Thermodynamic and Scaling Limits of the non-Gaussian Membrane Model

Abstract

We characterize the behavior of a random discrete interface $\varphi$ on $[-L,L{]}^d\cap {\mathbb{Z}}^d$ with energy $\sum V(\Delta \varphi (x))$ as $L\to \infty$, where $\Delta$ is the discrete Laplacian and $V$ is a uniformly convex, symmetric, and smooth potential. The interface $\varphi$ is called the non-Gaussian membrane model. By analyzing the Helffer-Sjöstrand representation associated to $\Delta \varphi$, we provide a unified approach to continuous scaling limits of the rescaled and interpolated interface in dimensions $d=2,3$, Gaussian approximation in negative regularity spaces for all $d\ge 2$, and the infinite volume limit in $d\ge 5$. Our results generalize some of those of arXiv:1801.05663.

39 pages

References in corpus (3)

• Maximum and entropic repulsion for a Gaussian membrane model in the critical dimension
• Scaling limits of $(1+1)$-dimensional pinning models with Laplacian interaction
• Quantitative hydrodynamic limits of the Langevin dynamics for gradient interface models