This entire book is, in one way or another, devoted to a single process: the random walk. As we will see, the rules that control the random walk are simple, even when we add elaborations that turn out to have considerable significance. However, as often occurs in mathematics and the physical sciences, the consequences of simple rules are far from elementary. We will also discover that random walks, as interesting as they are in themselves, provide a basis for the understanding of a wide range of phenomena. This is true in part because random walk processes are relevant to so many processes in such a wide range of contexts. It also follows from the fact that the solution of the random walk problem requires the use of so many of the mathematical techniques that have been developed and applied in contemporary twentieth-century physics. We'll start out simply, but it won't be long before we enounter aspects to the problem that invite – indeed require – intense scrutiny.

We begin our investigations by looking at the random walk in its most elementary manifestation. The reader may find that most of what follows in this chapter is familiar material. It is, nevertheless, useful to read through it. For one thing, review is always helpful. More importantly, connections that are hinted at in the early portions of this book will play an important role in later discussion.

The simplest walk

In the simplest example of a random walk the walker is confined to a straight line. This kind of walk is called, appropriately enough, a one-dimensional walk. In this case, steps take the walker in one direction or the other.