Random matrices: overcrowding estimates for the spectrum

TL;DR

This paper investigates overcrowding estimates for eigenvalues and singular values of random matrices, demonstrating long-range separation and near-optimal bounds even with discrete entries and perturbations.

Contribution

It introduces new methods for overcrowding estimates that work under broad conditions, including arbitrary perturbations and discrete distributions.

Findings

Long-range separation persists under arbitrary perturbations.

Nearly optimal bounds are achieved for overcrowding estimates.

Methods are effective for matrices with discrete entry distributions.

Abstract

We address overcrowding estimates for the singular values of random iid matrices, as well as for the eigenvalues of random Wigner matrices. We show evidence of long range separation under arbitrary perturbation even in matrices of discrete entry distributions. In many cases our method yields nearly optimal bounds