Scaling limit of a drainage network model on perturbed lattice

TL;DR

This paper investigates a drainage network model on a randomly perturbed lattice, demonstrating that it forms a single tree and converges to the Brownian web under diffusive scaling, extending understanding of stochastic networks beyond standard models.

Contribution

It introduces a novel perturbed lattice model for drainage networks and proves convergence to the Brownian web, advancing the theory of stochastic network scaling limits.

Findings

The network forms a.s. a single tree.

Under diffusive scaling, the network converges to the Brownian web.

The model extends stochastic network analysis beyond Poisson point processes.

Abstract

Study of random networks generally requires the nodes to be independently and uniformly distributed such as a Poisson point process. In this work, we venture beyond this standard paradigm and investigate a stochastic forest obtained from a drainage network model constructed on a randomly perturbed subset of $\mathbb{Z}^2$, where both horizontal and vertical perturbations are given by exponentially decaying unbounded discrete random variables and vertical perturbations are allowed in the upward direction only. We show that the resultant stochastic network is a single tree a.s. We further establish that as a collection of paths, under diffusive scaling the resultant network converges to the Brownian web.