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 realized 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 τ-functions for Painlevé systems, allowing for the former to be characterized as the solution of certain nonlinear equations. We consider the random matrix ensembles for which the nonlinear equation is the σ form of PVI. Known results are reviewed, as is t..