Painleve IV and degenerate Gaussian Unitary Ensembles

TL;DR

This paper studies Gaussian Unitary Ensembles with prescribed eigenvalue multiplicities, deriving joint eigenvalue densities and connecting recurrence coefficients to Painleve IV equations, revealing new links between random matrix theory and special functions.

Contribution

It introduces a novel analysis of GUE with eigenvalue multiplicities, linking orthogonal polynomial recurrence coefficients to Painleve IV equations and generalized Hermite polynomials.

Findings

Recurrence coefficients satisfy Painleve IV equations.

Eigenvalue densities are derived for ensembles with prescribed multiplicities.

Connections established between random matrix ensembles and special functions.

Abstract

We consider those Gaussian Unitary Ensembles where the eigenvalues have prescribed multiplicities, and obtain joint probability density for the eigenvalues. In the simplest case where there is only one multiple eigenvalue t, this leads to orthogonal polynomials with the Hermite weight perturbed by a factor that has a multiple zero at t. We show through a pair of ladder operators, that the diagonal recurrence coefficients satisfy a particular Painleve IV equation for any real multiplicity. If the multiplicity is even they are expressed in terms of the generalized Hermite polynomials, with t as the independent variable.