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

We study the point process $W$ in $\mathbb{R}^d$ obtained by adding an independent Gaussian vector to each $\mathbb{Z}^d$. Our main concern is asymptotic size of fluctuations linear statistics large volume limit, defined as \[ N(h,R) = \sum_{w\in W} h\left(\frac{w}{R}\right), \] where $h\in \left(L^1\cap L^2\right)(\mathbb{R}^d)$ a test function and $R\to \infty$. will also consider stationary counter-part $W$, all perturbations random which uniformly distributed on $[0,1]^d$ Gaussians. focus two examples interest, when $h$ either smooth or indicator convex set with boundary whose curvature does not vanish.