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The number of unbounded components in the Poisson Boolean model of continuum percolation in hyperbolic space. (English) Zbl 1136.82010

In the paper the Poisson Boolean model of continuum percolation with balls of fixed radius \(R\) in \(n\)-dimensional hyperbolic space \({\mathbb H}^n\) is considered. The author shows that there are intensities for which there are almost surely infinitely many unbounded components in the covered region if \(R\) is big enough. In \({\mathbb H}^2\) he also shows the existence of three distinct phases regarding the number of unbounded components, for any \(R\).

MSC:

82B21 Continuum models (systems of particles, etc.) arising in equilibrium statistical mechanics
82B43 Percolation
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