Outlier eigenvalues for non-Hermitian polynomials in independent i.i.d. matrices and deterministic 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 ∗-distribution, when N goes to infinity, towards a tuple a in some ${\mathcal{C}}^{\ast }$-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 $P(0,A)$.