Real-space renormalization of the self-avoiding walk by a linear transformation

Abstract

We show how the Niemeijer-Van Leeuwen real-space renormalization can be used to study the self-avoiding walk on a lattice. We first establish that the de Gennes-des Cloizeaux equivalence between this problem and an $n$-component spin system (as $n\to 0$) remains valid if one takes discrete-valued cubically symmetric spins. A transformation for the self-avoiding walk is then obtained by letting $n$ tend to zero in the transformation for the spin system. In the $n\to 0$ limit the interaction constants of the spin system are shown to correspond to the weights of the elementary segments of a self-avoiding walk. A general technique for practical renormalization calculations in the $n\to 0$ limit is given. As an example we consider a linear transformation depending upon two parameters, and apply it to a triangular lattice in second-order cumulant approximation. The results agree well with data from other sources. Our findings concerning the parameter dependence of linear renormalization transformations confirm and extend those of Bell and Wilson based on the Gaussian model.