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

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 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.

Ancestral lineages in spatial population models with local regulation

“…The results in [8] can again be interpreted as an averaging statement about the percolation cluster: apart from a change of variance, it behaves as the full lattice (for which convergence to the Brownian web was proved in [36]), i.e. the effect of the 'holes' in the cluster vanishes on a large scale.…”

“…By Theorem 2.1 and space-time stationarity, any X (x,n) converges to a Brownian motion under diffusive rescaling. As shown in [8], in spatial dimension d = 1, the collection of all these paths converges after diffusive rescaling as in Theorem 2.1 in distribution to the Brownian web. Informally, this limit object describes an infinite system of coalescing Brownian motions starting from all space-time points in R × R. One may then apply our convergence result to investigate the behaviour of interfaces in the discrete time contact process analogously to [36,Theorem 7.6 and Remark 7.7], as observed in [8, p. 1051].…”

“…Studying two (or more) copies of the walk on C, especially when one stipulates that they coalesce as soon as they meet, is also very natural from the point of view of larger samples. In fact, this is exactly the device that is made use of in [8] and it also plays a key role in the proof of (2.20) below. We will however not spell out the details here and instead refer to [6,8].…”

“…In fact, this is exactly the device that is made use of in [8] and it also plays a key role in the proof of (2.20) below. We will however not spell out the details here and instead refer to [6,8].…”

The stepping stone model in a random environment and the effect of local heterogneities on isolation by distance patterns

“…For this we have to study the coalescence of pairs of random walks in a random environment. This was done in a particular setting in [BGS18], where the authors consider a family of random walks following the model in [BCDG13] which coalesce whenever they find themselves in the same site. They show that, adequately rescaled, the resulting process converges in distribution to the Brownian web (i.e.…”