CONVERGENCE TO THE THERMODYNAMIC LIMIT FOR RANDOM-FIELD RANDOM SURFACES

• 成果类型: Article
• 署名作者: Dario, Paul
• 署名单位: Universite Gustave-Eiffel; Universite Paris-Est-Creteil-Val-de-Marne (UPEC)
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/22-AAP1844
• 发表日期: 2023
• 页码: 1173-1195
• 关键词: gradient gibbs measures; Scaling Limit; interfaces; models; homogenization; perturbations; transitions; uniqueness; systems; decay

摘要

We study random surfaces with a uniformly convex gradient interaction in the presence of quenched disorder taking the form of a random independent external field. Previous work on the model has focused on proving existence and uniqueness of infinite-volume gradient Gibbs measures with a given tilt and on studying the fluctuations of the surface and its discrete gradient.In this work, we focus on the convergence of the thermodynamic limit, establishing convergence of the finite-volume distributions with Dirich-let boundary conditions to translation-covariant (gradient) Gibbs measures. Specifically, it is shown that, when the law of the random field has finite sec-ond moment and is symmetric, the distribution of the gradient of the surface converges in dimensions d >= 4 while the distribution of the surface itself con-verges in dimensions d >= 5. Moreover, a power-law upper bound on the rate of convergence in Wasserstein distance is obtained. The results partially an-swer a question discussed by Cotar and Kulske (Ann. Appl. Probab. 22 (2012) 1650-1692).