The generating function for the Bessel point process and a system of coupled Painlev\'{e} V equations

TL;DR

This paper derives a system of coupled Painlevé V equations for the joint probability generating function of occupancy numbers in the Bessel point process, extending previous results and enabling analysis of various probabilistic quantities.

Contribution

It generalizes Tracy and Widom's result to multiple intervals, expressing the generating function via coupled Painlevé V equations derived from a Riemann-Hilbert problem.

Findings

Expressed the generating function as a Fredholm determinant.

Derived coupled Painlevé V equations from a Lax pair.

Applied results to gap probabilities and Hankel determinants.

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 Painlev\'{e} 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.