y in r the image γ y lies in the future of y. If r is past oriented, the singularity is positive if and only if γ y lies in the past of y for every y in r . Remark 3.20. As far as we know, this kind of singular line is not considered in physics literature. However, it is a very natural extension of the notion of massive particles.
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It sounds to us natural to call these singularities, representing particles faster than light, tachyons, which can be positive or negative, depending on their influence on lightrays. Remark 3.21. Space-like singularity of any (even) degree 2k can be constructed as kbranched cover of a space-like singularity of degree 2. In other words, they are obtained by identifying P1 and P2 , but now seen as the boundaries of a wedge turning k times around c. 3.7.3. Misner singularities. Let S1 , S2 be two space-like half-planes with common boundary as appearing in the second version of definition of tachyons in the previous section, with S2 lying in the future of S1 . Now, instead of removing the intersection V between the future of S1 and the past of S2 , keep it and remove the other part (the main part!) of AdS3 . Glue its two boundary components S1 , S2 by an AdS-isometry fixing c pointwise. The reader will easily convince himself that the resulting spacetime contains a space-like line of degree 0, i.e. what we have called a Misner singular line (see Fig. 3). The reason of this terminology is that this kind of singularity is often considered, or mentioned2 , in papers dedicated to gravity in dimension 2 + 1, maybe most of the time in the Minkowski background, but also in the AdS background. They are attributed to Misner who considered the 3 + 1-dimensional analog of this spacetime (for example, the glueing is called “Misner identification” in [DS93]; see also [GL98]). 3.7.4. BTZ-like singularities. Consider the same data (c, c+ , P1 , P2 ) used for the description of tachyons, i.e. space-like singularities, but now remove D, and glue the boundaries P1 , P2 of W by a hyperbolic element γ0 fixing every point in c. The resulting space is a manifold B containing two singular lines, that we abusively still denote c and c+ , and is locally AdS3 outside c, c+ (see Fig. 2). Observe that every point of B lies in the past of the singular line corresponding to c+ and in the future of the singular line corresponding to c. It follows easily that c is a BTZ-like past singularity, and that c+ is a BTZ-like future singularity. Remark 3.22. Let E be the open domain in AdS3 , intersection between the future of c and the past of c+ . Observe that W \ P1 is a fundamental domain for the action on E of the group γ0 generated by γ0 . In other words, the regular part of B is isometric to the quotient E/γ0 . This quotient is precisely a static BTZ black-hole as first introduced by Bañados, Teitelboim and Zanelli in [BTZ92] (see also [Bar08a,Bar08b]). It is homeomorphic to the product of the annulus by the real line. The singular spacetime B is obtained by adjoining to this BTZ black-hole two singular lines: this follows that B is homeomorphic to the product of a 2-sphere with the real line in which c+ and c can be naturally considered respectively as the future singularity and the past singularity. This is the explanation of the “BTZ-like” terminology. More details will be given in Sect. 7.3. Remark 3.23. This kind of singularity appears in several papers in the physics literature. We point out among them the excellent paper [HM99] where Gott’s construction quoted above is adapted to the AdS case, and where a complete and very subtle description of singular AdS-spacetimes interpreted as the creation of a BTZ black-hole by a pair of light-like particles, or by a pair of massive particles is provided. In our terminology, these spacetimes contains three singularities: a pair of light-like or time-like positive singular lines, and a BTZ-like future singularity. These examples show that even if all 2 Essentially because of their main feature pointed out in Sect. 3.6: they are surrounded by CTC.
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the singular lines are causal, in the sense of Definition 3.16, a singular spacetime may exhibit big CCC due to a more global phenomenon. 3.7.5. Light-like and extreme BTZ-like singularities. The definition of a light-like singularity is similar to that of space-like singularities of degree 2 (tachyons), but starts with the choice of a light-like geodesic c in AdS3 . Given such a geodesic, we consider another light-like geodesic c+ in the future of c, and two disjoint time-like totally geodesic annuli P1 , P2 with boundary c ∪ c+ . More precisely, consider pairs of space-like geodesics (cn , c+n ) as those appearing in the description of tachyons, contained in time-like planes Q n1 , Q n2 , so that cn converge to the light-like geodesic c. Then, c+n converge to a light-like geodesic c+ , whose past extremity in the boundary of AdS3 coincide with the future extremity of c. The time-like planes Q n1 , Q n2 converge to time-like planes Q 1 , Q 2 containing c and c+ . Then Pi is the annulus bounded in Q i by c and c+ . Glue the boundaries P1 and P2 of the component D of AdS3 \ (P1 ∪ P2 ) contained in the future of c by an isometry of AdS3 fixing every point in c (and in c+ ): the resulting space is a singular AdS-spacetime, containing two singular lines, abusely denoted by c, c+ . As in the case of tachyons, we can see that these singular lines have degree 2, but they are light-like instead of space-like. The line c is called positive, and c+ is negative. Similarly to what happens for tachyons, there is an alternative way to construct lightlike singularities: let L be one of the two light-like half-planes bounded by c. Cut AdS3 along L, and glue back by an isometry γ fixing pointwise c: the result is a singular spacetime containing a light-like singularity of degree 2. Finally, extreme BTZ-like singularities can be described in a way similar to what we have done for (non extreme) BTZ-like singularities. As a matter of fact, when we glue the wedge W between P1 and P2 we obtain a (static) extreme BTZ black-hole as described in [BTZ92] (see also [Bar08b, Sect. 3.2, Sect. 10.3]). Further comments and details are left to the reader. Remark 3.24. Light-like singularities of degree 2 appear very frequently in physics, where they are called wordlines, or cosmic strings, of massless particles, or even sometimes “photons” ([DS93]). Remark 3.25. As in the case of tachyons (see Remark 3.21) one can construct light-like singularities of any degree 2k by considering a wedge turning k times around c before glueing its boundaries. Remark 3.26. A study similar to what has been done in Remark 3.18 shows that positive photons attract lightrays, whereas negative photons have a repelling behavior. Remark 3.27. However, there is no positive/negative dichotomy for BTZ-like singularities, extreme or not. Remark 3.28. From now on, we allow ourselves to qualify HS-singularities according to the nature of the associated AdS-singular lines: an elliptic HS-singularity is a (massive) particle, a space-like singularity is a tachyon, positive or negative, etc... Remark 3.29. Let [ p1 , p2 ] be an oriented arc in ∂H2+ , and for every x in H2+ consider the elliptic singularity (with positive mass) obtained by removing the wedge composed of geodesic rays issued from x and with extremity in [ p1 , p2 ], and glueing back by an
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elliptic isometry. Move x until it reaches a point x∞ in ∂H2 \ [ p1 , p2 ]. It provides a continuous deformation of an elliptic singularity to a light-like singularity, which can be continued further into dS2 by a continuous sequence of space-like singularities. Observe that the light-like (resp. space-like) singularities appearing in this continuous family are positive (resp. have positive mass). 3.8. Positive HS-surfaces. Among singular lines, i.e. “particles”, we can distinguish the ones having an attracting behavior on lightrays (see Remark 3.17, 3.18, 3.26): Definition 3.30. A HS-surface, an interaction or a singular line is positive if all spacelike and light-like singularities of degree ≥ 2 therein are positive, and if all time-like singularities have a cone angle less than 2π . 4. Particle Interactions and Convex Polyhedra This short section briefly describes a relationship between interactions of particles in 3-dimensional AdS manifolds, HS-structure on the sphere, and convex polyhedra in HS3 , the natural extension of the hyperbolic 3-dimensional by the de Sitter space. Convex polyhedra in HS3 provide a convenient way to visualize a large variety of particle interactions in AdS manifolds (or more generally in Lorentzian 3-manifolds). This section should provide the reader with a wealth of examples of particle interactions – obtained from convex polyhedra in HS3 – exhibiting various interesting behaviors. It should then be easier to follow the classification of positive causal HS-surfaces in the next section. The relationship between convex polyhedra and particle interactions might however be deeper than just a convenient way to construct examples. It appears that many, and possibly all, particle interactions in an AdS manifold satisfying some natural conditions correspond to a unique convex polyhedron in HS3 . This deeper aspect of the relationship between particle interactions and convex polyhedra is described in Sect. 4.5 only in a special case: interactions between only massive particles and tachyons. It appears likely that it extends to a more general context, however it appears preferable to restrict those considerations here to a special case which, although already exhibiting interesting phenomena, avoids the technical complications of the general case. 4.1. The space HS3 . The definition used above for HS2 can be extended as it is to higher dimensions. So HS3 is the space of geodesic rays starting from 0 in the four-dimensional Minkowski space R3,1 . It admits a natural action of S O0 (1, 3), and has a decomposition in 5 components: • The “upper” and “lower” hyperbolic components, denoted by H+3 and H−3 , corresponding to the future-oriented and past-oriented time-like rays. On those two components the angle between geodesic rays corresponds to the hyperbolic metric on H 3. • The domain d S3 composed of space-like geodesic rays. • The two spheres ∂ H+3 and ∂ H−3 which are the boundaries of H+3 and H−3 , respectively. We call Q their union. There is a natural projective model of HS3 in the double cover of RP3 – we have to use the double cover because HS3 is defined as a space of geodesic rays, rather than as a
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Fig. 5. Three types of polyhedra in HS3
space of geodesics containing 0. This model has the key feature that the connected components of the intersections of the projective lines with the de Sitter/hyperbolic regions correspond to the geodesics of the de Sitter/hyperbolic regions. Note that there is a danger of confusion with the notations used in [Sch98], since the ˜ 3 there, while the space HS3 in [Sch98] space which we call HS3 here is denoted by HS is the quotient of the space HS3 considered here by the antipodal action of Z/2Z. 4.2. Convex polyhedra in HS3 . In all this section we consider convex polyhedra in HS3 but will always suppose that they do not have any vertex on Q. We now consider such a polyhedron, calling it P. The geometry induced on the boundary of P depends on its position relative to the two hyperbolic components of HS3 , and we can distinguish three types of polyhedra (Fig. 5). • polyhedra of hyperbolic type intersect one of the hyperbolic components of HS3 , but not the other. We find for instance in this group: – the usual, compact hyperbolic polyhedra, entirely contained in one of the hyperbolic components of HS3 , – the ideal or hyperideal hyperbolic polyhedra, – the duals of compact hyperbolic polyhedra, which contain one of the hyperbolic components of HS3 in their interior. • polyhedra of bi-hyperbolic type intersect both hyperbolic components of HS3 , • polyhedra of compact type are contained in the de Sitter component of HS3 . The terminology used here is taken from [Sch01].
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We will see below that polyhedra of bi-hyperbolic type play the simplest role in relation to particle interactions: they are always related to the simpler interactions involving only massive particles and tachyons. Those of hyperbolic type are (sometimes) related to particle interactions involving a BTZ-type singularity. Polyhedra of compact type are the most exotic when considered in relation to particle interactions and will not be considered much here, for reasons which should appear clearly below. 4.3. Induced HS-structures on the boundary of a polyhedron. We now consider the geometric structure induced on the boundary of a convex polyhedron in HS3 . Those geometric structures have been studied in [Sch98,Sch01], and we will partly rely on those references, while trying to make the current section as self-contained as possible. Note however that the notion of HS metric used in [Sch98,Sch01] is more general than the notion of HS-structure considered here. In fact the geometric structure induced on the boundary of a convex polyhedron P ⊂ HS3 is an HS-structure in some, but not all, cases, and the different types of polyhedra behave differently in this respect. 4.3.1. Polyhedra of bi-hyperbolic type. This is the simplest situation: the induced geometric structure is always a causal positive singular HS-structure. The geometry of the induced geometric structure on those polyhedra is described in [Sch01], under the condition that there there is no vertex on the boundary at infinity of the two hyperbolic components of HS3 . The boundary of P can be decomposed in three components: • A “future” hyperbolic disk D+ := ∂ P ∩ H+3 , on which the induced metric is hyperbolic (with cone singularities at the vertices) and complete. • A “past” hyperbolic disk D− = ∂ P ∩ H−3 , similarly with a complete hyperbolic metric. • A de Sitter annulus, also with cone singularities at the vertices of P. In other terms, ∂ P is endowed with an HS-structure. Moreover all vertices in the de Sitter part of the HS-structure have degree 2. A key point is that the convexity of P implies directly that this HS-structure is positive: the cone angles are less than 2π at the hyperbolic vertices of P, while the positivity condition is also satisfied at the de Sitter vertices. This can be checked by elementary geometric arguments or can be found in [Sch01, Def. 3.1 and Thm. 1.3]. 4.3.2. Polyhedra of hyperbolic type. In this case the induced geometric structure is sometimes a causal positive HS-structure. The geometric structure on those polyhedra is described in [Sch98], again when P has no vertex on ∂ H+3 ∪ ∂ H−3 . Figure 6 shows on the left an example of polyhedron of hyperbolic type for which the induced geometric structure is not an HS-structure, since the upper face (in gray) is a space-like face in the de Sitter part of HS3 , so that it is not modelled on HS2 . The induced geometric structure on the boundary of the polyhedron shown on the right, however, is a positive causal HS-structure. At the upper and lower vertices, this HS-structure has degree 0. The three “middle” vertices are contained in the hyperbolic part of the HS-structure, and the positivity of the HS-structure at those vertices follows from the convexity of the polyhedron.
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Fig. 6. Two polyhedra of hyperbolic type
Fig. 7. Two polyhedra of compact type
4.3.3. Polyhedra of compact type. In this case too, the induced geometric structure is also sometimes a causal HS-structure. On the left side of Fig. 7 we find an example of a polyhedron of compact type on which the induced geometric structure is not an HS-structure – the upper face, in gray, is a space-like face in the de Sitter component of HS3 . On the right side, the geometric structure on the boundary of the polyhedron is a positive causal HS-structure. All faces are time-like faces, so that they are modelled on HS2 . The upper and lower vertices have degree 0, while the three “middle” vertices have degree 2, and the positivity of the HS-structure at those points follows from the convexity of the polyhedron (see [Sch01]).
4.4. From a convex polyhedron to a particle interaction. When a convex polyhedron has on its boundary an induced positive causal HS-structure, it is possible to consider the interaction corresponding to this HS-structure.
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This interaction can be constructed from the HS-structure by a warped product metric construction. It can also be obtained as in Sect. 2, by noting that each open subset of the regular part of the HS-structure corresponds to a cone in Ad S3 , and that those cones can be glued in a way corresponding to the gluing of the corresponding domains in the HS-structure. The different types of polyhedra – in particular the examples in Fig. 7 and Fig. 6 – correspond to different types of interactions. 4.4.1. Polyhedra of bi-hyperbolic type. For those polyhedra the hyperbolic vertices in H+3 (resp. H−3 ) correspond to massive particles leaving from (resp. arriving at) the interaction. The de Sitter vertices, at which the induced HS-structure has degree 2, correspond to tachyons. 4.4.2. Polyhedra of hyperbolic type. In the example on the right of Fig. 6, the upper and lower vertices correspond, through the definitions in Sect. 3, to two future BTZ-type singularities (or two past BTZ-type singularities, depending on the time orientation). The three middle vertices correspond to massive particles. The interaction corresponding to this polyhedron therefore involves two future (resp. past) BTZ-type singularities and three massive particles. The interactions corresponding to polyhedra of hyperbolic type can be more complex, in particular because the topology of the intersection of the boundary of a convex polyhedron with the de Sitter part of HS3 could be a sphere with an arbitrary number of disks removed. Those interactions can involve future BTZ-type singularities and massive particles, but also tachyons. 4.4.3. Polyhedra of compact type. The interaction corresponding to the polyhedron at the right of Fig. 7 is even more exotic. The upper vertex corresponds to a future BTZ-type singularity, the lower to a past BTZ-type singularity, and the three middle vertices correspond to tachyons. The interaction therefore involves a future BTZ-type singularity, a past BTZ-type singularity, and three tachyons. 4.5. From a particle interaction to a convex polyhedron. This section describes, in a restricted setting, a converse to the construction of an interaction from a convex polyhedron in HS3 . We show below that, under an additional condition which seems to be physically relevant, an interaction can always be obtained from a convex polyhedron in HS3 . Using the relation described in Sect. 2 between interactions and positive causal HS-structures, we will relate convex polyhedra to those HS-structures rather than directly to interactions. This converse relation is described here only for simple interactions involving massive particles and tachyons. 4.5.1. A positive mass condition. The additional condition appearing in the converse relation is natural in view of the following remark. Remark 4.1. Let M be a singular AdS manifold, c be a cone singularity along a time-like curve, with positive mass (angle less than 2π ). Let x ∈ c and let L x be the link of M at x, and let γ be a simple closed space-like geodesic in the de Sitter part of L x . Then the length of γ is less than 2π .
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Proof. An explicit description of L x follows from the construction of the AdS metric in the neighborhood of a time-like singularity, as seen in Sect. 2. The de Sitter part of this link contains a unique simple closed geodesic, and its length is equal to the angle at the singularity. So it is less than 2π . In the sequel we consider a singular HS-structure σ on S 2 , which is the link of an interaction involving massive particles and tachyons. This means that σ is positive and causal, and moreover: • it has two hyperbolic components, D− and D+ , on which σ restricts to a complete hyperbolic metric with cone singularities, • any future-oriented inextendible time-like line in the de Sitter region of σ connects the closure of D− to the closure of D+ . Definition 4.2. σ has positive mass if any simple closed space-like geodesic in the de Sitter part of (S 2 , σ ) has length less than 2π . This notion of positivity of mass for an interaction generalizes the natural notion of positivity for time-like singularities. 4.5.2. A convex polyhedron from simpler interactions. Theorem 4.3. Let σ be a positive causal HS-structure on S 2 , such that • it has two hyperbolic components, D− and D+ , on which σ restricts to a complete hyperbolic metric with cone singularities, • any future-oriented inextendible time-like line in the de Sitter region of σ connects the closure of D− to the closure of D+ . Then σ is induced on a convex polyhedron in HS3 if and only if it has positive mass. If so, this polyhedron is unique, and it is of bi-hyperbolic type. Proof. This is a direct translation of [Sch01, Thm. 1.3] (see in particular case D.2).
The previous theorem is strongly related to classical statements on the induced metrics on convex polyhedra in the hyperbolic space, see [Ale05]. 4.5.3. More general interactions/polyhedra. As mentioned above we believe that Theorem 4.3 might be extended to wider situations. This could be based on the statements on the induced geometric structures on the boundaries of convex polyhedra in HS3 , as studied in [Sch98,Sch01]. 5. Classification of Positive Causal HS-Surfaces In all this section denotes a closed (compact without boundary) connected positive causal HS-surface. It decomposes in three regions: • Photons: a photon is a point corresponding in every HS-chart to points in ∂H2± . Observe that a photon might be singular, i.e. corresponds to a light-like singularity (a lightlike singularity of degree one, a cuspidal singularity, or an extreme BTZ-like singularity). The set of photons, denoted P(), or simply P in the non-ambiguous situations, is the disjoint union of a finite number of isolated points (extreme BTZ-like singularities or cuspidal singularities) and of a compact embedded one dimensional manifold, i.e. a finite union of circles.
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• Hyperbolic regions: They are the connected components of the open subset H2 () of corresponding to the time-like regions H2± of HS2 . They are naturally hyperbolic surfaces with cone singularities. There are two types of hyperbolic regions: the future and the past ones. The boundary of every hyperbolic region is a finite union of circles of photons and of cuspidal (parabolic) singularities. • De Sitter regions: They are the connected components of the open subset dS2 () of corresponding to the time-like regions dS2 of HS2 . Alternatively, they are the connected components of \ P that are not hyperbolic regions. Every de Sitter region is a singular dS surface, whose closure is compact and with boundary made of circles of photons and of a finite number of extreme parabolic singularities. 5.1. Photons. Let C be a circle of photons. It admits two natural RP1 -structures, which may not coincide if C contains light-like singularities. Consider a closed annulus A in containing C so that all HS-singularities in A lie in C. Consider first the hyperbolic side, i.e. the component A H of A \ C comprising time-like elements. Reducing A if necessary we can assume that A H is contained in one hyperbolic region. Then every path starting from a point in C has infinite length in A H , and conversely every complete geodesic ray in A H accumulates on an unique point in C. In other words, C is the conformal boundary at ∞ of A H . Since the conformal boundary of H2 is naturally RP1 and that hyperbolic isometries are restrictions of real projective transformations, C inherits, as a conformal boundary of A H , a RP1 -structure that we call RP1 -structure on C from the hyperbolic side. Consider now the component A S in the de Sitter region adjacent to C. It is is foliated by the light-like lines. Actually, there are two such foliations (for more details, see 5.3 below). An adequate selection of this annulus ensures that the leaf space of each of these foliations is homeomorphic to the circle - actually, there is a natural identification between this leaf space and C: the map associating to a leaf its extremity. These foliations are transversely projective: hence they induce a RP1 -structure on C. This structure is the same for both foliations, we call it RP1 -structure on C from the de Sitter side. In order to sustain this claim, we refer to [Mes07, § 6]: first observe that C can be slightly pushed inside A S onto a space-like simple closed curve (take a loop around C following alternatively past oriented light-like segments in leaves of one of the foliations, and future oriented segments in the other foliation; and smooth it). Then apply [Mes07, Prop. 17]. If C contains no light-like singularity, the RP1 -structures from the hyperbolic and de Sitter sides coincide. But it is not necessarily true if C contains light-like singularities. Actually, the holonomy from one side is obtained by composing the holonomy from the other side by parabolic elements, one for each light-like singularity in C. Observe that in general even the degrees may not coincide. 5.2. Hyperbolic regions. Every component of the hyperbolic region has a compact closure in . It follows easily that every hyperbolic region is a complete hyperbolic surface with cone singularities (corresponding to massive particles) and cusps (corresponding to cuspidal singularities) and that is of finite type, i.e. homeomorphic to a compact surface without boundary with a finite set of points removed. Proposition 5.1. Let C be a circle of photons in , and H the hyperbolic region adjacent to C. Let H¯ be the open domain in comprising H and all cuspidal singularities contained in the closure of H . Assume that H¯ is not homeomorphic to the disk. Then, as a RP1 -circle defined by the hyperbolic side, the circle C is hyperbolic of degree 0.
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Proof. The proposition will be proved if we find an annulus in H containing no singularity and bounded by C and a simple closed geodesic in H . Indeed, the holonomy of the RP1 -structure of C coincides then with the holonomy of the RP1 -structure of the closed geodesic, and it is well-known that closed geodesics in hyperbolic surfaces are hyperbolic. Further details are left to the reader. Since we assume that H¯ is not a disk, C represents a non-trivial free homotopy class in H . Consider absolutely continuous simple loops in H freely homotopic to C in H ∪C. Let L be the length of one of them. There are two compact subsets K ⊂ K ⊂ H¯ such that every loop of length ≤ 2L containing a point in the complement of K stays outside K and is homotopically trivial. It follows that every loop freely homotopic to C of length ≤ L lies in K : by Ascoli and semi-continuity of the length, one of them has minimal length l0 (we also use the fact that C is not freely homotopic to a small closed loop around a cusp of H , details are left to the reader). It is obviously simple, and it contains no singular point, since every path containing a singularity can be shortened (observe that since is positive, cone angles of hyperbolic singular points are less than 2π ). Hence it is a closed geodesic. There could be several such closed simple geodesics of minimal length, but they are two-by-two disjoint, and the annulus bounded by two such minimal closed geodesics must contain at least one singularity since there is no closed hyperbolic annulus bounded by geodesics. Hence, there is only a finite number of such minimal geodesics, and for one of them, c0 , the annulus A0 bounded by C and c0 contains no other minimal closed geodesic. If A0 contains no singularity, the proposition is proved. If not, for every r > 0, let A(r ) be the set of points in A0 at distance < r from c0 , and let A (r ) be the complement of A(r ) in A0 . For small values of r , A(r ) contains no singularity. Thus, it is isometric to the similar annulus in the unique hyperbolic annulus containing a geodesic loop of length l0 . This remark holds as long as A(r ) is regular. Denote by l(r ) the length of the boundary c(r ) of A(r ). Let R be the supremum of positive real numbers r0 such that for every r < r0 every essential loop in A (r ) has length ≥ l(r ). Since A0 contains no closed geodesic of length ≤ l0 , this supremum is positive. On the other hand, let r1 be the distance between c0 and the singularity x1 in A0 nearest to c0 . We claim that r1 > R. Indeed: near x1 the surface is isometric to a hyperbolic disk D centered at x1 with a wedge between two geodesic rays l1 , l2 issued from x1 of angle 2θ removed. Let be the geodesic ray issued from x1 made of points at equal distance from l1 and from l2 . Assume by contradiction r1 ≤ R. Then, c(r1 ) is a simple loop, containing x1 and minimizing the length of loops inside the closure of A (r1 ). Singularities of cone angle 2π − 2θ < π cannot be approached by length minimizing closed loops, hence θ ≤ π/2. Moreover, we can assume without loss of generality that c(r ) near x1 is the projection of a C 1 -curve cˆ in D orthogonal to at x1 , and such that the removed wedge between l1 , l2 , and the part of D projecting into A(r ) are on opposite sides of this curve. For every > 0, let y1 , y2 be the points at distance from x1 in respectively l1 , l2 . Consider the geodesic i at equal distance from yi and x1 (i = 1, 2): it is orthogonal to li , hence not tangent to c. ˆ It follows that, for small enough, cˆ contains a point pi closer to yi than to x1 . Hence, c(r1 ) can be shortened by replacing the part between p1 and p2 by the union of the projections of the geodesics [ pi , yi ]. This shorter curve is contained in A (r1 ): contradiction. Hence R < r1 . In particular, R is finite. For small enough, the annulus A (R + ) contains an essential loop c of minimal length < l(R + ). Since it lies in A (R), this
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loop has length ≥ l(R). On the other hand, there is α > 0 such that any essential loop in A (R + ) contained in the α-neighborhood of c(R + ) has length ≥ l(R + ) > l(R). It follows that c is disjoint from c(R + ), and thus, is actually a geodesic loop. The annulus A bounded by c and c(R + ) cannot be regular: indeed, if it was, its union with A(R + ) would be a regular hyperbolic annulus bounded by two closed geodesics. Therefore, A contains a singularity. Let A1 be the annulus bounded by C and c : every essential loop inside A1 has length ≥ l(R) (since it lies in A (R)). It contains strictly less singularities than A0 . If we restart the process from this annulus, we obtain by induction an annulus bounded by C and a closed geodesic inside T with no singularity. 5.3. De Sitter regions. Let T be a de Sitter region of . We recall that is assumed to be positive, i.e. that all non-time-like singularities of non-vanishing degree have degree 2 and are positive. This last feature will be essential in our study (cf. Remark 5.5). Future oriented isotropic directions define two oriented line fields on the regular part of T , defining two oriented foliations. Since we assume that is causal, space-like singularities have degree 2, and these foliations extend continuously on singularities (but not differentially) as regular oriented foliations. Besides, in the neighborhood of every BTZ-like singularity x, the leaves of each of these foliations spiral around x. They thus define two singular oriented foliations F 1 , F 2 , where the singularities are precisely the BTZ-like singularities, i.e. hyperbolic time-like ones, and have degree +1. By Poincaré-Hopf index formula we immediately get: Corollary 5.2. Every de Sitter region is homeomorphic to the annulus, the disk or the sphere. Moreover, it contains at most two BTZ-like singularities. If it contains two such singularities, it is homeomorphic to the 2-sphere, and if it contains exactly one BTZ-like singularity, it is homeomorphic to the disk. Let c : R → L be a parametrization of a leaf L of F i , increasing with respect to the time orientation. Recall that the α-limit set (respectively ω-limit set) is the set of points in T which are limits of a sequence (c(tn ))(n∈N) , where (tn )(n∈N) is a decreasing (respectively an increasing) sequence of real numbers. By assumption, T contains no CCC. Hence, according to the Poincaré-Bendixson Theorem: Corollary 5.3. For every leaf L of F 1 or F 2 , oriented by its time orientation, the α-limit set (resp. ω-limit set) of L is either empty or a past (resp. future) BTZ-like singularity. Moreover, if the α-limit set (resp. ω-limit set) is empty, the leaf accumulates in the past (resp. future) direction to a past (resp. future) boundary component of T that is a point in a circle of photons, or a extreme BTZ-like singularity. Proposition 5.4. Let be a positive, causal singular HS-surface. Let T be a de Sitter component of adjacent to a hyperbolic region H along a circle of photons C. If the completion H¯ of H is not homeomorphic to the disk, then either T is a disk containing exactly one BTZ-like singularity, or the boundary of T in is the disjoint union of C and one extreme BTZ-like singularity. Proof. If T is a disk, we are done. Hence we can assume that T is homeomorphic to the annulus. Reversing the time if necessary we also can assume that H is a past hyperbolic component. Let C be the other connected boundary component of T , i.e. its future boundary. If C is an extreme BTZ-like singularity, the proposition is proved. Hence we are reduced to the case where C is a circle of photons.
