Phase transitions for a class of gradient fields

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.