Random directed forest and the Brownian web

TL;DR

This paper studies a random directed graph on a lattice where vertices connect to the nearest open vertex above them, showing it forms a tree in low dimensions and converges to the Brownian web when scaled in two dimensions.

Contribution

It proves the structure of the graph as a tree in dimensions 2 and 3, and demonstrates convergence of its paths to the Brownian web in two dimensions.

Findings

The graph is a tree for d=2 and 3.

For d≥4, the graph consists of disjoint trees.

In 2D, scaled paths converge to 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 $\mathbb{u} := (\mathbb{u}(1), \mathbb{u}(2),...,\mathbb{u}(d))$ 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 $d\geq 4$. In addition for $d=2$, we show that when properly scaled, family of its paths converges in distribution to the Brownian web.