Scaling limit of the Aldous-Broder chain on regular graphs: the transient regime

TL;DR

This paper proves that the rescaled Aldous-Broder chain converges to the root growth with re-grafting process on regular graphs, extending previous results on the continuum random tree and spanning trees.

Contribution

It establishes the convergence of the rescaled Aldous-Broder chain to the RGRG on regular graphs, with specific conditions and scaling factors.

Findings

Convergence of the Aldous-Broder chain to RGRG in the Gromov-Hausdorff topology.

Conditions on graph sequences for convergence on regular graphs.

Probabilistic scaling factors for time and edge lengths.

Abstract

The continuum random tree is the scaling limit of the uniform spanning tree on the complete graph with $N$ vertices. The Aldous-Broder chain on a graph $G=(V,E)$ is a discrete-time stochastic process with values in the space of rooted trees whose vertex set is a subset of $V$ which is stationary under the uniform distribution on the space of rooted trees spanning $G$. In Evans, Pitman and Winter (2006) the so-called root growth with re-grafting process (RGRG) was constructed. Further it was shown that the suitable rescaled Aldous-Broder chain converges to the RGRG weakly with respect to the Gromov-Hausdorff topology. It was shown in Peres and Revelle (2005) that (up to a dimension depending constant factor) the continuum random tree is, with respect to the Gromov-weak topology, the scaling limit of the uniform spanning tree on $\mathbb{Z}_N^d$, $d\ge 5$. This result was recently strengthens in Archer, Nachmias and Shalev (2024) to convergence with respect to the Gromov-Hausdorff-weak topology, and therefore also with respect to the Gromov-Hausdorff topology. In the present paper we show that also the suitable rescaled Aldous-Broder chain converges to the RGRG weakly with respect to the Gromov-Hausdorff topology when initially started in the trivial rooted tree. We give conditions on the increasing graph sequence under which the result extends to regular graphs and give probabilistic expressions scales at which time has to be speed up and edge lengths have to be scaled down.