The circular law for sparse non-Hermitian matrices

TL;DR

This paper proves that the eigenvalue distribution of certain sparse non-Hermitian matrices converges to the circular law under specific sparsity conditions, extending to directed Erdős-Rényi graphs.

Contribution

It establishes the circular law for sparse non-Hermitian matrices with new bounds on the smallest singular value, covering a range of sparsity levels.

Findings

Empirical spectral distribution converges to the circular law for p_n=( )

Almost sure convergence when np_n > ( )

Extension to adjacency matrices of directed Erd
53s-Re9nyi graphs

Abstract

For a class of sparse random matrices of the form $A_n =(\xi_{i,j}\delta_{i,j})_{i,j=1}^n$, where $\{\xi_{i,j}\}$ are i.i.d.~centered sub-Gaussian random variables of unit variance, and $\{\delta_{i,j}\}$ are i.i.d.~Bernoulli random variables taking value $1$ with probability $p_n$, we prove that the empirical spectral distribution of $A_n/\sqrt{np_n}$ converges weakly to the circular law, in probability, for all $p_n$ such that $p_n=\omega({\log^2n}/{n})$. Additionally if $p_n$ satisfies the inequality $np_n > \exp(c\sqrt{\log n})$ for some constant $c$, then the above convergence is shown to hold almost surely. The key to this is a new bound on the smallest singular value of complex shifts of real valued sparse random matrices. The circular law limit also extends to the adjacency matrix of a directed Erd\H{o}s-R\'{e}nyi graph with edge connectivity probability $p_n$.