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.

phase transitions

exceptional lines

continuum percolation

hyperbolic geometry


I. Benjamini

Johan Jonasson

Chalmers, Mathematical Sciences, Mathematics

University of Gothenburg

O. Schramm

Johan Tykesson

Chalmers, Mathematical Sciences, Mathematical Statistics

University of Gothenburg


1980-0436 (ISSN)

Vol. 6 323-342

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