## Guenter Last (30/10/15)

Math Colloquium

### Speaker: Guenter Last

Title: Second order properties of the Boolean model and the Gilbert graph

Location: V-157, VRII.

Time: Friday, October 30, at 15:00-16:00.

### Abstract:

The Boolean model is a fundamental model of stochastic geometry and continuum percolation. It is a random subset of Euclidean space that arises as the union of random convex grains, independently centered around the points of a stationary Poisson process. The restriction of the Boolean model to a convex and compact observation 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, derived with the so-called Stein-Malliavin method. If time permits we will also discuss some cluster properties of the Gilbert graph which is a close relative of the Boolean model.

Large parts of the talk are based on joint work with Daniel Hug and

Matthias Schulte.