  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 $l_N$.
