Visibility to infinity in the hyperbolic plane, despite obstacles
Journal article, 2009
Suppose that Z is a random closed subset of the hyperbolic plane H-2, whose law is invariant under isometrics of H-2. We prove that if the probability that Z contains a fixed ball of radius 1 is larger than some universal constant p0 < 1, then there is positive probability that Z contains (hi-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 as. no lines i Z. We also show that if the decay of f (r) is not too fast, then there are as. lines in Z. In the case of the Poisson Boolean model with balls of fixed radius R we characterize the critical intensity for the as. 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 space of lines in H-2.