Affiliation:
1. Centre for Mathematical Sciences Cambridge University Cambridge UK
2. Department of Mathematics University of Connecticut Storrs Connecticut USA
Abstract
AbstractA necessary and sufficient condition is established for the strict inequality between the critical probabilities of site percolation on a one‐ended, quasi‐transitive, plane graph and on its matching graph . When is transitive, strict inequality holds if and only if is not a triangulation. The basic approach is the standard method of enhancements, but its implementation has complexity arising from the non‐Euclidean (hyperbolic) space, the study of site (rather than bond) percolation, and the generality of the assumption of quasi‐transitivity. This result is complementary to the work of the authors (“Hyperbolic site percolation,” arXiv:2203.00981) on the equality , where is the critical probability for the existence of a unique infinite open cluster. It implies for transitive, one‐ended that , with equality if and only if is a triangulation.