# Outlier eigenvalues for non-Hermitian polynomials in independent i.i.d. matrices and deterministic matrices

## TL;DR

This paper studies the behavior of outlier eigenvalues in large non-Hermitian random matrices formed by noncommutative polynomials of independent i.i.d. and deterministic matrices, identifying conditions for their asymptotic convergence.

## Contribution

It introduces conditions under which outlier eigenvalues of non-Hermitian polynomial matrices match those of the deterministic part, extending understanding of spectral outliers in complex random matrix models.

## Key findings

- Eigenvalues outside the spectrum of $P(c,a)$ are characterized asymptotically.
- Provided a sufficient condition for outliers to match eigenvalues of $P(0,A)$.
- Analyzed the spectral behavior of non-Hermitian polynomials in large random matrices.

## Abstract

We consider a square random matrix of size $N$ of the form $P(Y,A)$ where $P$ is a noncommutative polynomial, $A$ is a tuple of deterministic matrices converging in $\ast$-distribution, when $N$ goes to infinity, towards a tuple $a$ in some $\mathcal{C}^*$-probability space and $Y$ is a tuple of independent matrices with i.i.d. centered entries with variance $1/N$. We investigate the eigenvalues of $P(Y,A)$ outside the spectrum of $P(c,a)$ where $c$ is a circular system which is free from $a$. We provide a sufficient condition to guarantee that these eigenvalues coincide asymptotically with those of $P(0,A)$.