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Asymptotics of Visibility in the Hyperbolic Plane

Published online by Cambridge University Press:  22 February 2016

Johan Tykesson*
Affiliation:
Weizmann Institute of Science, Rehovot
Pierre Calka*
Affiliation:
Université de Rouen
*
Current address: Department of Mathematics, Uppsala University, Box 480, 75106 Uppsala, Sweden. Email address: [email protected]
∗∗ Postal address: Laboratoire de Mathématiques Raphaël Salem, UMR 6085 CNRS-Université de Rouen, Avenue de l'Université, BP 12 Technopôle du Madrillet, F76801 Saint-Etienne-du-Rouvray, France. Email address: [email protected]
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Abstract

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At each point of a Poisson point process of intensity λ in the hyperbolic plane, center a ball of bounded random radius. Consider the probability Pr that, from a fixed point, there is some direction in which one can reach distance r without hitting any ball. It is known (see Benjamini, Jonasson, Schramm and Tykesson (2009)) that if λ is strictly smaller than a critical intensity λgv thenPr does not go to 0 as r → ∞. The main result in this note shows that in the case λ=λgv, the probability of reaching a distance larger than r decays essentially polynomially, while if λ>λgv, the decay is exponential. We also extend these results to various related models and we finally obtain asymptotic results in several situations.

Type
Stochastic Geometry and Statistical Applications
Copyright
© Applied Probability Trust 

Footnotes

Research supported by a postdoctoral grant from the Swedish Research Council.

References

Ballani, F. (2006). On second-order characteristics of germ–grain models with convex grains. Mathematika 53, 255285.Google Scholar
Benjamini, I., Jonasson, J., Schramm, O. and Tykesson, J. (2009). Visibility to infinity in the hyperbolic plane, despite obstacles. ALEA Latin Amer. J. Prob. Math. Statist. 6, 323342.Google Scholar
Calka, P. (2002). The distributions of the smallest disks containing the Poisson-Voronoi typical cell and the Crofton cell in the plane. Adv. Appl. Prob. 34, 702717.CrossRefGoogle Scholar
Calka, P., Michel, J. and Porret-Blanc, S. (2011). Asymptotics of the visibility function in the Boolean model. Preprint. Available http://arxiv.org/abs/0905.4874v2.Google Scholar
Cannon, J. W., Floyd, W. J., Kenyon, R. and Parry, W. R. (1997). Hyperbolic geometry. In Flavors of Geometry (Math. Sci. Res. Inst. Publ. 31), Cambridge University Press, pp. 59115.Google Scholar
Hall, P. (1988). Introduction to the Theory of Coverage Processes. John Wiley, New York.Google Scholar
Heinrich, L. (1998). Contact and chord length distribution of a stationary Voronoı˘ tessellation. Adv. Appl. Prob. 30, 603618.Google Scholar
Herman, I., Melançon, G. and Marshall, M. S. (2000). Graph visualization and navigation in information visualization: a survey. IEEE Trans. Visual. Comput. Graphics 6, 2443.Google Scholar
Janson, S. (1986). Random coverings in several dimensions. Acta Math. 156, 83118.Google Scholar
Jonasson, J. (2008). Dynamical circle covering with homogeneous Poisson updating. Statist. Prob. Lett. 78, 24002403.Google Scholar
Kahane, J.-P. (1985). Some Random Series of Functions (Camb. Stud. Adv. Math. 5), 2nd edn. Cambridge University Press.Google Scholar
Kahane, J.-P. (1990). Recouvrements aléatoires et théorie du potentiel. Colloq. Math. 60/61, 387411.Google Scholar
Kahane, J.-P. (1991). Produits de poids aléatoires indépendants et applications. In Fractal Geometry and Analysis (Montreal, PQ, 1989; NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci. 346), Kluwer, Dordrecht, pp. 277324.Google Scholar
Last, G. and Schassberger, R. (2001). On the second derivative of the spherical contact distribution function of smooth grain models. Prob. Theory Relat. Fields 121, 4972.Google Scholar
Lovász, L. (2009). Very large graphs. In Current Developments in Mathematics, Int. Press, Somerville, MA, pp. 67128.Google Scholar
Lyons, R. (1996). Diffusions and random shadows in negatively curved manifolds. J. Funct. Anal. 138, 426448.Google Scholar
Meester, R. and Roy, R. (1996). Continuum Percolation (Cambr. Tracts Math. 119). Cambridge University Press.Google Scholar
Porret-Blanc, S. (2007). Sur le caractère borné de la cellule de Crofton des mosaï ques de géodésiques dans le plan hyperbolique. C. R. Math. Acad. Sci. Paris 344, 477481.Google Scholar
Santaló, L. A. (1976). Integral Geometry and Geometric Probability (Encyclopedia Math. Appl. 1). Addison-Wesley, Reading, MA.Google Scholar
Tykesson, J. (2007). The number of unbounded components in the Poisson Boolean model of continuum percolation in hyperbolic space. Electron. J. Prob. 12, 13791401.Google Scholar
Ungar, A. A. (2008). Analytic Hyperbolic Geometry and Albert Einstein's Special Theory of Relativity. World Scientific, Hackensack, NJ.Google Scholar