A drainage network with dependence and 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 $d\ge 4$. Also, there is no bi-infinite path in the graph almost surely for $d\ge 2$. Subsequently, we prove that for $d=2$ the diffusive scaling of this system converges in distribution to the Brownian web.

Journal of Statistical Physics, volume 189, Article number: 14 (2022)

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
• Coalescing Brownian flows: A new approach