We are almost ready to fully exploit the connection, established in earlier chapters, between the statistics of a self-avoiding random walk and the statistical mechanics of a magnet near the phase transition from its paramagnetic and ferromagnetic states. Because of the mathematical similarity between the two systems, we will be able to make use of an array of calculational strategies that, collectively, represent realizations of the renormalization group. This generic method for the study of systems with long-range correlations has fundamentally altered the way in which physicists view the world around them. The method is so powerful and so widespread in its application, that it seems worthwhile to do a little more than simply explain how to use it in the present context. This chapter consists of a discussion of the philosophy underlying the renormalization group and of a general description of the way in which it is applied. We will finish off by taking the reader through a simple calculation that is relevant to random walks and the associated magnetic system. Then, we will generalize the method to encompass a wide class of systems, the O(n) model being one of them. In the next chaper, the reader will be subjected to a full-blown introduction to the method, as it applies to the self-avoiding walk. Those already familiar with the renormalization group may wish to skip directly to Chapter 12.

Scale invariance in mathematics and nature

The notion of scale invariance is not exactly new. A famous poem by Jonathan Swift goes as follows: