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On the Distribution of the Length of the Longest Increasing Subsequence of Random Permutations

Jinho Baik·Percy Deift·Kurt Johansson
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Abstract

The authors consider the length, $l_N$, of the length of the longest increasing subsequence of a random permutation of $N$ numbers. The main result in this paper is a proof that the distribution function for $l_N$, suitably centered and scaled, converges to the Tracy-Widom distribution [TW1] of the largest eigenvalue of a random GUE matrix. The authors also prove convergence of moments. The proof is based on the steepest decent method for Riemann-Hilbert problems, introduced by Deift and Zhou in 1993 [DZ1] in the context of integrable systems. The applicability of the Riemann-Hilbert technique depends, in turn, on the determinantal formula of Gessel [Ge] for the Poissonization of the distribution function of $l_N$.

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BibTeXRIS

Jinho Baik, Percy Deift, Kurt Johansson. 1999-03-26. On the Distribution of the Length of the Longest Increasing Subsequence of Random Permutations. https://arxiv.org/abs/math/9810105

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