Planar Site Percolation, End Structure, and the Benjamini-Schramm Conjecture

Let $G$ be an infinite, connected, locally finite planar graph and consider i.i.d.\ Bernoulli$(p)$ site percolation. Write $p_c^{\mathrm{s}\mathrm{i}\mathrm{t}\mathrm{e}}(G)$ and $p_u^{\mathrm{s}\mathrm{i}\mathrm{t}\mathrm{e}}(G)$ for the critical and uniqueness thresholds. Using a well–separated Freudenthal embedding $G\hookrightarrow {\mathbb{S}}^2$, we introduce a cycle–separation equivalence on ends and associated ``directional’’ thresholds $p_{c,F}^{\mathrm{s}\mathrm{i}\mathrm{t}\mathrm{e}}(G)$. When the set of end–equivalence classes is countable, we show that $p_c^{\mathrm{s}\mathrm{i}\mathrm{t}\mathrm{e}}(G)={\inf }_F{p}_{c,F}^{\mathrm{s}\mathrm{i}\mathrm{t}\mathrm{e}}(G)$ and that for every $p\in (\frac{1}{2},1-p_c^{\mathrm{s}\mathrm{i}\mathrm{t}\mathrm{e}}(G))$ there are almost surely infinitely many infinite open clusters. Combined with the $0\mathrm{/}\mathrm{\infty }$ theorem of Glazman–Harel–Zelesko for $p\le \frac{1}{2}$, this yields non–uniqueness throughout the full coexistence interval $(p_c^{\mathrm{s}\mathrm{i}\mathrm{t}\mathrm{e}}(G),1-p_c^{\mathrm{s}\mathrm{i}\mathrm{t}\mathrm{e}}(G))$, and hence $p_u^{\mathrm{s}\mathrm{i}\mathrm{t}\mathrm{e}}(G)\ge 1-p_c^{\mathrm{s}\mathrm{i}\mathrm{t}\mathrm{e}}(G)$ in this setting. This resolves the extension problem posed by Glazman–Harel–Zelesko for the upper half of the coexistence regime under a natural countability hypothesis. In contrast, for graphs with uncountably many end–equivalence classes we give criteria guaranteeing infinitely many infinite clusters above criticality, and we construct an explicit locally finite planar graph of minimum degree at least $7$ for which $p_u^{\mathrm{s}\mathrm{i}\mathrm{t}\mathrm{e}}(G)<1-p_c^{\mathrm{s}\mathrm{i}\mathrm{t}\mathrm{e}}(G)$. Consequently, the Benjamini–Schramm conjecture (Conjecture 7 in \cite{bs96}) that planarity together with minimal vertex degree at least 7 forces infinitely many infinite clusters for all $p\in (p_c,1-p_c)$ does not hold in full generality. Our proofs combine a cutset characterization of $p_c^{\mathrm{s}\mathrm{i}\mathrm{t}\mathrm{e}}$ with a planar alternating–arm exploration organized by an end–adapted boundary decomposition.