Percolation on self-dual polygon configurations

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.

Expanded; 73 pages, 24 figures

References in corpus (3)

• Exact bond percolation thresholds in two dimensions
• Percolation on dual lattices with k-fold symmetry
• Equality of bond percolation critical exponents for pairs of dual lattices

Cited by in corpus (9)

• On the scaling limits of planar percolation
• Inhomogeneous bond percolation on square, triangular and hexagonal lattices
• Universality for bond percolation in two dimensions
• The critical manifolds of inhomogeneous bond percolation on bow-tie and checkerboard lattices
• Delocalization of the height function of the six-vertex model
• Rigorous Confidence Intervals on Critical Thresholds in 3 Dimensions
• Percolation on Hypergraphs with Four-Edges
• The computation of generalized percolation critical polynomials by the deletion-contraction algorithm
• Logarithmic variance for the height function of square-ice