Semicircle law for generalized Curie-Weiss matrix ensembles at subcritical temperature

Transcription

1 Seicircle law for generalized Curie-Weiss atrix ensebles at subcritical teperature Werner Kirsch Fakultät für Matheatik und Inforatik FernUniversität in Hagen, Gerany Thoas Kriecherbauer Matheatisches Institut Universität Bayreuth, Gerany Abstract In [13] Hochstättler, Kirsch, and Warzel showed that the seicircle law holds for generalized Curie-Weiss atrix ensebles at or above the critical teperature. We extend their result to the case of subcritical teperatures for which the correlations between the atrix entries are stronger. Nevertheless, one ay use the concept of approxiately uncorrelated ensebles that was first introduced in [13]. In order to do so one needs to reove the average agnetization of the entries by an appropriate odification of the enseble that turns out to be of rank 1 thus not changing the liiting spectral easure. 1 Introduction Hochstättler, Kirsch, and Warzel proved in [13] the seicircle law for ensebles of real syetric atrices where the upper triangular part is filled by what they called approxiately uncorrelated rando variables [13, Def. 4] (see also Definition 5 below). An iportant otivation for introducing this notion is that collections of rando variables with values in {, 1} that are distributed according to the Curie-Weiss law at or above the critical teperature are approxiately uncorrelated and thus the seicircle law holds for the corresponding atrix ensebles. It is the ain goal of the present paper to show how one ay use the concept of approxiately uncorrelated rando variables to prove a seicircle law also for subcritical teperatures. 1

2 In order to state our result precisely we need a few definitions. Curie- Weiss rando variables ξ 1,..., ξ M, also called spins, take values in {, 1} with probability P M β (ξ 1 = x 1,..., ξ M = x M ) = Z β,m e β M ( M i=1 x i) where Z β,m denotes the noralization constant. By E M β we denote the expectation with respect to P M β. The paraeter β 0 is interpreted in physics as inverse teperature, β = 1 T. Inforation on the physical eaning of the Curie-Weiss odel can be found in [6, 6]. If β = 0 the rando variables ξ i are independent while for β > 0 there is a positive correlation between the ξ i that grows with β. At the critical inverse teperature β = 1 a phase transition occurs. While at and above the critical teperature (β 1) the average spin 1 M M i=1 ξ i converges in distribution to the Dirac easure δ 0, the average spin converges to 1 (δ (β) + δ (β) ) below the critical teperature (β > 1), where = (β) (0, 1) is called the average agnetization. It can be defined as the (unique) strictly positive solution of tanh(β) =. (1) For a proof of this fact see e. g. [6] or [15]. The following observation is fundaental for the analysis of [13] and also for the present paper: Curie- Weiss distributed rando variables are of de Finetti type, i.e. they can be represented as an average of independently distributed rando variables. More precisely, for t [, 1] we denote the probability easures P (M) t = M j=1 P (1) t as the M-fold product of P (1) t with P (1) t ({1}) = 1 (1) (1 + t) and P t ({}) = 1 (1 t). () If M is clear fro the context we write P t instead of P (M) t. By E t resp. E (M) t we denote the corresponding expectation. For any function φ on {, 1} M we have ( ) E M β φ(x 1,..., X M ) = 1 ) e MF β (t)/ E t (φ(x 1,..., X M ) Z 1 t dt, with F β (t) := 1 ( 1 β ln 1 + t ) + ln(1 t ), t (, 1), (3) 1 t and noralization constant Z := e MF β(t)/ dt. 1 t In this sense P M (M) β is a weighted t average over all P t. This fact can be proved using the so called Hubbard-Stratonovich transforation (see [13] or [15] and references therein). The above considerations otivate the following definition of generalized Curie-Weiss ensebles introduced in [13, Def. 9].

