Hyperbolic site percolation
Abstract
Several results are presented for site percolation on quasitransitive, 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 quasitransitive 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 in 1964. 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 latticepairs", 2022), 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) 7182) in the case of quasitransitive graphs.
 Publication:

arXiv eprints
 Pub Date:
 March 2022
 DOI:
 10.48550/arXiv.2203.00981
 arXiv:
 arXiv:2203.00981
 Bibcode:
 2022arXiv220300981G
 Keywords:

 Mathematics  Probability;
 Mathematical Physics;
 60K35;
 82B43
 EPrint:
 v1 of this post has been split into two parts, of which the current v2 is the first part. The second part will be posted separately on arxiv