Percolation on self-dual polygon configurations

TL;DR

This paper proves that self-duality conditions in a broad class of planar percolation models imply criticality, extending previous results to models with dependent bonds and less symmetry, using new mathematical techniques.

Contribution

It generalizes the connection between self-duality and criticality to models with dependent bonds and weaker symmetry, introducing new tools like a generalized Harris's Lemma.

Findings

Self-duality implies criticality in generalized percolation models.

Developed a generalized Harris's Lemma for partially ordered sets.

Proved a Russo-Seymour-Welsh type lemma under minimal symmetry.

Abstract

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.