Random directed forest and the Brownian web

Abstract

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 $u:=(u(1),u(2),...,u(d))$ is connected by an edge to another open vertex which has the minimum $L_1$ 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 $d\ge 4$. In addition for $d=2$, we show that when properly scaled, family of its paths converges in distribution to the Brownian web.

References in corpus (4)

• The Brownian web: Characterization and convergence
• The 2d-Directed Spanning Forest is almost surely a tree
• Semi-infinite paths of the 2d-Radial Spanning Tree
• Convergence to the Brownian Web for a generalization of the drainage network model

Cited by in corpus (1)

• Scaling limit of the radial Poissonian web