Propagation of singular behavior for Gaussian perturbations of random matrices

TL;DR

This paper investigates how Gaussian perturbations affect the eigenvalue distribution of large Hermitian random matrices, revealing propagation of singular behaviors and universal microscopic correlations.

Contribution

It demonstrates the propagation of singular eigenvalue behaviors under Gaussian perturbations and establishes universality of microscopic correlations near singular points.

Findings

Singular behavior propagates macroscopically for small Gaussian perturbations.

Microscopic eigenvalue correlations remain universal near singular points.

Results extend to nonintersecting Brownian paths with multiple time points.

Abstract

We study the asymptotic behavior of the eigenvalues of Gaussian perturbations of large Hermitian random matrices for which the limiting eigenvalue density vanishes at a singular interior point or vanishes faster than a square root at a singular edge point. First, we show that the singular behavior propagates macroscopically for sufficiently small Gaussian perturbations, and we describe the macroscopic eigenvalue behavior for Gaussian perturbations of critical size. Secondly, for sufficiently small Gaussian perturbations of unitary invariant random matrices, we prove that the microscopic eigenvalue correlations near the singular point are described by the same limiting kernel as in the unperturbed case. We also interpret our results in terms of nonintersecting Brownian paths with random starting positions, and we establish multi-time generalizations of the microscopic results.