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Consider the d dimensional lattice Z d where each vertex is open or closed with probability p or 1 − p respectively. An open vertex u := (u(1),u(2),... ,u(d)) is connected by an edge to another open vertex which has the minimum L1 distance among all the open vertices with x(d) > u(d). 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.

Random directed forest and the Brownian web