Discrete gap probabilities and discrete Painlevé equations

Abstract

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].