The Oxford Handbook of Random Matrix Theory
The Oxford Handbook of Random Matrix Theory
Gernot Akemann, Fakultät für Physik, Universität Bielefeld, Postfach 100131, D-33501 Bielefeld, Germany, akemann@physik.uni-bielefeld.de
Jinho Baik, 530 Church Street, Department of Mathematics, University of Michigan, Ann Arbor, MI 48109-1043, USA, baik@umich.edu
Philippe Di Francesco, Institut de Physique Théorique, CEA/Saclay, Orme des Merisiers, F-91191 Gif-sur-Yvette Cedex, France, philippe.di-francesco@cea.fr
Cite
Abstract
This handbook showcases the major aspects and modern applications of random matrix theory (RMT). It examines the mathematical properties and applications of random matrices and some of the reasons why RMT has been very successful and continues to enjoy great interest among physicists, mathematicians and other scientists. It also discusses methods of solving RMT, basic properties and fundamental objects in RMT, and different models and symmetry classes in RMT. Topics include the use of classical orthogonal polynomials (OP) and skew-OP to solve exactly RMT ensembles with unitary, and orthogonal or symplectic invariance respectively, all at finite matrix size; the supersymmetric and replica methods; determinantal point processes; Painlevé transcendents; the fundamental property of RMT known as universality; RNA folding; two-dimensional quantum gravity; string theory; and the mathematical concept of free random variables. In addition to applications to mathematics and physics, the book considers broader applications to other sciences, including economics, engineering, biology, and complex networks.
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Front Matter
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Part I Introduction
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Part II Properties of random matrix theory
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·3·
Symmetry classes
Martin R. Zirnbauer
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4
Spectral statistics of unitary ensembles
Greg W. Anderson
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·5·
Spectral statistics of orthogonal and symplectic ensembles
Mark Adler
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·6·
Universality
Arno Kuijlaars
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·7·
Supersymmetry
Thomas Guhr
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8
Replica approach in random matrix theory
Eugene Kanzieper
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9
Painlevé transcendents
Alexander R. Its
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10
Random matrix theory and integrable systems
Pierre van Moerbeke
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11
Determinantal point processes
Alexei Borodin
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12
Random matrix representations of critical statistics
Vladimir Kravtsov
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13
Heavy-tailed random matrices
Zdzislaw Burda andJerzy Jurkiewicz
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14
Phase transitions
Giovanni Cicuta andLuca Molinari
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15
Two-matrix models and biorthogonal polynomials
Marco Bertola
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16
Chain of matrices, loop equations, and topological recursion
Nicolas Orantin
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17
Unitary integrals and related matrix models
Alexei Morozov
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18
Non-Hermitian ensembles
Boris Khoruzhenko andHans-Jurgen Sommers
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19
Characteristic polynomials
Edouard Brezin andSinobu Hikami
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20
Beta ensembles
Peter Forrester
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21
Wigner matrices
Gerard Ben Arous andAlice Guionnet
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22
Free probability theory
Roland Speicher
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23
Random banded and sparse matrices
Thomas Spencer
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·3·
Symmetry classes
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Part III Applications of random matrix theory
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24
Number theory
Jon Keating andNina Snaith
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25
Random permutations and related topics
Grigori Olshanski
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26
Enumeration of maps
Jeremie Bouttier
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27
Knot theory and matrix integrals
Paul Zinn-Justin andJean-Bernard Zuber
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28
Multivariate statistics
Noureddine El Karoui
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29
Algebraic geometry and matrix models
Leonid Chekhov
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30
Two-dimensional quantum gravity
Ivan Kostov
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31
String theory
Marcos Marino
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32
Quantum chromodynamics
Jac Verbaarschot
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·33·
Quantum chaos and quantum graphs
Sebastian Muller andMartin Sieber
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·34·
Resonance scattering of waves in chaotic systems
Yan Fyodorov andDmitry Savin
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·35·
Condensed matter physics
Carlo W. J. Beenakker
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·36·
Classical and quantum optics
Carlo W. J. Beenakker
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·37·
Extreme eigenvalues of Wishart matrices: application to entangled bipartite system
Satya N. Majumdar
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38
Random growth models
Patrik L. Ferrari andHerbert Spohn
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39
Random matrices and Laplacian growth
Anton Zabrodin
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40
Financial applications of random matrix theory: a short review
Jean-Phillipe Bouchaud andMarc Potters
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41
Asymptotic singular value distributions in information theory
Antonia Tulino andSergio Verdu
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42
Random matrix theory and ribonucleic acid (RNA) folding
Graziano Vernizzi andHenri Orland
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43
Complex networks
Geoff Rodgers andTaro Nagao
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24
Number theory
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End Matter
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