Thermodynamic and Scaling Limits of the non-Gaussian Membrane Model

We characterize the behavior of a random discrete interface $\varphi$ on $[-L,L{]}^d\cap {\mathbb{Z}}^d$ with energy $\sum V(\mathrm{\Delta }\varphi (x))$ as $L \to \infty$,where$\Delta$isthediscreteLaplacianand$V$ is a uniformly convex, symmetric, and smooth potential. The interface $\varphi$ is called the non-Gaussian membrane model. By analyzing the Helffer-Sj"ostrand representation associated to $\mathrm{\Delta }\varphi$, 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.