Gaussian Fluctuation for the Number of Particles in Airy, Bessel, Sine, and Other Determinantal Random Point Fields

Transcription

1 Journal of Statistical Physics, Vol. 00, Nos. 34, 000 Gaussian Fluctuation for the Number of Particles in Airy, Bessel, Sine, and Other Determinantal Random Point Fields Alexander B. Soshnikov Received December 3, 999 We prove the Central Limit Theorem (CLT) for the number of eigenvalues near the spectrum edge for certain Hermitian ensembles of random matrices. To derive our results, we use a general theorem, essentially due to Costin and Lebowitz, concerning the Gaussian fluctuation of the number of particles in random point fields with determinantal correlation functions. As another corollary of the CostinLebowitz Theorem we prove the CLT for the empirical distribution function of the eigenvalues of random matrices from classical compact groups. KEY WORDS: Determinantal random point fields; central limit theorem; random matrices; Airy and Bessel kernels; classical compact groups.. INTRODUCTION AND FORMULATION OF RESULTS Random Hermitian matrices were introduced in mathematical physics by Wigner in the fifties [Wig, Wig]. The main motivation of pioneers in this field was to obtain a better understanding of the statistical behavior of energy levels of heavy nuclei. An archetypical example of random matrices is the Gaussian Unitary Ensemble (G.U.E.) which can be defined by the probability distribution on a space of n-dimensional Hermitian matrices as P(dA)=const n } e &n Trace A da (.) Here da is the Lebesgue measure on n -parameters set [Re a ij,i< jn; Ima ij,i< jn, a ii,in] (.) Department of Mathematics, California Institute of Technology, Pasadena, California 95, and Department of Mathematics, University of California, Davis, Davis, California Plenum Publishing Corporation

2 49 Soshnikov and const n =(?n) &n } n(n&) is a normalization constant. (.) implies that matrix entries (.) area independent Gaussian random variables N(0, (+$ ij )8n). It is well known that the G.U.E. is the only ensemble of Hermitian random matrices (up to a trivial rescaling) that satisfies both of the following properties: () probability distribution P(dA) is invariant under unitary transformation A U & AU, U # U(n) () matrix entries up from the diagonal are independent random variables (see [Me, Chap. ]). The n eigenvalues, all real, of a Hermitian matrix A will be denoted by *, *,..., * n. For the formulas for their joint distribution density p n (*,..., * n ) and k-point correlation functions \ n, k (*,..., * k ) we refer to [Me]. One has p n (*,..., * n ) =const$ n } ` i< jn * i &* j } exp \&n } : n \ n, k (*,..., * k ):= n! (n&k)! p n (*,..., * n ) d* k+ }}}d* n R n&k i= * i (.3) + =det(k n (* i, * j )) k i, j= (.4) where K n (x, y) is a projection kernel, and n& K n (x, y)=- n } : l=0 l (x)= (&)l? 4 }( l } l!) } exp \ x l (- n x)} l (- n } y) (.5) + } d l dx l exp(&x ) (.6) l=0,,..., are WeberHermite functions. The global behavior of eigenvalues is governed by the celebrated semicircle law, which states that the empirical distribution function of the eigenvalues weakly converges to a nonrandom (Wigner) distribution: F n (*)= n *[* i*] ww w F(*)= n * \(x) dx (.7) &

3 Determinantal Random Point Fields 493 with probability one ([Wig, Wig]) where the spectral density \ is given by \(t)={? - &t, 0, t t > (.8) To study the local behavior of eigenvalues near an arbitrary point in the spectrum x # [&, ], one has to consider rescaling * i =x+ y i, i=,..., k (.9) \ n, (x) and study the rescaled k-point correlation functions R n.k ( y,..., y k ):=(\ n, (x)) &k } \ n, k (*,..., * k ) (.0) The biggest interest is paid to the asymptotics of rescaled correlation functions when n goes to infinity. For G.U.E. the answer can be obtained from the PlancherelRotach asymptotic formulas for Hermite polynomials [PR]: lim R n, k ( y,..., y k )=\ k ( y,..., y k )=det(k( y i, y j )) k i, j= (.) n The kernel K actually also depends on x but in a very simple way. It can be represented as A( y)}a$(z)&a(z) A$( y) K( y, z)= y&z (.) where for all x < the function A is just sin(?y)?, and for x=\ it is Ai(\y)=? cos 0 \ 3 t3 \yt dt (.3) + The function defined by (.3) is known as the Airy function and the kernel (.)(.3) is known as the Airy kernel (see [Me, TW, F]). The limiting correlation functions (.)(.3) determine a random point field on the real line, i.e., probability measure on the Borel _-algebra of the space of locally finite configurations, 0=[ =(y i ) i # Z : \T>0*[y i : y i <T]<] (.4)

4 494 Soshnikov The distribution of random point field is uniquely defined by the generating function k (z,..., z k ; I,..., I k )=E ` where I j, j=,..., k, are disjoint intervals on the real line, & j =*[y i # I j ]= *(I j ), the number of particles in I j, and k # Z +. It follows from the general theory of existence and uniqueness for random point fields [L, L], that if K( y, z) is locally bounded than determinantal correlation functions uniquelly determine random point field assuming that such random point field exists. The generating function (z,..., z k ) is given by Fredholm determinant of the intergal operator in L (R ): k (z,..., z k )=det($(x& y)+ : j= j= z & j j (z j &) } K(x, y)}/ Ij ( y)) (.5) where / Ij is an indicator of I j. In particular these results are applicable to the Airy kernel (.)(.3). We shall call the corresponding random point field the Airy random point field. For one-level density formulas (.)(.3) produce \ ( y)=&y }(Ai) ( y)+(ai$( y)) (.6) The asymptotic expansion of the Airy function is well known (see [Ol]). One can deduce from it? \ ( y)t{ y &cos(4 } y 3 3) +0( y &5 ) 4? } y 7 96 }?y } exp(&4y3 3) as as y & y +. (.7) \ satisfies the third order differential equation \ $$$( y)=&\ ( y)+4y } \$ ( y) (.8) One can think about the one-point correlation function as a level density, since for any interval I/R we have E*(I)= I \ ( y) dy. It follows from (.7) that E*((&T, +)) is finite for any T and E*((&T, +))t (T 3 3?)+0 () when T goes to +. The last formula means that E*((&T, +))&(T 3 3?) stays bounded for large positive T. Let us denote & (T ):=*[y i >&T]=*((&T, +)), & k (T ):=*((&kt, &(k&) T ])), k=, 3,... Theorem establishes the Central Limit Theorem for & k (T ), k.

