riemann–hilbert approach to gap probabilities for the bessel process》.pdf

Physica D 295–296 (2015) 103–121
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Physica D
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Riemann–Hilbert approach to gap probabilities for the Bessel process
Manuela Girotti
Department of Mathematics and Statistics, Concordia University, 1455 de Maisonneuve Ouest, Montréal, Québec, Canada, H3G 1M8
h i g h l i g h t s
• Gap probabilities for the single and multi-time Bessel process are considered.
• Fredholm determinants of such processes are related to Riemann–Hilbert problems.
• In the single-time case the Fredholm determinant is related to Painlevé 3 equation.
a r t i c l e i n f o a b s t r a c t
Article history: We consider the gap probability for the Bessel process in the single-time and multi-time case. We prove
Received 7 September 2013 that the scalar and matrix Fredholm determinants of such process can be expressed in terms of deter-
Received in revised form minants of integrable kernels in the sense of Its–Izergin–Korepin–Slavnov and thus related to suitable
10 December 2014 Riemann–Hilbert problems. In the single-time case, we construct a Lax pair formalism and we derive a
Accepted 23 December 2014
Painlevé III equation related to the Fredholm determinant.
Available online 1 January 2015 ©
Communicated by . Miller 2015 Elsevier . All rights reserved.
Keywords:
Determinantal point processes
Riemann–Hilbert problem
Integrable kernel
1. Introduction
Many models in Mathematical Physics rely on the notion of a determinantal random point process (DPP). A few examples are offered
by the