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

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

1998-10-16

arxiv.org/abs/math/9810105v2

J. Amer. Math. Soc. 12 (1999), no. 4, 1119--1178

Mathematics

Combinatorics

60 pages, 14 figures, AMS-LaTeX, typo correstions, new references

Scientific paper

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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