Results 1 
7 of
7
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 ..."
Abstract

Cited by 22 (9 self)
 Add to MetaCart
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
Circular law, extreme singular values and potential theory, arXiv:0705.3773v2 [math.PR
, 2007
"... Abstract. 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 ..."
Abstract

Cited by 18 (0 self)
 Add to MetaCart
Abstract. 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. 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 ..."
Abstract

Cited by 11 (1 self)
 Add to MetaCart
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 ..."
Abstract

Cited by 10 (0 self)
 Add to MetaCart
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
Random matrices: A general approach for the least singular value problem, preprint
"... Abstract. 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 es ..."
Abstract

Cited by 5 (4 self)
 Add to MetaCart
Abstract. 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. 1.
RANDOM MATRICES: UNIVERSALITY OF ESDS AND THE CIRCULAR LAW
, 2008
"... Given an n × n complex matrix A, let ..."