On $\beta=6$ Tracy-Widom distribution and the second Calogero-Painlev\'e system

Abstract

The Calogero-Painlev\'e systems were introduced in 2001 by K. Takasaki as a natural generalization of the classical Painlev\'e equations to the case of the several Painlev\'e ``particles'' coupled via the Calogero type interactions. In 2014, I. Rumanov discovered the remarkable fact that a particular case of the second Calogero-Painlev\'e II equation describes the Tracy-Widom distribution function for the general beta-ensembles with even values of the parameter beta. Most recently, in 2017 work of M. Bertola, M. Cafasso, and V. Rubtsov, it was proven that all Calogero-Painlev\'e systems are Lax integrable, and hence their solutions admit a Riemann-Hilbert representation. This important observation has opened the door to rigorous, based on the Deift-Zhou nonlinear steepest descent method, asymptotic analysis of the Calogero-Painlev\'e equations. This 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 $\beta =1, 2, 4.$ In this work we shall start an asymptotic analysis of the Calogero-Painlev\'e system with a special focus on the Calogero-Painlev\'e system corresponding to $\beta = 6$ Tracy-Widom distribution function. The principle technical challenge is the implementation of the nonlinear steepest descent approach beyond the $2\times 2$ matrix dimension of the corresponding Riemann-Hilbert problem; in our case, it is $6\times 6$.