Limiting Spectral Distribution of Sum of Unitary and Orthogonal Matrices

TL;DR

This paper proves that the eigenvalue distribution of sums of large independent Haar unitary and orthogonal matrices converges to a specific free probability measure, extending previous results and relaxing certain technical conditions.

Contribution

It establishes convergence of eigenvalue measures for sums of Haar unitary and orthogonal matrices to the Brown measure of their free sum, and relaxes conditions on the Stieltjes transform.

Findings

Eigenvalue measures converge to the Brown measure for large matrices.

Results apply to both unitary and orthogonal matrices.

The approach relaxes previous technical assumptions.

Abstract

We show that the empirical eigenvalue measure for sum of $d$ independent Haar distributed $n$-dimensional unitary matrices, converge for $n \to \infty$ to the Brown measure of the free sum of $d$ Haar unitary operators. The same applies for independent Haar distributed $n$-dimensional orthogonal matrices. As a byproduct of our approach, we relax the requirement of uniformly bounded imaginary part of Stieltjes transform of $T_n$ that is made in [Guionnet, Krishnapur, Zeitouni; Theorem 1].