Matthias Erbar (University of Bielefeld)
Optimal transport, stationary point processes, and log gases

Abstract: In this talk I would like give an overview of the topics and aims of the project "Optimal transport for stationary point processes" within the SPP "Random geometric systems". I will give a brief introduction of the classical theory of optimal transport for finite measures and will outline how we envision to develop a counterpart to this theory in the setting of stationary point processes. I will also present first results in this direction from joint work with M. Huesmann and T. Lebl\'e which apply ideas from optimal transport in the study of log gases. The one-dimensional log gas in finite volume is a system of particles interacting via a repulsive logarithmic potential and confined by some external field. When the number of particles goes to infinity, their macroscopic empirical distribution approaches a deterministic limit shape. When zooming in one sees microscopic fluctuations around this limit which are described in the limit by a stationary point process, the Sine_\beta process constructed by Valko and Virag. We show that this process can be characterized as the unique minimizer of a renormalized free energy by leveraging strict convexity properties of this functional along interpolations built via optimal transport.

Zakhar Kabluchko (University of Münster)
Random beta tessellations

We shall define and study several families of random tessellations of the Euclidean space, the sphere and the hyperbolic space that generalize the classical Poisson-Voronoi and Poisson-Delaunay tessellations. The talk is based on a joint work with Anna Gusakova and Christoph Thäle.