Mingo J.A., Speicher R. Free Probability and Random Matrices

Springer Science+Business Media LLC, 2017. — 343 p. — (Fields Institute Monographs 35) — ISBN: 1493969412.This volume opens the world of free probability to a wide variety of readers. From its roots in the theory of operator algebras, free probability has intertwined with non-crossing partitions, random matrices, applications in wireless communications, representation theory of large groups, quantum groups, the invariant subspace problem, large deviations, subfactors, and beyond.
This book puts a special emphasis on the relation of free probability to random matrices, but also touches upon the operator algebraic, combinatorial, and analytic aspects of the theory.
The book serves as a combination textbook/research monograph, with self-contained chapters, exercises scattered throughout the text, and coverage of important ongoing progress of the theory. It will appeal to graduate students and all mathematicians interested in random matrices and free probability from the point of view of operator algebras, combinatorics, analytic functions, or applications in engineering and statistical physics.Asymptotic Freeness of Gaussian Random Matrices
The Free Central Limit Theorem and Free Cumulants
Free Harmonic Analysis
Asymptotic Freeness for Gaussian, Wigner, and Unitary Random Matrices
Fluctuations and Second Order Freeness
Free Group Factors and Freeness
Free Entropy X: The Microstates Approach via Large Deviations
Free Entropy X*: The Non-microstates Approach via Free Fisher Information
Operator-Valued Free Probability Theory and Block Random Matrices
Deterministic Equivalents, Polynomials in Free Variables, and Analytic Theory of Operator-Valued Convolution.
Brown Measure
Solutions to Exercises