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

@article{Dzhamay2019qRacahEA,
  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},
  year={2019}
}
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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References

SHOWING 1-10 OF 28 REFERENCES
Discrete gap probabilities and discrete Painlevé equations
We prove that Fredholm determinants of the form det(1-K_s), where K_s is the restriction of either the discrete Bessel kernel or the discrete {}_2F_1 kernel to {s,s+1,...}, can be expressed through…
The hunting for the discrete Painlev\'e VI is over
We present the discrete, q-, form of the Painleve VI equation written as a three-point mapping and analyse the structure of its singularities. This discrete equation goes over to P_{VI} at the…
Moduli spaces of q-connections and gap probabilities
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…
Fredholm determinants, differential equations and matrix models
AbstractOrthogonal polynomial random matrix models ofN×N hermitian matrices lead to Fredholm determinants of integral operators with kernel of the form (ϕ(x)ψ(y)−ψ(x)ϕ(y))/x−y. This paper is…
Log-gases on quadratic lattices via discrete loop equations and q-boxed plane partitions
Geometric Aspects of Painlev\'e Equations
In this paper a comprehensive review is given on the current status of achievements in the geometric aspects of the Painlev\'e equations, with a particular emphasis on the discrete Painlev\'e…
The Jacobi polynomial ensemble and the Painlevé VI equation
We consider the Jacobi polynomial ensemble of n×n random matrices. We show that the probability of finding no eigenvalues in the interval [−1,z] for a random matrix chosen from the ensemble, viewed…
Orthogonal polynomial ensembles in probability theory
We survey a number of models from physics, statistical mechanics, probability theory and combinatorics, which are each described in terms of an {it orthogonal polynomial ensemble}. The most prominent…
q-Distributions on boxed plane partitions
We introduce elliptic weights of boxed plane partitions and prove that they give rise to a generalization of MacMahon’s product formula for the number of plane partitions in a box. We then focus on…
Gap probabilities in the finite and scaled Cauchy random matrix ensembles
The probabilities for gaps in the eigenvalue spectrum of finite N?N random unitary ensembles on the unit circle with a singular weight, and the related Hermitian ensembles on the line with Cauchy…
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