Geometric & Functional Analysis GAFA

, Volume 10, Issue 4, pp 702–731

On the distribution of the length of the second row of a Young diagram under Plancherel measure

  • J. Baik
  • P. Deift
  • K. Johansson

DOI: 10.1007/PL00001635

Cite this article as:
Baik, J., Deift, P. & Johansson, K. GAFA, Geom. funct. anal. (2000) 10: 702. doi:10.1007/PL00001635

Abstract.

We investigate the probability distribution of the length of the second row of a Young diagram of size N equipped with Plancherel measure. We obtain an expression for the generating function of the distribution in terms of a derivative of an associated Fredholm determinant, which can then be used to show that as \( N \to \infty \) the distribution converges to the Tracy—Widom distribution [TW1] for the second largest eigenvalue of a random GUE matrix. This paper is a sequel to [BDJ], where we showed that as \( N \to \infty \) the distribution of the length of the first row of a Young diagram, or equivalently, the length of the longest increasing subsequence of a random permutation, converges to the Tracy—Widom distribution [TW1] for the largest eigenvalue of a random GUE matrix.

Copyright information

© Birkhäuser Verlag, Basel 2000

Authors and Affiliations

  • J. Baik
    • 1
  • P. Deift
    • 2
  • K. Johansson
    • 3
  1. 1.Courant Institute of Mathematics, NYU, 251 Mercer Street, New York, NY 10012, USA, e-mail: biak@cims.nyu.eduUS
  2. 2.Courant Institute of Mathematics, NYU, 251 Mercer Street, New York, NY 10012, USA, and Institute of Advanced Study, Princeton, NJ 0850, USA, e-mail: dieft@cims.nyu.eduUS
  3. 3.Department of Mathematics, Royal Institute of Technology, S-100 44 Stockholm, Sweden, e-mail: kurtj@math.kth.seSE