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

TL;DR

This paper proves that the distribution of the length of the longest increasing subsequence in a random permutation, when properly scaled, converges to the Tracy-Widom distribution, linking combinatorics with random matrix theory.

Contribution

The authors establish the convergence of the distribution and moments of the LIS length to the Tracy-Widom distribution using Riemann-Hilbert analysis, providing a rigorous proof of this asymptotic behavior.

Findings

Distribution of LIS length converges to Tracy-Widom distribution

Moments of LIS length also converge

Method relies on Riemann-Hilbert problem analysis

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 $l_N$.