Random Matrix Theory and the Sixth Painlevé Equation

@inproceedings{Forrester2006RandomMT,
title={Random Matrix Theory and the Sixth Painlev{\'e} Equation},
author={Peter J. Forrester and Nicholas S. Witte},
year={2006}
}

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 τ-functions for Painlevé systems, allowing for the… CONTINUE READING