5 Determinantal Random Point Fields 495 Theorem. The variance of & k (T ) grows logarithmically Var & k (T )t? } log T+0 () and the sequence of normalized random variables (& k (T )&E& k (T )) - Var & k (T ) converges in distribution to the centalized gaussian random sequence [! k ] with the covariance function E! k! l =$ k, l & $ k, l+& k, l&. Remark. The first result about Gaussian fluctuation of the number of particles in random matrix model was established by Costin and Lebowitz [CL] for the kernel (sin?(x& y))?(x& y). See Section for a more detailed discussion. Remark. Basor and Widom [BaW] recently proved the Central Limit Theorem for a large class of smooth linear statistics + f( y i=& it ) where f satisfies some decay and differentiality conditions. Similar results for smooth linear statistics in other random matrix ensembles were proven in [Sp], [DS], [Jo], [SiSo], [KKP], [SiSo], [Ba], [BF], [BdMK], [Br]; see also [So] for the results about global distribution of spacings. Another class of random Hermitian matrices, called Laguerre ensemble, was introduced by Bronk in [Br]. This one is the ensemble of positive n_n Hermitian matrices. Any positive Hermitian matrix H can be represented as H=AA*, where A is some complex valued n_n matrix and A* is its conjugate. The distribution on such matrices is defined as P(dH)=const" n } exp(&n } Trace A } A*) } [det(aa*)] : da (.9) where :>& and da is Lebesgue measure on n -dimensional space of complex matrices. The joint distribution of n (positive) eigenvalues of H is given by p n (*,..., * n )=const n $$$ } exp \&n } : n i= * i+ } ǹ i= * : i } ` i< jn (* i &* j ) (.0) The Vandermonde factor in (.0) implies that correlation functions still have the determinantal form (.4) with the kernel n& K n (x, y)=n } : l=0, l (nx)}, l (ny) (.)

6 496 Soshnikov where the sequence [, l (x)] is obtained by orthonormalizing the sequence [x k } x : } e &x ] on (0, +). The limiting level density is supported on [0, ] and given by the formula \ (x)=? x& }(&x) (.) (It is not surprising to see that (.) is the density of a square of the Wigner random variable!) PlancherelRotach type asymptotics for Laguerre polynomials [E], [PR] imply that by scaling the kernel K n (x, y) in the bulk of the spectrum, we obtain the sine kernel (sin?(x& y))?(x& y), and by scaling at x= (``soft edge''), we obtain the Airy kernel. As we already know, the same kernels appear after rescaling in G.U.E. This feature, called universality of local correlations, has been established recently for a variety of ensembles of Hermitian random matrices (see [PS], [BI], [DKMVZ], [Jo], [So], [BZ]). To scale the kernel at the ``hard edge'' x=0, we need an asymptotic formula of Hilb's type (see [E]), which leads to lim n 4n n\ K x 4n, y 4n+ =J :(- x)}-y } J$ : (- y)&- xj$ : (- x)}j : (- y) (x& y) (.3) where J : is the Bessel function of order : [F, TW, Ba]. The kernel (.3) is also known to appear at hard edges in the Jacobi ensemble [NW]. For a quick reference, we note that in the Jacobi case, a sequence [, l (x)] from (.) is obtained by orthonormalizing [x k (&x) : (+x) ; ]. The random point field on [0, +] with the determinantal correlation functions defined by (.3) will be referred to as the Bessel random point field. There is a general belief among people working in random matrix theory that in the same way as the sine kernel appears to be a universal limit in the bulk of the spectrum for random Hermitian matrices, Airy and Bessel kernels are universal limits at the soft and hard edge of the spectrum. The next theorem establishes the CLT for & k (T )=*(((k&) T, kt ]), k=,,... Theorem. Let &(T ) be the number of particles in (0, T ) for the Bessel random point field. Then E& k (T )t? T (k &(k&) )+0 () Var & k (T )t 4? log T+0 ()

7 Determinantal Random Point Fields 497 and the sequence of the the normalized random variable (& k (T )&E& k (T )) - Var & k (T ) converges in distribution to the gaussian random sequence from the Theorem. Theorems and, as well as similar results for the random fields arising from the classical compact groups (see Section 4) are the corollaries of the general result about determinantal random point fields, which is essentially due to Costin and Lebowitz. Recently a number of discrete determinantal random point fields appeared in two-dimensional growth models [Jo3, Jo5], asymptotics of Plancherel measures on symmetric groups and the representation theory of the infinite symmetric group [BO, BO, BOO, Jo4, Ok, Ok]. If one can show the infinite growth of the variance of the number of particles in these models (the goal which is probably attainable since the asymptotics of the discrete orthogonal polynomials arising in some of these problems are known) the CostinLebowitz theorem should work there as well. The rest of the paper is organized as follows. We discuss the general (CostinLebowitz) theorem in Section. Theorems and will be proven in Sections 3 and 4. In the Bessel case, we will see that the kernel (sin?(x& y)?(x& y))\(sin?(x+ y)?(x+ y)) naturally appears in our considerations. We recall in Section 4 that the sine kernel also appears in the limiting distribution of eigenvalues in unitary group and the even and odd sine kernels appear in the distribution of eigenvalues in orthogonal and symplectic groups and then prove the Gaussian fluctuation for the number of eigenvalues in these models in Theorem 36.. THE CENTRAL LIMIT THEOREM FOR DETERMINANTAL RANDOM POINT FIELDS Let [P t ] t # R + be a family of random point fields on Rd such that their correlation functions have determinantal form at the r.h.s. of (.) with Hermitian kernels K t ( y, z), and [I t ] a collection of Borel subsets t # R Rd. + We denote by A t an integral operator on I t with the kernel K t ( y, z), A t : L (I t ) L (I t ), by & t the number of particles in I t, & t =*(I t ), and by E t, Var t the mathematical expectation and variance with respect to the probability distribution of the random field P t. In many applications the random point field P t, and therefore the kernel K t will be the same for all t. In such situations I t will be expanding. Theorem (O. Costin, J. Lebowitz). Let A t =K t } / It be a family of trace class Hermitian operators associated with determinantal random

