Brownian Web in the Scaling Limit of Supercritical Oriented Percolation in Dimension 1+1

TL;DR

This paper proves that the collection of rightmost infinite open paths in supercritical oriented percolation converges to the Brownian web after rescaling, confirming a conjecture and revealing independence properties of exploration clusters.

Contribution

It establishes the convergence of rightmost infinite open paths in supercritical oriented percolation to the Brownian web, confirming Wu and Zhang's conjecture.

Findings

Convergence of paths to the Brownian web after rescaling

Paths can be approximated by exploration clusters

Exploration clusters evolve independently before intersecting

Abstract

We prove that, after centering and diffusively rescaling space and time, the collection of rightmost infinite open paths in a supercritical oriented percolation configuration on the space-time lattice Z^2_{even}:={(x,i) in Z^2: x+i is even} converges in distribution to the Brownian web. This proves a conjecture of Wu and Zhang. Our key observation is that each rightmost infinite open path can be approximated by a percolation exploration cluster, and different exploration clusters evolve independently before they intersect.