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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 ..."
Random matrices: The circular law
, 2008
"... Let x be a complex random variable with mean zero and bounded variance σ². Let Nn be a random matrix of order n with entries being i.i.d. 1 copies of x. Let λ1,..., λn be the eigenvalues of ..."
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Cited by 22 (9 self)
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Let x be a complex random variable with mean zero and bounded variance σ². Let Nn be a random matrix of order n with entries being i.i.d. 1 copies of x. Let λ1,..., λn be the eigenvalues of
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.
Invertibility of random matrices: unitary and orthogonal transformation
 Journal of the AMS
"... Abstract. We show that a perturbation of any fixed square matrix D by a random unitary matrix is well invertible with high probability. A similar result holds for perturbations by random orthogonal matrices; the only notable exception is when D is close to orthogonal. As an application, these result ..."
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Cited by 2 (1 self)
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Abstract. We show that a perturbation of any fixed square matrix D by a random unitary matrix is well invertible with high probability. A similar result holds for perturbations by random orthogonal matrices; the only notable exception is when D is close to orthogonal. As an application, these results completely eliminate a hardtocheck condition from the Single Ring Theorem by Guionnet, Krishnapur and Zeitouni. Contents
RANDOM MATRICES: UNIVERSALITY OF ESDS AND THE CIRCULAR LAW
, 2008
"... Given an n × n complex matrix A, let ..."
Local Circular Law for Random Matrices
"... The circular law asserts that the spectral measure of eigenvalues of rescaled random matrices without symmetry assumption converges to the uniform measure on the unit disk. We prove a local version of this law at any point z away from the unit circle. More precisely, if z  − 1  � τ for arbitra ..."
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The circular law asserts that the spectral measure of eigenvalues of rescaled random matrices without symmetry assumption converges to the uniform measure on the unit disk. We prove a local version of this law at any point z away from the unit circle. More precisely, if z  − 1  � τ for arbitrarily small τ> 0, the circular law is valid around z up to scale N −1/2+ε for any ε> 0 under the assumption that the distributions of the matrix entries satisfy a uniform subexponential decay condition.
The local circular law II: the edge case
"... In the first part of this article [8], we proved a local version of the circular law up to the finest scale N −1/2+ε for nonHermitian random matrices at any point z ∈ C with z  − 1 > c for any c> 0 independent of the size of the matrix. Under the main assumption that the first three moments of ..."
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In the first part of this article [8], we proved a local version of the circular law up to the finest scale N −1/2+ε for nonHermitian random matrices at any point z ∈ C with z  − 1 > c for any c> 0 independent of the size of the matrix. Under the main assumption that the first three moments of the matrix elements match those of a standard Gaussian random variable after proper rescaling, we extend this result to include the edge case z  − 1 = o(1). Without the vanishing third moment assumption, we prove that the circular law is valid near the spectral edge z  − 1 = o(1) up to scale N −1/4+ε.
SMALL PROBABILITY, INVERSE THEOREMS, AND APPLICATIONS
"... Abstract. Let ξ be a real random variable with mean zero and variance one and A = {a1,..., an} be a multiset in R d. The random sum SA: = a1ξ1 + · · · + anξn where ξi are iid copies of ξ is of fundamental importance in probability and its applications. We discuss the small ball problem, the aim o ..."
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Abstract. Let ξ be a real random variable with mean zero and variance one and A = {a1,..., an} be a multiset in R d. The random sum SA: = a1ξ1 + · · · + anξn where ξi are iid copies of ξ is of fundamental importance in probability and its applications. We discuss the small ball problem, the aim of which is to estimate the maximum probability that SA belongs to a ball with given small radius, following the discovery made by LittlewoodOfford and Erdős almost 70 years ago. We will mainly focus on recent developments that characterize the structure of those sets A where the small ball probability is relatively large. Applications of these results include full solutions or significant progresses of many open problems in different areas. Contents 1. LittlewoodOfford and Erdős estimates 2 2. High dimensional extenstions 3 3. Refinements by restrictions on A 5