AG Stochastische Geometrie (Wintersemester 2018/19)
- Dozent*in: Prof. Dr. Günter Last, Prof. Dr. Daniel Hug
- Veranstaltungen: Seminar (0127500)
- Semesterwochenstunden: 2
|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: firstname.lastname@example.org||Seminarleitung||Prof. Dr. Daniel Hug|
|Sprechstunde: Nach Vereinbarung.|
|Zimmer 2.051 Kollegiengebäude Mathematik (20.30)|
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.
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 between the d-dimensional lattice and a stationary Poisson process (or a determinantal point process) with intensity . The matched points from form a stationary and ergodic (under lattice shifts) point process with intensity $1$ with many interesting properties. For close to 1 the point process very much resembles a Poisson process, while for it approaches the lattice. Moreover, 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 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.
10.15 Uhr Problem Session
9.45 Uhr Hermann Thorisson (University of Iceland, Reykjavik)
Shift-coupling, invariant sets and Cesaro asymptotics
Abstract: Let and be two random elements (e.g. processes or measures) acted on by a group . Let denote the shift of the origin to a location . Say that and admit shift-coupling if there exists (possibly after extension) a random location in such that has the same distribution as ,
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.
10.15 Uhr Moritz Otto
Poisson process approximation of thinnings of stationary point processes
9.45 Uhr Günter Last
Exponentielle Dekorrelation subkritischer Gibbsscher Partikelprozesse
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.
9.45 Uhr Norbert Henze
Ein Poissonscher Grenzwertsatz für die Anzahl von Nächst-Nachbar-Kugeln mit großem Wahrscheinlichkeitsinhalt
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.