A drainage network with dependence and the Brownian web

TL;DR

This paper analyzes a coalescing random walk system on integer lattices, revealing its geometric structure across dimensions and demonstrating convergence to the Brownian web in two dimensions.

Contribution

It characterizes the path geometry and proves the diffusive scaling limit converges to the Brownian web in two dimensions.

Findings

Paths form a single tree in 2D and 3D, multiple trees in higher dimensions

No bi-infinite paths exist for dimensions ≥ 2

Diffusive scaling in 2D converges to the Brownian web

Abstract

We study a system of coalescing random walks on the integer lattice $\mathbb{Z}^{d}$ in which the walk is oriented in the $d$-th direction and follows certain specified rules. We first study the geometry of the paths and show that, almost surely, the paths from a graph consisting of just one tree for dimentions $d=2,3$ and infinitely many disjoint trees for dimensions $d\geq 4$. Also, there is no bi-infinite path in the graph almost surely for $d\geq 2$. Subsequently, we prove that for $d=2$ the diffusive scaling of this system converges in distribution to the Brownian web.