MARK KAC SEMINAR

Poisson-Voronoi tessellations are one of the central models studied in stochastic geometry with a considerable history, that are widely used to model physical, biological and technological systems. They are random dissections of d-dimensional space, obtained as follows. With each point of a Poisson point process (a certain, natural model of a random, countable subset of d-dimensional space) we associate its Voronoi cell, consisting of all points of the ambient space that are closer to it than to any other point of the Poisson process.

In the Poisson-Voronoi percolation model, each cell of the Poisson-Voronoi tessellation is coloured black with probability p and white with probability 1-p (independently of all other cells). We are now interested in the "component structure" of the black set. We say that percolation occurs if there exists an infinite, black connected cluster of cells.

While Poisson-Voronoi tessellations and Voronoi percolation have traditionally been studied on Euclidean space, it is of course entirely possible to define these models on other ambient geometric spaces. A natural first choice of an alternative space is d-dimensional hyperbolic space. Perhaps surprisingly this leads to spectacularly different behaviour from the Euclidean case. By now standard methods show that in the Euclidean case there is never more than one infinite cluster. In contrast, in the hyperbolic case there are choices of the parameters for which there are infinitely many infinite clusters (with probability one).

During the talk, after the appropriate introductions and background, I plan to present some recent results on the behaviour of the model as a function of its parameters and the underlying ambient geometric space. No prior knowledge on Poisson point processes or hyperbolic geometry will be assumed.

(based on joint work with Benjamin Hansen and Matthias Irlbeck.)

In the first part of the talk, I will introduce branching Brownian motion, a classical spatial branching process, and the Fisher–Kolmogorov–Petrovsky–Piskunov (F-KPP) equation, a reaction-diffusion equation that admits traveling wave solutions. I will then explain in detail the duality connecting these two objects.

In the second part, I will show how this duality can be used to study transition fronts of inhomogeneous F-KPP equations and to investigate the extremal process of the corresponding inhomogeneous branching Brownian motion. This part of the talk is based on joint work with A. Drewitz and P. Hanigk.nderstand the extremal process in the dual inhomogeneous branching Brownian motion. The second part of the talk is based on joint work with A. Drewitz and P. Hanigk.