Brown measure and asymptotic freeness of elliptic and related matrices

TL;DR

This paper demonstrates that independent elliptic matrices become freely independent in the limit and computes their Brown measure, revealing the spectral distribution of their products.

Contribution

It establishes asymptotic freeness of elliptic matrices with deterministic matrices and computes the Brown measure of elliptic element products.

Findings

Elliptic matrices converge to freely independent elliptic elements.

Elliptic matrices are asymptotically free with deterministic matrices.

Brown measure of the product matches the limiting spectral distribution.

Abstract

We show that independent elliptic matrices converge to freely independent elliptic elements. Moreover, the elliptic matrices are asymptotically free with deterministic matrices under appropriate conditions. We compute the Brown measure of the product of elliptic elements. It turns out that this Brown measure is same as the limiting spectral distribution.