Moduli spaces of q-connections and gap probabilities

TL;DR

This paper demonstrates that the one-interval gap probability in the q-Hahn orthogonal polynomial ensemble can be described using solutions to the asymmetric q-Painleve V equation, connecting orthogonal polynomials, q-connections, and integrable systems.

Contribution

It introduces a novel approach linking q-connections on the Riemann sphere to the q-Painleve V equation for the most general q-Hahn ensemble.

Findings

Expressed gap probability via q-Painleve V solutions.

Derived a new q-difference equation in Sakai's hierarchy.

Calculated the Lax pair for the equation.

Abstract

Our goal is to show that the one-interval gap probability for the q-Hahn orthogonal polynomial ensemble can be expressed through a solution of the asymmetric q-Painleve V equation. The case of the q-Hahn ensemble we consider is the most general case of the orthogonal polynomial ensembles that have been studied. Our approach is based on the analysis of q-connections on the Riemann sphere with a particular singularity structure. It requires a new derivation of a q-difference equation of Sakai's hierarchy of type A_{2}^{(1)}. We also calculate its Lax pair.