## Electronic Journal of Probability

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

Johan Tykesson

#### 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 $\lambda \in [\lambda_u, \infty)$.

#### Article information

Source
Electron. J. Probab., Volume 12 (2007), paper no. 51, 1379-1401.

Dates
Accepted: 4 November 2007
First available in Project Euclid: 1 June 2016

https://projecteuclid.org/euclid.ejp/1464818522

Digital Object Identifier
doi:10.1214/EJP.v12-460

Mathematical Reviews number (MathSciNet)
MR2354162

Zentralblatt MATH identifier
1136.82010

Subjects
Primary: 82B21: Continuum models (systems of particles, etc.)

Rights

#### Citation

Tykesson, Johan. The number of unbounded components in the Poisson Boolean model of continuum percolation in hyperbolic space. Electron. J. Probab. 12 (2007), paper no. 51, 1379--1401. doi:10.1214/EJP.v12-460. https://projecteuclid.org/euclid.ejp/1464818522

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