Limiting spectral distribution of sum of unitary and orthogonal matrices
Amir Dembo (Stanford University)
Abstract
We show that the empirical eigenvalue measure for sum of $d$ independent Haar distributed $n$-dimensional unitary matrices, converge for $n \rightarrow \infty$ 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].
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Pages: 1-19
Publication Date: August 10, 2013
DOI: 10.1214/ECP.v18-2466
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