Convergence of simple random walks on random discrete trees to Brownian motion on the continuum random tree

TL;DR

This paper proves that simple random walks on large discrete trees converge to Brownian motion on the continuum random tree, with results applicable to Galton-Watson trees, establishing a key link between discrete and continuous stochastic processes.

Contribution

It demonstrates the convergence of simple random walks on discrete trees to Brownian motion on the continuum random tree, including both quenched and annealed versions, covering Galton-Watson trees.

Findings

Convergence of discrete random walks to Brownian motion on CRT

Results hold for Galton-Watson trees conditioned on size

Includes both quenched and annealed convergence

Abstract

In this article it is shown that the Brownian motion on the continuum random tree is the scaling limit of the simple random walks on any family of discrete $n$-vertex ordered graph trees whose search-depth functions converge to the Brownian excursion as $n\to\infty$. We prove both a quenched version (for typical realisations of the trees) and an annealed version (averaged over all realisations of the trees) of our main result. The assumptions of the article cover the important example of simple random walks on the trees generated by the Galton-Watson branching process, conditioned on the total population size.