8 498 Soshnikov point fields [P t ] such that Var t & t =Trace(A t &A t ) goes to infinity as t +. Then the distribution of the normalized random variable (& t &E t & t )- Var t & t with respect to the random point field P t weakly converges to the normal law N(0, ). Remark 3. The result has been proven by Costin and Lebowitz t when d=, K t (x, y)=(sin?(x& y))?(x& y) for any t and I t ww (see [CL]). The original paper contains a remark, due to Widom, that the result holds for more general kernels. Remark 4. There is a general result that a (locally) trace class Hermitian operator K defines a determinantal random point field iff 0K (see [So4] or, for a slightly weaker version, [Ma]). The idea of the proof is very clear and consists of two parts. Let us denote the lth cumulant of & t by C l (& t ). We remind that by definition : l= C l (iz) l l!=log(e t exp(iz& t )) Lemma. The following recursive relation holds for any l: l& C l (& t )=(&) l }(l&)! Trace(A t &A l )+ : t s= : sl C s (& t ) (.) where : sl,sl&, are some combinatorial coefficients (irrelevant for our purposes). The proof can be found in [CL] or [So, Section ]; (of course one has to replace everywhere (sin?(x& y))?(x& y) by K t (x, y)). For the convinience of the reader we sketch the main ideas here. We start by introducing the Ursell (cluster) functions: and, in general, r (x )=\(x ), r (x, x )=\ (x, x )&\ (x ) \(x ) k r k (x,..., x k )= : m= : G m (&) m& (m&)! ` j= r Gj (x (G j )) (.) where G is a partition of indices [,,..., k] into m subgroups G,..., G m, and x (G j ) stands for the collection of x i with indices in G j.

9 Determinantal Random Point Fields 499 It appears that the integral of k-point Ursell function r k (x,..., x k ) over dk-dimensional cube I t _}}}I t is equal to the linear combination of C j (& t ), j=,..., k. Namely, let us denote T k (& t )= It }}} It r k (x,..., x k ) dx }}}dx k Then : k C k (iz) k k!= : k= (exp(z)&) k T k (& t )k! (.3) Taking into account that for the determinantal random point fields T k (& t )=(&) k }(k&)! Trace(A t ) k the last two equations imply (.). The next lemma allows us to estimate Trace(A t &A l t ). Lemma. 0Trace(A t &A l t )(l&) } Trace(A t &A t ). The proof is elementary: 0Trace(A t &A l t )=l& Trace(A j j= t j& &At & } Trace(A t &A )(l&) } Trace(A t t&a ). K t l& j= &A j+ t ) As a corollary of the lemmas we have C l (& t )=0(C (& t )) for any l. Since C (& t )=Trace(A t &A ) ww t t +, we conclude that for l>, C l ((& t &E& t )- Var t & t )=C l (& t )((C (& t )) l t ) ww 0. At the same time the first two cumulants of the normalized random variable are 0 and, respectively. The convergence of cumulants implies the convergence of moments to the moments of N(0, ). The theorem is proven. K To generalize the CostinLebowitz theorem to the case of several intervals we consider I (m) t, m=,..., s, and define & (m) t =*(I (m) t ). The equation : C k,..., k s (iz ) k k!}}}(iz s ) k s ks!=log(e t exp(i(z & () t +}}}+z s & (s) t ))) (.4) defines the joint cumulants of & (m) t 's. Proposition. C k,..., k s (& () t,..., & t (s)) is equal to a linear combination of the traces Trace K t } / (}}}) I t } K t } / (}}}) I t }}}K t } / (}}}) I t

10 500 Soshnikov with some combinatorial coefficients (irrelevant for our purposes), such that for any nonzero k j at least one indicator in each term of the linear combination is the indicator of I ( j) t. The proof immediately follows from the analogue of (.4) for the case of several intervals. In the next section we will apply these results to prove Theorem. 3. PROOF OF THEOREM For the most part of the section we will study the case of one interval (&T, +). We start by recalling the asymptotic expansion of Airy function for large positive and negative y (see [Ol]). A i ( y )t e&?z? } y } : 4 A$ i ( y )t y 4 } e &?z A i (& y )t s=0 } :? s=0? } y 4 } { cos \?z+? 4+ } : +sin \?z+? 4+ } : (&) s } u s z s (3.) (&) s } v s z s (3.) s=0 s=0 (&) s } u s z s (&) s } u s+ z s+= (3.3) 4 y A$ i (& y )t? } { sin \?z+? 4+ } : (&) s+ } v s z s s=0 &cos \?z+? 4+ } : s=0 (&) s+ } v s+ z s+= (3.4) where z=(3?) y } y ; u 0 =v 0 =, and u s =((s+) } (s+3) } }}} } (6s&))(6 }?) } s!, v s =&((6s+)(6s&)) u s, s. In particular, as a consequence of (3.)(3.4) one has (.7). It follows from (3.)(3.4) together with the boundedness of A i ( y), A$ i ( y) on any compact set that for any fixed a # R all moments of *((a, +)) are finite. Therefore it is enough to establish the CLT for *((&T, a)). We choose a=&(3?) 3 ( y=&(3?) 3 corresponds to z=&). We are going to show that the conditions of the theorem from Section are satisfied by / (&T, a) K } / (&T, a), where, as above, this notation is reserved for the integral operator with the kernel / (&T, a) (x)k(x, y)}/ (&T, a) ( y). Lemma 3. class. K 0/ (&T, a) K } / (&T, a) and / (&T, a) K } / (&T, a) is trace

