Thin-shell concentration for zero cells of stationary Poisson mosaics

2018/09/11 by O'Reilly, Eliza
#FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.1809.04134

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 ℝⁿ for large dimensions 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(2πe)^-(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.

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