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Asymptotic Eigenvalue Distribution of Random Matrices and Free Stochastic Analysis

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Random Matrices and Iterated Random Functions

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 53))

Abstract

This is a survey on some recent work on free stochastic calculus and free Malliavin calculus. It is hoped that these theories will in the long run provide us with tools for qualitative descriptions of the asymptotic eigenvalue distribution of selfadjoint polynomials of independent Gaussian random matrices. The main concrete results center around the free Fourth Moment Theorem, which says that for a sequence of random variables which are constrained to live in a fixed free chaos, the convergence to the semicircle distribution can be controlled by the convergence of the second and the fourth moments.

I thank Octavio Arizmendi for the preparation of the figures.

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Authors and Affiliations

  1. Saarland University, Fachrichtung Mathematik, 151150, 66041, Saarbrücken, Germany

    Roland Speicher

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  1. Roland Speicher
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Correspondence to Roland Speicher .

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Editors and Affiliations

  1. Westphalian Wilhelms University Münster Institute of Mathematics and Informatics, Münster, Germany

    Gerold Alsmeyer

  2. Westphalian Wilhelms University Münster Institute for Mathematical Statistics, Münster, Germany

    Matthias Löwe

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Speicher, R. (2013). Asymptotic Eigenvalue Distribution of Random Matrices and Free Stochastic Analysis. In: Alsmeyer, G., Löwe, M. (eds) Random Matrices and Iterated Random Functions. Springer Proceedings in Mathematics & Statistics, vol 53. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-38806-4_2

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