Continuum percolation at and above the uniqueness treshold on homogeneous spaces
Abstract
Description
We consider the Poisson Boolean model of continuum percolation on a homogeneous Riemannian manifold $M$. Let $lambda$ be intensity of the Poisson process in the model and let $lambda_u$ be the infimum of the set of intensities that a.s. produce a unique unbounded component. We show that above $λ_u$ there is a.s. a unique unbounded component. We also study what happens at $λ_u$ for some spaces. In particular, if $M$ is the product of the hyperbolic disc and the real line, then at $λ_u$ there is a.s. not a unique unbounded component. The results are inspired by results for Bernoulli bond percolation on graphs due to Haggstrom, Peres and Schonmann.
16 pages, corrections made
16 pages, corrections made