Uniqueness of gradient Gibbs measures with disorder

• 成果类型: Article
• 署名作者: Cotar, Codina; Kuelske, Christof
• 署名单位: University of London; University College London; Ruhr University Bochum
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-014-0580-x
• 发表日期: 2015
• 页码: 587-635
• 关键词: logarithmic sobolev inequalities; continuous symmetry; systems; interfaces; models; decay

摘要

We consider-in a uniformly strictly convex potential regime-two versions of random gradient models with disorder. 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. We assume a general distribution on the disorder with uniformly-bounded finite second moments. It is well known that for gradient models without disorder there are no Gibbs measures in infinite volume in dimension , while there are shift-invariant gradient Gibbs measures describing an infinite-volume distribution for the gradients of the field, as was shown by Funaki and Spohn (Commun Math Phys 185:1-36, 1997). Van Enter and Kulske proved in (Ann Appl Probab 18(1):109-119, 2008) that adding a disorder term as in model (A) prohibits the existence of such gradient Gibbs measures for general interaction potentials in . In Cotar and Kulske (Ann Appl Probab 22(5):1650-1692, 2012) we proved the existence of shift-covariant random gradient Gibbs measures for model (A) when , the disorder is i.i.d and has mean zero, and for model (B) when and the disorder has a stationary distribution. In the present paper, we prove existence and uniqueness of shift-covariant random gradient Gibbs measures with a given expected tilt and with the corresponding annealed measure being ergodic: for model (A) when and the disordered random fields are i.i.d. and symmetrically-distributed, and for model (B) when and for any stationary disorder-dependence structure. We also compute for both models for any gradient Gibbs measure constructed as in Cotar and Kulske (Ann Appl Probab 22(5):1650-1692, 2012), when the disorder is i.i.d. and its distribution satisfies a Poincar, inequality assumption, the optimal decay of covariances with respect to the averaged-over-the-disorder gradient Gibbs measure.