The limit empirical spectral distribution of complex matrix polynomials

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.

arXiv admin note: text overlap with arXiv:2005.07501

References in corpus (3)

• Asymptotic distribution of complex zeros of random analytic functions
• On the spectrum of sum and product of non-hermitian random matrices
• The limit empirical spectral distribution of Gaussian monic complex matrix polynomials