Indistinguishability of collections of trees in the uniform spanning forest

TL;DR

This paper proves a broad indistinguishability theorem for collections of trees in the uniform spanning forest of c^d, showing properties are almost surely shared by all or none of the trees, extending previous results.

Contribution

It generalizes the indistinguishability theorem from individual trees to collections of trees in the uniform spanning forest, under certain graph conditions.

Findings

All or none of the k-tuples of trees share a property c^d USF

Applicable to graphs with Liouville property and one-ended components

Extends previous indistinguishability results to collections of trees.

Abstract

We prove the following indistinguishability theorem for $k$-tuples of trees in the uniform spanning forest of $\mathbb{Z}^d$: Suppose that $\mathscr{A}$ is a property of a $k$-tuple of components that is stable under finite modifications of the forest. Then either every $k$-tuple of distinct trees has property $\mathscr{A}$ almost surely, or no $k$-tuple of distinct trees has property $\mathscr{A}$ almost surely. This generalizes the indistinguishability theorem of the author and Nachmias (2016), which applied to individual trees. Our results apply more generally to any graph that has the Liouville property and for which every component of the USF is one-ended.