Wegner estimate and upper bound on the eigenvalue condition number of non-Hermitian random matrices

TL;DR

This paper establishes near-optimal bounds on the local eigenvalue density and the expected eigenvalue condition number for a broad class of non-Hermitian random matrices, advancing understanding of their spectral stability.

Contribution

It proves a Wegner estimate and bounds on the eigenvalue condition number for non-Hermitian matrices of the form X+A, improving previous bounds and providing new tail estimates for small singular values.

Findings

Eigenvalue density bounded by N^{1+o(1)}

Expected eigenvalue condition number bounded by N^{1+o(1)}

Improved bounds over previous results

Abstract

We consider $N\times N$ non-Hermitian random matrices of the form $X+A$, where $A$ is a general deterministic matrix and $\sqrt{N}X$ consists of independent entries with zero mean, unit variance, and bounded densities. For this ensemble, we prove (i) a Wegner estimate, i.e. that the local density of eigenvalues is bounded by $N^{1+o(1)}$ and (ii) that the expected condition number of any bulk eigenvalue is bounded by $N^{1+o(1)}$; both results are optimal up to the factor $N^{o(1)}$. The latter result complements the very recent matching lower bound obtained in [15] (arXiv:2301.03549) and improves the $N$-dependence of the upper bounds in [5,6,32] (arXiv:1906.11819, arXiv:2005.08930, arXiv:2005.08908). Our main ingredient, a near-optimal lower tail estimate for the small singular values of $X+A-z$, is of independent interest.