Integrability, Random Matrices and Painlev\'e Transcendents

TL;DR

This paper explores the connection between eigenvalue gap probabilities in unitary matrix ensembles and Painlevé transcendents, extending Tracy and Widom's formalism to express these probabilities through special functions.

Contribution

It extends earlier work by expressing eigenvalue gap probabilities and auxiliary quantities in terms of Painlevé transcendents for classical weights.

Findings

Eigenvalue gap probabilities are represented via Painlevé transcendents.

Extended formalism includes additional auxiliary quantities.

Provides a unified framework for classical weight functions.

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 Painlev\'e transcendents.