Towards the bulk universality of non-Hermitian random matrices

TL;DR

This paper investigates the universality of local eigenvalue statistics in large non-Hermitian random matrices, demonstrating that they match those of the Ginibre ensemble in the bulk, by analyzing the Hermitized resolvent.

Contribution

It reduces the problem of bulk universality for non-Hermitian matrices to a microscopic regime analysis of the Hermitized resolvent, simplifying the overall proof.

Findings

Eigenvalue statistics in the bulk are universal and match Ginibre ensemble.

The microscopic regime of the Hermitized resolvent governs bulk universality.

Other regimes are shown to be negligible in the analysis.

Abstract

We consider the non-Hermitian analogue of the celebrated Wigner-Dyson-Mehta bulk universality phenomenon, i.e. that in the bulk the local eigenvalue statistics of a large random matrix with independent, identically distributed centred entries are universal, in particular they asymptotically coincide with those of the Ginibre ensemble in the corresponding symmetry class. In this paper we reduce this problem to understanding a certain microscopic regime for the Hermitized resolvent in Girko's formula by showing that all other regimes are negligible.