Scaling limit for a class of gradient fields with nonconvex potentials

We consider gradient fields $(\phi_x:x\in \mathbb{Z}^d)$ whose law takes the Gibbs--Boltzmann form $Z^{-1}\exp\{-\sum_{< x,y>}V(\phi_y-\phi_x)\}$, where the sum runs over nearest neighbors. We assume that the potential $V$ admits the representation $$V(\eta):=-\log\int\varrho({d}\kappa)\exp\biggl[-{1/2}\kappa\et a^2\biggr],$$ where $\varrho$ is a positive measure with compact support in $(0,\infty)$. 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.