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arXiv math-ph/0203049math-phmath.MP
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{ \bf $τ$ -正交和辛矩阵系综间隙概率的函数估计}

{\bf $τ$-Function Evaluation of Gap Probabilities in Orthogonal and Symplectic Matrix Ensembles}

P. J. Forrester, N. S. Witte

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摘要：

最近有人强调，所有已知的精确评估间隙概率的经典酉矩阵合奏，事实上是$τ$ -功能，某些Painlevé系统。我们表明，所有的精确评估的间隙概率为经典的正交矩阵合奏，无论是已知的或可从现有的文献，同样是$τ$ -某些Painlevé系统的功能。在辛矩阵合奏的情况下，所有精确的评价，无论是已知的或可从现有的文献中推导出，被确定为两个$τ$ -功能，这两个对应于满足相同的微分方程，不同的只是在边界条件的Hamilton的平均值。此外，这两个$τ$ -函数的乘积给出了相应酉对称情况下的间隙概率，而这些$τ$ -函数之一是相应正交对称情况下的间隙概率。

英文摘要：

It has recently been emphasized that all known exact evaluations of gap probabilities for classical unitary matrix ensembles are in fact $τ$-functions for certain Painlevé systems. We show that all exact evaluations of gap probabilities for classical orthogonal matrix ensembles, either known or derivable from the existing literature, are likewise $τ$-functions for certain Painlevé systems. In the case of symplectic matrix ensembles all exact evaluations, either known or derivable from the existing literature, are identified as the mean of two $τ$-functions, both of which correspond to Hamiltonians satisfying the same differential equation, differing only in the boundary condition. Furthermore the product of these two $τ$-functions gives the gap probability in the corresponding unitary symmetry case, while one of those $τ$-functions is the gap probability in the corresponding orthogonal symmetry case.

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