潘勒韦方程的$τ$ -函数理论在随机矩阵PIV、PII和GUE中的应用
Application of the $τ$-Function Theory of Painlevé Equations to Random Matrices: PIV, PII and the GUE
摘要：
Tracy和Widom分别根据PIV和PII超越性评估了有限和缩放无限GUE的最大特征值的累积分布。我们概括这些结果的评价$\tilde{E}_N(λ;a) := \Big < \prod_{l=1}^N χ_{(-\infty, λ]}^{(l)} (λ- λ_l)^a \Big>$，其中$ χ_{(-\infty, λ]}^{(l)} = 1$为$λ_l \in (-\infty, λ]$和$ χ_{(-\infty, λ]}^{(l)} = 0$，否则，和平均值是相对于联合特征值分布的GUE，以及评价$F_N(λ;a) := \Big < \prod_{l=1}^N (λ- λ_l)^a \Big >$。特别感兴趣的是$\tilde{E}_N(λ;2)$和$F_N(λ;2)$，以及它们的缩放极限，它们分别给出了最大特征值和密度的分布。我们的研究结果是通过应用冈本PDMATHAMT-函数理论的PIV和PII，为此，我们给出了一个自我包含的介绍，根据最近的工作，Noumi和山田。我们指出，同样的方法可以用来研究数量$\tilde{E}_N(λ;a)$和$F_N(λ;a)$的其他经典矩阵系综。
英文摘要：
Tracy and Widom have evaluated the cumulative distribution of the largest eigenvalue for the finite and scaled infinite GUE in terms of a PIV and PII transcendent respectively. We generalise these results to the evaluation of $\tilde{E}_N(λ;a) := \Big < \prod_{l=1}^N χ_{(-\infty, λ]}^{(l)} (λ- λ_l)^a \Big>$, where $ χ_{(-\infty, λ]}^{(l)} = 1$ for $λ_l \in (-\infty, λ]$ and $ χ_{(-\infty, λ]}^{(l)} = 0$ otherwise, and the average is with respect to the joint eigenvalue distribution of the GUE, as well as to the evaluation of $F_N(λ;a) := \Big < \prod_{l=1}^N (λ- λ_l)^a \Big >$. Of particular interest are $\tilde{E}_N(λ;2)$ and $F_N(λ;2)$, and their scaled limits, which give the distribution of the largest eigenvalue and the density respectively. Our results are obtained by applying the Okamoto $τ$-function theory of PIV and PII, for which we give a self contained presentation based on the recent work of Noumi and Yamada. We point out that the same approach can be used to study the quantities $\tilde{E}_N(λ;a)$ and $F_N(λ;a)$ for the other classical matrix ensembles.
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