8 - Properties of the random walk: introduction to scaling

Summary

We now have had an introduction to the random walk, and there has been a discussion of some of the most useful methods that can be utilized in the analysis of the process. The unifying theme of this chapter is the introduction of scaling arguments to provide insights into the behavior of the walker under a variety of circumstances. First, we will address in more depth the notion of universality of ordinary random walk statistics. Then, we will discuss the (mathematical) sources of non-Gaussian statistics. Finally, we will develop a few simple but central scaling results by looking once more at the path integral formulation of the random walk. In particular, using simple scaling arguments, we will be able to provide heuristic arguments leading to Flory's formula for the influence of self-avoidance on the spatial extent of a random walker's path.

Universality

Notions of universality play an important role in discussions of the statistics of the random walk.We have already seen a version of universality in Chapter 2, in which it was shown that the statistics of a long walk are Gaussian, regardless of details of the rules governing the walk, as long as the individual steps taken by the walker do not carry it too far. In the remainder of this section, the idea of universality will be developed in a way that will allow us to apply it to the random walk when conditions leading to Gaussian behavior do not apply.

As it turns out, universality in random walk statistics has a close mathematical and conceptual connection to fundamental properties of a system undergoing a special type of phase transition.