---
abstract: |
    We consider the Poisson Boolean model of continuum percolation on a homogeneous space $M$. Let $\lambda$ be the intensity of the underlying Poisson process. Let $\lambda_u$ be the infimum of the set of intensities that a.s. produce a unique unbounded component. First we show that if $\lambda>\lambda_u$ then there is a.s. a unique unbounded component at $\lambda$. Then we let $M=\hyp\times
      \R$ and show that at $\lambda_u$ there is a.s. not a unique unbounded component. These results are continuum analogies of theorems by Häggström, Peres and Schonmann.
address: 'Department of Mathematics, Division of Mathematical Statistics, Chalmers University of Technology and Göteborg University, S-412 96 '
author:
- 'Johan H. Tykesson'
title: Continuum percolation at and above the uniqueness treshold on homogeneous spaces
---

Introduction and results
========================

In this paper we show continuum analogies to some theorems concerning the uniqueness phase in the theory of independent bond and site percolation on graphs. Before turning to our results, we review these theorems.

Let $G=(V,E)$ be an infinite transitive graph with vertex set $V$ and edge set $E$. Keep each edge with probability $p$ and delete it otherwise, independently for all edges. We call this independent bond percolation on 