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

Abstract

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 ${\tilde{E}}_N(\lambda ;a):=⟨{\prod }_{l=1}^N{\chi }_{(-\infty ,\lambda ]}^{(l)}(\lambda -{\lambda }_l{)}^a⟩$, 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 $F_N(\lambda ;a):=⟨{\prod }_{l=1}^N(\lambda -{\lambda }_l{)}^a⟩$. 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.

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

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