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SS 2016

10.05.2016 um 12:30 Uhr in Raum 69/E15

Jens Grygierek (Universität Osnabrück)

Poisson meets Poisson: A second order Poincaré inequality

14.06.2016 um 12:00 Uhr in Raum 69/E15

Gilles Bonnet (Universität Osnabrück)

Small cells

21.06.2016 um 12:00 Uhr in Raum 69/E15

Nicolas Chenavier (Université du Littoral Côte d'Opale)

Stretch factor of long paths in a planar Poisson-Delaunay triangulation

Let $X_n$ be a homogeneous Poisson point process of intensity $n$ in the plane and let $p$ and $q$ be two (deterministic) points in the plane. The point process $X:=X_n\cup\{p,q\}$ generates the so-called Delaunay triangulation DT(X) associated with $X$. This graph is a triangulation of the plane such that there is no points of $X$ in the interiors of the circumdisks of the triangles in $DT(X)$. In this talk, we investigate the length of the smallest path in the Delaunay triangulation starting from $p$ and going to $q$ as the intensity $n$ of the Poisson point process $X_n$ goes to infinity.

29.06.2016 um 12:00 Uhr in Raum 69/118

Raphael Lachièze-Rey (Université Paris Descartes)

Covariograms, Euler characteristic, and random excursions

It is known that for a measurable set F of the plane, its perimeter can be expressed as the limit as t goes to 0 of the renormalised covariogram |(F+tu)\F|/t, after averaging over the set of unit vectors u. When F is a random set, it allows one to express the mean perimeter of F uniquely in function of its second-order marginals. It turns out that there is a similar formula for the Euler characteristic in dimension 2, which then allows one to express the mean Euler characteristic of a sufficiently regular set F in function of its third-order marginals. We applied this formula to Gaussian fields and shot noise processes excursions, and improved the conclusions or hypotheses of some related results.