Random Unitary Matrices, Permutations and Painleve
Abstract
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.
Cite
@article{arxiv.math/9811154,
title = {Random Unitary Matrices, Permutations and Painleve},
author = {Craig A. Tracy and Harold Widom},
journal= {arXiv preprint arXiv:math/9811154},
year = {2009}
}
Comments
21 pages, 1 figure. Revised paper simplifies the statement of Theorem 1 and adds some additional references