Eigenvalue Gaps of Random Perturbations of Large Matrices

TL;DR

This paper investigates the eigenvalue gaps of large matrices perturbed by random symmetric matrices with subgaussian entries, providing probabilistic bounds that ensure simple spectra with high probability.

Contribution

It introduces combinatorial tools to control eigenvalue gaps and improves existing probability bounds for the simplicity of the spectrum of perturbed matrices.

Findings

Eigenvalue gaps are controlled using combinatorial methods.

The spectrum of the perturbed matrix is simple with high probability.

Probability bounds are improved compared to previous results.

Abstract

The current work applies some recent combinatorial tools due to Jain to control the eigenvalue gaps of a matrix $M_n = M + N_n$ where $M$ is deterministic, symmetric with large operator norm and $N_n$ is a random symmetric matrix with subgaussian entries. One consequence of our tail bounds is that $M_n$ has simple spectrum with probability at least $1 - \exp(-n^{2/15})$ which improves on a result of Nguyen, Tao and Vu in terms of both the probability and the size of the matrix $M$.