Interlacements and the Wired Uniform Spanning Forest

TL;DR

This paper extends the Aldous-Broder algorithm using random interlacements to generate and analyze the wired uniform spanning forest (WUSF) on infinite graphs, proving one-endedness under certain conditions and providing counterexamples.

Contribution

It introduces a novel algorithm for WUSF generation on infinite graphs using random interlacements and establishes new results on the structure of WUSF components.

Findings

Every WUSF component is one-ended almost surely under certain conditions.

The number of excessive ends in the WUSF is non-random.

Counterexample shows one-endedness is not preserved by rough isometries.

Abstract

We extend the Aldous-Broder algorithm to generate the wired uniform spanning forests (WUSFs) of infinite, transient graphs. We do this by replacing the simple random walk in the classical algorithm with Sznitman's random interlacement process. We then apply this algorithm to study the WUSF, showing that every component of the WUSF is one-ended almost surely in any graph satisfying a certain weak anchored isoperimetric condition, that the number of `excessive ends' in the WUSF is non-random in any graph, and also that every component of the WUSF is one-ended almost surely in any transient unimodular random rooted graph. The first two of these results answer positively two questions of Lyons, Morris and Schramm, while the third extends a recent result of the author. Finally, we construct a counterexample showing that almost sure one-endedness of WUSF components is not preserved by rough isometries of the underlying graph, answering negatively a further question of Lyons, Morris and Schramm.