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Fig. 8. Regularization of a tachyon and a light-like singularity
According to Corollary 5.3 every leaf of F 1 or F 2 is a closed line joining the two boundary components of T . For every singularity x in T , or every light-like singularity in C, let L x be the future oriented half-leaf of F 1 emerging from x. Assume that L x does not contain any other singularity. Cut along L x : we obtain a singular dS2 -surface T ∗ admitting in its boundary two copies of L x . Since L x accumulates to a point in C it develops in dS2 into a geodesic ray touching ∂H2 . In particular, we can glue the two copies of L x in the boundary of T ∗ by an isometry fixing their common point x. For the appropriate choice of this glueing map, we obtain a new dS2 -spacetime where x has been replaced by a regular point: we call this process, well defined, regularization at x (see Fig. 8). After a finite number of regularizations, we obtain a regular dS2 -spacetime T (in particular, if a given leaf of F 1 initially contains several singularities, they are eliminated during the process one after the other). Moreover, all these surgeries can actually be performed on T ∪ C ∪ H : the de Sitter annulus A can be glued to H ∪ C, giving rise to a HS-surface containing the circle of photons C disconnecting the hyperbolic region H from the regular de Sitter region T (however, the other boundary component C has been modified and does not match anymore the other hyperbolic region adjacent to T ). Moreover, the circle of photons C now contains no light-like singularity, hence its RP1 -structure from the de Sitter side coincides with the RP1 -structure from the hyperbolic side. According to Proposition 5.1 this structure is hyperbolic of degree 0: it is the quotient of an interval I of RP1 by a hyperbolic element γ0 , with no fixed point inside I . Denote by F 1 , F 2 the isotropic foliations in T . Since we performed the surgery along half-leaves of F 1 , leaves of F 1 are still closed in T . Moreover, each of them accumulates at a unique point in C: the space of leaves of F 1 is identified with C. Let 1 be the lifting of F 1 . Recall that dS2 is be the universal covering of T , and let F T 1 1 naturally identified with RP × RP \ D, where D is the diagonal. The developing map 1 into a fiber {∗} × RP1 . Besides, as → RP1 × RP1 \ D maps every leaf of F D:T affine lines, they are complete affine lines, meaning that they still develop onto the entire and the geodesic {∗} × (RP1 \ {∗}). It follows that D is a homeomorphism between T
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Fig. 9. The domain W and its quotient T
open domain W = I × RP1 \ D, i.e. the region in dS2 bounded by two γ0 -invariant isotropic geodesics. Hence T is isometric to the quotient of W by γ0 , which is well understood (see Fig. 9; it has been more convenient to draw the lift W in the region in 1 × RP 1 between the graph of the identity map and the translation δ, a region which RP is isomorphic to the universal cover of RP1 × RP1 \ D). Hence the foliation F 2 admits two compact leaves. These leaves are CCC, but it is not yet in contradiction with the fact that is causal, since the regularization might create such CCC. The regularization procedure is invertible and T is obtained from T by positive surgeries along future oriented half-leaves of F 1 , i.e. obeying the rules described in Remark 3.19. We need to be more precise: pick a leaf L 1 of F 1 . It corresponds to a vertical line in W depicted in Fig. 9. We consider the first return f map from L 1 to L 1 along future oriented leaves of F 2 : it is defined on an interval ] − ∞, x∞ [ of L 1 , where −∞ corresponds to the end of L 1 accumulating on C. It admits two fixed points x1 < x2 < x∞ , corresponding to the two compact leaves of F 2 . The former is attracting
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Fig. 10. First return maps. The identification maps along lines above time-like and light-like singularities compose the almost horizontal broken arcs which are contained in leaves of F2
and the latter is repelling. Let L 1 be a leaf of F 1 corresponding, by the reverse surgery, to L 1 . We can assume without loss of generality that L 1 contains no singularity. Let f be the first return map from L 1 into itself along future oriented leaves of F 2 (see Fig. 10). There is a natural identification between L 1 and L 1 , and since all light-like singularities and tachyons in T ∪ C are positive, the deviation of f with respect to f is in the past direction, i.e. for every x in L 1 ≈ L 1 we have f (x) ≤ f (x) (it includes the case where x is not in the domain of definition of f , in which case, by convention, f (x) = ∞). In particular, f (x2 ) ≤ x2 . It follows that the future part of the oriented leaf of F 2 through x2 is trapped below its portion between x2 , f (x2 ). Since it is closed, and not compact, it must accumulate on C. But it is impossible since future oriented leaves near C exit from C, intersect a space-like loop, and cannot go back because of orientation considerations. The proposition is proved. Remark 5.5. In Proposition 5.4 the positivity hypothesis is necessary. Indeed, consider a regular HS-surface made of one annular past hyperbolic region connected to one annular future hyperbolic region by two de Sitter regions isometric to the region T = W/γ0 appearing in the proof of Proposition 5.4. Pick up a photon x in the past boundary of one of these de Sitter components T , and let L be the leaf of F 1 accumulating in the past to x. Then L accumulates in the future to a point y in the future boundary component. Cut along L, and glue back by a parabolic isometry fixing x and y. The main argument in the proof above is that if this surgery is performed in the positive way, so that x and y become positive tachyons, then the resulting spacetime still admits two CCC, leaves of the foliation F 2 . But if the surgery is performed in the negative way, with a sufficiently
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big parabolic element, the closed leaves of F 2 in T are destroyed, and every leaf of the new foliation F 2 in the new singular surface joins the two boundary components of the de Sitter region, which is therefore causal. Theorem 5.6. Let be a singular causal positive HS-surface, homeomorphic to the sphere. Then, it admits at most one past hyperbolic component, and at most one future hyperbolic component. Moreover, we are in one of the following mutually exclusive situations: (1) Causally regular case: There is a unique de Sitter component, which is an annulus connecting one past hyperbolic region homeomorphic to the disk to a future hyperbolic region homeomorphic to the disk. (2) Interaction of black holes or white holes: There is no past or no future hyperbolic region, and every de Sitter region is a either a disk containing a unique BTZ-like singularity, or a disk with an extreme BTZ-like singularity removed. (3) Big Bang and Big Crunch: There is no de Sitter region, and only one hyperbolic region, which is a singular hyperbolic sphere - if the time-like region is a future one, the singularity is called a Big Bang; if the time-like region is a past one, the singularity is a Big Crunch. (4) Interaction of a white hole with a black hole: There is no hyperbolic region. The surface contains one past BTZ-like singularity and one future BTZ-like singularity these singularities may be extreme or not. Remark 5.7. This theorem, despite the terminology inspired from cosmology, has no serious pretention of relevance for physics. However these appelations have the advantage to provide a reasonable intuition on the geometry of the interaction. For example, in what is called a Big Bang, the spacetime is entirely contained in the future of the singularity, and the singular lines can be seen as massive particles or “photons” emitted by the initial singularity. Actually, it is one of few examples suggesting that the prescription of the surface to be a sphere could be relaxed: whereas it seems hard to imagine that the spacetime could fail to be a manifold at a singular point describing a collision of particles, it is nevertheless not so hard, at least for us, to admit that the topology of the initial singularity may be more complicated, as it is the case in the regular case (see [ABB+ 07]). Proof. If the future hyperbolic region and the past hyperbolic region is not empty, there must be a de Sitter annulus connecting one past hyperbolic component to a future hyperbolic component. By Proposition 5.4 these hyperbolic components are disks: we are in the causally regular case. If there is no future hyperbolic region, but one past hyperbolic region, and at least one de Sitter region, then there cannot be any annular de Sitter component connecting two hyperbolic regions. Hence, the closure of each de Sitter component is a closed disk. It follows that there is only one past hyperbolic component: is an interaction of black holes. Similarly, if there is a de Sitter region, a future hyperbolic region but no past hyperbolic region, is an interaction of white holes. The remaining situations are the cases where has no de Sitter region, or no hyperbolic region. The former case corresponds obviously to the description (3) of Big Bang or Big Crunch , and the latter to the description (4) of an interaction between one black hole and one white hole. Remark 5.8. It is easy to construct singular hyperbolic spheres, i.e. Big Bang or Big Crunch: take for example the double of a hyperbolic triangle. The existence of interactions of a white hole with black hole is slightly less obvious. Consider the HS-surface
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m associated to the BTZ black hole Bm . It can be described as follows: take a point p in dS2 , let d1 , d2 be the two projective circles in HS containing p, its opposite − p, and tangent to ∂H2± . It decomposes HS2 in four regions. One of these components, that we denote by U , contains the past hyperbolic region H2− . Then, m is the quotient of U by the group generated by a hyperbolic isometry γ0 fixing p, − p, d1 and d2 . Let x1 , x2 be the points where d1 , d2 are tangent to ∂H2− , and let I1 , I2 be the connected components of ∂H2− \ {x1 , x2 }. We select the index so that I1 is the boundary of the de Sitter component T1 of U containing p. Now let q be a point in T1 so that the past of q in T1 has a closure in U containing a fundamental domain J for the action of γ0 on I1 . Then there are two time-like geodesic rays starting from q and accumulating at points in I1 which are extremities of a subinterval containing J . These rays project in m onto two time-like geodesic rays l1 and l2 starting from the projection q¯ of q. These rays admit a first intersection point q¯ in the past of q. ¯ Let l1 , l2 be the subintervalls in respectively l1 , l2 with extremities q, ¯ q¯ : their union is a circle disconnecting the singular point p¯ from the boundary of the de Sitter component. Remove the component of \ (l1 ∪ l2 ) adjacent to this boundary. If q¯ is well-chosen, l1 and l2 have the same proper time. Then we can glue one to the other by a hyperbolic isometry. The resulting spacetime is as required an interaction between a BTZ black hole corresponding to p¯ with a white hole corresponding to q¯ - it contains also a tachyon of positive mass corresponding to q. ¯ 6. Global Hyperbolicity In previous sections, we considered local properties of AdS manifolds with particles. We already observed in Sect. 3.6 that the usual notions of causality (causal curves, future, past, time functions...) available for regular Lorentzian manifolds still hold. In this section, we consider the global character of causal properties of AdS manifolds with particles. The main point presented here is that, as long as no interaction appears, global hyperbolicity is still a meaningful notion for singular AdS spacetimes. This notion will be necessary in Sect. 7, as well as in the continuation of this paper [BBS10] (see also the final part of [BBS09]). The content of this section is presented in the AdS setting. We believe that most results could be extended to Minkowski or de Sitter singular manifolds. In all this section M denotes a singular AdS manifold admitting as singularities only massive particles and no interaction. The regular part of M is denoted by M ∗ . Since we will consider other Lorentzian metrics on M, we need a denomination for the singular AdS metric : we denote it g0 . 6.1. Local coordinates near a singular line. Causality notions only depend on the conformal class of the metric, and AdS is conformally flat. Hence, AdS spacetimes and flat spacetimes share the same local causal properties. Every regular AdS spacetime admits an atlas for which local coordinates have the form (z, t), where z describes the unit disk D in the complex plane, t the interval ] − 1, 1[ and such that the AdS metric is conformal to: −dt 2 + |dz|2 . For the singular case considered here, any point x lying on a singular line l (a massive particle of mass m), the same expression holds, but we have to remove a wedge
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{2απ < Arg(z) < 2π } where α = 1 − m is positive, and to glue the two sides of this wedge. Consider the map z → ζ = z 1/α : it sends the disk D with a wedge removed onto the entire disk, and is compatible with the glueing of the sides of the wedge. Hence, a convenient local coordinate system near x is (ζ, t) where (ζ, t) still lies in D×] − 1, 1[. The singular AdS metric is then, in these coordinates, conformal to (1 − m)2
|dζ |2 − dt 2 . |ζ |2m
In these coordinates, future oriented causal curves can be parametrized by the time coordinate t, and satisfies ζ (t) 1 . ≤ |ζ |m 1−m Observe that all these local coordinates define a differentiable atlas on the topological manifold M for which the AdS metric on the regular part is smooth. 6.2. Achronal surfaces. Usual definitions in regular Lorentzian manifolds still apply to the singular AdS spacetime M: Definition 6.1. A subset S of M is achronal (resp. acausal) if there is no non-trivial time-like (resp. causal) curve joining two points in S. It is only locally achronal (resp. locally acausal) if every point in S admits a neighborhood U such that the intersection U ∩ S is achronal (resp. acausal) inside U . Typical examples of locally acausal subsets are space-like surfaces, but the definition above also includes non-differentiable “space-like” surfaces, with only Lipschitz regularity. Lipschitz space-like surfaces provide actually the general case if one adds the edgeless assumption : Definition 6.2. A locally achronal subset S is edgeless if every point x in S admits a neighborhood U such that every causal curve in U joining one point of the past of x (inside U ) to a point in the future (in U ) of x intersects S. In the regular case, closed edgeless locally achronal subsets are embedded locally Lipschitz surfaces. More precisely, in the coordinates (z, t) defined in Sect. 6.1, they are graphs of 1-Lipschitz maps defined on D. This property still holds in M, except the locally Lipschitz property which is not valid anymore at singular points, but only a weaker weighted version holds: closed edgeless acausal subsets containing x corresponds to Hölder functions f : D →] − 1, 1[ differentiable almost everywhere and satisfying: dζ f <
|ζ |−m . 1−m
Go back to the coordinate system (z, t). The acausal subset is then the graph of a 1-Lipschitz map ϕ over the disk minus the wedge. Moreover, the values of ϕ on the boundary of the wedge must coincide since they have to be sent one to the other by the rotation performing the glueing. Hence, for every r < 1: ϕ(r ) = ϕ(r ei2απ ) .
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We can extend ϕ over the wedge by defining ϕ(r eiθ ) = ϕ(r ) for 2απ ≤ θ ≤ 2π . This extension over the entire D \ {0} is then clearly 1-Lipschitz. It therefore extends at 0. We have just proved: Lemma 6.3. The closure of any closed edgeless achronal subset of M ∗ is a closed edgeless achronal subset of M. Definition 6.4. A space-like surface S in M is a closed edgeless locally acausal subset whose intersection with the regular part M ∗ is a smooth embedded space-like surface. 6.3. Time functions. As in the regular case, we can define time functions as maps T : M → R which are strictly increasing along any future oriented causal curve. For nonsingular spacetimes the existence is related to stable causality : Definition 6.5. Let g, g be two Lorentzian metrics on the same manifold X . Then, g dominates g if every causal tangent vector for g is time-like for g . We denote this relation by g ≺ g . Definition 6.6. A Lorentzian metric g is stably causal if there is a metric g such that g ≺ g , and such that (X, g ) is chronological, i.e. admits no periodic time-like curve. Theorem 6.7 (See [BEE96]). A Lorentzian manifold (M, g) admits a time function if and only if it is stably causal. Moreover, when a time function exists, then there is a smooth time function. Remark 6.8. In Sect. 6.1 we defined some differentiable atlas on the manifold M. For this differentiable structure, the null cones of g0 degenerate along singular lines to half-lines tangent to the “singular” line (which is perfectly smooth for the selected differentiable atlas). Obviously, we can extend the definition of domination to the more general case g0 ≺ g, where g0 is our singular metric and g a smooth regular metric. Therefore, we can define the stable causality in this context: g0 is stably causal if there is a smooth Lorentzian metric g which is chronological and such that g0 ≺ g . Theorem 6.7 is still valid in this more general context. Indeed, there is a smooth Lorentzian metric g such that g0 ≺ g ≺ g , which is stably causal since g is dominated by the achronal metric g . Hence there is a time function T for the metric g, which is still a time function for g0 since g0 ≺ g: causal curves for g0 are causal curves for g. Lemma 6.9. The singular metric g0 is stably causal if and only if its restriction to the regular part M ∗ is stably causal. Therefore, (M, g0 ) admits a smooth time function if and only if (M ∗ , g0 ) admits a time function. Proof. The fact that (M ∗ , g0 ) is stably causal as soon as (M, g0 ) is stably causal is obvious. Let us assume that (M ∗ , g0 ) is stably causal: let g be a smooth chronological Lorentzian metric on M ∗ dominating g0 . On the other hand, using the local models around singular lines, it is easy to construct a chronological Lorentzian metric g on a tubular neighborhood U of the singular locus of g0 (the fact that g is chronological implies that the singular lines are not periodic). Actually, by reducing the tubular neighborhood U and modyfing g therein, one can assume that g dominates g on U . Let U be a smaller tubular neighborhood of the singular locus such that U ⊂ U , and let a, b be a partition of unity subordinate to U , M \ U . Then g1 = ag + bg is a smooth Lorentzian metric dominating g0 . Moreover, we also have g1 ≺ g on M ∗ . Hence any time-like curve for g1 can be slightly perturbed to a time-like curve for g avoiding the singular lines. It follows that (M, g0 ) is stably causal.
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6.4. Cauchy surfaces. Definition 6.10. A space-like surface S is a Cauchy surface if it is acausal and intersects every inextendible causal curve in M. Since a Cauchy surface is acausal, its future I + (S) and its past I − (S) are disjoint. Remark 6.11. The regular part of a Cauchy surface in M is not a Cauchy surface in the regular part M ∗ , since causal curves can exit the regular region through a time-like singularity. Definition 6.12. A singular AdS spacetime is globally hyperbolic if it admits a Cauchy surface. Remark 6.13. We defined Cauchy surfaces as smooth objects for further requirements in this paper, but this definition can be generalized for non-smooth locally achronal closed subsets. This more general definition leads to the same notion of globally hyperbolic spacetimes, i.e. singular spacetimes admitting a non-smooth Cauchy surface also admits a smooth one. Proposition 6.14. Let M be a singular AdS spacetime without interaction and with singular set reduced to massive particles. Assume that M is globally hyperbolic. Then M admits a time function T : M → R such that every level T −1 (t) is a Cauchy surface. Proof. This is a well-known theorem by Geroch in the regular case, even for general globally hyperbolic spacetimes without compact Cauchy surfaces ([Ger70]). But, the singular version does not follow immediately by applying this regular version to M ∗ (see Remark 6.11). Let l be an inextendible causal curve in M. It intersects the Cauchy surface S, and since S is achronal, l cannot be periodic. Therefore, M admits no periodic causal curve, i.e. is acausal. Let U be a small tubular neighborhood of S in M, such that the boundary ∂U is the union of two space-like hypersurfaces S− , S+ with S− ⊂ I − (S), S+ ⊂ I + (S), and such that every inextendible future oriented causal curve in U starts from S− , intersects S and then hits S + . Any causal curve starting from S− leaves immediately S− , crosses S at some point x , and then cannot cross S anymore. In particular, it cannot go back in the past of S since S is acausal, and thus, does not reach S− anymore. Therefore, S− is acausal. Similarly, S+ is acausal. It follows that S± are both Cauchy surfaces for (M, g0 ). For every x in I + (S− ) and every past oriented g0 -causal tangent vector v, the past oriented geodesic tangent to (x, v) intersects S. The same property holds for tangent vector (x, v ) nearby. It follows that there exists on I + (S− ) a smooth Lorentzian metric g1 such that g0 ≺ g1 and such that every inextendible past oriented g1 -causal curve attains S. Furthermore, we can select g1 such that S is g1 -space-like, and such that every future oriented g1 -causal vector tangent at a point of S points in the g0 -future of S. It follows that future oriented g1 -causal curves crossing S cannot come back to S: S is acausal, not only for g0 , but also for g1 . We can also define g2 in the past of S+ so that g0 ≺ g2 , every inextendible future oriented g2 -causal curve attains S, and such that S is g2 -acausal. We can now interpolate in the common region I + (S− ) ∩ I − (S+ ), getting a Lorentzian metric g on the entire M such that g0 ≺ g ≺ g1 on I + (S− ), and g0 ≺ g ≺ g2 on I − (S+ ). Observe that even if it is not totally obvious that the metrics gi can be selected continuous, we have enough room to pick such a metric g in a continuous way.
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Let l be a future oriented g -causal curve starting from a point in S. Since g ≺ g1 , this curve is also g1 -causal as long as it remains inside I + (S− ). But since S is acausal for g1 , it implies that l cannot cross S anymore: hence l lies entirely in I + (S). It follows that S is acausal for g . By construction of g1 , every past-oriented g1 -causal curve starting from a point inside I + (S) must intersect S. Since g ≺ g1 the same property holds for g -causal curves. Using g2 for points in I + (S− ), we get that every inextendible g -causal curve intersects S. Hence, (M, g ) is globally hyperbolic. According to Geroch’s Theorem in the regular case, there is a time function T : M → R whose levels are Cauchy surfaces. The proposition follows, since g0 -causal curves are g -causal curves, implying that g -Cauchy surfaces are g0 -Cauchy surfaces and that g -time functions are g0 -time functions. Corollary 6.15. If (M, g0 ) is globally hyperbolic, there is a decomposition M ≈ S × R, where every level S ×{∗} is a Cauchy surface, and very vertical line {∗}×R is a singular line or a time-like line. Proof. Let T : M → R be the time function provided by Proposition 6.14. Let X be minus the gradient (for g0 ) of T : it is a future oriented time-like vector field on M ∗ . Consider also a future oriented time-like vector field Y on a tubular neighborhood U of the singular locus: using a partition of unity as in the proof of Lemma 6.9, we can construct a smooth time-like vector field Z = aY + bX on M tangent to the singular lines. The orbits of the flow generated by Z are time-like curves. The global hyperbolicity of (M, g0 ) ensures that each of these orbits intersect every Cauchy surface, in particular, the level sets of T . In other words, for every x in M the Z -orbit of x intersects S at a point p(x). Then the map F : M → S × R defined by F(x) = ( p(x), T (x)) is the desired diffeomorphism between M and S × R. 6.5. Maximal globally hyperbolic extensions. From now we assume that M is globally hyperbolic, admitting a compact Cauchy surface S. In this section, we prove the following facts, well-known in the case of regular globally hyperbolic solutions to the Einstein equation ([Ger70]): there exists a maximal extension, which is unique up to isometry. Definition 6.16. An isometric embedding i : (M, S) → (M , S ) is a Cauchy embedding if S = i(S) is a Cauchy surface of M . Remark 6.17. If i : M → M is a Cauchy embedding then the image i(S ) of any Cauchy surface S of M is also a Cauchy surface in M . Indeed, for every inextendible causal curve l in M , every connected component of the preimage i −1 (l) is an inextendible causal curve in M, and thus intersects S. Since l intersects i(S) in exactly one point, i −1 (l) is connected. It follows that the intersection l ∩ i(S ) is non-empty and reduced to a single point: i(S ) is a Cauchy surface. Therefore, we can define Cauchy embeddings without reference to the selected Cauchy surface S. However, the natural category is the category of marked globally hyperbolic spacetimes, i.e. pairs (M, S). Lemma 6.18. Let i 1 : (M, S) → (M , S ), i 2 : (M, S) → (M , S ) be two Cauchy embeddings into the same marked globally hyperbolic singular AdS spacetime (M , S ). Assume that i 1 and i 2 coincide on S. Then, they coincide on the entire M.
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Proof. If x , y are points in M sufficiently near to S , say, in the future of S , then they are equal if and only if the intersections I − (x )∩ S and I − (y )∩ S are equal. Apply this observation to i 1 (x), i 2 (x) for x near S: we obtain that i 1 , i 2 coincide in a neighborhood of S. Let now x be any point in M. Since there is only a finite number of singular lines in M, there is a time-like geodesic segment [y, x], where y lies in S, and such that [y, x[ is contained in M ∗ (x may be singular). Then x is the image by the exponential map of some ξ in Ty M. Then i 1 (x), i 2 (x) are the image by the exponential map of respectively d y i 1 (ξ ), d y i 2 (ξ ). But these tangent vectors are equal, since i 1 = i 2 near S. Lemma 6.19. Let i : M → M be a Cauchy embedding into a singular AdS spacetime. Then, the image of i is causally convex, i.e. any causal curve in M admitting extremities in i(M) lies inside i(M). Proof. Let l be a causal segment in M with extremities in i(M). We extend it as an ˆ Let l be a connected component of lˆ ∩ i(M): it is an ininextendible causal curve l. extendible causal curve inside i(M). Thus, its intersection with i(S) is non-empty. But lˆ ∩ i(S) contains at most one point: it follows that lˆ ∩ i(M) admits only one connected component, which contains l. Corollary 6.20. The boundary of the image of a Cauchy embedding i : M → M is the union of two closed edgeless achronal subsets S + , S − of M , and i(M) is the intersection between the past of S + and the future of S − . Each of S + , S − might be empty, and is not necessarily connected. Proof. This is a general property of causally convex open subsets: S + (resp. S − ) is the set of elements in the boundary of i(M) whose past (resp. future) intersects i(M). The proof is straightforward and left to the reader. Definition 6.21. (M, S) is maximal if every Cauchy embedding i : M → M into a singular AdS spacetime is onto, i.e. an isometric homeomorphism. Proposition 6.22. (M, S) admits a maximal singular AdS extension, i.e. a Cauchy ˆ with S) embedding into a maximal globally hyperbolic singular AdS spacetime ( M, out interaction. Proof. Let M be the set of Cauchy embeddings i : (M, S) → (M , S ). We define on M the relation (i 1 , M1 , S1 ) (i 2 , M2 , S2 ) if there is a Cauchy embedding i : (M1 , S1 ) → (M2 , S2 ) such that i 2 = i ◦ i 1 . It defines a preorder on M. Let M be the space of Cauchy embeddings up to isometry, i.e. the quotient space of the equivalence relation identifying (i 1 , M1 , S1 ) and (i 2 , M2 , S2 ) if there is an isometric homeomorphism i : (M1 , S1 ) → (M2 , S2 ) such that i 2 = i ◦ i 1 . Then induces on M a preorder relation, that we still denote by . Lemma 6.18 ensures that is a partial order (if (i 1 , M1 , S1 ) (i 2 , M2 , S2 ) and (i 2 , M2 , S2 ) (i 1 , M1 , S1 ), then M1 and M2 are isometric and represent the same element of M). Now, any totally ordered subset A of M admits an upper bound in A: the inverse limit of (representants of) the elements of A. By the Zorn Lemma, we obtain that M contains a maximal element. Any representant in M) of this maximal element is a maximal extension of (M, S). Remark 6.23. The proof above is sketchy: for example, we did not justify the fact that the inverse limit is naturally a singular AdS spacetime. This is however a straightforward verification, the same as in the classical situation, and is left to the reader.
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Proposition 6.24. The maximal extension of (M, S) is unique up to isometry. 2 , S2 ) be two maximal extensions of (M, S). Consider the set of 1 , S1 ), ( M Proof. Let ( M globally hyperbolic singular AdS spacetimes (M , S ) for which there is a commutative diagram as below, where arrows are Cauchy embeddings.
Reasoning as in the previous proposition, we get that this set admits a maximal element: there is a marked extension (M , S ) of (M, S), and Cauchy embeddings ϕi : i which cannot be simultaneously extended. M → M as the union of ( M 1 , S1 ) and ( M 2 , S2 ), identified along their respective Define M embedded copies of (M , S ), through ϕ := ϕ2 ◦ ϕ1−1 , equipped with the quotient topol is Hausdorff. Assume not: there is a point x1 in ogy. The key point is to prove that M M1 , a point x2 in M2 , and a sequence yn in M such that ϕi (yn ) converges to xi , but It means that yn does not such that x1 and x2 do not represent the same element of M. converge in M , and that xi is not in the image of ϕi . Let Ui be small neighborhoods in i of xi . M i (cf. Corollary 6.20). Denote by Si+ , Si− the upper and lower boundaries of ϕi (M ) in M + Up to time reversal, we can assume that x1 lies in S1 : it implies that all the ϕ1 (yn ) lies in I − (S1+ ), and that, if U1 is small enough, U1 ∩ I − (x1 ) is contained in ϕ1 (M ). It is an open subset, hence ϕ extends to some AdS isometry ϕ between U1 and U2 (reducing the Ui if necessary). Therefore, every ϕi can be extended to isometric embeddings ϕ i of a spacetime M containing M , so that ϕ2 = ϕ ◦ ϕ1. We intend to prove that xi and Ui can be chosen such that Si is a Cauchy surface in ϕ i (M ) = ϕ i (M ) ∪ Ui . Consider past oriented causal curves, starting from x1 , and contained in S1+ . They are partially ordered by the inclusion. According to the Zorn Lemma, there is a maximal causal curve l1 satisfying all these properties. Since S1+ is disjoint from S1 , and since every inextendible causal curve crosses S, the curve l1 is not inextendible: it has a final endpoint y1 belonging to S1+ (since S1+ is closed). Therefore, any past oriented causal curve starting from y1 is disjoint from S1+ (except at the starting point y1 ). We have seen that ϕ can be extended over in a neighborhood of x1 : this extension 2 starting from x2 and contained in maps the initial part of l1 onto a causal curve in M + S2 . By compactness of l1 , this extension can be performed along the entire l1 , and the image is a causal curve admitting a final point y2 in S2+ . The points y1 and y2 are not Replacing xi by yi , we can thus separated one from the other by the topology of M. assume that every past oriented causal curve starting from xi is contained in I − (Si+ ). It follows that, once more reducing Ui if necessary, inextendible past oriented causal curves starting from points in Ui and in the future of Si+ intersects Si+ before escaping
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from Ui . In other words, inextendible past oriented causal curves in Ui ∪ I − (Si+ ) are i , and therefore, intersect Si . As required, Si is a also inextendible causal curves in M Cauchy surface in Ui ∪ ϕi (M ). Hence, there is a Cauchy embedding of (M, S) into some globally hyperbolic spacetime (M , S ), and Cauchy embeddings ϕ i : (M , S ) → ϕi (M ) ∪ Ui , which are related by some isometry ϕ : ϕ1 (M ) ∪ U1 → ϕ2 (M ) ∪ U2 : ϕ2 = ϕ ◦ ϕ1. It is a contradiction with the maximality of (M , S ). Hence, we have proved that M 1 , M 2 induce a singular is Hausdorff. It is a manifold, and the singular AdS metrics on M Observe that S1 and S2 projects in M onto the same space-like surface AdS metric on M. Without loss of generality, we can assume that S. Let l be any inextendible curve in M. 1 in M. Then every connected component of l ∩ W1 l intersects the projection W1 of M 1 . It follows that l intersects is an inextendible causal curve in W1 ≈ M S. Finally, if some causal curve links two points in S, then it must be contained in W1 since globally hyperbolic open subsets are causally convex. It would contradict the acausality of S1 1 . inside M is globally hyperbolic, and that The conclusion is that M S is a Cauchy surface in i into M is a Cauchy embedding. Since M i is a M. In other words, the projection of M 1 and M 2 are isometric. maximal extension, these projections are onto. Hence M Remark 6.25. The uniqueness of the maximal globally hyperbolic AdS extension is no longer true if we allow interactions. Indeed, in the next section we will see how, given some singular AdS spacetime without interaction, to define a surgery near a point in a singular line, introducing some collision or interaction at this point. The place where such a surgery can be performed is arbitrary. However, the uniqueness of the maximal globally hyperbolic extension holds in the case of interactions, if one stipulates that no new interactions can be introduced. The point is to consider the maximal extension in the future of a Cauchy surface in the future of all interactions, and the maximal extension in the past of a Cauchy surface contained in the past of all interactions. This point, along with other aspects of the global geometry of moduli spaces of AdS manifolds with interacting particles, is further studied in [BBS10]. 7. Global Examples The main goal of this section is to construct examples of globally hyperbolic singular AdS manifolds with interacting particles, so we go beyond the local examples constructed in Sect. 2. In a similar way examples of globally hyperbolic flat or de Sitter space-times with interacting particles can be also constructed. Sections 7.1 and 7.2 are presented in the AdS setting, but can presumably largely be extended to the Minkowski or de Sitter setting. The next two parts, however, are more specifically AdS and an extension to the Minkowski or de Sitter context is less clear. 7.1. An explicit example. Let S be a hyperbolic surface with one cone point p of angle θ . Denote by μ the corresponding singular hyperbolic metric on S. Let us consider the Lorentzian metric on S × (−π/2, π/2) given by h = −dt 2 + cos2 t μ, where t is the real parameter of the interval (−π/2, π/2).
(2)
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We denote by M(S) the singular spacetime (S × (−π/2, π/2), h). Lemma 7.1. M(S) is an Ad S spacetime with a particle corresponding to the singular line { p} × (−π/2, π/2). The corresponding cone angle is θ . Level surfaces S × {t} are orthogonal to the singular locus. Proof. First we show that h is an Ad S metric. The computation is local, so we can assume S = H2 . Thus we can identify S to a geodesic plane in Ad S3 . We consider Ad S3 as embedded in R2,2 , as mentioned in the Introduction. Let n be the normal direction to S, then we can consider the normal evolution F : S × (−π/2, π/2) (x, t) → cos t x + sin tn ∈ Ad S3 . The map F is a diffeomorphism onto an open domain of Ad S3 and the pull-back of the Ad S3 -metric takes the form (2). To prove that { p} × (−π/2, π/2) is a conical singularity of angle θ , take a geodesic plane P in Pθ orthogonal to the singular locus. Notice that P has exactly one cone point p0 corresponding to the intersection of P with the singular line of Pθ (here Pθ is the singular model space defined in Subsect. 3.7). Since the statement is local, it is sufficient to prove it for P. Notice that the normal evolution of P \ { p0 } is well-defined for any t ∈ (−π/2, π/2). Moreover, such evolution can be extended to a map on the whole P × (−π/2, π/2) sending { p0 } × (−π/2, π/2) onto the singular line. This map is a diffeomorphism of P × (−π/2, π/2) with an open domain of Pθ . Since the pull-back of the Ad S-metric of Pθ on (P \ { p0 }) × (−π/2, π/2) takes the form (2) the statement follows. Let T be a triangle in H S 2 , with one vertex in the future hyperbolic region and two vertices in the past hyperbolic region. Doubling T , we obtain a causally regular HS-sphere with an elliptic future singularity at p and two elliptic past singularities, q1 , q2 . Let r be the future singular ray in e(). For a given > 0 let p be the point at distance from the interaction point. Consider the geodesic disk D in e() centered at p , orthogonal to r and with radius . The past normal evolution n t : D → e() is well-defined for t ≤ . In fact, if we restrict to the annulus A = D \ D /2 , the evolution can be extended for t ≤ for some > (Fig. 11). Let us set U = {n t ( p) | p ∈ D , t ∈ (0, )}, = {n t ( p) | p ∈ D \ D /2 , t ∈ (0, )}. Notice that the interaction point is in the closure of U . It is possible to contruct a neighborhood of the interaction point p0 such that • U ∪ ⊂ ⊂ U ∪ ∪ B( p0 ) where B( p0 ) is a small ball around p0 ; • admits a foliation in achronal disks (D(t))t∈(0, ) such that (1) D(t) = n t (D ) for t ≤ , (2) D(t) ∩ t = n t (D \ D /2 ) for t ∈ (0, ), (3) D(t) is orthogonal to the singular locus. Consider now the space M(S) as in the previous lemma. For small the disk D embeds in M(S), sending p to ( p, 0).
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Fig. 11. Construction of a singular tube with an interaction of two particles
Let us identify D with its image in M(S). The normal evolution on D in M(S) is well-defined for 0 < t < π/2 and in fact coincides with the map n t (x, 0) = (x, t). It follows that the map F : (D \ D /2 ) × (0, ) → , defined by F(x, t) = n t (x) is an isometry (Fig. 11). Thus if we glue (S \ D /2 ) × (0, ) to by identifying D \ D /2 to via F we get a spacetime with particles Mˆ = (S \ D /2 ) × (0, ) ∪ F that easily verifies the following statement. Proposition 7.2. There exists a locally Ad S3 manifold with particles Mˆ such that (1) topologically, Mˆ is homeomorphic to S × R, ˆ two particles collide producing one particle only, (2) in M, ˆ (3) M admits a foliation by spacelike surfaces orthogonal to the singular locus. We say that Mˆ is obtained by a surgery on M = S × (0, ). 7.2. Surgery. In this section we get a generalization of the construction explained in the previous section. In particular we show how to do a surgery on a spacetime with conical singularity in order to obtain a spacetime with collision more complicated than that described in the previous section. Lemma 7.3. Let be a causally regular HS-sphere containing only elliptic singularities. Suppose that the circle of photons C+ bounding the future hyperbolic part of carries an elliptic structure of angle θ . Then e() \ (I + ( p0 ) ∪ I − ( p0 )) embeds in Pθ ( p0 denotes the interaction point of e()).
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Proof. Let D be the de Sitter part of , Notice that e(D) = e() \ (I + ( p0 ) ∪ I − ( p0 )). To prove that e(D) embeds in Pθ it is sufficient to prove that D is isometric to the de Sitter part of the HS sphere θ that is the link of a singular point of Pθ . Such de Sitter surface is the quotient of d˜S 2 under an elliptic transformation of S˜O(2, 1) of angle θ . So the statement is equivalent to proving that the developing map d : D˜ → d ˜S2 is a diffeomorphism. Since d ˜S2 is simply connected and d is a local diffeomorphism, it is sufficient to prove that d is proper. As in Sect. 5, d˜S 2 can be completed by two lines of photons, say R+ , R− that are ˜ 1. projectively isomorphic to RP Consider the left isotropic foliation of d˜S 2 . Each leaf has an α-limit in R− and an ω-limit on R+ . Moreover every point of R− (resp. R+ ) is an α-limit (resp. ω-limit) of exactly one leaf of each foliation. Thus we have a continuous projection ι L : d ˜S2 ∪ R− ∪ R+ → R+ , obtained by sending a point x to the ω-limit of the leaf of the left foliation through it. The map ι L is a proper submersion. Since D does not contain singularities, we have an analogous proper submersion, ιL : D˜ ∪ C˜ − ∪ C˜ + → C˜ + , where C˜ + , C˜ − are the universal covering of the circle of photons of . By the naturality of the construction, the following diagram commutes d D˜ ∪ C˜ − ∪ C˜ −−−−→ d ˜S2 ∪ R− ∪ R+ ⏐ ⏐ ⏐ ⏐ ιL ιL
C˜ +
d
−−−−→
R˜ + .
The map d|C˜ + is the developing map for the projective structure of C+ . By the hypothesis, we have that d|C˜ + is a homeomorphism, so it is proper. Since the diagram is commutative and the fact that ι L and ιL are both proper, one easily proves that d is proper. Remark 7.4. If is a causally regular HS-sphere containing only elliptic singularities, the map ιL : C˜ − → C˜ + induces a projective isomorphism ι¯ : C− → C+ . Definition 7.5. Let M be a singular spacetime homeomorphic to S × R and let p ∈ M. A neighborhood U of p is said to be cylindrical if • U is topologically a ball; • ∂± C := ∂U ∩ I ± ( p) is a spacelike disk; • there are two disjoint closed spacelike slices S− , S+ homeomorphic to S such that S− ⊂ I − (S+ ) and I ± ( p) ∩ S± = ∂± C. Remark 7.6. • If a spacelike slice through p exists then cylindrical neighborhoods form a fundamental family of neighborhoods. • There is an open retract M of M whose boundary is S− ∪ S+ .
