Online NETWORKS Seminar Talk: Hyperbolic Voronoi percolation by Tobias Müller
Speaker: Tobias Müller (Groningen University)
Title: Hyperbolic Voronoi percolation
I will discuss percolation on the Voronoi tessellation generated by a homogeneous Poisson point process on the hyperbolic plane.
That is, to each point z of a constant intensity Poisson point process Z on the hyperbolic plane we assign its Voronoi cell -- the region consisting of all points that are closer to z than to any other z' in Z -- and we colour each cell black with probability p and white with probability 1-p, independently of the colours of all other cells. We say that percolation occurs if there is an infinite connected cluster of black cells.
Hyperbolic Poisson-Voronoi percolation was first studied by Benjamini and Schramm about twenty years ago. Their results show that there are spectacular differences with the corresponding model in the Euclidean plane.
I will sketch joint work with my recently graduated doctoral student Ben Hansen that resolves a conjecture and an open question, posed by Benjamini and Schramm, on the behaviour of the ``critical probability for percolation'' as a function of the intensity parameter of the underlying Poisson process.
(Unlike in Euclidean Poisson-Voronoi percolation, this critical value depends on the intensity of the underlying Poisson process.)
Based on joint work with Benjamin Hansen.