Gap Probabilities in the Laguerre Unitary Ensemble and Discrete Painlevé Equations

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 Painlevé equations. On one hand, this gives us one more detailed example of the appearance of discrete Painlevé 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 Painlevé equations.

arXiv comment16 pages. arXiv admin note: text overlap with arXiv:1910.10981

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• Geometric Aspects of Painlevé Equations
• Linearization And Solutions Of The Discrete Painlevé-III Equation
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Cited by in corpus (3)

• Different Hamiltonians for differential Painlevé equations and their identification using a geometric approach
• On the recurrence coefficients for the $q$-Laguerre weight and discrete Painlevé equations
• Special solutions of a discrete Painlevé equation for quantum minimal surfaces