Brazilian Journal of Probability and Statistics

Hausdorff dimension of visible sets for well-behaved continuum percolation in the hyperbolic plane

Christoph Thäle

Full-text: Open access


Let ${\mathcal{Z}}$ be a so-called well-behaved percolation, that is, a certain random closed set in the hyperbolic plane, whose law is invariant under all isometries; for example, the covered region in a Poisson Boolean model. In terms of the $\alpha$-value of ${\mathcal{Z}}$, the Hausdorff-dimension of the set of directions is determined in which visibility from a fixed point to the ideal boundary of the hyperbolic plane is possible within ${\mathcal{Z}}$. Moreover, the Hausdorff-dimension of the set of (hyperbolic) lines through a fixed point contained in ${\mathcal{Z}}$ is calculated. Thereby several conjectures raised by Benjamini, Jonasson, Schramm and Tykesson are confirmed.

Article information

Braz. J. Probab. Stat., Volume 28, Number 1 (2014), 73-82.

First available in Project Euclid: 5 February 2014

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Boolean model continuum percolation fractal geometry Hausdorff-dimension hyperbolic geometry


Thäle, Christoph. Hausdorff dimension of visible sets for well-behaved continuum percolation in the hyperbolic plane. Braz. J. Probab. Stat. 28 (2014), no. 1, 73--82. doi:10.1214/12-BJPS194.

Export citation


  • Benedetti, R. and Petrino, C. (2008). Lectures on Hyperbolic Geometry. Berlin: Springer.
  • Benjamini, I., Jonasson, J., Schramm, O. and Tykesson, J. (2009). Visibility to infinity in the hyperbolic plane, despite obstacles. ALEA Lat. Am. J. Probab. Math. Stat. 6, 323–342.
  • Benjamini, I. and Schramm, O. (2001). Percolation in the hyperbolic plane. J. Am. Math. Soc. 14, 487–507.
  • Calka, P., Michel, J. and Porret-Blanc, S. (2010). Asymptotics of the visibility function in the Boolean model. Available at arXiv:0905.4874 [math.PR].
  • Calka, P. and Tykesson, J. (2011). Asymptotics of visibility in the hyperbolic plane. Available at arXiv:1012.5220 [math.PR].
  • Falconer, H. (2003). Fractal Geometry. Mathematical Foundations and Applications, 2nd ed. Chichester: Wiley.
  • Federer, H. (1969). Geometric Measure Theory. New York: Springer.
  • Dawson, D. A. and Hochberg, K. J. (1979). The carrying dimension of a stochastic measure diffusion. Ann. Probab. 7, 693–703.
  • Kallenberg, O. (1983). Random Measures, 3rd ed. Berlin: Akademie Verlag.
  • Kallenberg, O. (2002). Foundations of Modern Probability, 2nd ed. New York: Springer.
  • Lalley, S. (2011). Percolation clusters in hyperbolic tessellations. Geom. Funct. Anal. 11, 971–1030.
  • Meester, R. and Roy, R. (1996). Continuum Percolation. Cambridge: Cambridge Univ. Press.
  • Molchanov, I. (2005). Theory of Random Sets. London: Springer.
  • Ramsay, A. and Richtmyer, R. D. (2010). Introduction to Hyperbolic Geometry. New York: Springer.
  • Tykesson, J. (2007). The number of unbounded components in the Poisson Boolean model of continuum percolation in hyperbolic space. Electron. J. Probab. 12, 1379–1401.
  • Zähle, U. (1984). Random fractals generated by random cutouts. Math. Nachr. 116, 27–52.
  • Zähle, U. (1988). The fractal character of localizable measure-valued processes, III. Fractal carrying sets of branching diffusions. Math. Nachr. 138, 293–311.