The limiting spectral law for sparse iid matrices

Abstract

Let $A$ be an $n\times n$ matrix with iid entries where $A_{ij}\sim \mathrm{Ber}(p)$ is a Bernoulli random variable with parameter $p=d/n$. We show that the empirical measure of the eigenvalues converges, in probability, to a deterministic distribution as $n\to \infty$. This essentially resolves a long line of work to determine the spectral laws of iid matrices and is the first known example for non-Hermitian random matrices at this level of sparsity.

70 pages. Forum of Math Pi, to appear