- The Oxford Handbook of Random Matrix Theory
- Dedication
- Foreword
- Detailed Contents
- List of Contributors
- Introduction and guide to the handbook
- History – an overview
- Symmetry classes
- Spectral statistics of unitary ensembles
- Spectral statistics of orthogonal and symplectic ensembles
- Universality
- Supersymmetry
- Replica approach in random matrix theory
- Painlevé transcendents
- Random matrix theory and integrable systems
- Determinantal point processes
- Random matrix representations of critical statistics
- Heavy-tailed random matrices
- Phase transitions
- Two-matrix models and biorthogonal polynomials
- Chain of matrices, loop equations, and topological recursion
- Unitary integrals and related matrix models
- Non-Hermitian ensembles
- Characteristic polynomials
- Beta ensembles
- Wigner matrices
- Free probability theory
- Random banded and sparse matrices
- Number theory
- Random permutations and related topics
- Enumeration of maps
- Knot theory and matrix integrals
- Multivariate statistics
- Algebraic geometry and matrix models
- Two-dimensional quantum gravity
- String theory
- Quantum chromodynamics
- Quantum chaos and quantum graphs
- Resonance scattering of waves in chaotic systems
- Condensed matter physics
- Classical and quantum optics
- Extreme eigenvalues of Wishart matrices: application to entangled bipartite system
- Random growth models
- Random matrices and Laplacian growth
- Financial applications of random matrix theory: a short review
- Asymptotic singular value distributions in information theory
- Random matrix theory and ribonucleic acid (RNA) folding
- Complex networks
- Index

## Abstract and Keywords

This article discusses the first four decades of the history of random matrix theory (RMT), that is, until about 1990. It first considers Niels Bohr's formulation of the concept of the compound nucleus, which is at the root of the use of random matrices in physics, before analysing the development of the theory of spectral fluctuations. In particular, it examines the Wishart ensemble; Dyson's classification leading to the three canonical ensembles — Gaussian Orthogonal Ensemble (GOE), Gaussian Unitary Ensemble (GUE), and Gaussian Symplectic Ensemble (GSE); and the breaking of a symmetry or an invariance. It also describes how random matrix models emerged from quantum physics, more specifically from a statistical approach to the strongly interacting many-body system of the atomic nucleus. The article concludes with an overview of data on nuclear resonances, many-body theory, chaos, number theory, scattering theory, replica trick and supersymmetry, disordered solids, and interacting fermions and field theory.

Keywords: random matrix theory (RMT), Niels Bohr, compound nucleus, random matrices, spectral fluctuation, quantum physics, nuclear resonance, many-body theory, chaos, number theory

Oriol Bohigas, Laboratoire de Physique Théorique et Modeles Statistiques, bâtiment 100, Université Paris-Sud, Centre scientifique d’Orsay, 15 rue Georges Clemenceau, 91405 Orsay cedex, France, bohigas@lptms.u-psud.fr

Hans A. Weidenmüller, Max-Planck-Institut für Kernphysik, Saupfercheck-weg 1, D-69117 Heidelberg, Germany, Hans.Weidenmueller@mpi-hd.mpg.de

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- The Oxford Handbook of Random Matrix Theory
- Dedication
- Foreword
- Detailed Contents
- List of Contributors
- Introduction and guide to the handbook
- History – an overview
- Symmetry classes
- Spectral statistics of unitary ensembles
- Spectral statistics of orthogonal and symplectic ensembles
- Universality
- Supersymmetry
- Replica approach in random matrix theory
- Painlevé transcendents
- Random matrix theory and integrable systems
- Determinantal point processes
- Random matrix representations of critical statistics
- Heavy-tailed random matrices
- Phase transitions
- Two-matrix models and biorthogonal polynomials
- Chain of matrices, loop equations, and topological recursion
- Unitary integrals and related matrix models
- Non-Hermitian ensembles
- Characteristic polynomials
- Beta ensembles
- Wigner matrices
- Free probability theory
- Random banded and sparse matrices
- Number theory
- Random permutations and related topics
- Enumeration of maps
- Knot theory and matrix integrals
- Multivariate statistics
- Algebraic geometry and matrix models
- Two-dimensional quantum gravity
- String theory
- Quantum chromodynamics
- Quantum chaos and quantum graphs
- Resonance scattering of waves in chaotic systems
- Condensed matter physics
- Classical and quantum optics
- Extreme eigenvalues of Wishart matrices: application to entangled bipartite system
- Random growth models
- Random matrices and Laplacian growth
- Financial applications of random matrix theory: a short review
- Asymptotic singular value distributions in information theory
- Random matrix theory and ribonucleic acid (RNA) folding
- Complex networks
- Index