- 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 reviews the theory of random matrices with eigenvalues distributed in the complex plane and more general ‘beta ensembles’ (logarithmic gases in 2D). It first considers two ensembles of random matrices with complex eigenvalues: ensemble C of general complex matrices and ensemble N of normal matrices. In particular, it describes the Dyson gas picture for ensembles of matrices with general complex eigenvalues distributed on the plane. It then presents some general exact relations for correlation functions valid for any values of N and β before analysing the distribution and correlations of the eigenvalues in the large N limit. Using the technique of boundary value problems in two dimensions and elements of the potential theory, the article demonstrates that the finite-time blow-up (a cusp–like singularity) of the Laplacian growth with zero surface tension is a critical point of the normal and complex matrix models.

Keywords: random matrices, complex plane, beta ensemble, Dyson gas, correlation function, eigenvalue, N limit, Laplacian growth, matrix model

Anton Zabrodin, Institute of Biochemical Physics, 4 Kosygina st., 119991 Moscow, Russia, and Institute of Theoretical and Experimental Physics (ITEP), B. Cheremushkinskaya, 25, Moscow 117259, Russia, zabrodin@itep.ru

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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