Finite-energy infinite clusters without anchored expansion

Abstract

Hermon and Hutchcroft have recently proved the long-standing conjecture that in $Bernoulli(\mathit{p})$ bond percolation on any nonamenable transitive graph G, at any $\mathit{p}>{\mathit{p}}_{\mathit{c}}(\mathit{G})$, the probability that the cluster of the origin is finite but has a large volume n decays exponentially in n. A corollary is that all infinite clusters have anchored expansion almost surely. They have asked if these results could hold more generally, for any finite energy ergodic invariant percolation. We give a counterexample, an invariant percolation on the 4-regular tree.