Painlev\'{e} V and the distribution function of discontinuous linear statistics in the Laguerre Unitary Ensembles

TL;DR

This paper links the distribution function of a specific discontinuous linear statistic in Laguerre Unitary Ensembles to a fifth Painlevé transcendent, providing new insights into their mathematical structure.

Contribution

It introduces a proof of ladder operators for orthogonal polynomials with discontinuous weights and derives the associated Painlevé V equation for the statistic.

Findings

Characteristic function expressed via Painlevé V

Derived nonlinear difference equations for auxiliary quantities

Established connection between discontinuous statistics and Painlevé equations

Abstract

In this paper we study the characteristic or generating function of a certain discontinuous linear statistics of the Laguerre unitary ensembles and show that this is a particular fifth Painl\'eve transcendant in the variable $t,$ the position of the discontinuity. The proof of the ladder operators adapted to orthogonal polynomial with discontinuous weight announced sometime ago is presented here, followed by the non-linear difference equations satisfied by two auxiliary quantities and the derivation of the Painl\'eve equation.