# Coalescing directed random walks on the backbone of a 1 +1-dimensional oriented percolation cluster converge to the Brownian web

## TL;DR

This paper proves that coalescing directed random walks on the backbone of a supercritical oriented percolation cluster in 1+1 dimensions converge to the Brownian web after rescaling, revealing large-scale diffusive behavior.

## Contribution

It demonstrates convergence of coalescing random walks on a percolation backbone to the Brownian web, extending understanding of scaling limits in random environments.

## Key findings

- Convergence to the Brownian web after diffusive rescaling.
- Tail bounds on meeting times of walkers.
- Percolation cluster behaves like the full lattice at large scales.

## Abstract

We consider the backbone of the infinite cluster generated by supercritical oriented site percolation in dimension 1 +1. A directed random walk on this backbone can be seen as an "ancestral line" of an individual sampled in the stationary discrete-time contact process. Such ancestral lineages were investigated in [BCDG13] where a central limit theorem for a single walker was proved. Here, we consider infinitely many coalescing walkers on the same backbone starting at each space-time point. We show that, after diffusive rescaling, the collection of paths converges in distribution to the Brownian web. Hence, we prove convergence to the Brownian web for a particular system of coalescing random walks in a dynamical random environment. An important tool in the proof is a tail bound on the meeting time of two walkers on the backbone, started at the same time. Our result can be interpreted as an averaging statement about the percolation cluster: apart from a change of variance, it behaves as the full lattice, i.e. the effect of the "holes" in the cluster vanishes on a large scale.