11 Determinantal Random Point Fields 50 The kernel K( y, y )=(A i ( y )}A$ i ( y )&A$ i ( y )}A i ( y ))( y & y ) was obtained from K n (x, x )=- n } n& l=0 l (- nx )} l (- n } x ) after rescaling x i =+(y i n 3 ), i=,, and taking the limit n. The convergence is uniform on compact sets. As a projection operation K n satisfies 0K n. We immediately conclude that / (&T, a) K } / (&T, a) satisfies the same inequalities and since the kernel is continuous and non-negative definite the operator is trace class (see, e.g., [GK] or [RS, Section XI.4]). Now the main step of the proof consists of Proposition. Var \* \ &T, & \ 3? t? log T+0 () Proof. We introduce the chana5 of variables z i = 3? y i } y i (3.5) and agree to use the notations Q(z, z ), q k (z,..., z k ), k=,,..., for the kernel and k-point correlation function of the new random point field obtained after the change of coordinates. It follows from (.7) that q (z)t +(cos?z6?z)+0 (z & ) for z &, so we see that the configuration (z i ) is equally spaced at &. The kernel Q(z, z ) is defined by? Q(z, z )= y 4 } y } K( y, y 4 )? = y 4 } y } A i( y )}A$ i ( y )&A$ i ( y )}A i ( y ) 4 y & y Formulas (3.3)(3.4) allow us to represent Q as the sum of six kernels Q (i), i=,..., 6, with the known asymptotic expansions: Q(z, z )= 6 i= Q(i) (z, z ), where Q () (z, z )t 3? } } sin?(z z 3 &z ) &z3 } { : m, n=0 (&) m+n } u m } v n }(z &m&3 } z &n +z &n } z &m&3 ) = (3.6)

12 50 Soshnikov Q () (z, z )t 3? } } cos?(z z 3 &z 3 +z ) } { : m, n=0 (&) m+n } u m } v n }(z &n } z &m&3 &z &m&3 } z &n ) = (3.7) Q (3) (z, z )t 3? } } cos z 3 &z 3 \?z +? 4+ } cos \?z +? 4+ } { : m, n=0 (&) m+n+ } u m } v n+ }(z &m&3 } z &n& &z &n& } z &m&3 ) = (3.8) Q (4) (z, z )t 3? } } sin z 3 &z3 \?z +? 4+ } sin \?z +? 4+ } { : m, n=0 (&) m+n } u m+ } v n }(z &m&43 } z &n &z &n } z &m&43 ) = (3.9) Q (5) (z, z )t 3? } } sin?(z z 3 &z ) &z3 } { : m, n=0 (&) m+n+ } u m+ } v n+ }(z &m&43 } z &n& +z &n& } z &m&43 ) = (3.0) Q (6) (z, z )t 3? } } cos?(z z 3 +z ) &z3 } { : m, n=0 (&) m+n } u m+ } v n+ }(z &n& } z &m&43 &z &m&43 } z &n& ) = (3.)

13 Determinantal Random Point Fields 503 We denote by Q (i) m, n (z, z ) the (m, n)th term in the asymptotic expansion of Q (i) (z, z ). Then 0, 0(z, z )= sin?(z &z ) } Q () z 3 &z3 = sin?(z &z )?(z &z ) 3? }(z&3 +z &3 ) } z3 +z 3 } z 3 +z 3 3}z 3 } z 3 We note that near the diagonal Q () 0, 0 is essentially the sine kernel. We also will need Q () 0, 0(z, z )= cos?(z +z )?(z +z ) } z3 &z3 } z 3 +z3 3}z 3 } z 3 Let us define S(z, z )=Q () 0, 0 (z, z )+Q () 0, 0 (z, z ), U(z, z )=Q(z, z )& S(z, z ). Lemma 4. & & (Q () (z 0, 0, z )) dz dz =L& &L &L 3? log L+0 () (3.) Proof. The integral can be written as 9 L L \ sin?(z &z ) }?(z &z ) + \z3 +z3 } z 3 +z3 dz z 3 } z + 3 dz = 9 } L& sin?u 0 \?u + } L&u _\ z+u 3 ++ z + \ z+u+ z 3 We represent the inner integral as L&u _\ & dz du z+u 3+ } 3+ z + \z+u z + \ z+u+ z 3+ } \ z+u+ z 3 +3 & dz =I (u)+i (u)+3 } (L&u&) (3.3) with I (u)= L&u \ I (u)= L&u \ z+u } z + \z+u dz z + z+u+ z 3 + } \ z+u+ z 3 dz

14 504 Soshnikov To calculate I (u) we introduce the change of variables t=((z+u)z) 3. Then We have I (u)= (+u)3 (t +t)} \ u 3+ $ dt (L(L&u)) 3 &t =((+u) 3 + } (+u) 3 )}(&) & \\ L&u+ L 3 + } \ L&u+ L 3 + }(&L+u) +u (+u)3 (t+) } dt (3.4) (L(L&u)) 3 t 3 & and (t+) } (L(L&u)) 3 t 3 & dt u } (+u)3 =u } (+u)3 (L(L&u)) } \ t& & t+ t +t++ dt = 4u 3 } log[(+u)3 &]& 4u 3 log _\ L&u+ L 3 & & & u 3 log[(+u)3 +(+u) 3 +] + u 3 log _\ L&u+ L 3+ \ L&u+ L 3 + & (3.5) 3 3 I (u)=&(+u) 3 &(+u) 3 +L } L+ \&u +L } L+ \&u + 4u 3 log[(+u)3 &]& 4u 3 log _\ L&u+ L 3 & & + u 3 log _\ L&u+ L 3+ \ L&u+ L 3 + & & u 3 log[(+u)3 +(+u) 3 +] (3.6)

