# Maximum and shape of interfaces in 3D Ising crystals

## TL;DR

This paper analyzes the maximum height and shape of interfaces in 3D Ising crystals at low temperatures, establishing laws of large numbers and central limit theorems for interface properties, and generalizing results to higher dimensions.

## Contribution

It provides a detailed probabilistic description of the interface's maximum, shape, and fluctuations in the 3D Ising model, extending understanding beyond previous tractable models.

## Key findings

- Maximum interface height scales as (2/α_β) log n
- Interface decomposes into a linear number of weakly-dependent increments
- Surface area, volume, and tip location obey CLTs with linear variance

## Abstract

Dobrushin (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 $M_n / \log n \to 2/\alpha_\beta$ as $n\to\infty$, 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 $d\geq 3$.