Characterization of the heavy-cluster nonuniqueness phase

Characterize when the heavy-cluster threshold is strictly below the uniqueness threshold for Bernoulli percolation on a quasi-transitive graph with a nonunimodular automorphism subgroup, including whether this strict inequality occurs exactly under the corresponding weighted nonamenability condition.

Background

For nonunimodular graphs, the paper notes that the critical threshold is strictly below the heavy-cluster threshold. The unresolved issue is the analogous comparison between the heavy-cluster threshold and the uniqueness threshold. The authors state that this question parallels the classical problem of determining when the critical threshold is below the uniqueness threshold, and identify weighted amenability or amenability of the acting group as the relevant geometric condition.

References

On the other hand, understanding when $p_h(G,\Gamma)<p_u(G)$ remains open and is largely parallel to the question of when $p_c(G)<p_u(G)$. While nonunimodular graphs are necessarily nonamenable, the relevant geometric condition remains to be the amenability of the group $\Gamma$ or, equivalently, the weighted-amenability of the graph.

— $β$-Skewed Maximal Spanning Forests  (2609.11845 - Chatterjee et al., 10 Sep 2026) in Section 3.2, paragraph beginning “A celebrated result of Hutchcroft”