Nonlinear Aspects Of The Renormalization Group

We consider the problem of calculating the nonuniversal parameters entering into the scaling laws of renormalized quantities. We examine the renormalization group transformation of Dyson's hierarchical model and a simplified version of the model. We introduce the hierarchical model and explain the use of approximations which aid in extracting numerical information about the susceptibility. We then use these approximations to relate the hierarchical model to a simplified, one- parameter model. In the simplified model, we can express the magnetic susceptibility in terms of dual expansions, one about the high-temperature fixed point, and the other about the critical point. With the introduction of two dual scaling variables which transform covariantly under the renormalization group transformation, we are able to calculate all of the nonuniversal quantities. Extensions of these techniques to multi-parameter models, such as the hierarchical model, are discussed