Limiting Spectral Distribution of Sum of Unitary and Orthogonal Matrices

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].

17 pages; changes in the presentation style, several minor changes in proofs

References in corpus (1)

• Random regularization of Brown spectral measure