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

TL;DR

This paper introduces quantitative versions of indistinguishability in factor of IID percolations on nonamenable Cayley graphs, establishing properties like 'sparse implies thin' and analyzing their implications for cluster structure and ergodicity.

Contribution

It defines and proves the equivalence of quantitative indistinguishability properties, extending classical results with entropy methods and applying them to various group and graph settings.

Findings

(qI) and (qSI) are equivalent to 'sparse implies thin' property (SiT).

(SiT) holds for free groups but fails for non-exact groups.

High-degree FIID percolations have a single dense cluster with high probability.

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 $\eta$-non-hyperfinite clusters, there are at most $M(G,\eta)<\infty$ (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 $\eta$-non-hyperfinite clusters has density at least $c(G,\eta)>0$. 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 Jard\'on-S\'anchez, Mellick, Poulin, and Wr\'obel 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 $2+\delta$ must have density at least $c(d,\delta)>0$. In the uniform random d-regular graph $G_{n,d}$, this holds for every subgraph of average degree at least $2+\delta$.