Dobrushin (1972) showed that the interface of a 3D Ising model with plus boundary conditions above the xy-plane and minus below is rigid (has $O(1)$ fluctuations) at low enough temperatures. We study the large deviations of this interface in a cube of side-length $n$, and obtain a law of large numbers for its maximum height $M_n$: for every inverse-temperature $\beta$ large enough, $M_n/\log n$ converges in probability to $h_\beta$ as $n\to\infty$.

We further show that on the large deviation event that the interface connects the origin to a height $H$, it consists of a 1D spine that behaves like a random walk in that it decomposes into asymptotically-stationary, weakly-dependent increments each of which has an exponential tail. These results generalize to every dimension $d\geq 3$.

Joint work with Eyal Lubetzky.