Hyperbolic Site Percolation

Abstract

ABSTRACT Several results are presented for site percolation on quasi‐transitive, planar graphs with one end, when properly embedded in either the Euclidean or hyperbolic plane. If is a matching pair derived from some quasi‐transitive mosaic , then , where is the critical probability for the existence of an infinite cluster, and 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 (Thm. 3.8, Journal of the American Mathematical Society 14 (2001): 487–507) for transitive bond percolation. It follows that , where denotes the matching graph of . In particular, and hence, when is amenable we have . When combined with the main result of the companion paper by the same authors ( Random Structures & Algorithms (2024)), we obtain for transitive that the strict inequality holds if and only if 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 matters, the results reported here answer positively two conjectures of Benjamini and Schramm (Conj. 7, 8, Electronic Communications in Probability 1 (1996): 71–82) in the case of quasi‐transitive graphs.