Abstract
We consider gradient fields (ϕx : x∈ℤd) whose law takes the Gibbs–Boltzmann form Z−1exp{−∑〈x, y〉V(ϕy−ϕx)}, 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.
Citation
Marek Biskup. Herbert Spohn. "Scaling limit for a class of gradient fields with nonconvex potentials." Ann. Probab. 39 (1) 224 - 251, January 2011. https://doi.org/10.1214/10-AOP548
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