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February 2014 Hausdorff dimension of visible sets for well-behaved continuum percolation in the hyperbolic plane
Christoph Thäle
Braz. J. Probab. Stat. 28(1): 73-82 (February 2014). DOI: 10.1214/12-BJPS194

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.

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Christoph Thäle. "Hausdorff dimension of visible sets for well-behaved continuum percolation in the hyperbolic plane." Braz. J. Probab. Stat. 28 (1) 73 - 82, February 2014. https://doi.org/10.1214/12-BJPS194

Information

Published: February 2014
First available in Project Euclid: 5 February 2014

zbMATH: 1293.60094
MathSciNet: MR3165429
Digital Object Identifier: 10.1214/12-BJPS194

Rights: Copyright © 2014 Brazilian Statistical Association

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Vol.28 • No. 1 • February 2014
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