## THE NUMBER OF UNBOUNDED COMPONENTS IN THE POISSON BOOLEAN MODEL OF CONTINUUM PERCOLATION IN HYPERBOLIC SPACE (2007)

Citations: | 3 - 2 self |

### BibTeX

@MISC{Tykesson07thenumber,

author = {Johan Tykesson},

title = {THE NUMBER OF UNBOUNDED COMPONENTS IN THE POISSON BOOLEAN MODEL OF CONTINUUM PERCOLATION IN HYPERBOLIC SPACE},

year = {2007}

}

### OpenURL

### Abstract

Abstract. We consider the Poisson Boolean model of continuum percolation with balls of fixed radius R in n-dimensional hyperbolic space H n. Let λ be the intensity of the underlying Poisson process, and let NC denote the number of unbounded components in the covered region. For the model in any dimension we show that there are intensities such that NC = ∞ a.s. if R is big enough. In H 2 we show a stronger result: for any R there are two intensities λc and λu where 0 < λc < λu < ∞, such that NC = 0 for λ ∈ [0, λc], NC = ∞ for λ ∈ (λc, λu) and NC = 1 for λ ∈ [λu, ∞).

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