Longest increasing path within the critical strip

TL;DR

This paper investigates the behavior of the longest increasing path within a narrow strip around the diagonal in a Poisson point process, revealing a transition from Tracy-Widom to Gaussian fluctuations as the strip width shrinks.

Contribution

It extends previous work by analyzing the maximal increasing path confined to a strip of width less than the critical $n^{2/3}$, deriving new expectation, variance, and distribution results.

Findings

Maximal path length expectation: $2n - n^{1- ext{width exponent}+o(1)}$

Variance scales as $n^{1 - rac{ ext{width exponent}}{2}+o(1)}$

Distribution converges to Gaussian for strip width less than $n^{2/3}$

Abstract

A Poisson point process of unit intensity is placed in the square $[0,n]^2$. An increasing path is a curve connecting $(0,0)$ with $(n,n)$ which is non-decreasing in each coordinate. Its length is the number of points of the Poisson process which it passes through. Baik, Deift and Johansson proved that the maximal length of an increasing path has expectation $2n-n^{1/3}(c_1+o(1))$, variance $n^{2/3}(c_2+o(1))$ and that it converges to the Tracy-Widom distribution after suitable scaling. Johansson further showed that all maximal paths have a displacement of $n^{\frac23+o(1)}$ from the diagonal with probability tending to one as $n\to \infty$. Here we prove that the maximal length of an increasing path restricted to lie within a strip of width $n^{\gamma}, \gamma<\frac23$, around the diagonal has expectation $2n-n^{1-\gamma+o(1)}$, variance $n^{1 - \frac{\gamma}{2}+o(1)}$ and that it converges to the Gaussian distribution after suitable scaling.