15 Determinantal Random Point Fields 505 In a similar way in order to calculate I (u)= +u\ L z&u } z + \z&u dz z + we consider the change of variables t=((z&u)z) 3. Then I (u)= (L(L&u))&3 (+u) &3 (t +t)} \ u &t 3+ $ dt 3 3 = \\L&u + } L + \L&u L + + } L &((+u) &3 + } (+u) &3 )}(+u) +u } (L(L&u))&3 (t+) } (+u) &3 t 3 & dt 3 3 =&(+u) 3 &(+u) 3 +L } L+ \&u +L } L+ \&u u } log _ & \ L&u & 3 u } log _\ L&u 3 + L + \L&u 3 L + & &4 3 u } log[&(+u)&3 ] 3 L + + & + 3 u } log[(+u)&3 +(+u) &3 +]. (3.7) It follows from (3.3) that & & (Q () (z 0, 0, z )) dz dz &L &L We note that = 9 } L& 0 \ L& 0 \ sin?u }(I?u + (u)+i (u)+3l&3u&3) du. (3.8) sin?u } Ldu=?u + L +0 (), (3.9) L& 0 (sin?u)?u du=? log L+0 (), (3.0)

16 506 Soshnikov L& 3 sin?u } L } 0 \?u + L+ \&u du L& 0 \ =L } L& 0 \ sin?u?u + u } \& 3 L +0 \ u L ++ du = L & 6? log L+0 (), (3.) 3 sin?u } L }?u + L+ \&u du = L & 3? log L+0 (). (3.) The combined contribution of all other terms to (3.8) is 0 (). Indeed, L& 0 \ L& 0 \ L& 0 \ L& 0 \ sin?u }(+u)?u + 3 =0(), sin?u }(+u)?u + 3 =0(), sin?u }?u + 4u 3 log[(+u)3 &] du& L& sin?u 0 \?u + } u 3 } log[(+u)3 +(+u) 3 +] du = 3? L& sin?u 0? u } log ((+u) _ 3 &) (+u) 3 +(+u) +& du 3 = u 3? L& 0 L& 0 \ sin?u }0(u &3 ) du=0(), u sin?u }?u + 4u 3 log[&(+u)&3 ] du=0(), sin?u }?u + u 3 log[(+u)&3 +(+u) &3 +] du=0().

17 Determinantal Random Point Fields 507 The last expression to consider is & L& 0 \ sin?u }?u u } log _\ L&u+ L 3& & } 3 u } log _\ L&u+ L 3 + \ L&u+ L } log \ L&u _& L + 3& du& L& 0 \ } log 3+ _\L&u L + \L&u L + = 3? L& 0 3+ & du+ L& 0 \ sin?u?u & du sin?u _(&((L&u)L)3 ) } log u ((L(L&u)) 3 &) _ ((L(L&u))3 +(L(L&u)) 3 +) ((L&u)L) 3 +((L&u)L) 3 +)& du = 3? L& 0 sin?u } log \ L u L&u+ du = 3? } L& u } \ &log \ &u L++ du+0 () = 3? } L& = 3? } : k= u } : k } \ u k du+0() L+ k= k +0 ()=0(). du+ L& 0 \ } 3 u sin?u?u + sin?u?u + } 4 3 u Combining all the above integrals and looking specifically for the contributions from (3.9)(3.) we obtain & & (Q () (z 0, 0, z )) dz dz &L &L = 9\ 3}L &3 }? log L+L & 6? } log L+ } L & } 3? } log L + L & 3? log L+ } L & } 6? log L+0 () + =L& 3? log L+0 ().

18 508 Soshnikov In the next lemma we evaluate the integrals involving Q () 0, 0. Lemma 5. (a) & &L & &L (Q() 0, 0, z )) dz dz =(8? ) log L+0(). (b) & &L & &L Q() 0, 0, z )}Q () 0, 0, z ) dz dz =0(). Proof. The integral in part (a) can be written as 9 L L \ cos?(z +z ) }?(z +z ) + \z3 = 9 L+ \ We denote the inner integral by I 3 (u) := u _\ =} u &z3 } z 3 z 3 } z 3 +z3 + dz dz cos?u?u + } u _\ u&z+ z 3 3 &+ \u&z dz du+0(). z + & u&z+ z 3 & \ u&z+ z \u&z & z + \u&z + z + & dz \ u&z 3 3 & z + \u&z dz+(u&) z + =} 0 (t &t)} \ u (u&) 3 t ++ $ dt+(u&) 3 =6u } (u&)3 0 = u \+6 0 (t &t)}t (t 3 +) dt+(u&) t 4 &t 3 (t 3 +) dt + +0 (u 3 ). Taking into account 0 ((t4 &t 3 )(t 3 +) ) dt=0, we obtain I 3 (u)= u+0 (u 3 ), and 9 L+ cos?u }(u+0(u \?u + 3 )) du= 8? log L+0 ()

19 Determinantal Random Point Fields 509 Now let us consider the integral in part (b): 9 L L cos?(z +z ) } sin?(z &z )?(z +z )? }(z &z ) \z3 _ +z3 } z 3 +z3 = 9 } L z 3 } z + 3 cos?u?u } u& 0 } \ z 3 &z3 } z 3 z 3 } z + 3 +z3 dz dz 3 sin?v } 3+?v u&v+ u+v+ _\u+v \u&v + dv du & It is not difficult to see that because of the oscillations of trigonometric functions this integral is of order of constant. To show this we write the inner integral as I 4 (u)+i 5 (u)+i 6 (u), where I 4 (u)= u& 0 I 5 (u)= u& 0 I 6 (u)= u& 0 3 sin?v }?v u&v+ \u+v dv 3 sin?v }?v u&v+ \u+v dv sin?v?v Integration by parts gives I 6 (u)= +0 (u & ). Consider now I 4 (u)= u&u4 0 \ + u& u&u dv sin?v?v + } \ u+v 3 dv u&v+ 4\ sin?v 3 u&v+?v + } \ u+v dv (3.3) The second integral in (3.3) is 0 (u 4 } u & } u 3 )=0 (u & ). As for the first one we introduce /([v] even), an indicator of the set where the integer part of v is even, and write the integral in the following form u&u4 0 sin?v } /([v] even)? } \ v } \ u+v 3 & u&v+ v+\ u+v+ 3 u&v&+ + dv+0 (u & )

