3 problems

Let $G$ be an amenable transitive graph, let $\lambda>0$, and let $\mathcal{C}_{\mathbf{0}}$ denote the set of vertices reached from the origin in the infinite-lifespan frog model.…

Equivalence conjecture. For amenable graphs $G$, these four assertions are equivalent.

Subexponential finite-cluster decay conjecture. There is a sequence $\alpha_n=o(n)$ such that, for every $p>p_c(G)$,