On Percolation of Two-Dimensional Hard Disks

Abstract

We consider the hard-core model in R 2 , in which a random set of non-intersecting unit disks is sampled with an intensity parameter λ. Given ε > 0 we consider the graph in which two disks are adjacent if they are at distance ≤ ε from each other. We prove that this graph, G, is highly connected when λ is greater than a certain threshold depending on ε. Namely, given a square annulus with inner radius L 1 and outer radius L 2 , the probability that the annulus is crossed by G is at least 1 − C exp(−cL 1 ). As a corollary we prove that a Gibbs state admits an infinite component of G if the intensity λ is large enough, depending on ε.

Continuum Percolation in a Nonstabilizing Environment

“…Another line of research is concerned with generalizations towards using other stationary point processes as the underlying set of vertices in the network. Let us mention for example the continuum-percolation results for Gibbs point processes in [Mür75,Stu13,Jan16,CD14,Mag18], for repelling point processes in [GKP16,BY14], for negatively associated point processes in [BY13], or general stationary point process [MR96,Gou09]. A particularly interesting class of stationary point process, for which continuum percolation can be investigated, is given by Cox point processes.…”