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

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 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)