Convergence of the Laws of Non-Hermitian Sums of Projections

TL;DR

This paper proves that the spectral distribution of certain non-Hermitian random matrices converges to the Brown measure of a sum of free Hermitian operators, extending non-Hermitian spectral analysis.

Contribution

It establishes the convergence of empirical spectral distributions of non-Hermitian sums of projections to the Brown measure, connecting random matrix models with free probability.

Findings

Spectral distributions converge to Brown measure of the sum

Use of Hermitization technique for non-Hermitian matrices

Extension of spectral convergence results to matrices with at most 2 atoms

Abstract

We consider the random matrix model $X_n = P_n + i Q_n$, where $P_n$ and $Q_n$ are independently Haar-unitary rotated Hermitian matrices with at most $2$ atoms in their spectra. Let $(M, \tau)$ be a tracial von Neumann algebra and let $p, q \in (M, \tau)$, where $p$ and $q$ are Hermitian and freely independent. Our main result is the following convergence result: if the law of $P_n$ converges to the law of $p$ and the law of $Q_n$ converges to the law of $q$, then the empirical spectral distributions of the $X_n$ converges to the Brown measure of $X = p + i q$. To prove this, we use the Hermitization technique introduced by Girko, along with the algebraic properties of projections to prove the key estimate. We also prove a converse statement by using the properties of the Brown measure of $X$.