Continuum Percolation in non-Euclidean Spaces

Doctoral thesis, 2008

In this thesis we first consider the Poisson Boolean model of continuum percolation in $n$-dimensional hyperbolic space ${\mathbb H}^n$. Let $R$ be the radius of the balls in the model, and $\lambda$ the intensity of the underlying Poisson process. We show that if $R$ is large enough, then there is an interval of intensities such that there are infinitely many unbounded components in the covered region. For $n=2$, more refined results are obtained. We then consider the model on some more general spaces. For a large class of homogeneous spaces, it is established that if $\lambda$ is such that there is a.s.\ a unique unbounded component in the covered region, then this is also the case for any $\lambda_1>\lambda$. In ${\mathbb H}^2\times{\mathbb R}$ it is proved that if $\lambda$ is critical for a.s.\ having a unique unbounded component in the covered region, then there is a.s.\ not a unique unbounded component. Finally, we consider another aspect of continuum percolation in ${\mathbb H}^2$. We show that in the Poisson Boolean model, there are intensities for which infinite geodesics are contained in unbounded components of the covered region. This is also shown for the vacant region, as well as for a larger class of continuum percolation models. We also consider some dynamical models.