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

2017/09/21 by Christophe Charlier, Charlier, Christophe, Antoine Doeraene +1
Mathematics · #Advanced Mathematical Identities #FOS: Physical sciences #Mathematical Physics (math-ph) #Mathematical functions and polynomials #Random Matrices and Applications

paper · pdf · doi:10.48550/arxiv.1709.07365

openalex publication_date 2017/09/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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× n Hankel determinants with a Laguerre weight possessing several jumps discontinuities near the hard edge.

Related

• Riemann-Hilbert approach to gap probabilities for the Bessel process
• Discrete gap probabilities and discrete Painleve equations
• Applications in random matrix theory of a PIII' τ-function sequence from Okamoto's Hamiltonian formulation
• The generating function for the Airy point process and a system of\n coupled Painlev 'e II equations
• Gap probability at the hard edge for random matrix ensembles with pole singularities in the potential