The Renormalized Tree Approximation: a Simple Illustration of the Theory

Abstract

In this chapter we illustrate the power of the theory with a discussion of the tree approximation, which yields most simple expressions for the scaling functions. We thus can study the effect of renormalization and RG flow without struggling with the technical complexities of higher loop orders. We will see the results of scaling theory and blob model emerging naturally from our form of the renormalized theory, and we will encounter the characteristic features of crossover scaling functions. Clearly we should not expect that the tree approximation is very accurate quantitatively — though it performs not too bad. (In [BL86] a set of critical ratios was evaluated in renormalized tree approximation, with reasonable success.) However, almost all the qualitative aspects of the theory can be studied here.