Discrete Painlev\'e equations and random matrix averages

TL;DR

This paper uses Painlevé equations to derive recurrence relations for random matrix averages over unitary groups, connecting discrete Painlevé equations with eigenvalue distribution computations.

Contribution

It introduces a novel approach linking Painlevé systems to random matrix averages, deriving recurrences involving auxiliary quantities satisfying discrete Painlevé equations.

Findings

Derived recurrences for random matrix averages using Painlevé theory

Connected auxiliary quantities to discrete Painlevé equations

Demonstrated convergence to limiting eigenvalue distributions

Abstract

The $\tau$-function theory of Painlev\'e systems is used to derive recurrences in the rank $n$ of certain random matrix averages over U(n). These recurrences involve auxilary quantities which satisfy discrete Painlev\'e equations. The random matrix averages include cases which can be interpreted as eigenvalue distributions at the hard edge and in the bulk of matrix ensembles with unitary symmetry. The recurrences are illustrated by computing the value of a sequence of these distributions as $n$ varies, and demonstrating convergence to the value of the appropriate limiting distribution.