Event Type:

Department Colloquium

Date/Time:

Monday, September 28, 2015 - 16:00 to 17:00

Location:

KIdder 350

Guest Speaker:

Institution:

Karlsruhe Institute of Technology

Abstract:

A stationary Poisson process (in Euclidean space) is a fundamental random process that scatters isolated points in a purely random manner. Its probabilistic properties are completely characterized by the mean number of points lying in a set of unit volume. In this lecture we shall consider a stationary Poisson point process whose points are independently marked with random convex grains. The Boolean model is then the union of all these grains. This random closed set is a fundamental model of stochastic geometry and continuum percolation. When restricted to a convex and compact observation window the Boolean model is a finite union of convex sets. Therefore it makes sense to talk about its intrinsic volumes as volume, surface content, and Euler characteristic. In this lecture we shall first discuss classical formulae for the densities (normalized expectations) of these intrinsic volumes. Then we proceed with studying asymptotic covariances for growing observation window. These covariances can be expressed in terms of curvature measures associated with a typical grain. In the two-dimensional isotropic case the formulae become surprisingly explicit. We also present a multivariate central limit theorem including Berry-Esseen bounds. If time permits we will discuss the closely related Gilbert graph.

The talk is based on joint work with Daniel Hug and Matthias Schulte.

Host: