Stochastic geometry in non-Euclidean spaces
Description Phase 2
In this project we aim to extend the definition and study of some models for random tessellations from Euclidean setting to non-Euclidean spaces and, in particular, to the spaces of constant sectional curvature. We will concentrate our study on two types of random tessellations, namely random Laguerre tessellation generalizing the classical Poisson-Voronoi model and Poisson hyperplane tessellation, which are well understood in the Euclidean set-up. We will show that analogous constructions can be performed in constant curvature spaces, leading to a variety of new non-Euclidean random tessellation models. We will study the geometry of the typical and weighted typical cells of these tessellations and the probabilistic behavior of various geometrical functionals, such as the cell intensities, volume and f-vector of the typical and zero cell. Additionally, we extend the study of geometric extremes to constant curvature spaces and develop general plug-in results for Poisson point process convergence building on the Malliavin-Stein machinery on the Poisson space. These results will be applied to study the geometric extreme values (e.g. maximal vertex degree, in-radii of the typical cell) of non-Euclidean random tessellation models. Our project contributes to a deeper understanding allowing to distinguish those properties of a random geometric system that are universal from the ones which are sensitive to the underlying geometry, especially to the curvature of the space. In particular, we analyze the dependence on the curvature parameter of the models and their geometric quantities with a particular focus on the analyticity as a function of the curvature.