- 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 interaction between random matrix theory (RMT) and integrable theory, leading to ordinary and partial differential equations (PDEs) for the eigenvalue distribution of random matrix models of size n and the transition probabilities of non-intersecting Brownian motion models, for finite n and for n → ∞. It first provides an overview of the connection between the theory of orthogonal polynomials and the KP-hierarchy in integrable systems before examining matrix models and the Virasoro constraints. It then considers multiple orthogonal polynomials, taking into account non-intersecting Brownian motions on ℝ (Dyson’s Brownian motions), a moment matrix for several weights, Virasoro constraints, and a PDE for non-intersecting Brownian motions. It also analyses critical diffusions, with particular emphasis on the Airy process, the Pearcey process, and Airy process with wanderers. Finally, it describes the Tacnode process, along with kernels and p-reduced KP-hierarchy.

Keywords: random matrix theory (RMT), integrable theory, Brownian motion, orthogonal polynomial, KP-hierarchy, random matrix model, Virasoro constraints, critical diffusion, Airy process, Pearcey process

Pierre van Moerbeke, Département de Mathématique, Université Catholique de Louvain, Bât. M. de Hemptinne, Chemin du Cyclotron, 2, 1348 Louvain-la-Neuve, Belgium, Pierre.Vanmoerbeke@uclouvain.be

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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