Finite-energy infinite clusters without anchored expansion

TL;DR

This paper discusses the properties of percolation clusters on nonamenable graphs, showing that certain expansion properties do not hold universally by providing a counterexample on a 4-regular tree.

Contribution

It demonstrates that the previously conjectured properties of infinite clusters do not extend to all finite energy ergodic invariant percolations, countering earlier assumptions.

Findings

Exponential decay of finite clusters in nonamenable graphs

Counterexample on the 4-regular tree showing failure of anchored expansion

Limits of previous percolation conjectures

Abstract

Hermon and Hutchcroft have recently proved the long-standing conjecture that in Bernoulli(p) bond percolation on any nonamenable transitive graph G, at any p > p_c(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.