The number of unbounded components in the Poisson Boolean model of continuum percolation in hyperbolic space

Abstract

We consider the Poisson Boolean model of continuum percolation with balls of fixed radius $R$ in $n$-dimensional hyperbolic space $H^n$. Let $\lambda$ be the intensity of the underlying Poisson process, and let $N_C$ denote the number of unbounded components in the covered region. For the model in any dimension we show that there are intensities such that $N_C=\infty$ a.s. if $R$ is big enough. In $H^2$ we show a stronger result: for any $R$ there are two intensities $\lambda_c$ and $\lambda_u$ where $0< \lambda_c < \lambda _u < \infty$, such that$N_C=0$ for $\lambda \in [0,\lambda_c]$, $N_C=\infty$ for $\lambda \in (\lambda_c,\lambda_u)$ and $N_C=1$ for $\lambda \in [\lambda_u, \infty)$.