Local and global geometry of the 2D Ising interface in critical prewetting

Abstract

Consider the Ising model at low temperatures and positive external field λ on an $\mathit{N}\times \mathit{N}$ box with Dobrushin boundary conditions that are plus on the north, east and west boundaries and minus on the south boundary. If $\mathit{\lambda }=0$, the interface separating the plus and minus phases is diffusive, having $\mathit{O}(\sqrt{\mathit{N}})$ height fluctuations, and the model is fully wetted. Under an order one field, the interface fluctuations are $\mathit{O}(1)$, and the interface is only partially wetted, being pinned to its southern boundary. We study the critical prewetting regime of ${\mathit{\lambda }}_{\mathit{N}}\downarrow 0$, where the height fluctuations are expected to scale as ${\mathit{\lambda }}^{-1/3}$ and the rescaled interface is predicted to converge to the Ferrari–Spohn diffusion. Velenik (Probab. Theory Related Fields 129 (2004) 83–112) identified the order of the area under the interface up to logarithmic corrections. Since then, more refined features of such interfaces have only been identified in simpler models of random walks under area tilts.

In this paper we resolve several conjectures of Velenik regarding the refined features of the Ising interface in the critical prewetting regime. Our main result is a sharp bound on the one-point height fluctuation, proving ${\mathit{e}}^{-\mathrm{\Theta }({\mathit{x}}^{3/2})}$ upper tails reminiscent of the Tracy–Widom distribution, capturing a tradeoff between the locally Brownian oscillations and the global field effect. We further prove a concentration estimate for the number of points above which the interface attains a large height. These are used to deduce various geometric properties of the interface, including the order and tails of the area it confines and the polylogarithmic prefactor governing its maximum height fluctuation. Our arguments combine classical inputs from the random-line representation of the Ising interface with novel local resampling and coupling schemes.