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"Integrable" gap probabilities for the Generalized Bessel process

Banff International Research Station for Mathematical Innovation and Discovery

We consider the gap probability for the Generalized Bessel process, a determinantal point process which arises as critical limiting kernel near the hard edge of the spectrum of a certain random matrix ensemble. We prove that such probability can be expressed in terms of the Fredholm determinant of a suitable Its-Izergin-Korepin-Slavnov integrable operator and linked in a canonical way to Riemann-Hilbert problem. Starting from the RH problem, we can construct a Lax pair and we can link the gap probability to the Painleve III hierarchy. Moreover, we are able to construct a system of two coupled Hamiltonians which can be hopefully identified with the 2-dimensional Garnier system LH(2+3). The talk is based on some previous results and an on-going project with Dr. Mattia Cafasso.

Mathematics; Ordinary differential equations; Dynamical systems and ergodic theory; Dynamical systems

video/mp4

Author affiliation: Université Catholique de Louvain (UCL)

2017-04-05

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/61111

Unreviewed

http://creativecommons.org/licenses/by-nc-nd/4.0/