Ergodicity and 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.
Keywords
Cite
@article{arxiv.1210.1548,
title = {Ergodicity and indistinguishability in percolation theory},
author = {Sébastien Martineau},
journal= {arXiv preprint arXiv:1210.1548},
year = {2014}
}