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40
Nonasymptotic theory of random matrices: extreme singular values
 PROCEEDINGS OF THE INTERNATIONAL CONGRESS OF MATHEMATICIANS
, 2010
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Random matrices: Universality of esds and the circular law
, 2009
"... Given an n × n complex matrix A, let ..."
PDEs for the Gaussian ensemble with external source and the Pearcey distribution
 MOTIONS, INTEGRABLE SYSTEMS AND ORTHOGONAL POLYNOMIALS 395
, 2007
"... ..."
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
Bulk universality for Wigner hermitian matrices with subexponential decay
"... Abstract. We consider the ensemble of n × n Wigner hermitian matrices H = (hℓk)1≤ℓ,k≤n that generalize the Gaussian unitary ensemble (GUE). The matrix elements hkℓ = ¯ hℓk are given by hℓk = n −1/2 (xℓk + √ −1yℓk), where xℓk, yℓk for 1 ≤ ℓ < k ≤ n are i.i.d. random variables with mean zero and vari ..."
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Cited by 12 (5 self)
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Abstract. We consider the ensemble of n × n Wigner hermitian matrices H = (hℓk)1≤ℓ,k≤n that generalize the Gaussian unitary ensemble (GUE). The matrix elements hkℓ = ¯ hℓk are given by hℓk = n −1/2 (xℓk + √ −1yℓk), where xℓk, yℓk for 1 ≤ ℓ < k ≤ n are i.i.d. random variables with mean zero and variance 1/2, yℓℓ = 0 and xℓℓ have mean zero and variance 1. We assume the distribution of xℓk, yℓk to have subexponential decay. In [3], four of the authors recently established that the gap distribution and averaged kpoint correlation of these matrices were universal (and in particular, agreed with those for GUE) assuming additional regularity hypotheses on the xℓk, yℓk. In [7], the other two authors, using a different method, established the same conclusion assuming instead some moment and support conditions on the xℓk, yℓk. In this short note we observe that the arguments of [3] and [7] can be combined to establish universality of the gap distribution and averaged kpoint correlations for all Wigner matrices (with subexponentially decaying entries), with no extra assumptions. 1.
From the LittlewoodOfford problem to the Circular Law: Universality of the spectral distribution of random matrices
 BULL. AMER. MATH. SOC
, 2009
"... The famous circular law asserts that if Mn is an n×n matrix with iid complex entries of mean zero and unit variance, then the empirical spectral distribution of the normalized matrix 1 √ Mn converges both in probability and n almost surely to the uniform distribution on the unit disk {z ∈ C: z  ≤1 ..."
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Cited by 11 (1 self)
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The famous circular law asserts that if Mn is an n×n matrix with iid complex entries of mean zero and unit variance, then the empirical spectral distribution of the normalized matrix 1 √ Mn converges both in probability and n almost surely to the uniform distribution on the unit disk {z ∈ C: z  ≤1}. After a long sequence of partial results that verified this law under additional assumptions on the distribution of the entries, the circular law is now known to be true for arbitrary distributions with mean zero and unit variance. In this survey we describe some of the key ingredients used in the establishment of the circular law at this level of generality, in particular recent advances in understanding the LittlewoodOfford problem and its inverse.
Central limit theorem for linear eigenvalue statistics of random matrices with . . .
, 2009
"... We consider n × n real symmetric and Hermitian Wigner random matrices n −1/2 W with independent (modulo symmetry condition) entries and the (null) sample covariance matrices n −1 X ∗ X with independent entries of m × n matrix X. Assuming first that the 4th cumulant (excess) κ4 of entries of W and X ..."
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Cited by 9 (0 self)
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We consider n × n real symmetric and Hermitian Wigner random matrices n −1/2 W with independent (modulo symmetry condition) entries and the (null) sample covariance matrices n −1 X ∗ X with independent entries of m × n matrix X. Assuming first that the 4th cumulant (excess) κ4 of entries of W and X is zero and that their 4th moments satisfy a Lindeberg type condition, we prove that linear statistics of eigenvalues of the above matrices satisfy the central limit theorem (CLT) as n → ∞, m → ∞, m/n → c ∈ [0, ∞) with the same variance as for Gaussian matrices if the test functions of statistics are smooth enough (essentially of the class C 5). This is done by using a simple “interpolation trick ” from the known results for the Gaussian matrices and the integration by parts, presented in the form of certain differentiation formulas. Then, by using a more elaborated version of the techniques, we prove the CLT in the case of nonzero excess of entries again for essentially C 5 test function. Here the variance of statistics contains an additional term proportional to κ4. The proofs of all limit theorems follow essentially the same scheme.
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
Dyson’s nonintersecting Brownian motions with a few outliers
 Comm. Pure Appl. Math., online
, 2008
"... 1 A constrained Brownian motion with a few outliers 8 2 The existence of the rAiry process and the rAiry kernel 15 3 An integrable deformation of Gaussian random ensemble with ..."
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Cited by 6 (3 self)
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1 A constrained Brownian motion with a few outliers 8 2 The existence of the rAiry process and the rAiry kernel 15 3 An integrable deformation of Gaussian random ensemble with