20 50 Soshnikov The absolute value of the last expression is estimated from above by u&u4 0 sin?v? = u&u4 0 + u&u4 0 =0 (u &6 ) } }(u&v)3 &(u&v&) 3 }(v+) \v dv v }(v+) } (u&v) 3 }(u&v&) 3+ sin?v } \ (u&v) 3 &(u&v&) 3 dv? (v+) } (u&v) 3 }(u&v&) 3+ sin?v } \? v }(v+) } (u&v) 3) dv Therefore I 4 (u)=0 (u &6 ), and similarly I 5 (u)=0 (u & ). As a result 9 L sin?u?u (I 4(u)+I 5 (u)+i 6 (u)) du = 8 L sin?u?u du+ 9 L sin?u?u }0 (u & ) du=0() Lemma 5 is proven. K As a result of the last two lemmas we have & & &L & S (z, z ) dz dz =L& log L+ 3? 8? log L+0 () =L& 8? log L+0 () Remember that S was defined as S(z, z )=Q () 0, 0 (z, z )+Q () 0, 0 (z, z ). To finish the proof of the CLT for *(&T, +) we just need to show that the remainder term U(z, z )=Q(z, z )&S(z, z ) is negligible in the following sense: Lemma 6. (a) & &L & &L (z, z ) dz dz =0() (b) & &L & &L, z )}S(z, z ) dz dz =0() (c) & &L () Proof. We shall establish part (a). The estimates (b) and (c) can be obtained in a similar manner. Repeating the calculations of the last two

21 Determinantal Random Point Fields 5 lemmas, it is easy to see that for any fixed indices (i, m, n) such that (i&)(i&)+m+n>0, we have & & (Q (i) (z m, n, z )) dz dz =0() &L &L Let us now choose N to be sufficiently large and write U(z, z )= U N (z, z )+V N (z, z ), where U N (z, z )= : i= : 0<m+nN 6 Q (i) (z m, n, z )+ : i=3 : 0m+N Q (i) m, n (z, z ) We see that & &L & &L (U N(z, z )) dz dz =0 (). Asymptotic formulas (3.6)(3.) imply that V N (z, z ) const N z 3 &z3 }(z&n +z &N n ) const N z &z } 3 }(z3 +z3 )}(z&n +z &N ) (3.4) It follows from (3.4) that if we choose N then z &z (z ) (V n (z, z )) dz dz =0 () (3.5) (The integration in (3.5) is over the subset of [, L]_[, L]). Indeed, to estimate the integral over z we write z &z (z )\ z (z &z ) } z N dz + L L z 4 (z &z ) + } z3 } z 4N dz z 3 (z &z ) + } z&4 dz =0 (z &43 ) Integrating over z we arrive at (3.5). To integrate V N near the diagonal we observe that kernels Q(z, z ), Q (i) (z m, n, z ) are bounded in [, +)_ [, +); therefore there exists some const$ N such that V N (z, z ) const$ N and z &z (z ) (V N (z, z )) dz dz + (( } const$ N )z ) dz =0(). Lemma 6 is proven. K

22 5 Soshnikov Taking L=(3?) T 3 we deduce from Lemmas 46 and (3.5) that Var \ \* y i # \&T, & \ 3? = &(3?)3 &T K( y, y) dy& &(3?)3 &T &(3?)3 K ( y, y ) dy dy &T = 3? T 3 +0()& 3? T 3 + 8? log \ 3+ 3? T +0 () =? log T+0 () This finishes the proof of Proposition as well as the proof of the CLT for *(&T, +). In a very similar way one proves the CLT for arbitrary & k (T )= *((&kt, &(k&) T )), k>. To prove the result for the joint distribution of [& k (T )] we note that the decay of K(x, y) off the diagonal implies C, (& k (T ), & l (T ))=Cov(& k (T ), & l (T )) This together with implies that Var \ : k l= = &(k&) T &kt (l&) T &lt K (x, x ) dx dx =&Trace / [&lt, &(l&) T ) } K } / [&kt, &(k&) T ) } K =0 () if k&l > & l (T ) +=(? ) log T+0 () Var(& k (T ))=(? ) log T+0 (), k=,,... C, (& k (T ), & l (T ))=Cov(& k (T ), & l (T )) for k&l =. Therefore as T. E & k(t )&E& k (T ) - Var & k (T ) =&(4? ) log T+0 () } & l(t )&E& l (T ) $ k, l & $ k, l& & $ k, l+ - Var & l (T )

