Poisson percolation on the oriented square lattice

TL;DR

This paper investigates the limiting shape and boundary fluctuations of the cluster in Poisson percolation on an oriented square lattice, extending previous work to the oriented case with variable edge rates.

Contribution

It characterizes the limiting shape and boundary fluctuations of the percolation cluster in the oriented lattice with decreasing edge rates, extending prior non-oriented results.

Findings

The cluster's density at height y approximates the homogeneous percolation probability.

The boundary exhibits specific fluctuation behaviors.

The shape converges to a deterministic limit.

Abstract

In Poisson percolation each edge becomes open after an independent exponentially distributed time with rate that decreases in the distance from the origin. As a sequel to our work on the square lattice, we describe the limiting shape of the component containing the origin in the oriented case. We show that the density of occupied sites at height $y$ in the cluster is close to the percolation probability in the corresponding homogeneous percolation process, and we study the fluctuations of the boundary.