One of the advantages of studying random network models on the infinite plane is that it is possible to observe sharp phase transitions. Informally, a phase transition is defined as a phenomenon by which a small change in the local parameters of a system results in an abrupt change of its global behaviour, which can be observed over an infinite domain. We shall see in subsequent chapters how these phenomena observed on the infinite plane are a useful indication of the behaviour in a finite domain. For now, however, we stick with the analysis on the infinite plane.

The random tree; infinite growth

We start by making a precise statement on the possibility that the branching process introduced in Chapter 1 grows forever. This is trivially true when the offspring distribution is such that P(Xi ≥ 1) = 1, i.e., when each node in the tree has at least one child. However, it is perhaps less trivial that for generic offspring distribution it is still possible to have an infinite growth if and only if E(Xi) = μ > 1.

Theorem 2.1.1When μ ≤ 1 the branching process does not grow forever with probability one, except when P(X = 1) = 1. When μ > 1, the branching process grows forever with positive probability.

The proof of Theorem 2.1.1 uses generating functions, so we start by saying a few words about these. Generating functions are a very convenient tool for all sorts of computations that would be difficult and tedious without them. These computations have to do with sums of random variables, expectations and variances.