THE SCALING LIMIT OF THE MEMBRANE MODEL

• 成果类型: Article
• 署名作者: Cipriani, Alessandra; Dan, Biltu; Hazra, Rajat Subhra
• 署名单位: Delft University of Technology; Indian Statistical Institute; Indian Statistical Institute Kolkata
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/19-AOP1351
• 发表日期: 2019
• 页码: 3963-4001
• 关键词: (1+1)-dimensional pinning models; entropic repulsion; FIELDS; localization; maximum

摘要

On the integer lattice, we consider the discrete membrane model, a random interface in which the field has Laplacian interaction. We prove that, under appropriate rescaling, the discrete membrane model converges to the continuum membrane model in d >= 2. Namely, it is shown that the scaling limit in d = 2, 3 is a Holder continuous random field, while in d >= 4 the membrane model converges to a random distribution. As a by-product of the proof in d = 2, 3, we obtain the scaling limit of the maximum. This work complements the analogous results of Caravenna and Deuschel (Ann. Probab. 37 (2009) 903-945) in d = 1.