Maximum height of 3D Ising interfaces

Dobrushin (1972) showed that, at low enough temperatures, the interface of the 3D Ising model - the random surface separating the plus and minus phases above and below the $xy$-plane - is localized: it has $O(1)$ height fluctuations above a fixed point, and its maximum height $M_n$ on a box of side length $n$ is $O_P(\log n)$. We study this interface and derive a shape theorem for its "pillars" conditionally on reaching an atypically large height. We use this to analyze the maximum height $M_n$ of the interface, and prove that at low temperature $M_n/\log n$ converges to $c\beta$ in probability. Furthermore, the sequence $(M_n - E[M_n])_{n\geq 1}$ is tight, and even though this sequence does not converge, its subsequential limits satisfy uniform Gumbel tails bounds.
Joint work with Reza Gheissari.