Fluctuations of Linear Statistics for Gaussian Perturbations of the Lattice ${\mathbb {Z}}^d$

Abstract

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

$$\begin{aligned} N(h,R) = \sum _{w\in W} h\left( \frac{w}{R}\right) , \end{aligned}$$

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.