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Zbl 1042.82019
Forrester, P. J.; Witte, N. S.
Application of the $\tau$-function theory of Painlevé equations to random matrices: PIV, PII and the GUE. (English)
[J] Commun. Math. Phys. 219, No. 2, 357-398 (2001). ISSN 0010-3616; ISSN 1432-0916/e

Summary: {\it C. A. Tracy} and {\it H. Widom} [Commun. Math. Phys. 159, 151--174 (1994; Zbl 0789.35152))] have evaluated the cumulative distribution of the largest eigenvalue for the finite and scaled infinite GUE in terms of a PTV and PII transcendent respectively. We generalise these results to the evaluation of $$\widetilde E_N(\lambda:a): =\left\langle \prod^N_{l=1} \chi^{(l)}_{(-\infty, \lambda]} (\lambda-\lambda_l)^a \right\rangle,$$ where $\chi^{(l)}_{(-\infty, \lambda]} =1$ for $\lambda_l\in(-\infty,\lambda]$ and $\chi^{(l)}_{(-\infty, \lambda]}=0$ otherwise, and the average is with respect to the joint eigenvalue distribution of the GUE, as well as to the evaluation of $$F_N(\lambda;a): = \left\langle \prod^N_{l=1}(\lambda-\lambda_l)^a \right\rangle.$$ Of particular interest are $\widetilde E_N(\lambda;2)$ and $F_N(\lambda;2)$, and their scaled limits, which gives 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 [cf, {\it K. Okamoto}, [Studies on the Painlevé Equations. III. Second and Fourth Painlevé Equations, $P_{II}$ and $P_{IV}$. Math. Ann. 275, 221--255 (1986; Zbl 0589.58008)], for which we give a self contained presentation based on the recent work of {\it M. Noumi} and {\it Y. Yamada} [Nagoya Math. J. 153, 53--86 (1999; Zbl 0932.34088)]. We point out that the same approach can be used to study the quantities $\widetilde E_N(\lambda;a)$ and $F_N(\lambda;a)$ for the other classical matrix ensembles.

MSC 2000:: *82B41 Random walks, etc. (statistical mechanics)
15A52 Random matrices
34M55 Painlevé and other special equations

Citations: Zbl 0789.35152; Zbl 0589.58008; Zbl 0932.34088

Cited in: Zbl 1056.15023

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