Spectrum occupies pseudospectrum for random matrices with diagonal deformation and variance profile

TL;DR

This paper studies the eigenvalue distribution of large non-Hermitian random matrices with variance profiles and diagonal deformations, showing their spectrum aligns with the pseudospectrum and identifying the limiting measure as a Brown measure.

Contribution

It establishes the convergence of eigenvalue distributions to a limiting density and precisely characterizes the support as the pseudospectrum, linking it to Brown measures of deformed operators.

Findings

Eigenvalue distribution converges to a limiting density.

Support of the density coincides with the pseudospectrum.

Limiting spectral measure is identified as a Brown measure.

Abstract

We consider $n\times n$ non-Hermitian random matrices with independent entries and a variance profile, as well as an additive deterministic diagonal deformation. We show that their empirical eigenvalue distribution converges to a limiting density as $n$ tends to infinity and that the support of this density in the complex plane exactly coincides with the $\varepsilon$-pseudospectrum in the consecutive limits $n \to \infty$ and $\varepsilon \to 0$. The limiting spectral measure is identified as the Brown measure of a deformed operator-valued circular element with the help of [arXiv:2409.15405].