33 research outputs found
Graph diameter in long-range percolation
We study the asymptotic growth of the diameter of a graph obtained by adding
sparse "long" edges to a square box in . We focus on the cases when an
edge between and is added with probability decaying with the Euclidean
distance as when . For we show
that the graph diameter for the graph reduced to a box of side scales like
where . In particular, the
diameter grows about as fast as the typical graph distance between two vertices
at distance . We also show that a ball of radius in the intrinsic metric
on the (infinite) graph will roughly coincide with a ball of radius
in the Euclidean metric.Comment: 17 pages, extends the results of arXiv:math.PR/0304418 to graph
diameter, substantially revised and corrected, added a result on volume
growth asymptoti
Visibility to infinity in the hyperbolic plane, despite obstacles
Suppose that 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 contains a fixed ball of radius 1 is larger than some
universal constant , then there is positive probability that 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 be the probability that a line segment
of length is contained in such a set . We show that if decays
fast enough, then there are almost surely no lines in . We also show that if
the decay of is not too fast, then there are almost surely lines in .
In the case of the Poisson Boolean model with balls of fixed radius 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