Thin-shell concentration for zero cells of stationary Poisson mosaics

Abstract

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 $n$. Assuming the cell intensity of the random mosaic scales like $e^{n{\rho }_n}$, where ${\lim }_{n\to \infty }{\rho }_n=\rho$, $|Y|$ is on the order of $\sqrt{n}$ for large $n$. For the Poisson-Voronoi mosaic, we show that $|Y|/\sqrt{n}$ concentrates to $e^{-\rho }(2\pi e{)}^{-\frac{1}{2}}$ 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.

20 pages