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

TL;DR

This work advances the understanding of phase transitions in Bernoulli site percolation on infinite planar graphs by introducing a Freudenthal-embedding–based end-equivalence framework and an FCA methodology that couples cutset analysis with end-directed multi-arm explorations. It establishes that when end-equivalence classes are countable, the critical threshold equals the infimum of end-directed thresholds and nonuniqueness persists across the upper half of the coexistence interval, yielding $p_u^{\mathrm{site}}(G)\ge 1-p_c^{\mathrm{site}}(G)$. In the uncountable regime, the authors give conditions guaranteeing infinitely many infinite clusters and construct a planar graph with $p_u^{\mathrm{site}}(G)<1-p_c^{\mathrm{site}}(G)$, disproving a general Benjamini–Schramm conjecture under minimal degree 7. The paper also develops a robust analytic toolset via the $\phi$-functional and boundary-armed events to control end-directed connectivity, with potential extensions to other planar models. Overall, the results delineate sharp boundaries between countable and uncountable end-structures and confirm that local finiteness is essential for global coexistence statements in planar percolation.

Abstract

Let $G$ be an infinite, connected, locally finite planar graph and consider i.i.d.\ Bernoulli$(p)$ site percolation. Write $p_c^{\mathrm{site}}(G)$ and $p_u^{\mathrm{site}}(G)$ 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 $p^{\mathrm{site}}_{c,F}(G)$. When the set of end--equivalence classes is countable, we show that $p_c^{\mathrm{site}}(G)=\inf_F p^{\mathrm{site}}_{c,F}(G)$ and that for every $p\in\bigl(\tfrac12,\,1-p_c^{\mathrm{site}}(G)\bigr)$ there are almost surely infinitely many infinite open clusters. Combined with the $0/\infty$ theorem of Glazman--Harel--Zelesko for $p\le \tfrac12$, this yields non--uniqueness throughout the full coexistence interval $\bigl(p_c^{\mathrm{site}}(G),\,1-p_c^{\mathrm{site}}(G)\bigr)$, and hence $p_u^{\mathrm{site}}(G)\ge 1-p_c^{\mathrm{site}}(G)$ 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 $p_u^{\mathrm{site}}(G)<1-p_c^{\mathrm{site}}(G)$. 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 $p_c^{\mathrm{site}}$ with a planar alternating--arm exploration organized by an end--adapted boundary decomposition.

Theorems & Definitions (78)

• Definition 1.1: Critical and uniqueness thresholds
• Conjecture 1.3
• Conjecture 1.4
• Definition 1.5: Planar graph
• Remark 1.6
• Theorem 1.7
• Definition 2.1
• Proposition 2.2
• Lemma 2.3
• proof