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January 2011 Scaling limit for a class of gradient fields with nonconvex potentials
Marek Biskup, Herbert Spohn
Ann. Probab. 39(1): 224-251 (January 2011). DOI: 10.1214/10-AOP548

Abstract

We consider gradient fields (ϕx : x∈ℤd) whose law takes the Gibbs–Boltzmann form Z−1exp{−∑x, yV(ϕyϕx)}, where the sum runs over nearest neighbors. We assume that the potential V admits the representation

V(η):=−log∫ϱ(dκ)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

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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

Information

Published: January 2011
First available in Project Euclid: 3 December 2010

zbMATH: 1222.60076
MathSciNet: MR2778801
Digital Object Identifier: 10.1214/10-AOP548

Subjects:
Primary: 60F05 , 60K35 , 82B41

Keywords: Gaussian free field , Gradient fields , Scaling limit

Rights: Copyright © 2011 Institute of Mathematical Statistics

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Vol.39 • No. 1 • January 2011
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