Indistinguishability for recurrent clusters

TL;DR

The paper develops a general grafting framework to prove indistinguishability of infinite clusters in group-invariant percolation without requiring transience, by exploiting a weakened finite-energy property that permits moving infinite branches between clusters. It formalizes the tools (definitions, smallness notions, and a robust technical lemma) and applies the method to several models—interchange process, loop O(n), biased corner percolation, and Poisson zoo—demonstrating indistinguishability of infinite clusters under broad invariant conditions. A key contribution is showing that not essentially tail properties cannot distinguish infinite clusters on Cayley graphs, broadening ergodicity results to a wide class of invariant systems. The work thus provides a versatile, broadly applicable approach to cluster ergodicity beyond traditional end-transience assumptions, with concrete implications for multiple interacting systems in probability and statistical physics.

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.

Figures (9)

• Figure 1: On the left, the interchange process on a path of length 4; in red the trajectory of particle 1. On the right, the two longest cycles in the interchange process on $[\![1,25]\!]^2$ with $\beta=1$.
• Figure 2: Two loop $O(n)$ samples in finite volume, $n=2$ on the left, $n=10$ on the right. These pictures are only meant to give an idea of the model in the simplest case, the two-dimensional hexagonal lattice, but here the infinite volume measure is not believed to have infinite cycles.
• Figure 3: On the left, corner percolation with $(p,q)=(0.2,0.8)$. It contains almost surely infinite paths that have an asymptotic slope of 1. On the right, $(p,q)=(0.5,0.5)$. In that case there are almost surely only finite paths.
• Figure 4: On the left, a Poisson Zoo where the animals are boxes of exponentially distributed size. On the right, the animals are random walks of exponentially distributed length.
• Figure 5: In the setup of \ref{['cor: technical on graphs']}, the configurations $\eta$ and $\eta'$ only differ in $S$. In the first, $K_0$ is part of the cluster $K$, in the second, it is part of $K'$.

Theorems & Definitions (29)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Lemma 2.1
• Proposition 2.2
• Remark 2.3
• Lemma 2.4: Mass transport principle
• Lemma 2.5