Percolation and connectivity in AB random geometric graphs

Abstract

Given two independent Poisson point processes Φ⁽¹⁾, Φ⁽²⁾ in R^d, 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 2r 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 τn 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.