Planar site percolation on semi-transitive graphs

Abstract

Semi-transitive graphs, defined in \cite{hps98} as examples where ``uniform percolation" holds whenever $p>p_c$, are a large class of graphs more general than quasi-transitive graphs. Let $G$ be a semi-transitive graph with one end which can be properly embedded into the plane with uniformly bounded face degree for finite faces and minimal vertex degree at least 7. We show that $p_u^{site}(G) +p_c^{site}(G_*)=1$, where $G_*$ denotes the matching graph of $G$. This fulfils and extends an observation of Sykes and Essam in 1964 (\cite{SE64}) to semi-transitive graphs.