Percolation on dual lattices with k-fold symmetry

Details Percolation on dual lattices with k-fold symmetry Percolation on dual lattices with k-fold symmetry

2006-06-07

arxiv.org/abs/math/0606149v2

Random Structures and Algorithms 32 (2008), 463--472.

Mathematics

Probability

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

Scientific paper

10.1002/rsa.20205

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.

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