TIGHTNESS AND TAILS OF THE MAXIMUM IN 3D ISING INTERFACES

• 成果类型: Article
• 署名作者: Gheissari, Reza; Lubetzky, Eyal
• 署名单位: New York University
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/20-AOP1459
• 发表日期: 2021
• 页码: 732-792
• 关键词: large deviations; sos surfaces; fluctuations; coexistence; phases; limit; MODEL

摘要

Consider the 3D Ising model on a box of side length n with minus boundary conditions above the xy-plane and plus boundary conditions below it. At low temperatures, Dobrushin (1972) showed that the interface separating the predominantly plus and predominantly minus regions is localized: its height above a fixed point has exponential tails. Recently, the authors proved a law of large numbers for the maximum height M-n of this interface: for every beta large, M-n/log n -> c beta in probability as n -> infinity. Here, we show that the laws of the centered maxima (M-n - E[M-n])(n >= 1) are uniformly tight. Moreover, even though this sequence does not converge, we prove that it has uniform upper and lower Gumbel tails (exponential right tails and doubly exponential left tails). Key to the proof is a sharp (up to O (1) precision) understanding of the surface large deviations. This includes, in particular, the shape of a pillar that reaches near-maximum height, even at its base, where the interactions with neighboring pillars are dominant.