## Brazilian Journal of Probability and Statistics

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

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

#### Abstract

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

**Source**

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

**Dates**

First available in Project Euclid: 5 February 2014

**Permanent link to this document**

https://projecteuclid.org/euclid.bjps/1391611338

**Digital Object Identifier**

doi:10.1214/12-BJPS194

**Mathematical Reviews number (MathSciNet)**

MR3165429

**Zentralblatt MATH identifier**

1293.60094

**Keywords**

Boolean model continuum percolation fractal geometry Hausdorff-dimension hyperbolic geometry

#### Citation

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. https://projecteuclid.org/euclid.bjps/1391611338