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AG Stochastische Geometrie (Wintersemester 2018/19)

Seminar: Freitag 9:45-11:15 SR 2.58
Seminarleitung Prof. Dr. Günter Last
Sprechstunde: nach Vereinbarung.
Zimmer 2.001, Sekretariat 2.056 Kollegiengebäude Mathematik (20.30)
Email: guenter.last@kit.edu
Seminarleitung Prof. Dr. Daniel Hug
Sprechstunde: Nach Vereinbarung.
Zimmer 2.051 Kollegiengebäude Mathematik (20.30)
Email: daniel.hug@kit.edu

Studierende und Gäste sind jederzeit herzlich willkommen. Wenn nicht explizit anders unten angegeben, finden alle Vorträge im Seminarraum 2.58 im Mathebau (Gebäude 20.30) statt.

Freitag, 19.10.2018

10.15 Uhr Daniel Hug

Anisotropic splitting tessellations in spherical space

Dienstag, 30.10.2018 (im Rahmen der AG Stochastik)

15.45 Uhr Günter Last

Hyperuniform stable matchings of point processes

Abstract: Stable matchings were introduced in a seminal paper by Gale and Shapley (1962) and play an important role in economics. Following closely Holroyd, Pemantle, Peres and Schramm (2009), we first discuss a few basic properties of stable matchings between two discrete point sets (resp. point processes) in $\R^d$, where the points prefer to be close to each other. In the second part of the talk we consider a stable matching \tau between the d-dimensional lattice and a stationary Poisson process (or a determinantal point process) with intensity \alpha>1. The matched points from \Psi form a stationary and ergodic (under lattice shifts) point process \Psi^\tau with intensity $1$ with many interesting properties. For \alpha close to 1 the point process \Psi very much resembles a Poisson process, while for \alpha\to\infty it approaches the lattice. Moreover, \Psi^\tau is hyperuniform, that is, the variance of the number of points in an increasing window grows much slower than the volume. Furthermore, the point process \Psi^\tau is number rigid, that is the number of points in a bounded set is almost surely determined by the points in the complement of that set. These properties are in sharp contrast to a Poisson process. The talk is based on joint work with M. Klatt and D. Yogeshwaran.

Freitag, 6.11.2018

10.15 Uhr Problem Session

Freitag, 23.11.2018

9.45 Uhr Hermann Thorisson (University of Iceland, Reykjavik)

Shift-coupling, invariant sets and Cesaro asymptotics

Abstract: Let X and Y be two random elements (e.g. processes or measures) acted on by a group G. Let \theta_t denote the shift of the origin to a location t \in G . Say that X and Y admit shift-coupling if there exists (possibly after extension) a random location T in G such that \theta_T X has the same distribution as Y,

$ \theta_T X \overset{D}{=} Y\qquad \text{ (shift-coupling) }.$

Shift-coupling has turned out in resent years to be a useful tool in Palm theory. In this talk we consider some basic shift-coupling theory which involves invariant sets and Cesaro total variation asymptotics. The proofs rely mainly on common components of measures and on transfer of random elements.

Freitag, 30.11.2018

10.15 Uhr Moritz Otto

Poisson process approximation of thinnings of stationary point processes

Freitag, 18.1.2019

9.45 Uhr Günter Last

Exponentielle Dekorrelation subkritischer Gibbsscher Partikelprozesse

Freitag, 25.1.2019

9.45 Uhr Sabine Jansen (LMU München)

Cluster expansions for Gibbs point processes

Abstract: Gibbs point processes form an important class of models in statistical mechanics, stochastic geometry and spatial statistics. A notorious difficulty is that many quantities cannot be computed explicitly; for example, the intensity measure of a Gibbs point process is a highly non-trivial function of the intensity of the underlying Poisson point process. As a partial way out, physicists and mathematical physicists have long worked with perturbation series, called cluster expansions.

The talk presents some recent results on cluster expansions for pairwise repulsive interactions and explains connections with generating functions of trees, branching processes, Boolean percolation, and diagrammatic expansions of second-order U-statistics.

Freitag, 1.2.2019

9.45 Uhr Norbert Henze

Ein Poissonscher Grenzwertsatz für die Anzahl von Nächst-Nachbar-Kugeln mit großem Wahrscheinlichkeitsinhalt

Freitag, 8.2.2019

9.45 Uhr Ecaterina Sava-Huss (TU Graz)

Growth models and the fractals they produce

Abstract: In this talk, I will focus on several cluster growth models based on particles moving around according to some rule (that can be either random or deterministic) and aggregating. Describing the limit shape of the cluster these particles produce is one of the main questions one would like to answer. I will consider the following models: internal DLA, the rotor model and the divisible sandpile model and I will present several results on the limit shape. In particular, I will present a limit shape universality result on the Sierpinski gasket graph, and conclude with some possible research directions on other fractal graphs. The results are based on collaborations with J. Chen, W. Huss, and A. Teplyaev.