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 R n 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 ) 1 2 approaches one for large n. Assuming the cell intensity of the random mosaic scales like e n ρ n , where lim n → ∞ ⁡ ρ n = 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 ( 2 π e ) − 1 2 as n increases, and for a stationary and isotropic Poisson hyperplane mosaic, we show there is a range ( R l , R u ) 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.