Maximum and Shape of Interfaces in 3D Ising Crystals

• Published in: Communications on pure and applied mathematics Vol. 75; no. 12; pp. 2575 - 2684
• Main Authors: Gheissari, Reza, Lubetzky, Eyal
• Format: Journal Article
• Language: English
• Published: Melbourne John Wiley & Sons Australia, Ltd 01.12.2022; John Wiley and Sons, Limited
• Subjects: Asymptotic properties; Boundary conditions; Deviation; Interfaces; Ising model; Low temperature; Probability theory; Random walk; Three dimensional models
• ISSN: 0010-3640; 1097-0312
• DOI: 10.1002/cpa.21960

Dobrushin in 1972 showed that the interface of a 3D Ising model with minus boundary conditions above the xy‐plane and plus below is rigid (has O(1) fluctuations) at every sufficiently low temperature. Since then, basic features of this interface—such as the asymptotics of its maximum—were only identified in more tractable random surface models that approximate the Ising interface at low temperatures, e.g., for the (2+1)D solid‐on‐solid model. Here we study the large deviations of the interface of the 3D Ising model in a cube of side length n with Dobrushin's boundary conditions, and in particular obtain a law of large numbers for Mn, its maximum: if the inverse temperature β is large enough, then Mn/logn → 2/αβ as n → ∞, in probability, where αβ is given by a large‐deviation rate in infinite volume. We further show that, on the large‐deviation event that the interface connects the origin to height h, it consists of a 1D spine that behaves like a random walk in that it decomposes into a linear (in h) number of asymptotically stationary, weakly dependent increments that have exponential tails. As the number T of increments diverges, properties of the interface such as its surface area, volume, and the location of its tip, all obey CLTs with variances linear in T. These results generalize to every dimension d ≥ 3. © 2020 Wiley Periodicals LLC.