The circular law for random matrices

TL;DR

This paper proves the circular law for eigenvalues of large random matrices with independent entries, including sparse matrices, showing their eigenvalue distribution converges to uniform on the unit disc.

Contribution

It extends the circular law to a broader class of sparse matrices without requiring a density for entry distributions.

Findings

Eigenvalue distribution converges to uniform on the unit disc.

Valid for matrices with finite moments greater than two.

Includes sparse matrices in the analysis.

Abstract

We consider the joint distribution of real and imaginary parts of eigenvalues of random matrices with independent 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 a finite moment of order larger than two and consider the case of sparse matrices. The results are based on previous work of Bai, Rudelson and the authors extending those results to a larger class of sparse matrices.