Duke Mathematical Journal

Discrete gap probabilities and discrete Painlevé equations

Alexei Borodin

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We prove that Fredholm determinants of the form $\det(1-K\sb s)$, where $K\sb s$ is the restriction of either the discrete Bessel kernel or the discrete $\sb 2F\sb 1$-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 $e\sp {\eta(\zeta+\zeta\sp {-1})}$ and $(1 +\sqrt {\xi}\zeta)\sp z(1 +\sqrt {\xi}/\zeta)\sp {z\sp \prime}$.

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 in [BD].

Article information

Duke Math. J., Volume 117, Number 3 (2003), 489-542.

First available in Project Euclid: 26 May 2004

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Zentralblatt MATH identifier

Primary: 39Axx: Difference equations {For dynamical systems, see 37-XX; for dynamic equations on time scales, see 34N05}
Secondary: 35Q15: Riemann-Hilbert problems [See also 30E25, 31A25, 31B20] 37K10: Completely integrable systems, integrability tests, bi-Hamiltonian structures, hierarchies (KdV, KP, Toda, etc.)


Borodin, Alexei. Discrete gap probabilities and discrete Painlevé equations. Duke Math. J. 117 (2003), no. 3, 489--542. doi:10.1215/S0012-7094-03-11734-2. https://projecteuclid.org/euclid.dmj/1085598403

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