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Percolation and Connectivity in AB Random Geometric Graphs
Ist Teil von
Advances in applied probability, 2012-03, Vol.44 (1), p.21-41
Erscheinungsjahr
2012
Quelle
Alma/SFX Local Collection
Beschreibungen/Notizen
Given two independent Poisson point processes Φ
(1)
, Φ
(2)
in
, the
AB
Poisson Boolean model is the graph with the points of Φ
(1)
as vertices and with edges between any pair of points for which the intersection of balls of radius 2
r
centered at these points contains at least one point of Φ
(2)
. This is a generalization of the
AB
percolation model on discrete lattices. We show the existence of percolation for all
d
≥ 2 and derive bounds for a critical intensity. We also provide a characterization for this critical intensity when
d
= 2. To study the connectivity problem, we consider independent Poisson point processes of intensities
n
and τ
n
in the unit cube. The
AB
random geometric graph is defined as above but with balls of radius
r
. We derive a weak law result for the largest nearest-neighbor distance and almost-sure asymptotic bounds for the connectivity threshold.