Integrability, Random Matrices and Painlevé Transcendents

Abstract

The probability that an interval $I$ is free of eigenvalues in a matrix ensemble with unitary symmetry is given by a Fredholm determinant. When the weight function in the matrix ensemble is a classical weight function, and the interval $I$ includes an endpoint of the support, Tracy and Widom have given a formalism which gives coupled differential equations for the required probability and some auxilary quantities. We summarize and extend earlier work by expressing the probability and some of the auxilary quantities in terms of Painlevé transcendents.