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.
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https://hal-ens-lyon.archives-ouvertes.fr/ensl-00738628
Contributor : Sébastien Martineau <>
Submitted on : Monday, October 27, 2014 - 4:44:28 PM
Last modification on : Thursday, January 11, 2018 - 6:12:31 AM
Long-term archiving on : Wednesday, January 28, 2015 - 11:51:42 AM

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  • HAL Id : ensl-00738628, version 3
  • ARXIV : 1210.1548

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Sébastien Martineau. Ergodicity and indistinguishability in percolation theory. 2014. ⟨ensl-00738628v3⟩

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