SCALING LIMIT FOR A CLASS OF GRADIENT FIELDS WITH NONCONVEX POTENTIALS

• 成果类型: Article
• 署名作者: Biskup, Marek; Spohn, Herbert
• 署名单位: University of California System; University of California Los Angeles; University of South Bohemia Ceske Budejovice; Technical University of Munich
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/10-AOP548
• 发表日期: 2011
• 页码: 224-251
• 关键词: phi interface model; quenched invariance-principles; bounded random conductances; random-walk; percolation clusters; perturbations

摘要

We consider gradient fields (phi(x) : x is an element of Z(d)) whose law takes the Gibbs-Boltzmann form Z(-1) exp{-Sigma(< x,y >) V(phi(y) - phi(x))}, where the sum runs over nearest neighbors. We assume that the potential V admits the representation V(eta) := -log integral rho(dk)exp[-1/2 kappa eta(2)], where rho is a positive measure with compact support in (0, infinity). Hence, the potential V is symmetric, but nonconvex in general. While for strictly convex V's, the translation-invariant, ergodic gradient Gibbs measures are completely characterized by their tilt, a nonconvex potential as above may lead to several ergodic gradient Gibbs measures with zero tilt. Still, every ergodic, zero-tilt gradient Gibbs measure for the potential V above scales to a Gaussian free field.