9 problems

Let $G$ be a planar graph, and consider i.i.d. Bernoulli site percolation at parameter $p=\tfrac{1}{2}$. Benjamini–Schramm Conjecture 8. If percolation occurs almost surely, meanin…

Consider minimal spanning forests on ${\mathbb Z}^d$, and let the number of infinite clusters refer to the clusters of the relevant minimal spanning forest. Newman–Stein conjecture…

Consider Bernoulli percolation on a transitive graph in a parameter regime where there are almost surely infinitely many infinite clusters. Two infinite clusters are said to have i…

Let $A_0=3\sqrt{3}/2$, and let an infinite planar cluster consist of equal-area regions of area $A_0$. Let $E$ be the area of its complement, with $0\leq E\leq\infty$. A cluster is…

Let $\mu$ be a translation-invariant, ergodic percolation measure on $\mathbb{Z}^2$ that almost surely has infinitely many infinite clusters. Let $\mu_{\operatorname{hp}}$ denote i…

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

Let $d\geq 3$ and consider the percolation with constant freezing (PCF) process on $\mathbbm{Z}^d$, with edge-opening rate $1$ and freezing rate $\alpha>0$. Multiple infinite clust…

Let $p$ be the percolation parameter, and let $p_\mathrm c$ and $p_\mathrm u$ denote the critical and uniqueness thresholds, respectively. Let $p_{1/2}\in(p_\mathrm c,p_\mathrm u)$…

Nonuniqueness conjecture. For every nonamenable transitive graph, there is some value of $p$ for which there is more than one infinite cluster almost surely.