Planar site percolation on semi-transitive graphs

TL;DR

This paper investigates site percolation on semi-transitive planar graphs, establishing a key relation between critical probabilities of the graph and its matching graph, extending classical percolation results.

Contribution

It proves a new relation between critical percolation thresholds for semi-transitive graphs and their matching graphs, generalizing previous observations.

Findings

Proves that $p_u^{site}(G) + p_c^{site}(G_*)=1$ for certain semi-transitive graphs.

Extends classical percolation results to a broader class of graphs.

Fulfills and generalizes an observation from 1964 to 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.