On β=6 Tracy-Widom distribution and the second Calogero-Painlevé system

Abstract

The Calogero-Painlevé systems were introduced in 2001 by K. Takasaki as a natural generalization of theclassical Painlevé equations to the case of the several Painlevé “particles” coupled via the Calogero type interactions. In 2014, I. Rumanov discovered a remarkable fact that a particular case of the second Calogero–Painlevé IIequation describes the Tracy-Widom distribution function for the general beta-ensembles with the even valuesof parameter beta. Most recently, in 2017 work of M. Bertola, M. Cafasso, and V. Rubtsov, it was proven that all Calogero-Painlevé systems are Lax integrable, and hence their solutions admit a Riemann-Hilbert representation. This important observation has opened the door to rigorous asymptotic analysis of the Calogero Painlevé equations which in turn yields the possibility of rigorous evaluation of the asymptotic behavior of the Tracy-Widom distributions for the values of beta beyond the classical β = 1,2,4. In this work we shall start an asymptotic analysis of the Calogero-Painlevé system with a special focus on the Calogero-Painlevé system corresponding to β = 6 Tracy-Widom distribution function.