3 Definition 1 Suppose α > 0, β 0 and let F β, P (M) t be defined as in (3), () above. We define a probability easure on R N by P N α F β N = P (N ) t dν N (t), dν N (t) = 1 αfβ(t)/ e N Z N 1 t dt, where Z N is chosen such that the de Finetti easure ν N becoes a probability easure (see Proposition 8 c) for noralizability of ν N ). The corresponding atrix enseble (X N ) N is then defined as follows. Pick N(N+1) different coponents of the P N α F β - distributed rando vector to fill the N upper triangular part of X N R N N. The reaining entries X N (i, j), 1 j < i N are then deterined by the syetry X N (i, j) = X N (j, i). Observe that P N α F β is invariant under perutations of coponents. Thus N the resulting atrix enseble does neither depend on the choice of the N(N+1) coponents (out of N ) nor on the order in which the upper triangular part of X N is filled. The so generated rando atrix enseble (X N ) N is called a generalized P N α F β - Curie-Weiss enseble. N Reark Note that it is only in the case α = that the N real rando variables fro which the entries of the atrix X N are chosen are indeed Curie-Weiss distributed. The generalization thus consists of allowing for α which still reains in the fraework of exchangeable rando variables. However, for α > one ay view the enseble to be generated by selecting the atrix entries fro a collection of N α Curie-Weiss distributed rando spins ±1. Observe that the definition in [13] is even ore general than Definition 1. It also allows to replace F β by ore general functions F (cf. Reark 1). Let P be the probability easure underlying soe atrix enseble (A N ) N of real syetric atrices A N R N N. Denote by λ 1... λ N the eigenvalues of A N. Then we call σ N := 1 N N i=1 δ λ i the eigenvalue distribution easure of A N. We say that the seicircle law holds for the enseble (A N ) N, if σ N converges weakly in probability to σ sc as N, i.e. for all bounded continuous functions f : R R and for all ɛ > 0 we have ( ) li P N f(x) dσ N (x) f(x) dσ sc (x) > ɛ = 0, where the seicircle σ sc is the Borel easure on R with support [, ] and (Lebesgue-) density 1 π 4 x at x [, ]. The following result is proved in [13]. Theore 3 (Theore 31 and Corollary 8 in [13]) Let (X N ) N be a generalized P N α F β - Curie-Weiss enseble. Then the seicircle law holds for N X N / N if either 3

4 (i) β [0, 1) and α 1 or (ii) β = 1 and α is satisfied. The purpose of the present paper is to extend this result to subcritical teperatures β > 1. Our ain result reads: Theore 4 Let (X N ) N be a generalized P N α F β - Curie-Weiss enseble N with α 1 and β > 1. Denote by = (β) the unique strictly positive solution of the equation tanh(β) = (cf. (1)). Then the seicircle law holds for A N := X N / N(1 ), i.e. for all bounded continuous functions f : R R and for all ɛ > 0 we have ( ) li α N PN F β f(x) dσ N N (x) f(x) dσ sc (x) > ɛ = 0, where σ N denotes the eigenvalue distribution easure of A N. As entioned above, Theore 3 was proved in [13] by introducing the concept of approxiately uncorrelated rando variables. For ensebles of real syetric atrices this ay be forulated as follows (cf. [13, Def. 4]). Definition 5 Let (Ω, F, P) denote the probability space for an enseble (X N ) N of real syetric N N atrices. We say that the entries are approxiately uncorrelated if ( l E X N (i ν, j ν ) p ρ=1 ) X N (u ρ, v ρ ) C l,p N l and for every l N there is a sequence a l,n converging to 0 as N with ( l E X N (i ν, j ν ) ) 1 a l,n (5) for all sequences (i 1, j 1 ),..., (i l, j l ) which are pairwise disjoint and disjoint to the sequences (u 1, v 1 ),..., (u p, v p ) with N-independent constants C l,p. By sequence we ean that all indices (i s, j s ), (u s, v s ) ay depend on N and that for each value of N they belong to the set {(i, j) : 1 i j N}. In [13, Theore 5] it is shown that conditions (4) and (5) are well suited to apply the original ideas of Wigner [7, 8] and Grenander [1] to prove the seicircle law via the ethod of oents (see also [, 3, 18, 1, ] and the onographs [1, 4, 3, 5]). Matrix ensebles with correlated entries have already been considered in [5, 7, 8, 10, 11, 14, 19, 4]. See [13] and the recent survey [16] for ore inforation on these results. (4) 4

