2006/06/30 by Béla Bollobás, Bela Bollobas, Oliver Riordan · 1 citation
Mathematics · Physics and Astronomy · #Random Matrices and Applications #Stochastic processes and statistical mechanics #Theoretical and Computational Physics #math.CO #math.PR #msc:60K35 #msc:82B43
paper · pdf · doi:10.1002/rsa.20205
published as Random Structures and Algorithms 32 (2008), 463--472. · 11 pages, 1 figure. Revised with applications added; to appear in Random Structures and Algorithms
arxiv created 2007/02/06 · openalex publication_date 2008/02/21 · arxiv updated 2009/12/01 · openalex created_date 2019/06/27 · openalex updated_date 2026/07/28
Abstract 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