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

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 lineage” of an individual sampled in the stationary discrete-time contact process. Such ancestral lineages were investigated in Birkner et al. (2013) 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 (under the averaged law) 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.