Ergodicity and indistinguishability in percolation theory

TL;DR

This paper investigates the relationship between ergodicity and indistinguishability of infinite clusters in percolation theory, introducing a stronger form of indistinguishability and analyzing its implications in Bernoulli percolation.

Contribution

It introduces the concept of strong indistinguishability, linking it to strong ergodicity, and explores its validity in Bernoulli percolation, expanding the theoretical understanding.

Findings

Strong indistinguishability holds in Bernoulli percolation.

An invariant percolation without insertion-tolerance satisfies indistinguishability but not strong indistinguishability.

The paper clarifies the connection between ergodicity and cluster indistinguishability in percolation theory.

Abstract

This paper explores the link between the ergodicity of the clus-ter equivalence relation restricted to its infinite locus and the indis-tinguishability of infinite clusters. It is an important element of the dictionary connecting orbit equivalence and percolation theory. This note starts with a short exposition of some standard material of these theories. Then, the classic correspondence between ergodicity and in-distinguishability is presented. Finally, we introduce a notion of strong indistinguishability that corresponds to strong ergodicity, and obtain that this strong indistinguishability holds in the Bernoulli case. We also define an invariant percolation that is not insertion-tolerant, sat-isfies the Indistinguishability Property and does not satisfy the Strong Indistinguishability Property.