AG Stochastische Geometrie (Wintersemester 2015/16)
- 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)|
9.45 Uhr: Matthias Schulte:
Limit theorems for random geometric graphs
A random geometric graph is constructed by connecting two points of a Poisson process in a compact convex set whenever their distance does not exceed a prescribed distance. The aim of this talk is to investigate the asymptotic behaviour of the total edge length or, more general, sums of powers of the edge lengths of this random graph as the intensity of the underlying Poisson process is increased and the threshold for connecting points is adjusted. Depending on the interplay of these two parameters one obtains limit theorems where the limiting distribution can be Gaussian, compound Poisson or stable. This talk is based on joint work with Laurent Decreusefond, Matthias Reitzner and Christoph Thäle.
9.45 Uhr: Daniel Hug:
Über ein(ig)e Ungleichung(en) für konvexe Körper
9.45 Uhr: Mir-Omid Haji-Mirsadeghi (Sharif University of Technology)
Mass transports between stationary random measures
The problem of finding a transport kernel between (samples of) stationary random measures was initiated by Hermann Thorisson with a necessary and sufficient condition for two random measures to be obtained by a random translation from each other, which is called a shift coupling. A particular example is when the second random measure is the Palm version of the first one. In the case of point processes a number of algorithms are presented to construct a shift coupling, which we will review. In particular, we are interested in the algorithm presented by Hoffman, C., Holroyd, A. E., & Peres, Y. (2006) which is called stable marriage of Poisson and Lebesgue. This algorithm extends the Gale-Shapley stable marriage algorithm and the notion of stability to a continuum setting. We then generalize it for arbitrary random measures. For this, we limit ourselves to 'mild' transport kernels, which is a special case of capacity constrained transport kernels. We give a definition of stability of mild transport kernels and introduce a construction algorithm inspired by the Gale-Shapley stable marriage algorithm.
9:45 Uhr: Dusan Pokorny (Charles University Prague)
Kinematic formulas for sets defined by differences of convex functions
A real function is called delta-convex if it can be expressed as a difference of two convex functions. A subset of a Euclidean space is called WDC if it is a sublevel set of a delta-convex function at a weakly regular value. The sets of positive reach, for instance, form a strict subclass of WDC sets. In the talk, the existence and properties of Federer's curvature measures, namely the validity of kinematic formulas, for the class of WDC sets will be discussed. The presented results are a joint work with Jan Rataj and Joseph Fu.
9:45 Uhr: Julia Hörrmann (Ruhr-Universität Bochum)
A random cell splitting scheme on the sphere
A random recursive cell splitting scheme of the 2-dimensional unit sphere is considered, which is the spherical analogue of the STIT tessellation process from Euclidean stochastic geometry. First-order moments are computed for a large array of combinatorial and metric parameters of the induced splitting tessellations by means of martingale methods combined with tools from spherical integral geometry. The findings are compared with those in the Euclidean case, making thereby transparent the influence of the curvature of the underlying space. Moreover, the capacity functional is computed and the point process that arises from the intersection of a splitting tessellation with a fixed great circle is characterized.
This talk is basewd on joint work with Christian Deuss and Christoph Thäle.
9:45 Uhr: Michael Schrempp
Zur Asymptotik des maximalen Abstandes zufälliger Punkte in d-dimensionalen Ellipsoiden
9:45 Uhr: Sebastian Ziesche
Perkolation auf zufälligen Mosaiken
9:45 Uhr: Tobias Müller (Utrecht University)
The critical probability for confetti percolation equals 1/2
In the confetti percolation model, or two-coloured dead leaves model, radius one disks arrive on the plane according to a space-time Poisson process. Each disk is colored black with probability and white with probability . This defines a two-colouring of the plane, where the color of a point of the plane is determined by the last disk to arrive that covers it. In this work we show that the critical probability for confetti percolation equals . That is, if then a.s. there is an unbounded curve in the plane all of whose points are black; while if then a.s. all connected components of the set of black points are bounded. This answers a question of Benjamini and Schramm. The proof makes use of earlier work by Hirsch and an asymmetric version of a "sharp threshold" result of Bourgain.
9:45 Uhr: Kai Krokowski (Ruhr-Universität Bochum)