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

1999/01/26 by Jinho Baik, Baik, Jinho, Percy Deift +3 · 2 citations
Mathematics · Physics and Astronomy · #Advanced Algebra and Geometry #Advanced Combinatorial Mathematics #Combinatorics (math.CO) #Exactly Solvable and Integrable Systems (nlin.SI) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Random Matrices and Applications #math-ph #math.CO #math.MP #nlin.SI #solv-int

paper · pdf · doi:10.48550/arxiv.math/9901118

25 pages, AMS-LaTex file

arxiv created 1999/01/26 · openalex publication_date 1999/01/26 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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→∞ the distribution converges to the Tracy-Widom distribution [TW] for the second largest eigenvalue of a random GUE matrix. This paper is a sequel to [BDJ], where we showed that as N→∞ 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 [TW] for the largest eigenvalue of a random GUE matrix.

Cited by

• Random Matrices and Random Permutations
• Infinite wedge and random partitions

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