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Corollary 7.7. Let be a HS-sphere as in Lemma 7.3. Given an Ad S spacetime M homeomorphic to S × R containing a particle of angle θ , let us fix a point p on it and suppose that a spacelike slice through p exists. There is a cylindrical neighborhood C of p and a cylindrical neighborhood C0 of the interaction point p0 in e() such that C \ (I + ( p) ∪ I − ( p)) is isometric to C0 \ (I + ( p0 ) ∪ I − ( p0 )). Take an open deformation retract M ⊂ M with spacelike boundary such that ∂± C ⊂ ∂ M . Thus let us glue M \(I + ( p)∪ I − ( p)) and C0 by identifying C \(I + ( p)∪ I − ( p)) to C0 ∩e(D). In this way we get a spacetime Mˆ homeomorphic to S ×R with an interaction point modelled on e(). We say that Mˆ is obtained by a surgery on M . The following proposition is a kind of converse to the previous construction. Proposition 7.8. Let Mˆ be a spacetime with conical singularities homeomorphic to S × R containing only one interaction between particles. Suppose moreover that a neighborhood of the interaction point is isometric to an open subset in e(), where is a HS-surface as in Lemma 7.3. Then a subset of Mˆ is obtained by a surgery on a spacetime without interaction. Proof. Let p0 be the interaction point. There is an HS-sphere as in Lemma 7.3 such that a neighborhood of p0 is isometric to a neighborhood of the vertex of e(). In particular there is a small cylindrical neighborhood C0 around p0 . According to Lemma 7.3, for a suitable cylindrical neighborhood C of a singular point p in Pθ we have C \ (I + ( p) ∪ I − ( p)) ∼ = C0 \ (I + ( p0 ) ∪ I − ( p0 )). Taking the retract M of Mˆ such that ∂± C0 is in the boundary of M , the spacetime M \ (I + ( p0 ) ∪ I − ( p0 )) can be glued to C via the above identification. We get a spacetime M with only one singular line. Clearly the surgery on M of C0 produces M . 7.3. Spacetimes containing BTZ-type singularities. In this section we describe a class of spacetimes containing BTZ-type singularities. We use the projective model of Ad S geometry, that is the Ad S3,+ . From Subsect. 2.2, Ad S3,+ is a domain in RP3 bounded by the double ruled quadric Q. Using the double family of lines Ll , Lr we identify Q to RP1 × RP1 so that the isometric action of Isom0,+ = P S L(2, R) × P S L(2, R) on Ad S3 extends to the product action on the boundary. We have seen in Sect. 2.2 that gedesics of Ad S3,+ are projective segments whereas geodesics planes are the intersection of Ad S3,+ with projective planes. The scalar product of R2,2 induces a duality between points and projective planes and between projective lines. In particular points in Ad S3 are dual to spacelike planes and the dual of a spacelike geodesic is still a spacelike geodesic. Geometrically, every timelike geodesic starting from a point p ∈ Ad S3 orthogonally meets the dual plane at time π/2, and points on the dual plane can be characterized by the property to be connected to p be a timelike geodesic of length π/2. Analogously, the dual line of a line l is the set of points that be can be connected to every point of l by a timelike geodesic of length π/2. Now, consider two hyperbolic transformations γ1 , γ2 ∈ P S L(2, R) with the same translation length. There are exactly 2 spacelike geodesics l1 , l2 in Ad S3 that are invariant under the action of (γ1 , γ2 ) ∈ P S L(2, R)× P S L(2, R) = Isom0,+ . Namely, if x + (c)
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denotes the attractive fixed point of a hyperbolic transformation c ∈ P S L(2, R), l2 is the line in Ad S3 joining the boundary points (x + (γ1 ), x + (γ2 )) and (x + (γ1−1 ), x + (γ2−1 )). On the other hand l1 is the geodesic dual to l2 , the endpoints of l1 are (x + (γ1 ), x + (γ2−1 )) and (x + (γ1−1 ), x + (γ2 )). Points of l1 are fixed by (γ1 , γ2 ) whereas it acts by pure translation on l2 . The union of the timelike segments with the past end-point on l2 and the future end-point on l1 is a domain 0 in Ad S3,+ invariant under (γ1 , γ2 ). The action of (γ1 , γ2 ) on 0 is proper and free and the quotient M0 (γ1 , γ2 ) = 0 /(γ1 , γ2 ) is a spacetime homeomorphic to S 1 ×R2 . There exists a spacetime with singularities Mˆ 0 (γ1 , γ2 ) such that M0 (γ1 , γ2 ) is isometric to the regular part of Mˆ 0 (γ1 , γ2 ) and it contains a future BTZ-type singularity. Define Mˆ 0 (γ1 , γ2 ) = (0 ∪ l1 )/(γ1 , γ2 ). To show that l1 is a future BTZ-type singularity, let us consider an alternative description of Mˆ 0 (γ1 , γ2 ). Notice that a fundamental domain in 0 ∪l1 for the action of (γ1 , γ2 ) can be constructed as follows. Take on l2 a point z 0 and put z 1 = (γ1 , γ2 )z 0 . Then consider the domain P that is the union of a timelike geodesic joining a point on the segment [z 0 , z 1 ] ⊂ l2 to a point on l1 . P is clearly a fundamental domain for the action with two timelike faces. Mˆ 0 (γ1 , γ2 ) is obtained by gluing the faces of P. We now generalize the above constructions as follows. Let us fix a surface S with some boundary component and negative Euler characteristic. Consider on S two hyperbolic metrics μl and μr with geodesic boundary such that each boundary component has the same length with respect to those metrics. Let h l , h r : π1 (S) → P S L(2, R) be the corresponding holonomy representations. The pair (h l , h r ) : π1 (S) → P S L(2, R) × P S L(2, R) induces an isometric action of π1 (S) on Ad S3 . In [Bar08a,Bar08b,BKS06] it is proved that there exists a convex domain in AdS3,+ invariant under the action of π1 (S) and the quotient M = /π1 () is a strongly causal manifold homeomorphic to S × R. For the convenience of the reader we sketch the construction of referring to [Bar08a,Bar08b] for details. The domain can be defined as follows. First consider the limit set defined as the closure of the set of pairs (x + (h l (γ )), x + (h r (γ ))) for γ ∈ π1 (S). is a π1 (S)-invariant subset of ∂ Ad S3,+ and it turns out that there exists a spacelike plane P disjoint from . So we can consider the convex hull K of in the affine chart RP3 \ P. K is a convex subset contained in Ad S3,+ . For any peripheral loop γ , the spacelike + −1 + −1 + + geodesic cγ joining (x (h l (γ )), x (h r (γ ))) to (x (h l (γ )), x (h r (γ ))) is contained in ∂ K and ∪ cγ disconnects ∂ K into components called the future boundary, ∂+ K , and the past boundary, ∂− K . One then defines as the set of points whose dual plane is disjoint from K . We have (1) the interior of K is contained in . (2) ∂ is the set of points whose dual plane is a support plane for K . (3) ∂ has two components: the past and the future boundary. Points dual to support planes of ∂− K are contained in the future boundary of , whereas points dual to support planes of ∂+ K are contained in the past boundary of . (4) Let A be the set of triples (x, v, t), where t ∈ [0, π/2], x ∈ ∂− K and v ∈ ∂+ is a point dual to some support plane of K at x. We consider the normal evolution map : A → Ad S3,+ , where (x, v, t) is the point on the geodesic segment joining x to v at distance t from x. In [BB09b] the map is shown to be injective (Figs. 12, 13).
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Fig. 12. The region P is bounded by the dotted triangles, whereas M0 (γ1 , γ2 ) is obtained by gluing the faces of P
Proposition 7.9. There exists a manifold with singularities Mˆ such that (1) The regular part of Mˆ is M. (2) There is a future BTZ-type singularity and a past BTZ-type singularity for each boundary component of M.
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Fig. 13. The segment r (c) projects to a BTZ-type singularity for M
Proof. Let c ∈ π1 (S) be a loop representing a boundary component of S and let γ1 = h l (c), γ2 = h r (c). By hypothesis, the translation lengths of γ1 and γ2 are equal, so, as in the previous example, there are two invariant geodesics l1 and l2 . Moreover the geodesic l2 is contained in and is in the boundary of the convex core K of . By [BKS06,BB09a], there exists a face F of the past boundary of K that contains l2 . The dual point of such a face,
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say p, lies in l1 . Moreover a component of l1 \ { p} contains points dual to some support planes of the convex core containing l2 . Thus there is a ray r = r (c) in l1 with vertex at p contained in ∂+ (and similarly there is a ray r− = r− (c) contained in l1 ∩ ∂− ). Now let U (c) be the union of timelike segments in with past end-point in l2 and future end-point in r (c). Clearly U (c) ⊂ (γ1 , γ2 ). The stabilizer of U (c) in π1 (S) is the group generated by (γ1 , γ2 ). Moreover we have • for some a ∈ π1 (S) we have a · U (c) = U (aca −1 ), • if d is another peripheral loop, U (c) ∩ U (d) = ∅. (The last property is a consequence of the fact that the normal evolution of ∂− K is injective – see property (4) before Proposition 7.9.) So if we put Mˆ = ( ∪ r (c) ∪ r− (c))/π1 (S), then a neighborhood of r (c) in Mˆ is isometric to a neighborhood of l1 in M(γ1 , γ2 ), and is thus a BTZ-type singularity (and analogously r− (c) is a white hole singularity). 7.4. Surgery on spacetimes containing BTZ-type singularities. Now we illustrate how to get spacetimes ∼ = S × R containing two particles that collide producing a BTZ-type singularity. Such examples are obtained by a surgery operation similar to that implemented in Sect. 7.2. The main difference with that case is that the boundary of these spacetimes is not spacelike. Let M be a spacetime ∼ = S × R containing a BTZ-type singularity l of mass m and fix a point p ∈ l. Let us consider a HS-surface containing a BTZ-type singularity p0 of mass m and two elliptic singularities q1 , q2 . A small disk 0 around p0 is isomorphic to a small disk in the link of the point p ∈ l. (As in the previous section, one can construct such a surface by doubling a triangle in H S 2 with one vertex in the de Sitter region and two vertices in the past hyperbolic region.) Let B be a ball around p and B be the intersection of B with the union of segments starting from p with velocity in . Clearly B embeds in e(), moreover there exists a small disk 0 around the vertex of e() such that e(0 ) ∩ B0 is isometric to the image of B in B0 . Now = ∂ B \ B is a disk in M. So there exists a topological surface S0 in M such that • S0 contains ; • S0 ∩ B = ∅; • M \ S0 is the union of two copies of S × R. Notice that we do not require S0 to be spacelike. Let M1 be the component of M \ S0 that contains B. Consider the spacetime Mˆ obtained by gluing M1 \ (B \ B ) to B0 , identifying B to its image in B0 . Clearly Mˆ contains two particles that collide giving a BH singularity and topologically Mˆ ∼ = S ×R. References [ABB+ 07] [Ale05]
Andersson, T., Barbot, T., Benedetti, R., Bonsante, F., Goldman, W.M., Labourie, F., Scannell, K.P., Schlenker, J.M.: Notes on a paper of mess. Geom. Dedicata 126, 47–70 (2007) Alexandrov, A.D.: Convex polyhedra. Springer Monographs in Mathematics. Berlin: SpringerVerlag, 2005, translated from the 1950 Russian edition by N. S. Dairbekov, S. S. Kutateladze, A. B. Sossinsky, with comments and bibliography by V. A. Zalgaller and Appendices by L. A. Shor and Yu. A. Volkov
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[Bar08a] [Bar08b] [BB09a] [BB09b] [BBES03] [BBS09] [BBS10] [BEE96] [BKS06] [Bro04] [BS09a] [BS09b] [BS10] [BTZ92] [Car03] [CFGO94] [DS93] [Ger70] [GL98] [Gol10] [Got91] [Gra93] [HE73] [HK98] [HM99] [KS07] [Mes07] [MM] [Sch98] [Sch01] [Ste94]
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Barbot, T.: Causal properties of AdS-isometry groups. i. causal actions and limit sets. Adv. Theor. Math. Phys. 12(1), 1–66 (2008) Barbot, T.: Causal properties of AdS-isometry groups. ii. BTZ multi-black-holes. Adv. Theor. Math. Phys. 12(6), 1209–1257 (2008) Benedetti, R., Bonsante, F.: (2 + 1) Einstein spacetimes of finite type. In: Handbook of Teichmüller theory. Vol. II, Volume 13 of IRMA Lect. Math. Theor. Phys. Zürich: Eur. Math. Soc., 2009, pp. 533–609 Benedetti, R., Bonsante, F.: Canonical Wick rotations in 3-dimensional gravity. Mem. Amer. Math. Soc., 198, Providence, RI: Amer. Math. Soc., 2009 Brock, J., Bromberg, K., Evans, R., Souto, J.: Tameness on the boundary and Ahlfors’ measure conjecture. Publ. Math. Inst. Hautes Études Sci. 98, 145–166 (2003) Barbot, T., Bonsante, F., Schlenker, J.-M.: Collisions of particles in locally AdS spacetimes. http://arXiv.org/abs/0905.1823v1 [math.DG], 2009 Barbot, T., Bonsante, F., Schlenker, J.-M.: Collisions of particles in locally AdS spacetimes II. Moduli of globally hyperbolic spaces. Work in progress, 2010 Beem J.K., Ehrlich P.E., Easley K.L.: Global Lorentzian Geometry. New York: Marcel Dekker, 1996 Bonsante, F., Krasnov, K., Schlenker, J.-M.: Multi black holes and earthquakes on Riemann surfaces with boundaries. Int. Math. Res. Not. 2011(3), 487–552 (2011) Bromberg, K.: Hyperbolic cone-manifolds, short geodesics, and Schwarzian derivatives. J. Amer. Math. Soc., 17(4), 783–826 (electronic), (2004) Bonsante, F., Schlenker, J.-M.: AdS manifolds with particles and earthquakes on singular surfaces. Geom. Funct. Anal. 19(1), 41–82 (2009) Bonsante, F., Schlenker, J.-M.: Fixed points of compositions of earthquakes. http://arXiv.org/ abs/0812.3471v3 [math.GT], 2010 Bonsante, F., Schlenker, J.-M.: Maximal surfaces and the universal Teichmüller space. Invent. Math. 182(2), 279–333 (2010) Bañados, M., Teitelboim, C., Zanelli, J.: Black hole in three-dimensional spacetime. Phys. Rev. Lett. 69(13), 1849–1851 (1992) Carlip, S.: Quantum Gravity in 2+ 1 Dimensions. Cambridge: Cambridge University Press, 2003 Carroll, S.M., Farhi, E., Guth, A.H., Olum, K.D.: Energy-momentum restrictions on the creation of gott time machines. Phys. Rev. D 50, 6190–6206 (1994) Deser, S., Steif, A.R.: No Time Machines from Lightlike Sources in 2 + 1 Gravity. In: B. L. Hu, M. P. Ryan Jr., C. V. Vishveshwara, Ed., Directions in General Relativity: Papers in Honor of Charles Misner, Volume 1, 1993, Cambridge: Cambridge Univ. Press, 2005, p. 78ff Geroch, R.: Domain of dependence. J. Math. Phys. 11(2), 437–449 (1970) Gott, J.R., Li-Xin, Li.: Can the universe create itself? Phys. Rev. Lett. 58(2), 02351 (1998) Goldman, W.M.: Locally homogeneous geometric manifolds. In: Proceedings of the International Congress of Mathematicians (Hyderabad, India, 2010), New Delhi: Hindustan Boook Agency, 2010, pp. 717–744 also available at http://arXiv.org/abs/1003.2759v2 [mathDG], 2010 Gott, J.R.: Closed timelike curves produced by pairs of moving cosmic strings: exact solutions. Phys. Rev. Lett. 66(2), 1126–1129 (1991) Grant, J.D.E.: Cosmic strings and chronology protection. Phys. Rev. D 47, 2388 (1993) Hawking, S.W., Ellis, G.F.R.: The large scale structure of space-time. Cambridge Monographs on Mathematical Physics, No. 1. London: Cambridge University Press, 1973 Hodgson, C.D., Kerckhoff, S.P.: Rigidity of hyperbolic cone-manifolds and hyperbolic Dehn surgery. J. Diff. Geom. 48, 1–60 (1998) Holst, S., Matschull, H.J.: The anti-de Sitter Gott universe: a rotating BTZ wormhole. Class. Quant. Grav. 16(10), 3095–3131 (1999) Krasnov, K., Schlenker, J.-M.: Minimal surfaces and particles in 3-manifolds. Geom. Dedicata 126, 187–254 (2007) Mess, G.: Lorentz spacetimes of constant curvature. Geom. Dedicata 126, 3–45 (2007) Mazzeo, R., Montcouquiol, G.: Infinitesimal rigidity of cone-manifolds and the Stoker problem for hyperbolic and Euclidean polyhedra. J. Diff. Geom. 37(3), 525–576 (2011) Schlenker, J.-M.: Métriques sur les polyèdres hyperboliques convexes. J. Diff. Geom. 48(2), 323–405 (1998) Schlenker, J.-M.: Convex polyhedra in Lorentzian space-forms. Asian J. Math. 5(2), 327–363 (2001) Steif, A.R.: Multiparticle solutions in 2+1 gravity and time machines. Int. J. Mod. Phys. D 3(1), 277–280 (1994)
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’t Hooft, G.: The evolution of gravitating point particles in 2 + 1 dimensions. Class. Quant. Grav. 10(5), 1023–1038 (1993) ’t Hooft, G.: Quantization of point particles in (2 + 1)-dimensional gravity and spacetime discreteness. Class. Quant. Grav. 13(5), 1023–1039 (1996) Thurston, W.P.: Shapes of polyhedra and triangulations of the sphere. Geom. Topo. Mono. 1(1), 511–549 (1998) Weiss, H.: The deformation theory of hyperbolic cone-3-manifolds with cone-angles less than 2π . http://arXiv.org/abs/0904.4568v2 [math.DG], 2009
Communicated by P.T. Chru´sciel
Commun. Math. Phys. 308, 201–225 (2011) Digital Object Identifier (DOI) 10.1007/s00220-011-1317-7
Communications in
Mathematical Physics
Energy Transfer in a Fast-Slow Hamiltonian System Dmitry Dolgopyat1 , Carlangelo Liverani2 1 Department of Mathematics, University of Maryland, 4417 Mathematics Bldg, College Park, MD 20742,
USA. E-mail: [email protected]
2 Dipartimento di Matematica, II Università di Roma (Tor Vergata), Via della Ricerca Scientifica,
00133 Roma, Italy. E-mail: [email protected] Received: 19 October 2010 / Accepted: 13 April 2011 Published online: 10 September 2011 – © Springer-Verlag 2011
Abstract: We consider a finite region of a lattice of weakly interacting geodesic flows on manifolds of negative curvature and we show that, when rescaling the interactions and the time appropriately, the energies of the flows evolve according to a nonlinear diffusion equation. This is a first step toward the derivation of macroscopic equations from a Hamiltonian microscopic dynamics in the case of weakly coupled systems. 1. Introduction One of the central problems in the study of non-equilibrium statistical physics is the derivation of transport equations for conserved quantities, in particular energy transport, from first principles, (see [6], and references therein, or [30], for a more general discussion on the derivation of macroscopic equations from microscopic dynamics). Lately several results have appeared trying to bring new perspective to the above problem in a collective effort to attack the problem from different points of views. Let us just mention, as examples, papers considering stochastic models [3–5], approaches starting from kinetic equations or assuming extra hypotheses [2,7,26] or papers trying to take advantage of the point of view and results developed in the field of Dynamical Systems [8,9,13–16,29]. This paper belongs to the latter category but it is closely related to results obtained for stochastic models (e.g., [25]). We consider a microscopic dynamics determined by a (classical) Hamiltonian describing a finite number of weakly interacting strongly chaotic systems and we explore the following strategy to derive a macroscopic evolution: first one looks at times for which we have an effective energy exchange between interacting systems, then takes the limit for the strength of the interaction going to zero and hopes to obtain a self-contained equation describing the evolution of the energies only. We call such an equation mesoscopic since most of the degrees of freedom have been averaged out. Second, one performs on such a mesoscopic equation a thermodynamic limit to obtain a macroscopic evolution. In particular, one can consider a scaling limit of the diffusive type in order to
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obtain a nonlinear heat equation as in the case of the so-called hydrodynamics limit for particle systems, see [22,31] for more details. A similar strategy has been carried out, at a heuristic level, in [19,20]. The first step of such a program is accomplished in this paper. It is interesting to note that the mesoscopic equation that we obtain seems to have some very natural and universal structure since it holds also when starting from different models. Indeed, essentially the same equation is obtained in [25] for a system of coupled nonlinear oscillators in the presence of an energy preserving randomness. In addition, such an equation is almost identical to the one studied in [31] apart from the necessary difference that the diffusion is a degenerate one. Indeed, since it describes the evolution of energies, and energies are positive, the diffusion coefficients must necessarily be zero when one energy is zero. Since, due to the weak interaction, the energies vary very slowly, once the time is rescaled so that the energies evolve on times of order one all the other variables will evolve extremely fast. Thus our result is an example of averaging theory for slow-fast systems. Yet, in our case the currents have zero average which means that standard averaging theory (such as, e.g. [18]) cannot suffice. It is necessary to look at longer times when the fluctuations play a fundamental role. The study of such longer times can in principle be accomplished thanks to the theory developed in [12]. Unfortunately, the results in [12] do not apply directly and we are forced to a roundabout in order to obtain the wanted result. Not surprisingly, the trouble takes place at low energies. We have thus to investigate with particular care the behavior of the system at low energies. In particular, we prove that the probability for any particle to reach zero energy, in the relevant time scale, tends to zero. The structure of the paper is as follows: Sect. 2 contains the precise description of the microscopic model and the statement of the results. Sect. 3 describes the logic of the proof at a non-technical level and points out the technical difficulties that must be overcome to make the argument rigorous. In the following section we show how to modify the dynamics at low energies in such a way that existing results can be applied. Then, in Sect. 5, we investigate the modified dynamics and show that its accumulation points satisfy a mesoscopic equation of the wanted type. In Sect. 6 we compute explicitly the properties of the coefficients of the limit equation for the modified dynamics and in Sect. 7 we use this knowledge to show that the equation has a unique solution, hence the modified process converges to this solution. In Sect. 8 we discuss the limit equation for the original dynamics under the condition that no particle reaches zero in finite time. The fact that this condition holds in our model is proven in Sect. 9. The paper ends with two appendices. In the first, for reader convenience, some known results from the averaging theory for systems with hyperbolic fast motion are restated in a way suitable for our needs. The second appendix contains some boring, but essential, computations.
2. The Model and the Result For d ∈ N, we consider a lattice Zd and a finite connected region ⊂ Zd . Associated to each site in we have the cotangent bundle T ∗ M of a C ∞ compact Riemannian d-dimensional manifold M of strictly negative curvature and the associated geodesic flow g t . We have then the phase space M = (T ∗ M) and we designate a point as (qx , px ), x ∈ . It is well known that the geodesic flows is a Hamiltonian flow. If we define i : T ∗ M → T M to be the natural isomorphism defined by w(v) = i(w), vG ,
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G being the Riemannian metric, then the Hamiltonian reads1 H0 = x∈ 21 px2 and the symplectic form is given by ω = dq ∧ d p.2 Thus, given x ∈ , the equations of motion take the form (see [27, Sect. 1] for more details) q˙ x = i( px ) , ˜ x , px ) , p˙ x = F(q
(2.1)
where the F˜ is homogeneous in the px of degree two. Note that, by the Hamiltonian structure, ex := 21 px2 is constant in time for each x ∈ . It is then natural to use the 1
variables (qx , vx , ex ), where vx := ( px2 )− 2 i( px ) belongs to the unit tangent bundle T 1 M of M.3 We have then the equations q˙ x = 2ex vx , (2.2) v˙ x = 2ex F(qx , vx ) , e˙x = 0, where F is homogeneous of second degree in vx . Next we want to introduce a small energy exchange between particles. To describe such an exchange we introduce a symmetric, non-constant, function (potential) V ∈ C ∞ (M 2 , R) and, for each ε > 0, consider the flow gεt determined by the Hamiltonian Hε = x∈ 21 px2 + 2ε |x−y|=1 V (qx , q y ), that is by the equations q˙ x = i( px ) , ˜ x , px ) − ε dqx V (qx , q y ). p˙ x = F(q |y−x|=1
Or, alternatively,4 q˙ x = 2ex vx , ε v˙ x = 2ex F(qx , vx ) + √ 2ex e˙x = −ε 2ex L x V,
{vx L x V − ∇qx V (qx , q y )},
(2.3)
|y−x|=1
|x−y|=1
where ∇V, wG = d V (w) and L x = vx ∂qx + F(qx , vx )∂vx
(2.4)
denotes the generator associated to the geodesic flow of the x particle on T1 M. 1 By p 2 we mean i( p ), i( p ) ˜ = i ∗ (G). x x G(qx ) = px , px G˜ , where G x 2 To be more precise, given the canonical projection π(q, p) = q, first define the one form, on T (T ∗ M), 1 1 ω(q, p) (ξ ) = p(dπ(ξ )). Then ω := −dω . Given coordinates q on U ⊂ M and using the coordinates p for ∗ the one form p = i pi dq i ∈ T M, we have ω1 = i pi dq i and ω = i dq i ∧ d pi , as stated. 3 Clearly e is the (kinetic) energy of the geodesic flow at x. x 4 In the interacting case one could choose to include the interaction in the energy and define eε := 1 p 2 + x 2 x ε ε |x−y|=1 V (q x , q y ). This is the choice made in [25]. Yet, in the present context |ex − ex | ≤ |V |∞ ε, hence 4 the actual choice is irrelevant in the limit ε → 0 and ex turns out to be computationally simpler.
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We will consider random initial conditions of the following type E( f (q(0), v(0)) = f (q, v)ρ(q, v)dm, ∀ f ∈ C 0 ((T1 M) , R), (T1 M)
(2.5)
ex (0) = E x > 0,
where m is the Riemannian measure on (T1 M) and ρ ∈ C 1 . Since the currents L x V have zero average with respect to the microcanonical measure, one expects that it will take a time of order ε−2 in order to see a change of energy of order one. It is then natural to introduce the process ex (ε−2 t) and to study the convergence of such a process in the limit ε → 0. Our main result is the following. Theorem 1. Provided d ≥ 3, the process {ex (ε−2 t)} defined by (2.3) with initial conditions (2.5) converges to a random process {Ex (t)} with values in R + which satisfies the stochastic differential equation √ dEx = a(Ex , E y )dt + 2β(Ex , E y )d Bx y , |x−y|=1 |x−y|=1 (2.6) Ex (0) = E x > 0, where Bx y are standard Brownian motions which are independent except that Bx y = −B yx . The coefficients have the following properties: β is symmetric and a is antisymmetric; β ∈ C 0 ([0, ∞)2 , R+ ) and β(a, b)2 = abG(a, b), where G ∈ C ∞ ((0, ∞)2 , R+ ) ∩ 3 C 1 ((0, ∞) × [0, ∞), R+ ) and G(a, 0) = A(2a)− 2 for some A > 0. Moreover, a = (∂Ex − ∂E y )β 2 +
d − 2 −1 (Ex − E y−1 )β 2 . 2
(2.7)
In addition, (2.6) has a unique solution and the probability for one energy to reach zero in finite time is zero. Remark 2.1. A direct computation shows that the measures with density h β = d 2 −1 −β Ex e are invariant for the above process for each β ∈ R+ . Indeed, using x∈ Ex (2.7), we can write the generator of the process (2.6) in the simple form L=
1 2h 0
(∂Ex − ∂E y )h 0 β 2 (∂Ex − ∂E y )
|x−y|=1
from which the reversibility of the generator is evident. Remark 2.2. The case d = 2 is harder because the second term in (2.7) (which otherwise would give the main contribution at small energies) is zero. We believe the result to be still true,5 but a much more detailed (and messy) analysis of (2.6) is needed to establish it. As this would considerably increase the length of Sect. 9 without adding anything really substantial to the paper, we do not pursue such matter. 5 That is the fact that zero is unreachable.
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Remark 2.3. Note that if we could apply [31] to perform the hydrodynamics limit, then we would obtain the heat equation. Unfortunately, (2.6) does not satisfy the hypotheses of Varadhan’s Theorem on several accounts, the most relevant being that the domain where the diffusion takes place is not all the space and a, β vanish on the boundary of the domain. This is unavoidable as the energy is naturally bounded from below. Nevertheless, the results of this paper can be considered as a first step along the bumpy road to obtaining the heat equation from a purely mechanical deterministic model.6 Remark 2.4. As a last remark, let us comment on the choice of Zd . This is done just to simplify notations: our arguments are of a local nature, hence the structure of Zd does not play any role in the proof. In particular, one can prove, with exactly the same arguments, the following extension of our result. Consider a loopless symmetric directed graph G determined by the collection of its vertexes V (G) and the collection of its directed edges E(G).7 At each vertex v ∈ V (G) we associate a mixing geodesic flow as before; consider then the Hamiltonian 1 ε pv2 + V (qe1 , qe2 ). Hε = 2 2 v∈V (G)
(e1 ,e2 )∈E(G)
We then have the exact analogues8 of Theorem 1 for the variables {Ev }v∈V (G) with the only difference that the limiting equation now reads √ dEv = a(Ev , Ew )dt + 2β(Ev , Ew )d B(v,w) (2.8) (v,w)∈E(G) (v,w)∈E(G) Ev (0) = E v > 0, where again for each e ∈ E(G), the Be are independent standard Brownian motions apart form the fact that B(v,w) = −B(w,v) . An interesting application of the above remark is the case where G is a complete graph (i.e. E(G) = {(v1 , v2 ) : v1 , v2 ∈ V (G)}) in which case all particles interact with each other. The rest of the paper is devoted to proving Theorem 1. Before going into details we explain exactly how the various results we are going to derive are collected together to prove the theorem. Proof of Theorem 1. Fix T > 0 and let Pε be the probability measure, on the space −2 C 0 ([0, T ], R + ), associated to the process {ex (ε t)}t∈[0,T ] defined by (2.3), Pε,δ to the one defined by (4.1), P˜ δ the one associated to the process {e z(t) } with z(t) defined by (7.2) and P the one defined by (2.6). Also, let δ = {τδ ≥ T }, where τδ = inf{t ∈ R+ : min x∈ Ex (t) ≤ δ}. By construction, for each F ∈ C 0 , EPε (F1 δ ) = EPε,δ (F1 δ ), EP˜ δ (F 1 δ ) = EP (F 1 δ ). 6 One could object that geodesic motion in negative curvature is not really mechanical. Yet, it is possible to construct a bona fide mechanical system which motion is equivalent to a geodesic flow in negative curvature [23]. In any case, by Maupertuis’ principle, any Hamiltonian system can be viewed as a geodesic flow, possibly on a non-compact manifold. 7 Directed means that the edges e ∈ E(G) are ordered pairs (e , e ), e ∈ V (G), which is interpreted 1 2 i as an edge going from e1 to e2 . Symmetric means that if (e1 , e2 ) ∈ E(G), then (e2 , e1 ) ∈ E(G). Loopless means that, for each a ∈ V (G), (a, a) ∈ E(G). This abstract setting reduces to the previous one if we choose V (G) = Zd and E(G) = {(x, y) ∈ Zd × Zd : |x − y| = 1}. 8 In particular the condition d ≥ 3 refers to the manifolds M, not to the lattice or graph.
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Proposition 7.4 implies that Pε,δ ⇒ P˜ δ and, since δ is a continuity set for P˜ δ , limε→0 Pε,δ ( δ ) = P˜ δ ( δ ) = P( δ ). Next, Lemma 9.1, based on estimate (8.1), tells us that limδ→0 P˜ δ ( cδ ) = 0. Thus lim lim Pε ( cδ ) = 0.
δ→0 ε→0
Hence Pε ⇒ P. The information on the coefficients follows by collecting (8.3), (2.7) (proven in Lemma 8.1), Lemmata 6.1 and 6.3. Finally, the uniqueness follows from standard results on SDE and the unreachability of zero (Lemma 9.1). 3. Heuristic Let us give a sketch of the argument where we ignore all the technical difficulties and perform some daring formal computations. If we could apply [12, Theorem 7] to Eq. (2.3) we would obtain a limiting process characterized by an equation that, after some algebraic manipulations detailed in Sect. 7, reads9 √ dEx = a(Ex , E y )dt + 2β(Ex , E y )d Bx y , (3.1) |x−y|=1
|x−y|=1
where β(Ex , E y ) = β(E y , Ex ) is symmetric and Bx y = −B yx are independent standard Brownian motions. The marginal of the Gibbs measure on the energy variables reads d −1 dμβ = Ex2 e−β Ex dEx =: h β ∧x dEx , x
for each β ∈ [0, ∞). Hence we expect such a measure to be invariant for (3.1). Even more, on physical grounds (see Lemma 7.1) one expects the process (3.1) to be reversible with respect to these measures. A straightforward computation shows that the generator associated to the above SDE reads 1 L= a x y ∂E x + β 2x y (∂Ex − ∂E y )2 , 2 |x−y|=1
|x−y|=1
where a x y = a(Ex , E y ), β x y = β(Ex , E y ). The adjoint with respect to μ0 reads d + 1 −1 ∗ −1 2 2 −a x y + (Ex − E y )β x y + (∂Ex − ∂E y )β x y ∂Ex L = 2 |x−y|=1
+ +
1 2
β 2x y (∂Ex − ∂E y )2 −
|x−y|=1
1 2h 0
1 h0
∂Ex (h 0 a x y )
|x−y|=1
(∂Ex − ∂E y )2 (h 0 β x y ).
|x−y|=1
Computing what it means, L = L∗ implies (2.7). Remark 3.1. Note that, as expected, a x y = −a yx . Thus d
x
Ex = 0.
9 See Appendix A for a precise statement of the results in [12] relevant to our purposes.
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Going to a bit less vague level of analysis, one must notice that since Ex ≥ 0, the diffusion equation (3.1) must be degenerate at zero, also it is not clear how regular the coefficients a, β are. Hence, a priori, it is not even obvious that such an equation has a solution and, if so, if such a solution is unique. To investigate such an issue it is necessary to obtain some information on the behavior of the coefficients at low energies. To this end one can use the explicit formula given in [12, Theorem 7] for the diffusion coefficient. This allows to verify that the coefficients are smooth away from zero. An explicit, but lengthy, computation yields, for Ex ≤ E y ,
3 AEx β 2x y = + O Ex2 E y−1 2E y (3.2)
√ Ex Ad , ax y = +O Ey 2 2E y see Lemma 8.1 for details. Thus, in particular, a x y Ex = d2 β 2x y + o(β 2x y ). We will see in Sect. 9 that such a relation, provided d > 2, suffices to prove that the set {(Ex ) : x Ex = 0} is unreachable and hence to insure that Eq. (3.1) has a unique solution. In the rest of the paper we show how to make rigorous the above line of reasoning. 4. A Modified Dynamics Since the geodesic flows on manifolds of strictly negative curvature enjoy exponential decay of correlations [11,24] we are in a setting very close to the one in [12], i.e. we have a slow-fast system in which the fast variables have strong mixing properties. Unfortunately, the perturbation to the geodesic flows in (2.3) is not small when ex = O(ε), so at low energies one is bound to loose control on the statistical properties of the dynamics. The only easy way out would be to prove that the limit system spends very little time in configurations in which one particle has low energy.10 If this were the case, then one could first introduce a modified system in which one offsets the bad behavior at small energies and then tries to remove the cutoff by showing that, in the limit process, the probability to reach very small energies is small. We will pursue precisely such a strategy. We now define the process. Since our equations are Hamiltonian with modified ε 1 2 Hamiltonian H = p + x x∈ 2 |x−y|=1 V (q x , q y ), the simplest approach is to 2 modify the kinetic part of the Hamiltonian making it homogeneous of degree one at low velocities and decreasing correspondingly the interaction at low energies. More precisely,given any two functions ϕ, φ ∈ C ∞ (R+ \{0}, R), consider the Hamiltonians ε Hϕ,φ = x∈ ϕ(ex ) + 2 |x−y|=1 φ(ex )φ(e y )V (qx , q y ), which yield the equations of motion φ (ex )φ(e y )V (qx , q y )i( px ), q˙ x = ϕ (ex )i( px ) + ε |x−y|=1
˜ x , px ) + ε p˙ x = ϕ (ex ) F(q
˜ x , px ) φ (ex )φ(e y )V (qx , q y ) F(q
|x−y|=1 10 To investigate low energy situations directly for the coupled geodesic flows seems extremely hard: when the kinetic energy is comparable with the potential energy all kinds of uncharted behaviors, including coexistence of positive entropy and elliptic islands, could occur!
