Painleve Representation of Tracy-Widomβ Distribution for β=6

In Rumanov (J Math Phys 56:013508, 2015), we found explicit Lax pairs for the soft edge of beta ensembles with even integer values of . Using this general result, the case is further considered here. This is the smallest even , when the corresponding Lax pair and its relation to Painlev, II (PII) have not been known before, unlike cases and 4. It turns out that again everything can be expressed in terms of the Hastings-McLeod solution of PII. In particular, a second order nonlinear ordinary differential equation (ODE) for the logarithmic derivative of Tracy-Widom distribution for involving the PII function in the coefficients is found, which allows one to compute asymptotics for the distribution function. The ODE is a consequence of a linear system of three ODEs for which the local singularity analysis yields series solutions with exponents in the set 4/3, 1/3 and -2/3.