  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
$d\geq 3$.
