Percolation on dual lattices with k-fold symmetry

Abstract

Zhang found a simple, elegant argument deducing the non-existence of an infinite open cluster in certain lattice percolation models (for example, p=1/2 bond percolation on the square lattice) from general results on the uniqueness of an infinite open cluster when it exists; this argument requires some symmetry. Here we show that a simple modification of Zhang's argument requires only 2-fold (or 3-fold) symmetry, proving that the critical probabilities for percolation on dual planar lattices with such symmetry sum to 1. Like Zhang's argument, our extension applies in many contexts; in particular, it enables us to answer a question of Grimmett concerning the anisotropic random cluster model on the triangular lattice.

11 pages, 1 figure. Revised with applications added; to appear in Random Structures and Algorithms

References in corpus (2)

• The Random-Cluster Model
• A short proof of the Harris-Kesten Theorem

Cited by in corpus (6)

• Percolation thresholds on 2D Voronoi networks and Delaunay triangulations
• Percolation on self-dual polygon configurations
• Complex systems: features, similarity and connectivity
• Factor-of-iid Schreier decorations of lattices in Euclidean spaces
• Rigorous Confidence Intervals on Critical Thresholds in 3 Dimensions
• Non-Coexistence of Infinite Clusters in Two-Dimensional Dependent Site Percolation