
Contents
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Abstract Abstract
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22.1 Introduction 22.1 Introduction
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22.2 The moment method for several random matrices and the concept of freeness 22.2 The moment method for several random matrices and the concept of freeness
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22.3 Basic definitions 22.3 Basic definitions
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22.4 Combinatorial theory of freeness 22.4 Combinatorial theory of freeness
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22.5 Free harmonic analysis 22.5 Free harmonic analysis
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22.5.1 Sums of free variables: the R-transform 22.5.1 Sums of free variables: the R-transform
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22.5.2 Products of free variables: the S-transform 22.5.2 Products of free variables: the S-transform
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22.5.3 The free central limit theorem 22.5.3 The free central limit theorem
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22.5.4 Free Poisson distribution and Wishart matrices 22.5.4 Free Poisson distribution and Wishart matrices
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22.6 Second-order freeness 22.6 Second-order freeness
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22.7 Operator-valued free probability theory 22.7 Operator-valued free probability theory
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22.8 Further free-probabilistic aspects of random matrices 22.8 Further free-probabilistic aspects of random matrices
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22.9 Operator algebraic aspects of free probability 22.9 Operator algebraic aspects of free probability
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22.9.1 Operator algebraic models for freeness 22.9.1 Operator algebraic models for freeness
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Free group factors Free group factors
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Creation and annihilation operators on full Fock spaces Creation and annihilation operators on full Fock spaces
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22.9.2 Free entropy 22.9.2 Free entropy
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22.9.3 Other operator algebraic applications of free probability theory 22.9.3 Other operator algebraic applications of free probability theory
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Acknowledgements Acknowledgements
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References References
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22 Free probability theory
Get accessRoland Speicher, Universität des Saarlandes, Fachrichtung Mathematik, Postfach 151150, 66041 Saarbrücken, Germany, [email protected]
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Published:08 August 2018
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Abstract
This article focuses on free probability theory, which is useful for dealing with asymptotic eigenvalue distributions in situations involving several matrices. In particular, it considers some of the basic ideas and results of free probability theory, mostly from the random matrix perspective. After providing a brief background on free probability theory, the article discusses the moment method for several random matrices and the concept of freeness. It then gives some of the main probabilistic notions used in free probability and introduces the combinatorial theory of freeness. In this theory, freeness is described in terms of free cumulants in relation to the planar approximations in random matrix theory (RMT). The article also examines free harmonic analysis, second-order freeness, operator-valued free probability theory, further free-probabilistic aspects of random matrices, and operator algebraic aspects of free probability.
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