Quantitative indistinguishability and sparse and dense clusters in factor of IID percolations

2025/12/08 by Csóka, Endre, Mester, Péter, Pete, Gábor · 2 citations
#Combinatorics (math.CO) #Dynamical Systems (math.DS) #FOS: Mathematics #Group Theory (math.GR) #Probability (math.PR)

paper · doi:10.48550/arxiv.2512.07740

Abstract

Chifan-Ioana (2010) implies that, for any factor of IID percolation on any nonamenable Cayley graph G, there is a countable set of (strong) indistinguishability classes for non-hyperfinite clusters. We introduce quantitative strengthenings, called (qI) and (qSI): for η-non-hyperfinite clusters, there are at most M(G,η)<∞ (strong) indistinguishability classes, for any FIID percolation. We first show that (qI) and (qSI) for any G are equivalent to the ``sparse implies thin'' property (SiT): any FIID percolation with η-non-hyperfinite clusters has density at least c(G,η)>0. Also, (SiT) is independent of the finite generating set of a group. We prove, using entropy inequalities, that (SiT) holds for free groups, even for weak FIIDs. On the other hand, recent work of Jardón-Sánchez, Mellick, Poulin, and Wróbel implies that (SiT) fails for weak FIIDs on non-exact, i.e., not property (A) groups. Furthermore, (SiT) implies that the Bernoulli graphing over any non-hyperfinite FIID cluster is strongly ergodic, and that indistinguishability for non-hyperfinite FIID clusters is equivalent to strong indistinguishability. These results follow from the work of Chifan-Ioana for every nonamenable Cayley graph, but with non-probabilistic proofs. We also prove, again using entropy inequalities, this time for all nonamenable Cayley graphs, that any FIID percolation with high enough expected degree must have a density close to 1, and there must be a single indistinguishability class of such clusters. On Kazhdan groups, there must be a single such cluster. Our results have finite counterparts: in any large girth d-regular graph sequence, any FIID subgraph of average degree at least 2+δ must have density at least c(d,δ)>0. In the uniform random d-regular graph G_n,d, this holds for every subgraph of average degree at least 2+δ.

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Cited by

• Indistinguishability for recurrent clusters
• Infinitesimal containment and sparse factors of iid

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