Nonexistence of random gradient Gibbs measures in continuous interface models in d=2

• 成果类型: Article
• 署名作者: van Enter, Aernout C. D.; Kulske, Christof
• 署名单位: University of Groningen
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/07-AAP446
• 发表日期: 2008
• 页码: 109-119
• 关键词: transitions

摘要

We consider statistical mechanics models of continuous spins in a disordered environment. These models have a natural interpretation as effective interface models. It is well known that without disorder there are no interface Gibbs measures in infinite volume in dimension d = 2, while there are gradient Gibbs measures describing an infinite-volume distribution for the increments of the field, as was shown by Funaki and Spohn. In the present paper we show that adding a disorder term prohibits the existence of such gradient Gibbs measures for general interaction potentials in d = 2. This nonexistence result generalizes the simple case of Gaussian fields where it follows from an explicit computation. In d = 3 where random gradient Gibbs measures are expected to exist, our method provides a lower bound of the order of the inverse of the distance on the decay of correlations of Gibbs expectations w.r.t. the distribution of the random environment.