论文标题
晶格$ \ mathbb {z}^d $的高斯扰动线性统计的波动
Fluctuations of linear statistics for Gaussian perturbations of the lattice $\mathbb{Z}^d$
论文作者
论文摘要
我们研究了通过在$ \ Mathbb {z}^d $中添加独立的高斯矢量来获得的$ \ mathbb {r}^d $中的$ w $。我们主要关注的是线性统计量波动的渐近尺寸，以大量限制（定义为 \ [ n（h，r）= \ sum_ {w \ in w} h \ left（\ frac {w} {r} {r} \ right）， \] 其中$ h \ in \ left（l^1 \ cap l^2 \ right）（\ mathbb {r}^d）$是测试功能，$ r \ to \ infty $。我们还将考虑通过添加所有扰动来获得的流程$ W $的固定反向零件，该随机向量均匀分布在$ [0,1]^d $上，并且独立于所有高斯人。当测试功能$ h $要么平滑，要么是凸面设置的指示函数时，我们的曲率不会消失时，我们专注于两个主要示例。
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 \[ N(h,R) = \sum_{w\in W} h\left(\frac{w}{R}\right), \] where $h\in \left(L^1\cap L^2\right)(\mathbb{R}^d)$ is a test function and $R\to \infty$. 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.
