Random Unitary Matrices, Permutations and Painlevé
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.
 Publication:

Communications in Mathematical Physics
 Pub Date:
 1999
 DOI:
 10.1007/s002200050741
 arXiv:
 arXiv:math/9811154
 Bibcode:
 1999CMaPh.207..665T
 Keywords:

 Mathematics  Combinatorics;
 Mathematics  Probability;
 Nonlinear Sciences  Exactly Solvable and Integrable Systems;
 05A15;
 47B35;
 60C05;
 82B23
 EPrint:
 21 pages, 1 figure. Revised paper simplifies the statement of Theorem 1 and adds some additional references