23 Determinantal Random Point Fields 53 To take care of the joint cumulants of higher order it is enough to prove Lemma 7. Let at least two indices in (k,..., k s ) are non-zero. Then C k,..., k s (& (T ),..., & s (T ))=0 (log T ) Proof. According to Proposition Lemma 7 follows from Lemma 8. Trace / l K/ l K }}}K/ ls K/ l =0 (log T ) where / lj are the indicators of the intervals (&l j T,&(l j &) T ], l j # Z + and at least two intervals are disjoint. Proof. This has been already established for s=. Let s>. Since not all indices coincide by cyclicity of the trace we may assume l {l. Now if l =l 3 we can use the positivity of / l K/ l K/ l to write Trace(/ l K/ l K/ l K/ l4 }}}K/ ls K/ l ) Trace(/ l K/ l K/ l )}&K/ l4 }}}K/ ls K/ l & Trace(/ l K/ l K/ l ) where we used &K&, &/ lj &. Since Trace(/ l K/ l K/ l )=0 (log T ) Lemma 8 is proven when l =l 3 {l. If l {l 3 one more trick is needed. Let us denote D =/ l K/ l, D =K/ l3 K }}}/ ls K/ l Then Trace(/ l K/ l K/ l3 }}}/ ls K/ l ) =Trace(D D )(Trace(B B *)) (Trace(D D*)) (see [RS, Vol. I, Section VI.6]). As before Trace(D D *)=Trace(/ l K/ l K/ l )=0 (log T ) To obtain a similar bound for Trace(D D *) we define <ps as the maximal index such that l p {l. Since we assume in Lemma 8 that there are at least two different indices, such p always exists. Then Trace(K/ l3 K }}}/ lp K/ l }}}K/ l }}}K/ l )}(K/ l3 K }}}/ lp K/ l }}}K/ l }}}K/ l )* =Trace(K/ l3 K }}}/ lp K/ l }}}K/ l }}}K/ l ) }(/ l K }}}/ l K }}}K/ l K/ lp }}}K/ l3 K )

24 54 Soshnikov Using the identity Trace(D D )=Trace(D D ) where D =K/ l3 K }}}/ lp& K D =/ lp K/ l K }}}/ l K/ l }}}/ l K/ lp K/ lp& K }}}K/ l3 K we can rewrite and estimate the r.h.s. as Trace(/ lp K/ l }}}/ l K }}}/ l K/ lp )}(K/ lp& }}}K/ l3 KK/ l3 K }}}/ lp& K ) Trace(/ lp K/ l }}}K/ l K }}}/ l K/ lp )}&K/ lp& }}}K/ l3 KK/ l3 K }}}/ lp& K& Here we used the positivity of / lp K/ l K/ l }}}K/ l }}}K/ l. The norm of the last factor again is not greater than. Finally, Trace(/ lp K/ l K/ l }}}K/ l }}}/ l K/ lp )=Trace(/ l K/ lp K/ l }}}K/ l }}}K/ l ) Trace(/ l K/ lp K/ l )}=0(log T ) Here we also used cyclicity of the trace. Combining the estimates for Trace D D * and Trace D D* we finish the proof of the Lemmas 7 and 8. K It follows from Lemma 7 that the higher joint cumulants of the normalized random variables go to zero which implies that the limiting distribution function is Gaussian with the known covariance function. Theorem is proven. K 4. PROOF OF THEOREM AND SIMILAR RESULTS FOR THE CLASSICAL COMPACT GROUPS The Bessel kernel has the form (see Section ) K( y, y )= J :(- y )}- y } J$ : (- y )&- y } J$ : (- y )}J : (- y ) ( y & y ) The level density is given by y, y #(0,+), :>& (4.) \ ( y)=k( y, y)= 4 J :(- y) & 4 J :+(- y)}j :& (- y) (4.) The asymptotic formula for large y is well known in the case of Bessel functions (see asymptotic expansion in (4.5) below). In particular, one can see that \ ( y)t for y + (4.3)? - y

25 Determinantal Random Point Fields 55 The last formula suggests to make the (unfolding) change of variables z i =- y i?, i=,. The kernel Q corresponding to the new evenly spaced random point field is given by Q(z, z )=?y 4 } y 4 } K( y, y ) =z z } J :(?z )}?z } J$ : (?z )&?z } J$ : (?z )}J : (?z ) z &z (4.4) The asymptotic expansion of the Bessel function at infinity is given by (see, for example, [Ol]) where J : (z)t \?z+ } \ _cos z& :?& 4? + } : (&) s } A s(:) z s s=0 &sin \z& :?& 4? + } : s=0 (&) s } A s+(:) (4.5) z & s+ A 0 (:)=, A s (:)= (4: & )}(4: &3 )}}}(4: &(s&) ) s!}8 s (4.6) Similarly, J$ : (z)t \?z+ } _ &sin \ z& :?& 4? + } : (&) s } B s(:) z s s=0 &cos \z& :?& 4? + } : s=0 (&) s } B s+(:) (4.7) z & s+ The coefficients B s (:) can be obtained from (4.5)(4.6); for example, B 0 (:)=. It follows from (4.5)(4.7) that for :=\ Q(z, z )= sin?(z &z ) + sin?(z +z &:&)?(z &z )?(z +z ) (4.8) which can be further simplified as sin?(z &z ) sin?(z +z )?(z &z )?(z +z )

26 56 Soshnikov For general values of : a small remainder term appears at the r.h.s. of (4.8). To write the asymptotic expansion, we represent Q(z, z ) as the sum of six kernels: Q(z, z )= 6 i= Q(i) (z, z ), where Q () (z, z )t sin?(z &z )?(z &z) } : (&) n+m } A n (:) B m (:) n, m=0 }(z &n } z &m +z &m } z &n ) (4.9) Q () (z, z )t sin?(z +z &:&)?(z &z) } : (&) n+m } A n (:) n, m=0 } B m (:)}(z &n } z &m &z &m } z &n ) (4.0) Q (3) (z, z )t cos(?z &(?:)&(?4)) } cos(?z &(?:)&(?4))?(z &z ) } : n, m=0 (&) n+m+ } A n (:)}B m+ (:) }(z &n } z &m +z &m } z &n ) (4.) Q (4) (z, z )t sin(?z &(?:)&(?4)) } sin(?z &(?:)&(?4))?(z &z ) } : n, m=0 (&) n+m } A n+ (:)}B m (:) }(z &&n } z &m +z &m } z &&n ) (4.) Q (5) (z, z )t sin?(z &z )?(z &z) } : (&) n+m+ } A n+ (:)}B m+ (:) n, m=0 }(z &&n } z &m +z &m } z &&n ) (4.3) Q (6) (z, z )t sin?(z +z &:&)?(z &z) } : (&) n+m+ } A n+ (:) n, m=0 } B m+ (:)}(z &&n } z &m &z &m } z &&n ) (4.4) The analysis of (4.9)(4.4) is very similar to Section 3. One can see that the only contribution to the leading term of the variance comes from

