EXISTENCE OF RANDOM GRADIENT STATES

• 成果类型: Article
• 署名作者: Cotar, Codina; Kuelske, Christof
• 署名单位: Ruhr University Bochum
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/11-AAP808
• 发表日期: 2012
• 页码: 1650-1692
• 关键词: models; interfaces; convexity

摘要

We consider two versions of random gradient models. In model A the interface feels a bulk term of random fields while in model B the disorder enters through the potential acting on the gradients. It is well known that for gradient models without disorder there are no Gibbs measures in infinite-volume in dimension d = 2, while there are gradient Gibbs measures describing an infinite-volume distribution for the gradients of the field, as was shown by Funaki and Spohn. Van Enter and Kulske proved that adding a disorder term as in model A prohibits the existence of such gradient Gibbs measures for general interaction potentials in d = 2. In the present paper we prove the existence of shift-covariant gradient Gibbs measures with a given tilt u is an element of R-d for model A when d >= 3 and the disorder has mean zero, and for model B when d >= 1. When the disorder has nonzero mean in model A, there are no shift-covariant gradient Gibbs measures for d >= 3. We also prove similar results of existence/nonexistence of the surface tension for the two models and give the characteristic properties of the respective surface tensions.