1998/11/30 by Craig A. Tracy, Harold Widom · 3 citations
Computer Science · Mathematics · Physics and Astronomy · #Advanced Combinatorial Mathematics #Bayesian Methods and Mixture Models #Random Matrices and Applications #math.CO #math.PR #msc:05A15 #msc:47B35 #msc:60C05 #msc:82B23 #nlin.SI #solv-int
paper · pdf · doi:10.1007/s002200050741
published as Commun. Math. Phys. 207 (1999), 665-685 · 21 pages, 1 figure. Revised paper simplifies the statement of Theorem 1 and adds some additional references
arxiv created 1999/05/12 · openalex publication_date 1999/11/01 · arxiv updated 2009/11/30 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28
This paper is concerned with certain connections between the ensemble of n x 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 Painleve functions in the answers to apparently unrelated questions. Among the results is a representation in terms of a Painleve V function for the characteristic function of tr(U) and (using recent results of Baik, Deift and Johansson) an expression in terms of a Painleve II function for the limiting distribution of the length of the longest increasing subsequence in the hyperoctahedral group.