Hyperbolic site percolation
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2022
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| author | Grimmett, Geoffrey R. Li, Zhongyang |
| author_facet | Grimmett, Geoffrey R. Li, Zhongyang |
| contents | Several results are presented for site percolation on quasi-transitive, planar graphs $G$ with one end, when properly embedded in either the Euclidean or hyperbolic plane. If $(G_1,G_2)$ is a matching pair derived from some quasi-transitive mosaic $M$, then $p_u(G_1)+p_c(G_2)=1$, where $p_c$ is the critical probability for the existence of an infinite cluster, and $p_u$ is the critical value for the existence of a unique such cluster. This fulfils and extends to the hyperbolic plane an observation of Sykes and Essam(1964), and it extends to quasi-transitive site models a theorem of Benjamini and Schramm (Theorem 3.8, J. Amer. Math. Soc. 14 (2001) 487--507) for transitive bond percolation.
It follows that $p_u (G)+p_c (G_*)=p_u(G_*)+p_c(G)=1$, where $G_*$ denotes the matching graph of $G$. In particular, $p_u(G)+p_c(G)\ge 1$ and hence, when $G$ is amenable we have $p_c(G)=p_u(G) \ge \frac12$. When combined with the main result of the companion paper by the same authors ("Percolation critical probabilities of matching lattice-pairs", Random Struct. Alg. 2024), we obtain for transitive $G$ that the strict inequality $p_u(G)+p_c(G)> 1$ holds if and only if $G$ is not a triangulation.
A key technique is a method for expressing a planar site percolation process on a matching pair in terms of a dependent bond process on the corresponding dual pair of graphs. Amongst other things, the results reported here answer positively two conjectures of Benjamini and Schramm (Conjectures 7 and 8, Electron. Comm. Probab. 1 (1996) 71--82) in the case of quasi-transitive graphs. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2203_00981 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Hyperbolic site percolation Grimmett, Geoffrey R. Li, Zhongyang Probability Mathematical Physics 60K35, 82B43 Several results are presented for site percolation on quasi-transitive, planar graphs $G$ with one end, when properly embedded in either the Euclidean or hyperbolic plane. If $(G_1,G_2)$ is a matching pair derived from some quasi-transitive mosaic $M$, then $p_u(G_1)+p_c(G_2)=1$, where $p_c$ is the critical probability for the existence of an infinite cluster, and $p_u$ is the critical value for the existence of a unique such cluster. This fulfils and extends to the hyperbolic plane an observation of Sykes and Essam(1964), and it extends to quasi-transitive site models a theorem of Benjamini and Schramm (Theorem 3.8, J. Amer. Math. Soc. 14 (2001) 487--507) for transitive bond percolation. It follows that $p_u (G)+p_c (G_*)=p_u(G_*)+p_c(G)=1$, where $G_*$ denotes the matching graph of $G$. In particular, $p_u(G)+p_c(G)\ge 1$ and hence, when $G$ is amenable we have $p_c(G)=p_u(G) \ge \frac12$. When combined with the main result of the companion paper by the same authors ("Percolation critical probabilities of matching lattice-pairs", Random Struct. Alg. 2024), we obtain for transitive $G$ that the strict inequality $p_u(G)+p_c(G)> 1$ holds if and only if $G$ is not a triangulation. A key technique is a method for expressing a planar site percolation process on a matching pair in terms of a dependent bond process on the corresponding dual pair of graphs. Amongst other things, the results reported here answer positively two conjectures of Benjamini and Schramm (Conjectures 7 and 8, Electron. Comm. Probab. 1 (1996) 71--82) in the case of quasi-transitive graphs. |
| title | Hyperbolic site percolation |
| topic | Probability Mathematical Physics 60K35, 82B43 |
| url | https://arxiv.org/abs/2203.00981 |