Abstract:
This paper is concerned with certain connections between the ensemble of n×n unitary matrices – specifically the characteristic function of the random variable tr(U) – and combinatorics – specifically Ulam's problem concerning the distribution of the length of the longest increasing subsequence in permutation groups – and the appearance of Painlevé functions in the answers to apparently unrelated questions. Among the results is a representation in terms of a Painlevé V function for the characteristic function of tr(U) and (using recent results of Baik, Deift and Johansson) an expression in terms of a Painlevé II function for the limiting distribution of the length of the longest increasing subsequence in the hyperoctahedral groups.
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Received: 2 December 1998 / Accepted: 12 May 1999
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Tracy, C., Widom, H. Random Unitary Matrices, Permutations and Painlevé. Comm Math Phys 207, 665–685 (1999). https://doi.org/10.1007/s002200050741
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DOI: https://doi.org/10.1007/s002200050741