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…

Let $Q^d$ be the $d$-dimensional hypercube, let $V_p$ denote the vertices retained in the site-percolation model, and let $x$ be the constant such that the unique giant component o…

Fix constants $0<\alpha<1$ and $c_1,c_2>0$, and let $\epsilon>0$ be sufficiently small. Let $C_1=C_1(\epsilon,\alpha,c_1,c_2)>0$ and $C_2=C_2(\epsilon,\alpha,c_1,c_2)>0$ be suitabl…