Free probability and random matrices. (English) Zbl 1387.60005
Fields Institute Monographs 35. Toronto: The Fields Institute for Research in the Mathematical Sciences; New York, NY: Springer (ISBN 978-1-4939-6941-8/hbk; 978-1-4939-6942-5/ebook). xiv, 336 p. (2017).
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. On the backside of the book you can read:
“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.”
Contents: Introduction; 1. Asymptotic freeness of Gaussian random matrices; 2. The free central limit theorem and free cumulants; 3. Free harmonic analysis; 4. Asymptotic freeness for Gaussian, Wigner, and unitary; 5. Fluctuations and second-order freeness; 6. Free group factors and freeness; 7. Free entropy \(\chi\): the microstates approach via large deviations; 8. Free entropy \(\chi^*\): The non-microstates approach via Fisher information; 9. Operator-valued free probability theory and block random matrices; 10. Deterministic equivalents, polynomials in free variables, and analytic theory of operator-valued convolution; 11. Brown measure; 12. Solutions of exercises; References; Index of exercises: Index.
“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.”
Contents: Introduction; 1. Asymptotic freeness of Gaussian random matrices; 2. The free central limit theorem and free cumulants; 3. Free harmonic analysis; 4. Asymptotic freeness for Gaussian, Wigner, and unitary; 5. Fluctuations and second-order freeness; 6. Free group factors and freeness; 7. Free entropy \(\chi\): the microstates approach via large deviations; 8. Free entropy \(\chi^*\): The non-microstates approach via Fisher information; 9. Operator-valued free probability theory and block random matrices; 10. Deterministic equivalents, polynomials in free variables, and analytic theory of operator-valued convolution; 11. Brown measure; 12. Solutions of exercises; References; Index of exercises: Index.
Reviewer: Ludwig Paditz (Dresden)
MSC:
| 60-02 | Research exposition (monographs, survey articles) pertaining to probability theory |
| 60F05 | Central limit and other weak theorems |
| 60F10 | Large deviations |
| 60G15 | Gaussian processes |
| 15B52 | Random matrices (algebraic aspects) |
| 60B20 | Random matrices (probabilistic aspects) |
