$q$-Racah ensemble and $q$-P$\left(E_7^{(1)}/A_{1}^{(1)}\right)$ discrete Painlev\'e equation
Mathematical Physics
2019-04-08 v2 Algebraic Geometry
math.MP
Probability
Abstract
The goal of this paper is to investigate the missing part of the story about the relationship between the orthogonal polynomial ensembles and Painlev\'e equations. Namely, we consider the -Racah polynomial ensemble and show that the one-interval gap probabilities in this case can be expressed through a solution of the discrete -P equation. Our approach also gives a new Lax pair for this equation. This Lax pair has an interesting additional involutive symmetry structure.
Keywords
Cite
@article{arxiv.1903.06159,
title = {$q$-Racah ensemble and $q$-P$\left(E_7^{(1)}/A_{1}^{(1)}\right)$ discrete Painlev\'e equation},
author = {Anton Dzhamay and Alisa Knizel},
journal= {arXiv preprint arXiv:1903.06159},
year = {2019}
}