On the Open Question of The Tracy-Widom Distribution of \beta-Ensemble With \beta=6

TL;DR

This paper completely determines the Tracy-Widom distribution for the eta-ensemble with eta=6 by analyzing a specific nonlinear ODE and solving an open problem about the existence of global smooth solutions, confirming the distribution's properties.

Contribution

The authors solve the open question of the existence of a global smooth solution to a key ODE for eta=6, establishing the Tracy-Widom distribution in this case.

Findings

Confirmed the existence of a unique global smooth solution with specific boundary conditions.

Parameterization of solutions by (c_1, c_2) and identification of the unique solution corresponding to (0,0).

Proved the solution is bounded and satisfies the boundary conditions of the Bloemendal-Virág equation.

Abstract

We determine completely the Tracy-Widom distribution for Dyson's \beta-ensemble with \beta=6. The problem of the Tracy-Widom distribution of \beta-ensemble for general \beta>0 has been reduced to find out a bounded solution of the Bloemendal-Vir\'ag equation with a specified boundary. Rumanov proposed a Lax pair approach to solve the Bloemendal-Vir\'ag equation for even integer \beta. He also specially studied the \beta=6 case with his approach and found a second order nonlinear ordinary differential equation (ODE) for the logarithmic derivative of the Tracy-Widom distribution for \bea=6. Grava et al. continued to study \beta=6 and found Rumanov's Lax pair is gauge equivalent to that of Painlev\'e II in this case. They started with Rumanov's basic idea and came down to two auxiliary functions {\alpha}(t) and q_2(t), which satisfy a coupled first-order ODE. The open question by Grava et al. asks whether a global smooth solution of the ODE with boundary condition {\alpha}(\infty)=0 and q_2(\infty)=1 exists. By studying the linear equation that is associated with q_2 and {\alpha}, we give a positive answer to the open question. Moreover, we find that the solutions of the ODE with {\alpha}(\infty)=0 and q_2(\infty)=1 are parameterized by c_1 and c_2 . Not all c_1 and c_2 give global smooth solutions. But if (c_1, c_2) \in R_{smooth}, where R_{smooth} is a large region containing (0,0), they do give. We prove the constructed solution is a bounded solution of the Bloemendal-Vir\'ag equation with the required boundary condition if and only if (c_1,c_2)=(0,0).