Thermodynamic and scaling limits of the non-Gaussian membrane model

Abstract

We characterize the behavior of a random discrete interface ϕ on ${[-\mathit{L},\mathit{L}]}^{\mathit{d}}\cap {\mathbb{Z}}^{\mathit{d}}$ with energy $\sum \mathit{V}(\mathrm{\Delta }\mathit{\varphi }(\mathit{x}))$ as $\mathit{L}\to \infty$, where Δ is the discrete Laplacian and V is a uniformly convex, symmetric, and smooth potential. The interface ϕ is called the non-Gaussian membrane model. By analyzing the Helffer–Sjöstrand representation, associated to $\mathrm{\Delta }\mathit{\varphi }$, we provide a unified approach to continuous scaling limits of the rescaled and interpolated interface in dimensions $\mathit{d}=2,3$, Gaussian approximation in negative regularity spaces for all $\mathit{d}\ge 2$, and the infinite volume limit in $\mathit{d}\ge 5$.