Order of the variance in the discrete Hammersley process with boundaries

TL;DR

This paper analyzes the variance order of the last passage time in a lattice Hammersley process with Bernoulli environment and boundaries, revealing different scaling behaviors along characteristic directions and outside a cone.

Contribution

It establishes the variance order in the boundary Hammersley process, showing $N^{2/3}$ scaling along characteristic directions and linear or constant scaling outside, due to shape function edges.

Findings

Variance along characteristic directions is of order $N^{2/3}$.

Variance outside the cone is $O(N)$ in the boundary model and $O(1)$ in the non-boundary model.

Shape function edges cause the change in variance behavior.

Abstract

We discuss the order of the variance on a lattice analogue of the Hammersley process with boundaries, for which the environment on each site has independent, Bernoulli distributed values. The last passage time is the maximum number of Bernoulli points that can be collected on a piecewise linear path, where each segment has strictly positive but finite slope. We show that along characteristic directions the order of the variance of the last passage time is of order $N^{2/3}$ in the model with boundary. These characteristic directions are restricted in a cone starting at the origin, and along any direction outside the cone, the order of the variance changes to $O(N)$ in the boundary model and to $O(1)$ for the non-boundary model. This behaviour is the result of the two flat edges of the shape function.