Finite size corrections relating to distributions of the length of longest increasing subsequences

TL;DR

This paper investigates finite size corrections to the distribution of the longest increasing subsequences in various models, linking them to random matrix theory and providing both analytical and numerical insights.

Contribution

It extends previous results by analyzing the transition from hard to soft edge, deriving correction terms in terms of Fredholm operators and Painlevé transcendents, and providing numerical evidence for permutation models.

Findings

Leading correction proportional to $z^{-2/3}$ for certain models

Numerical evidence of $N^{-1/3}$ correction in permutation models

Functional forms derived in terms of Fredholm operators and Painlevé transcendents

Abstract

Considered are the large $N$, or large intensity, forms of the distribution of the length of the longest increasing subsequences for various models. Earlier work has established that after centring and scaling, the limit laws for these distributions relate to certain distribution functions at the hard edge known from random matrix theory. By analysing the hard to soft edge transition, we supplement and extend results of Baik and Jenkins for the Hammersley model and symmetrisations, which give that the leading correction is proportional to $z^{-2/3}$, where $z^2$ is the intensity of the Poisson rate, and provides a functional form as derivates of the limit law. Our methods give the functional form both in terms of Fredholm operator theoretic quantities, and in terms of Painlev\'e transcendents. For random permutations and their symmetrisations, numerical analysis of exact enumerations and simulations gives compelling evidence that the leading corrections are proportional to $N^{-1/3}$, and moreover provides an approximation to their graphical forms.