Konstantin Recke

Poisson-Voronoi percolation in higher rank

Poisson-Voronoi percolation is a continuum percolation model that can be defined on any metric space (M,d) with an infinite Radon measure μ as follows. For λ>0, consider a Poisson point process of intensity λ⋅μ and associate to each point of the process its Voronoi cell, i.e., the set of all points in M closer to this point than to any other point of the process. For p∈(0,1), independently color each cell black with probability p and consider the union of black cells. For fixed λ>0, define the uniqueness threshold p_u(λ) to be the infimal value of p such that a unique unbounded path-connected component exists. In this talk, we will discuss the behavior of p_u(λ) as λ→0 in the case that M is the symmetric space of a connected higher rank semisimple real Lie group with property (T). In this case, the behavior turns out to be fundamentally different from all situations in which Poisson-Voronoi percolation has previously been studied. Joint work with Jan Grebík (Leipzig).