Local limit theorems for a directed random walk on the backbone of a supercritical oriented percolation cluster

TL;DR

This paper proves local limit theorems for a directed random walk on the backbone of a supercritical oriented percolation cluster in high dimensions, showing convergence of transition probabilities under both annealed and quenched measures.

Contribution

It establishes annealed and quenched local limit theorems for the random walk, introducing a new invariant measure related to the environment viewed from the particle.

Findings

Quenched transition probabilities converge to annealed probabilities reweighted by an invariant density.

Existence and uniqueness of an invariant measure for the environment viewed from the particle.

Concentration properties of the invariant measure.

Abstract

We consider a directed random walk on the backbone of the supercritical oriented percolation cluster in dimensions $d+1$ with $d \ge 3$ being the spatial dimension. For this random walk we prove an annealed local central limit theorem and a quenched local limit theorem. The latter shows that the quenched transition probabilities of the random walk converge to the annealed transition probabilities reweighted by a function of the medium centred at the target site. This function is the density of the unique measure which is invariant for the point of view of the particle, is absolutely continuous with respect to the annealed measure and satisfies certain concentration properties.