Brownian web in the scaling limit of supercritical oriented percolation in dimension 1 + 1

Abstract

We prove that, after centering and diffusively rescaling space and time, the collection of rightmost infinite open paths in a supercritical oriented percolation configuration on the space-time lattice Z 2 even := {(x, i) ∈ Z 2 : x + i is even} converges in distribution to the Brownian web. This proves a conjecture of Wu and Zhang [WZ08]. Our key observation is that each rightmost infinite open path can be approximated by a percolation exploration cluster, and different exploration clusters evolve independently before they intersect.

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

“…We invoke condition pE 1 1 q because because in our model, paths π z1 and π z2 can cross each other without coalescing. In this respect, our scenario is different from that in Sarkar and Sun [2013].…”

“…3. Sarkar and Sun [2013] considered the system of rightmost paths on an oriented (bond) percolation cluster and showed that it converges to the Brownian web after suitable centering and rescaling. Thus, in Sarkar and Sun [2013], walkers move to the right whenever possible (and in particular they cannot cross each other) whereas in our set-up, the walks pick uniformly among the allowed neighbors.…”

Perturbations of Supercritical Oriented Percolation and Sticky Brownian Webs

“…The main tool used in [13] is the notion of a percolation exploration cluster, which we briefly recall here. For every z = (x, t) ∈ Z 2 even , the percolation exploration cluster C z := (C z (n)) n≥t consists of a set of sequentially explored edges such that for each time n > t, a minimal set of edges C z (n) before that C z is bounded between the paths γ z and ρ z , and furthermore, γ z and ρ z converge to the same Brownian motion after proper centering and scaling.…”

“…As in [13], we denote by (ρ(T i ), T i ) i∈N the successive break points along ρ := ρ o . More precisely, the T i 's correspond to the successive times at which γ := γ o and ρ coincide.…”

“…In [13], Sun and Sarkar showed that after suitable centering and rescaling, the set of paths Γ = {γ z : z ∈ K} converges to a family of coalescing Brownian motions, known as the Brownian web [3,16,14] and which can also be obtained as the "universal" scaling limit of coalescing random walks [10]. See Fig 1. More precisely, for a, b, > 0, let S a,b, (x, t) := b (x − at), 2 t , (x, t) ∈ R 2 , (1.1)…”