LA Probability Forum

USC, Kaprielian (KAP) 414

This talk is concerned with the geometry of the (wired) Uniform Spanning Tree (UST) on infinite graphs, and in particular with a property called one-endedness, which means that removing a finite subset does not produce multiple infinite clusters. It was conjectured by Aldous and Lyons (2006) that the UST is one-ended on all non-trivial stationary random graphs. For unimodular random rooted graphs, we show that there is an equiva- lence between the one-endedness of the UST and (a) the existence and uniqueness of the potential kernel, (b) existence and unique- ness of the harmonic measure from infinity, and (c) a new anchored Harnack inequality. Building on this equivalence and extending the relation between recurrent potential theory and the UST, we prove the conjecture of Aldous and Lyons for recurrent stationary graphs.

This is based on joint works with Nathanaël Berestycki and Tom Hutchcroft.