Percolation and connectivity in \(AB\) random geometric graphs (Q2879905)

Motivated by the so-called AB percolation processes, the paper defines and considers a special clas of random geometric graphs called the AB model. These are random geometric graphs on the \(d\)-dimensional Euclidean space which are defined through two independent Poisson processes. One of them provides the points which are the vertices of the random graph, whereas the other is used as a means of determining whether two given points from the first process are joined by an edge.NEWLINENEWLINEMore specifically, two points of the first process are joined if there is a point from the second process that that is within distance \(2r\) from each one of them. This is the AB Poisson Boolean model. The AB random geometric graph is defined in a similar way, except that the metric that is used is the toroidal metric, which is used in order to eliminate any boundary effects, as both sets of points are located on the unit cube. This as well as the intensities of the two processes are the parameters that determine the random graph. Regarding, the AB Poisson Boolean model, given the intensity of the first model, the existence of a critical intensity for the second process is shown above which percolation occurs. For the AB random geometric graph, a critical value for \(r\) is determined above which the random graph is connected.