The Limit Empirical Spectral Distribution of Gaussian Monic Complex Matrix Polynomials

We define the empirical spectral distribution (ESD) of a random matrix polynomial with invertible leading coefficient, and we study it for complex n x n Gaussian monic matrix polynomials of degree k. We obtain exact formulae for the almost sure limit of the ESD in two distinct scenarios: (1) n -> infinity with k constant and (2) k -> infinity with n constant. The main tool for our approach is the replacement principle by Tao, Vu and Krishnapur. Along the way, we also develop some auxiliary results of potential independent interest: We slightly extend a result by Burgisser and Cucker on the tail bound for the norm of the pseudoinverse of a nonzero mean matrix, and we obtain several estimates on the singular values of certain structured random matrices.