Indistinguishability for recurrent clusters

TL;DR

This paper presents a new framework for proving the indistinguishability of infinite clusters in various invariant percolation models, removing previous transience restrictions and applying to broader classes of random graphs.

Contribution

It introduces a general method for showing cluster indistinguishability under weaker conditions, applicable to diverse models and graph structures.

Findings

Indistinguishability of infinite clusters in the interchange process.

Applicability to the loop O(n) model on amenable Cayley graphs.

Indistinguishability of clusters for any 'not essentially tail' property.

Abstract

We introduce a general framework to show the indistinguishability of infinite clusters (ergodicity of the cluster subrelation) in group-invariant percolation processes with a weaker version of the finite energy property: the possibility of moving infinite branches from one infinite cluster to another. Crucially, this removes the necessity for the infinite clusters to be transient, present in most previous works. Our method also applies to more general random graphs, whenever a stationary sequence of vertices is definable. We use this to show the indistinguishability of infinite clusters (or permutation cycles) in the interchange process (a.k.a.~random stirring process), the loop $O(n)$ model on amenable Cayley graphs, biased corner percolation on $\mathbb{Z}^2$, and the Poisson Zoo process. Finally, we show that infinite clusters in any invariant process on a Cayley graph are indistinguishable for any ``not essentially tail'' property, i.e., properties that depend only on the local structure of the cluster.