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The Calogero-Painlev\'e systems were introduced in 2001 by K. Takasaki as anatural generalization of the classical Painlev\'e equations to the case of theseveral Painlev\'e "particles" coupled via the Calogero type interactions. In2014, I. Rumanov discovered a remarkable fact that a particular case of thesecond Calogero-Painlev\'e II equation describes the Tracy-Widom distributionfunction for the general beta-ensembles with the even values of parameter beta.Most recently, in 2017 work of M. Bertola, M. Cafasso, and V. Rubtsov, it wasproven that all Calogero-Painlev\'e systems are Lax integrable, and hence theirsolutions admit a Riemann-Hilbert representation. This important observationhas opened the door to rigorous asymptotic analysis of the Calogero-Painlev\'eequations which in turn yields the possibility of rigorous evaluation of theasymptotic behavior of the Tracy-Widom distributions for the values of betabeyond the classical $\beta =1, 2, 4.$ In this work we shall start anasymptotic analysis of the Calogero-Painlev\'e system with a special focus onthe Calogero-Painlev\'e system corresponding to $\beta = 6$ Tracy-Widomdistribution function.  Some technical details were omitted here and will be presented in the nextversion of the text.

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