Discrete Gap Probabilities and Discrete Painlevé Equations

We prove that Fredholm determinants of the form det(1−Ks), where Ks is the restriction of either the discrete Bessel kernel or the discrete 2 F1-kernel to {s, s + 1, . . . }, 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 −1) and (1 + √ ξζ )z(1 + √ ξ/ζ )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 in [BD].