Random Matrix Theory and the Sixth Painlevé Equation

Abstract

A feature of certain ensembles of random matrices is that the corresponding measure is invariant under conjugation by unitary matrices. Study of such ensembles realised by matrices with Gaussian entries leads to statistical quantities related to the eigenspectrum, such as the distribution of the largest eigenvalue, which can be expressed as multidimensional integrals or equivalently as determinants. These distributions are well known to be $\tau$-functions for Painlevé systems, allowing for the former to be characterised as the solution of certain nonlinear equations. We consider the random matrix ensembles for which the nonlinear equation is the $\sigma$ form of \PVI. Known results are reviewed, as is their implication by way of series expansions for the distributions. New results are given for the boundary conditions in the neighbourhood of the fixed singularities at $t=0,1,\infty$ of $\sigma$\PVI displayed by a generalisation of the generating function for the distributions. The structure of these expansions is related to Jimbo's general expansions for the $\tau$-function of $\sigma$\PVI in the neighbourhood of its fixed singularities, and this theory is itself put in its context of the linear isomonodromy problem relating to \PVI.

Dedicated to the centenary of the publication of the Painlevé VI equation in the Comptes Rendus de l'Academie des Sciences de Paris by Richard Fuchs in 1905

References in corpus (6)

• Fredholm Determinants, Differential Equations and Matrix Models
• Application of the $\tau$-Function Theory of Painlevé Equations to Random Matrices: PIV, PII and the GUE
• Gap probabilities in the finite and scaled Cauchy random matrix ensembles
• Calculation of some determinants using the s-shifted factorial
• Boundary conditions associated with the Painlevé III' and V evaluations of some random matrix averages
• Discrete Painlevé equations for a class of \PVI $\tau$-functions given as U(N) averages

Cited by in corpus (15)

• The recurrence coefficients of semi-classical Laguerre polynomials and the fourth Painlevé equation
• Quantum Painleve-Calogero Correspondence
• Painlevé VI and Hankel determinants for the generalized Jacobi Weight
• On determinant representation and integrability of Nekrasov functions
• The lowest eigenvalue of Jacobi random matrix ensembles and Painlevé VI
• Finite size corrections in random matrix theory and Odlyzko's data set for the Riemann zeros
• Moment determinants as isomonodromic tau functions
• Resurgence, Painleve Equations and Conformal Blocks
• Scaled limit and rate of convergence for the largest eigenvalue from the generalized Cauchy random matrix ensemble
• Painlevé VI, Painlevé III and the Hankel Determinant Associated with a Degenerate Jacobi Unitary Ensemble
• On tau functions for orthogonal polynomials and matrix models
• On a class of elliptic orthogonal polynomials and their integrability
• Paley-Wiener spaces with vanishing conditions and Painlevé VI transcendents
• Integrable Differential Systems for Deformed Laguerre-Hahn Orthogonal Polynomials
• Boundary Conditions for Scaled Random Matrix Ensembles in the Bulk of the Spectrum