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Random matrices: Universality of esds and the circular law
, 2009
"... Given an n × n complex matrix A, let ..."
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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 60 (14 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 53 (7 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.
The Circular Law for Random Matrices
"... We consider the joint distribution of real and imaginary parts of eigenvalues of random matrices with independent real entries with mean zero and unit variance. We prove the convergence of this distribution to the uniform distribution on the unit disc without assumptions on the existence of a densit ..."
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Cited by 46 (4 self)
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We consider the joint distribution of real and imaginary parts of eigenvalues of random matrices with independent real entries with mean zero and unit variance. We prove the convergence of this distribution to the uniform distribution on the unit disc without assumptions on the existence of a density for the distribution of entries. We assume that the entries have moment of order E Xjk  2 ϕ(x), with some positive function ϕ(x) which is growing of order (log(1 + x)) 7, or that they are sparsely nonzero. The results are based on and extend previous work of Bai, Rudelson and the authors. 1
Circular law, extreme singular values and potential theory
, 2007
"... Consider the empirical spectral distribution of complex random n×n matrix whose entries are independent and identically distributed random variables with mean zero and variance 1/n. In this paper, via applying potential theory in the complex plane and analyzing extreme singular values, we prove th ..."
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Cited by 43 (1 self)
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Consider the empirical spectral distribution of complex random n×n matrix whose entries are independent and identically distributed random variables with mean zero and variance 1/n. In this paper, via applying potential theory in the complex plane and analyzing extreme singular values, we prove that this distribution converges, with probability one, to the uniform distribution over the unit disk in the complex plane, i.e. the well known circular law, under the finite fourth moment assumption on matrix elements.
Random matrices: A general approach for the least singular value problem
, 2008
"... Let x be a complex random variable with mean zero and bounded variance. Let Nn be the random matrix of size n whose entries are iid copies of x and M be an arbitrary matrix. We give a general estimate for the least singular value of the matrix Mn: = M + Nn. In various special cases, our estimate e ..."
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Cited by 6 (4 self)
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Let x be a complex random variable with mean zero and bounded variance. Let Nn be the random matrix of size n whose entries are iid copies of x and M be an arbitrary matrix. We give a general estimate for the least singular value of the matrix Mn: = M + Nn. In various special cases, our estimate extends or refines previous known results.
THE CIRCULAR LAW PROOF OF THE REPLACEMENT PRINCIPLE
, 2009
"... It was conjectured in the early 1950s that the empirical spectral distribution (ESD) of an n × n matrix whose entries are independent and identically distributed with mean zero and variance one, normalized by a factor of 1√ n converges to the uniform distribution over the unit disk on the complex pl ..."
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It was conjectured in the early 1950s that the empirical spectral distribution (ESD) of an n × n matrix whose entries are independent and identically distributed with mean zero and variance one, normalized by a factor of 1√ n converges to the uniform distribution over the unit disk on the complex plane, which is called the circular law. The goal of this thesis is to prove the so called Replacement Principle introduced by Tao and Vu which is a crucial step in their recent proof of the circular law in full generality. It gives a general criterion for the difference of the ESDs of two normalised random matrices 1√ n An, 1√nBn to converge to 0.
RANDOM MATRICES: UNIVERSALITY OF ESDS AND THE CIRCULAR LAW
, 2008
"... Given an n × n complex matrix A, let ..."