Indistinguishability of components of random spanning forests

TL;DR

This paper proves that in certain random spanning forests on Cayley graphs, the infinite components are indistinguishable by invariant properties, and classifies the possible number of components under specific conditions.

Contribution

It establishes the indistinguishability of infinite components in free uniform and minimal spanning forests on Cayley graphs, extending results to unimodular random graphs and percolations with weak insertion tolerance.

Findings

Infinite components are indistinguishable by invariant properties.

Number of components can only be 0, 1, or infinite.

Results apply to a broad class of percolations and unimodular random graphs.

Abstract

We prove that the infinite components of the Free Uniform Spanning Forest of a Cayley graph are indistinguishable by any invariant property, given that the forest is different from its wired counterpart. Similar result is obtained for the Free Minimal Spanning Forest. We also show that with the above assumptions there can only be 0, 1 or infinitely many components. These answer questions by Benjamini, Lyons, Peres and Schramm. Our methods apply to a more general class of percolations, those satisfying "weak insertion tolerance", and work beyond Cayley graphs, in the more general setting of unimodular random graphs.