# On the outlying eigenvalues of a polynomial in large independent random matrices

## TL;DR

This paper studies the asymptotic behavior of eigenvalues of polynomials in large independent random matrices, identifying conditions under which outliers occur due to spikes and extending previous results to more general polynomials.

## Contribution

It extends the understanding of eigenvalue outliers for polynomials of large random matrices, using free probability tools, beyond the classical sum case.

## Key findings

- Eigenvalues of polynomial random matrices converge to a deterministic measure.
- Outliers are characterized in terms of spikes and free probability subordination functions.
- Results apply to both Hermitian and Wigner matrices.

## Abstract

Given a selfadjoint polynomial $P(X,Y)$ in two noncommuting selfadjoint indeterminates, we investigate the asymptotic eigenvalue behavior of the random matrix $P(A\_N,B\_N)$, where $A\_N$ and $B\_N$ are independent Hermitian random matrices and the distribution of $B\_N$ is invariant under conjugation by unitary operators. We assume that the empirical eigenvalue distributions of $A\_N$ and $B\_N$ converge almost surely to deterministic probability measures $\mu $ and $\nu$, respectively. In addition, the eigenvalues of $A\_N$ and $B\_N$ are assumed to converge uniformly almost surely to the support of $\mu$ and $\nu,$ respectively, except for a fixed finite number of fixed eigenvalues (spikes) of $A\_N$. It is known that almost surely the empirical distribution of the eigenvalues of $P(A\_N,B\_N)$ converges to a certain deterministic probability measure $\eta$ (sometimes denoted $\eta=P^\square(\mu,\nu)$) and, when there are no spikes, the eigenvalues of $P(A\_N,B\_N)$ converge uniformly almost surely to the support of $\eta$. When spikes are present, we show that the eigenvalues of $P(A\_N,B\_N)$ still converge uniformly to the support of $\eta$, with the possible exception of certain isolated outliers whose location can be determined in terms of $\mu,\nu,P$, and the spikes of $A\_N$. We establish a similar result when $B\_N$ is replaced by a Wigner matrix. The relation between outliers and spikes is described using the operator-valued subordination functions of free probability theory. These results extend known facts from the special case in which $P(X,Y)=X+Y$.