The limit empirical spectral distribution of complex matrix polynomials

We study the empirical spectral distribution (ESD) for complex [Formula: see text] matrix polynomials of degree [Formula: see text] under relatively mild assumptions on the underlying distributions, thus highlighting universality phenomena. In particular, we assume that the entries of each matrix coefficient of the matrix polynomial have mean zero and finite variance, potentially allowing for distinct distributions for entries of distinct coefficients. We derive the almost sure limit of the ESD in two distinct scenarios: (1) [Formula: see text] with [Formula: see text] constant and (2) [Formula: see text] with [Formula: see text] bounded by [Formula: see text] for some [Formula: see text]; the second result additionally requires that the underlying distributions are continuous and uniformly bounded. Our results are universal in the sense that they depend on the choice of the variances and possibly on [Formula: see text] (if it is kept constant), but not on the underlying distributions. The results can be specialized to specific models by fixing the variances, thus obtaining matrix polynomial analogues of results known for special classes of scalar polynomials, such as Kac, Weyl, elliptic and hyperbolic polynomials.