The discrete Painlev\'{e} XXXIV hierarchy arising from the gap probability distributions of Freud random matrix ensembles

TL;DR

This paper links the discrete Painlevé XXXIV hierarchy to gap probability distributions in Freud random matrix ensembles, revealing a novel appearance of this hierarchy in Random Matrix Theory through orthogonal polynomial analysis.

Contribution

It demonstrates the emergence of the discrete Painlevé XXXIV hierarchy from Freud ensemble gap probabilities using ladder operators, a novel connection in Random Matrix Theory.

Findings

Derived difference equations for orthogonal polynomial recurrence coefficients.

Identified the discrete Painlevé XXXIV hierarchy in the context of Freud ensembles.

Established relationships between gap probabilities, orthogonal polynomial coefficients, and recurrence coefficients.

Abstract

We consider the symmetric gap probability distributions of certain Freud unitary ensembles. This problem is related to the Hankel determinants generated by the Freud weights supported on the complement of a symmetric interval. By using Chen and Ismail's ladder operator approach, we obtain the difference equations satisfied by the recurrence coefficients for the orthogonal polynomials with the discontinuous Freud weights. We find that these equations, with a minor change of variables, are the discrete Painlev\'{e} XXXIV hierarchy proposed by Cresswell and Joshi [{\em J. Phys. A: Math. Gen.} {\bf 32} ({1999}) {655--669}]. This is the first time that the discrete Painlev\'{e} XXXIV hierarchy appears in the study of Random Matrix Theory. We also derive the differential-difference equations for the recurrence coefficients and show the relationship between the logarithmic derivative of the gap probabilities, the nontrivial leading coefficients of the monic orthogonal polynomials and the recurrence coefficients.