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Free Probability and Random Matrices

  • Book
  • © 2017

Overview

Authors:
  1. James A. Mingo

    1. Department of Mathematics and Statistics, Queen’s University, Kingston, Canada

    View author publications

    You can also search for this author in PubMed   Google Scholar

  2. Roland Speicher

    1. FB Mathematik, Universität des Saarlandes, Saarbrücken, Germany

    View author publications

    You can also search for this author in PubMed   Google Scholar

  • Covers a variety of different facets of free probability, giving a flavor of the breadth of the subject
  • Features exercises scattered throughout the text
  • Showcases basic ideas and results in order to focus on their relation
  • Includes supplementary material: sn.pub/extras

Part of the book series: Fields Institute Monographs (FIM, volume 35)

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Table of contents (12 chapters)

  1. Front Matter

    Pages i-xiv
    Download chapter PDF

  2. Asymptotic Freeness of Gaussian Random Matrices

    • James A. Mingo, Roland Speicher
    Pages 1-21

  3. The Free Central Limit Theorem and Free Cumulants

    • James A. Mingo, Roland Speicher
    Pages 23-49

  4. Free Harmonic Analysis

    • James A. Mingo, Roland Speicher
    Pages 51-91

  5. Asymptotic Freeness for Gaussian, Wigner, and Unitary Random Matrices

    • James A. Mingo, Roland Speicher
    Pages 93-120

  6. Fluctuations and Second Order Freeness

    • James A. Mingo, Roland Speicher
    Pages 121-158

  7. Free Group Factors and Freeness

    • James A. Mingo, Roland Speicher
    Pages 159-174

  8. Free Entropy χ: The Microstates Approach via Large Deviations

    • James A. Mingo, Roland Speicher
    Pages 175-194

  9. Free Entropy χ ∗: The Non-microstates Approach via Free Fisher Information

    • James A. Mingo, Roland Speicher
    Pages 195-223

  10. Operator-Valued Free Probability Theory and Block Random Matrices

    • James A. Mingo, Roland Speicher
    Pages 225-247

  11. Deterministic Equivalents, Polynomials in Free Variables, and Analytic Theory of Operator-Valued Convolution

    • James A. Mingo, Roland Speicher
    Pages 249-262

  12. Brown Measure

    • James A. Mingo, Roland Speicher
    Pages 263-280

  13. Solutions to Exercises

    • James A. Mingo, Roland Speicher
    Pages 281-318

  14. Back Matter

    Pages 319-336
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Keywords

About this book

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.

Reviews

“This book is an excellent survey, respectively introduction, into recent developments in free probability theory and its applications to random matrices. The authors superbly guide the reader through a number of important examples and present a carefully selected list of 207 relevant publications.” (Ludwig Paditz, zbMATH 1387.60005, 2018)

Authors and Affiliations

  • Department of Mathematics and Statistics, Queen’s University, Kingston, Canada

    James A. Mingo

  • FB Mathematik, Universität des Saarlandes, Saarbrücken, Germany

    Roland Speicher

Bibliographic Information

  • Book Title: Free Probability and Random Matrices

  • Authors: James A. Mingo, Roland Speicher

  • Series Title: Fields Institute Monographs

  • DOI: https://doi.org/10.1007/978-1-4939-6942-5

  • Publisher: Springer New York, NY

  • eBook Packages: Mathematics and Statistics, Mathematics and Statistics (R0)

  • Copyright Information: Springer Science+Business Media LLC 2017

  • Hardcover ISBN: 978-1-4939-6941-8Published: 26 June 2017

  • Softcover ISBN: 978-1-4939-8346-9Published: 27 July 2018

  • eBook ISBN: 978-1-4939-6942-5Published: 24 June 2017

  • Series ISSN: 1069-5273

  • Series E-ISSN: 2194-3079

  • Edition Number: 1

  • Number of Pages: XIV, 336

  • Number of Illustrations: 45 b/w illustrations

  • Topics: Probability Theory and Stochastic Processes, Functional Analysis

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