Adam Timar: Indistinguishable clusters in random spanning forests

It was proved by Lyons and Schramm that the infinite components of Bernoulli percolation on a Cayley graph are indistinguishable. This means that any invariantly defined property either holds for every infinite component or for none of them. Indistinguishability of clusters is the same as the ergodicity of the cluster equivalence relation. The perhaps most important invariant random spanning forests of a Cayley graph are the Uniform Spanning Forest (USF) and the Minimal Spanning Forest (MSF). We show that the free versions of these forests satisfy indistinquishability whenever they differ from their wired counterparts. This question was asked by Benjamini, Lyons, Peres and Schramm.