5 problems

Let $G$ be a non-amenable unimodular vertex-transitive graph, and consider the infinite-lifespan frog model. Write $\lambda_{\mathrm{c}}(\mathrm{recurrence})$ for the critical dens…

Let $G$ be a transitive non-amenable graph with critical and uniqueness thresholds satisfying $p_c<p_u$, and fix a vertex $o$. Let $K_o(\omega_p)$ be the open cluster of $o$ in Ber…

Strict inequality conjecture. If $\Gamma$ is non-amenable and quasi-transitive, then

Let $G$ be a transitive non-amenable graph, let $\omega_p$ be Bernoulli site percolation, and let $C^\infty(\omega_p)$ be the set of vertices belonging to infinite clusters. Consid…

Non-amenability conjecture. Let $\mu$ be the measure of the trace of a transient BRW on a URG. Then $\mu$ is non-amenable.