Large deviations and continuum limit in the 2D Ising model

• 成果类型: Article
• 署名作者: Pfister, CE; Velenik, Y
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s004400050139
• 发表日期: 1997
• 页码: 435-506
• 关键词: order large deviations; phase-separation line; correlation inequalities; bernoulli percolation; surface-tension; ferromagnet; transitions; coexistence; systems; bounds

摘要

We study the 2D Ising model in a rectangular box Lambda(L) of linear size O(L). We determine the exact asymptotic behaviour of the large deviations of the magnetization Sigma(t is an element of Lambda L) sigma(t) when L --> infinity for values of the parameters of the model corresponding to the phase coexistence region, where the order parameter m* is strictly positive. We study in particular boundary effects due to an arbitrary real-valued boundary magnetic field. Using the self-duality of the model a large part of the analysis consists in deriving properties of the covariance function [sigma(0)sigma(t)], as \t\ --> infinity, at dual values of the parameters of the model. To do this analysis we establish new results about the high-temperature representation of the model. These results are valid for dimensions D greater than or equal to 2 and up to the critical temperature. They give a complete non-perturbative exposition of the high-temperature representation. We then study the Gibbs measure conditioned by {\Sigma(t is an element of Lambda L) sigma(t) -m\Lambda(L)\\ less than or equal to \Lambda(L)\L-c}, with 0 < c < 1/4 and -m* < m < m*. We construct the continuum limit of the model and describe the limit by the solutions of a variational problem of isoperimetric type.