Probability and Statistics Seminar: Voronoï percolation in the hyperbolic plane

  • Date: –11:15
  • Location: Ångströmlaboratoriet, Lägerhyddsvägen 1 64119
  • Lecturer: Benjamin Thomas Hansen, University of Groningen
  • Organiser: Matematiska institutionen
  • Contact person: Xing Shi Cai
  • Seminarium

Abstract:  We consider site percolation on Voronoï cells generated by a Poisson point process on the hyperbolic plane H2. Each cell is coloured black independently with probability p, otherwise the cell is coloured white. Benjamini and Schramm proved the existence of three phases:

For p ∈ [0, pc] all black clusters are bounded and there is a unique infinite white cluster. For p ∈ (pc, pu), there are infinitely many unbounded black and white clusters. For p ∈ [pu, 1] there is a unique infinite black cluster and all white clusters are bounded. They also showed pu = 1 − pc. The critical values pc and pc depend on the intensity of the Poisson point process. We prove that pc tends to 1/2 as the intensity tends to infinity. This confirms a conjecture of Benjamini and Schramm.

Joint work with Tobias Müller.