Circular law, Extreme Singular values and Potential theory

Abstract

Consider the empirical spectral distribution of complex random $n\times n$ 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.

20 pages, a revised version

Cited by in corpus (5)

• Random matrices: Universality of ESDs and the circular law
• Random Matrices: The circular Law
• On one generalization of the elliptic law for random matrices
• The Littlewood-Offord Problem and invertibility of random matrices
• From the Littlewood-Offord problem to the Circular Law: universality of the spectral distribution of random matrices