Riemann-Hilbert approach to gap probabilities for the Bessel process

2013/06/24 by Girotti, Manuela
#Exactly Solvable and Integrable Systems (nlin.SI) #FOS: Physical sciences #Mathematical Physics (math-ph)

paper · doi:10.48550/arxiv.1306.5663

Abstract

We consider the gap probability for the Bessel process in the single-time and multi-time case. We prove that the scalar and matrix Fredholm determinants of such process can be expressed in terms of determinants of integrable kernels à la Its-Izergin-Korepin-Slavnov and thus related to suitable Riemann-Hilbert problems. In the single-time case, we construct a Lax pair formalism and we derive a Painlevé III equation related to the Fredholm determinant.

Related

• The generating function for the Bessel point process and a system of coupled Painlevé V equations
• Discrete gap probabilities and discrete Painleve equations
• The asymptotics a Bessel-kernel determinant which arises in Random Matrix Theory
• Gap probability at the hard edge for random matrix ensembles with pole singularities in the potential
• Large gap asymptotics at the hard edge for product random matrices and\n Muttalib-Borodin ensembles