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

   
@article{EJP460,
	author = {Johan Tykesson},
	title = {The number of unbounded components in the Poisson Boolean model of continuum percolation in hyperbolic space},
	journal = {Electron. J. Probab.},
	fjournal = {Electronic Journal of Probability},
	volume = {12},
	year = {2007},
	keywords = {continuum percolation; phase transitions; hyperbolic space},
	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 $\lambda \in  [\lambda_u, \infty)$.},
	pages = {no. 51, 1379-1401},
	issn = {1083-6489},
	doi = {10.1214/EJP.v12-460},    
        url = {http://ejp.ejpecp.org/article/view/460}}