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Poisson statistics for the largest eigenvalues of Wigner random matrices with heavy tails
 Elec. Commun. Probab
, 2004
"... We study large Wigner random matrices in the case when the marginal distributions of matrix entries have heavy tails. We prove that the largest eigenvalues of such matrices have Poisson statistics. 1 Introduction and Formulation of Results. The main goal of this paper is to study the largest eigenva ..."
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Cited by 14 (0 self)
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We study large Wigner random matrices in the case when the marginal distributions of matrix entries have heavy tails. We prove that the largest eigenvalues of such matrices have Poisson statistics. 1 Introduction and Formulation of Results. The main goal of this paper is to study the largest eigenvalues of Wigner real symmetric and Hermitian random matrices in the case when the matrix entries have heavy tails of distribution. We remind that a real symmetric Wigner random matrix is defined as a square symmetric n × n matrix with i.i.d. entries up from the diagonal
A generalization of the Lindeberg principle
 Annals Probab
, 2006
"... We present a generalization of Lindeberg’s method of proving the central limit theorem to encompass general smooth functions (instead of just sums) and dependent random variables. The technique is then used to obtain an invariance result for smooth functions of exchangeable random variables. As an i ..."
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Cited by 12 (0 self)
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We present a generalization of Lindeberg’s method of proving the central limit theorem to encompass general smooth functions (instead of just sums) and dependent random variables. The technique is then used to obtain an invariance result for smooth functions of exchangeable random variables. As an illustrative application of this theorem, we then establish “convergence to Wigner’s law ” for eigenspectra of matrices with exchangeable random entries. 1 Introduction and
A simple invariance theorem
, 2004
"... This is an old article (from May 2004), that will probably not be published, because a much improved paper with new results is in preparation. Still, I decided to put it in the archive because there are some things of interest here (in particular, the section on the SK model) which will not appear ..."
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Cited by 7 (1 self)
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This is an old article (from May 2004), that will probably not be published, because a much improved paper with new results is in preparation. Still, I decided to put it in the archive because there are some things of interest here (in particular, the section on the SK model) which will not appear in the new paper. We present a simple extension of Lindeberg’s argument for the Central Limit Theorem to get a general invariance result. We apply the technique to prove results from random matrix theory, spin glasses, and maxima of random fields. 1 Introduction and
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.
Concentration of the Spectral Measure for Large Random Matrices with Stable Entries
, 2007
"... We derive concentration inequalities for functions of the empirical measure of large random matrices with infinitely divisible entries and, in particular, stable ones. We also give concentration results for some other functionals of these random matrices, such as the largest eigenvalue or the larges ..."
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Cited by 1 (0 self)
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We derive concentration inequalities for functions of the empirical measure of large random matrices with infinitely divisible entries and, in particular, stable ones. We also give concentration results for some other functionals of these random matrices, such as the largest eigenvalue or the largest singular value.
Corrections To The Green Functions Of Random Matrices With Independent Entries
, 1995
"... We propose a general approach to the construction of 1=N corrections to the Green function GN (z) of the ensembles of random real symmetric and Hermitean N \Theta N matrices with independent entries H k;l . By this approach we study the correlation function CN (z 1 ; z 2 ) of normalized trace N \G ..."
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We propose a general approach to the construction of 1=N corrections to the Green function GN (z) of the ensembles of random real symmetric and Hermitean N \Theta N matrices with independent entries H k;l . By this approach we study the correlation function CN (z 1 ; z 2 ) of normalized trace N \Gamma1 Tr GN assuming that the average of jH k;l j 5 is bounded. We found that to the leading order CN (z 1 ; z 2 ) = N \Gamma2 F (z 1 ; z 2 ), where F (z 1 ; z 2 ) depends only on the second and the fourth moments of H k;l . For the correlation function of the density of energy levels we obtain an expression which, in the scaling limit depends only on the second moment of H k;l . This can be viewed as a support to the universality conjecture of random matrix theory 1. Random N \Theta N matrices with independent entries were introduced by E.Wigner [1]. The majority of rigorous results for these matrices concerns the proof of convergence of the density ae N (E) of their eigenvalues to t...
Results
, 2003
"... 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 (or real symmetric) matrix weakly converge to the distributions est ..."
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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 (or real symmetric) matrix weakly converge to the distributions established by Tracy and Widom in G.U.E. (G.O.E.) cases.
Fluct. Noise Lett. 2008.08:L341L348. Downloaded from www.worldscientific.com by UNIVERSITY OF PARMA Biblioteca Politecnica on 10/08/12. For personal use only. FLUCTUATIONS INDUCE TRANSITIONS IN FRUSTRATED SPARSE NETWORKS
"... With the aim of describing a general benchmark for several complex systems, we analyze, by means of statistical mechanics, a sparse network with random competitive interactions among dichotomic variables pasted on the nodes. The model is described by an infinite series of order parameters (the multi ..."
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With the aim of describing a general benchmark for several complex systems, we analyze, by means of statistical mechanics, a sparse network with random competitive interactions among dichotomic variables pasted on the nodes. The model is described by an infinite series of order parameters (the multioverlaps) and has two tunable degrees of freedom: the noise level and the connectivity (the averaged number of links). We show that there are no multiple transition lines, one for every order parameter, as a naive approach would suggest, but just one corresponding to ergodicity breaking. We explain this scenario within a novel and simple mathematical technique via a driving mechanism such that, as the first order parameter (the two replica overlap) becomes different from zero due to a real second order phase transition (with properly associated diverging rescaled fluctuations), it enforces all the other multioverlaps toward positive values thanks to the strong correlations which develop among themselves and the two replica overlap at the critical line.