Gap Probabilities in the Laguerre Unitary Ensemble and Discrete Painlev\'e Equations

TL;DR

This paper explores the connection between recurrence relations in the Laguerre Unitary ensemble and discrete Painlevé equations, illustrating a method to reduce complex recurrences to canonical forms within orthogonal polynomial theory.

Contribution

It demonstrates how a specific recurrence relation for the Laguerre Unitary ensemble can be derived from Sakai's geometric theory and reduced to a discrete Painlevé equation, showcasing a novel application of this reduction method.

Findings

Recurrence relations can generate ladder operators for the Laguerre Unitary ensemble.

Discrete Painlevé equations naturally appear in the context of orthogonal polynomials.

The reduction procedure effectively simplifies complex recurrences to canonical forms.

Abstract

In this paper we study a certain recurrence relation, that can be used to generate ladder operators for the Laguerre Unitary ensemble, from the point of view of Sakai's geometric theory of Painlev\'e equations. On one hand, this gives us one more detailed example of the appearance of discrete Painlev\'e equations in the theory of orthogonal polynomials. On the other hand, it serves as a good illustration of the effectiveness of a recently proposed procedure on how to reduce such recurrences to some canonical discrete Painlev\'e equations.