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

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.