The modification of random walk statistics resulting from the imposition of constraints on the walkers will be repeatedly visited in this book. In fact, several chapters will be dedicated to the treatment of walkers that are forbidden to step on points they have previously visited. This kind of constraint represents the influence of a walk on itself. As we will see, calculations of the properties of the self-avoiding random walk require the use of fairly sophisticated techniques. The payoff is the ability to extract the fundamental properties of a model that accurately represents the asymptotic statistics of important physical systems, particularly long linear polymers.

The discussion referred to above will take place in later chapters. Here, we focus on other constraints, which embody the influence of a static environment on a walker. In the first part of this chapter, we will address the effect on a random walker of the requirement that it remain in a specified region of space. Specifically, we will look at walkers confined to a half-space, a linear segment in one dimension and a spherical volume in three dimensions. Then we will look at the case of walkers that are confined to the region outside a spherical volume. This case turns out to be relevant to issues relating to the intake of nutrients and the excretion of wastes by a simple, stationary organism, such as a cell. In fact, the final section of this chapter utilizes random walk statistics to investigate the limits on the size of a cell that survives by ingesting nutrients that diffuse to it from its surroundings.