非凸位势梯度模型协方差的衰减性
Decay of covariance for gradient models with non-convex potential
摘要：
我们考虑格点$Z^d$上的梯度模型。这些模型作为界面的有效模型，也被称为连续伊辛模型。界面的高度由具有能量的随机场建模，该能量是二次相互作用的非凸扰动。我们对该模型的逆温度$β$下倾斜边界条件$u$的Gibbs测度感兴趣。本文对梯度场的协方差进行了精细的分析。我们表明，吉布斯分布的协方差同意与高斯自由场的协方差的条款，以更快的代数率衰减。关键的工具是在[BBS15a]中开发的重正化群方法扩展到可观测量。
英文摘要：
We consider gradient models on the lattice $Z^d$. These models serve as effective models for interfaces and are also known as continuous Ising models. The height of the interface is modelled by a random field with an energy which is a non-convex perturbation of the quadratic interaction. We are interested in the Gibbs measure with tilted boundary condition $u$ at inverse temperature $β$ of this model. In this paper we present a fine analysis of the covariance of the gradient field. We show that the covariances of the Gibbs distribution agree with the covariance of the Gaussian free field up to terms which decay at a faster algebraic rate. The key tool is the extension of the renormalisation group method to observables as developed in [BBS15a].
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