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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.
Physical Applications of Freeness
"... Introduction The mathematical concept `freeness' was introduced by Voiculescu about 15 years ago in order to get some insight into the structure of the von Neumann algebras of the free groups. The starting idea of Voiculescu was to separate the notion of freeness from that concrete problem and to c ..."
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Cited by 3 (0 self)
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Introduction The mathematical concept `freeness' was introduced by Voiculescu about 15 years ago in order to get some insight into the structure of the von Neumann algebras of the free groups. The starting idea of Voiculescu was to separate the notion of freeness from that concrete problem and to consider it on its own sake. One leading principle of his investigations was to consider freeness as a noncommutative analogue to the classical notion `independence of random variables' This approach was very successful. Gradually it became clear that freeness has indeed a very interesting structure of its own and that there exist a lot of links to other fields. One of the main links of freeness to an apriori unrelated context is the description of random matrices via freeness. This has opened the possibility to use freeness as an exact mathematical concept for the description of physical models and approximations which build (explicitely or implicitly) on random matrices. One model of
Random Matrices in 2D, Laplacian Growth and Operator Theory
, 805
"... Abstract. Since it was first applied to the study of nuclear interactions by Wigner and Dyson, almost 60 years ago, Random Matrix Theory (RMT) has developed into a field of its own whithin applied mathematics, and is now essential to many parts of theoretical physics, from condensed matter to high e ..."
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Abstract. Since it was first applied to the study of nuclear interactions by Wigner and Dyson, almost 60 years ago, Random Matrix Theory (RMT) has developed into a field of its own whithin applied mathematics, and is now essential to many parts of theoretical physics, from condensed matter to high energy. The fundamental results obtained so far rely mostly on the theory of random matrices in one dimension (the dimensionality of the spectrum, or equilibrium probability density). In the last few years, this theory has been extended to the case where the spectrum is twodimensional, or even fractal, with dimensions between 1 and 2. In this article, we review these recent developments and indicate some physical problems where the theory can be applied.