Bulk Universality for Sparse Complex non-Hermitian Random Matrices

TL;DR

This paper establishes the universality of local eigenvalue statistics in the bulk for a broad class of sparse complex non-Hermitian random matrices, extending previous results to matrices with minimal moment conditions.

Contribution

It proves bulk universality for sparse complex non-Hermitian matrices with minimal moment assumptions, including those with entries as products of Bernoulli variables and complex random variables.

Findings

Universality holds for matrices with $r$-th absolute moment decaying as $N^{-1-(r-2)\epsilon}$.

Extends universality to matrices with $4+\epsilon$ moments via truncation.

Introduces a sparse multi-resolvent local law for products of resolvents.

Abstract

We prove that the local eigenvalue statistics in the bulk for complex random matrices with independent entries whose $r$-th absolute moment decays as $N^{-1-(r-2)\epsilon}$ for some $\epsilon>0$ are universal. This includes sparse matrices whose entries are the product of a Bernouilli random variable with mean $N^{-1+\epsilon}$ and an independent complex-valued random variable. By a standard truncation argument, we can also conclude universality for complex random matrices with $4+\epsilon$ moments. The main ingredient is a sparse multi-resolvent local law for products involving any finite number of resolvents of the Hermitisation and deterministic $2N\times2N$ matrices whose $N\times N$ blocks are multiples of the identity.