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−ε
φ(ex )φ(e y ) dqx V,
|y−x|=1
e˙x = −ε
φ(ex )φ(e y ) dqx V (i( px )),
|y−x|=1
with F˜ as in (2.1).11 Which, in the variables (qx , vx , ex ), reads q˙ x =
2ex ϕ (ex )vx + ε 2ex φ (ex )φ(e y )V (qx , q y )vx , |x−y|=1
v˙ x = ϕ (ex ) 2ex F(qx , vx ) + ε
φ (ex )φ(e y ) 2ex V (qx , q y )F(qx , vx )
|x−y|=1
φ(ex )φ(e y ) φ(ex )φ(e y ) ∇ V + v d V (v ) √ √ qx x qx x , 2e 2e x x |y−x|=1 |y−x|=1 e˙x = − φ(ex )φ(e y ) 2ex dqx V (vx ), −
(4.1)
|y−x|=1
with F as in (2.2). φ(e )φ(e ) d 2 Since dt vx = ε(vx2 − 1) |y−x|=1 √x 2e y dqx V (vx ), the manifold vx2 = 1 is an x invariant manifold for the Eqs. (4.1), thus such equations determine a flow in the variables 1 (ξx , ex ) = (qx , vx , ex ) ∈ T M × R+ . Finally, we chose ϕ = ϕδ and φ = φδ such that, for all δ > 0,
s ϕδ (s) = √ 2 δs
1 1 if s ≥ δ = √s ; φδ (s) = √ if s ≤ 8δ ϕδ (s) δ
if s ≥ δ if s ≤ 8δ ,
(4.2)
where φδ is increasing. We denote the solution of the above equations (4.1) with initial conditions (ξ, e) by (ξ ε,δ (t), eε,δ (t)). Our goal is to apply [12, Thm. 7] to the flow (ξ ε,δ (t), eε,δ (t)), see Appendix A for a simplified statement (Thm. A.1) adapted to our needs. Before discussing the applicability of this Theorem, there is one last issue we need to take care of: the equation for e is clearly degenerate at low energies; this is related to the fact that the energies in (4.1) are strictly positive for all times if they are strictly positive at time zero.12 This may create a problem in the limiting process that is bound to have a degenerate diffusion coefficient. To handle this problem it turns out to be much more convenient to use the variables z x = ln ex . In these new variables we finally have the equations we are looking for 11 By d V we mean the differential of the function V (·, q ) for any fixed q . qx y y 12 Indeed, the equation for the energy can be written, near zero, as e˙ = −εe G(e , ξ ), where G is a x x =x
t bounded function, hence the solution has the form ex (t) = ex (0)e−ε 0 G(e=x (s),ξ(s))ds .
Energy Transfer in a Fast-Slow Hamiltonian System
ε 2
q˙ x = ωδ (z x )vx +
ζδ (z x )φδ (e z y )V (qx , q y )vx ,
|x−y|=1
v˙ x = ωδ (z x )F(qx , vx ) +
209
ε 2
ζδ (z x )φδ (e z y )V (qx , q y )F(qx , vx )
|x−y|=1
ε − zx e 2 φδ (e z x )φδ (e z y )∇qx V (qx , q y ) −√ 2 |y−x|=1 zx ε vx e− 2 φδ (e z x )φδ (e z y )L x V (ξx , ξ y ), +√ 2 |y−x|=1 √ − zx z˙ x = −ε 2 e 2 φδ (e z x )φδ (e z y )L x V (ξx , ξ y ),
(4.3)
|y−x|=1
where L x is as in Eq. (2.4) and ωδ (z) = ζδ (z) =
√ √
2e
z 2
ϕδ (e z )
√ = √
2e
z 2
φδ (e z )
=
z
if z ≥ ln δ , if z ≤ ln δ − ln 8
2e 2 2δ
if z ≥ ln δ if z ≤ ln δ − ln 8.
0 √1 2δ
(4.4)
√ Remark 4.1. Note that we can chose ωδ ≥ δ and ζδ ≥ 0 decreasing.13 In addition, it is possible to arrange that |ωδ |Cr (I L ,R) ≤ Cr e L , where I L = (−∞, 2L), and |ζδ |Cr (R,R) ≤ 1
Cr δ − 2 , for each r ∈ N, L , δ ∈ R+ . We will assume such properties in the following.
Since the total energy is conserved, we can consider Eqs. (4.3) on the set (T 1 M) × (−∞, L] for some L > 0. Hence, by the above remark together with (4.2), the vector field in (4.3) has bounded C r norm, as a function of x, z, ε, for each r ∈ N. Let f˜δ (ξ, z, ε, δ) = ξ ε,δ (1), Fε,δ (ξ, z) = (ξ ε,δ (1), z ε,δ (1)), and √ 1 − z x (τ ) δ A x (ξ, z, ε) = − 2 e 2 φδ (e z x (τ ) )φδ (e z y (τ ) )L x V (ξxε,δ (τ ), ξ yε,δ (τ ))dτ, (4.5) 0 |x−y|=1
then Fε,δ (ξ, z) =
f˜δ (ξ, z, ε), z + ε Aδ (ξ, z, ε) .
(4.6)
Lemma 4.2. Setting F˜δ (x, z, ε) = Fε,δ (x, z) we have, for each δ ∈ (0, 1), L > 0, F˜δ ∈ C ∞ ((T 1 M) × (−∞, L] × [0, 1]), and Aδ (·, ·, ε)Cr ((T 1 M) ×(−∞,L] ) ≤ Cr,δ , for each r ∈ N, ε ∈ [0, 1]. In addition, for each β ∈ R+ , the probability measure
˜ dμδ,ε,β = Z˜ β−1 e−β Hδ,ε + x ε H˜ δ,ε (q, ν, z) = ϕδ (e z x ) + 2 x∈
d 2 zx
dqdvdz, φδ (e z x )φ(e z y )V (qx , q y ),
|x−y|=1
13 Indeed,
φδ (s) = 1 −
δ min{s,δ}
Remark that once ζδ is chosen all the functions are fixed.
ζδ (ln x) d x. √ 2x
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D. Dolgopyat, C. Liverani
is invariant for Fε,δ . Moreover, for each z¯ ∈ Rd and sub-manifold z¯ := {z x = z¯ x }, the Dynamical System (z¯ , F0,δ ) has a unique SRB measure μz¯ . Proof. The first part of the statement follows from Remark 4.1 and subsequent comments together with standard results of existence of solutions and smooth dependence on the initial data from O.D.E.. The bound on Aδ is then immediate from formula (4.5). By the Hamiltonian nature of Eqs. (4.1) the measures dμδ,β = Z β−1 e−β Hϕδ ,φδ dqdp , are invariant for the associated dynamics for each β > 0. By changing variables we obtain the statement of the Lemma. Finally, calling μ˜ the Riemannian measure on T 1 M we have that μz¯ = ⊗|| μ˜ is a SRB measure for the map ξ → f˜δ (ξ, z, 0), which turns out to be the product of the time ωδ (z x ) maps of the geodesic flow on T 1 M. The uniqueness of the SRB follows by the mixing of the geodesic flows [1] and the fact that the product of mixing systems is mixing. 5. Existence of the Limit: δ > 0 We are finally ready to consider the limit ε → 0, for the modified dynamics. Proposition 5.1. For each δ ∈ (0, 1) there exists εδ > 0 such that the Dynamical System defined by (4.6) satisfies the hypotheses of Theorem A.1 for ε ∈ [0, εδ ]. Hence, the family z ε,δ (ε−2 t) is tight and its weak accumulation points are a solution of the Martingale problem associated to the stochastic differential equation dz δx = axδ (z δ )dt + σxδy (z δ )d B y , y (5.1) δ z x (0) = z¯ x , where (σ δ )2x y (z) =
Aδx (( f˜δ )n (ξ, z, 0)Aδy (ξ, z, 0)dμz
1 n∈Z (T M) +∞
=2
dt
−∞
|x−w|=1 |y−w |=1
φδ (e z x )φδ (e z w )φδ (e z y )φδ (e z w ) e
z x +z y 2
×E L x V (ξx0,δ (t), ξw0,δ (t)) · L y V (ξ y , ξw ) .
(5.2)
Here E is the expectation with respect to μz and a δ C 0 + (σ δ )2 C 1 < ∞. Proof. First of all notice that the hypotheses on the smoothness of Fε,δ and the boundedness of Aδ are insured by Lemma 4.2. Next, notice that F0,δ (ξ, z) = ( f zδ (ξ ), z) with f zδ (ξ )x = g ωδ (z x ) (ξx ), where g t is the geodesic flow on the unit tangent bundle T 1 M, thus the f zδ are FAE.14 14 FAEs are defined in Appendix A. In our case, the abelian action is the one determined by the geodesic flows themselves, ×i∈ g ti .
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211
Also we have that μz (Aδ (·, z, 0)) = 0. This follows by considering the transformat associated to (4.3) tion (q, v) = (q, −v). Indeed ∗ μz = μz while, the flow δ,ε −t t satisfies δ,ε ◦ = ◦ δ,ε . On the other hand, using the antisymmetry of L x V with respect to vx , √ 1 − zx τ Aδx ((ξ ), z, 0) = − 2 e 2 φδ (e z x )φδ (e z y )L x V ◦ δ,0 ◦ (ξ )dτ √ = 2
0 |x−y|=1
1
0 |x−y|=1
zx
−τ e− 2 φδ (e z x )φδ (e z y )L x V ◦ δ,0 (ξ )dτ
−1 (ξ ), z, 0). = −Aδx (δ,0 −1 Thus μz (Aδ (·, z, 0)) = μz (Aδ ((·), z, 0)) = −μz (Aδ (δ,0 (·), z, 0)) = −μz (Aδ (·, z, 0)), by the invariance of the measure. √ The last thing to check is the uniform decay of correlation. Since ωδ ≥ δ, the results in [11,24] imply15 that the f z are FAE with uniform exponential decay of correlation. In fact, in Theorem A.1 the decay of correlations is meant in a very precise technical sense. To see that the results in [24] imply the wanted decay we must translate them into the language of standard pairs in which Theorem A.1 is formulated. Let us start by stating the result in [24]: let g a be the time a map of the geodesic flow on the unit tangent bundle. For each smooth function A let As = A∞ + ∂ s A∞ , where ∂ s is the derivative in the weak stable direction. Then there exists C, c > 0 such that, for each z and ρ, A ∈ C 1 , holds true E(ρ · A ◦ g˜ an ) − E(A)E(ρ) ≤ CρC 1 As e−can . (5.3)
Since, setting f zδ (ξ ) = f˜δ (ξ, z, 0), f zδ = ×x g ωδ (z x ) , and ωδ is uniformly bounded from below, for E(A) = 0, it follows (suppressing, to ease notation, the superscript δ)16 E(ρ · A ◦ f n ) ≤ C|| ρC 1 As e−can . (5.4) z To see that this is stronger than needed, consider a standard pair = (D, ρ).17 One can smoothly foliate a ε neighborhood of D and define a probability density ρε supported in it such that ρε C 1 ≤ Cε−2 , while ρε C 1 ≤ C when ρε is restricted to a leaf of the foliation. Thanks to the α-Hölder regularity and the absolute continuity of the weak stable foliation, one can take ρε so that |E (A) − E(ρε A)| ≤ Cεα As . Accordingly, E (A ◦ f n ) ≤ E(ρε · A ◦ f n ) + Cεα A ◦ f n s ≤ C ε−2 e−can + εα As z z z αcan
≤ Ce− 2+α AC 1 , 15 [11] proves the exponential decay of correlations for geodesic flows on negatively curved surfaces, [24] extends the results to any negatively curved manifold. 16 Just note that one can write E(ρ · A ◦ f n ) = E(E(ρ · A ◦ f n | ξ y =x )) and that the relevant norms of z z ρξ y=x (ξx ) = ρ(ξx , ξ y=x ) and Aξ y=x (ξx ) = A(ξx , f zn (ξ y=x )) are bounded by the full norms of ρ and A. Aξ ◦ f˜n ). Proceeding in such a way one One can then apply (5.3) to E(ρ · A ◦ f zn | ξ y=x ) = E(ρξ y =x
y =x
ωδ (z x )
variable at a time yields the result. 17 Recall that D is a manifold of fixed size close to the strong unstable one and ρ a smooth density on it.
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D. Dolgopyat, C. Liverani can
where, in the last equality, we have chosen ε = e− 2+α . Thus, all the hypotheses of Theorem A.1 are satisfied and (5.2) follows by a direct computation. By Theorem A.1(b), in order to prove that z ε,δ (ε−2 t) has a limit it suffices to prove that (5.1) has a unique solution. This would follow by standard results if we knew that a δ is locally Lipschitz. In fact, [12] provides also an explicit formula for a δ . Unfortunately this formula is much more complex than the formula for the variance and is quite difficult to investigate. We will avoid a direct computation of aδ and we will instead use the knowledge of the invariant measure to determine it. Before doing that a deeper understanding of the variance is required. 6. Computing the Variance Let g t be the geodesic flow on the unit cotangent bundle of M. As already noted, for each function h, h(ξx0,δ (t)) = h ◦ g ωδ (z x )t (ξx ) for all x ∈ . For convenience let us set x := ωδ (z x ). Also, it turns out to be useful to define two functions of two variables: consider two geodesic flows on T 1 M, let (ξ, η) be the variables of the two flows respectively, E the expectation with respect to the Riemannian volume on (T 1 M)2 and L 1 , L 2 the generators associated to the geodesic flow of ξ and η respectively, then we define ρ, ρ˜ : R2 → R by ∞ ρ(a, b) := dt E L 1 V (g at (ξ ), g bt (η)) · L 1 V (ξ, η) , −∞ (6.1) ∞ ρ(a, ˜ b) := dt E L 1 V (g at (ξ ), g bt (η)) · L 2 V (ξ, η) . −∞
Also, it is convenient to define ρx y := ρ(ωδ (z x ), ωδ (z y )), ρ˜x y := ρ(ω ˜ δ (z x ), ωδ (z y )).
(6.2)
Indeed, the understanding of the variance will be reduced shortly to understanding the properties of ρx y . Here is a list of relevant properties whose proof can be found in Appendix B. Lemma 6.1. The function ρ˜ is non-positive and C ∞ for a, b > 0. In addition, for each a, b, λ > 0 we have ρ(a, ˜ b) = ρ(b, ˜ a) and ρ(λa, λb) = λ−1 ρ(a, b). Finally, a ρ(a, ˜ b) = − b ρ(a, b). Remark 6.2. Note that the previous lemma implies a 2 ρ(a, b) = b2 ρ(b, a). Lemma 6.3. There exists A, B > 0 such that, for all a, b > 0, 2 3 ρ(a, b) − A b ≤ B ab . a 3 + b3 a 5 + b5 Finally, for all a, b > 0, |∂a ρ(a, b)| ≤
B ab2 ; a 5 + b5
a∂a ρ(a, b) + b∂b ρ(a, b) = −ρ(a, b).
We are now in the position to derive a helpful formula for the variance.
Energy Transfer in a Fast-Slow Hamiltonian System
213
Lemma 6.4. The following formula holds true ⎧ −z 2e x |x−w|=1 {φδ (e z x )φδ (e z w )}2 ρxw ⎪ ⎪ ⎨ (σ δ )2x y (z) = −2e−z y φδ (e z x )φδ (e z y )3 ρx y ⎪ ⎪ ⎩ 0
if x = y if |x − y| = 1 if |x − y| > 1.
Proof. Remembering (5.2), given any two couples of neighboring sites x, w, y, w we want to compute
∞
−∞
dt E L x V (g x t (ξx ), g w t (ξw )) · L y V (ξ y , ξw ) .
In fact, remembering the properties of the transformation in the proof of Lemma 5.1, it suffices to compute the integral on [0, ∞). Since E(vx | q=x , v=x ) = 0, it follows that the above integral is different from zero only if x = y or x = w and w = y. On the other hand if x = y, since g at × g bt is a mixing flow for each a, b > 0, we can write
∞
0
d dt E x−1 V (g x t (ξx ), g w t (ξw )) dt w x t w t L w V (g (ξx ), g (ξw )) · L x V (ξx , ξw ) − x
= x−1 E (V (qx , qw )) E (L x V (ξx , ξw )) − x−1 E (V (qx , qw ) · L x V (ξx , ξw )) w ∞ − dt E L w V (g x t (ξx ), g w t (ξw )) · L x V (ξx , ξw ) x 0 w ∞ dt E L w V (g x t (ξx ), g w t (ξw )) · L x V (ξx , ξw ) = −δw,w x ∞ 0 dt E L x V (g x t (ξx ), g w t (ξw )) · L x V (ξx , ξw ) . = δw,w 0
Thus, remembering (4.2), (4.4) and that x = ωδ (z x ), σx2x = 2e−z x
φδ (e z x )2 φδ (e z w )2 ρxw ,
|x−w|=1
and σx2y = 0 if |x − y| > 1. If |x − y| = 1, then (remembering the symmetry of the potential and using Lemma 6.1) σx2y = 2φδ (e z x )2 φδ (e z y )2 e−
z x +z y 2
ρ˜x y = −2e−z y φδ (e z x )φδ (e z y )3 ρx y .
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D. Dolgopyat, C. Liverani
7. The Limit Equation (δ > 0): Structure Having gained a good knowledge on the variance we are ready to write the limit equation in a more explicit and convenient form. We introduce standard Brownian motions Bx y indexed by oriented edges, so that the motions associated to different non-oriented edges are independent and Bx y = −B yx . Considering the Gaussian processes Wx := |x−y|=1 βx y (z)Bx y we have ⎧ 2 ⎪ for x = y ⎨ |x−w|=1 βxw (z) t E(Wx (t)W y (t) | z) = −βx y (z)β yx (z) t for |x − y| = 1 ⎪ ⎩0 for |x − y| > 1. We set18 βx y (z) =
√
zx √ 2e− 2 φδ (e z x )φδ (e z y ) ρx y ,
(7.1)
hence, remembering Lemmata 6.4, 6.1 and Eqs. (6.2), (4.4), (4.2), ⎧ 2 ⎪ if x = y ⎨ |x−w|=1 βxw δ 2 (σ )x y (z) = −βx y β yx if |x − y| = 1 ⎪ ⎩0 if |x − y| > 1. Then, we can write (5.1) as dz δx = axδ (z δ )dt +
βx y (z δ ) d Bx y .
(7.2)
|x−y|=1
Let L be the operator in the Martingale problem associated to the diffusion defined by (5.1). Lemma 7.1. If the manifold M is d dimensional, then for each β > 0,
e
d zx x 2 z x −βϕδ (e )
dz
is an invariant measure for the process defined by (7.2). In addition, the process (7.2) is reversible. That is, calling Eβ the expectation with respect to the above invariant measure, Eβ (ϕLh) = Eβ (hLϕ) for each smooth real function ϕ, h. Proof. Recall that Lemma 4.2 gives the invariant measures of the original Dynamical System. In particular , for each ψ ∈ C 0 (R|| , R), |μδ,ε,β (ψ(z ε,δ (ε−2 t))) − μδ,0,β (ψ(z ε,δ (ε−2 t)))| ≤ Cε|ψ|∞ . Thus |μδ,0,β (ψ(z ε,δ (ε−2 t))) − μδ,0,β (ψ(z ε,δ (0)))| ≤ 2Cε|ψ|∞ . 18 This is well defined since ρ x,y ≥ 0 by Lemma 6.1.
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Taking the limit ε → 0 along any subsequence leading to an accumulation point we see that μδ,0,β is an invariant measure for the process (5.1). The claim of the lemma now follows by taking the marginal of μδ,0,β in the variables z. In the same manner, using the same notation as in the proof of Lemma 5.1, for each continuous function ψ, g and converging sequence z εk ,δ (εk−2 t) we have ε−2 t
−ε−2 t
Eβ (ψ(z(t))g(z)) = lim μδ,εk ,β (g · ψ ◦ εkk,δ ) = lim μδ,εk ,β (ψ · g ◦ εk ,δk ) k→∞
k→∞ ε−2 t
= lim μδ,εk ,β (ψ ◦ · g ◦ ◦ εkk,δ ) = Eβ (g ◦ (z(t))ψ ◦ (z)). k→∞
Since g, ψ are functions of the z only, it follows that g ◦ = g, ψ ◦ = ψ and Eβ (ψ(z(t))g(z)) = Eβ (g(z(t))ψ(z)). Differentiating with respect to t at t = 0 yields the lemma. Lemma 7.2. The drift axδ has the form ∂z x e−z x φδ (e z x )2 φδ (e z y )2 ρx y − ∂z y e−z y φδ (e z x )φδ (e z y )3 ρx y axδ = |x−y|=1
+
d 2
e−z x φδ (e z x )2 φδ (e z y )2 − e−z y φδ (e z x )φδ (e z y )3 ρx y .
|x−y|=1
Proof. The idea to compute the axδ is very simple: first compute L and L∗ and then check what the reversibility condition implies. The operator associated to the diffusion (5.1) is given by L=
axδ ∂z x +
x
1 δ 2 (σ )x y ∂z x ∂z y . 2 x,y
The adjoint L∗ with respect to the invariant measures in Lemma 7.1 can then be computed by integrating by parts. Setting x (z) := d2 − βφδ (e z x )−1 we have {∂z x axδ + axδ x }ψ − axδ ∂z x ψ L∗ ψ = − x
x
1 + ∂z x ∂z y (σ δ )2x y + 2x ∂z y (σ δ )2x y + x y (σ δ )2x y + δx y ∂z x x (σ δ )2x y ψ 2 xy 1 δ 2 ∂z y (σ δ )2x y + y (σ δ )2x y ∂z x ψ + + (σ )x y ∂z x ∂z y ψ. 2 xy xy This implies axδ =
1 ∂z y (σ δ )2x y + y (σ δ )2x y 2 y
and the lemma follows by direct algebraic computations using Lemma 6.4. The next result is an obvious fact that is nevertheless of great importance.
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Lemma 7.3. The function H :=
x
ϕδ (e z x ) is constant in time.
Proof. It is useful to notice that, setting ψx := By Ito’s formula we have dH =
ψx ax dt +
ez x φδ (e z x )
κ x y d Bx y +
|x−y|=1
x
and κx y = ψx βx y , κx y = κ yx .
1 ∂z ψ x 2 x
βx2y dt.
|x−y|=1
The second term is zero by the antisymmetry of Bx y , thus (using Lemma 7.2 and the symmetry of κx y again) dH =
1 2
ψx ∂z x βx2y − ψ y ∂z x βx y β yx dt |x−y|=1
d + 2
1 ψx−1 − ψ y−1 κx2y dt + 2
|x−y|=1
βx2y ∂z ψx dt = 0.
|x−y|=1
We conclude with the main result of this section. Proposition 7.4. For each δ > 0 the family z ε,δ (ε−2 t) converges weakly, for ε → 0, to the process z(t) determined by the SDE (7.2). Proof. From Lemma 7.2 and Lemma 6.1 it follows that a δ ∈ C ∞ ; this, together with the boundedness and convergence results established in Lemma 5.1 and the standard results on the uniqueness of the solution of the SDE, imply that all the accumulation points of z ε,δ (ε−2 t) must coincide, hence the proposition. 8. The Limit Equation (δ = 0): Properties and Stopping Times It is natural to consider the stopping time τδ := inf{t ∈ R+ : min x∈ z x ≤ ln δ}. In addition, Lemma 7.3 suggests the convenience of going back to the more physical process Ex (t) = ϕδ (e z x (t∧τδ ) ) = e z x (t∧τδ ) . Lemma 8.1. For each t ≤ τδ , the process Ex satisfies the SDE, √ dEx = a(Ex , E y )dt + 2β(Ex , E y )d Bx y , y
where a, β ∈ C ∞ ((0, ∞)2 , R) are respectively anti-symmetric and symmetric functions that satisfy (2.7), (3.2). In addition, if d ≥ 3, then for each constant d − 1 + 8B A M ≥ max 1, , d −2 if E y > MEx , then a(Ex , E y )Ex ≥ β(Ex , E y )2 .
(8.1)
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217
Proof. By Ito’s formula and (7.2) we have19 ⎤ ⎡ 1 dEx = ⎣e z x ax + e z x βx2y ⎦ dt + e z x β x y d Bx y . 2 |x−y|=1
(8.2)
|x−y|=1
Using (7.1), (6.2), (4.4) and Lemma 6.1 we can write e z x βx y =
√ 2Ex ρ( 2Ex , 2E y ) =: 2β(Ex , E y ).
(8.3)
Lemma 7.2, Eqs. (4.2), (6.2) and (4.4) yield ax =
! " d −2 ∂Ex ρ − ∂E y ρ + 2
|x−y|=1
Ex−1 − E y−1 ρ. |x−y|=1
Using Eq. (8.2) we finally obtain (2.7) and from Lemma 6.3 follows (3.2). Moreover, by Lemma 6.3, 1 ∂a ρ( 2Ex , 2E y ), 2Ex 1 ρ( 2Ex , 2E y ) + 2Ex ∂a ρ( 2Ex , 2E y ) =− 2E y
∂Ex ρx y = √ ∂E y ρ x y
=−
β(Ex , E y )2 Ex − ∂E x ρ x y . 2Ex E y Ey
Hence d −2 2 d −2 β − Ex E y−1 β 2 Ex a(Ex , E y ) = β 2 + Ex2 ∂Ex ρx y − Ex2 ∂E y ρx y + 2 2 Ex d − 1 Ex d β(Ex , E y )2 + Ex2 1 + ∂E x ρ x y . − = 2 2 Ey Ey The regularity of the coefficients follows from the previous results and some algebraic computations. At last, for E y > MEx , Ex a(Ex , E y ) ≥
d −1 d B(1 + M −1 ) Ex − β(Ex , E y )2 − . 1 2 2M 2M (2E y ) 2
On the other hand β(Ex , E y ) = Ex ρ( 2Ex , 2E y ) ≥ Ex 2
#
$ √ A B 2Ex AEx − , ≥ 2E y 2E y 4 2E y
(8.4)
from which the lemma follows. 19 Here we suppress the δ-dependence since we stop the motion before seeing the region in which the dynamics has been modified.
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9. The Limit Equation (δ = 0): Unreachability of Zero Energy Our last task it to prove that the stopping time τδ tends to infinity when δ tends to zero or, in other words, energy zero is unreachable for the limit equation. Fix any T > 0. For each subset ⊂ let us define the energy of the cluster E := x∈ Ex . Also, for each δ > 0, n ∈ {1, . . . , ||}, let us define the stopping times τδn := inf{t ∈ [0, ∞) : ∃ ⊂ , || = n, E (t) ≤ δ} ∧ T. Note that τδ1 = τδ ∧ T , where τδ is defined at the beginning of Sect. 8. Lemma 9.1. Let P be the measure associated to the process (2.6), then lim P τδ1 < T = 0. δ→0
Proof. We will prove that for each η > 0 and n ∈ {1, . . . ||} there exists δn = δn (η), such that % & P τδnn < T ≤ 2−n η. The proof is by (backward) induction. The case n = || follows by the energy conservation by choosing δ|| < E2 . Next, suppose the statement true for n + 1 ≤ ||. It is convenient to define, for each ⊂ the stopped process Eˆ (t) = E (t ∧ τδn+1 ) and the set = {τδn+1 ≥ T }. Then, n+1 n+1 for each 0 < δ < δn+1 , we have % & % & P τδn < T ≤ P τδn < T ∩ + 2−(n+1) η ⎛ ⎞ ⎜* ⎟ −(n+1) inf Eˆ (t) ≤ δ ⎟ ≤ P⎜ η. ⎝ ⎠+2 ⊂ ||=n
t∈[0,T ]
It thus suffices to show that there exists δn ≤ δn+1 such that, for each ⊂ , || = n, we have
−1 || ≤ 2−(||+n+1) η ≤ P inf Eˆ (t) ≤ δn 2−(n+1) η. t∈[0,T ] n Let us fix ⊂ , || = n. δn+1 n+1 Observe that if holds but E (t) ≤ M+1 then E y ≥ Mδ M+1 ≥ ME ≥ MEx for all y ∈ and x ∈ . In the following we will choose M as in the statement of Lemma 8.1. Next, we define the process Y = ln E which satisfies √ a(Ex , E y )E − β(Ex , E y )2 −1 dt + 2β(Ex , E y )E d Bx y , (9.1) dY = 2 2E (x,y)∈B() where B() = {(x, y) ∈ 2 : x ∈ , y ∈ , |x − y| = 1}.
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219
Observe that by Corollary 8.1 the drift is positive, indeed a(Ex , E y )E − β(Ex , E y )2 a(Ex , E y )Ex − β(Ex , E y )2 ≥ ≥ 0. 2E2 2E2 (x,y)∈B() (x,y)∈B() δn+1 we have, for some constant C > 0, In addition, arguing as in (8.4), if E (t) ≤ M+1 # $ √ Ex B 2Ex A C −2 2 E β(Ex , E y ) ≤ 2 2 + (9.2) ≤ 3. 2E y E 2E y E2
Therefore Y (t ∧ τδn+1 ) ≥ Y (0) + n+1
t∧τδn+1
n+1
0
(x,y)∈B()
√ 2β(Ex , E y )E−1 d Bx y =: M(t).
Note that M is a Martingale. Let τ∗ = inf{t : M(t) ≤ ln δn+1 } ∧ T . Consider the new . = M(t) − M(t ∧ τ∗ ) and the stopping time martingale M(t) . ≥ − 1 ln δn+1 } ∧ T. . ≤ ln δn − ln δn+1 or M(t) τˆ = inf{t : M(t) 2 Setting p = P({M(τˆ ) = ln δn }) we obtain 1 0 ≤ p(ln δn − ln δn+1 ) − (1 − p) ln δn+1 , 2 which implies P({M(τˆ ) = ln δn }) ≤
ln δn+1 . 2 ln δn − ln δn+1
α , α > 1 to be chosen later. The probability that M, starting from ln δ Set δn = δn+1 n+1 reaches ln δn before reaching 21 ln δn+1 is smaller than (2α − 1)−1 . Accordingly, the probability that the martingale reaches ln δn before downcrossing L times the interval [ln δn+1 , 21 ln δn+1 ] is smaller than 1 − (1 − (2α − 1)−1 ) L ≤ α −1 L. On the other hand by Doob’s inequality the expectation of the number of downcrossing √ is bounded by 2 1 + ). Since M − 1 ln δ E((M − ln δ ) ≥ 0 implies E ≥ δn+1 , by (9.2), n+1 n+1 −1 2 2 ln δn+1
1 −3 ln δn+1 )+ ) ≤ Cδn+14 follows, 2 for some constant C independent on ε. From this it immediately follows that the proba−1 bility to have more than L downcrossing is less than L −1 δn+1 . In conclusion,
−1 ≤ C(α −1 L + L −1 δn+1 P inf Eˆ (t) ≤ δn ), E((M −
t∈[0,T ]
−1 which yields the wanted estimate by first choosing L 2 = αδn+1 and then setting α = −1 2||+2n+4 −2 20 C 2 δn+1 2 η . −1 20 Note that δ ∼ δ Cδn+1 for some constant C. So, for large , δ is absurdly small. Yet, this suffices for n 1 n+1
our purposes.
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Corollary 9.2. The set {∃x : Ex = 0} is inaccessible for the limiting equation. Acknowledgements. We thank Gabriel Paternain for suggesting to us reference [10]. C.Liverani acknowledges the partial support of the European Advanced Grant Macroscopic Laws and Dynamical Systems (MALADY) (ERC AdG 246953). Both authors warmly thank CIRM, Marseille, that fostered the beginning of this work by financing our research in pairs. Finally, it is a pleasure to thank the Fields Institute, Toronto, where the paper was finished.
Appendix A. An Averaging Theorem In this appendix, for reader convenience, we recall [12, Thm. 7] stating it in reduced generality but in a form directly applicable to our setting. Let M be a C ∞ Riemannian manifold, z ∈ Rd and f z ∈ Diff ∞ (M, M) a family of partially hyperbolic diffeomorphisms.21 We say that { f z } is a family of Anosov elements (FAE) if there exists Abelian actions gz,t , t ∈ Rdc , where dc = dim E c , such that f z ◦ gz,t = gz,t ◦ f z and span{∂ti gz,t } = E c . Next, we need to discuss decay of correlations that in [12] is meant in a very precise technical sense. The basic concept is the one of standard pairs. For the present purposes a standard pair can be taken to be a probability measure determined by the couple = (D, ρ), where D is a C 2 dim(E u )-dimensional manifold D close to the strong unstable manifold and a smooth function ρ ∈ C 1 (D, R+ ) such that D ρ = 1.22
We set E (A) = D Aρ. The point is that it is possible to choose a set of manifolds D of uniform bounded diameter and curvature such that, for each D ∈ , f z D can be covered by a fixed number of elements of . For each C > 0 we consider the set E 1 = {(D, ρ) : D ∈ , ρC 1 (D, R) ≤ C} and let E 2 be the convex hull of E 1 in the space of probability measures. It is easy to check that one can choose and C such that for all ∈ E 1 there exists nz a family {i } ⊂ E 1 such that E (A ◦ f z ) = i=1 ciz Ei (A). In addition one can insure that any measure with C 1 density with respect to the Riemannian volume belongs to the weak closure of E 2 (see [12] for more details). We say that the family { f z } has uniform exponential decay of correlations if there exists C1 , C2 > 0 such that, for each z ∈ Rd there exists probability measures μz such that for each n ∈ N, standard pair ∈ E 1 and functions A ∈ C 1 (M, R), E (A ◦ f n ) − μz (A) ≤ C1 e−C2 n |A|C 1 holds. z Consider now the function F ∈ C ∞ (M × Rd × R+ , M × Rd ), F(x, z, ε) = ( f˜(x, z, ε), z + ε A(x, z, ε)),
(A.1)
and the associated dynamical systems Fε (x, z) = F(x, z, ε), such that f˜(x, z, 0) = f z (x). Let (xnε (x, z), z nε (x, z)) := Fεn (x, z). Then for each g ∈ C r (M, R+ ), μ(g) = 1 we can define the measure μg (h) := μ(g · h) and consider the Dynamical Systems (Fε , M × Rd ) with initial conditions z = z 0 and x distributed according to the measure 21 By this we mean that, for each fixed z, at each point x ∈ M the tangent space of T M can be written x as E u (x) ⊕ E c (x) ⊕ E s (x), where the splitting is invariant with respect to the dynamics, i.e. dx f E ∗ (x) = E ∗ ( f (x)) for ∗ ∈ {u, c, s}. In addition, there exists constants λ1 ≤ λ2 < λ3 ≤ λ4 < λ5 ≤ λ6 , with λ2 , λ−1 5 < 1, such that λ1 ≤ α(d f | E s ) ≤ d f | E s ≤ λ2 , λ3 ≤ α(d f | E c ) ≤ d f | E c ≤ λ4 and λ5 ≤ α(d f | E u ) ≤ d f | E u ≤ λ6 , where α(A) = A−1 −1 . 22 The integral is with respect to the volume form on D induced by the Riemannian metric.
