Visibility to infinity in the hyperbolic plane, despite obstacles

Details Visibility to infinity in the hyperbolic plane, despite obstacles Visibility to infinity in the hyperbolic plane, despite obstacles

2008-07-21

arxiv.org/abs/0807.3308v1

Mathematics

Probability

Scientific paper

Suppose that $Z$ is a random closed subset of the hyperbolic plane $\H^2$, whose law is invariant under isometries of $\H^2$. We prove that if the probability that $Z$ contains a fixed ball of radius 1 is larger than some universal constant $p<1$, then there is positive probability that $Z$ contains (bi-infinite) lines. We then consider a family of random sets in $\H^2$ that satisfy some additional natural assumptions. An example of such a set is the covered region in the Poisson Boolean model. Let $f(r)$ be the probability that a line segment of length $r$ is contained in such a set $Z$. We show that if $f(r)$ decays fast enough, then there are almost surely no lines in $Z$. We also show that if the decay of $f(r)$ is not too fast, then there are almost surely lines in $Z$. In the case of the Poisson Boolean model with balls of fixed radius $R$ we characterize the critical intensity for the almost sure existence of lines in the covered region by an integral equation. We also determine when there are lines in the complement of a Poisson process on the Grassmannian of lines in $\H^2$.

Affiliated with

Also associated with

No associations

Rating

Visibility to infinity in the hyperbolic plane, despite obstacles does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Visibility to infinity in the hyperbolic plane, despite obstacles, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Visibility to infinity in the hyperbolic plane, despite obstacles will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-72962