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Asymptotic freeness almost everywhere for random matrices
 Acta Sci. Math. (Szeged
, 2000
"... Voiculescu’s asymptotic freeness result for random matrices is improved to the sense of almost everywhere convergence. The asymptotic freeness almost everywhere is first shown for standard unitary matrices based on the computation of multiple moments of their entries, and then it is shown for rather ..."
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Voiculescu’s asymptotic freeness result for random matrices is improved to the sense of almost everywhere convergence. The asymptotic freeness almost everywhere is first shown for standard unitary matrices based on the computation of multiple moments of their entries, and then it is shown for rather general unitarily invariant selfadjoint random matrices (in particular, standard selfadjoint Gaussian matrices) by applying the first result to the unitary parts of their diagonalization. Biunitarily invariant nonselfadjoint random matrices are also treated via polar decomposition.
On the Law of Addition of Random Matrices
 COMMUNICATIONS IN MATHEMATICAL PHYSICS
, 2000
"... Normalized eigenvalue counting measure of the sum of two Hermitian (or real symmetric) matrices An and Bn rotated independently with respect to each other by the random unitary (or orthogonal) Haar distributed matrix Un (i.e. An + U ∗ n BnUn) is studied in the limit of large matrix order n. Converg ..."
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Normalized eigenvalue counting measure of the sum of two Hermitian (or real symmetric) matrices An and Bn rotated independently with respect to each other by the random unitary (or orthogonal) Haar distributed matrix Un (i.e. An + U ∗ n BnUn) 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 An and Bn is obtained and studied.
A simple approach to global regime of the random matrix theory
 Mathematical Results in Statistical Mechanics. Singapore: World Scientific
, 1999
"... Abstract. We discuss a method of the asymptotic computation of moments of the normalized eigenvalue counting measure of random matrices of large order. The method is based on the resolvent identity and on some formulas relating expectations of certain matrix functions and the expectations including ..."
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Abstract. We discuss a method of the asymptotic computation of moments of the normalized eigenvalue counting measure of random matrices of large order. The method is based on the resolvent identity and on some formulas relating expectations of certain matrix functions and the expectations including their derivatives or, equivalently, on some simple formulas of the perturbation theory. In the framework of this unique approach we obtain functional equations for the Stieltjes transforms of the limiting normalized eigenvalue counting measure and the bounds for the rate of convergence for the majority known random matrix ensembles. 1.
Free Probability Theory And Random Matrices
"... Free probability theory originated in the context of operator algebras, however, one of the main features of that theory is its connection with random matrices. Indeed, free probability can be considered as the theory providing concepts and notations, without relying on random matrices, for dealing ..."
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Free probability theory originated in the context of operator algebras, however, one of the main features of that theory is its connection with random matrices. Indeed, free probability can be considered as the theory providing concepts and notations, without relying on random matrices, for dealing with the limit N !1 of N Nrandom matrices. One of the basic approaches to free probability, on which I will concentrate in this lecture, is of a combinatorial nature and centers around socalled free cumulants. In the spirit of the above these arise as the combinatorics (in leading order) of N Nrandom matrices in the limit N = 1. These free cumulants are multilinear functionals which are dened in combinatorial terms by a formula involving noncrossing partitions. I will present the basic denitions and properties of noncrossing partitions and free cumulants and outline its relations with freeness and random matrices. As examples, I will consider the problems of calculating the eigenvalue distribution of the sum of randomly rotated matrices and of the compression (upper left corner) of a randomly rotated matrix. 1. Random matrices and freeness Free probability theory, due to Voiculescu, originated in the context of operator algebras, however, one of the main features of that theory is its connection with random matrices. Indeed, free probability can be considered as the theory providing concepts and notations, without relying on random matrices, for dealing with the limit N ! 1 of N Nrandom matrices. Let us consider a sequence (AN ) N2N of selfadjoint N Nrandom matrices AN . In which sense can we talk about the limit of these matrices ? Of course, such a limit does not exist as a 11matrix and Research supported by a grant of NSERC, Canada. Lectures at the ...
Free Calculus
, 1998
"... .4> F(H) := C\Omega \Phi M n1 H\Omega n ; where\Omega is a distinguished unit vector, called vacuum. 2) The vacuum expectation is the state A 7! h\Omega ; A\Omega i: 1 2 ROLAND SPEICHER 3) For each f 2 H we define the (left) annihilation operator l(f) and the (left) creation operator ..."
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.4> F(H) := C\Omega \Phi M n1 H\Omega n ; where\Omega is a distinguished unit vector, called vacuum. 2) The vacuum expectation is the state A 7! h\Omega ; A\Omega i: 1 2 ROLAND SPEICHER 3) For each f 2 H we define the (left) annihilation operator l(f) and the (left) creation operator l (f) by l(f)\Omega = 0 l(f)f 1\Omega \Delta \Delta \Delta\Omega f n = hf; f 1 if 2\Omega \Delta \Delta \Delta\Omega f n and l (f)f<F