On the Law of Addition of Random Matrices


Normalized eigenvalue counting measure of the sum of two Hermitian (or real symmetric) matrices AnA_{n} and BnB_{n} rotated independently with respect to each other by the random unitary (or orthogonal) Haar distributed matrix UnU_{n} (i.e. An+Unβˆ—BnUnA_{n}+U_{n}^{\ast}B_{n}U_{n}) is studied in the limit of large matrix order nn. 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 AnA_{n} and BnB_{n} is obtained and studied. Keywords: random matrices, eigenvalue distributionComment: 41 pages, submitted to Commun. Math. Phy

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