Scaling limit for a class of gradient fields with nonconvex potentials

Abstract

We consider gradient fields (ϕ_x : x∈ℤ^d) whose law takes the Gibbs–Boltzmann form Z⁻¹exp{−∑_〈x, y〉V(ϕ_y−ϕ_x)}, where the sum runs over nearest neighbors. We assume that the potential V admits the representation

V(η):=−log∫ϱ(d κ)exp[−½κη²],

where ϱ 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.