No abstract

Consider the d-dimensional lattice Z d where each vertex is "open" or "closed" with probability p or 1 − p, respectively. An open vertex v is connected by an edge to the closest open vertex w such that the dth co-ordinates of v and w satisfy w(d) = v(d) − 1. In case of nonuniqueness of such a vertex w, we choose any one of the closest vertices with equal probability and independently of the other random mechanisms. It is shown that this random graph is a tree almost surely for d = 2 and 3 and it is an infinite collection of distinct trees for d ≥ 4. In addition, for any dimension, we show that there is no bi-infinite path in the tree and we also obtain central limit theorems of (a) the number of vertices of a fixed degree ν and (b) the number of edges of a fixed length l.

It is shown that this random graph is a tree almost surely for d = 2 and 3 and it is an infinite collection of disjoint trees for d ≥ 4. In addition, for d = 2, we show that when properly scaled, the family of its paths converge in distribution to the Brownian web.

No abstract

No abstract

No 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.

No abstract