Asymptotic expansions relating to the distribution of the length of longest increasing subsequences

TL;DR

This paper derives an asymptotic expansion for the distribution of the longest increasing subsequence length in random permutations, providing explicit correction terms and new analytic de-Poissonization methods.

Contribution

It introduces explicit finite-size correction terms expressed via derivatives of the GUE Tracy-Widom distribution and develops an analytic de-Poissonization approach based on growth conditions.

Findings

Explicit correction terms for the length distribution are derived.

An alternative analytic de-Poissonization method is proposed.

Asymptotic expansions of related distributions are obtained.

Abstract

We study the distribution of the length of longest increasing subsequences in random permutations of $n$ integers as $n$ grows large and establish an asymptotic expansion in powers of $n^{-1/3}$. Whilst the limit law was already shown by Baik, Deift and Johansson to be the GUE Tracy-Widom distribution $F$, we find explicit analytic expressions of the first few finite-size correction terms as linear combinations of higher order derivatives of $F$ with rational polynomial coefficients. Our proof replaces Johansson's de-Poissonization, which is based on monotonicity as a Tauberian condition, by analytic de-Poissonization of Jacquet and Szpankowski, which is based on growth conditions in the complex plane; it is subject to a tameness hypothesis concerning complex zeros of the analytically continued Poissonized length distribution. In a preparatory step an expansion of the hard-to-soft edge transition law of LUE is studied, which is lifted into an expansion of the Poissonized length distribution for large intensities. Finally, expansions of Stirling-type approximations and of the expected value and variance of the length distribution are given.