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Register a new userOn the Open Question of The Tracy-Widom Distribution of beta-Ensemble With beta=6
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Yuqi Li
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We determine completely the Tracy-Widom distribution for Dysons 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-Virag equation with a specified boundary. Rumanov proposed a Lax pair approach to solve the Bloemendal-Virag 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 Rumanovs Lax pair is gauge equivalent to that of Painleve II in this case. They started with Rumanovs 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-Virag equation with the required boundary condition if and only if (c_1,c_2)=(0,0).
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