Energy Transfer in a Fast-Slow Hamiltonian System
221
μg . We can then view z nε as a random variable, clearly E(ψ(z nε )) = μg (ψ˜ ◦ Fεn ), where ˜ ψ(x, z) = ψ(z). Theorem A.1 ([12]). Let F, Fε , f z be defined as in (A.1) and subsequent lines. Let f z be FAE with uniform exponential decay of correlation. Suppose that there exists ε0 , Cr ∈ R+ such that supε≤ε0 A(·, ·, ε)Cr ≤ Cr and μz (A(·, z, 0)) = 0 for all z. Also assume that z 0ε = z ∗ and x0ε has a smooth distribution on M as described above, then ε a) The family {z tε −2 } is tight.
b) There exist functions σ 2 ∈ C 1 (Rd , S L(d, Rd )), σ 2 > 0, a ∈ C 0 (Rd , Rd ) such that ε the accumulation points of {z tε −2 } are a solution of the Martingale problem associated to the diffusion dz = adt + σ d B, z(0) = z ∗ , d are independent standard Brownian motions and where {Bi }i=1 ∞
σ 2 (z) =
n=−∞ M
A(x, z, 0) ⊗ A( f zn x, z, 0)μz (d x).
Moreover aC 0 + σ 2 C 1 < ∞. Appendix B. The Properties of ρ x y Here we collect, a bit boring, proofs of the Lemmata concerning ρx y . Proof of Lemma 6.1. The non-negativity follows from the fact that the quantity is an autocorrelation, see footnote 24 for details. By definition
˜ b) ∂an ∂bm ρ(a,
m at bt dt t n+m E (L n+1 L V ) ◦ g ⊗ g · L V 2 1 2 −∞ ∞ m at bt . = (−1)n+m dt t n+m E (L n+1 1 L2 V ) · L2V ◦ g ⊗ g
=
∞
−∞
Applying (5.3) to the above formula yields ˜ b)| ≤ Cn,m |∂an ∂bm ρ(a,
∞
dt t n+m e−c min{a,b}t ≤ Cn,m min{a, b}−n−m−1 .
0
This proves the smoothness of ρ. ˜ To continue, consider ρ(λa, ˜ λb) =
∞
−∞
dt E (L 1 V ) ◦ g aλt ⊗ g bλt · L 2 V = λ−1 ρ(a, ˜ b)
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D. Dolgopyat, C. Liverani
by the change of variables t → λt. The symmetry follows by a change of variables as well. Finally, ∞ ρ(a, ˜ b) = dt E (L 1 V ) ◦ g at ⊗ g bt · L 2 V −∞ ∞ = dt E L 1 V · (L 2 V ) ◦ g at ⊗ g bt −∞ ∞ d −1 =b E L 1 V · V ◦ g at ⊗ g bt dt dt −∞ ∞ a − dt E L 1 V · (L 1 V ) ◦ g at ⊗ g bt . b −∞ The lemma follows then by the mixing of g at ⊗ g bt (being the product of two mixing flows) and the definition of ρ. To continue it is useful to define and study the function (τ ) := ρ(τ, 1). Lemma B.1. There exists A, B > 0 and D ∈ R such that (τ ) − A ≤ Bτ , ∀τ > 0, 1 + τ3 1 + τ5 | (τ ) − Dτ | ≤ Bτ 2 , ∀τ ∈ (0, 1], | (τ ) + 3Aτ −4 | ≤ Bτ −5 , ∀τ ≥ 1. Proof. Let us start by assuming τ ≤ 1. By setting V (q1 ) = E(V | q1 , v1 ), and taking care of adding and subtracting that is needed to write convergent integrals, ∞ (τ ) = 2 dt E L 1 V · L 1 V ◦ g τ t ⊗ g t ) 0 ∞ dt E L 1 V · L 1 V ◦ id ⊗ g t − E((L 1 V )2 ) =2 0 ∞ τt +2 dt ds E L 1 V · L 21 V ◦ g s ⊗ g t − E(L 1 V · L 21 V ◦ g s ) 0 0 ∞ dt E(L 1 V · L 1 V ◦ g τ t ). +2 0
The third term here vanishes since it is the variance of a coboundary. That is, ∞ ∞ d E(L 1 V · V ◦ g t ) = 0. dt E(L 1 V · L 1 V ◦ g τ t ) = τ −1 dt dt 0 0 Also, setting V˜ = V − V , ∞ τt dt ds E L 1 V · L 21 V ◦ g s ⊗ g t − E(L 1 V · L 21 V ◦ g s ) 0
∞ ∞ ∞0 ∞ ds dt E L 1 V˜ · L 21 V˜ ◦ g s ⊗ g t = O ds e−ct dt = O(τ ), = 0
τ −1 s
0
τ −1 s
Energy Transfer in a Fast-Slow Hamiltonian System
223
where we have used (5.3) after conditioning with respect to q1 , v1 . Thus23 ∞ (τ ) = 2 dt E ∂q1 V · ∂q1 V ◦ id ⊗ g t − E((∂q1 V )2 ) 0
+O(τ ) = A + O(τ ).
(B.1)
The fact that A > 0 follows from general theory of mixing flows combined with cocycle rigidity of geodesic flows [10,21].24 Next, consider the case τ ≥ 1. By Lemma 6.1 we have (τ ) = ρ(τ, 1) = τ −1 ρ(1, τ −1 ) = −τ −2 ρ(τ ˜ −1 , 1) = τ −3 (τ −1 ).
(B.2)
A −4 ). This readily implies the first part of the lemma. Thus (τ ) = 1+τ 3 + O(τ Let us compute the derivative ∞ (τ ) = dt t E L 1 V · L 21 V ◦ g τ t ⊗ g t 2 0 ∞ τt = dt t ds E L 1 V · L 31 V ◦ g s ⊗ g t − E L 1 V · L 31 V ◦ g s 0 0 ∞ τt + dt t ds E L 1 V · L 31 V ◦ g s 0 ∞ ∞ 0 ∞ t = ds dt t E L 1 V˜ · L 31 V˜ ◦ g s ⊗ g t + dt 2 E L 1 V · L 21 V ◦ g t τ 0 τ −1 s 0 ∞ ∞ ∞ E L 1 V · L 1 V ◦ gt 3 ˜ t ˜ = ds dt t E L 1 V · L 1 V ◦ id ⊗ g − dt −1 τ2 0 0 ∞ τ s ∞ s + ds dt t dr E L 1 V˜ · L 41 V˜ ◦ gr ⊗ g t 0
τ −1 s
0
23 Here we use the fact that E(v ⊗ v | q , η) = 1. 1 1 1 24 Indeed, for each T > 0 and f ∈ C ∞ , E( f ) = 0,
⎛ 2 ⎞ T T T t f ◦ g dt ⎠ = 2 dt (T − t)E( f ◦ g t · f ) = 2T dt E( f ◦ g t · f ) + O(1). 0 ≤ E ⎝ 0 0 0
Thus the autocorrelation must be non negative. If it is zero then 0T f ◦ g t dt has uniformly bounded L 2 norm. This implies that there exists a weakly converging subsequence to some L 2 function h such that E(h) = 0. It is easy to check that such a function is smooth in the stable direction (just compare with the average on stable manifolds) and, for each smooth function ϕ, E(h Lϕ) = −E( f ϕ). Thus E(h L n ϕ) = −E( f L n−1 ϕ) = (−1)n E(L n−1 f ϕ), which implies L n h ∈ L 2 , i.e. h is smooth along weak-stable leaves. Next, letting (q, ν) = (q, −ν), we have E( f ϕ) = E(h ◦ · Lϕ), that is E((h + h ◦ )Lϕ) = 0 for each smooth ϕ. In turns, this implies h = −h ◦ a.s.. Indeed, given ρ ∈ L 2 , if E(ρ) = 0 and E(ρ Lϕ) = 0 for all smooth ϕ, then one can choose smooth ρn that converges to ρ in L 2 , thus Lρn converges weakly to zero, but then there exist convex combinations ρ˜n of the {ρm }m≤n such that L ρ˜n converges to zero strongly (since the weak closure of a convex set agrees with its strong closure) and, since L is a closed operator on L 2 , it follows that ρ is in the domain of L and Lρ = 0. In addition, the ergodicity of the flow implies that the only L 2 , zero average, solution of Lρ = 0 is ρ = 0. Finally, since h is smooth along the weak-stable foliation and h ◦ is smooth along the unstable foliation, then h has a continuos version by the absolute continuity of the foliations and is smooth by [17], hence Lh = f . That is, if the autocorrelation is zero, then f is a smooth coboundary. At last, the claim follows since a smooth function of the coordinates only which is a coboundary must be iden ∞ tically zero, [10, Corollary 1.4]. Accordingly, −∞ dt E ∂q1 V (q1 , q2 )∂q1 V (q1 , g t (q2 , v2 )) | q1 , v1 must be strictly positive for positive measure set of q1 otherwise, by the symmetry of the potential, the potential would be constant.
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=−
∞
ds
0
= −τ 0
dt t E
τ −1 s ∞
+O
∞
L 21 V˜
·
L 21 V˜
ds τ −1 s 2 e−csτ
−1
◦ id ⊗ g
t
2 ) E L 1 V )E(V − E( L 1 (V 2 ) − τ2
0 ∞
dt t 2 E L 21 V˜ · L 21 V˜ ◦ id ⊗ g t + O(τ 2 ) =: Dτ + O(τ 2 ).
On the other hand, differentiating (B.2) yields, for τ large, (τ ) = −3τ −4 (τ −1 ) − τ −5 (τ −1 ) = −3Aτ −4 + O(τ −5 ), which completes the proof of the lemma. Remark B.2. Note that (0) is not defined as the corresponding integral diverges. Nevertheless, we can set (0) = A by continuity. Proof of Lemma 6.3. Note that, by Lemma 6.1, ρ(a, b) = b−1 ρ(ab−1 , 1) = b−1 (ab−1 ). Hence the lemma follows from Lemma B.1. References 1. Anosov, D.V., Sinai, Ja.G.: Certain smooth ergodic systems. Uspehi Mat. Nauk 22(5), 107–172 (1967) 2. Aoki, K., Lukkarinen, J., Spohn, H.: Energy transport in weakly anharmonic chains. J. Stat. Phys. 124(5), 1105–1129 (2006) 3. Basile, G., Bernardin, C., Olla, S.: Thermal conductivity for a momentum conservative model. Commun. Math. Phys. 287(1), 67–98 (2009) 4. Basile, G., Olla, S., Spohn, H.: Energy transport in stochastically perturbed lattice dynamics. Arch. Rat. Mech. Anal. 195(1), 171–203 (2010) 5. Bernardin, C.: Thermal conductivity for a noisy disordered harmonic chain. J. Stat. Phys. 133(3), 417–433 (2008) 6. Bonetto, F., Lebowitz, J.L., Rey-Bellet, L.: Fourier’s law: a challenge to theorists. In: Mathematical Physics 2000, Imp. Coll. Press, London, 2000, pp. 128–150 7. Bricmont, J., Kupiainen, A.: Towards a derivation of Fourier’s law for coupled anharmonic oscillators. Commun. Math. Phys. 274(3), 555–626 (2007) 8. Bricmont, J., Kupiainen, A.: Random walks in space time mixing environments. J. Stat. Phys. 134(5-6), 979–1004 (2009) 9. Collet, P., Eckmann, J.-P.: A model of heat conduction. Commun. Math. Phys. 287(3), 1015–1038 (2009) 10. Croke C.B., Sharafutdinov V.A.: Spectral rigidity of a compact negatively curved manifold. Topology 37(6), 1265–1273 (1998) 11. Dolgopyat, D.: On decay of correlations in Anosov flows. Ann. Math. 147, 357–390 (1998) 12. Dolgopyat, D.: Averaging and Invariant measures. Moscow Math. J. 5, 537–576 (2005) 13. Dolgopyat, D., Keller, G., Liverani, C.: Random walk in Markovian environment. Ann. Probab. 36, 1676– 1710 (2008) 14. Dolgopyat, D., Liverani, C.: Random walk in deterministically changing environment. Lat. Am. J. Probab. Math. Stat. 4, 89–116 (2008) 15. Dolgopyat, D., Liverani, C.: Non-perturbative approach to random walk in Markovian environment. Elect. Commun. Prob. 14, 245–251 (2009) 16. Eckmann, J.-P., Young, L.-S.: Nonequilibrium energy profiles for a class of 1-D models. Commun. Math. Phys. 262(1), 237–267 (2006) 17. Journé, J.-L.: A regularity lemma for functions of several variables. Rev. Mat. Iberoamericana 4(2), 187–193 (1988) 18. Freidlin, M.I., Wentzell, A.D. Random perturbations of dynamical systems. Translated from the 1979 Russian original by Joseph Szcs. 2nd edition. Fundamental Principles of Mathematical Sciences, 260. New York: Springer-Verlag, 1998
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19. Gaspard, P., Gilbert, T.: Heat conduction and Fourier’s law in a class of many particle dispersing billiards. New J. Phys. 10, 103004 (2008) 20. Gaspard, P., Gilbert, T.: Heat conduction and Fourier’s law by consecutive local mixing and thermalization. Phys. Rev. Lett. 101, 020601 (2008) 21. Guillemin, V., Kazhdan, D.: Some inverse spectral results for negatively curved n-manifolds. Proc. Sympos. Pure Math., XXXVI. Providence, RI: Amer. Math. Soc., 1980, pp. 153–180 22. Guo, M.Z., Papanicolaou, G.C., Varadhan, S.R.S.: Nonlinear diffusion limit for a system with nearest neighbor interactions. Commun. Math. Phys. 118(1), 31–59 (1988) 23. Hunt, T.J., MacKay, R.S.: Anosov parameter values for the triple linkage and a physical system with a uniformly chaotic attractor. Nonlinearity 16(4), 1499–1510 (2003) 24. Liverani, C.: On Contact Anosov flows. Ann. of Math. 159(3), 1275–1312 (2004) 25. Liverani, C., Olla, S.: Toward the Fourier law for a weakly interacting anharmonic crystal. preprint http:// arXiv.org/ans/1006.2900v1 [math.PR], 2010 26. Lukkarinen, J., Spohn, H.: Anomalous energy transport in the FPU-beta chain. Commun. Pure Appl. Math. 61(12), 1753–1786 (2008) 27. Paternain, G.P.: Geodesic flows. Progress in Mathematics, 180. Boston, MA: Birkhäuser Boston, Inc., 1999 28. Revuz, D., Yor, M.: Continuous Martingales and Brownian Motion, 3rd edition. Fundamental Principles of Mathematical Sciences, 293. Berlin: Springer-Verlag, 1999 29. Ruelle, D.: A mechanical model for Fourier’s Law of the heat conduction, http://arXiv.org/abs/1102.5488 [nlin.CD], 2011 30. Spohn, H.: Large scale dynamics of interacting particles. Berlin, New York: Springer-Verlag, 1991 31. Varadhan, S.R.S.: Nonlinear diffusion limit for a system with nearest neighbor interactions-II. Pitman Res. Notes Math. Ser. 283. Harlow: Longman Sci. Tech., 1993, pp. 75–128 Communicated by H. Spohn
Commun. Math. Phys. 308, 227–279 (2011) Digital Object Identifier (DOI) 10.1007/s00220-011-1322-x
Communications in
Mathematical Physics
Non-Intersecting Squared Bessel Paths: Critical Time and Double Scaling Limit A. B. J. Kuijlaars1 , A. Martínez-Finkelshtein2,3 , F. Wielonsky4 1 Department of Mathematics, Katholieke Universiteit Leuven, Celestijnenlaan 200B, 3001 Leuven, Belgium.
E-mail: [email protected]
2 Department of Statistics and Applied Mathematics, University of Almería, Almería, Spain.
E-mail: [email protected]
3 Instituto Carlos I de Física Teórica y Computacional, Granada University, Granada, Spain 4 Laboratoire d’Analyse, Topologie et Probabilités, Université de Provence, 39 Rue Joliot Curie,
13453 Marseille Cedex 20, France. E-mail: [email protected] Received: 4 November 2010 / Accepted: 18 March 2011 Published online: 6 September 2011 – © Springer-Verlag 2011
Abstract: We consider the double scaling limit for a model of n non-intersecting squared Bessel processes in the confluent case: all paths start at time t = 0 at the same positive value x = a, remain positive, and are conditioned to end at time t = 1 at x = 0. After appropriate rescaling, the paths fill a region in the t x–plane as n → ∞ that intersects the hard edge at x = 0 at a critical time t = t ∗ . In a previous paper, the scaling limits for the positions of the paths at time t = t ∗ were shown to be the usual scaling limits from random matrix theory. Here, we describe the limit as n → ∞ of the correlation kernel at critical time t ∗ and in the double scaling regime. We derive an integral representation for the limit kernel which bears some connections with the Pearcey kernel. The analysis is based on the study of a 3 × 3 matrix valued Riemann-Hilbert problem by the Deift-Zhou steepest descent method. The main ingredient is the construction of a local parametrix at the origin, out of the solutions of a particular third-order linear differential equation, and its matching with a global parametrix. Contents 1.
2.
3. 4.
Introduction and Main Results . . . . . . . . . 1.1 Introduction . . . . . . . . . . . . . . . . 1.2 Statement of results . . . . . . . . . . . . 1.3 Riemann-Hilbert problem . . . . . . . . . First and Second Transformation . . . . . . . 2.1 The first transformation . . . . . . . . . . 2.2 The Riemann surface . . . . . . . . . . . 2.3 Modified ζ functions . . . . . . . . . . . 2.4 The λ-functions . . . . . . . . . . . . . . 2.5 Second transformation of the RH problem Third Transformation of the RH Problem . . . Global Parametrix . . . . . . . . . . . . . . .
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5.
Local Parametrices . . . . . . . . . . . . . . . . . . 5.1 Parametrix P around q . . . . . . . . . . . . . 5.2 Parametrix Q around 0: required properties . . 5.3 Reduction to constant jumps . . . . . . . . . . 5.4 Definition of α (z; τ ) . . . . . . . . . . . . . 5.5 Asymptotics of α . . . . . . . . . . . . . . . 5.6 Definition and properties of f (z) and τ (z) . . . 5.7 Definition and properties of the prefactor E n (z) 6. Fourth Transformation of the RH Problem . . . . . 7. Final Transformation . . . . . . . . . . . . . . . . . 8. The Limiting Kernel . . . . . . . . . . . . . . . . . 8.1 Expression for the critical kernel . . . . . . . . 8.2 Proof of Theorem 1.2 . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . .
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1. Introduction and Main Results 1.1. Introduction. We considered in [28] a model of n non-intersecting squared Bessel paths in the confluent case. In this model, all paths start at time t = 0 at the same positive value x = a > 0 and end at time t = 1 at x = 0. Our aim was to study the asymptotic behavior of the model as n → ∞. The positions of the squared Bessel paths at any given time t ∈ (0, 1) are a determinantal point process with a correlation kernel that is built out of the transition probability density function of the squared Bessel process. In [28] we found that, after appropriate scaling, the paths fill out a region in the t x plane that we described explicitly. Initially, the paths stay away from the hard edge at x = 0. At a certain critical time t ∗ the smallest paths come to the hard edge and then remain close to it, as can be seen in Fig. 1. In [28] we also proved the local scaling limits of the correlation kernel as n → ∞, that are typical from random matrix theory. Thus we find the sine kernel in the bulk 2.5
2
x
1.5
1
0.5
0
0
0.2
0.4
0.6
0.8
1
t
Fig. 1. Numerical simulation of 50 rescaled non-intersecting squared Bessel paths with a = 1. Bold lines are the boundaries of the domain filled out by the paths as their number increases
Non-Intersecting Squared Bessel Paths: Critical Time and Double Scaling Limit
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and the Airy kernel at the soft edges, which includes the lower boundary of the limiting domain for t < t ∗ . For t > t ∗ , we find the Bessel kernel at the hard edge 0, see [28, Thms. 2.7–2.9]. In this paper we consider the critical time t = t ∗ . We describe the transition from the Airy kernel to the Bessel kernel by means of a new one-parameter family of limiting kernels that arise as limiting kernels around the critical time. This soft-to-hard edge transition is different from previously studied ones in [6] or [9], but is related to the one in [7]. We consider the squared Bessel process with parameter α > −1, with transition probabality density ptα given by, see [8,26,29], √ xy 1 y α/2 −(x+y)/(2t) , x > 0, y ≥ 0, ptα (x, y) = e Iα 2t x t (1.1) yα α −y/(2t) e pt (0, y) = , y > 0, (2t)α+1 (α + 1) where Iα denotes the modified Bessel function of the first kind of order α, Iα (z) =
∞ k=0
(z/2)2k+α . k! (k + α + 1)
(1.2)
A remarkable theorem of Karlin and McGregor [24] describes the distribution of n independent non-intersecting copies of a one-dimensional diffusion process at any given time t in terms of its transition probabilities. In the case of the squared Bessel process, with all starting points at time 0 in a > 0 and all ending points at a later time T > 0 in 0, the theorem implies that the positions of the paths at time t ∈ (0, T ) have the joint probability density P(x1 , . . . , xn ) =
1 det[ f j (xk )] j,k=1,...,n det[g j (xk )] j,k=1,...,n Zn
(1.3)
on (R+ )n , with functions f 2 j−1 (x) = x j−1 ptα (a, x), f 2 j (x) =
x j−1 ptα+1 (a, x), x j−1 − 2(T −t)
g j (x) = x
e
,
j = 1, . . . , n 1 := n/2 ,
(1.4)
j = 1, . . . , n 2 := n − n 1 ,
(1.5)
j = 1, . . . , n,
(1.6)
see [28, Prop. 2.1]. The constant Z n is a normalizing constant which is taken so that (1.3) defines a probability density function on (R+ )n . Formula (1.3) is characteristic of a biorthogonal ensemble [3]. It is known that (1.3) n , defines a determinantal point with correlation kernel K n (x, y; t, T ) = n (x, y) = K K
n
f j (x) A−1
k, j
j,k=1
where A−1 k, j is the (k, j)th entry of the inverse of the matrix A= 0
∞
f j (x)gk (x)d x
. j,k=1,...,n
gk (y),
(1.7)
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This means that P(x1 , . . . , xn ) =
1 n (xi , x j ) det K i, j=1,...,n n!
(1.8)
and for each m = 1, . . . , n − 1,
∞
∞
n! n (xi , x j ) ··· P(x1 , . . . , xn )d xm+1 · · · d xn = det K . i, j=1,...,m (n − m)! 0 0 (1.9) Determinantal processes arise naturally in probability theory, see e.g. [23,30]. The connection with models of non-intersecting paths is well-known see [19, Chap. 10] and references therein. Non-intersecting squared Bessel paths and related continuous models with a wall are studied in [25–27,29,31]. Non-intersecting discrete random walks with a wall are considered in the recent papers [4,5,7,32]. As in [28] we introduce a time rescaling t →
t , 2n
T →
1 , 2n
and we consider the rescaled kernels n x, y; t , 1 , K n (x, y; t) = e−n(x−y)/(1−t) K 2n 2n
x, y > 0,
0 < t < 1, (1.10)
that depend on the variable t. The prefactor e−n(x−y)/(1−t) does not affect the correlation functions (1.9). We define w1,n , w2,n on [0, ∞) by √ 2n ax nx Iα , w1,n (x) = x α/2 exp − t (1 − t) t (1.11) √ 2n ax nx w2,n (x) = x (α+1)/2 exp − Iα+1 , t (1 − t) t as in [28, Eq. (2.20)]. Then the kernel (1.10) is expressed in terms of a RH problem. Indeed we have ⎛ ⎞ 1 −1 1 0 w1,n (y) w2,n (y) Y+ (y)Y+ (x) ⎝0⎠ , (1.12) K n (x, y; t) = 2πi(x − y) 0 where Y is the solution of the following matrix valued Riemann-Hilbert problem, see [28]: RH problem 1.1. Find Y : C\[0, ∞) → C3×3 such that 1. Y is analytic in C\[0, ∞). 2. On the positive real axis, Y possesses continuous boundary values Y+ (from the upper half plane) and Y− (from the lower half plane), and ⎞ ⎛ 1 w1,n (x) w2,n (x) 1 0 ⎠, x > 0. (1.13) Y+ (x) = Y− (x) ⎝0 0 0 1
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3. Y (z) has the following behavior at infinity: ⎞ ⎛ z n 0 0 1 ⎝ 0 z −n 1 0 ⎠ , z → ∞, z ∈ C\[0, ∞), (1.14) Y (z) = I + O z −n 0 0 z 2 where n 1 = n/2 and n 2 = n/2 . 4. Y (z) has the following behavior near the origin, as z → 0, z ∈ C\[0, ∞): ⎧ ⎛ ⎞ 1 h(z) 1 ⎨ |z|α , if −1 < α < 0, α = 0, (1.15) Y (z) = O ⎝1 h(z) 1⎠ , with h(z) = log |z|, if ⎩ 1, 1 h(z) 1 if 0 < α. The O condition in (1.15) is to be taken entry-wise. This RH problem has a unique solution given in terms of multiple orthogonal polynomials for the modified Bessel weights (1.11). It was proven in [28, Prop. 2.3 and Thm. 2.4] that in this scaling there is a critical time a t∗ = (1.16) a+1 depending only on the starting value a. For every t ∈ (0, 1), we have that 1 K n (x, x; t) = ρ(x; t) n→∞ n lim
exists, where the limiting density ρ(x; t) is supported on an interval [ p(t), q(t)] with p(t) > 0 if t < t ∗ and p(t) = 0 if t ≥ t ∗ . The results of [28] were obtained from a steepest descent analysis of the above RH problem for values of t = t ∗ . In this paper we develop the steepest descent analysis at the critical time. 1.2. Statement of results. The main result of our paper is the following theorem. Theorem 1.2. Let K n be the correlation kernel (1.12) for the positions of the rescaled non-intersecting squared Bessel paths with parameter α > −1, starting at a > 0 at time 0 and ending at zero at time 1. Let t ∗ = a/(a + 1) as in (1.16) and c∗ = t ∗ (1 − t ∗ ) =
a . (a + 1)2
Then we have, for every fixed τ ∈ R, and x, y > 0, ∗ c x c∗ y ∗ c∗ τ c∗ = K αcrit (x, y; τ ), Kn , ;t − √ lim n→∞ n 3/2 n 3/2 n 3/2 n where K αcrit is the kernel K αcrit (x, y; τ ) =
1 (2πi)2
t∈
s∈
t α τ/t+1/(2t 2 )−τ/s−1/(2s 2 ) xt−ys dtds . e e sα s−t
(1.17)
(1.18)
(1.19)
The contours and in (1.19) are as in Fig. 2. The fractional powers s α and t α in (1.19) are defined with a branch cut on the positive semi-axis, i.e., 0 < arg s, arg t < 2π .
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Fig. 2. The contours of integration and used in the definition of the critical kernel (1.19). The contour consists of a closed loop in the left half-plane tangent to the origin and is oriented clockwise. The contour
is an unbounded loop oriented counterclockwise and encircling
We prove Theorem 1.2 by an asymptotic analysis of the RH problem 1.1 by means of the steepest descent analysis of Deift and Zhou, as we did in [28] for the non-critical times. At a certain stage in the analysis we have to construct a local parametrix at the origin x = 0 (the hard edge). This was done in [28] with the Bessel parametrix. We also had to construct an Airy parametrix at another position (a soft edge). In the critical case that we are considering in this paper this other position coincides with the origin. The coalescing of the soft edge with the hard edge leads to the construction of a new local parametrix at the origin. The construction uses a new model Riemann-Hilbert problem that we describe in the next subsection. The functions that appear in the model RH problem ultimately lead to the expression (1.19) for the limiting kernels.
1.3. Riemann-Hilbert problem. The model RH problem is defined on the contour shown in Fig. 3. It consists of the six rays arg z = 0, ±π/4, ±3π/4, oriented from left to right. RH problem 1.3. Let α > −1 and τ ∈ C. The RH problem is to find α = α (·; τ ) : C\ → C3×3 such that 1. α is analytic in C\ . 2. α has boundary values on each part of \{0} satisfying α,+ (z; τ ) = α,− (z; τ )Jα (z),
z ∈ ,
(1.20)
where the jump matrices Jα are shown in Fig. 3. 3. Let ω = e2πi/3 and θk (z) = θk (z; τ ) =
3 2k 2/3 ω z + ωk τ z 1/3 , 2
k = 1, 2, 3.
(1.21)
Non-Intersecting Squared Bessel Paths: Critical Time and Double Scaling Limit
233
Fig. 3. Contour and jump matrices Jα in the RH problem for α
Then as z → ∞, we have ⎞⎛ ⎛ 1/3 0 0 ω ω2 i z −α/3 ⎝z ⎠ ⎝ 0 1 0 1 1 α (z; τ ) = √ 3 0 0 z −1/3 ω2 ω ⎛ 0 eθ1 (z;τ ) I + O(z −1/3 ) ⎝ 0 eθ2 (z;τ ) 0 0
⎞ ⎛ απi/3 ⎞ 1 0 0 e 1⎠ ⎝ 0 e−απi/3 0⎠ 1 0 0 1 ⎞ 0 (1.22) 0 ⎠ , Im z > 0, eθ3 (z;τ )
and
⎞⎛ 2 ⎞ ⎛ −απi/3 ⎞ ⎛ 1/3 0 0 0 0 ω −ω 1 e i z −α/3 ⎝z 0 1 0 ⎠ ⎝ 1 −1 1⎠ ⎝ 0 α (z; τ ) = √ eαπi/3 0⎠ −1/3 2 3 0 0 z ω −ω 1 0 0 1 ⎛ θ (z;τ ) ⎞ 0 0 e2 I + O(z −1/3 ) ⎝ 0 (1.23) 0 ⎠ , Im z < 0. eθ1 (z;τ ) 0 0 eθ3 (z;τ )
4. As z → 0 we have
⎛
⎞ zα 0 0 α (z; τ ) ⎝ 0 z α 0⎠ = O(1), 0 0 1 ⎛ ⎞ 1 0 0 α (z; τ ) ⎝0 z α 0⎠ = O(1), 0 0 1 ⎛ ⎞ 1 0 0 α (z; τ ) ⎝0 z α 0 ⎠ = O(1), 0 0 zα
0 < | arg z| < π/4,
(1.24)
π/4 < | arg z| < 3π/4,
(1.25)
3π/4 < | arg z| < π.
(1.26)
Note that the parameter τ appears in (1.21) and in the asymptotic conditions (1.22) and (1.23) of the RH problem. We prove the following.
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A. B. J. Kuijlaars, A. Martínez-Finkelshtein, F. Wielonsky
Theorem 1.4. Let α > −1 and τ ∈ C. The RH problem 1.3 for α has a unique solution with det α (z; τ ) = z −α , The critical kernel (1.19) satisfies 1 −1 K αcrit (x, y; τ ) = 2πi(x − y)
1
z ∈ C\ .
(1.27)
⎛ ⎞ 1 ⎝ 0 −1 1⎠ , (1.28) (y; τ ) (x; τ ) α,+ α,+ 0
for x, y > 0 and τ ∈ R. The uniqueness statement in Theorem 1.4 follows from standard arguments where one first proves (1.27). The existence of a solution follows from an explicit construction of α , given in Proposition 5.2, in terms of solutions of the third order ODE, zp + αp − τ p − p = 0. A particular solution of this equation is given by
2 p(z) = t α−3 eτ/t e1/(2t ) e zt dt,
(1.29)
(1.30)
where is the closed contour shown in Fig. 2. The inverse matrix −1 α is built out of solutions of the adjoint equation zq + (3 − α)q − τ q + q = 0 which has the special solution q(z) =
s −α e−τ/s e−1/(2s ) e−zs ds 2
(1.31)
(1.32)
where is also shown in Fig. 2. In terms of these functions the kernel (1.19), (1.28), can also be written as 2πi(x − y)K αcrit (x, y; τ )
= q (y) − (α − 2)q (y) − τ q(y) p(x) + −yq (y) + (α − 1)q(y) p (x) +yq(y) p (x).
(1.33)
The formula (1.33) shows that K αcrit is an integrable kernel in the sense of Its et al. [22], see also [14,20]. For y = x the right-hand side of (1.33) is the bilinear concomitant [1,21] which is constant for any two solutions p and q of the differential equations (1.29), (1.31), and which turns out to be zero for the two particular solutions (1.30) and (1.32). Remark 1.5. There are solutions of the differential equations (1.29) and (1.31) that can be written as integrals of Bessel functions. In particular, we have that
+∞ √ 2 p(z) = z −α/2 u α/2 e−τ u−u /2 Jα (2 zu) du and 0 (1.34)
+i∞ √ α/2 −α/2 τ v+v 2 /2 v e Jα (2 zv) dv, q(z) = z −i∞
Non-Intersecting Squared Bessel Paths: Critical Time and Double Scaling Limit
235
are solutions of (1.29) and (1.31), respectively, where Jα is the Bessel function of the first kind of order α. Based on a similarity with formulas by Desrosiers and Forrester [18, Prop. 5] for a perturbed chiral GUE, we suspect that it should be possible to write an alternative expression for the critical kernel in (1.19) in terms of the functions (1.34), namely
u∈R+
u α/2 v∈i R v
e
v2 u 2 2 − 2 +τ v−τ u
√ dudv √ . Jα (2 xu)Jα (2 yv) u−v
Unfortunately, we have not been able to make this identification. Observe however that for α = −1/2 the double integral √ above reduces√to the so-called symmetric Pearcey √ kernel K(σ1 ; σ2 ; η), with σ1 = x 2 / 2, σ2 = y 2 / 2, η = 2τ . The correlation kernel K(σ1 ; σ2 ; η) =
2 π 2i
u∈C
x∈R+
e−ηx
2 +ηu 2 −x 4 +u 4
cos(σ1 x) cos(σ2 u)
ud xdu , u2 − x 2
where C is the contour in C consisting of rays from ∞eiπ/4 to 0 to ∞e−iπ/4 , was introduced by Borodin and Kuan in [7]; the authors point out the possible connection with the non-intersecting Bessel paths in the critical regime, as it seems to be the case.
