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


 
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1. Title Title of document The number of unbounded components in the Poisson Boolean model of continuum percolation in hyperbolic space
 
2. Creator Author's name, affiliation, country Johan Harald Tykesson; Chalmers University of Technology
 
3. Subject Discipline(s)
 
3. Subject Keyword(s) continuum percolation; phase transitions; hyperbolic space
 
3. Subject Subject classification 82B21; 82B43
 
4. Description 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 $\lambda \in [\lambda_u, \infty)$.
 
5. Publisher Organizing agency, location
 
6. Contributor Sponsor(s) Swedish Natural Science Research Council
 
7. Date (YYYY-MM-DD) 2007-11-04
 
8. Type Status & genre Peer-reviewed Article
 
8. Type Type
 
9. Format File format PDF
 
10. Identifier Uniform Resource Identifier http://ejp.ejpecp.org/article/view/460
 
10. Identifier Digital Object Identifier 10.1214/EJP.v12-460
 
11. Source Journal/conference title; vol., no. (year) Electronic Journal of Probability; Vol 12
 
12. Language English=en
 
14. Coverage Geo-spatial location, chronological period, research sample (gender, age, etc.)
 
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