Scaling limit for a class of gradient fields with nonconvex potentials

We consider gradient fields $({\varphi }_x:x\in {\mathbb{Z}}^d)$ whose law takes the Gibbs–Boltzmann form $Z^{-1}\exp \{-{\sum }_{<x,y>}V({\varphi }_y-{\varphi }_x)\}$, where the sum runs over nearest neighbors. We assume that the potential $V$ admits the representation $$V(\eta ):=-\log \int \varrho (d\kappa )\exp [-1\mathrm{/}2\kappa \text{et}{a}^2],$$ where $\varrho$ is a positive measure with compact support in $(0,\mathrm{\infty })$. Hence, the potential $V$ is symmetric, but nonconvex in general. While for strictly convex $V$’s, the translation-invariant, ergodic gradient Gibbs measures are completely characterized by their tilt, a nonconvex potential as above may lead to several ergodic gradient Gibbs measures with zero tilt. Still, every ergodic, zero-tilt gradient Gibbs measure for the potential $V$ above scales to a Gaussian free field.