In this chapter we present a few basic results which will be used repeatedly in the subsequent development of the subject. The last few sections of this chapter will be devoted to obtain some inequalities, which are the continuum version of similar theorems in the discrete percolation models. Some proofs here will be obtained by a suitable discretisation and approximation, thus we will be making use of the corresponding results in discrete percolation. The first two sections are devoted to the concepts of ergodicity, coupling and scaling. The techniques of the proofs are hardly needed in the rest of the book, so it is quite possible to read the statements of the main results and move on to the next chapter.

Ergodicity

In this section, we review some results from classical ergodic theory and apply this theory to stationary point processes. The account on ergodic theory will be fairly short; we restrict ourselves to those results which we need in this book. More information about ergodicity and stationary point processes can be found in the book of Daley and Vere-Jones (1988). For a general account on ergodic theory, we refer to the books by Krengel (1985) and Petersen (1983). It will be very convenient here to use a slightly different notation than in the rest of the book in order to clearly see stationary point processes from the viewpoint of measure-preserving transformations (which we introduce in the next paragraph).

In this chapter we discuss the properties of the vacant components in the Poisson Boolean model. Unlike the occupied components in the Poisson Boolean model, where the structure of the occupied regions arises because of placing balls around the Poisson points, the vacancy structure arises in the negative sense, i.e., in the absence of any ball covering a point. This lack of a structure to describe directly the vacancy configuration is a limitation due to which it is often harder to establish results concerning vacancy.

In the study of percolation on discrete graphs, the vacancy configuration is usually thought of as the ‘dual’ of the occupancy structure. In that sense we shall occasionally refer to the vacant region as the dual of the occupied region. This nomenclature is more informal than exact, because in the discrete percolation models, the dual structure has a legitimate construct of its own, rather than being just an appendage of the occupied structure.

We shall define critical densities via vacancy and show that in two dimensions, when the radii are bounded, λ∗H = λ∗T/sub>, where these notations have the same meaning in the vacancy as they had (without the superscript) in the occupancy. In addition, we shall show that in two dimensions, the critical densities arising from the occupancy agree with that arising from vacancy. We shall also establish a uniqueness result for the vacant component as in Section 3.6 of Chapter 3.

This is the first book completely devoted to continuum percolation. The idea to write this book came up after we noticed that even specialists working in the larger area of spatial random processes were unaware about the current state of the art of continuum percolation. Although stochastic geometers have extensively studied the Boolean model, which is one of the most common models of continuum percolation, their focus has been on geometric and statistical aspects rather than on percolation-theoretical issues.

Initially, we planned to write a review article, but it became clear very quickly that it would be impossible to cover even the most basic results in such a review. Also it became apparent that it would be impossible to include in one volume all available results of a subject this size and still expanding. Therefore, we decided on a book which would give attention to all major issues and techniques without necessarily pushing them to the frontier of today's knowledge. When there is more to say on a specific subject than is found here, we provide the appropriate references for further reading.

Continuum percolation models are easily described verbally, but unlike discrete percolation models, their formal mathematical construction is not completely straightforward. In fact, many people (the authors included) have been quite careless with these constructions in the literature. The setup we have chosen in this book is probably the simplest rigorous construction which allows us to use all the ergodic theory we want.

Motivation for continuum models

Many phenomena in physics, chemistry and biology can be modelled by spatial random processes where the randomness is in the geometry of the space rather than in the random behaviour or motion of an object in a deterministic setting. As typical examples of the phenomena we have in mind, consider the spread of a disease in an orchard where the trees are arranged in a grid, and where the disease spreads from an infected tree to its neighbouring trees. In this example, the owner of the orchard is interested in the probability that a particular disease will eventually kill all the trees in the orchard. Another example is the process of the ground getting wet during a period of rain. The randomness here is the place where the raindrops fall on the ground and the size of the wetted region per raindrop. Finally, consider the spread of a disease in a forest. The infection is transmitted from one tree to another, which need not be in the vicinity of the infected tree. This is more likely to happen when the trees are closely spaced than when they are far apart. The collection of infected trees forms a random subset of trees in the forest.

The geometric structure of the first example is discrete, whereas in the next two examples, although the number of raindrops or trees is countable, the position of either is in the continuous space. A rigorous mathematical model to describe the first example is the standard discrete percolation model.