5 Theore 3 follows fro [13, Theore 5] by verifying that under conditions (i) and (ii) the enseble P N α F β has approxiately uncorrelated entries. For β > 1 the situation is different. For exaple, condition (4) is N violated for any α > 0, because (cf. [13, proof of Proposition 35]) li N EN α F β N ( XN (1, 1)X N (1, ) ) = > 0, (6) where, again, denotes the average agnetization (1). Indeed, by the independence of X N (1, 1) and X N (1, ) with respect to P t we conclude E t (X N (1, 1)X N (1, )) = E t (X N (1, 1))E t (X N (1, )) = t. Moreover, and this is the crucial difference fro the case β 1, the function F β has two iniizers ± (cf. Proposition 8) so that for large values of N we have ν N 1 (δ + δ ) fro which (6) follows. Proposition 7 stated below can be used to ake this arguent rigorous. In order to prove Theore 4 we use an idea that has already been introduced in the proof of Proposition 3 in [13]. For the oent let us proceed heuristically and assue ν N = 1 (δ + δ ). Therefore we only need to consider the atrix ensebles related to P ±. These fall into the class of real syetric atrices with i.i.d. entries X N (i, j), i j that are distributed according to 1 [(1 ± )δ 1 + (1 )δ ]. Consequently E ± (X N (i, j)) = ± and E ± (XN (i, j)) = 1. Subtracting the ean and dividing by the standard deviation leads to standard Wigner ensebles X N,± := 1 1 (X N E N ), E N (i, j) := 1 for all 1 i, j N. (7) By the classical results of Wigner [7, 8] the eigenvalue distribution easures of X N± / N converge to the seicircle law σ sc when considered with respect to the probability easures P ±. Now we need to relate X N,± back to X N. To this end, observe that X N / N(1 ) is a rank 1 perturbation of both X N,± / N and we can therefore expect (see Lea 11) that the eigenvalue distribution easures of X N / N(1 ) converge to the seicircle law as well. What is still issing is an indicator when to replace the P N α F β - distributed X N N by X N,+ and when by X N,. As we will see by soe basic large deviations arguent in Proposition 9 the su of the entries is an efficient choice for that. Define S N := X N (i, j), X N,+ := 1 {SN >0}, X N, := 1 {SN 0} = 1 X N,+, 1 i j N (8) 1 Y N,± := (X N E N )X N,±, 1 Y N := Y N,+ + Y N,. (9) The core of the proof of Theore 4 is to show that the entries of Y N are approxiately uncorrelated. Since Y N X N / 1 has rank 1 our ain result is then a consequence of [13, Theore 5] and Lea 11. 5

6 We also have results on the largest and second largest singular value of generalized Curie-Weiss atrices [17], which copleent and iprove on [13]. The plan of the paper is as follows. In Section we use Laplace s ethod to ake the starting point of our heuristic arguent, ν N 1 (δ + δ ) for large values of N, precise. The reaining arguents for the proof of our ain result Theore 4 are gathered in Section 3. Acknowledgeent The second author would like to thank the Lehrgebiet Stochastics at the FernUniversität in Hagen, where ost of the work was accoplished, for support and great hospitality. The authors are grateful to Michael Fleerann for valuable suggestions. Analysis of the de Finetti easures In this section we state and prove in Lea 6 the precise version of the forula ν N 1 (δ + δ ) for the de Finetti easure that we used in the heuristic arguents at the end of the Introduction. Lea 6 Assue α > 0, β > 1 and let ν N ν N,α,β be given as in Definition 1. Recall also the definition of the average agnetization (β) in (1). Then the following holds: a) There exist nubers C, δ > 0 such that for all N N : ν N ([, α ]) Ce δn b) For all l N there exist C l > 0 such that for all N N : c) li N t l dν N (t) C l N αl (1 + t) l dν N (t) = 1 (1 ) l for all l N. Lea 6 is proved at the end of this section using Laplace s ethod. The following version is a special case of [0, Theore 7.1] (cf. [13, Proposition 4]). Proposition 7 (Laplace ethod [0]) Suppose F : (, 1) R is differentiable, φ : (, 1) R is continuous and for soe < a < b 1 we have 1. inf t [c,b) F (t) > F (a) for all c (a, b). 6

