The limit empirical spectral distribution of complex matrix polynomials

TL;DR

This paper investigates the asymptotic behavior of the empirical spectral distribution of complex matrix polynomials, providing exact formulas for their limits in different growth regimes of matrix size and polynomial degree.

Contribution

It derives explicit formulas for the limiting spectral distribution of complex matrix polynomials in two key asymptotic regimes, advancing understanding of their spectral properties.

Findings

Exact limit formulas for ESD as n -> ∞ with fixed k

Exact limit formulas for ESD as k -> ∞ with bounded n

Utilization of logarithmic potential and singular value estimates

Abstract

We study the empirical spectral distribution (ESD) for complex n x n matrix polynomials of degree k. We obtain exact formulae for the almost sure limit of the ESD in two distinct scenarios: (1) n -> \infty with k constant and (2) k -> \infty with n bounded by O(k^P) for some P>0. The main tools used are the logarithmic potential of some measure related to the matrix polynomial, and some classical estimates on the singular values of full random matrices with i.i.d. entries.