Scaling limit for a class of gradient fields with nonconvex potentials

Abstract

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 \[V(η):=-\log\int\varrho({d}κ)\exp\biggl[-{1/2}κ\et a^2\biggr],\] 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.

Published in at http://dx.doi.org/10.1214/10-AOP548 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)

References in corpus (5)

• Quenched invariance principles for random walks with random conductances
• Localization and delocalization of random interfaces
• Invariance principle for the random conductance model with unbounded conductances
• Anomalous heat-kernel decay for random walk among bounded random conductances
• Phase coexistence of gradient Gibbs states

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