P29
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.