Boundaries of sine kernel universality for Gaussian perturbations of Hermitian matrices

TL;DR

This paper investigates the conditions under which eigenvalues of Gaussian-perturbed Hermitian matrices exhibit sine kernel universality, focusing on how initial data density and rigidity influence the time to universality.

Contribution

It explicitly characterizes the time scale for sine kernel universality in terms of initial data properties, especially when the initial density vanishes inside the support.

Findings

Time to universality increases with faster vanishing density.

Less rigidity in initial points delays the onset of universality.

Provides explicit descriptions of the universality boundary in terms of initial data.

Abstract

We explore the boundaries of sine kernel universality for the eigenvalues of Gaussian perturbations of large deterministic Hermitian matrices. Equivalently, we study for deterministic initial data the time after which Dyson's Brownian motion exhibits sine kernel correlations. We explicitly describe this time span in terms of the limiting density and rigidity of the initial points. Our main focus lies on cases where the initial density vanishes at an interior point of the support. We show that the time to reach universality becomes larger if the density vanishes faster or if the initial points show less rigidity.