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Asymptotics of visibility in the hyperbolic plane. (English) Zbl 1276.60012
Summary: At each point of a Poisson point process of intensity \(\lambda \) in the hyperbolic plane, center a ball of bounded random radius. Consider the probability \(P_{r}\) that, from a fixed point, there is some direction in which one can reach distance \(r\) without hitting any ball. It is known [Benjamini et al., “Visibility to infinity in the hyperbolic plane, despite obstacles”, ALEA, Lat. Am. J. Probab. Math. Stat. 6, 323–342 (2009)] that, if \(\lambda \) is strictly smaller than a critical intensity \(\lambda _{ gv}\), then \(P_{r}\) does not go to zero as \(r \rightarrow \infty \). The main result in this note shows that, in the case \(\lambda =\lambda _{gv}\), the probability of reaching a distance larger than \(r\) decays essentially polynomially, while, if \(\lambda >\lambda _{gv}\), the decay is exponential. We also extend these results to various related models, and we finally obtain asymptotic results in several situations.

60D05 Geometric probability and stochastic geometry
60G55 Point processes (e.g., Poisson, Cox, Hawkes processes)
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[1] Ballani, F. (2006). On second-order characteristics of germ-grain models with convex grains. Mathematika 53 , 255-285. · Zbl 1127.60009 · doi:10.1112/S0025579300000139
[2] 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 , 323-342. · Zbl 1276.82012
[3] 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 , 702-717. · Zbl 1029.60002 · doi:10.1239/aap/1037990949
[4] 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. · Zbl 1177.60011
[5] 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. 59-115. · Zbl 0899.51012 · www.msri.org
[6] Hall, P. (1988). Introduction to the Theory of Coverage Processes . John Wiley, New York. · Zbl 0659.60024
[7] Heinrich, L. (1998). Contact and chord length distribution of a stationary Voronoĭ tessellation. Adv. Appl. Prob. 30 , 603-618. · Zbl 0922.60020 · doi:10.1239/aap/1035228118
[8] Herman, I., Melançon, G. and Marshall, M. S. (2000). Graph visualization and navigation in information visualization: a survey. IEEE Trans. Visual. Comput. Graphics 6 , 24-43.
[9] Janson, S. (1986). Random coverings in several dimensions. Acta Math. 156 , 83-118. · Zbl 0597.60014 · doi:10.1007/BF02399201
[10] Jonasson, J. (2008). Dynamical circle covering with homogeneous Poisson updating. Statist. Prob. Lett. 78 , 2400-2403. · Zbl 1157.60008 · doi:10.1016/j.spl.2008.03.001
[11] Kahane, J.-P. (1985). Some Random Series of Functions (Camb. Stud. Adv. Math. 5 ), 2nd edn. Cambridge University Press. · Zbl 0571.60002
[12] Kahane, J.-P. (1990). Recouvrements aléatoires et théorie du potentiel. Colloq. Math. 60 /61, 387-411. · Zbl 0728.60053
[13] 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. 277-324. · Zbl 0737.60016
[14] 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 , 49-72. · Zbl 0993.60010 · doi:10.1007/s004400100138
[15] Lovász, L. (2009). Very large graphs. In Current Developments in Mathematics , Int. Press, Somerville, MA, pp. 67-128. · Zbl 1179.05100
[16] Lyons, R. (1996). Diffusions and random shadows in negatively curved manifolds. J. Funct. Anal. 138 , 426-448. · Zbl 0877.58058 · doi:10.1006/jfan.1996.0071
[17] Meester, R. and Roy, R. (1996). Continuum Percolation (Cambr. Tracts Math. 119 ). Cambridge University Press.
[18] 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 , 477-481. · Zbl 1125.60006 · doi:10.1016/j.crma.2007.02.018
[19] Santaló, L. A. (1976). Integral Geometry and Geometric Probability (Encyclopedia Math. Appl. 1 ). Addison-Wesley, Reading, MA. · Zbl 0342.53049
[20] Tykesson, J. (2007). The number of unbounded components in the Poisson Boolean model of continuum percolation in hyperbolic space. Electron. J. Prob. 12 , 1379-1401. · Zbl 1136.82010 · doi:10.1214/EJP.v12-460 · eudml:128881
[21] Ungar, A. A. (2008). Analytic Hyperbolic Geometry and Albert Einstein’s Special Theory of Relativity . World Scientific, Hackensack, NJ. · Zbl 1147.83004
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