Applications in random matrix theory of a PIII’ τ-function sequence from Okamoto’s Hamiltonian formulation
Applications in random matrix theory of a PIII’ τ-function sequence from Okamoto’s Hamiltonian formulation
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DOI：
10.1142/s2010326322500149
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发表时间：
2019-09
期刊：
影响因子：
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通讯作者：
D. Dai;P. Forrester;Shuai‐Xia Xu
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文献类型：
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作者：
D. Dai;P. Forrester;Shuai‐Xia Xu
我们考虑由特征值倒数之和组成的拉盖尔酉系综 (LUE) 的奇异线性统计量。据观察，该统计量的指数生成函数可以写为托普利茨行列式，其中的条目以特定的[公式：参见文本]贝塞尔函数给出。早期的研究已经确定了相同的行列式，但在非负整数拉盖尔参数的情况下，将[公式：参见文本]贝塞尔函数替换为[公式：参见文本]贝塞尔函数，与LUE广义间隙概率的硬边缘缩放限制相关。我们证明，由这两个 Bessel 函数的任意线性组合形成的 Toeplitz 行列式在冈本 Painlevé III 的哈密顿公式中作为 [公式：参见文本] 函数序列出现[公式：参见文本]，因此两个 Toeplitz 行列式的对数导数满足相同的 [公式：参见文本]-形式 Painlevé III[公式：参见文本] 微分方程，给出了可以从以下观察到的事实的解释较早的结果。此外，还给出了对生成函数的这种表征与其在[公式：参见文本]极限中的表征之间关系的一些见解，无论是拉盖尔参数[公式：参见文本]固定，还是[公式：参见文本]（后一种情况与维格纳时间延迟统计分布的应用相关）。
We consider the singular linear statistic of the Laguerre unitary ensemble (LUE) 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 [Formula: see text] Bessel functions. Earlier studies have identified the same determinant, but with the [Formula: see text] Bessel functions replaced by [Formula: see text] Bessel functions, as relating to the hard edge scaling limit of a generalized gap probability for the LUE, 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 [Formula: see text]-function sequence in Okamoto’s Hamiltonian formulation of Painlevé III[Formula: see text], and consequently the logarithmic derivative of both Toeplitz determinants satisfies the same [Formula: see text]-form Painlevé III[Formula: see text] 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 [Formula: see text] limit, both with the Laguerre parameter [Formula: see text] fixed, and with [Formula: see text] (this latter circumstance being relevant to an application to the distribution of the Wigner time delay statistic), are given.