TABLE OF CONTENTS We now want to treat the interference network model. As we shall see, there are some additional technical difficulties to show the phase transition in this model, due to its infinite range dependence structure. Furthermore, the model presents a number of interesting properties besides the phase transition. We start this section by stating two technical theorems that will turn out to be useful in the proofs.
The first theorem shows a property of the boolean model, namely that in the supercritical regime there are with high probability paths that cross a large box in the plane.
Theorem 2.7.1 Consider a supercritical boolean model of radius r and density ? > ? c . For any 0 < ? < 1, let R ?n be a rectangle of sides 
on the plane. Let 
denote the event of a left to right crossing inside the rectangle, that is, the existence of a connected component of Poisson points of R ? n, such that each of the two smaller sides of R ? n has at least a point of the component within distance r from it. We have
A proof of this theorem can be found in the book by Meester and Roy (1996) (Corollary 4.1); see also Penrose (2003). We shall prove a similar property in the context of discrete percolation in Chapter 5.
The second result we mention is known as Campbell s theorem. We prove a...
