Percolation on self-dual polygon configurations
Recently, Scullard and Ziff noticed that a broad class of planar percolation models are self-dual under a simple condition that, in a parametrized version of such a model, reduces to a single equation. They state that the solution of the resulting equation gives the critical point. However, just as in the classical case of bond percolation on the square lattice, self-duality is simply the starting point: the mathematical difficulty is precisely showing that self-duality implies criticality. Here we do so for a generalization of the models considered by Scullard and Ziff. In these models, the states of the bonds need not be independent; furthermore, increasing events need not be positively correlated, so new techniques are needed in the analysis. The main new ingredients are a generalization of Harris’s Lemma to products of partially ordered sets, and a new proof of a type of Russo-Seymour- Welsh Lemma with minimal symmetry assumptions.
Bolyai Society Mathematical Studies
Bollobás, B., & Riordan, O. (2010). Percolation on self-dual polygon configurations. Bolyai Society Mathematical Studies, 21, 131-217. https://doi.org/10.1007/978-3-642-14444-8_3