The local limit of uniform spanning trees

TL;DR

This paper proves that as the degree of regular graphs increases, the local structure of uniform spanning trees converges to a Poisson(1) branching process conditioned to survive, with extensions to nearly regular graphs.

Contribution

It establishes the local limit of uniform spanning trees on high-degree regular graphs and extends results to almost regular graphs and quenched versions.

Findings

Local limit of uniform spanning trees converges to Poisson(1) branching process

Results hold for regular graphs with degree tending to infinity

Extensions to almost regular graphs and quenched versions

Abstract

We show that the local limit of the uniform spanning tree on any finite, simple, connected, regular graph sequence with degree tending to infinity is the Poisson(1) branching process conditioned to survive forever. An extension to "almost" regular graphs and a quenched version are also given.