Ends in Uniform Spanning Forests

TL;DR

This paper extends the understanding of the structure of wired spanning forests by relaxing symmetry assumptions, showing that trees in such forests typically have only one end under broader conditions related to isoperimetric profiles.

Contribution

It introduces a new condition based on isoperimetric profiles that ensures trees in wired spanning forests have only one end, removing the need for transitivity and unimodularity.

Findings

Trees in wired spanning forests have only one end under broader conditions.

The new isoperimetric condition generalizes previous symmetry assumptions.

The results apply to a wider class of graphs beyond transitive unimodular ones.

Abstract

It has hitherto been known that in a transitive unimodular graph, each tree in the wired spanning forest has only one end a.s. We dispense with the assumptions of transitivity and unimodularity, replacing them with a much broader condition on the isoperimetric profile that requires just slightly more than uniform transience.