We consider gradient fields on \({\mathbb {Z}}^d\) for potentials V that can be expressed as

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 \(d\ge 4\), for q close to one or for \(p(1-p)\) sufficiently small.

我们考虑\（{\ mathbb {Z}} ^ d \）上的势场V的梯度场，它可以表示为

这种表示使我们能够将随机电导类型模型与零倾斜的梯度场相关联。我们研究了这种随机电导模型，并证明了在某些情况下相关不等式，对偶性质和吉布斯测度的唯一性。然后我们表明，对于电势V，随机电导模型的吉布斯测度与具有零倾斜度的梯度吉布斯测度之间存在密切关系。基于这些结果，我们可以提供关于二维二维遍历零倾斜梯度Gibbs测度的非唯一性的新证明。与该结果的第一个证明相反，我们依赖于平面对偶性，并且不使用反射正性。此外，我们展示了遍历零倾斜梯度的吉布斯度量值对于p和q几乎所有值的唯一性在维度\（d \ ge 4 \）中，对于q接近于1或对于\（p（1-p）\）足够小。