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2007 The number of unbounded components in the Poisson Boolean model of continuum percolation in hyperbolic space
Johan Tykesson
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Electron. J. Probab. 12(none): 1379-1401 (2007). DOI: 10.1214/EJP.v12-460

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

Citation

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Johan Tykesson. "The number of unbounded components in the Poisson Boolean model of continuum percolation in hyperbolic space." Electron. J. Probab. 12 1379 - 1401, 2007. https://doi.org/10.1214/EJP.v12-460

Information

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

zbMATH: 1136.82010
MathSciNet: MR2354162
Digital Object Identifier: 10.1214/EJP.v12-460

Subjects:
Primary: 82B21
Secondary: 82B43

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