Application of the τ-function theory of Painlevé equations to random matrices: PVI , the JUE, CyUE, cJUE and scaled limits
Application of the τ-function theory of Painlevé equations to random matrices: PVI , the JUE, CyUE, cJUE and scaled limits
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Painlevé 方程的 τ 函数理论在随机矩阵中的应用：PVI、JUE、CyUE、cJUE 和缩放极限
DOI：
10.1017/s0027763000008801
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发表时间：
2002
影响因子：
0.8
通讯作者：
N. Witte
中科院分区：
文献类型：
--
作者：
P. Forrester;N. Witte
Okamoto在Gauss超几何方程解的基础上，得到了PVI系统的τ-函数序列，它表示为双Wronskian行列式。从高斯超几何方程的积分解开始，我们表明行列式可以重新表示为多维积分，而这些又可以通过Jacobi酉系综（JUE）和Cauchy酉系综的本征值概率密度函数的平均值来识别。（CyUE）（后者相当于圆雅可比酉系综（cJUE））。因此，这些平均值，这取决于四个连续参数和离散参数N，可以被描述为二阶二次方程的解决方案所满足的哈密顿PVI理论。我们表明，哈密顿量也满足离散PV方程相关的方程，从而提供了另一种表征的差分方程。在cJUE的情况下，考虑了谱奇异性的尺度极限，给出了某个四参数平均值的σ形式的广义超越PV的估计.应用于计算圆形酉系综（CUE）及其相应的尺度系综的间距分布，给出了比已知公式更简洁的公式;在缩放的拉盖尔正交系综（LOE）中的硬边缘间隙概率的表达式（参数a为非负整数）和拉盖尔辛系综（LSE）（参数a为偶数非负整数）分别作为辛群和正交群上的有限维组合积分，用于某些定向渗流模型中末次通过时间的累积分布函数的计算;对参数α = 0的有限LOE和LSE中最大本征值的τ函数计算，以及二维Ising模型中对角-对角自旋-自旋关联的表征。
Abstract Okamoto has obtained a sequence of τ-functions for the PVI system expressed as a double Wronskian determinant based on a solution of the Gauss hypergeometric equation. Starting with integral solutions of the Gauss hypergeometric equation, we show that the determinant can be re-expressed as multidimensional integrals, and these in turn can be identified with averages over the eigenvalue probability density function for the Jacobi unitary ensemble (JUE), and the Cauchy unitary ensemble (CyUE) (the latter being equivalent to the circular Jacobi unitary ensemble (cJUE)). Hence these averages, which depend on four continuous parameters and the discrete parameter N, can be characterised as the solution of the second order second degree equation satisfied by the Hamiltonian in the PVI theory. We show that the Hamiltonian also satisfies an equation related to the discrete PV equation, thus providing an alternative characterisation in terms of a difference equation. In the case of the cJUE, the spectrum singularity scaled limit is considered, and the evaluation of a certain four parameter average is given in terms of the general PV transcendent in σ form. Applications are given to the evaluation of the spacing distribution for the circular unitary ensemble (CUE) and its scaled counterpart, giving formulas more succinct than those known previously; to expressions for the hard edge gap probability in the scaled Laguerre orthogonal ensemble (LOE) (parameter a a non-negative integer) and Laguerre symplectic ensemble (LSE) (parameter a an even non-negative integer) as finite dimensional combinatorial integrals over the symplectic and orthogonal groups respectively; to the evaluation of the cumulative distribution function for the last passage time in certain models of directed percolation; to the τ-function evaluation of the largest eigenvalue in the finite LOE and LSE with parameter a = 0; and to the characterisation of the diagonal-diagonal spin-spin correlation in the two-dimensional Ising model.
影响因子：
2.4
作者：
H. Sakai
通讯作者：
H. Sakai
仿射韦尔群和 Painleve；
DOI：
--
发表时间：
2019
期刊：
影响因子：
--
作者：
Kagawa;K. and Otani;M.;中本敦浩;Takao Suzuki
通讯作者：
Takao Suzuki