The generating function for the Bessel point process and a system of coupled Painlevé V equations

Abstract

We study the joint probability generating function for $k$ occupancy numbers on disjoint intervals in the Bessel point process. This generating function can be expressed as a Fredholm determinant. We obtain an expression for it in terms of a system of coupled Painlevé V equations, which are derived from a Lax pair of a Riemann-Hilbert problem. This generalizes a result of Tracy and Widom [24], which corresponds to the case $k=1$. We also provide some examples and applications. In particular, several relevant quantities can be expressed in terms of the generating function, like the gap probability on a union of disjoint bounded intervals, the gap between the two smallest particles, and large $n$ asymptotics for $n\times n$ Hankel determinants with a Laguerre weight possessing several jumps discontinuities near the hard edge.

27 pages, 2 figures