We study the point process W in \({\mathbb {R}}^d\) obtained by adding an independent Gaussian vector to each point in \({\mathbb {Z}}^d\). Our main concern is the asymptotic size of fluctuations of the linear statistics in the large volume limit, defined as

where \(h\in \left( L^1\cap L^2\right) ({\mathbb {R}}^d)\) is a test function and \(R\rightarrow \infty \). We will also consider the stationary counter-part of the process W, obtained by adding to all perturbations a random vector which is uniformly distributed on \([0,1]^d\) and is independent of all the Gaussians. We focus on two main examples of interest, when the test function h is either smooth or is an indicator function of a convex set with a smooth boundary whose curvature does not vanish.

我们研究\（{\ mathbb {R}} ^ d \中的点过程W，该过程是通过向\（{\ mathbb {Z}} ^ d \）中的每个点添加独立的高斯向量而获得的。我们主要关心的是在大容量限制下线性统计量波动的渐近大小，定义为

其中\（h \ in \ left（L ^ 1 \ cap L ^ 2 \ right）（{\ mathbb {R}} ^ d）\）是一个测试函数，\（R \ rightarrow \ infty \）。我们还将考虑过程W的平稳对立部分，该过程是通过将一个随机矢量加到所有扰动上而获得的，该随机矢量均匀分布在\（[0,1] ^ d \）上，并且独立于所有高斯分布。当测试函数h是平滑的或者是具有光滑边界且曲率不消失的凸集的指示函数时，我们关注两个主要的感兴趣示例。