Circular law, extreme singular values and potential theory

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. © 2009 Elsevier Inc. All rights reserved.