3 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…

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…