Visibility to infinity in the hyperbolic plane, despite obstacles

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dc.creatorBenjamini, Itai
dc.creatorJonasson, Johan
dc.creatorSchramm, Oded
dc.creatorTykesson, Johan
dc.date2008-07-21
dc.date.accessioned2026-07-07T09:51:52Z
dc.date.available2026-07-07T09:51:52Z
dc.descriptionSuppose 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 $\H^2$.
dc.identifierhttps://arxiv.org/abs/0807.3308
dc.identifierhttp://arxiv.org/abs/0807.3308
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/165395
dc.subjectProbability
dc.subjectMathematical Physics
dc.subject82B43; 82B27; 82B21
dc.titleVisibility to infinity in the hyperbolic plane, despite obstacles
dc.typetext

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