Maximum and Shape of Interfaces in 3D Ising Crystals

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 M-n, its maximum: if the inverse temperature beta is large enough, then M-n/logn -> 2/alpha(beta) as n -> infinity, in probability, where alpha(beta) 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. (c) 2020 Wiley Periodicals LLC