  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.
