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Dobrushin (1972) showed that the interface of a 3D Ising model with minusboundary conditions above the $xy$-plane and plus below is rigid (has$O(1)$-fluctuations) at every sufficiently low temperature. Since then, basicfeatures of this interface -- such as the asymptotics of its maximum -- wereonly identified in more tractable random surface models that approximate theIsing 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 acube of side-length $n$ with Dobrushin's boundary conditions, and in particularobtain a law of large numbers for $M_n$, its maximum: if theinverse-temperature $\beta$ is large enough, then $M_n / \log n \to2/\alpha_\beta$ as $n\to\infty$, in probability, where $\alpha_\beta$ is givenby a large deviation rate in infinite volume.  We further show that, on the large deviation event that the interfaceconnects the origin to height $h$, it consists of a 1D spine that behaves likea random walk, in that it decomposes into a linear (in $h$) number ofasymptotically-stationary weakly-dependent increments that have exponentialtails. As the number $T$ of increments diverges, properties of the interfacesuch as its surface area, volume, and the location of its tip, all obey CLTswith variances linear in $T$. These results generalize to every dimension$d\geq 3$.

Maximum and shape of interfaces in 3D Ising crystals