Scaling limit of a drainage network model on a perturbed lattice

Abstract

Abstract The study of random networks generally requires that vertices be independently and uniformly distributed over disjoint regions 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 random subset of double struck upper Z squared Z 2 $\mathbb{Z}^2$ , where points in double struck upper Z squared Z 2 $\mathbb{Z}^2$ are randomly perturbed such that both horizontal and vertical perturbations are given by exponentially decaying discrete random variables with unbounded support and edges between two such points are specified according to the distance between them. We restrict vertical perturbations in the upward direction only. Here we establish that (i) the resultant stochastic network is a single tree almost surely and (ii) as a collection of paths, under diffusive scaling the resulting network converges to the Brownian web.