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Universality Of The Local Eigenvalue Statistics For A Class Of Unitary Invariant Random Matrix Ensembles
, 1997
"... The paper is devoted to the rigorous proof of the universality conjecture of the random matrix theory, according to which the limiting eigenvalue statistics of n \Theta n random matrices within spectral intervals of the order O(n \Gamma1 ) is determined by the type of matrices (real symmetric, Her ..."
Abstract

Cited by 55 (4 self)
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The paper is devoted to the rigorous proof of the universality conjecture of the random matrix theory, according to which the limiting eigenvalue statistics of n \Theta n random matrices within spectral intervals of the order O(n \Gamma1 ) is determined by the type of matrices (real symmetric, Hermitian or quaternion real) and by the density of states. We prove this conjecture for a certain class of the Hermitian matrix ensembles that arose in the quantum field theory and have the unitary invariant distribution defined by a certain function (the potential in the quantum field theory) satisfying some regularity conditions. Key words: random matrices, local asymptotic regime, universality conjecture, orthogonal polynomial technique. 1 Introduction. Problem and results. The random matrix theory (RMT) has been extensively developed and used in a number of areas of theoretical and mathematical physics. In particular the theory provides quite satisfactory description of fluctuations in s...
SPECTRAL AND PROBABALISTIC ASPECTS OF MATRIX MODELS
, 1995
"... The paper deals with the eigenvalue statistics of n n random Hermitian matrices as n!1.Weconsider a certain class of unitary invariant matrix probability distributions which havebeen actively studied in recent years in the quantum eld theory (QFT). These ensembles are natural extensions of the arche ..."
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The paper deals with the eigenvalue statistics of n n random Hermitian matrices as n!1.Weconsider a certain class of unitary invariant matrix probability distributions which havebeen actively studied in recent years in the quantum eld theory (QFT). These ensembles are natural extensions of the archetype Gaussian ensemble well known and widely studied in the eld called random matrix theory (RMT) and having applications in a number of areas of physics and mathematics. Our goal is to analyze the QFT motivated matrix ensembles from the point of view of the RMT. We consider the normalized counting functions of matrix eigenvalues (NCF), discuss the RMT content ofvarious physical results (limiting form of the NCF, the eigenvalue spacing distribution, etc.), present rigorous versions and extensions some of them and other rigorous results, and discuss open mathematical problems, conjectures, and links with other areas. 1