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arXiv · 1812.00522

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

Yuqi Li
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Abstract

We determine completely the Tracy-Widom distribution for Dyson's β-ensemble with β=6. The problem of the Tracy-Widom distribution of β-ensemble for general β>0 has been reduced to find out a bounded solution of the Bloemendal-Virág equation with a specified boundary. Rumanov proposed a Lax pair approach to solve the Bloemendal-Virág equation for even integer β. He also specially studied the β=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 β=6 and found Rumanov's Lax pair is gauge equivalent to that of Painlevé II in this case. They started with Rumanov's basic idea and came down to two auxiliary functions α(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 α(\infty)=0 and q_2(\infty)=1 exists. By studying the linear equation that is associated with q_2 and α, we give a positive answer to the open question. Moreover, we find that the solutions of the ODE with α(\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-Virág equation with the required boundary condition if and only if (c_1,c_2)=(0,0).

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Yuqi Li. 2018-12-03. On the Open Question of The Tracy-Widom Distribution of β-Ensemble With β=6. https://arxiv.org/abs/1812.00522

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