Random directed forest and the Brownian web

Consider the $d$ dimensional lattice ${\mathbb{Z}}^d$ where each vertex is open or closed with probability $p$ or $1-p$ respectively. An open vertex $\mathbb{u}:=(\mathbb{u}(1),\mathbb{u}(2),\mathrm{.}\mathrm{.}\mathrm{.},\mathbb{u}(d))$ is connected by an edge to another open vertex which has the minimum $L_1$ distance among all the open vertices with $\mathbb{x}(d)>\mathbb{u}(d)$. 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 $d\ge 4$. In addition for $d=2$, we show that when properly scaled, family of its paths converges in distribution to the Brownian web.