TABLE OF CONTENTS We now turn to consider models in the continuum plane. One nice thing about these models is that we can use results from the discrete random networks to derive some of their percolation properties.
We start by considering the Poisson nearest neighbour model, where each point of a planar Poisson point process X is connected to its k nearest neighbours. As we have seen in Chapter 1, the density of the Poisson process does not play a role here, since by changing the unit of length we can vary the density of the process without changing the connections. In this sense, the model is scale free and we can assume the density to be as high as we want. Note also that in contrast to the previous models, in this case there is no independence between connections, as the existence of an edge in the network depends on the positions of all other points in the plane.
We call U the event that there exists an unbounded connected component in the resulting nearest neighbour random network. As in the discrete percolation model, this event can have only probability zero or one. To see this, note that the existence of an infinite cluster is invariant under any translation of coordinates on the plane, and by ergodicity (see Appendix A.3) this implies that it can have only probability zero or one. This, of course, requires as to show that the model...
