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September 2008 Approximate zero-one laws and sharpness of the percolation transition in a class of models including two-dimensional Ising percolation
J. van den Berg
Ann. Probab. 36(5): 1880-1903 (September 2008). DOI: 10.1214/07-AOP380

Abstract

One of the most well-known classical results for site percolation on the square lattice is the equation pc+pc*=1. In words, this equation means that for all values ≠ pc of the parameter p, the following holds: either a.s. there is an infinite open cluster or a.s. there is an infinite closed “star” cluster. This result is closely related to the percolation transition being sharp: below pc, the size of the open cluster of a given vertex is not only (a.s.) finite, but has a distribution with an exponential tail. The analog of this result has been proven by Higuchi in 1993 for two-dimensional Ising percolation (at fixed inverse temperature β<βc) with external field h, the parameter of the model.

Using sharp-threshold results (approximate zero-one laws) and a modification of an RSW-like result by Bollobás and Riordan, we show that these results hold for a large class of percolation models where the vertex values can be “nicely” represented (in a sense which will be defined precisely) by i.i.d. random variables. We point out that the ordinary percolation model obviously belongs to this class and we also show that the Ising model mentioned above belongs to it.

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J. van den Berg. "Approximate zero-one laws and sharpness of the percolation transition in a class of models including two-dimensional Ising percolation." Ann. Probab. 36 (5) 1880 - 1903, September 2008. https://doi.org/10.1214/07-AOP380

Information

Published: September 2008
First available in Project Euclid: 11 September 2008

zbMATH: 1155.60044
MathSciNet: MR2440926
Digital Object Identifier: 10.1214/07-AOP380

Subjects:
Primary: 60K35
Secondary: 82B43

Rights: Copyright © 2008 Institute of Mathematical Statistics

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Vol.36 • No. 5 • September 2008
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