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 , where for and otherwise, and the average is with respect to the joint eigenvalue distribution of the GUE, as well as to the evaluation of . Of particular interest are 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 and F N (λ;a) for the other classical matrix ensembles.
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Received: 27 June 2000 / Accepted: 8 December 2000
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Forrester, P., Witte, N. Application of the τ-Function Theory¶of Painlevé Equations to Random Matrices:¶PIV, PII and the GUE. Commun. Math. Phys. 219, 357–398 (2001). https://doi.org/10.1007/s002200100422
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DOI: https://doi.org/10.1007/s002200100422