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学术报告二十五:Applications in random matrix theory of a PIII' $\tau$-function sequence from Okamoto's Hamiltonian formulation

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时间:2020-06-16 16:30 

数学与统计学院学术报告[2020] 025

(高水平大学建设系列报告378)

 

报告题目:Applications in random matrix theory of a PIII' $\tau$-function sequence from Okamoto's Hamiltonian formulation   

报告人:代丹教授 (香港城市大学

报告时间:202006179001000

直播平台及链接: 腾讯会议485 353 797

报告内容:We consider the singular linear statistic of the Laguerre unitary ensemble consisting of the sum of the reciprocal of the eigenvalues. It is observed that the exponential generating function for this statistic can be written as a Toeplitz determinant with entries given in terms of particular K-Bessel functions. Earlier studies have identified the same determinant, but with the K-Bessel functions replaced by I-Bessel functions, as relating to the hard edge scaling limit of a generalized gap probability for the Laguerre unitary ensemble, in the case of non-negative integer Laguerre parameter. We show that the Toeplitz determinant formed from an arbitrary linear combination of these two Bessel functions occurs as a $\tau$-function sequence in Okamoto's Hamiltonian formulation of Painleve III', and consequently the logarithmic derivative of both Toeplitz determinants satisfies the same $\sigma$-form Painleve III' differential equation, giving an explanation of a fact which can be observed from earlier results. In addition, some insights into the relationship between this characterization of the generating function, and its characterization in the $n \to \infty$ limit, both with the Laguerre parameter $\alpha$ fixed, and with $\alpha=n$ (this latter circumstance being relevant to an application to the distribution of the Wigner time delay statistic), are given.

This is a joint work with Peter J. Forrester (University of Melbourne, Australia) and Shuai-Xia Xu (Sun Yat-sen University, China)

报告人简历:

代丹博士,2002年本科毕业于复旦大学数学系,2006年于香港城市大学获博士学位,之后赴比利时荷语鲁汶大学从事博士后研究。2008年起,代博士开始在香港城市大学工作,历任助理教授,副教授。代博士主要从事渐近分析,特殊函数以及随机矩阵研究,在Advances in MathematicsCommunications in Mathematical PhysicsSIAM Journal on Mathematical AnalysisStudies in Applied MathematicsNonlinearity等杂志发表多篇论文,并主持香港研究资助局多项研究项目。


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