Circular law for random matrices with unconditional log-concave distribution

TL;DR

This paper proves that for large random matrices with log-concave, unconditional, and isotropic distributions, the eigenvalues follow the circular law, extending classical results to more general dependent entries.

Contribution

It establishes the circular law for matrices with non-i.i.d. entries drawn from a log-concave, unconditional, isotropic distribution, broadening the scope of previous results.

Findings

Empirical spectral distribution converges to the uniform law on the unit disc.

Validates circular law for matrices with dependent entries.

Extends classical results to non-i.i.d. log-concave distributions.

Abstract

We explore the validity of the circular law for random matrices with non i.i.d. entries. Let A be a random n \times n real matrix having as a random vector in R^{n^2} a log-concave isotropic unconditional law. In particular, the entries are uncorellated and have a symmetric law of zero mean and unit variance. This allows for some dependence and non equidistribution among the entries, while keeping the special case of i.i.d. standard Gaussian entries. Our main result states that as n goes to infinity, the empirical spectral distribution of n^{-1/2}A tends to the uniform law on the unit disc of the complex plane.