Repulsion of determinantal point processes and stationary Poisson tessellations in high dimensions

In this dissertation, new results on stochastic geometric models in high dimensional space are presented. We first concentrate on a particular class of repulsive point processes called determinantal point processes (DPPs). We establish a coupling of a DPP and its reduced Palm version showing the repulsive effect of a point of the point process. This is used for discussing the degree of repulsiveness in DPPs, including Ginibre point processes and other specific parametric models for DPPs. We then study this repulsion for stationary DPPs in high dimensional Euclidean space. It is shown that for many families of DPPs, a typical point has no repulsive effect with high probability for large space dimension n. It is also proved that for some DPPs there exists an R* such that the repulsive effect occurs at a distance of [square root] nR* with high probability for large n. This R* is interpreted as the asymptotic reach of repulsion of the DPP. Examples of DPPs exhibiting this behavior are presented and an application to high dimensional Boolean models is given. The second half of this dissertation examines zero cells of stationary Poisson tessellations. First, a stationary stochastic geometric model is proposed for analyzing one-bit data compression. The data is assumed to be an unconstrained stationary set, and each data point is compressed using one bit with respect to each hyperplane in a stationary and isotropic Poisson hyperplane tessellation. Size metrics of the zero cell of the tessellation are studied to determine how the intensity of hyperplanes must scale with dimension to ensure sufficient separation of different data by the hyperplanes or sufficient proximity of the data compressed together. The results have direct implications in compressive sensing and source coding. We then study the concentration of the norm of a random vector Y uniformly sampled in the centered zero cell of a stationary random tessellation in high dimensions. It is shown that for a stationary and isotropic Poisson-Voronoi tessellation, [mathematical equation] approaches one as the dimension approaches infinity. For a stationary and isotropic Poisson hyperplane tessellation, we prove that [mathematical equation] will be within a fixed range (R [subscript ℓ], R [subscript u]) with probability approaching one as dimension n tends to infinity.