Benjamini–Schramm nonamenability conjecture for the nonuniqueness phase

Prove or disprove that, for every quasi-transitive graph, the interval between the critical and uniqueness thresholds for Bernoulli percolation is nonempty if and only if the graph is nonamenable.

Background

The paper discusses the classical question of when Bernoulli percolation on a quasi-transitive graph has infinitely many infinite clusters. It records the Benjamini–Schramm conjecture that this nonuniqueness phase is nontrivial exactly for nonamenable graphs. The forward implication has been established, but the converse remains known only in particular cases, so the conjecture is not resolved in the stated generality.

References

Benjamini and Schramm conjectured that for a quasi-transitive graph $G$, this phase is not empty if and only if $G$ is nonamenable Conjecture 6. Recall that we say that the graph is amenable if the Cheeger constant $\Phi_V(G)=0$, where

— $β$-Skewed Maximal Spanning Forests  (2609.11845 - Chatterjee et al., 10 Sep 2026) in Section 3.2, paragraph following the definition of the Cheeger constant

Finally, in Conjecture 6.2, Terlov and Timar conjectured that for a quasi-transitive graph $G$, $p_h(G,\Gamma)<p_u(G)$ if and only if $G$ is $w_\Gamma$-nonamenable, and showed the forward direction (analogous to the result of ).

— $β$-Skewed Maximal Spanning Forests  (2609.11845 - Chatterjee et al., 10 Sep 2026) in Section 3.2, paragraph beginning “Finally, in \cite[Conjecture 6.2]{TT25}”