Let $G$ be an infinite, connected planar graph that is properly embedded in $\RR^2$ and has minimal vertex degree at least $7$. We prove that for i.i.d.\ $\mathrm{ Bernoulli}(p)$ site percolation on $G$, for every \[ p \in \bigl(p_c^{ \mathrm{ site}}(G),\, 1-p_c^{ \mathrm{ site}}(G)\bigr), \] there are almost surely infinitely many infinite open $(1$-)$\,$clusters. In particular, we show that $p_c^{ \mathrm{ site}}(G)<\tfrac12$, so the non-uniqueness interval is nonempty. This verifies Conjecture~7 of Benjamini and Schramm~\cite{ bs96} for this class of properly embedded planar graphs. The main new ingredient is an explicit construction of trees embedded in $G$ with controlled local geometry (degrees $3$ and $4$ away from the root). These trees yield non-uniqueness near $p=\tfrac12$ and, via an embedded forest of pairwise disjoint trees, imply exponential decay of two-point connection probabilities in the matching graph $G_*$. To treat the complementary regime $p>\tfrac12$, we introduce a ``binary-tree'' version of uniform percolation that yields stability of infinite clusters under edge additions without any bounded-degree assumption. Beyond planarity, we also obtain a general two-point connectivity inequality on arbitrary infinite, connected locally finite graphs, which implies a non-transitive criterion for the existence of infinitely many infinite clusters in the regime $p_c<p<p_{ \mathrm{ conn}}$.
Categories: math.PR math.CO