Electronic Journal of Probability

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

Johan Tykesson

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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

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

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 82B21: Continuum models (systems of particles, etc.)
Secondary: 82B43: Percolation [See also 60K35]

continuum percolation phase transitions hyperbolic space

This work is licensed under aCreative Commons Attribution 3.0 License.


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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