论文标题
具有非凸电势的梯度模型的协方差衰减
Decay of covariance for gradient models with non-convex potential
论文作者
论文摘要
我们考虑晶格$ z^d $上的梯度模型。这些模型是接口的有效模型，也称为连续的ISING模型。界面的高度是由具有二次相互作用的非凸透扰动的能量的随机场建模的。我们对此模型的反向温度$β$在倾斜边界条件$ u $的吉布斯度量感兴趣。在本文中，我们对梯度场的协方差进行了很好的分析。我们表明，Gibbs分布的协方差与高斯自由场的协方差达成一致，直到以更快的代数速率衰减的术语。关键工具是将重量法化组方法扩展到[BBS15A]中开发的可观察物。
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].
