Deformed single ring theorems

TL;DR

This paper extends the single ring theorem using free probability to describe the eigenvalue distribution of deformed random matrices, removing regularity assumptions and establishing local convergence results.

Contribution

It generalizes the single ring theorem by removing regularity assumptions and proves local eigenvalue distribution convergence for deformed matrices.

Findings

Eigenvalue distribution converges to the Brown measure of a free operator.

Removal of regularity assumptions in the single ring theorem.

Establishment of local convergence on optimal scale.

Abstract

Given a sequence of deterministic matrices $A = A_N$ and a sequence of deterministic nonnegative matrices $\Sigma=\Sigma_N$ such that $A\to a$ and $\Sigma\to \sigma$ in $\ast$-distribution for some operators $a$ and $\sigma$ in a finite von Neumann algebra $\mathcal{A}$. Let $U =U_N$ and $V=V_N$ be independent Haar-distributed unitary matrices. We use free probability techniques to prove that, under mild assumptions, the empirical eigenvalue distribution of $U\Sigma V^*+A$ converges to the Brown measure of $T+a$, where $T\in\mathcal{A}$ is an $R$-diagonal operator freely independent from $a$ and $\vert T\vert$ has the same distribution as $\sigma$. The assumptions can be removed if $A$ is Hermitian or unitary. By putting $A= 0$, our result removes a regularity assumption in the single ring theorem by Guionnet, Krishnapur and Zeitouni. We also prove a local convergence on optimal scale, extending the local single ring theorem of Bao, Erd\H{o}s and Schnelli.