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.
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Submitted: March 2000, Revised version: April 2000.
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Baik, J., Deift, P. & Johansson, K. On the distribution of the length of the second row of a Young diagram under Plancherel measure . GAFA, Geom. funct. anal. 10, 702–731 (2000). https://doi.org/10.1007/PL00001635
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DOI: https://doi.org/10.1007/PL00001635