On universality of local edge regime for the deformed Gaussian Unitary Ensemble

TL;DR

This paper proves the universality of local eigenvalue statistics at the spectral edge for a class of deformed Gaussian ensembles, extending understanding of eigenvalue behavior in complex random matrix models.

Contribution

It establishes universality results for local eigenvalue statistics at the spectral edge of deformed GUE matrices under broad conditions.

Findings

Universality holds near the spectral edge for the deformed GUE ensemble.

Results apply to matrices with a wide class of deterministic or random initial matrices.

Convergence of local eigenvalue statistics to universal limits is demonstrated.

Abstract

We consider the deformed Gaussian ensemble $H_n=H_n^{(0)}+M_n$ in which $H_n^{(0)}$ is a hermitian matrix (possibly random) and $M_n$ is the Gaussian unitary random matrix (GUE) independent of $H_n^{(0)}$. Assuming that the Normalized Counting Measure of $H_n^{(0)}$ converges weakly (in probability if random) to a non-random measure $N^{(0)}$ with a bounded support and assuming some conditions on the convergence rate, we prove universality of the local eigenvalue statistics near the edge of the limiting spectrum of $H_n$.