Differential equations for deformed Laguerre polynomials

TL;DR

This paper explores differential equations related to deformed Laguerre polynomials, connecting Painleve V solutions with eigenvalue distributions in random matrix theory using isomonodromic deformations and ladder operators.

Contribution

It introduces a novel approach linking Painleve V solutions to Laguerre ensemble averages through two differential equation theories, enhancing understanding of eigenvalue statistics.

Findings

Eigenvalue spacing distribution expressed via Painleve V

Generating functions linked to Painleve V solutions

Comparison of isomonodromic and ladder operator methods

Abstract

The distribution function for the first eigenvalue spacing in the Laguerre unitary ensemble of finite size may be expressed in terms of a solution of the fifth Painleve transcendent. The generating function of a certain discontinuous linear statistic of the Laguerre unitary ensemble can similarly be expressed in terms of a solution of the fifth Painleve equation. The methodology used to derive these results rely on two theories regarding differential equations for orthogonal polynomial systems, one involving isomonodromic deformations and the other ladder operators. We compare the two theories by showing how either can be used to obtain a characterization of a more general Laguerre unitary ensemble average in terms of the Hamiltonian system for Painleve V.