Percolation on self-dual polygon configurations
| Autor: | Bollobas, Bela, Riordan, Oliver |
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| Rok vydání: | 2010 |
| Předmět: | |
| Zdroj: | Bolyai Society Mathematical Studies, Volume 21 (2010), Pages 131-217 |
| Druh dokumentu: | Working Paper |
| DOI: | 10.1007/978-3-642-14444-8_3 |
| Popis: | 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. Comment: Expanded; 73 pages, 24 figures |
| Databáze: | arXiv |
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