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

2000; Birkhäuser; Volume: 10; Issue: 4 Linguagem: Inglês

10.1007/pl00001635

1420-8970

Jinho Baik, Percy Deift, Kurt Johansson,

Advanced Algebra and Geometry

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.