Finite-energy infinite clusters without anchored expansion

Pete, Gábor and Timár, Ádám (2021) Finite-energy infinite clusters without anchored expansion. Bernoulli Journal, 27 (4). pp. 2353-2361. ISSN 1350-7265, ESSN: 1573-9759

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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 > pc(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.

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