7 . As t a we have F (t) = F (a) + P (t a) κ + O ( (t a) κ+1 ) (10) φ(t) = Q (t a) λ + O ( (t a) λ ) (11) where κ, λ and P are positive constants, Q is a real constant, and (10) is differentiable. 3. For x R sufficiently large the function t e x F (t) / φ(t) belongs to L 1 (a, b). Then as x the integral I(x) := b a e x F (t) / φ(t) dt satisfies I (x) Q ( ) ( ) λ λ κ Γ κ e x F (a)/ κ xp A(x) where A(x) B(x) eans li x B(x) = 1 and Γ denotes the Gaa function. Next we suarize those properties of the function F β that are used in the proof of Lea 6. In view of Proposition 7 we need to analyze the onotonicity properties of F β. In addition we provide an estiate that is useful to establish integrability of ν N near the endpoints t = ±1. Proposition 8 Assue β > 1 and let F β, (β) be defined as in (1). Then: a) F β C 3 (, 1) is an even function, i.e. F β (t) = F β ( t) for all t (, 1). b) F β < 0 on (0, ), F β > 0 on (, 1), and F β () > 0. c) For all t (, 1) we have e F β(t)/ e 9β/ (1 t ). Proof. Stateent a) is obvious. Stateent b) follows fro a straight forward coputation for which it is useful to observe that ln 1+t 1 t = Artanh(t) and F β (t) = β(1 t ) (Artanh(t) βt). Because of the evenness of F β it suffices to prove stateent c) for 0 t < 1 only. Set X := ln(1 t) 0. Since ln(1 + t) 0 we have F β (t) = 1 8β (X + ln(1 + t)) (ln(1 + t) X) 8β X 1 X. 1 Using in addition that 8β X 3 X + 9 β 0 the clai follows. We now have all ingredients to verify Lea 6. Proof (Lea 6). We begin by evaluating the asyptotic behavior of the noring constant Z N = e N α F β (t)/ φ(t) dt with φ(t) = 1 1 t. 7

8 In order to apply Proposition 7 we need to split the doain of integration into those four regions where F β is onotone. Due to the evenness of F β (Proposition 8 a) it suffices to consider the integrals over [, 0] and [, 1). In both cases the paraeters for condition. of Proposition 7 are κ =, P = F β 1. and 3. of Proposition 7 are satisfied because of stateents b) and c) of Proposition 8. Hence ( )/ = F β ()/ > 0, λ = 1, and Q = 1/(1 ). Conditions Z N Γ( 1 ) 1 ( 4 F β ()N α ) 1 e N α F β ()/. (1) Keeping the integrand fixed one ay apply Proposition 7 also to the integral over [, 0] (now κ = 1, P = F β ( /), λ = 1, Q = 1/(1 (/) )). We arrive at e N α F β (t)/ dt 1 t Γ(1) 1 ( ) F β ( )N α e N α F β ( )/ (13) Taking the quotient of (13) and (1) one easily derives stateent a) with, say, δ = 1 (F β( ) F β()) > 0. In order to obtain the reaining clais set φ 1 (t) := t l /(1 t ) resp. φ (t) := (1 + t) l /(1 t ). Applying Proposition 7 to the intervals [, ] and [, 1) we find e N α F β (t)/ φ 1 (t) dt e N α F β (t)/ φ (t) dt l+1 Γ( ) ( 1 Γ( 1 ) ( (1 ) 1 l 4 F β ()N α 4 F β ()N α ) l+1 e N α F β ()/ ) 1 e N α F β ()/ and stateents b) and c) follow fro (1). 3 Proof of the Main Result We begin the proof with large deviations estiates that deonstrate the efficiency of the indicators X N,± defined in (8). They are iediate consequences of Hoeffding s inequality. Nevertheless, we provide a short proof for the convenience of the reader. Proposition 9 Let P t and S N be defined as in (), (8). For a (0, 1) denote q a := 1 4 log(1 a ) > 0. Then the following estiates hold true. a) For all t [a, 1] : P (N ) t (S N 0) e qan. b) For all t [, a] : P (N ) t (S N > 0) e qan. 8

