Consider the random surface given by the interface separating the plus and minus phases in a low-temperature Ising model in dimensions d ≥ 3. Dobrushin (1972) famously showed that in cubes of side-length n the horizontal interface is rigid, typically exhibiting orderone height fluctuations. We study the large deviations of this interface and obtain a shape theorem for its pillar, conditionally on it reaching an atypically large height. We use this to analyze the law of the maximum height of the interface, Mn: we prove that for every β large, Mn/ log n → cβ in probability, and (Mn−E[Mn])n forms a tight sequence. Moreover, even though the centered sequence does not converge, all its subsequential limits satisfy uniform Gumbel tail bounds.
This is joint work with Eyal Lubetzky.