On $β=6$ Tracy-Widom distribution and the second Calogero-Painlevé system
Alexander Its, Andrei Prokhorov
TL;DR
The work advances a rigorous program to derive the right- and left-tail asymptotics of the β=6 Tracy–Widom distribution by embedding the problem into a 6×6 Riemann–Hilbert problem derived from the Calogero–Painlevé II system. It develops a full nonlinear steepest-descent analysis, constructing global and local parametrices from parabolic-cylinder, Airy, Bessel, and confluent-hypergeometric models, and reduces the problem to a small-norm RH problem to extract leading and subleading terms of the CP eigenvalues. The results connect the CP dynamics with the Bloemendal–Virág framework for Fβ, enabling a route to the Borot–Nadal-type left-tail asymptotics for β=6 and offering a blueprint for extending rigorous tails to other even β. The study also delineates open questions about solvability, uniqueness, and the precise constant term χ, outlining a path toward a complete rigorous proof of the β=6 conjecture and its β-generalization.
Abstract
The Calogero-Painlevé systems were introduced in 2001 by K. Takasaki as a natural generalization of the classical Painlevé equations to the case of the several Painlevé ``particles'' coupled via the Calogero type interactions. In 2014, I. Rumanov discovered the remarkable fact that a particular case of the second Calogero-Painlevé 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-Painlevé 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-Painlevé 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 $β=1, 2, 4.$ 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. 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 $6\times 6$.