Scaling limit for a class of gradient fields with nonconvex potentials

We consider gradient fields (φ{symbol}_x: x ∈ Z^d) whose law takes the Gibbs-Boltzmann form Z^-1 exp{-∑_(x,y) V (φ{symbol}_y - φ{symbol}_x)}, where the sum runs over nearest neighbors. We assume that the potential V admits the representation V (η):=-log ∫ e (dκ) exp[-1/2 κη²], where e is a positive measure with compact support in (0, ∞). 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.