# Continuum random tree as the scaling limit for a drainage network model: a Brownian web approach

## TL;DR

This paper proves that the tributary structure of Howard's drainage model, conditioned on survival, converges to a continuum random tree, confirming a prediction by Aldous using a Brownian web approach.

## Contribution

It establishes the convergence of scaled tributary structures to a continuum random tree using dual processes and Brownian web techniques, extending prior predictions.

## Key findings

- Scaled tributaries converge to a continuum random tree
- The limiting tree differs slightly from earlier predictions
- Dual process convergence to Brownian web is key

## Abstract

We consider the tributary structure of Howard's drainage model studied by Gangopadhyay et. al. Conditional on the event that the tributary survives up to time $n$, we show that, as a sequence of random metric spaces, scaled tributary converges in distribution to a continuum random tree with respect to Gromov Hausdorff topology. This verifies a prediction made by Aldous for a simpler model (where paths are independent till they coalesce) but for a different conditional set up. The limiting continuum tree is slightly different from what was surmised earlier. Our proof uses the fact that there exists a dual process such that the original network and it's dual jointly converge in distribution to the Brownian web and it's dual.