Konstantin Recke, Poisson–Voronoi percolation in higher rank скачать в хорошем качестве

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 pu(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 pu(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). We will describe a phenomenon fundamentally different from all situations in which Poisson-Voronoi percolation has previously been studied. Our approach builds on a recent breakthrough of Fraczyk, Mellick and Wilkens (2023) and provides an alternative proof strategy for Gaboriau’s fixed price problem. As a further application, we give a new class of examples of non-amenable Cayley graphs that admit factor of iid bond percolations with a unique infinite cluster and small expected degree, answering a question inspired by Hutchcroft and Pete (2020). Joint work with Jan Grebik (Leipzig).