Thin-shell concentration for zero cells of stationary Poisson mosaics

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 Rn for large dimension n. For a stationary and isotropic Poisson-Voronoi mosaic, Y has a radial and log-concave distribution, implying that |Y|/E(|Y|2)12 approaches one for large n. Assuming the cell intensity of the random mosaic scales like enρn, where limn→∞⁡ρn=ρ, |Y| is on the order of n for large n. For the Poisson-Voronoi mosaic, we show that |Y|/n concentrates to e−ρ(2πe)−12 as n increases, and for a stationary and isotropic Poisson hyperplane mosaic, we show there is a range (Rℓ,Ru) such that |Y|/n will be within this range with high probability for large n. The rates of convergence are also computed in both cases.