A Drainage Network with Dependence and the Brownian Web

Abstract

We study a system of coalescing random walks on the integer lattice Zd 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 form a graph consisting of just one tree for dimensions d= 2 , 3 and infinitely many disjoint trees for dimensions d≥ 4. Also, there is no bi-infinite path in the graph almost surely for d≥ 2. Subsequently, we prove that for d= 2 the diffusive scaling of this system converges in distribution to the Brownian web. © 2022, The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature.