Visibility to infinity in the hyperbolic plane, despite obstacles.

*(English)*Zbl 1276.82012Summary: Suppose that \(\mathcal Z\) is a random closed subset of the hyperbolic plane \(\mathbb H^2\), whose law is invariant under isometries of \(\mathbb H^2\). We prove that if the probability that \(\mathcal Z\) contains a fixed ball of radius 1 is larger than some universal constant \(p_0 < 1\), then there is positive probability that \(\mathcal Z\) contains (bi-infinite) lines.

We then consider a family of random sets in \(\mathbb H^2\) 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 \(\mathcal Z\). We show that if \(f (r)\) decays fast enough, then there are a.s. no lines in \(\mathcal Z\). We also show that if the decay of \(f (r)\) is not too fast, then there are a.s. lines in \(\mathcal Z\). In the case of the Poisson Boolean model with balls of fixed radius \(R\) we characterize the critical intensity for the a.s. 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 \(\mathbb H^2\).

We then consider a family of random sets in \(\mathbb H^2\) 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 \(\mathcal Z\). We show that if \(f (r)\) decays fast enough, then there are a.s. no lines in \(\mathcal Z\). We also show that if the decay of \(f (r)\) is not too fast, then there are a.s. lines in \(\mathcal Z\). In the case of the Poisson Boolean model with balls of fixed radius \(R\) we characterize the critical intensity for the a.s. 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 \(\mathbb H^2\).