The phase transitions of the planar random-cluster and Potts models with are sharp

• 成果类型: Article
• 署名作者: Duminil-Copin, Hugo; Manolescu, Ioan
• 署名单位: University of Geneva
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-015-0621-0
• 发表日期: 2016
• 页码: 865-892
• 关键词: ising-model; percolation

摘要

We prove that random-cluster models with on a variety of planar lattices have a sharp phase transition, that is that there exists some parameter below which the model exhibits exponential decay and above which there exists a.s. an infinite cluster. The result may be extended to the Potts model via the Edwards-Sokal coupling. Our method is based on sharp threshold techniques and certain symmetries of the lattice; in particular it makes no use of self-duality. Part of the argument is not restricted to planar models and may be of some interest for the understanding of random-cluster and Potts models in higher dimensions. Due to its nature, this strategy could be useful in studying other planar models satisfying the FKG lattice condition and some additional differential inequalities.