We study the concentration of the norm of a random vector $Y$ uniformly sampled in the centered zero cell of two types of stationary and isotropic random mosaics in $\mathbb{R}^n$ for large dimensions $n$. For a stationary and isotropic Poisson-Voronoi mosaic, $Y$ has a radial and log-concave distribution, implying that $|Y|/\mathbb{E}(|Y|^2)^{\frac{1}{2}}$ approaches one for large $n$. If we assume that the centroids of the mosaic have intensity scaling like $e^{n \lambda}$, then $|Y|$ is on the order of $\sqrt{n}$ for large $n$. For the Poisson-Voronoi mosaic, we show that $|Y|/\sqrt{n}$ concentrates to $e^{-\lambda}(2\pi e)^{-\frac{1}{2}}$ as $n$ increases, and for a stationary and isotropic Poisson hyperplane mosaic, we show there is a range $(R_{\ell}, R_u)$ such that ${|Y|}/{\sqrt{n}}$ will be within this range with high probability for large $n$. The rates of convergence are also computed in both cases.

中文翻译：

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静止泊松镶嵌体零单元的薄壳浓度

我们研究了在 $\mathbb{R}^n$ 中的两种类型的静止和各向同性随机镶嵌的中心零单元中均匀采样的随机向量 $Y$ 的范数的集中，用于大维度 $n$。对于平稳且各向同性的 Poisson-Voronoi 马赛克，$Y$ 具有径向和对数凹面分布，这意味着 $|Y|/\mathbb{E}(​​|Y|^2)^{\frac{1}{2 }}$ 接近一个大 $n$。如果我们假设马赛克的质心具有类似 $e^{n \lambda}$ 的强度缩放，那么 $|Y|$ 对于大 $n$ 的数量级为 $\sqrt{n}$。对于 Poisson-Voronoi 马赛克，我们证明 $|Y|/\sqrt{n}$ 集中到 $e^{-\lambda}(2\pi e)^{-\frac{1}{2}}$随着 $n$ 的增加，对于一个静止且各向同性的泊松超平面马赛克，我们表明存在一个范围 $(R_{\ell}, R_u)$ 使得 ${|Y|}/{\sqrt{n}}$ 将在此范围内，对于大 $n$ 的可能性很高。在这两种情况下也计算收敛率。