Scaling limits for simple random walks on random ordered graph trees

TL;DR

This paper extends the scaling limit results for simple random walks on random ordered graph trees, showing convergence to Brownian motion on the limiting real tree under broader conditions, including certain infinite variance Galton-Watson trees.

Contribution

It generalizes previous results by establishing diffusion limits for a wider class of random trees with nonatomic volume measures and polynomial volume bounds.

Findings

Random walks on certain infinite variance Galton-Watson trees converge to Brownian motion on stable trees.

The results apply to trees with nonatomic volume measures supported on leaves.

The scaling limits are characterized by the properties of the volume measure and the convergence of search-depth processes.

Abstract

Consider a family of random ordered graph trees $(T_n)_{n\geq 1}$, where $T_n$ has $n$ vertices. It has previously been established that if the associated search-depth processes converge to the normalised Brownian excursion when rescaled appropriately as $n\rightarrow\infty$, then the simple random walks on the graph trees have the Brownian motion on the Brownian continuum random tree as their scaling limit. Here, this result is extended to demonstrate the existence of a diffusion scaling limit whenever the volume measure on the limiting real tree is nonatomic, supported on the leaves of the limiting tree, and satisfies a polynomial lower bound for the volume of balls. Furthermore, as an application of this generalisation, it is established that the simple random walks on a family of Galton-Watson trees with a critical infinite variance offspring distribution, conditioned on the total number of offspring, can be rescaled to converge to the Brownian motion on a related $\alpha$-stable tree.