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Ergodicity and indistinguishability in percolation theory



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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Dates and versions

ensl-00738628 , version 1 (04-10-2012)
ensl-00738628 , version 2 (19-05-2014)
ensl-00738628 , version 3 (27-10-2014)



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