$q$-Racah ensemble and $q$-P$\left(E_7^{(1)}/A_{1}^{(1)}\right)$ discrete Painlev\'e equation

  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: Mathematical Physics},
The goal of this paper is to investigate the missing part of the story about the relationship between the orthogonal polynomial ensembles and Painleve equations. Namely, we consider the $q$-Racah polynomial ensemble and show that the one-interval gap probabilities in this case can be expressed through a solution of the discrete $q$-P$\left(E_7^{(1)}/A_{1}^{(1)}\right)$ equation. Our approach also gives a new Lax pair for this equation. This Lax pair has an interesting additional involutive… 
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