dorsal/arxiv
View SchemaPlanar Site Percolation, End Structure, and the Benjamini-Schramm Conjecture
| Authors | Zhongyang Li |
|---|---|
| Categories | |
| ArXiv ID | 2601.09958vv1 |
| URL | https://arxiv.org/abs/2601.09958 |
| License | http://creativecommons.org/licenses/by/4.0/ |
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.
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"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\u0027\u0027 thresholds $p^{\\mathrm{site}}_{c,F}(G)$.\n 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.\n 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 $p_u^{\\mathrm{site}}(G)\u003c1-p_c^{\\mathrm{site}}(G)$. 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.\n Our proofs combine a cutset characterization of $p_c^{\\mathrm{site}}$ with a planar alternating--arm exploration organized by an end--adapted boundary decomposition.",
"arxiv_id": "2601.09958",
"authors": [
"Zhongyang Li"
],
"categories": [
"math.PR",
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"math.CO",
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],
"license": "http://creativecommons.org/licenses/by/4.0/",
"title": "Planar Site Percolation, End Structure, and the Benjamini-Schramm Conjecture",
"url": "https://arxiv.org/abs/2601.09958",
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