Continuum percolation at and above the uniqueness treshold on homogeneous spaces

TL;DR

This paper investigates the behavior of continuum percolation on homogeneous Riemannian manifolds, establishing conditions for the uniqueness of unbounded components at and above a critical intensity threshold.

Contribution

It proves that above the critical intensity, there is almost surely a unique unbounded component, and explores the nature of this threshold on specific spaces like the hyperbolic disc times the real line.

Findings

Uniqueness of unbounded component above the threshold

Existence of multiple unbounded components at the threshold in certain spaces

Extension of percolation results to Riemannian manifolds

Abstract

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 $\lambda_u$ there is a.s. a unique unbounded component. We also study what happens at $\lambda_u$ for some spaces. In particular, if $M$ is the product of the hyperbolic disc and the real line, then at $\lambda_u$ 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.