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.