Spectral properties of polynomials in independent Wigner and deterministic matrices

TL;DR

This paper investigates the spectral behavior of polynomials in independent Wigner and deterministic matrices, showing almost sure absence of eigenvalues away from certain supports and establishing strong asymptotic freeness.

Contribution

It proves the almost sure spectral gap for Hermitian polynomials and demonstrates strong asymptotic freeness between Wigner and deterministic matrices.

Findings

Eigenvalues almost surely do not appear outside the computed support

Established strong asymptotic freeness of Wigner and deterministic matrices

Results are derived using free probability tools

Abstract

On the one hand, we prove that almost surely, for large dimension, there is no eigenvalue of a Hermitian polynomial in independent Wigner and deterministic matrices, in any interval lying at some distance from the supports of a sequence of deterministic probability measures, which is computed with the tools of free probability. On the other hand, we establish the strong asymptotic freeness of independent Wigner matrices and any family of deterministic matrices with strong limiting distribution.