Approximate zero-one laws and sharpness of the percolation transition in a class of models including two-dimensional ising percolation

• 成果类型: Article
• 署名作者: van den Berg, J.
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/07-AOP380
• 发表日期: 2008
• 页码: 1880-1903
• 关键词: critical probability; PHASE-TRANSITION; threshold; inequalities; plane

摘要

One of the most well-known classical results for site percolation on the square lattice is the equation p(c) + p(c)* = 1. In words, this equation means that for all vaues not equal p(c) of the Parameter p. the following holds: either a.s. there is all 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 p(c), 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 beta < beta(c)) 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 Bollobas and Riordan. we show that these results hold for a large class of percolation models where the vertex values Call 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.