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

TL;DR

This paper determines the Hausdorff dimension of visibility sets in well-behaved continuum percolation models within the hyperbolic plane, confirming several existing conjectures.

Contribution

It provides explicit formulas for the Hausdorff dimensions of visibility sets and lines in hyperbolic percolation, advancing understanding of geometric properties in these models.

Findings

Hausdorff dimension of visibility sets is characterized in terms of the α-value.

Explicit calculation of the Hausdorff dimension of lines through a fixed point in Z.

Confirmation of several conjectures by Benjamini, Jonasson, Schramm, and Tykesson.

Abstract

Let Z be a so-called well-behaved percolation, i.e. 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. The Hausdorff-dimension of the set of directions is determined in terms of the $\alpha$-value of Z in which visibility from a fixed point to the ideal boundary of the hyperbolic plane is possible within Z. Moreover, the Hausdorff-dimension of the set of (hyperbolic) lines through a fixed point contained in Z is calculated. Thereby several conjectures raised by Benjamini, Jonasson, Schramm and Tykesson are confirmed.