Communications in Mathematical Physics

, Volume 207, Issue 3, pp 665–685

Random Unitary Matrices, Permutations and Painlevé

  • Craig A. Tracy
  • Harold Widom

DOI: 10.1007/s002200050741

Cite this article as:
Tracy, C. & Widom, H. Comm Math Phys (1999) 207: 665. doi:10.1007/s002200050741


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.

Copyright information

© Springer-Verlag Berlin Heidelberg 1999

Authors and Affiliations

  • Craig A. Tracy
    • 1
  • Harold Widom
    • 2
  1. 1.Department of Mathematics and Institute of Theoretical Dynamics, University of California, Davis,¶CA 95616, USA. E-mail: tracy@itd.ucdavis.eduUS
  2. 2.Department of Mathematics, University of California, Santa Cruz, CA 95064, USA.¶E-mail: widom@math.ucsc.eduUS