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Large deviations and stochastic calculus for large random matrices
, 2004
"... Large random matrices appear in different fields of mathematics and physics such as combinatorics, probability theory, statistics, operator theory, number theory, quantum field theory, string theory etc... In the last ten years, they attracted lots of interests, in particular due to a serie of math ..."
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Cited by 13 (0 self)
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Large random matrices appear in different fields of mathematics and physics such as combinatorics, probability theory, statistics, operator theory, number theory, quantum field theory, string theory etc... In the last ten years, they attracted lots of interests, in particular due to a serie of mathematical breakthroughs allowing for instance a better understanding of local properties of their spectrum, answering universality questions, connecting these issues with growth processes etc. In this survey, we shall discuss the problem of the large deviations of the empirical measure of Gaussian random matrices, and more generally of the trace of words of independent Gaussian random matrices. We shall describe how such issues are motivated either in physics/combinatorics by the study of the socalled matrix models or in free probability by the definition of a noncommutative entropy. We shall show how classical large deviations techniques can be used in this context. These lecture notes are supposed to be accessible to non probabilists and non freeprobabilists.
On the Law of Addition of Random Matrices
 COMMUNICATIONS IN MATHEMATICAL PHYSICS
, 2000
"... Normalized eigenvalue counting measure of the sum of two Hermitian (or real symmetric) matrices An and Bn rotated independently with respect to each other by the random unitary (or orthogonal) Haar distributed matrix Un (i.e. An + U ∗ n BnUn) is studied in the limit of large matrix order n. Converg ..."
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Cited by 6 (1 self)
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Normalized eigenvalue counting measure of the sum of two Hermitian (or real symmetric) matrices An and Bn rotated independently with respect to each other by the random unitary (or orthogonal) Haar distributed matrix Un (i.e. An + U ∗ n BnUn) is studied in the limit of large matrix order n. Convergence in probability to a limiting nonrandom measure is established. A functional equation for the Stieltjes transform of the limiting measure in terms of limiting eigenvalue measures of An and Bn is obtained and studied.
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
Estimates for moments of random matrices with Gaussian elements, Séminaire de Probabilités XLI
 Lecture Notes in Mathematics
, 2008
"... We describe an elementary method to get nonasymptotic estimates for the moments of hermitian random matrices whose elements are gaussian independent random variables. As the basic example, we consider the GUE matrices. Immediate applications include GOE and the ensemble of gaussian skewsymmetric h ..."
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Cited by 5 (2 self)
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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. As the basic example, we consider the GUE matrices. Immediate applications include GOE and the ensemble of gaussian skewsymmetric hermitian matrices (GSE). The estimates we derive are asymptotically exact with respect to the first terms of 1/Nexpansions of the moments and the covariance terms. The expressions obtained show that GSE represents one more universality class different from those determined by GUE and GOE, respectively.
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.
1 Large deviations and stochastic calculus
, 2004
"... Large random matrices appear in different fields of mathematics and physics such as combinatorics, probability theory, statistics, number theory, operator theory, quantum field theory, string theory etc... In the last ten years, they attracted lots of interests, in particular due to a serie of mathe ..."
Abstract
 Add to MetaCart
Large random matrices appear in different fields of mathematics and physics such as combinatorics, probability theory, statistics, number theory, operator theory, quantum field theory, string theory etc... In the last ten years, they attracted lots of interests, in particular due to a serie of mathematical breakthroughs allowing for instance a better understanding of local properties of their spectrum, answering universality questions, connecting these issues with growth processes etc. In this survey, we shall discuss the problem of the large deviations of the empirical measure of Gaussian random matrices, and more generally of the trace of words of independent Gaussian random matrices. We shall describe how such issues are motivated either in physics/combinatorics by the study of the socalled matrix models or in free probability by the definition of a noncommutative entropy. We shall show how classical large deviations techniques can be used in this context. These lecture notes are supposed to be accessible to non probabilists and non freeprobabilists.
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 ..."
Abstract
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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
ON SPECTRAL NORM OF LARGE BAND RANDOM MATRICES A.Khorunzhy
, 2004
"... We consider the ensemble of N ×N hermitian random matrices H (N,b) whose entries are equal to zero outside of the band of width b along the principal diagonal. Inside this band the entries {Hij, i ≤ j} are given by independent identically distributed gaussian random variables with zero mean value an ..."
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We consider the ensemble of N ×N hermitian random matrices H (N,b) whose entries are equal to zero outside of the band of width b along the principal diagonal. Inside this band the entries {Hij, i ≤ j} are given by independent identically distributed gaussian random variables with zero mean value and variance v 2 /b. We study asymptotic behavior of the spectral norm ‖H (N,b) ‖ in the limit N → ∞ when the band width b is much smaller than the matrix size N but also tends to infinity. Our main result is that if b/(log N) 3 → ∞, then limsup N,b ‖H (N,b) ‖ is bounded by 2v with probability 1. To prove this, we derive a system of recurrent relations for the moments M (N,b)