## Point Processes and Random Geometry

Mini-workshop at Ruhr University Bochum, October 7 and 8, 2013

**Short description**

The aim of this mini-workshop is to bring together researchers working on point-process-based models for spatial random geometric structures. In recent years, the analysis of such models has become an increasingly important part of probability theory and their theory has advanced significantly. During the 2-days workshop we will discuss some of these recent advances and hope to stimulate future research.

**Confirmed invited speakers**

- David Dereudre (Lille)
- Hans-Otto Georgii (Munich)
- Sabine Jansen (Bochum)
- Günter Last (Karlsruhe)
- Matthias Schulte (Karlsruhe)
- Hermann Thorisson (Reykjavik)
- Hans Zessin (Bielefeld)

**Program**

The durations of the presentations are expected to be 1h, including discussion.

- Monday, 09.30 - 10.30
**Günter Last**Percolation on stationary tessellations pdf - Monday, 10.30 - 11.00 coffee break
- Monday, 11.00 - 12.00
**Hans Zessin**Some new non-classical interacting point processes - Monday, 12.00 - 14.30 lunch break and time for discussion
- Monday, 14.30 - 15.30
**Hans-Otto Georgii**Existence of point processes via entropy bounds pdf - Monday, 15.30 - 16.00 coffee break
- Monday, 16.00 - 17.00
**David Dereudre**Gibbsian germ-grain models pdf - Monday, 17.00 - 18.00
**Sabine Jansen**Continuum percolation for Gibbsian point processes with attractive interactions pdf - joint culinary and musical evening
- Tuesday, 09.30 - 10.30
**Matthias Schulte**Second order properties and central limit theorems for geometric functionals of Boolean models pdf - Tuesday, 10.30 - 11.00 coffee break
- Tuesday, 11.00 - 12.00
**Hermann Thorisson**What is typical? pdf - Tuesday, 12.00 - 14.00 lunch break and end of the mini-workshop

**Abstracts**

Günter Last:

We consider a stationary face-to-face tessellation of R^d and introduce several percolation models by coloring some of the faces black in a consistent way. Our main model is cell percolation, where cells are declared black with probability p and white otherwise. We first derive some basic geometric mean values for the union Z of black faces. In the special case of cell percolation on a Voronoi tessellation we discuss the asymptotic covariances of intrinsic volumes. In particular we show that these covariances exist in the Poisson Voronoi case. In the two-dimensional case some of the formulas are rather explicit.

This talk is based on joint work with Eva Ochsenreither (Karlsruhe).

Hans Zessin:

By means of the cluster expansion method we show that a recent result of Poghosyan Ueltschi combined with one of Nehring yield a construction of point processes of classical statistical mechanics as well as processes related to the Ginibre Bose and Fermi gas of quantum mechanics.

Ref.: Nehring, Poghosyan, Zessin: On the construction of point processes in statistical mechanics. J. Math. Phys. 54, 063302 (2013)

Hans-Otto Georgii:

It is a standard recipe for proving the existence of Gibbsian or similar point processes to introduce an approximating sequence of such processes within bounded regions that exhaust the full state space.
By construction, it is then often clear that any limiting point of this approximating sequence is a point process of the required type. So one is left with with the problem of proving the existence of such limiting objects. In this talk we will argue that this can often be achieved conveniently by establishing uniform bounds on the relative entropies with respect to a suitable reference process.

David Dereudre:

The germ-grain models are built by unifying random convex sets (the grains) centered at the points (the germs) of a spatial point process. It is used for modelling random surfaces and interfaces, geometrical structures growing from germs, etc. When the grains are independent and identically distributed, and the germs are given by the locations of a Poisson point process, the germ-grain model is known as the Boolean model. Because of the independence properties of the Poisson process, the Boolean model is sometimes caricatural for the applications in Biology or Physics. So Gibbsian modifications, based on a morphological Hamiltonian, are considered in order to be more relevant. The classical Quermass-Hamiltonian is defined by the linear combination of the fundamental Minkowski functionals (area, perimeter and Euler-Poincare characteristic in dimension 2). However other Hamiltonians may be considered. The random cluster interaction, based on the number of connected components or the exclusion interaction, which penalizes the overlapping of convex sets with different types, are such examples. During the talk we will discuss three questions around these Gibbsian germ-grain models: Existence, Phase transition (and percolation of course), and the statistical inference issues.

Sabine Jansen:

We study the problem of continuum percolation in inﬁnite volume Gibbs measures for particles with an attractive pair potential. The focus is on low temperatures (large $\beta$). The main results are bounds on percolation thresholds $\rho_\pm(\beta)$ in terms of the density rather than the chemical potential or activity. In addition, we prove a variational formula for a large deviations rate function for cluster size distributions (J., König, Metzer 2011). This formula establishes a link with the Gibbs variational principle and a form of equivalence of ensembles, and allows us to combine knowledge on ﬁnite volume, canonical Gibbs measures with inﬁnite volume, grand-canonical Gibbs measures.

Matthias Schulte:

Let $Z$ be a Boolean model in $\mathbb{R}^d$ that is based on a stationary Poisson point process of compact and convex particles. For an additive, translation invariant, and locally bounded functional $\psi$ we are interested in the asymptotic behaviour of $\psi(Z\cap W)$, where $W$ is a compact and convex observation window. We compute the asymptotic covariance matrix and derive univariate and multivariate central limit theorems for increasing observation windows. For the special case of intrinsic volumes of an isotropic Boolean model the covariance formulas can be further simplified. The proofs make use of the Fock space representation and the Malliavin-Stein method.

This is joint work with D. Hug and G. Last.

Hermann Thorisson:

Let $\xi$ be a random measure on a locally compact second countable topological group and let $X$ be a random element in a measurable space on which the group acts. In the compact case, we give a natural definition of the concept that the origin is at a typical location for $X$ in the mass of $\xi$, and prove that when this holds the same is true on sets placed uniformly at random around the origin. This result motivates an extension of the concept of typicality to the locally compact case where it coincides with the concept of mass-stationarity. We then describe recent developments in Palm theory where these ideas play a central role.

This is joint work with Günter Last.

**Participants**

Mareen Beermann (Osnabrück), Sascha Bachmann (Osnabrück), David Dereudre (Lille), Christian Deuß (Bochum), Hanna Döring (Bochum), Peter Eichelsbacher (Bochum), Hans-Otto Georgii (Munich), Sabine Jansen (Bochum), Kai Krokowski (Bochum), Günter Last (Karlsruhe), Benjamin Nehring (Bochum), Anselm Reichenbachs (Bochum), Matthias Reitzner (Osnabrück), Matthias Schulte (Karlsruhe), Johammes Stemmeseder (Salzburg), Christoph Thäle (Bochum), Hermann Thorisson (Reykjavik), Hans Zessin (Bielefeld)

**Location**

The workshop will take place in NA 3/64, see here for a map of the campus.

**How to participate?**

If you are interested in participating at the mini-workshop *Point Processes and Random Geometry*, please send an email to Christoph Thäle before September 15, 2013. We can cover travel and accommodation expenses only for invited speakers. There is no registration fee.

**Organizers**

Peter Eichelsbacher, Benjamin Nehring and Christoph Thäle