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Percolation on dual lattices with k ‐fold symmetry
Author(s) -
Bollobás Béla,
Riordan Oliver
Publication year - 2008
Publication title -
random structures and algorithms
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.314
H-Index - 69
eISSN - 1098-2418
pISSN - 1042-9832
DOI - 10.1002/rsa.20205
Subject(s) - hexagonal lattice , lattice (music) , square lattice , uniqueness , mathematics , combinatorics , percolation (cognitive psychology) , percolation threshold , physics , statistical physics , condensed matter physics , quantum mechanics , mathematical analysis , ising model , neuroscience , antiferromagnetism , acoustics , electrical resistivity and conductivity , biology
Zhang found a simple, elegant argument deducing the nonexistence 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 two‐fold (or three‐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. © 2008 Wiley Periodicals, Inc. Random Struct. Alg., 2008