On the convergence of the drainage network with branching

TL;DR

This paper studies a drainage network model of coalescing and branching random walks, showing that under certain conditions, its scaling limit contains a Brownian Net, extending previous convergence results to include branching.

Contribution

It introduces a perturbed drainage network allowing low-probability branching and proves its weak limits contain a Brownian Net, suggesting convergence to this structure.

Findings

The perturbed drainage network is tight under diffusive scaling.

Weak limit points of the network contain a Brownian Net.

Conjecture that the limit is exactly the Brownian Net.

Abstract

The Drainage Network is a system of coalescing random walks, exhibiting long-range dependence before coalescence, introduced by Gangopadhyay, Roy, and Sarkar. Coletti, Fontes, and Dias proved its convergence to the Brownian Web under diffusive scaling. In this work, we introduce a perturbation of the system allowing branching of the random walks with low probabilities varying with the scaling parameter. When the branching probability is inversely proportional to the scaling parameter, we show that this drainage network with branching consists of a tight family such that any weak limit point contains a Brownian Net. We conjecture that the limit is indeed the Brownian Net.