Polynomials in Asymptotically Free Random Matrices

TL;DR

This paper discusses an algorithm for calculating the asymptotic eigenvalue distributions of polynomials in free variables and extends it to non-selfadjoint polynomials, relevant for large random matrices.

Contribution

It recalls and extends a recent algorithm to compute the Brown measure for non-selfadjoint polynomials in asymptotically free random matrices.

Findings

Algorithm effectively computes eigenvalue distributions for large random matrices.

Extension to non-selfadjoint polynomials broadens applicability.

Provides a framework for analyzing complex polynomial functions in free probability.

Abstract

Recent work of Belinschi, Mai and Speicher resulted in a general algorithm to calculate the distribution of any selfadjoint polynomial in free variables. Since many classes of independent random matrices become asymptotically free if the size of the matrices goes to infinity, this algorithm allows then also the calculation of the asymptotic eigenvalue distribution of polynomials in such independent random matrices. We will recall the main ideas of this approach and then also present its extension to the case of the Brown measure of non-selfadjoint polynomials.