Directed random walk on the backbone of an oriented percolation cluster

TL;DR

This paper studies a directed random walk on the backbone of an infinite oriented percolation cluster, proving laws of large numbers and central limit theorems both in annealed and quenched settings.

Contribution

It establishes the first quenched CLT for directed walks on percolation clusters using a novel joint renewal analysis.

Findings

Proves a law of large numbers for the walk.

Establishes an annealed central limit theorem.

Derives a quenched central limit theorem for almost all cluster realizations.

Abstract

We consider a directed random walk on the backbone of the infinite cluster generated by supercritical oriented percolation, or equivalently the space-time embedding of the ``ancestral lineage'' of an individual in the stationary discrete-time contact process. We prove a law of large numbers and an annealed central limit theorem (i.e., averaged over the realisations of the cluster) using a regeneration approach. Furthermore, we obtain a quenched central limit theorem (i.e.\ for almost any realisation of the cluster) via an analysis of joint renewals of two independent walks on the same cluster.