On the distribution of the length of the longest increasing subsequence of random permutations

Baik, Jinho; Deift, Percy; Johansson, Kurt

doi:10.1090/S0894-0347-99-00307-0

On the distribution of the length of the longest increasing subsequence of random permutations
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by Jinho Baik, Percy Deift and Kurt Johansson

J. Amer. Math. Soc. 12 (1999), 1119-1178

DOI: https://doi.org/10.1090/S0894-0347-99-00307-0

Published electronically: June 24, 1999

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Abstract:

The authors consider the length, $l_N$, 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 of the largest eigenvalue of a random GUE matrix. The authors also prove convergence of moments. The proof is based on the steepest descent method for Riemann-Hilbert problems, introduced by Deift and Zhou in 1993 in the context of integrable systems. The applicability of the Riemann-Hilbert technique depends, in turn, on the determinantal formula of Gessel for the Poissonization of the distribution function of $l_N$.

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