On the spectral edge of non-Hermitian random matrices

TL;DR

This paper proves universality of local eigenvalue statistics at the spectral edge for non-Hermitian random matrices with deterministic deformations, and shows the spectrum lacks outliers beyond natural fluctuation scales, ensuring deterministic eigenvalue counts.

Contribution

It establishes universality at the spectral edge for non-Hermitian matrices with deterministic perturbations and rules out large outliers under natural conditions.

Findings

Eigenvalue statistics near spectral edges are universal.

Spectrum has no outliers beyond fluctuation scale.

Eigenvalue counts in spectral components are deterministic.

Abstract

For general non-Hermitian random matrices $X$ and deterministic deformation matrices $A$, we prove that the local eigenvalue statistics of $A+X$ close to the typical edge points of its spectrum are universal. Furthermore, we show that under natural assumptions on $A$ the spectrum of $A+X$ does not have outliers at a distance larger than the natural fluctuation scale of the eigenvalues. As a consequence, the number of eigenvalues in each component of $\mathrm{Spec}(A+X)$ is deterministic.