27 Determinantal Random Point Fields 57 Q () 0, 0 (z, z )+Q () 0, 0 (z, z ) which is exactly the r.h.s. of (4.8). It can be shown by a straightforward calculation that L 0 L 0 L 0 (Q () 0, 0 (z, z)+q() (z, z)) dztl+0 0, 0 () (Q () (z 0, 0, z )+Q () (z 0, 0, z ))) dz dz tl&? log L+0 () Taking into account that L=T? we finish the proof. K The kernels (sin?(x& y)?(x& y))\(sin?(x+ y)?(x+ y)) are well known in Random Matrix Theory. For one, they are the kernels of restrictions of the sine-kernel integral operator to the subspaces of even and odd functions and play an important role in spacings distribution in G.O.E. and G.S.E. [Me]. They also appear as the kernels of limiting correlation functions in orthogonal and symplectic groups near *= [So]. Let us start with the even case. Consider the normalized Haar measure on SO(n). The eigenvalues of matrix M can be arranged in pairs: exp(i% ), exp(&i% ),..., exp(i% n ), exp(&i% n ), 0%, %,..., % n? (4.5) In the rescaled coordinates near the origin x i =(n&) } (% i?), i=,..., n, the k-point correlation functions are equal to R n, k (x,..., x k )=det \ sin?(x i &x j ) (n&) } sin(? }(x i &x j )(n&)) sin?(x) i+x j ) + (n&) } sin(?(x i +x j )(n&))+ i, j=,..., k (4.6) In the limit n the kernel in (4.6) becomes (sin?(x i &x j )?(x i &x j ))+ (sin?(x i +x j )?(x i +x j )). If we consider resealing near arbitrary 0<%<? the limiting kernel will be just the sine kernel. Let us now consider the SO(n+) case. The first n eigenvalues of M # SO(n+) can be arranged in pairs as in (4.5). The last one equals. In the resealed coordinates near %=0 x i = n% i?, i=,..., n

28 58 Soshnikov the k-point correlation functions are given by the formula R n, k (x,..., x k ) =det \ sin?(x i &x j ) n } sin(? }(x i &x j )n) & sin?(x i +x j ) n } sin(? }(x i +x j )n)+i, j=,..., k (4.7) In the limit n the kernel in (4.7) becomes (sin?(x i &x j )? }(x i &x j )) &(sin?(x i +x j )? }(x i +x j )). If we again consider rescaling near 0<%<?, the limiting kernel appears to be the sine kernel. The case of symplectic group is very similar. Let M # Sp(n). Rescaled k-point correlation functions are given by R n, k (x,..., x k )=det \ sin?(x i &x j ) (n+) } sin(?(x i &x j )(n+) sin?(x i +x j ) & (n+) } sin(?(x i +x j )(n+))+i, j=,..., k (4.8) One can then deduce the following result from the CostinLebowitz Theorem. Theorem 3. Consider the normalized Haar measure on SO(n) or Sp(n). Let % #[0,?), $ n be such that 0<$ n <?&%&= for some =>0 and n } $ n +. Denote by & n the number of eigenvalues in [%, %+$ n ]. We have E& n =(n?)}$ n +0(), Var & n ={? log(n } $ n)+0 ()? log(n } $ n)+0 () if %>0 if %=0 and the normalized random variable (& n &E& n )- Var & n converges in distribution to the normal law N(0, ). Proof. Let K n (x, y) be the kernel in (4.6), (4.7), or (4.8). We observe that 0/ I K n / I Id as a composition of projection, Fourier transform, another projection and inverse Fourier transform. To check the asymptotics of Var & n is an excercise which is left to the reader. K The case of several intervals is treated in a similar fasion.

29 Determinantal Random Point Fields 59 Theorem 4. Let $ n >0 be such that $ n 0, n$ n and & k, n = *((k&) $ n, k$ n ]). Then a sequence of normalized random variables (& k, n &E& k, n )- Var & k, n converges in distribution to the centalized gaussian sequence [! k ] k= ] with the covariance function E! k! l = {$ k, l& $ k, l+ & $ k, l&, $ 0, l &- $, l, if k>0, l>0 if %=0. Finally we discuss the unitary group U(n). The eigenvalues of matrix M can be written as: exp(i% ), exp(i% ),..., exp(i% n )}d, 0%, %,..., % n? (4.9) In the rescaled coordinates x i =n }(% i?), i=,..., n, the k-point correlation functions are equal to R n, k (x,..., x k )=det \ sin?(x i &x j ) n } sin(? }(x i &x j )n)+i, j=,..., k (4.0) In the limit n the kernel in (4.0) becomes the sine kernel. We finish with the analoques of the last two theorems for U(n). Theorem 5. Consider the normalized Haar measure on U(n). Let % #[0,?), $ n be such that 0<$ n <?&%&= for some =>0 and n } $ n +. Denote by & n the number of eigenvalues in [%, %+$ n ]. We have E& n =(n?)}$ n, Var & n =? log(n } $ n)+0 () and the normalized random variable (& n &E& n )- Var & n distribution to the normal law N(0, ). converges in Theorem 6. Let $ n >0 be such that $ n 0, n$ n and & k, n = *((k&) $ n, k$ n ]). Then a sequence of normalized random variables (& k, n &E& k, n )- Var & k, n converges in distribution to the centalized gaussian sequence [! k ] k= ] with the covariance function E! k! l =$ k, l & $ k, l+ & $ k, l& Remark 5. Results similar to Theorems 5 and 6 in the regime $ n =$>0 have been also established by K. Wieand [W].

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