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Discrete gap probabilities and discrete Painlevé equations. (English) Zbl 1034.39013

Summary: 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 \(\{s, s+1, \ldots\}\), can be expressed, respectively, through solutions of discrete Painlevé II (dPII) and Painlevé V (dPV) equations.
These Fredholm determinants can also be viewed as distribution functions of the first part of the random partitions distributed according to a Poissonized Plancherel measure and a \(z\)-measure, or as normalized Toeplitz determinants with symbols \(e^{\eta(\zeta+\zeta^{-1})}\) and \((1+\sqrt{\xi}\zeta)^z (1+\sqrt{\xi}/\zeta)^{z'}\).
The proofs are based on a general formalism involving discrete integrable operators and discrete Riemann-Hilbert problems. A continuous version of the formalism has been worked out by A. Borodin and P. Deift [Commun. Pure Appl. Math. 55, 1160–1230 (2002; Zbl 1033.34089)].

MSC:

39A20 Multiplicative and other generalized difference equations
35Q15 Riemann-Hilbert problems in context of PDEs
37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
39A12 Discrete version of topics in analysis

Citations:

Zbl 1033.34089
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References:

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