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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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

Permanent link to this document
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.)
Secondary: 82B43: Percolation [See also 60K35]

Keywords
continuum percolation phase transitions hyperbolic space

Rights
This work is licensed under aCreative Commons Attribution 3.0 License.

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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