Outlier eigenvalues for full rank deformed single ring random matrices

TL;DR

This paper studies the behavior of outlier eigenvalues in large deformed random matrices, identifying conditions for their stability and absence outside the support of the associated Brown measure.

Contribution

It provides new criteria for the stability and non-existence of outliers in full rank deformed single ring random matrices, extending previous results.

Findings

Outliers are stable under certain conditions outside the support.

No outliers are found inside the inner disk of the single ring.

Results generalize earlier work by Benaych-Georges and Rochet.

Abstract

Let $A_n$ be an $n \times n$ deterministic matrix and $\Sigma_n$ be a deterministic non-negative matrix such that $A_n$ and $\Sigma_n$ converge in $*$-moments to operators $a$ and $\Sigma$ respectively in some $W^*$-probability space. We consider the full rank deformed model $A_n + U_n \Sigma_n V_n,$ where $U_n$ and $V_n$ are independent Haar-distributed random unitary matrices. In this paper, we investigate the eigenvalues of $A_n + U_n\Sigma_n V_n$ in two domains that are outside the support of the Brown measure of $a +u \Sigma$. We give a sufficient condition to guarantee that outliers are stable in one domain, and we also prove that there are no outliers in the other domain. When $A_n$ has a bounded rank, the first domain is exactly the one outside the outer boundary of the single ring, and the second domain is the inner disk of the single ring. Our results generalize the results of Benaych-Georges and Rochet (Probab. Theory Relat. Fields, 2016).