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Universality at the edge of the spectrum in Wigner random matrices
, 1999
"... We prove universality at the edge for rescaled correlation functions of Wigner random matrices in the limit n → +∞. As a corollary, we show that, after proper rescaling, the 1st, 2nd, 3rd, etc. eigenvalues of Wigner random hermitian (resp. real symmetric) matrix weakly converge to the distributions ..."
Abstract

Cited by 101 (7 self)
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We prove universality at the edge for rescaled correlation functions of Wigner random matrices in the limit n → +∞. As a corollary, we show that, after proper rescaling, the 1st, 2nd, 3rd, etc. eigenvalues of Wigner random hermitian (resp. real symmetric) matrix weakly converge to the distributions established by Tracy and Widom in G.U.E. (G.O.E.) cases.
ON ASYMPTOTIC EXPANSIONS AND SCALES OF SPECTRAL UNIVERSALITY IN BAND RANDOM MATRIX ENSEMBLES
, 2000
"... We consider the family of ensembles of random symmetric N × N matrices H (N,b) of the bandtype structure with characteristic length b. The variance of the matrix entries H (N,b) (x, y) is proportional to u ( x−y b) with certain decaying function u(t) ≥ 0. In the limit of (relatively) narrow band w ..."
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Cited by 5 (0 self)
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We consider the family of ensembles of random symmetric N × N matrices H (N,b) of the bandtype structure with characteristic length b. The variance of the matrix entries H (N,b) (x, y) is proportional to u ( x−y b) with certain decaying function u(t) ≥ 0. In the limit of (relatively) narrow band width 1 ≪ b ≪ N, we derive explicit expressions for the first terms of 1/bexpansions of the average of the Green function N −1 Tr(H (N,b) −z) −1 and its correlation function as well. The expressions obtained show that there exist several scales of the universal forms of the spectral correlation function. These scales are determined by the rate of decrease of the function u(t). They coincide with those detected in theoretical physics for the localization length and densitydensity correlator in the bandtype random matrix ensembles. 1 Problem, motivation and results
A simple approach to global regime of the random matrix theory
 Mathematical Results in Statistical Mechanics. Singapore: World Scientific
, 1999
"... Abstract. We discuss a method of the asymptotic computation of moments of the normalized eigenvalue counting measure of random matrices of large order. The method is based on the resolvent identity and on some formulas relating expectations of certain matrix functions and the expectations including ..."
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Cited by 4 (1 self)
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Abstract. We discuss a method of the asymptotic computation of moments of the normalized eigenvalue counting measure of random matrices of large order. The method is based on the resolvent identity and on some formulas relating expectations of certain matrix functions and the expectations including their derivatives or, equivalently, on some simple formulas of the perturbation theory. In the framework of this unique approach we obtain functional equations for the Stieltjes transforms of the limiting normalized eigenvalue counting measure and the bounds for the rate of convergence for the majority known random matrix ensembles. 1.
Real symmetric random matrices and paths counting
, 2004
"... Exact evaluation of Tr < S p> is here performed for real symmetric matrices S of arbitrary order n, where the entries are independent identically distributed random variables, with an arbitrary probability distribution, up to some integer p. These polynomials provide useful information on the spectr ..."
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Exact evaluation of Tr < S p> is here performed for real symmetric matrices S of arbitrary order n, where the entries are independent identically distributed random variables, with an arbitrary probability distribution, up to some integer p. These polynomials provide useful information on the spectral density of the ensemble in the large n limit. They also are a straightforward tool to examine a variety of rescalings of the entries in the large n limit. 1
The moments of N ×N Hermitian random matrices HN are given by expression
, 2008
"... We describe an elementary method to get nonasymptotic estimates for the moments of Hermitian random matrices whose elements are Gaussian independent random variables. We derive a system of recurrent relations for the moments and the covariance terms and develop a triangular scheme to prove the recu ..."
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We describe an elementary method to get nonasymptotic estimates for the moments of Hermitian random matrices whose elements are Gaussian independent random variables. We derive a system of recurrent relations for the moments and the covariance terms and develop a triangular scheme to prove the recurrent estimates. The estimates we obtain are asymptotically exact in the sense that they give exact expressions for the first terms of 1/Nexpansions of the moments and covariance terms. As the basic example, we consider the Gaussian Unitary Ensemble of random matrices (GUE). Immediate applications include Gaussian Orthogonal Ensemble and the ensemble of Gaussian antisymmetric Hermitian matrices. Finally we apply our method to the ensemble of N ×N Gaussian Hermitian random matrices H (N,b) whose elements are zero outside of the band of width b. The other elements are taken from GUE; the matrix obtained is normalized by b −1/2. We derive the estimates for the moments