8 problems

Let $G$ be a graph equipped with a nonunimodular transitive automorphism group, and let $p_h$ and $p_u$ denote the Hausdorff and uniqueness thresholds, respectively. Suppose that t…

Let $G$ be a graph, let $\Gamma$ be a closed subgroup of $\operatorname{Aut}(G)$ acting transitively on $G$, and let $\mathbf{w}$ be the induced relative weight function. Let…

Let $G$ be a graph, let $\Gamma$ be a closed subgroup of $\operatorname{Aut}(G)$ acting transitively on $G$, and let $\mathbf{w}$ be the induced relative weight function. A cluster…

Let $G$ be a graph whose automorphism group contains a nonunimodular subgroup acting quasi-transitively. Let $p_c$, $p_h$, and $p_u$ denote respectively the critical, heavy-cluster…

FUSF heaviness and branching conjecture. Each tree in the $\operatorname{FUSF}$ is $\Gamma$-heavy and has branching number bigger than one almost surely.

Let $G$ be a quasi-transitive nonunimodular graph, let $\Gamma\subset\operatorname{Aut}(G)$ be quasi-transitive, and suppose $p_h(G,\Gamma)=p_u$. Consider Bernoulli($p$) percolatio…

Let $G$ be a connected, locally finite graph, and let $\Gamma\subseteq\operatorname{Aut}(G)$ be quasi-transitive and nonunimodular. Let $p_h(G,\Gamma)$ denote the heavy-cluster thr…

Let $G$ be a connected, locally finite graph, and let $\Gamma\subseteq\operatorname{Aut}(G)$ be quasi-transitive and nonunimodular. For each $\lambda$, let $p_c(\lambda)$ denote th…