Mean field theory and spin wave contributions

Now that we have established the link between self-avoiding random walks and the O(n) model of magnetism, it is appropriate to look again at the magnetic system in the limit n → 0. Of special interest to us is its behavior in the immediate vicinity of the critical point. We will make extensive use of the insights provided by the study of critical phenomena that have emerged over the past three decades. Our initial approach to the problem of the statistical mechanics of the O(n) model will be to utilize the mean field ideas developed by Landau and others. Mean field theory will then be enhanced by the introduction of fluctuations which will be analyzed in a low order spin wave theory. The insights gained by this approach will lead us to a clearer picture of the phase transition as it pertains to the random walk problem. Finally, a full renormalization group calculation will be used to elucidate the scaling properties of this model. This final step in the analysis will be accomplished in Chapter 12.

Outline of the chapter

In this chapter we make use of the connection between the O(n) model and the self-avoiding walk to discuss aspects of the statistics of such a walk. We begin by reviewing the relationship between the spin model described by the O(n) energy function and the self-avoiding walk. Our initial focus will be on self-avoiding walks that are confined to a finite volume of space.