Random simplicial tessellations: high-dimensional probabilistic behaviour of the typical cells
A tessellation in mathbb{R}^d is a locally finite collection of convex polytopes, which cover the space and have disjoin interior. In this talk we consider a few models of random simplicial tessellations, the so-called Delaunay tessellations, whose construction is based on Poisson point process. Among them is the classical Poisson-Delaunay tessellation.
The main object we are interested in is the typical cell of a random tessellation mathcal{D}. Intuitively, one can think of it as a randomly chosen polytope from the collection mathcal{D}, assuming that each polytope has the same chance to be chosen. Considering the volume of the typical cells of our models we derive the explicit formulas for the moments as well as probabilistic representation in term of independent gamma- and beta-distributed random variables. Moreover, we investigate the limiting probabilistic behaviour of the logarithmic volume of typical cell, when dimension d tends to infinity. In particular we establish central limit theorem and large deviation principle.