On the Law of Addition of Random Matrices

Abstract

Normalized eigenvalue counting measure of the sum of two Hermitian (or real symmetric) matrices $A_n$ and $B_n$ rotated independently with respect to each other by the random unitary (or orthogonal) Haar distributed matrix $U_n$ (i.e. $A_n+U_n^{\ast }{B}_n{U}_n$) is studied in the limit of large matrix order $n$. Convergence in probability to a limiting nonrandom measure is established. A functional equation for the Stieltjes transform of the limiting measure in terms of limiting eigenvalue measures of $A_n$ and $B_n$ is obtained and studied. Keywords: random matrices, eigenvalue distribution

arXiv comment41 pages, submitted to Commun. Math. Phys

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