9 Proof. Since P ±1 (S N = ± 1 N(N + 1)) = 1 we only need to consider t (, 1). Define λ(t) := 1 1 t log 1+t. Then E t (e λ(t)x N (1,1) ) = 1 eλ(t) (1 + t) + 1 e λ(t) (1 t) = 1 t. For t [a, 1) we have λ(t) < 0. Using in addition that the rando variables X N (i, j), 1 i j N, are independent and identically distributed with respect to the probability easure P t we obtain P t (S N 0) E t (e λ(t)s N ) = E t (e λ(t)x N (1,1) ) N(N+1) e qan. Siilarly, λ(t) > 0 for t (, a] and P t (S N > 0) E t (e λ(t)s N ) e qan. We are now ready to prove the lea which is the key for proving our ain result, i.e. to show that Y N defined in (9) has approxiately uncorrelated entries. Lea 10 Let (X N ) N be a generalized P N α F β - Curie-Weiss enseble N with α 1 and β > 1 and let Y N be defined as in (9) (see also (7)). Then (Y N ) N has approxiately uncorrelated entries with respect to the probability easure P N α F β (see Definition 5). N Proof. Let us first consider (5) and define (see (8), (9)) G N,± := X N,± Obviously, l Y N (i ν, j ν ) = X N,± l (X N,± Y N (i ν, j ν )) l = X N,± Y N,± (i ν, j ν ) = (1 ) l X N,± l (X N (i ν, j ν ) ) ( E N α F β l Y N N (i ν, j ν ) ) = E N α F β (G N N,+ ) + E N α F β (G N N, ) and E N α F β N (G N,± ) = (1 ) l E t (X N,± H N,± ) dν N (t), (14) with H N,± := l (X N(i ν, j ν ) ). Note that X N (i ν, j ν ) = 1 and that E t (X N (i ν, j ν )) = t. Since the index pairs (i 1, j 1 ),..., (i l, j l ) are assued to be pairwise disjoint we obtain E t (H N,± ) = (1 + t) l. (15) 9

10 In order to evaluate E N F β N α E t (X N,+ H N,+ ) dν N = (G N,+ ) we decopose + E t (X N,+ H N,+ ) dν N E t (X N,+ H N,+ ) dν N (16) E t (X N, H N,+ ) dν N + E t (H N,+ )dν N. Fro Proposition 9, fro Lea 6 a), and fro the trivial estiate H N,+ (1 + ) l we obtain constants q, C, δ > 0 such that for all N N: E t (X N,+ H N,+ ) dν N (1 + ) l e qn, E t (X N, H N,+ ) dν N (1 + ) l e qn, E t (X N,+ H N,+ ) dν N C(1 + ) l e δn α. Thus the first three suands on the right hand side of (16) all converge to zero as N tends to. By (14), (15) and Lea 6 c) we have proved li N EN α F β N (G N,+ ) = 1. By analogue arguents the expected value of G N, can be seen to converge to 1 as well and (5) is established. Estiate (4) can be proved in a siilar fashion. To this end redefine l p H N,± := (X N (i ν, j ν ) ) (X N (u ρ, v ρ ) ). ρ=1 The expected value in (4) equals ( (1 ) l+p 1 E t (X N,+ H N,+ ) dν N + Oberve that E t (H N,± ) = (t ) l Q ± (t) E t (X N, H N, ) dν N ). (17) with Q ± being a polynoial of degree p and Q ± (t) (1 + ) p for all t [, 1]. Using (16), the arguents thereafter, H N,+ (1 + ) l+p, and stateent b) of Lea 6 one ay establish the existence of constants C, δ, q > 0 such that for all N N : E t (X N,+ H N,+ ) dν N (1 + ) l+p (e qn + Ce δn α + C l N αl ). 10

11 Clearly, the sae estiate holds for E t(x N, H N, ) dν N and (4) is proved since we have assued α 1. The final observation we need is that rank 1 perturbations preserve the weak convergence (in probability) of the eigenvalue distribution easures (cf. [9]). Lea 11 Let (Ω, F, P) denote the probability space underlying two ensebles (A N ) N, (B N ) N of real syetric N N atrices that satisfy rank(a N (ω) B N (ω)) 1 for all ω Ω. Denote by σ N resp. by µ N the eigenvalue distribution easures of A N resp. B N. Assue furtherore that (µ N ) N converges weakly in probability to the seicircle law σ sc. Then (σ N ) N also converges weakly in probability to the seicircle law. Proof. The assuption on the rank of A N B N iplies that the eigenvalues of the two atrices interlace. Therefore we have for any interval I R and any ω Ω that #(eigenvalues of A N (ω) in I) #(eigenvalues of B N (ω) in I). (18) In order to prove Lea 11 we fix a bounded continuous function f : R R and nubers ε, γ > 0. We have to show that there exists N 0 such that for all N N 0 the estiate ( ) P f dσ N f dσ sc > ε < γ (19) holds. Choose a step function g = r i=1 α ix Ii (α i R, the intervals I i [ 4, 4] are pairwise disjoint) such that sup{ f(x) g(x) : x [ 4, 4]} in ( ε 6, 1). (0) Write f dσ N f dσ sc = 6 j=1 j, with 1 := 4 3 := 5 := 4 f dµ N f dσ sc, := 4 g dσ N g dµ N, 4 := 4 R\[ 4,4] f dσ N, (f g) dσ N, (g f) dµ N, 4 6 := f dµ N. R\[ 4,4] It suffices to show for each j = 1,..., 6 that there exists N j such that for all N N j we have P( j > ε 6 ) < γ 6 (1) 11

