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Visibility to infinity in the hyperbolic plane, despite obstacles. (English) Zbl 1276.82012
Summary: 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 $$\mathbb H^2$$.

##### MSC:
 82B21 Continuum models (systems of particles, etc.) arising in equilibrium statistical mechanics 82B43 Percolation
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