Abstract
We consider gradient fields on {mathbb {Z}}^d for potentials V that can be expressed as e-V(x)=pe-qx22+(1-p)e-x22.\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} e^{-V(x)}=pe^{-\\frac{qx^2}{2}}+(1-p)e^{-\\frac{x^2}{2}}. \\end{aligned}$$\\end{document}This representation allows us to associate a random conductance type model to the gradient fields with zero tilt. We investigate this random conductance model and prove correlation inequalities, duality properties, and uniqueness of the Gibbs measure in certain regimes. We then show that there is a close relation between Gibbs measures of the random conductance model and gradient Gibbs measures with zero tilt for the potential V. Based on these results we can give a new proof for the non-uniqueness of ergodic zero-tilt gradient Gibbs measures in dimension 2. In contrast to the first proof of this result we rely on planar duality and do not use reflection positivity. Moreover, we show uniqueness of ergodic zero tilt gradient Gibbs measures for almost all values of p and q and, in dimension dge 4, for q close to one or for p(1-p) sufficiently small.
Highlights
Gradient fields are a statistical mechanics model that can be used to model phase separation or, in the case of vector valued fields, solid materials
We show that the law of the κ-marginal can be related to a random conductance model
The key observation is that the κ-marginal of extended gradient Gibbs measures is given by the infinite volume limit of a strongly coupled random conductance model
Summary
Gradient fields are a statistical mechanics model that can be used to model phase separation or, in the case of vector valued fields, solid materials. For potentials of the form V = U + g where U is strictly convex and g ∈ Lq for some q ≥ 1 with sufficiently small norm the problem can be led back to the convex theory by integrating out some degrees of freedom This way many results from the convex case can be proved in particular uniqueness and existence of the Gibbs measure for every tilt and that the scaling limit is Gaussian [11,12,15]. Our techniques would apply to general ρ we restrict our attention to the simplest case where ρ is as in (1.5) and the potential can be written as e−Vp,q (x) For this class of potentials we show uniqueness of the ergodic zero tilt gradient Gibbs measures for almost all p and q and, in dimension d ≥ 4 for p(1 − p) or q − 1 small. Two technical proofs and some results about regularity properties of discrete elliptic equations are delegated to appendices
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