Eigenvalues of non-hermitian random matrices and Brown measure of non-normal operators: hermitian reduction and linearization method

TL;DR

This paper develops a rigorous mathematical framework for calculating the Brown measure of non-normal operators in free probability, using hermitian reduction, subordination, and linearization techniques, to understand eigenvalue distributions of non-hermitian random matrices.

Contribution

It introduces a rigorous approach combining hermitian reduction, subordination, and linearization to compute Brown measures for a broad class of non-normal operators in free probability.

Findings

Derived explicit Brown measures for polynomials in free variables

Extended known results to triangular-elliptic elements

Provided a rigorous mathematical foundation for hermitian reduction method

Abstract

We study the Brown measure of certain non-hermitian operators arising from Voiculescu's free probability theory. Usually those operators appear as the limit in *-moments of certain ensembles of non-hermitian random matrices, and the Brown measure gives then a canonical candidate for the limit eigenvalue distribution of the random matrices. A prominent class for our operators is given by polynomials in *-free variables. Other explicit examples include R-diagonal elements and elliptic elements, for which the Brown measure was already known, and a new class of triangular-elliptic elements. Our method for the calculation of the Brown measure is based on a rigorous mathematical treatment of the hermitian reduction method, as considered in the physical literature, combined with subordination ideas and the linearization trick.