FB 6 Mathematik/Informatik/Physik

Institut für Mathematik

# Hauptinhalt

## SS 2016

### 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.