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

Authors:
Jinho Baik, Percy Deift and Kurt Johansson

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

MSC (1991):
Primary 05A05, 15A52, 33D45, 45E05, 60F99

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

Published electronically:
June 24, 1999

MathSciNet review:
1682248

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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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Additional Information

**Jinho Baik**

Affiliation:
Courant Institute of Mathematical Sciences, New York University, New York, New York 10012

MR Author ID:
646186

Email:
baik@cims.nyu.edu

**Percy Deift**

MR Author ID:
56085

Email:
deift@cims.nyu.edu

**Kurt Johansson**

Affiliation:
Department of Mathematics, Royal Institute of Technology, S-100 44 Stockholm, Sweden

MR Author ID:
258098

Email:
kurtj@math.kth.se

Keywords:
Random permutations,
orthogonal polynomials,
Riemann-Hilbert problems,
random matrices,
steepest descent method

Received by editor(s):
July 20, 1998

Received by editor(s) in revised form:
March 30, 1999

Published electronically:
June 24, 1999

Additional Notes:
The authors would like to acknowledge many extremely useful and enlightening conversations with Persi Diaconis and Andrew Odlyzko. Special thanks are due to Andrew Odlyzko and Eric Rains for providing us with the results of their Monte Carlo simulations.

The work of the second author was supported in part by NSF grant #DMS-9500867.

The work of the third author was supported in part by the Swedish Natural Research Council (NFR)

Article copyright:
© Copyright 1999
American Mathematical Society