Planar site percolation via tree embeddings

Abstract: We prove that for any any infinite, connected, planar graph $G$ properly embedded in $\RR^2$ with minimal vertex degree at least 7, the i.i.d.~Bernoulli($p$) site percolation on $G$ a.s.~has infinitely many infinite 1-clusters and for any $p\in (p_c^{site},1-p_c^{site})$. Moreover, $p_c^{site}<\frac{1}{2}$, so the above interval is non-empty. This confirms a conjecture of Benjamini and Schramm in 1996 (Conjecture 7 in \cite{bs96}). The proof is based on a novel construction of embedded trees on such graphs, which not only proves the existence of infinitely many infinite clusters when $p$ is in a neighborhood of $\frac{1}{2}$, but also proves the exponential decay of point-to-point connection probabilities by constructing infinitely many trees separating two vertices as the distance of the two vertices goes to infinity.