Ergodicity and indistinguishability in percolation theory

Abstract

This paper explores the link between the ergodicity of the cluster equivalence relation restricted to its infinite locus and the indistinguishability 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 classical correspondence between ergodicity and indistinguishability 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, satises the Indistinguishability Property and does not satisfy the Strong Indistinguishability Property.