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