Quantitative indistinguishability and sparse and dense clusters in factor of IID percolations
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 $M(G,η)<\infty$ (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 $η$-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 $2+δ$.