SPP2265-Meeting on
Stochastic Geometry
and Point Processes 

The Poisson-Voronoi tessellation is a dissection of d-dimensional Euclidean space, obtained by assigning to each point z of a constant intensity Poisson point process its Voronoi cell C(z), consisting of all points in d-space that are at least as close to z as to any other point of the Poisson point process. The typical cell of the Poisson-Voronoi tessellation is the cell of the origin in the Voronoi tessellation of the Poisson point process with the origin added in. Its significance is that the behaviour of the typical cell accurately describes the "average behaviour of the cells of the Poisson-Voronoi tessellation" (in a sense that can be made precise).

Here we study the asymptotics of some parameters related to the shape of the cell, as the dimension grows. Results by earlier authors on this regime all indicate that the typical cell is in some sense "sphere like". In particular, when rescaled by the appropriate quantity (the d-th root of the intensity times the volume of the d-ball), the inradius, outradius, diameter and mean width all converge to explicit constants -- where the convergence is in probability, as the dimension d tends to infinity. In particular our results imply that the Hausdorff distance of the typical cell to any ball is in fact large. We can also present a partial result on the width, given a lower bound that is a multiplicative factor larger than the trivial lower bound of twice the inradius, and an upper bound that is a multiplicative factor smaller than the trivial upper bound of the diameter.

Joint work with Zakhar Kabluchko.

We study random simplices defined as convex hulls $[X_1, \ldots, X_k]$ of $k \leq d+1$ independent, but not necessarily identically distributed random vectors $X_1, \ldots , X_k$ in $\R^d$ as well as random parallelotopes defined as the Minkowski sums of the random segments $[0, X_1], \ldots , [0, X_k]$. For $\beta_i > -1$, each vector $X_i$ is beta distributed, that is, it has density

\begin{align*}
  f_{d,\beta_i} (x) = c_{d,\beta_i} \left( 1-\|x\|^2 \right)^{\beta_i}, \qquad \|x\| \leq 1.
\end{align*}

where $c_{d,\beta_i}$ is a suitable normalizing cosntant. In particular, we prove an explicit formula for the expected volume of such a beta simplex by using a transparent and geometric approach which connects simplices and parallelotopes. This explicit formula opens up many interesting applications that we will touch upon. The talk is based on a joint work with Zakhar Kabluchko and Christoph Thäle.

I will discuss (compound) Poisson process approximation for stabilizing statistics of a stationary strongly mixing point process. The main results are formulated in a Wasserstein distance and are based on a general bound on the total variation distance of a stationary point process and its Palm measure. The new findings are applied to minimal angles in the stationary Poisson-Delaunay triangulation. In this example, the asymptotic cluster size distribution is explicit and compound Poisson process approximation is established with an explicit convergence rate. The talk is based on joint work with Nicolas Chenavier. 

In the first part, we develop a theory of optimal transport for stationary random measures in $\mathbb{R}^d$ with a particular focus on stationary point processes. This provides us with a notion of geodesic distance between distributions of stationary random measures and induces a natural displacement interpolation be- tween them. In the setting of stationary point processes we leverage this transport distance to give a geometric interpretation for the evolution of infinite particle systems with stationary distribution. Namely, we characterise the evolution of infinitely many Brownian motions as the gradient flow of the specific relative entropy w.r.t. the Poisson process. Further, we establish displacement convexity of the specific relative entropy along optimal interpolations of point processes.

In the second part, we focus on the one-dimensional case and introduce a metric based on the gap distributions. We prove strict displacement convexity of the free energy with a quantitative formula for the gain. Consequences are uniqueness of minimizers and transport-inequalities. In the case of no interaction we present an EVI and a HWI inequality.