2. First and Second Transformation The steepest descent analysis consists of a sequence of transformations Y → X → U → T → S → R, which leads to a RH problem for R, normalized at infinity and with jump matrices that are close to the identity matrix if n is large. We start from the RH problem 1.1 for Y , stated in the Introduction. The RH problem depends on the parameters n and t. We assume that n is large, and t is close to the critical value t ∗ . Eventually we will take the double scaling limit n → ∞,
t → t ∗,
such that
√
n(t ∗ − t) = c∗ τ
remains fixed.
(2.1)
But throughout the transformations in Sects. 2–7, we assume that n and t are finite and fixed. The first transformation is the same as in [28]. 2.1. The first transformation. The first transformation Y → X is based on special properties of the modified Bessel functions that appear in the jump matrix (1.13) via the two weights (1.11). The result of the first transformation will be that the jump matrix on [0, ∞) is simplified at the expense of introducing jumps on (−∞, 0) and on two ± unbounded contours ± 2 that are shown in Fig. 4. The contours 2 are the boundaries of an unbounded lense around the negative real axis. Here and in the sequel, E i j denotes the 3 × 3 elementary matrix whose entries are all 0, except for the (i, j)th entry, which is 1.
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Fig. 4. Contour X = R ∪ ± 2 and jump matrices J X in the RH problem for X
√ √ Definition 2.1. We let y1 (z) = z (α+1)/2 Iα+1 (2 z) and y2 (z) = z (α+1)/2 K α+1 (2 z), where K α+1 is the modified Bessel function of second kind of order α + 1. Then we define X in terms of Y as follows: ⎞ 1 0 0 0 ⎠ X (z) = C1 Y (z) ⎝0 1 √ n a 0 0 t ⎛ ⎞⎛ ⎞ 1 0 0 1 0 0 2 α 2 ⎜ ⎟⎜ ⎟ t √ −z −α y1 n t 2az ⎟ ⎜0 0 2y2 n t 2az 0 ⎟ ×⎜ n a 2 2 ⎠⎝ ⎝ α ⎠ n az n az −α t z y1 t 2 0 −2y2 t 2 0 0 −2πi n √a ⎧ απi −α ⎪ ⎨ I − e z E 23 , for z in the upper part of the lens, × I + e−απi z −α E 23 , for z in the lower part of the lens, (2.2) ⎪ ⎩I elsewhere, ⎛
where C1 is some constant matrix, see [28, Eq. (3.12)] for its definition. Then X is the unique solution of the following RH problem, see [28, Sect. 3] for details. RH problem 2.2. 1. X is defined and analytic in C\ X where X = R ∪ ± 2. 2. On X we have the jump X + = X − JX ,
(2.3)
where the jump matrices J X are as in Fig. 4. 3. As z → ∞ we have X (z) =
I ⎛
1 ⎝0 0
⎛ ⎞ ⎞ 1 ⎛ 0 0 1 0 0 ⎜ ⎟ 1 1 √1 i ⎟ ⎠ ⎜ 0 √2 ⎝0 z (−1)n /4 0 +O 2 ⎝ ⎠ n z 0 0 z −(−1) /4 0 √1 i √1 2 2 ⎞⎛ n ⎞ z 0 √ 0 0 0 z α/2 0 ⎠ ⎝ 0 z −n/2 e−2n az/t 0 √ ⎠ . (2.4) −α/2 0 z 0 0 z −n/2 e2n az/t
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4. X (z) has the same behavior as Y (z) near the origin, see (1.15), as z → 0 from outside the lens around (−∞, 0]. If z → 0 within the lens around (−∞, 0], then ⎧ ⎛ ⎞ 1 |z|α 1 ⎪ ⎪ ⎪ ⎪ O ⎝1 |z|α 1⎠ if α < 0, ⎪ ⎪ ⎪ α ⎪ 1 |z| 1 ⎪ ⎪ ⎛ ⎞ ⎪ ⎪ 1 log |z| log |z| ⎨ X (z) = O ⎝1 log |z| log |z|⎠ (2.5) if α = 0, ⎪ ⎪ 1 log |z| log |z| ⎪ ⎪ ⎞ ⎛ ⎪ ⎪ 1 1 |z|−α ⎪ ⎪ ⎪ ⎪ if α > 0. O ⎝1 1 |z|−α ⎠ ⎪ ⎪ ⎩ 1 1 |z|−α 2.2. The Riemann surface. In the second transformation we are going to use certain functions that come from a Riemann surface. In [28, Sect. 4] we used the Riemann surface associated with the algebraic equation z=
1 − kζ , ζ (1 − t (1 − t)ζ )2
k = (1 − t)(t − a(1 − t)).
(2.6)
This equation was derived from a formal WKB analysis of the differential equation 2nz y (z) zy (z) + (2 + α) − t (1 − t) n2 z n(n − α − 2) an 2 n3 y − + 2 + (z) − y(z) = 0, (2.7) t (1 − t)2 t (1 − t) t2 t 2 (1 − t)2 see [11] and [28, Eq. (2.21)], that is satisfied by the multiple orthogonal polynomials associated with the weights (1.11). There are three inverse functions to (2.6), which behave as z → ∞ as 1 1 ζ1 (z) = + O 2 , z z √ 1 a 1 1 ζ2 (z) = − 1/2 − + O 3/2 , (2.8) t (1 − t) t z 2z z √ 1 a 1 1 ζ3 (z) = + 1/2 − + O 3/2 . t (1 − t) t z 2z z At critical time t = t ∗ , we have k = 0 and Eq. (2.6) reduces to z=
1 , ζ (1 − c∗ ζ )2
c∗ = t ∗ (1 − t ∗ ).
(2.9)
Then, the corresponding Riemann surface has two real branch points, 0 and q ∗ = 27c∗ /4 > 0, 0 being degenerate (of order 2), and q ∗ being simple. The point at infinity is also a simple branch point of the Riemann surface. In the present paper, we want to work with a Riemann surface R with a double branch point, even if t = t ∗ . Following the approach of [2] we do not consider (2.6) if t = t ∗ but instead consider a modified equation
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Fig. 5. Plot of function z = z(w) given by (2.10) for w ∈ R
z=
1 , w(1 − cw)2
(2.10)
with c some positive number and w a new auxiliary variable. The Riemann surface has simple branch points at q = 27c/4
(2.11)
and at infinity, and it has a double branch point at 0. The sheet structure of R can be readily visualized from Fig. 5 and is shown in Fig. 6. The sheets R1 and R2 are glued together along the cut 1 = [0, q] and the sheets R3 and R2 are glued together along the cut 2 = (0, ∞] in the usual crosswise manner. There are three inverse functions wk , k = 1, 2, 3, that behave as z → ∞ as 1 1 w1 (z) = + O 2 , z z 1 1 1 1 (2.12) + O 3/2 , w2 (z) = − √ 1/2 − c 2z z cz 1 1 1 1 w3 (z) = + √ 1/2 − + O 3/2 , c 2z z cz and which are defined and analytic on C\1 , C\(1 ∪ 2 ) and C\2 respectively. The algebraic function w = w(z) in (2.10) gives a bijection between the Riemann surface R and the extended complex w-plane. Figure 7 represents this mapping, along j = w j (R j ), j = 1, 2, 3 (the images of the corresponding sheets with the domains R of R) and the points wq = w2 (q) =
1 > 0, 3c
w∞ = w2 (∞) =
1 > 0. c
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239
Fig. 6. The Riemann surface R used in the steepest descent analysis. The origin is a double branch point for every t
We point out that w2+ ( j ), j = 1, 2, are analytic arcs that extend to infinity in the upper half plane, while w2− ( j ), j = 1, 2, are in the lower half plane. 2.3. Modified ζ functions. We next define modified ζ -functions with the same asymptotic behavior as z → ∞ as given in (2.8) up to order O(z −3/2 ). Definition 2.3. For k = 1, 2, 3 we define ζk = wk + pwk2 ,
(2.13)
where wk is given by (2.12) and (1 − t)2 c= 4
√ 2 2t a a c2 + + − c. and p = 2 4 1−t t (1 − t)
(2.14)
Note that for t = t ∗ , the critical time, we have c = c∗ = t ∗ (1 − t ∗ ) and p = 0, so that we recover in this case Eq. (2.9) and the ζ -functions defined in (2.8). Lemma 2.4. For c and p given by (2.14), the asymptotic behavior of functions ζk , k = 1, 2, 3, defined in (2.13), as z → ∞, is given by (2.8). Proof. This follows from direct computations using (2.12) and (2.13).
In what follows we need the behavior of the ζ -functions (2.13) near the origin. The following lemmas are analogous to Lemmas 3.2 and 3.3 in [2]. We put ω = e2πi/3 , as before.
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Fig. 7. Bijection between the Riemann surface R and the extended w-plane
Lemma 2.5. There exist analytic functions f 1 and g1 defined in a neighborhood U1 of the origin such that with z ∈ U1 and k = 1, 2, 3, wk (z) =
2k −1/3 ω z f 1 (z) + ωk z 1/3 g1 (z) +
2 3c 2 3c
ωk z −1/3 f 1 (z) + ω2k z 1/3 g1 (z) +
for Im z > 0, for Im z < 0,
(2.15)
In addition, we have f 1 (0) = c−2/3 , and f 1 (z) and g1 (z) are real for real z ∈ U1 . Proof. We put z = x 3 and w = (x y)−1 in (2.10) and obtain (x y − c)2 − y 3 = 0.
(2.16)
It has a solution y = y(x) that is analytic in a neighborhood U1 of 0 and y(0) = c2/3 > 0, y (0) = − 23 c1/3 < 0. Then, we can write xw(x) = 1/y(x) = f 1 (x 3 ) + x 2 g1 (x 3 ) + xh 1 (x 3 ),
(2.17)
with f 1 , g1 and h 1 analytic in U1 and f 1 (0) = c−2/3 > 0. Plugging this into (2.10), we find after some calculations that f 1 (z)g1 (z) =
1 2 z + c2 g13 (z)z 2 = 0, , c2 f 13 (z) − 1 + 2 9c 27c
(2.18)
and h 1 (z) = 2/(3c). Hence, there is a solution w = w(z) of (2.10) with w(z) = z −1/3 f 1 (z) + z 1/3 g1 (z) +
2 , 3c
for z ∈ U1 \(−∞, 0],
where we take the principal branches for the fractional powers. This solution is real for z real and positive, thus it coincides with w3 (z), which proves (2.15) for k = 3. Considering the two others solutions of (2.16), we get the expressions (2.15) for k = 1, 2 by analytic continuation. Since y(x) is real for real x, (2.17) implies that f 1 (z) and g1 (z) are also real when z is real. The next lemma describes the behavior of the ζ -functions at the origin.
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Lemma 2.6. There exist analytic functions f 2 and g2 defined in a neighborhood U2 of the origin such that for z ∈ U2 and k = 1, 2, 3, 2k −1/3 2 ω z f 2 (z) + ωk z −2/3 g2 (z) + 3t (1−t) for Im z > 0, ζk (z) = (2.19) 2 ωk z −1/3 f 2 (z) + ω2k z −2/3 g2 (z) + 3t (1−t) for Im z < 0. In addition, we have 4p , f 2 (0) = c−2/3 1 + 3c
g2 (0) = pc−4/3 ,
(2.20)
and f 2 (z) and g2 (z) are real for real z ∈ U2 . Proof. The proof follows from (2.13) and the previous lemma. It suffices to compute wk2 (z) by using (2.15) and the first identity in (2.18). Then, (2.19) follows from (2.13) if we set 4p 4p 2 f 1 (z) + pzg1 (z), zg1 (z) + p f 12 (z), (2.21) f 2 (z) = 1 + g2 (z) = 1 + 3c 3c and (2.20) follows from the value of f 1 (0) given in Lemma 2.5. The functions f 2 (z) and g2 (z) are real for real z ∈ U2 since f 1 (z) and g1 (z) are real for real z ∈ U1 . 2.4. The λ-functions. We next define the λ-functions as anti-derivatives of the modified ζ -functions (2.13). Definition 2.7. We define for k = 1, 2, 3,
λk (z) =
z
ζk (s)ds,
(2.22)
0+
where the path of integration starts at 0 on the upper half-plane (which is denoted by 0+ ) and is contained in C\(−∞, q] for k = 1, 2, and in C\(−∞, 0] for k = 3. By construction the functions λ1 and λ2 are analytic in C\(−∞, q] while λ3 is analytic in C\(−∞, 0]. From Lemma 2.4 and (2.22), it follows that, as z → ∞, 1 , λ1 (z) = log z + 1 + O z √ 1/2 2 az 1 z 1 − − log z + 2 + O 1/2 , λ2 (z) = t (1 − t) t 2 z √ 1/2 2 az 1 z 1 λ3 (z) = + − log z + 3 + O 1/2 , t (1 − t) t 2 z where k , k = 1, 2, 3, are certain constants of integration, and log z is the principal branch of the logarithm which is real on (0, +∞). Using the structure of the Riemann surface R and the residue calculation based on the expansion of ζ1 at infinity, see (2.8), we conclude that λ1− (0) = −2πi, λ2− (0) = 2πi, λ3− (0) = 0.
(2.23)
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Moreover, the following jump relations hold true: λ1± (x) = λ2∓ (x) − 2πi, λ3+ (x) = λ3− (x), λ1+ (x) = λ1− (x) + 2πi, λ2+ (x) = λ3− (x), λ2− (x) = λ3+ (x) + 2πi,
x ∈ 1 = (0, q), x ∈ 2 = (−∞, 0), x ∈ 2 .
(2.24)
The behavior of the λ-functions at the origin follows from Lemma 2.6 and (2.22). Lemma 2.8. There exist analytic functions f 3 and g3 defined in a neighborhood U3 of the origin such that for z ∈ U3 and k = 1, 2, 3, 3 2k 2/3 2z f 3 (z) + ωk z 1/3 g3 (z) + 3t (1−t) for Im z > 0, 2ω z (2.25) λk (z) = 3 k 2/3 2z 2k 1/3 λk− (0) + 2 ω z f 3 (z) + ω z g3 (z) + 3t (1−t) for Im z < 0, with λk− (0) given by (2.23). In addition, we have 4p , g3 (0) = 3g2 (0) = 3 pc−4/3 , f 3 (0) = f 2 (0) = c−2/3 1 + 3c
(2.26)
and both f 3 (z) and g3 (z) are real for real z ∈ U3 . Proof. Integrating (2.19) and taking into account that λk+ (0) = 0, we get (2.25) and (2.26). The fact that f 3 (z) and g3 (z) are real for real z ∈ U3 also follows from Lemma 2.6. The functions f 3 and g3 depend on t. We write f 3 (z; t) and g3 (z; t) if we want to emphasize the dependence on t. In order to control the jumps in the different RH problems that we are going to consider in the sequel, we need to compare the real parts of the λ-functions. Figure 8 shows, at the critical time t = t ∗ , the curves in the complex plane where the real parts of the λ-functions are equal. These curves are critical trajectories of the quadratic differentials (ζ j (z) − ζk (z))2 dz 2 . The curve Re λ2 (z) = Re λ3 (z) consists of the negative real axis. The curve Re λ1 (z) = Re λ3 (z) consists of two trajectories emanating from the origin, symmetric with respect to the real axis, and going to infinity. Finally, the curve Re λ1 (z) = Re λ2 (z) consists of 1 along with two symmetric trajectories emanating from the branch point q ∗ . At a non-critical time t = t ∗ , the configuration of the curves remains the same, except in a small neighborhood of the origin. Figures 9 and 10 show the new configurations in such a neighborhood. When t < t ∗ , the function ζ1 (z) − ζ2 (z) has an additional zero on 1 , which causes the appearance of a new loop around the origin where Re λ1 (z) = Re λ2 (z). Also, the curve Re λ1 (z) = Re λ3 (z) is shifted to the left and becomes a continuation of the loop as it intersects the negative real axis. Similarly, when t > t ∗ , the function ζ2 (z) − ζ3 (z) has an additional zero on 2 , which causes the appearance of a new loop around the origin where Re λ2 (z) = Re λ3 (z). Now, the curve Re λ1 (z) = Re λ3 (z) is shifted to the right and becomes a continuation of the loop as it intersects the positive real axis. In both cases, when t tends to t ∗ , the loop shrinks to the origin. Concerning the relative ordering of the real parts in a neighborhood of the real axis, the following lemma holds true. Lemma 2.9. (a) For z ∈ (q, +∞), we have Re λ1 (z) < Re λ2 (z).
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3
2
1
0
q*
0 −1
Reλ1 = Reλ2
−2
Reλ = Reλ 1
3
Reλ = Reλ 2
−3 −4
−3
3
−2
−1
0
1
2
3
4
Fig. 8. Curves Re λ j = Re λk at the critical time (here a = 1 and t = t ∗ = 0.5) 0.03
0.02
0.01
0
0
−0.01
Reλ1 = Reλ2
−0.02
Reλ = Reλ 1
3
Reλ2 = Reλ3 −0.03 −0.1
−0.08
−0.06
−0.04
−0.02
0
0.02
0.04
Fig. 9. Curves Re λ j = Re λk near the origin, before the critical time (here a = 1, t = 0.3 < t ∗ = 0.5)
(b) The open interval (0, q) has a neighborhood U1 in the complex plane such that for z ∈ U1 \(0, q) and outside of the additional loop around 0 when t = t ∗ , we have Re λ2 (z) < Re λ1 (z). (c) The open interval (−∞, 0) has a neighborhood U2 in the complex plane such that for z ∈ U2 \(−∞, 0) and outside of the additional loop around 0 when t = t ∗ , we have Re λ2 (z) < Re λ3 (z). The neighborhood U2 is unbounded and contains a full neighborhood of infinity.
244
A. B. J. Kuijlaars, A. Martínez-Finkelshtein, F. Wielonsky 0.025 0.02 0.015 0.01 0.005 0 0
−0.005 −0.01 −0.015
Reλ = Reλ 1
2
Reλ = Reλ
−0.02
1
3
Reλ2 = Reλ3
−0.025 −0.05
0
0.05
0.1
Fig. 10. Curves Re λ j = Re λk near the origin, after the critical time (here a = 1, t = 0.7 > t ∗ = 0.5)
Proof. This is similar to the proof of [28, Lemma 4.3]. When t = t ∗ , only the ordering of the real parts inside the additional loop is modified. See also [2, Lemma 4.2]. In the double scaling limit (2.1) that we are going to consider, we have that t − t ∗ is of order n −1/2 as n → ∞. Then, the special ordering of the real parts of the λ-functions inside the loop will not cause a problem because the (shrinking) disk around the origin where we are going to construct the local parametrix will be big enough to contain the loop for n large, see the proof of Lemma 6.3 below. 2.5. Second transformation of the RH problem. The goal of the second transformation X → U is to normalize the RH problem at infinity using the functions λ j from Sect. 2.4. Definition 2.10. Given X as in (2.2) we define ⎛ −nλ1 (z) 0 e z −n(λ2 (z)− t (1−t) ) ⎜ 0 e U (z) = C2 X (z) ⎝ 0
0
e
0 0
z −n λ3 (z)− t (1−t)
⎞ ⎟
⎠ ,
(2.27)
where C2 is some explicit constant matrix, see [28, Eq. (4.19)]. As a consequence of the assertion (c) in Lemma 2.9 we may (and do) assume that the contours ± 2 , defined above (and depicted in Fig. 4) lie in the neighborhood U2 of 2 where Re (λ2 − λ3 ) < 0 (except when it intersects the small loop near 0 when t = t ∗ , see Figs. 9 and 10). Using the jump relations (2.24) and other properties of the λ functions we find that U solves the following RH problem, see [28, Sect. 4] for details. RH problem 2.11. The matrix-valued function U (z) defined by (2.27) is the unique solution of the following RH problem.
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1. U (z) is analytic in C\ U , where U = X = R ∪ ± 2. 2. On U we have the jump U+ (z) = U− (z)JU (z),
z ∈ U
(2.28)
with jump matrices JU (z) given by ⎛ ⎞ 1 0 0 0 −|x|−α ⎠ , x ∈ 2 = (−∞, 0), JU (x) = ⎝0 0 |x|α 0 ⎞ ⎛ n(λ −λ ) (x) xα 0 e 2 1+ x ∈ 1 = (0, q), JU (x) = ⎝ 0 en(λ2 −λ1 )− (x) 0⎠ , 0 0 1 x ∈ (q, ∞), JU (x) = I + x α en(λ1 −λ2 )(x) E 12 , JU (z) = I + e±απi z −α en(λ2 −λ3 )(z) E 23 , z ∈ ± 2.
(2.29)
(2.30) (2.31) (2.32)
3. As z → ∞ we have U (z) =
⎛ ⎞⎛ 1 0 1 0 ⎜ ⎟ ⎜ 1 ⎜0 z 1/4 ⎜0 0 ⎟ I +O ⎝ ⎠⎝ z 0 0 0 z −1/4
0 √1 2 √1 i 2
0
⎞⎛
1
⎟⎜ √1 i ⎟ ⎜0 2 ⎠⎝ √1 0 2
0 z α/2 0
⎞
0
⎟ 0 ⎟ ⎠.
z −α/2
(2.33) 4. U (z) has the same behavior as X (z) at the origin, see (1.15) and (2.5). It follows from Lemma 2.9 that in the double scaling limit (2.1) the jump matrices in (2.31) and (2.32) tend to the identity matrix I as n → ∞ at an exponential rate, except when z lies inside the small loop around the origin, see Figs. 9 and 10. Moreover, (λ2 − λ1 )+ = −(λ2 − λ1 )− − 4πi is purely imaginary on 1 , so that the first two diagonal elements of the jump matrices in (2.30) are highly oscillatory if n is large. 3. Third Transformation of the RH Problem The third transformation U → T consists in opening a lens around 1 , see Fig. 11. It transforms the oscillatory entries in the jump matrix on 1 into exponentially small off-diagonal terms. Following [28, Sect. 5], we define T as follows. Definition 3.1. We define T (z) = U (z) I ∓ z −α en(λ2 −λ1 )(z) E 21 ,
(3.1)
for z in the domain bounded by ± 1 and 1 (the shaded region in Fig. 11), and we define T (z) = U (z) for z outside of the lens around 1 .
(3.2)
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± Fig. 11. Opening of lens around 1 . The contour T = R ∪ ± 1 ∪ 2 is the jump contour in the RH problem for T
± Let T = R ∪ ± 1 ∪ 2 be the union of the contours depicted in Fig. 11. Then, straightforward calculations yield that the matrix-valued function T is the solution of the following RH problem.
RH problem 3.2. The matrix-valued function T (z) satisfies 1. T is analytic in C\ T . 2. On T we have T+ = T− JT , where
⎛
1 JT (x) = ⎝0 0
0 0 |x|α
(3.3)
⎞ 0 −|x|−α ⎠ , 0
x ∈ 2 ,
(3.4)
JT (z) = I + e±απi z −α en(λ2 −λ3 )(z) E 23 , ⎛ ⎞ 0 xα 0 JT (x) = ⎝−x −α 0 0⎠ , 0 0 1
z ∈ ± 2,
(3.5)
x ∈ 1 ,
(3.6)
JT (z) = I + z −α en(λ2 −λ1 )(z) E 21 ,
z ∈ ± 1,
(3.7)
x ∈ (q, +∞).
(3.8)
α n(λ1 −λ2 )(x)
JT (x) = I + x e
E 12 ,
3. As z → ∞, we have T (z) =
⎛ ⎞⎛ 1 0 1 0 ⎜ ⎟ ⎜ 1 ⎜0 z 1/4 ⎜0 0 ⎟ I +O ⎝ ⎠⎝ z 0 0 0 z −1/4
0 √1 2 √1 i 2
0
⎞⎛
1
⎟⎜ √1 i ⎟ ⎜0 2 ⎠⎝ √1 0 2
0 z α/2 0
0
⎞
⎟ 0 ⎟ ⎠.
z −α/2
(3.9) 4. For −1 < α < 0, T (z) behaves near the origin like: ⎛ ⎞ 1 |z|α 1 ⎜ ⎟ T (z) = O ⎝1 |z|α 1⎠ , as z → 0. 1
|z|α
1
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For α = 0, T (z) behaves near the origin like: ⎛ ⎞ 1 log |z| 1 ⎜ ⎟ T (z) = O ⎝1 log |z| 1⎠ , as z → 0 outside the lenses around 2 and 1 , 1
log |z|
1
and
⎞ ⎧ ⎛ ⎪ 1 log |z| log |z| ⎪ ⎪ ⎜ ⎟ ⎪ ⎪ O ⎝1 log |z| log |z|⎠ , as z → 0 inside the lens around 2 , ⎪ ⎪ ⎪ ⎪ ⎨ 1 log |z| log |z| T (z) = ⎛ ⎞ ⎪ ⎪ log |z| log |z| 1 ⎪ ⎪ ⎪ ⎜ ⎟ ⎪ ⎪ O ⎝log |z| log |z| 1⎠ , as z → 0 inside the lens around 1 . ⎪ ⎪ ⎩ log |z| log |z| 1
For α > 0, T (z) behaves near the origin like: ⎛ ⎞ 1 1 1 T (z) = O ⎝1 1 1⎠ , as z → 0 outside the lenses around 2 and 1 , 1 1 1 and
T (z) =
⎧ ⎛ 1 ⎪ ⎪ ⎪ ⎝ ⎪ 1 O ⎪ ⎪ ⎪ 1 ⎨
1 1 1
⎛ −α ⎪ ⎪ |z| ⎪ ⎪ ⎪ ⎝|z|−α ⎪ O ⎪ ⎩ |z|−α
⎞ |z|−α |z|−α ⎠ , as z → 0 inside the lens around 2 , |z|−α 1 1 1
⎞ 1 1⎠ , as z → 0 inside the lens around 1 . 1
5. T (z) remains bounded as z → q. 4. Global Parametrix ± In the next step we ignore the jumps on ± 1 and 2 in the RH problem for T and we consider the following RH problem.
RH problem 4.1. Find Nα : C\(−∞, q] → C3×3 such that 1. Nα is analytic in C\(−∞, q]. 2. Nα has continuous boundary values on (−∞, 0) and (0, q), satisfying the following jump relations: ⎛ ⎞ 0 xα 0 Nα+ (x) = Nα− (x) ⎝−x −α 0 0⎠ , x ∈ (0, q), (4.1) 0 0 1 ⎛ ⎞ 1 0 0 0 −|x|−α ⎠ , x ∈ (−∞, 0). Nα+ (x) = Nα− (x) ⎝0 (4.2) α 0 |x| 0
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3. As z → ∞, Nα (z) =
⎛ ⎞ 1 ⎛ 1 0 0 ⎜ 1 ⎝0 z 1/4 0 0 ⎠⎜ I +O ⎝ z 0 0 z −1/4 0
0 √1 2 √1 i 2
0
⎞⎛
1
⎟⎜ √1 i ⎟ ⎜0 2 ⎠⎝ √1 0 2
0
0
z α/2 0
⎞
⎟ 0 ⎟ ⎠.
z −α/2
(4.3) 4. As z → q we have ⎛ |z − q|−1/4 ⎜ −1/4 Nα (z) = O ⎜ ⎝|z − q| |z − q|−1/4 5. As z → 0 we have
⎛
1 z α/3 Nα (z) ⎝0 0
0
z −α 0
|z − q|−1/4 |z − q|−1/4 |z
− q|−1/4
⎞ 0 0⎠ = Mα± z −1/3 + O(1), 1
1
⎞
⎟ 1⎟ ⎠.
(4.4)
1
±Im z > 0,
(4.5)
where Mα± is a rank one matrix. This RH problem is solved as in [28, Sect. 6] in terms of the branches wk , k = 1, 2, 3, of the algebraic function w, defined by (2.10), (2.12). Proposition 4.2. (a) The solution of the RH problem 4.1 for α = 0 is given by ⎛ ⎞ F1 (w1 (z)) F1 (w2 (z)) F1 (w3 (z)) N0 (z) = ⎝ F2 (w1 (z)) F2 (w2 (z)) F2 (w3 (z))⎠ , F3 (w1 (z)) F3 (w2 (z)) F3 (w3 (z))
(4.6)
where (w − w∞ )2 w(w − w ∗ ) , F (w) = K , 2 2 D(w)1/2 D(w)1/2 w(w − w∞ ) , F3 (w) = K 3 D(w)1/2 F1 (w) = K 1
(4.7)
with w ∗ = w∞ , K 1 , K 2 , K 3 certain non-zero constants that depend on a and t, and D(w) = (w − wq )(w − w∞ ).
(4.8)
The square root D(w)1/2 in (4.7) is defined with a cut along w2− (1 ) ∪ w2− (2 ), such that it is positive for real w > w∞ . (b) The solution of the RH problem 4.1 for general α is given by ⎛ αG (z) ⎞ e 1 0 0 Nα (z) = Cα N0 (z) ⎝ 0 (4.9) 0 ⎠, eαG 2 (z) αG 0 0 e 3 (z)
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where N0 (z) is given by (4.6), Cα is a constant matrix that depends on a, t and α, such that det Cα = 1 (see [28, Eq. (6.14)] for details), and the functions G j (z) are given by G j (z) = r j (w j (z)), z ∈ R j ,
j = 1, 2, 3,
(4.10)
r2 (w) = − log w − log(1 − cw),
1 , w∈R 2 , w∈R
(4.11)
r3 (w) = log(1 − cw) + iπ,
3 . w∈R
with r1 (w) = log(1 − cw),
The branches of the logarithms in (4.11) are chosen so that log(1 − cw) vanishes for w = 0 and has a branch cut along w2− (2 ) (cf. Fig. 7), and log w is the principal branch with a cut along (−∞, 0]. Proof. The fact that Nα satisfies items 1, 2, 3, and 4 of the RH problem 4.1 is proved as in [28, Sect. 6]. From (4.6), (4.7), and the behavior of the functions wk at 0 as given in Lemma 2.5 we obtain ⎛ ⎞ ⎧ K1 ⎪ ⎪ ⎪ ⎜ ⎟ 2 ⎪ −2/3 ⎪ c z −1/3 + O(1), z → 0, Im z > 0, ω K ω 1 ⎝ ⎠ 2 ⎪ ⎪ ⎪ ⎨ K3 ⎛ ⎞ N0 (z) = (4.12) ⎪ K1 ⎪ ⎪ ⎪ ⎜ ⎟ ⎪ ⎪ c−2/3 ⎝ K 2 ⎠ ω −ω2 1 z −1/3 + O(1), z → 0, Im z < 0, ⎪ ⎪ ⎩ K3 where, for Im z < 0, there is a minus sign in the second entry of the rowvector ω −ω2 1 because of the choice for the branch of the square root D(w)1/2 used in (4.7). This proves (4.5) for the case α = 0. For general α, we use that by (4.10) and (4.11) we have that e G 1 (z) = O(z −1/3 ),
e G 2 (z) = O(z 2/3 ),
e G 3 (z) = O(z −1/3 )
as z → 0. Then using (4.9) we find (4.5) also for this case.
It will be convenient in what follows to consider besides Nα also the matrix valued function ⎛ ⎞ 1 0 0 α (z) = z α/3 Nα (z) ⎝0 z −α 0⎠ , N (4.13) 0 0 1 which already appeared in (4.5). Lemma 4.3. The solution Nα of the RH problem 4.1 is unique and satisfies α (z) = 1, det Nα (z) = det N
z ∈ C\(−∞, q],
α is given by (4.13). where N As z → 0 we have both α (z) = O |z|−1/3 , and N α (z)−1 = O |z|−1/3 . N
(4.14)
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Proof. From the jump condition in the RH problem 4.1 it follows that det Nα (z) has an analytic continuation to C\{0, q}. The isolated singularities are removable by (4.4) and (4.5), where it is important that Mα± in (4.5) is of rank one. Thus det Nα (z) is an entire function, and since it tends to 1 as z → ∞ by (4.3) we conclude from Liouville’s theorem that det Nα (z) = 1. The uniqueness of the solution of the RH problem 4.1 now follows with similar arguments in a standard way. α (z) = 1. The behavior (4.14) From the definition (4.13) we then also find that det N for both Nα and its inverse finally follows from the condition (4.5) in the RH problem, α has determinant one. and the fact that N We need two more results that will be used later in Sect. 7. Lemma 4.4. For N0 defined in (4.9), we have ⎛ ⎞ 1 0 0 N0−1 = N0T ⎝0 0 −i ⎠ , z ∈ C\(−∞, q], 0 −i 0
(4.15)
where the superscript T denotes the matrix transpose. Proof. Observe from (4.1)–(4.2) that N0 and N0−T have the same jumps on 1 and 2 , so that N0 N0T is analytic in C\{0, q}. The singularies at 0 and q are removable because of (4.4) and (4.5), so that N0 N0T is entire. By (4.3), ⎛ ⎞ 1 0 0 1 , z → ∞, N0 (z)N0T (z) = ⎝0 0 i ⎠ + O z 0 i 0 and the assertion (4.15) follows by Liouville’s theorem.
As a consequence, we obtain the following corollary. Lemma 4.5. The constants K j from (4.7) satisfy the relation K 12 − 2i K 2 K 3 = 0.
(4.16)
Proof. From (4.15) we obtain ⎛
1 N0T (z) ⎝0 0
0 0 −i
⎞ 0 −i ⎠ N0 (z) = I. 0
Then insert the behavior (4.12) for both N0T (z) and N0 (z) and observe that the coefficient of z −2/3 must vanish. This yields ⎛ ⎞⎛ ⎞ 0 K1 1 0 K 1 K 2 K 3 ⎝0 0 −i ⎠ ⎝ K 2 ⎠ = 0, 0 −i 0 K3 which is (4.16).
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5. Local Parametrices The next step is the construction of local parametrices around the branch points q and 0. We shall be brief about the local parametrix P around q. The main issue will be the construction of the local parametrix Q around the origin. 5.1. Parametrix P around q. We build P in a fixed disk D(q ∗ , rq ) around q ∗ = 27c∗ /4 with some (small) radius rq > 0. For t sufficiently close to t ∗ we then have that q is in this neighborhood, and we ask that P should satisfy: 1. P is analytic on D(q ∗ , rq )\ T ; 2. P has the same jumps as T has on D(q ∗ , rq ) ∩ T , see (3.6)–(3.8); 3. as n → ∞, uniformly for |z − q ∗ | = rq . P(z) = Nα (z) I + O n −1
(5.1)
The construction of P is done in a standard way by means of Airy functions, see [13,15,16]. We will not give any details. 5.2. Parametrix Q around 0: required properties. The construction of the parametrix at the origin is the main novel ingredient in the present RH analysis. A similar problem has been previously solved in [2, Sect. 8], which serves as an inspiration for the approach we follow here. We want to define a matrix Q in a neighborhood D(0, r0 ) of the origin such that 1. Q is analytic on D(0, r0 )\ T , where T has been defined in Sect. 3, see also Fig. 11; 2. Q has the same jumps as T has on T ∩ D(0, r0 ), see (3.4)–(3.7). That is, we have Q + (z) = Q − (z)J Q (z), where J Q is given by ⎛
1 J Q (x) = ⎝0 0
0 0 |x|α
z ∈ T ∩ D(0, r ),
⎞ 0 −|x|−α ⎠ , x ∈ 2 ∩ D(0, r ), 0
(5.2)
(5.3)
J Q (z) = I + e±απi z −α en(λ2 −λ3 )(z) E 23 , z ∈ ± 2 ∩ D(0, r ), ⎛ ⎞ α 0 x 0 J Q (x) = ⎝−x −α 0 0⎠ , x ∈ 1 ∩ D(0, r ), 0 0 1
(5.4)
J Q (z) = I + z −α en(λ2 −λ1 )(z) E 21 , z ∈ ± 1 ∩ D(0, r ).
