Application of the $τ$-Function Theory of Painlevé Equations to Random Matrices: PIV, PII and the GUE

Details Application of the $τ$-Function Theory of Painlevé Equations to Random Matrices: PIV, PII and the GUE Application of the $τ$-Function Theory of Painlevé Equations to Random Matrices: PIV, PII and the GUE

2001-03-21

arxiv.org/abs/math-ph/0103025v1

Physics

Mathematical Physics

40 pages, Latex2e plus AMS and XY packages. to appear Commun. Math. Phys

Scientific paper

10.1007/s002200100422

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 $\tilde{E}_N(\lambda;a) := \Big < \prod_{l=1}^N \chi_{(-\infty, \lambda]}^{(l)} (\lambda - \lambda_l)^a \Big>$, where $ \chi_{(-\infty, \lambda]}^{(l)} = 1$ for $\lambda_l \in (-\infty, \lambda]$ and $ \chi_{(-\infty, \lambda]}^{(l)} = 0$ otherwise, and the average is with respect to the joint eigenvalue distribution of the GUE, as well as to the evaluation of $F_N(\lambda;a) := \Big < \prod_{l=1}^N (\lambda - \lambda_l)^a \Big >$. Of particular interest are $\tilde{E}_N(\lambda;2)$ and $F_N(\lambda;2)$, and their scaled limits, which give the distribution of the largest eigenvalue and the density respectively. Our results are obtained by applying the Okamoto $\tau$-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 $\tilde{E}_N(\lambda;a)$ and $F_N(\lambda;a)$ for the other classical matrix ensembles.

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