Tightness and tails of the maximum in 3D Ising interfaces

Abstract

Consider the 3D Ising model on a box of side length n with minus boundary conditions above the $\mathit{x}\mathit{y}$-plane and plus boundary conditions below it. At low temperatures, Dobrushin (1972) showed that the interface separating the predominantly plus and predominantly minus regions is localized: its height above a fixed point has exponential tails. Recently, the authors proved a law of large numbers for the maximum height ${\mathit{M}}_{\mathit{n}}$ of this interface: for every β large, ${\mathit{M}}_{\mathit{n}}/log\mathit{n}\to {\mathit{c}}_{\mathit{\beta }}$ in probability as $\mathit{n}\to \infty$.

Here, we show that the laws of the centered maxima ${({\mathit{M}}_{\mathit{n}}-\mathbb{E}[{\mathit{M}}_{\mathit{n}}])}_{\mathit{n}\ge 1}$ are uniformly tight. Moreover, even though this sequence does not converge, we prove that it has uniform upper and lower Gumbel tails (exponential right tails and doubly exponential left tails). Key to the proof is a sharp (up to $\mathit{O}(1)$ precision) understanding of the surface large deviations. This includes, in particular, the shape of a pillar that reaches near-maximum height, even at its base, where the interactions with neighboring pillars are dominant.