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Application of the τ-function theory of Painlevé equations to random matrices: PIV, PII and the GUE. (English) Zbl 1042.82019

Summary: C. A. Tracy and 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

E ˜ N (λ:a):= l=1 N χ (-,λ] (l) (λ-λ l ) a ,

where χ (-,λ] (l) =1 for λ l (-,λ] and χ (-,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):= l=1 N (λ-λ l ) a ·

Of particular interest are E ˜ N (λ;2) and F N (λ;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 τ-function theory of PIV and PII [cf, 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 M. Noumi and 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 E ˜ N (λ;a) and F N (λ;a) for the other classical matrix ensembles.

82B41Random walks, random surfaces, lattice animals, etc. (statistical mechanics)
15A52Random matrices (MSC2000)
34M55Painlevé and other special equations; classification, hierarchies