(5.6)
(5.5)
3. As z → 0, Q has the same behavior as T has, see item 4 in the RH problem 3.2. As we will see in Sect. 7, the radius r0 will actually depend on n, namely r0 = n −1/2 , so that the parametrix Q will be defined in a disk shrinking neighborhood as n → ∞. Note that we did not state a matching condition for Q. Usually one asks for a matching condition of the type Q(z) = Nα (z)(I + O(1/n κ )),
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as n → ∞, uniformly for z on the circle |z| = r0 , with some κ > 0. In the present situation we are not able to get such a matching condition. We will only be able to match Q(z) with Nα (z) up to a bounded factor. Hence, it will be necessary to introduce an additional transformation, defined globally in the complex plane, as a last step of the Riemann-Hilbert analysis, see Sect. 7. 5.3. Reduction to constant jumps. The jump condition (5.2) can be reduced to a condition with constant jump matrices as follows. We put n (z) = diag(1, z α , 1) diag e−nλ1 (z) , e−nλ2 (z) , e−nλ3 (z) , (5.7) with z α = |z|α eiα arg(z) and arg z ∈ (0, 2π ) is defined with the branch cut [0, +∞). Then the jump matrices J Q from (5.3)–(5.6) factorize as 0 J Q (z) = −1 n,− (z) J Q (z) n,+ (z),
where
⎛
1 J Q0 (x) = ⎝0 0
0 0
e−απi
0
(5.8)
⎞
−e−απi ⎠ , x ∈ 2 , 0
(5.9)
J Q0 (z) = I + e±απi E 23 , z ∈ ± 2, ⎛ ⎞ 0 1 0 J Q0 (x) = ⎝−1 0 0⎠ , x ∈ 1 , 0 0 1
(5.10)
J Q0 (z) = I + E 21 , z ∈ ± 1.
(5.12)
(5.11)
Observe that the jump matrices J Q0 agree with the jump matrices J in the RH problem 1.3 for α , see Fig. 3, except that the jump matrices J are on six infinite rays emanating from the origin. We will therefore look for Q in the form Q(z) = E n (z)α ( f n (z); τn (z))n (z),
(5.13)
where E n (z) and τn (z) are analytic in D(0, r0 ) and where f n is a conformal map on D(0, r0 ) that maps the contours T ∩ D(0, r0 ) into the six rays so that [0, r0 ) is mapped into the positive real axis. For any choice of conformal f n , and analytic E n and τn , the matrix valued Q defined by (5.13) will then satisfy the required jumps (5.2). 5.4. Definition of α (z; τ ). We next construct the matrix valued function α (z) = α (z; τ ) that solves the RH problem 1.3 stated in the Introduction. As already mentioned in the Introduction we use the following third order linear differential equation: zp + αp − τ p − p = 0,
(5.14)
with α > −1 and τ ∈ C. Then z = 0 is a regular singular point of this ODE with Frobenius indices 0, 1, and −α + 2. There are two linearly independent entire solutions, and one solution that branches at the origin.
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Fig. 12. The contours of integration j , j = 1, . . . 4, used in the definition of the functions p j . The contours 3 and 4 extend to infinity in the t-plane in a direction where Re(zt) < 0. This direction depends on z and is not necessarily along the imaginary axis, as may be suggested by the figure
Due to the special form of the ODE (5.14) (the coefficients are at most linear in z), it can be solved with Laplace transforms. We find solutions with an integral representation
2 p(z) = C t α−3 eτ/t e1/(2t ) e zt dt,
where is an appropriate contour in the complex t-plane and C is a constant. A basis of solutions of (5.14) can be chosen by selecting different contours . We will make use of four contours j , j = 1, 2, 3, 4, defined as follows, see also Fig. 12: (a) We let 1 be a simple closed contour passing through the origin, but otherwise lying in the right half-plane and which is tangent to the imaginary axis. 1 is oriented counterclockwise and we put
2 p1 (z) = t α−3 eτ/t e1/(2t ) e zt dt. (5.15) 1
We choose the branch of t α−3 = |t|α−3 ei(α−3) arg t with −π/2 < arg t < π/2. (b) 2 is the reflection of 1 in the imaginary axis, oriented clockwise. We put
2 p2 (z) = e−απi t α−3 eτ/t e1/(2t ) e zt dt. (5.16) 2
In (5.16) we define the branch of t α−3 with π/2 < arg t < 3π/2. (c) 3 is an unbounded contour in the upper half-plane that starts at infinity at an angle where Re (zt) < 0 as t → ∞, and ends at the origin along the positive imaginary axis. We put
2 p3 (z) = e−απi t α−3 eτ/t e1/(2t ) e zt dt. (5.17) 3
In (5.17) we take t α−3 with 0 < arg t < π .
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The condition Re (zt) < 0 is necessary to have convergence of the integral. This condition can be met with a contour that is in the upper half-plane if and only if −π/2 < arg z < 3π/2. Therefore p3 is well-defined and analytic in C\iR− , that is, with a branch cut along the negative imaginary axis. (d) 4 is similar to 3 , but in the lower half-plane. It is an unbounded contour in the lower half-plane starting at infinity at an angle where Re (zt) < 0 as t → ∞, and it ends at the origin along the negative imaginary axis. We put
2 απi p4 (z) = e t α−3 eτ/t e1/(2t ) e zt dt. (5.18) 4
In (5.18) the branch of t α−3 is defined with −π < arg t < 0. Then p4 is well-defined and analytic in C\iR+ , thus with a branch cut along the positive imaginary axis. With these definitions it is clear that p1 and p2 are entire functions, while p3 and p4 have a branch point at the origin. The four solutions are not linearly independent, but any three of them are. We define α in each of the sectors determined by the six rays: arg z = 0, ±π/4, ±3π/4, π , as shown in Fig. 3, as a Wronskian matrix using three of the functions p j . Definition 5.1. We define α in the six sectors as follows: ⎞ ⎛ (z) p (z) p (z) − p 4 3 1 2 eτ /6 ⎜ ⎟ α (z; τ ) = √ ⎝− p4 (z) p3 (z) p1 (z)⎠ , 0 < arg z < π/4, 2π − p4 (z) p3 (z) p1 (z) ⎛ ⎞ p2 (z) p3 (z) p1 (z) 2 /6 τ e ⎜ ⎟ α (z; τ ) = √ ⎝ p2 (z) p3 (z) p1 (z)⎠ , π/4 < arg z < 3π/4, 2π p2 (z) p3 (z) p1 (z) ⎞ ⎛ p2 (z) p3 (z) −e−απi p4 (z) 2 /6 τ e ⎟ ⎜ α (z; τ ) = √ ⎝ p2 (z) p3 (z) −e−απi p4 (z)⎠ , 3π/4 < arg z < π, 2π p2 (z) p3 (z) −e−απi p4 (z) ⎞ ⎛ p2 (z) p4 (z) eαπi p3 (z) 2 /6 τ e ⎟ ⎜ α (z; τ ) = √ ⎝ p2 (z) p4 (z) eαπi p3 (z)⎠ , −π < arg z < −3π/4, 2π p2 (z) p4 (z) eαπi p3 (z) ⎛ ⎞ (z) p (z) p (z) p 2 4 1 2 eτ /6 ⎜ ⎟ α (z; τ ) = √ ⎝ p2 (z) p4 (z) p1 (z)⎠ , −3π/4 < arg z < −π/4, 2π p2 (z) p4 (z) p1 (z) ⎛ ⎞ p3 (z) p4 (z) p1 (z) 2 /6 τ e ⎜ ⎟ α (z; τ ) = √ ⎝ p3 (z) p4 (z) p1 (z)⎠ , −π/4 < arg z < 0. 2π p3 (z) p4 (z) p1 (z)
(5.19)
(5.20)
(5.21)
(5.22)
(5.23)
(5.24)
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τ 2 /6
The scalar factor e√ is needed in (5.19)–(5.24) in order to have the exact asymptotic 2π behavior (1.22) in the RH problem 1.3. The functions p j clearly depend on α and τ , although we did not emphasize it in the notation. Proposition 5.2. Let α > −1 and τ ∈ C. Then α (z; τ ) as defined above satisfies the RH problem 1.3 stated in the Introduction. Proof. It is a tedious but straightforward check based on the integral representations (5.15)–(5.18) that α has the constant jumps on six rays in the complex z-plane as given in (1.20). The asymptotic properties (1.22)–(1.23) follow from a steepest descent analysis for the integrals defining the functions p j . We give more details about this in the next subsection, where we also describe the next term in the asymptotic expansions of (1.22)–(1.23) since we will need this later on. The behavior (1.24)–(1.26) at 0 follows from the behavior of the solutions p j of the ODE (5.14) at 0. Since p1 and p2 are entire solutions, they are bounded at 0. The solutions p3 and p4 satisfy p j (z) = O(z 2−α ),
p j (z) = O(z 1−α ),
p j (z) = O(z −α ),
j = 3, 4,
as z → 0, which can be found by analyzing the integral representations (5.17) and (5.18). This proves (1.24)–(1.26) in view of the definition of α in Definition 5.1. 5.5. Asymptotics of α . As before we define ω = e2πi/3 and θk as in (1.21). We also put
⎛ L α (z) = z
−α/3 ⎝
z 1/3 0 0
0 1 0
⎞ ⎛ απi/3 ⎞ ⎧⎛ ω ω2 1 0 0 e ⎪ ⎪ ⎪ 1 1⎠ ⎝ 0 ⎪ e−απi/3 0⎠ , ⎪⎝ 1 2 ⎪ ⎞ ⎪ ⎪ ω ω 1 0 0 1 ⎪ 0 ⎨ for Im z > ⎞ ⎛ −απi/3 ⎞ 0, 0 ⎠× ⎛ 2 ω −ω 1 0 0 e ⎪ −1/3 ⎪ z ⎪ ⎪ ⎝ 1 −1 1⎠ ⎝ 0 ⎪ eαπi/3 0⎠ , ⎪ ⎪ 2 ⎪ 0 0 1 ⎪ ⎩ ω −ω 1 for Im z < 0, (5.25)
where all fractional powers are defined with a branch cut along the negative real axis. Define also the constant matrices τ (τ 2 + 9α − 9) diag ω2 , ω, 1 27 ⎛ 0 iτ + √ diag ω−α/2 , ωα/2 , 1 ⎝ ω2 3 3 −ω2
Mα+ (τ ) =
−ω 0 ω
⎞ 1 −1⎠ diag ωα/2 , ω−α/2 , 1 , 0 (5.26)
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τ (τ 2 + 9α − 9) diag ω, ω2 , 1 27 ⎛ 0 iτ + √ diag ωα/2 , ω−α/2 , 1 ⎝ω 3 3 ω
Mα− (τ ) =
−ω2 0 ω2
⎞ −1 −1⎠ diag ω−α/2 , ωα/2 , 1 . 0 (5.27)
Lemma 5.3. Let α > −1 and τ ∈ C. Then we have, as z → ∞, ⎛ eθ1 (z) 0 i Mα+ (τ ) α (z; τ ) = √ L α (z) I + 1/3 + O(z −2/3 ) ⎝ 0 eθ2 (z) z 3 0 0 for Im z > 0, ⎛ eθ2 (z) 0 i Mα− (τ ) α (z; τ ) = √ L α (z) I + 1/3 + O(z −2/3 ) ⎝ 0 eθ1 (z) z 3 0 0
⎞ 0 0 ⎠,
eθ3 (z)
⎞
(5.28)
0 0 ⎠,
eθ3 (z)
for Im z < 0.
(5.29)
The expansions (5.28) and (5.29) are valid uniformly for τ in a compact subset of the complex plane. Proof. We apply the classical steepest descent analysis to the integral representations (5.15)–(5.18) of the functions p j . We set σ (t; z, τ ) = zt + τ/t + 1/(2t 2 ). The saddle points are solutions of ∂σ τ 1 = z − 2 − 3 = 0. ∂t t t
(5.30)
As z → ∞, while τ remains bounded, the three solutions to (5.30) have the following expansion: τ tk = tk (z; τ ) = ω2k z −1/3 + ωk z −2/3 + O(z −4/3 ), 3
k = 1, 2, 3,
(5.31)
and the corresponding values at the saddles are 3 2k 2/3 τ2 ω z + τ ωk z 1/3 − + O(z −1/3 ) 2 6 τ2 + O(z −1/3 ), = θk (z; τ ) − as z → ∞, 6
σ (tk (z; τ ); z, τ ) =
(5.32)
with θk introduced in (1.21). If Ck is the steepest descent path through the saddle point tk , we obtain from (5.32) and standard steepest descent arguments that
Ck
t α−3 eτ/t e1/(2t ) e zt dt = ± 2
2π 2 tkα−3 e−τ /6 eθk (z;τ ) (1 + O(z −1/3 )) ∂2σ − ∂t 2 (tk ; z, τ ) (5.33)
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Fig. 13. Deformation of the contours of integration 1 , 3 and 4 into the steepest descent paths for the integrals defining p1 , p3 and p4 in the case where 0 < arg z < π/4
as z → ∞, where the ± sign depends on the orientation of the steepest descent path. Plugging (5.31) into (5.33), and using the fact that 2 ∂ 2σ 3 = 4 1 + τt , ∂t 2 t 3 we obtain as z → ∞,
t Ck
α−3 τ/t 1/(2t 2 ) zt
e
e
e dt = ±
−2π −τ 2 /6 k −1/3 α−1 θk (z;τ ) e ω z 1 + O(z −1/3 ) . e 3 (5.34)
Consider now α (z) as defined in (5.19) for 0 < arg z < π/4. We see that only p1 , p3 and p4 play a role in this sector. The corresponding contours of integration can be deformed into the steepest descent paths through one of the saddle points as shown in Fig. 13. Hence, in the given sector, the functions p1 , p3 and p4 have an asymptotic behavior of the form (5.34) for some particular k and some choice of the ± sign, and multiplied by e±απi in case of p3 and p4 , see the formulas (5.15), (5.17), and (5.18). This will lead to the asymptotic expansion for the first row of (5.28) for 0 < z < π/4, except for the determination of Mα+ (τ ). The second and rows can be dealt with similarly. Here we have to consider the first and second derivatives of p1 , p3 , and p4 , which have similar integral representations (5.15), (5.17), and (5.18), but with α replaced by α + 1 and α + 2, respectively. The other sectors can be analyzed in a similar way. Tracing the behavior of the dominant saddle point we find that the asymptotic expression just obtained remains valid in the full upper half plane, while in the lower half plane we find (5.29), again up to the determination of Mα− (τ ). What remains is to obtain the constants Mα± (τ ) that appear in the O(z −1/3 ) term in the expansions (5.28) and (5.29). This can be done by calculating the next terms in the asymptotic expansion of the integrals. Alternatively, we can use the fact that α solves the first-order matrix-valued ODE ⎛ ⎞ 0 z 0 z ⎠ α (z). zα (z) = ⎝0 0 1 τ −α
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Substituting into this the asymptotic expansions for α and equating terms on both sides, we find after lengthy calculations (that were actually performed with the help of Maple) the formulas for Mα± (τ ) as given in (5.26) and (5.27). 5.6. Definition and properties of f (z) and τ (z). We will take the local parametrix Q in the form, see also (5.13), ⎛ −nλ (z) ⎞ e 1 0 0 2nz Q(z) = E n (z)α (n 3/2 f (z); n 1/2 τ (z)) ⎝ 0 0 ⎠ e 3t (1−t) , z α e−nλ2 (z) 0 0 e−nλ3 (z) (5.35) where E n is an analytic prefactor, f (z) is a conformal map defined in a neighborhood of z = 0 and τ (z) is analytic in z. Assuming that f maps the contour T ∩ D(0, r ) to the six rays such that f (z) is positive for positive real z, then it follows from the above construction that Q will satisfy the required jump condition. We are going to take f (z) and τ (z) in such a way that the exponential factors in (5.35) are cancelled. That is, we want analytic f (z) and τ (z) such that θk ( f (z); τ (z)) = λk (z) −
2z , 3t (1 − t)
k = 1, 2, 3,
(5.36)
for Im z > 0, while for Im z < 0, 2z − 2πi, 3t (1 − t) 2z + 2πi, θ2 ( f (z); τ (z)) = λ1 (z) − 3t (1 − t) 2z , θ3 ( f (z); τ (z)) = λ3 (z) − 3t (1 − t) θ1 ( f (z); τ (z)) = λ2 (z) −
(5.37) (5.38) (5.39)
where the functions θk were defined in (1.21). To define f (z) and τ (z) we use the functions f 3 (z; t) and g3 (z; t) from Lemma 2.8. Definition 5.4. We put f (z) = f (z; t) = z[ f 3 (z; t)]3/2 ,
τ (z) = τ (z; t) =
g3 (z; t) f 3 (z; t)1/2
,
(5.40)
where as usual we take the principal branches of the fractional powers. We write f (z; t) and g(z; t) in order to emphasize their dependence on the parameter t. Lemma 5.5. There exist r0 > 0 and δ > 0 such that for each t ∈ (t ∗ − δ, t ∗ + δ) we have that z → f (z; t) is a conformal mapping on the disk D(0, r0 ) and z → τ (z; t) is analytic on D(0, r0 ). The map z → f (z; t) is positive for positive real z and negative for negative real z. In addition, we have τ (z; t) = O(t − t ∗ ) + O(z)
as t → t ∗ and z → 0.
(5.41)
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Proof. Because of (2.26) we have that f 3 (0) > 0 and so f (z) defined by (5.40) is indeed a conformal map in a neigborhood of z = 0 which is positive for positive values of z. Also τ (z) is analytic in a neighborhood of z = 0. We have f 3 (z; t) = f 3 (z, t ∗ ) + O(t − t ∗ ), g3 (z; t) = g3 (z, t ∗ ) + O(t − t ∗ ), as t → t ∗ , (5.42) uniformly for z in a neighborhood of 0, and f 3 (z; t ∗ ) = (c∗ )−2/3 + O(z),
g3 (z; t ∗ ) = O(z) as z → 0.
(5.43)
This follows from the definitions (2.14) of c and p, Eq. (2.16), and the definitions of f j and g j , j = 1, 2, 3. Expansions (5.43) also use (2.26) and the fact that p = 0 when t = t ∗ . Then (5.41) is a consequence of the previous expansions and the definitions of f (z) and τ (z) in (5.40). 5.7. Definition and properties of the prefactor E n (z). The prefactor E n (z) in the definition of Q(z) in (5.35) should be analytic and chosen so that Q is close to Nα on |z| = n −1/2 . In view of the expansion of α given in Lemma 5.3, we set the following definition. We use r0 > 0 as given by Lemma 5.5 and we assume t ∈ (t ∗ − δ, t ∗ + δ). Definition 5.6. We define for z ∈ D(0, r0 )\R, ⎛
1 E n (z) = −i 3Nα (z) ⎝0 0 √
⎞ 0 3/2 0⎠ L −1 f (z)), α (n 1
0
z −α 0
(5.44)
where L α has been introduced in (5.25), and Nα is described in Sect. 4. Lemma 5.7. E n and E n−1 have an analytic continuation to D(0, r0 ). α defined in (4.13), Proof. Taking into account (4.1)–(4.2) we see that for N ⎛
0 α,+ (x) = N α,− (x) ⎝−1 N 0
1 0 0
⎞ 0 0⎠ 1
for x ∈ (0, r0 )
and ⎛
1 α,+ (x) = N α,− (x) ⎝0 N 0
0 0
e−απi
0
⎞
−e−απi ⎠ , 0
for x ∈ (−r0 , 0).
For L α we find the same jump matrices. Indeed, for x > 0, we have by the definition of L α ,
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⎛
⎞−1 ⎞⎛ 2 0 0 −ω 1 eαπi/3 ω −1 1⎠ L α,− (x)−1 L α,+ (x) = ⎝ 0 e−απi/3 0⎠ ⎝ 1 ω −ω2 1 0 0 1 ⎛ ⎞ ⎛ απi/3 ⎞ 0 0 ω ω2 1 e 1 1⎠ ⎝ 0 ×⎝ 1 e−απi/3 0⎠ 2 ω ω 1 0 0 1 ⎛ απi/3 ⎞⎛ ⎞ ⎛ απi/3 e 0 0 0 0 1 0 e =⎝ 0 e−απi/3 0⎠ ⎝−1 0 0⎠ ⎝ 0 e−απi/3 0 0 1 0 0 1 0 0 ⎛ ⎞ 0 1 0 = ⎝−1 0 0⎠ 0 0 1
⎞ 0 0⎠ 1
and for x < 0, ⎛
⎞−1 ⎞⎛ 2 0 0 −ω 1 eαπi/3 ω −1 1⎠ L α,− (x)−1 L α,+ (x) = e−2απi/3 ⎝ 0 e−απi/3 0⎠ ⎝ 1 ω −ω2 1 0 0 1 ⎞⎛ ⎛ ⎞ ⎞ ⎛ απi/3 ω 0 0 ω ω2 1 0 0 e 1 1⎠ ⎝ 0 × ⎝0 1 0 ⎠ ⎝ 1 e−απi/3 0⎠ 2 2 0 0 ω ω ω 1 0 0 1 ⎛ απi/3 ⎞⎛ ⎞ 0 0 e 1 0 0 = e−2απi/3 ⎝ 0 e−απi/3 0⎠ ⎝0 0 −1⎠ 0 1 0 0 0 1 ⎛ απi/3 ⎞ ⎛ ⎞ 1 0 0 e 0 0 0 −e−απi ⎠ . ×⎝ 0 e−απi/3 0⎠ = ⎝0 −απi 0 0 1 0 e 0 α and L α are the same. Since f is a conformal map on D(0, r0 ) with Thus, the jumps for N 3/2 f (z)) α (z)L −1 f (x) > 0 for x ∈ (0, r0 ) and f (x) < 0 for x < 0, it follows that N α (n has an analytic continuation to D(0, r0 )\{0}. As a result, we conclude that E n has an analytic continuation to D(0, r0 )\{0}. We show that the isolated singularity at the origin is removable. Indeed, by Lemma 4.3 we have α (z) = O(z −1/3 ) N
as z → 0,
and by the definition of L α , −1/3 ) z −α/3 L −1 α (z) = O(z
as z → 0.
Thus 3/2 α (z)L −1 z −α/3 N α (n
f (z)) =
n 3/2 f (z) z
= O(z −2/3 )
α/3
3/2 α (z) n 3/2 f (z)−α/3 L −1 f (z)) N α (n
as z → 0.
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Fig. 14. Jump contour S for the RH problem for S. The disk around 0 is shrinking with radius n −1/2 as n → ∞. The contour S is also the contour for the RH problem for R.
It follows that the singularity of the left hand side at z = 0 is removable and thus E n (z) is analytic in D(0, r0 ). α ≡ 1. From (5.25) we get Recall that by Lemma 4.3 we have that det Nα = det N that √ det L α (z) = 3i 3 z −α . Thus by (5.44), det E n (z) =
n 3/2 f (z) z
α ,
which is analytic and non-zero in a neighborhood of z = 0. Thus E n−1 (z) is analytic in the neighborhood D(0, r0 ) as well. This completes the proof of the lemma. Having defined f (z), τ (z) and E n (z) we then define the local parametrix Q as in formula (5.35). 6. Fourth Transformation of the RH Problem In the next transformation we define ⎧ ⎪ T (z)P(z)−1 , for z ∈ D(q ∗ , rq ), ⎪ ⎨ S(z) = T (z)Q(z)−1 , for z ∈ D(0, n −1/2 ), ⎪ ⎪ ⎩ T (z)Nα−1 (z), elsewhere,
(6.1)
where we use the matrix-valued functions Nα from (4.9), P constructed in Sect. 5 in the fixed neighborhood D(q ∗ , r ) of q ∗ , and Q given by (5.35) in the shrinking neighborhood D(0, n −1/2 ) of the origin. By construction, S(z) is piece-wise analytic and has jumps across the contour S shown in Fig. 14, with possible isolated singularities at 0 and q ∗ . The singularity at q ∗ is removable which follows from the properties of the Airy parametrix. We now check that the origin is not a singularity of S(z). Lemma 6.1. The singularity of S at z = 0 is removable.
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Proof. We give the proof for the case α > 0. Consider z → 0 with Im z > 0 and outside the lenses around 2 and 1 . By the RH problem 3.2 for T , we have that T remains bounded there. We show that Q −1 remains bounded as well. By (5.35), we have ⎛
Q −1 (z) = e
⎞ enλ1 (z) 0 0 ⎝ 0 0 ⎠ enλ2 (z) nλ 0 0 e 3 (z) ⎞ 0 0 z −α 0⎠ α (n 3/2 f (z); n 1/2 τ (z))−1 E n (z)−1 . 0 1
− 3t 2nz (1−t)
⎛
1 ⎝0 0
By Lemma 5.7 we know that E n (z)−1 is analytic and thus bounded as z → 0. Also the functions λ j are bounded as z → 0. Also ⎛
1 ⎝0 0
0
z −α 0
⎞ 0 0⎠ α (n 3/2 f (z); n 1/2 τ (z))−1 1
is bounded as z → 0 in the region under consideration because of condition 4. in the RH problem for α , see (1.25) and the fact that det α = z −α , see (1.27). We conclude that Q −1 remains bounded as z → 0 in the region in the upper half-plane outside of the lenses. The other regions can be treated in a similar way and the lemma follows. We find the following RH problem for S: RH problem 6.2. 1. S is analytic outside the contour S shown in Fig. 14. 2. On S there is a jump relation S+ (z) = S− (z)JS (z)
(6.2)
with jump matrix JS (z) given by JS (z) = Nα (z)P −1 (z), for |z − q ∗ | = rq , JS (z) = Nα (z)Q JS (z) =
−1
(z), for |z| = r0 = n
Nα (z)JT (z)Nα−1 (z),
−1/2
(6.3) ,
elsewhere on S .
(6.4) (6.5)
3. S(z) = I + O(z −1 ) as z → ∞. Recall now that we are interested in the limit (2.1) where n → ∞, t → t ∗ and = n 1/2 (t ∗ − t) remains fixed. The jump matrices JS in (6.3)–(6.5) depend on n and t, and we would like that they tend to the identity matrix in the double scaling limit (2.1). This turns out to be the case for the jump matrices (6.3) and (6.5). However, this is not the case for (6.4) as will be shown later on. We start with the good jumps. c∗ τ
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Lemma 6.3. In the limit (2.1) we have uniformly for |z − q ∗ | = rq , JS (z) = I + O n −1
(6.6)
and for some c > 0 (depending on α), 2/3 e−cn JS (z) = I + O uniformly for z ∈ S outside of the two circles. (6.7) 1 + |z| Proof. The behavior (6.6) follows from (6.3) and the matching condition (5.1) for P. In view of Lemma 2.9 and the asymptotic behavior of the λ functions we find that for some c > 0, Re (λ3 − λ2 )(z) ≥ c|z|1/2 , Re (λ1 − λ2 )(z) ≥ c, Re (λ2 − λ1 )(z) ≥ c|z|,
z ∈ ± 2 \D(0, 1), z∈ z∈
± 1 \(D(0, 1) ∪ (q ∗ + rq , ∞).
(6.8) ∗
D(q , rq )),
(6.9) (6.10)
According to (3.5), (6.5) and (6.8), for z ∈ ± 2 \D(0, 1),
1/2 JS (z) − I = e±απi z −α en(λ2 −λ3 )(z) Nα (z)E 23 Nα−1 (z) = O |z|−α e−cn|z|
for some c > 0. ∗ ∗ Analogous considerations on the lips ± 1 \(D(0, 1)∪ D(q , rq )) and on (q +rq , ∞), appealing to formulas (3.7)–(3.8) and (6.9)–(6.10), show that there exists some c > 0, such that 1/2 , z ∈ S \(D(0, 1) ∪ D(q ∗ , rq )). JS (z) = I + O |z||α| e−cn|z| What remains is to estimate JS (z) on the lips of the lenses near 0 for n −1/2 < |z| < 1. For t = t ∗ , we obtain from Lemma 2.8 that there exists a constant c1 > 0 such that (recall that p = 0 when t = t ∗ ) Re (λ3 − λ2 )(z; t ∗ ) ≥ c1 |z|2/3 , Re (λ1 − λ2 )(z; t ∗ ) ≥ c1 |z|2/3 ,
z ∈ ± 2 ∩ D(0, 1), z ∈ ± 1 ∩ D(0, 1).
Moreover, (2.25) and (5.42) imply that λ j (z, t) = λ j (z; t ∗ ) + z 1/3 O(t − t ∗ ) as t → t ∗ . Thus, Re (λ3 − λ2 )(z; t) ≥ c1 |z|2/3 − c2 |z|1/3 |t − t ∗ |, z ∈ ± 2 ∩ D(0, 1),
Re (λ1 − λ2 )(z; t) ≥ c1 |z|2/3 − c2 |z|1/3 |t − t ∗ |, z ∈ ± 1 ∩ D(0, 1), for some c2 > 0. Since t − t ∗ = O(n −1/2 ) we conclude that Re (λ3 − λ2 )(z; t) ≥ c3 n −1/3 , Re (λ1 − λ2 )(z; t) ≥ c1 n −1/3 ,
−1/2 z ∈ ± < |z| < 1, 2 ∩ D(0, 1), n −1/2 z ∈ ± < |z| < 1, 1 ∩ D(0, 1), n
for some positive constant c3 > 0 and n large enough. −1/2 , and using (4.13), we get Now, for z ∈ ± 2 ∩ D(0, 1), |z| > n α (z)E 23 N α−1 (z). JS (z) − I = e±απi en(λ2 −λ3 )(z) N
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α−1 (z) = O |z|−1/3 as |z| → 0, so that α (z) = O |z|−1/3 , N By Lemma 4.3, N 2/3 JS (z) − I = O e−cn −1/2 < |z| < 1. for some c > 0. Analogous conclusion is obtained on ± 1 ∩ D(0, 1), n Gathering both estimates and replacing them by a weaker uniform bound we obtain (6.7).
We next analyze the jump matrix (6.4) on |z| = n −1/2 again in the double scaling limit (2.1). We have JS = Nα Q −1 , where Nα is given by (4.9) and Q is given by (5.35). All notions that appear in these formulas depend on t or n (or both). For example, Nα depends on t since the endpoint q is varying with t, and tends to q ∗ as t → t ∗ . Indeed, q = q ∗ + O(t − t ∗ ). Also the matrix Cα from (4.9) and the constants K 1 , K 2 , K 3 from (4.7) depend on t and tend to limiting values corresponding to the value t ∗ at the same rate of O(t − t ∗ ) = O(n −1/2 ). We denote the limiting values with ∗ : Cα = Cα∗ + O(t − t ∗ ),
K j = K ∗j + O(t − t ∗ ),
j = 1, 2, 3,
(6.11)
and these quantities appear in the formula (6.12) below. Proposition 6.4. In the limit (2.1) we have that JS (z) = I − uniformly for |z| = n −1/2 , where ⎛ ∗⎞ K1 M∗α = Cα∗ ⎝ K 2∗ ⎠ K 1∗ K 3∗
h n (z; t) ∗ Mα + O(n −1/6 ) z
K 2∗
⎛ 1 K 3∗ ⎝0 0
0 0 −i
⎞ 0 −1 −i ⎠ Cα∗ 0
(6.12)
(6.13)
and h n (z; t) = τ (z; t)
nτ (z; t)2 + 9α . 9c∗
(6.14)
Proof. The local parametrix Q from (5.35) depends on both t and n. The functions λ j come from the Riemann surface and therefore depend on t, but only in a mild way. Of more importance is the dependence of the functions f (z) = f (z; t) and τ (z) = τ (z; t) on t, see Lemma 5.5. From (5.41) we have that n 1/2 τ (z; t) = O(1),
(6.15)
in the double scaling limit (2.1) uniformly for |z| = n −1/2 . [This is in fact the reason why we need the shrinking disk of radius n −1/2 .] We also have that n 1/2 f (z; t) remains bounded as |z| = n −1/2 . However, n 3/2 f (z; t) is growing in absolute value and is of order n uniformly for |z| = n −1/2 . Therefore we
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can apply the asymptotic formulas from Lemma 5.3 to α n 3/2 f (z); n 1/2 τ (z) and we find for Im z > 0, α n 3/2 f (z); n 1/2 τ (z) Mα+ (n 1/2 τ (z)) i 3/2 −2/3 + O(n ) = √ L α (n f (z)) I + 1/2 n f (z)1/3 3 ⎞ ⎛ 3/2 1/2 0 0 eθ1 (n f (z);n τ (z)) ⎟ ⎜ 3/2 1/2 ×⎝ ⎠ (6.16) 0 eθ2 (n f (z);n τ (z)) 0 3/2 1/2 θ (n f (z);n τ (z)) 3 0 0 e uniformly for |z| = n −1/2 , where we recall that θk (z; τ ), k = 1, 2, 3, also depend on τ , see (1.21). By (1.21) and (5.36) we actually have 2nz θk n 3/2 f (z); n 1/2 τ (z) = nθk ( f (z); τ (z)) = nλk (z) − 3t (1 − t) for k = 1, 2, 3, and Im z > 0, by our choice of f (z) and τ (z). Thus by (5.35) and (2.1) we have
⎛ 1 + (n 1/2 τ (z)) i M Q(z) = E n (z) √ L α (n 3/2 f (z)) I + α1/2 + O(n −2/3 ) ⎝0 n f (z)1/3 3 0
and then by inserting the definition (5.44) of E n (z), we obtain ⎛ ⎞ ⎛ 1 1 0 0 + (n 1/2 τ (z)) M Q(z) = Nα (z) ⎝0 z −α 0⎠ I + α1/2 + O(n −2/3 ) ⎝0 n f (z)1/3 0 0 0 1
0 zα 0
0 zα 0
⎞ 0 0⎠ , 1 ⎞ 0 0⎠ , 1 (6.17)
uniformly for |z| = n −1/2 with Im z > 0. For |z| = n −1/2 with Im z < 0 we obtain the same formula (6.17) but with Mα+ replaced by Mα− . Then for |z| = n −1/2 , α (z) JS−1 (z) = Q(z)Nα−1 (z) = I + N
Mα± (n 1/2 τ (z)) −1 N (z) + O(n −1/3 ), n 1/2 f (z)1/3 α
(6.18)
α . The entries of N α (z) and its inverse are of where we recall the definition (4.13) of N order |z|−1/3 = n 1/6 by (4.14). Therefore the error term has gone up from O(n −2/3 ) in (6.17) to O(n −1/3 ) in (6.18). α (z)Mα± (n 1/2 τ (z)) N α−1 (z) we encounter the following matrix: In the evaluation of N ⎛ ⎞ ⎛ ⎞ 0 K1 1 0 Mα = Cα ⎝ K 2 ⎠ K 1 K 2 K 3 ⎝0 0 −i ⎠ Cα−1 , (6.19) 0 −i 0 K3 which is a 3 × 3 rank one matrix depending on t, but not on z. The matrix Mα is in fact nilpotent, M2α = 0,
(6.20)
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which follows from (6.19) and the property (4.16) of the numbers K j . From (6.11) and (6.13) we also find that Mα = M∗α + O(t − t ∗ )
as t → t ∗ .
