The sparse circular law under minimal assumptions

TL;DR

This paper proves the circular law for eigenvalues of certain random matrices with Bernoulli entries under minimal assumptions, specifically when the average number of non-zero entries per row grows to infinity.

Contribution

It establishes the circular law for matrices with Bernoulli entries under the minimal condition that the expected number of non-zero entries per row tends to infinity.

Findings

Circular law holds under minimal assumptions on Bernoulli random matrices.

Eigenvalue distribution converges to uniform measure on the unit disc.

Results extend the applicability of the circular law to sparser matrices.

Abstract

The circular law asserts that the empirical distribution of eigenvalues of appropriately normalized $n\times n$ matrix with i.i.d. entries converges to the uniform measure on the unit disc as the dimension $n$ grows to infinity. Consider an $n\times n$ matrix $A_n=(\delta_{ij}^{(n)}\xi_{ij}^{(n)})$, where $\xi_{ij}^{(n)}$ are copies of a real random variable of unit variance, variables $\delta_{ij}^{(n)}$ are Bernoulli ($0/1$) with ${\mathbb P}\{\delta_{ij}^{(n)}=1\}=p_n$, and $\delta_{ij}^{(n)}$ and $\xi_{ij}^{(n)}$, $i,j\in[n]$, are jointly independent. In order for the circular law to hold for the sequence $\big(\frac{1}{\sqrt{p_n n}}A_n\big)$, one has to assume that $p_n n\to \infty$. We derive the circular law under this minimal assumption.