Dualities Useful in Bond Percolation

Abstract

We state four facts about dual pairs of graphs drawn in a plane. These facts pertain respectively to finite non-oriented, infinite non-oriented, finite oriented, and infinite oriented graphs. We do not include proofs of the former two facts (although we have them), but show that these facts are “evident” in some naive sense. Then we deduce the latter two facts from the former two ones. All of these facts can be used to obtain upper estimations for critical values of two-dimensional percolation models and we present three references and one example to illustrate this.