Hyperbolic site percolation

TL;DR

This paper investigates site percolation on quasi-transitive planar graphs in Euclidean and hyperbolic planes, establishing new relations between critical probabilities and extending known theorems to hyperbolic settings.

Contribution

It extends percolation critical probability relations to hyperbolic plane graphs and introduces a method linking site percolation to dependent bond processes.

Findings

Proves $p_u(G_1)+p_c(G_2)=1$ for matching pairs in hyperbolic plane.

Shows $p_c(G)=p_u(G) ext{ for amenable graphs, with } p_c(G) ext{ at least } 1/2.

Answers two conjectures of Benjamini and Schramm positively for quasi-transitive graphs.

Abstract

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.