Continuum percolation models in which each point of a two-dimensional Poisson point process is the centre of a disc of given (or random) radius $r$, have been extensively studied. In this paper, we consider the generalization in which a deterministic algorithm (given the points of the point process) places the discs on the plane, in such a way that each disc covers at least one point of the point process and that each point is covered by at least one disc. This gives a model for wireless communication networks, which was the original motivation to study this class of problems.
We look at the percolation properties of this generalized model, showing that an unbounded connected component of discs does not exist, almost surely, for small values of the density $\lambda$ of the Poisson point process, for any covering algorithm. In general, it turns out not to be true that unbounded connected components arise when $\lambda$ is taken sufficiently high. However, we identify some large families of covering algorithms, for which such an unbounded component does arise for large values of $\lambda$.
We show how a simple scaling operation can change the percolation properties of the model, leading to the almost sure existence of an unbounded connected component for large values of $\lambda$, for any covering algorithm.
Finally, we show that a large class of covering algorithms, which arise in many practical applications, can get arbitrarily close to achieving a minimal density of covering discs. We also construct an algorithm that achieves this minimal density.
"Covering algorithms, continuum percolation and the geometry of wireless networks." Ann. Appl. Probab. 13 (2) 722 - 741, May 2003. https://doi.org/10.1214/aoap/1050689601