12 Indeed, (19) then holds for all N N 0 := ax{n j : j = 1,..., 6}. For j =, 4 estiate (1) follows trivially fro (0) with N j = 1. The case j = 1 is a consequence of the assuption of Lea 11. Estiate (18) together with (0) iply 3 r i=1 α i σ N (I i ) µ N (I i ) r( f + 1) N < ε 6 by choosing N 3 sufficiently large. Next we treat j = 6. Using the assued weak convergence of (µ N ) N again and choosing an arbitrary continuous test function f 0 : R [0, 1] with f 0 [,] = 0 and f 0 R\[ 4,4] = 1 one ay obtain N 6 such that for all N N 6 : ( ) ε P µ N (R \ [ 4, 4]) > 1( f +1) P( f 0 dµ N > ε 1( f +1) ) < γ 6. () On the one hand this iplies P( 6 > ε 1 ) < γ 6 for all N N 6. On the other hand we ay use () to handle 5. It follows fro (18) that µ N (R \ [ 4, 4]) σ N (R \ [ 4, 4]) N < ε 1( f + 1) for all N Ñ5 with a suitable choice of Ñ5. Setting N 5 := ax(ñ5, N 6 ) then copletes the proof with the help of (). After all ingredients have been gathered we ay now conclude the proof of our ain result. Proof (Theore 4). We denote by σ N resp. µ N the eigenvalue distribution easures of A N := X N / N(1 ), resp. B N := Y N / N (see (9), (7) for the definition of Y N ). Since (Y N ) N is an approxiately uncorrelated schee of rando variables it follows fro [13, Theore 5] that (µ N ) N converges weakly in probability to the seicircle law σ sc. Since A N B N = N(1 ) (X + X )E N always has rank 1, the clai now follows fro Lea 11. Reark 1 Observe that in the proof of Theore 4 we did not use the special for of the function F β but only those properties that were collected in Proposition 8. Hence the result also holds for generalized P N α F N - Curie- Weiss ensebles (cf. [13, Definition 9]), if F : (, 1) R satisfies all the properties stated in Proposition 8. Of course, condition c) ay by replaced by the requireent that e xf (t)/ dt < for sufficiently 1 t large values of x, because this is the only use of property c). 1

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14 F. Gesztesy et al (Edts.), Partial Differential Equations, Matheatical Physics, and Stochastic Analysis. A Volue in Honor of Helge Holdens 60th Birthday, EMS Congress Reports [17] W. Kirsch, T. Kriecherbauer: Largest and second largest singular values of de Finetti rando atrices; in preparation [18] V. Marchenko, L. Pastur: Distribution of eigenvalues in certain sets of rando atrices. Math. USSR-Sbornik 1, (1967). [19] M. Löwe, K. Schubert: On the liiting spectral density of rando atrices filled with stochastic processes, to appear in: Rando Operators and Stochastic Equations, arxiv: [0] F. Olver: Asyptotics and special functions, Acadeic Press (1974). [1] L. Pastur: On the spectru of rando atrices, Theoret. and Math. Phys. 10 no. 1, (197). [] L. Pastur: Spectra of rando selfadjoint operators, Russian Math. Surveys 8, 1 67 (1973). [3] L. Pastur, M. Sherbina: Eigenvalue distribution of large rando atrices, Matheatical Surveys and Monographs 171, AMS (011). [4] J. Schenker, H. Schulz-Baldes: Seicircle law and freeness for rando atrices with syetries or correlations, Matheatical Research Letters 1, (005). [5] T. Tao: Topics in rando atrix theory, AMS (01). [6] C. Thopson: Matheatical Statistical Mechanics, Princeton University Press (1979). [7] E. Wigner: Characteristic vectors of bordered atrices with infinite diension, Ann. Math. 6, (1955). [8] E. Wigner: On the distribution of the roots of certain syetric atrices, Ann. Math. 67, (1958). Werner Kirsch Thoas Kriecherbauer werner.kirsch@fernuni-hagen.de thoas.kriecherbauer@uni-bayreuth.de 14