Quenched local limit theorem for a directed random walk on the backbone of a supercritical oriented percolation cluster for $d \ge 1$

TL;DR

This paper extends the quenched local limit theorem for a directed random walk on the backbone of a supercritical oriented percolation cluster, proving it holds for all spatial dimensions $d \, \ge \, 1$, not just $d \, \ge \, 3$.

Contribution

The authors prove that the quenched local limit theorem previously established for higher dimensions also applies to all dimensions $d \, \ge \, 1$, broadening its applicability.

Findings

Quenched local limit theorem holds for all $d \ge 1$.

Extension from $d \ge 3$ to all $d \ge 1$.

Supports the universality of the local limit behavior.

Abstract

In this work we extend the quenched local limit theorem obtained by the authors in [BBDS23]. More precisely, we consider a directed random walk on the backbone of the supercritical oriented percolation cluster in dimensions $d+1$ with $d\geq 1$ being the spatial dimension. In [BBDS23] an annealed local central limit theorem was proven for all $d\geq 1$ and a quenched local limit theorem under the assumption $d\geq 3$. Here we show that the latter result also holds for all $d \ge 1$.