Universality at weak and strong non-Hermiticity beyond the elliptic Ginibre ensemble

TL;DR

This paper investigates non-Gaussian extensions of the elliptic Ginibre ensemble with trace constraints, demonstrating universality of the interpolating kernel in weak non-Hermiticity and local Ginibre statistics in strong non-Hermiticity.

Contribution

It provides the first proof of universality for the interpolating kernel in non-Hermitian random matrices with trace constraints.

Findings

Universality of the interpolating kernel in weak non-Hermiticity.

Local Ginibre statistics in the bulk for strong non-Hermiticity.

Extension to non-Gaussian ensembles with correlated entries.

Abstract

We consider non-Gaussian extensions of the elliptic Ginibre ensemble of complex non-Hermitian random matrices by fixing the trace $\operatorname{Tr}(XX^*)$ of the matrix $X$ with a hard or soft constraint. These ensembles have correlated matrix entries and non-determinantal joint densities of the complex eigenvalues. We study global and local bulk statistics in these ensembles, in particular in the limit of weak non-Hermiticity introduced by Fyodorov, Khoruzhenko and Sommers. Here, the support of the limiting measure collapses to the real line. This limit was motivated by physics applications and interpolates between the celebrated sine and Ginibre kernel. Our results constitute a first proof of universality of the interpolating kernel. Furthermore, in the limit of strong non-Hermiticity, where the support of the limiting measure remains an ellipse, we obtain local Ginibre statistics in the bulk of the spectrum.