Phase transitions for a class of gradient fields

TL;DR

This paper studies phase transitions in gradient fields with a specific potential, linking them to a random conductance model, and establishes conditions for uniqueness and non-uniqueness of Gibbs measures.

Contribution

It introduces a novel approach connecting gradient fields with a random conductance model, providing new proofs and conditions for measure uniqueness.

Findings

Proves correlation inequalities and duality properties.

Establishes conditions for uniqueness and non-uniqueness of Gibbs measures.

Provides a new proof for non-uniqueness without reflection positivity.

Abstract

We consider gradient fields on $\mathbb{Z}^d$ for potentials $V$ that can be expressed as $$e^{-V(x)}=pe^{-\frac{qx^2}{2}}+(1-p)e^{-\frac{x^2}{2}}.$$ 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. Moreover, we 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 gradient Gibbs measures without using reflection positivity. We also show uniqueness of ergodic zero tilt gradient Gibbs measures for almost all values of $p$ and $q$ and, in dimension $d\geq 4$, for $q$ close to one or for $p(1-p)$ sufficiently small.