(6.21)
We first prove the following lemma. Lemma 6.5. We have for Im z > 0, ⎞ ⎛ ω2 0 0 α−1 (z) = 3c−4/3 Mα z −2/3 + O z −1/3 α (z) ⎝ 0 ω 0⎠ N N 0 0 1 and
⎞⎛ ⎛ −α/2 0 −ω 0 0 ω α (z) ⎝ 0 0 N ωα/2 0⎠ ⎝ ω2 0 0 1 −ω2 ω √ = −3 3ic−4/3 Mα z −2/3 + O z −1/3
⎞ ⎛ α/2 1 ω −1⎠ ⎝ 0 0 0
0
ω−α/2 0
(6.22)
⎞ 0 α−1 (z) 0⎠ N 1 (6.23)
as z → 0. Proof. We obtain (6.22) from (4.9), (4.12), Lemma 4.4, and the fact that ⎛ ⎞⎛ ⎞ 2 0 0 ω2 ω 2 ⎝ ⎠ ⎝ 0 ω 0 ω ⎠ = 3. ω 1 ω 0 0 1 1 From (4.9), we obtain that the left-hand side of (6.23) is equal to ⎛α ⎞ ⎞⎛ 0 −ω 1 0 0 ξ1 (z) ξ2α (z) 0 ⎠ ⎝ ω2 0 −1⎠ Cα N0 (z) ⎝ 0 α 0 0 ξ3 (z) −ω2 ω 0 ⎛ −α ⎞ ξ1 (z) 0 0 ×⎝ 0 ξ2−α (z) 0 ⎠ N0−1 (z)Cα−1 , 0 0 ξ3−α (z) with functions ξ1 (z) = ω−1/2 e G 1 (z) , ξ2 (z) = ω1/2 z −1 e G 2 (z) , ξ3 (z) = e G 3 (z) , with G j (z) = r j (w j (z)) defined in (4.10)–(4.11). Using these expressions and Lemma 2.5, we find the remarkable fact that ξ j (z) = 1 + O z 1/3 , z → 0. ξk (z) Then, using (4.12) and the fact that ⎛ 0 −ω 2 0 ω ω 1 ⎝ ω2 −ω2 ω we obtain (6.23).
⎞⎛ ⎞ 1 ω2 √ ⎠ ⎝ −1 ω ⎠ = −3 3i, 1 0
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We continue with the proof of Proposition 6.4. From the lemma and the formula (5.26) for Mα+ , we obtain α (z)Mα+ (n 1/2 τ (z)) N α−1 (z) N √ in 1/2 τ (z) −4/3 n 1/2 τ (z) nτ 2 (z) + 9α − 9 − 3 3i Mα z −2/3 + O z −1/3 = 3 c √ 27 3 3 2 τ (z)(nτ (z) + 9α) −2/3 −1/3 = n 1/2 . (6.24) M z + O z α 9c4/3 Inserting this into (6.18) we get for |z| = n −1/2 and Im z > 0, JS−1 (z) = I +
τ (z)(nτ 2 (z) + 9α) O(z −1/3 ) −2/3 M z + + O(n −1/3 ). (6.25) α 9c4/3 f (z)1/3 n 1/2 f (z)1/3
A similar analysis for Im z < 0 will show that the same formula (6.25) also holds for Im z < 0. (z −1/3 ) −1/6 ), where we use that f (z) is Note that for |z| = n −1/2 we have nO 1/2 f (z)1/3 = O(n a conformal map. In fact, by (2.26) and (5.40) f (z; t) = f (0; t)z + O(z 2 ) = (c∗ )−1 + O(t − t ∗ ) z + O(z 2 ), which implies that in the double scaling limit (2.1) f (z; t) = (c∗ )−1 z + O(n −1 ) for |z| = n −1/2 . Using also (6.20) we then obtain from (6.25) that JS (z) = I −
τ (z)(nτ 2 (z) + 9α) Mα z −2/3 + O(n −1/6 ), 9c∗ z 1/3
|z| = n −1/2 ,
(6.26)
which implies (6.12) in view of (6.21). This completes the proof of Proposition 6.4. Now recall that in the double scaling limit (2.1) we have that n 1/2 τ (z; t) remains bounded for |z| = n −1/2 , see also (6.15). Then also τ (z; t)(nτ 2 (z; t) + 9α) = O(1), c∗ z ∗ −1/2 , but and it follows that the term h n (z;t) z Mα in (6.12) remains bounded for |z| = n does not tend to 0 as n → ∞. Therefore the jump matrix JS on |z| = n −1/2 does not tend to the identity matrix as n → ∞.
7. Final Transformation We need one more transformation. What will help us in the final transformation is the 2 identity (6.20) for Mα , which also holds for the limiting value, namely M∗α = 0. The final transformation S → R is similar to the one in [17] and is defined as follows.
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Definition 7.1. With the notation (6.14) we define for z ∈ C\ S , h n (0; t) ∗ R(z) = S(z) I − Mα , |z| > n −1/2 , z h n (z; t) − h n (0; t) ∗ Mα , |z| < n −1/2 . R(z) = S(z) I + z
(7.1) (7.2)
Note that the transformation S → R is a global transformation which modifies S in every part of the complex plane. Then R satisfies the following RH problem on the contour R = S , see Fig. 14. RH problem 7.2. 1. R is defined and analytic in C\ R . 2. On R we have the jump R+ = R− J R with
J R (z) =
h n (0; t) ∗ Mα I+ z
for |z| = n −1/2 , and J R (z) =
I+
h n (z; t) − h n (0; t) ∗ JS (z) I + Mα , z
h n (0; t) ∗ Mα z
(7.3)
h n (0; t) ∗ JS (z) I − Mα , z
(7.4)
(7.5)
elsewhere on R . 3. R(z) = I + O(1/z) as z → ∞. All properties in the RH problem 7.2 follow easily fom the RH problem 6.2 for S and Definition (7.1)–(7.2). For (7.4) and (7.5) one also uses (6.20) and (6.21) which imply that for every constant γ , −1 = I + γ Mα∗ . I − γ M∗α Under the transformation S → R the jumps on the part of R outside of the circle |z| = n −1/2 are not essentially affected. We have the same estimates as in Lemma 6.3: Lemma 7.3. In the limit (2.1) we have uniformly for |z − q ∗ | = rq , J R (z) = I + O n −1 and J R (z) = I + O
e−cn 1 + |z|
2/3
(7.6)
uniformly for z ∈ R outside of the two circles.
(7.7)
Proof. By (6.6) and (7.5), for |z − q ∗ | = rq , h n (0; t) ∗ h n (0; t) ∗ −1 JR = I + I +O n Mα Mα I− z z h n (0; t) ∗ h n (0; t) ∗ −1 I− =I+ I+ Mα O n Mα z z = I + O n −1 . Analogous calculations yield (7.7). In both cases the fact that (M∗α )2 = 0 is crucial.
Non-Intersecting Squared Bessel Paths: Critical Time and Double Scaling Limit
Lemma 7.4. In the limit (2.1) we have J R (z) = I + O n −1/6
uniformly for |z| = n −1/2 .
269
(7.8)
Proof. Using (7.4) and (6.26) and the fact that (M∗α )2 = 0, direct calculation yields h n (z; t) − h n (0; t) ∗ h n (0; t) ∗ −1/6 Mα O n Mα , I+ J R (z) = I + I + z z and (7.8) follows.
As a result of the estimates on J R we may now conclude that in the double scaling limit (2.1), 1 as n → ∞ (7.9) R(z) = I + O n 1/6 (1 + |z|) uniformly for z ∈ C\ R . See [2, App. A] for arguments that justify this, also in a situation of varying contours. 8. The Limiting Kernel 8.1. Expression for the critical kernel. We start from (1.12), which gives the correlation kernel K n (x, y; t) in terms of the solution of the RH problem for Y . Following the transformation Y → X → U → T , we find that for x, y > 0 and x, y ∈ (0, q), ⎛ nλ (x) ⎞ e 1,+ −nλ (y) α −nλ (y) −1 1 K n (x, y; t) = −e 1,+ y e 2,+ 0 T+ (y)T+ (x) ⎝x −α enλ2,+ (x) ⎠. 2πi(x − y) 0 (8.1) For z inside the disk of radius n −1/2 , we have by (6.1), (5.35), (5.7), and (7.2), that T (z) = S(z)Q(z) τ (z; t)(nτ 2 (z; t) + 9α) − τ (0; t)(nτ 2 (0; t) + 9α) ∗ M = R(z) I − α 9c∗ z 2nz
×E n (z)α (n 3/2 f (z; t); n 1/2 τ (z; t))n (z)e 3t (1−t) . Thus, if 0 < x, y <
n −1/2 , 2n(x−y)
(8.2)
we get by plugging (8.2) into (8.1),
e 3t (1−t) 3/2 −1 1 0 −1 K n (x, y; t) = f (y; t); n 1/2 τ (y; t)) α,+ (n 2πi(x − y) τ (y; t)(nτ 2 (y; t) + 9α) − τ (0; t)(nτ 2 (0; t) + 9α) ∗ −1 Mα R −1 (y) ×E n (y) I + 9c∗ y τ (x; t)(nτ 2 (x; t) + 9α) − τ (0; t)(nτ 2 (0; t) + 9α) ∗ M ×R(x) I − α E n (x) 9c∗ x ⎛ ⎞ 1 ×α,+ (n 3/2 f (x; t); n 1/2 τ (x; t)) ⎝1⎠ , (8.3) 0
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which is an exact formula. Now we take the double scaling limit n → ∞, t → t ∗ such that c∗ τ = n 1/2 (t ∗ − t)
remains fixed.
(8.4)
We also replace x and y in (8.3) by xn =
c∗ x , n 3/2
yn =
c∗ y n 3/2
(8.5)
with x, y > 0 fixed. For n large enough we then have that xn and yn are less than n −1/2 . We study how the various factors in (8.3) behave in this limit. Lemma 8.1. Let x, y > be fixed. Then we have in the double scaling limit (8.4) with xn and yn given by (8.5) n 3/2 f (xn ; t) = x(1 + O(n −1/2 )),
n 3/2 f (yn ; t) = y(1 + O(n −1/2 )),
(8.6)
n 1/2 τ (yn ; t) = τ + O(n −1/2 )
(8.7)
and n 1/2 τ (xn ; t) = τ + O(n −1/2 ), as n → ∞. Proof. By (5.40), (5.42) and (5.43) we have 3/2 −2/3 f (z; t) = z[ f 3 (z; t)]3/2 = z c∗ + O(z) + O(t − t ∗ ) z
= ∗ 1 + O(z) + O(t − t ∗ ) c as z → 0 and t → t ∗ . This readily implies (8.6). Again by (5.40), (5.42) and (5.43) we have τ (z; t) =
1/3
g3 (z; t) = c∗ g3 (z; t) 1 + O(z) + O(t − t ∗ ) 1/2 f 3 (z; t)
as z → 0 and t → t ∗ . Then from the definitions in Lemmas 2.5, 2.6, and 2.8 it is not difficult to verify that g3 (z; t) is analytic in both arguments with g3 (0, t ∗ ) = 0, so that ∂g3 ∗ (0, t ) (t − t ∗ ) + O(z) + O(t − t ∗ )2 g3 (z, t) = ∂t as z → 0 and t → t ∗ . By (2.26) we have g3 (0; t) = 3 pc−4/3 and using the dependence of p and c on t as given in (2.14) we find, after some calculations, that −4/3 ∂g3 (0, t ∗ ) = − c∗ . ∂t Hence τ (z; t) = (c∗ )−4/3 (t ∗ − t) + O(z) + O((t − t ∗ )2 ) and −1 ∗ n 1/2 τ (xn ; t) = n 1/2 c∗ (t − t) + O(n −1 ) + n 1/2 O((t − t ∗ )2 ), which by (8.4) indeed leads to (8.7).
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Lemma 8.2. Under the same assumptions as in Lemma 8.1, τ (yn ; t)(nτ 2 (yn ; t) + 9α) − τ (0; t)(nτ 2 (0; t) + 9α) ∗ −1 E n−1 (yn ) I + M α R (yn ) 9c∗ yn τ (xn ; t)(nτ 2 (xn ; t) + 9α) − τ (0; t)(nτ 2 (0; t) + 9α) ∗ ×R(xn ) I − M α 9c∗ xn (8.8) ×E n (xn ) → I. Proof. For z = O(n −3/2 ) we have by Cauchy’s theorem and (7.9),
R(s) − I 1 ds = O(n 1/3 ) as n → ∞. R (z) = 2πi |s|=n −1/2 (s − z)2 Therefore R(xn ) − R(yn ) = O((xn − yn )n 1/3 ) = O(n −7/6 ) and so R −1 (yn )R(xn ) = I + R −1 (yn )(R(xn ) − R(yn )) = I + O(n −7/6 )
as n → ∞, (8.9)
where we use that R −1 (yn ) remains bounded as n → ∞, which also follows from (7.9). Let us write ρn (xn , t) :=
τ (xn ; t)(nτ 2 (xn ; t) + 9α) − τ (0; t)(nτ 2 (0; t) + 9α) 9c∗ xn
(8.10)
and similarly for ρn (yn , t). Then explicit calculations (done with the help of Maple) show that ρn (xn , t) =
τ 2 + 3α τ 3 (a − 2) − n −1/2 + O(n −1 ), ∗ 2 36(c ) 54(a + 1)(c∗ )2
(8.11)
and similarly for ρn (yn , t). Thus ρn (yn , t) − ρn (xn , t) = O(n −1 )
as n → ∞.
(8.12)
Using (8.9), (8.11), (8.12) and the fact that (M∗α )2 = 0, we see that (8.8) will follow from the following three estimates E n−1 (yn )E n (xn ) = I + O(n −1/2 ), E n−1 (yn )O(n −7/6 )E n (xn ) E n−1 (yn )M∗α E n (xn )
= O(n
−1/6
= O(n
1/2
(8.13)
),
(8.14)
),
(8.15)
as n → ∞. By (5.44), the analytic factor E n (z) depends on n mainly via the argument n 3/2 f (z; t) 3/2 f (z; t), of L −1 α . By (4.13), (5.25), and (5.44), we can factor out the dependence on n and we obtain E n (z) =
n 3/2 f (z; t) z
α/3 Fα (z) diag
n 3/2 f (z; t) z
−1/3 1
n 3/2 f (z; t) z
1/3 ,
(8.16)
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where
⎛ −απi/3 e 1 Fα (z) = √ Nα (z) ⎝ 0 i 3 0
0
eαπi/3 0
⎞⎛ 2 0 ω 0⎠ ⎝ ω 1 1
1 1 1
⎞ ⎛ −1/3 ω z ω2 ⎠ ⎝ 0 0 1
0 1 0
⎞ 0 0 ⎠ z 1/3 (8.17)
is analytic around z = 0 and depends on n in a very mild way only, namely via the α (z) on the endpoint q which is only slightly moving with n. dependence of N The scalar factor in (8.16) will appear in the products (8.13)–(8.15) in the form
n 3/2 f (xn ; t) yn xn n 3/2 f (yn ; t)
α/3
= 1 + O(n −1/2 ),
(8.18)
where the estimate follows from (8.5) and (8.6). Thus by (8.16), E n−1 (yn )E n (xn )
1/3 −1/3 n 3/2 f (yn ; t) n 3/2 f (yn ; t) 1 yn yn −1/3 1/3 n 3/2 f (xn ; t) n 3/2 f (xn ; t) −1 ×Fα (yn )Fα (xn ) diag , 1 xn xn
= (1 + O(n −1/2 )) diag
where Fα−1 (yn )Fα (xn ) = I + O(xn − yn ) = I + O(n −3/2 ). Since the two entries 3/2 1/3 1/3 3/2 n f (yn ; t) n f (xn ; t) and in the diagonal matrices grow like O(n 1/2 ) yn xn we find (8.13), where we also use (8.18). We similarly have E n−1 (yn )O(n −7/6 )E n (xn ) 1/3 −1/3 3/2 n f (yn ; t) n 3/2 f (yn ; t) −1/2 = (1+ O(n )) diag 1 yn yn 3/2 1/3 −1/3 n f (xn ; t) n 3/2 f (xn ; t) −1 −7/6 ×Fα (yn )O(n )Fα (xn ) diag . 1 xn xn
Since Fα−1 (yn ) and Fα (xn ) remain bounded as n → ∞, and the two diagonal matrices are O(n 1/2 ) we obtain the estimate (8.14). To prove the final estimate (8.15) we note that E n−1 (yn )M∗α E n (xn )
1/3 −1/3 3/2 n f (yn ; t) n 3/2 f (yn ; t) = (1 + O(n )) diag 1 yn yn −1/3 1/3 3/2 3/2 n f (xn ; t) n f (xn ; t) ×Fα−1 (yn )M∗α Fα (xn ) diag 1 xn xn −1/2
(8.19)
which would lead to O(n) as n → ∞ if we use the same estimates as above. However, by the form of the right hand side of (8.19), we see that it is only the (1, 3) entry
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E n−1 (yn )M∗α E n (xn ) that could grow like O(n). The other entries are O(n 1/2 ) as claimed in (8.15). We have by (8.19) and (8.17), E n−1 (yn )M∗α E n (xn )
1,3
1/3 n 3/2 f (yn ; t) f (xn ; t) Fα−1 (yn )M∗α Fα (xn ) = 1,3 xn yn ! 1/3 1/3 ! ! −1 ! ∗ n 3/2 f (yn ; t) = n 3/2 f (xn ; t) O !N α (yn )Mα Nα (x n )! ! ! ! −1 ! ∗ = O !N (y )M (x ) N ! , n α n α α n 3/2
1/3
α−1 (yn ) and N α (xn ) grow like n 1/2 , where in the last step we used (8.6). Both matrices N see (4.5). However α (xn ) = O(1) M∗α N
as n → ∞,
which follows from (4.13) and the fact that by (4.12), (6.19), ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ 1 0 0 K1 K1 M∗α Cα N0 (z) = Cα ⎝ K 2 ⎠ K 1 K 2 K 3 ⎝0 0 −i ⎠ c−2/3 ⎝ K 2 ⎠ 0 −i 0 K3 K3 −1/3 2 × ω + O(1) ω 1 z = O(1) as z → 0 α−1 (yn )M∗α N α (xn ) = because of the relation (4.16) satisfied by the constants K j . Thus N 1/2 O(n ) and (8.15) follows. This completes the proof of Lemma 8.2. From (8.3), (8.5) and Lemmas 8.1 and 8.2, we obtain ∗ c x c∗ y ∗ c∗ c∗ τ lim Kn , ; t − 1/2 = K αcrit (x, y; τ ), n→∞ n 3/2 n 3/2 n 3/2 n where K αcrit is given by (1.28). This proves Theorem 1.4. 8.2. Proof of Theorem 1.2. Let us analyze the expression (1.28) for the critical kernel. We define ⎛ ⎞ p2 p3 p1 ⎝ p2 p3 p1 ⎠ up α = p2 p3 p1 as an analytic matrix-valued function in C\iR− . It is the analytic continuation of the restriction of α to the upper sector π/4 < arg z < 3π/4 to the cut plane C\iR− . Then by the jump relations of α , see (1.20), we can rewrite (1.28) as ⎛ ⎞ 1 1 −1 ⎝ ⎠ 0 1 0 up K αcrit (x, y; τ ) = (y)up (8.20) α α (x) 0 , 2πi(x − y) 0
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for x, y > 0. Clearly, ⎛ ⎞ ⎛ ⎞ 1 p2 (x) ⎜ ⎟ ⎜ ⎟ up α (x) ⎝0⎠ = ⎝ p2 (x) ⎠ .
(8.21)
p2 (x)
0 up
The inverse of α is built out of solutions of the differential equation xq + (3 − α)q − τ q + q = 0,
(8.22)
which is, up to a sign, the adjoint of Eq. (5.14). Define the pairing [ p(x), q(y)] = (yq (y) − (α − 2)q (y) − τ q(y)) p(x) + (−yq (y) + (α − 1)q(y)) p (x) + yq(y) p (x),
(8.23)
and denote [ p, q](x) = [ p(x), q(x)] which is the bilinear concomitant. Then d [ p, q](x) = p(x) xq (x) + (3 − α)q (x) − τ q (x) + q(x) dx + q(x) x p (x) + αp (x) − τ p (x) − p(x) , which shows that if p and q satisfy the respective differential equations, then the bilinear concomitant [ p, q](x) is constant. up To find the inverse of α we need solutions, that we call q1 , q2 , and q3 , dual to p1 , p2 , p3 , satisfying [ p j , qk ] = δ j,k ,
j, k = 1, 2, 3.
(8.24)
The inverse matrix is then given by ⎛ zq2 (z)−(α − 2)q2 (z) − τ q2 (z) up −1 ⎜ (z) = ⎝ zq3 (z)−(α − 2)q3 (z)−τ q3 (z) α
−zq2 (z)+(α − 1)q2 (z) −zq3 (z)+(α − 1)q3 (z)
⎟ zq3 (z)⎠ ,
zq1 (z)−(α − 2)q1 (z)−τ q1 (z)
−zq1 (z)+(α − 1)q1 (z)
zq1 (z)
zq2 (z)
⎞
and
−1 0 1 0 up (y) = yq3 (y)−(α − 2)q3 (y)−τ q3 (y), −yq3 (y) α +(α − 1)q3 (y), yq3 (y) .
(8.25)
Hence, the solution q3 is the relevant one for the critical kernel: by (8.20)–(8.25), we get K crit (x, y) =
[ p2 (x), q3 (y)] . 2πi(x − y)
Before continuing, let us build the dual functions for p j . The solutions q j (z), j = 1, 2, 3, of (8.22) admit integral representations
2 t −α e−τ/t e−1/(2t ) e−zt dt. (8.26) q j (z) = C j
j
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275
Fig. 15. The contours of integration j , j = 1, 2, 3, (in bold) used in Definition (8.26) of the functions q j (z), when z is positive. Also shown the contours j , j = 1, 2, 3 (dashed lines)
When the variable z is positive, the contours j , j = 1, 2, 3, can be chosen as in Fig. 15; note that the integrals converge since the contours j approach the origin tangentially to the real axis and go to infinity along the positive real axis. We choose the main branch of t α in (8.26) with the cut in the t-plane along R+ , and allow 1 to go along the upper side of the cut. Observe that we may take the same branch cut in the definition of p2 in (5.16). In order to build an integral expression for the bilinear concomitant we need the following Lemma 8.3. Let p and q be solutions of (5.14) and (8.22) respectively, with integral expressions
p(z) = t −1 e V (t) e zt dt, q(z) = t −2 e−V (t) e−zt dt,
where and are one of the contours depicted in Fig. 15, respectively, and V (t) = (α − 2) log(t) +
τ 1 + . t 2t 2
(8.27)
Then (a)
t
t −2 e V (t) e zt dt = zp (z) + (α − 1) p (z) − τ p(z), t −3 e V (t) e zt dt = −τ zp (z) + (z − τ (α − 1)) p (z) + (α − 2 + τ 2 ) p(z),
−3 −V (t) −zt
e
e
dt = zq (z) + (2 − α) q (z) − τ q(z).
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(b)
[ p(x), q(y)] =
V (t) − V (s) (x − y)(t + s) − t −s s2
e V (t)−V (s) e xt−ys dtds. (8.28)
In particular,
[ p, q](x) =
V (t) − V (s) V (t)−V (s) x(t−s) e e dtds. t −s
(8.29)
(c) If the only point of intersection of and is the origin, then [ p, q](x) ≡ 0. (d) If and intersect transversally at z 0 = 0, and if the contours are oriented so that meets in z 0 on the “−”-side of , then [ p, q](x) ≡ 2πi. Proof. Let us denote
h(z) =
t −2 e V (t) e zt dt.
Then h (z) = p(z) = zp (z) + (α + 2) p (z) − τ p (z), where we have used the differential equation (5.14). Hence, h(z) = zp (z) + (α + 1) p (z) − τ p(z) + c, and in order to find the constant c we compute h(z) − (zp (z) + (α + 1) p (z) − τ p(z))
= t −2 − zt − (α + 1) − τ t −1 e V (t) e zt dt "
" V (t) zt V (t) zt V (t) zt " e − e e dt = te e " = 0, =− td e
due to the selection of the contour. This proves the first identity in (a). We leave the details of the proof of the remaining identities in (a) to the reader. Since V (t) − V (s) 1 1 1 1 1 2−α , = 3 + 2 2+ 3+ +τ + t −s t s t s ts ts t 2 s ts 2 the expression in (8.28) is obtained by direct substitution of those in (a) into the right hand side and comparison with (8.23). Furthermore, if and do not intersect, then
∂ 1 ∂ 1 0= + e V (t)−V (s) e x(t−s) dsdt ∂t t − s ∂s t − s
1 ∂ ∂ e V (t)+xt dsdt + e−V (s)−xs e V (t)+xt = ∂t t − s ∂s
1 −V (s)−xs −V (s)−xs V (t) + x V (t)+xt e e dsdt = − e dsdt × t −s t −s
−V (s) − x −V (s)−xs − e e V (t)+xt dsdt = −[ p, q](x), t −s
where we have used integration by parts and (8.28).
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On the other hand, if the only intersection of and is at the origin, we can perform the same calculation with and = \U (0), where U (0) = {z ∈ C : |z| < }. Taking a posteriori → 0 and observing that all integrands are strongly vanishing at the origin, we arrive at (c) also in this case. Assume finally that there exists a point z 0 = 0 such that ∩ = {z 0 }. Denote now
= \U (z 0 ). We have V (t) + x V (t)+xt e−V (s)−xs dsdt [ p, q](x) = lim e →0 t −s
−V (s) − x −V (s)−xs e + e V (t)+xt dsdt t −s
1 1 V (t)−V (s )+x(t−s ) V (t)−V (s )+x(t−s ) dt, e − e = lim →0 t − s t − s where s and s are the two points of intersection of with the circle |z − z 0 | = . We can deform the path of integration in such a way that it forms a small loop around z = s , picking up the reside of the integrand, and conclude that [ p, q](x) = 2πi. With account of Lemma 8.3 we define q j , j = 1, 2, 3, as in (8.26), with paths j specified in Fig. 15 and C1 =
1 eαπi , C2 = C3 = , 2πi 2πi
and conclude that condition (8.24) is satisfied. Let us turn to the equality (1.19); for that, let us define
dtds (x, y; z) = , K e V (t)−V (s) e xt−ys+(x−y) log(z) s−t t∈ s∈
(8.30)
(8.31)
where the contours are as described in Theorem 1.2. A straightforward computation shows that "
" ∂ dtds (x, y; 1) = (x − y) . = (x − y) K e V (t)−V (s) e xt−ys K (x, y; z)"" ∂z s−t t∈ s∈
z=1 On the other hand, the change of variables t → t − log(z) and s → s − log(z) in the (x, y; z) yields definition of K
dtds , e V (t−log(z))−V (s−log(z)) e xt−ys K (x, y; z) = s −t t∈ s∈
so that
"
" ∂ V (t) − V (s) V (t)−V (s) xt−ys " e = e dtds. K (x, y; z)" ∂z t −s t∈ s∈
z=1
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Thus, we get
V (t) − V (s) V (t)−V (s) xt−ys e dtds e t −s t∈ s∈
dtds = (x − y) . e V (t)−V (s) e xt−ys s−t t∈ s∈
From (8.28) where we plug in the previous identity, together with the value of C3 in (8.30) and the definition of p2 in (5.16), we obtain (1.19). Acknowledgements. ABJK and FW acknowledge the support of a Tournesol program for scientific and technological exchanges between Flanders and France, project code 18063PB. ABJK is supported by K.U. Leuven research grant OT/08/33, FWO-Flanders project G.0427.09 and G.0641.11, and by the Belgian Interuniversity Attraction Pole P06/02. A.M.-F. is supported in part by Junta de Andalucía grants FQM-229, P06-FQM-01735 and P09-FQM-4643. ABJK and A.M.-F. are also supported by the Ministry of Science and Innovation of Spain (project code MTM2008-06689-C02-01).
References 1. Bertola, M.: Biorthogonal polynomials for two-matrix models with semiclassical potentials. J. Approx. Theory 144, 162–212 (2007) 2. Bleher, P.M., Kuijlaars, A.B.J.: Large n limit of Gaussian random matrices with external source, part III, double scaling limit. Commun. Math. Phys. 270, 481–517 (2007) 3. Borodin, A.: Biorthogonal ensembles. Nucl. Phys. B 536, 704–732 (1998) 4. Borodin, A., Ferrari, P.L., Prähofer, M., Sasamoto, T., Warren, J.: Maximum of Dyson Brownian motion and non-colliding systems with a boundary. Electron. Commun. Probab. 14, 486–494 (2009) 5. Borodin, A., Ferrari, P.L., Sasamoto, T.: Two speed TASEP. J. Stat. Phys. 137, 936–977 (2009) 6. Borodin, A., Forrester, P.: Increasing subsequences and the hard-to-soft edge transition in matrix ensembles. J. Phys. A 36, 2963–2981 (2003) 7. Borodin, A., Kuan, J.: Random surface growth with a wall and Plancherel measures for O(∞). Commun. Pure Appl. Math. 63, 831–894 (2010) 8. Borodin, A., Salminen, P.: Handbook of Brownian motion: facts and formulae, Second edition. Basel: Birkhäuser, 1996 9. Claeys, T., Kuijlaars, A.B.J.: Universality in unitary random matrix ensembles when the soft edge meets the hard edge. In: Integrable systems and random matrices, Contemp. Math. 458, Providence, RI: Amer. Math. Soc., 2008, pp. 265–279 10. Coussement, E., Van Assche, W.: Asymptotics of multiple orthogonal polynomials associated with the modified Bessel functions of the first kind. J. Comput. Appl. Math. 153, 141–149 (2003) 11. Coussement, E., Van Assche, W.: Multiple orthogonal polynomials associated with the modified Bessel functions of the first kind. Constr. Approx. 19, 237–263 (2003) 12. Daems, E., Kuijlaars, A.B.J.: A Christoffel-Darboux formula for multiple orthogonal polynomials. J. Approx. Theory 130, 188–200 (2004) 13. Deift, P.: Orthogonal Polynomials and Random Matrices: a Riemann-Hilbert approach. Courant Lecture Notes in Mathematics, Vol. 3, Providence R.I: Amer. Math. Soc., 1999 14. Deift, P.: Integrable operators. In: Differential Operators and Spectral Theory, V. Buslaev et al., eds., Amer. Math. Soc. Transl. (2) 189, Providence, RI: Amer. Math. Soc., 1999, pp. 69-–84 15. Deift, P., Kriecherbauer, T., McLaughlin, K.T.-R., Venakides, S., Zhou, X.: Uniform asymptotics for polynomials orthogonal with respect to varying exponential weights and applications to universality questions in random matrix theory. Commun. Pure Appl. Math. 52, 1335–1425 (1999) 16. Deift, P., Kriecherbauer, T., McLaughlin, K.T.-R., Venakides, S., Zhou, X.: Strong asymptotics of orthogonal polynomials with respect to exponential weights. Commun. Pure Appl. Math. 52, 1491–1552 (1999) 17. Delvaux, S., Kuijlaars, A. B. J., Zhang, L.: Critical behavior of non-intersecting Brownian motions at a tacnode. Comm. Pure Appl. Math. 64, 1305–1383 (2011) 18. Desrosiers, P., Forrester, P.: A note on biorthogonal ensembles. J. Approx. Theory 152, 167–187 (2008) 19. Forrester, P.: Log-gases and Random Matrices. London Mathematical Society Monographs Series, 34. Princeton, NJ: Princeton University Press, 2010
Non-Intersecting Squared Bessel Paths: Critical Time and Double Scaling Limit
279
20. Harnad, J., Its, A.R.: Integrable Fredholm operators and dual isomonodromic deformations. Comm. Math. Phys. 226, 497-–530 (2002) 21. Ince, E.L.: Ordinary Differential Equations. New York: Dover Publications, 1944 22. Its, A.R., Izergin, A.G., Korepin, V.E., Slavnov, N.A.: Differential equations for quantum correlation functions. Internat. J. Modern Phys. B4, 1003-–1037 (1990) 23. Johansson, K.: Random matrices and determinantal processes. In: Mathematical Statistical Physics, A. Bovier et al., eds., Amsterdam: Elsevier, 2006, pp. 1–55 24. Karlin, S., McGregor, J.: Coincidence probabilities. Pacific J. Math. 9, 1141–1164 (1959) 25. Katori, M., Izumi, M., Kobayashi, N.: Two Bessel bridges conditioned never to collide, double Dirichlet series, and Jacobi theta function. J. Stat. Phys. 131, 1067–1083 (2008) 26. Katori, M., Tanemura, H.: Noncolliding processes, matrix-valued processes and determinantal processes. Sugaku Expositions 61, 225–247 (2009) 27. Katori, M., Tanemura, H.: Noncolliding squared Bessel processes. J. Stat. Phys 142, 592–615 (2011) 28. Kuijlaars, A.B.J., Martínez-Finkelshtein, A., Wielonsky, F.: Non-intersecting squared Bessel paths and multiple orthogonal polynomials for modified Bessel weights. Commun. Math. Phys. 286, 217–275 (2009) 29. König, W., O’Connell, N.: Eigenvalues of the Laguerre process as non-colliding squared Bessel processes. Elect. Commun. Probab. 6, 107–114 (2001) 30. Soshnikov, A.: Determinantal random point fields. Russ. Math. Surv. 55, 923–975 (2000) 31. Tracy, C.A., Widom, H.: Nonintersecting Brownian excursions. Ann. Appl. Prob. 17, 953–979 (2007) 32. Warren, J., Windridge, P.: Some examples of dynamics for Gelfand Tsetlin patterns. Electron. J. Probab. 14, 1745–1769 (2008) Communicated by H. Spohn
