- 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 history and modern theory of Painlevé transcendents, with particular emphasis on the Riemann–Hilbert method. In random matrix theory (RMT), the Painlevé equations describe either the eigenvalue distribution functions in the classical ensembles for finite N or the universal eigenvalue distribution functions in the large N limit. This article examines the latter. It first considers the main features of the Riemann–Hilbert method in the theory of Painlevé equations using the second Painlevé equation as a case study before analysing the two most celebrated universal distribution functions of RMT in terms of the Painlevé transcendents using the theory of integrable Fredholm operators as well as the Riemann–Hilbert technique: the sine kernel and the Airy kernel determinants.
Keywords: Painlevé transcendent, Riemann–Hilbert method, random matrix theory (RMT), Painlevé equation, integrable Fredholm operator, sine kernel, Airy kernel
Alexander R. Its, Department of Mathematical Sciences, Indiana University, 402 N. Blackford Street, LD270, Indianapolis, IN 46202-3216, USA, itsa@math.iupui.edu
Access to the complete content on Oxford Handbooks Online requires a subscription or purchase. Public users are able to search the site and view the abstracts and keywords for each book and chapter without a subscription.
Please subscribe or login to access full text content.
If you have purchased a print title that contains an access token, please see the token for information about how to register your code.
For questions on access or troubleshooting, please check our FAQs, and if you can''t find the answer there, please contact us.
- 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
