A direct renormalisation group approach for the self-avoiding walk

Self-avoiding walk on fractal complex networks: Exactly solvable cases

We study the self-avoiding walk on complex fractal networks called the (u,v)-flower by mapping it to the N-vector model in a generating function formalism. First, we analytically calculate the critical exponent ν and the connective constant by a renormalization-group analysis in arbitrary fractal dimensions. We find that the exponent ν is equal to the displacement exponent, which describes the speed of diffusion in terms of the shortest distance. Second, by obtaining an exact solution for the (u,u)-flower, we provide an example which supports the conjecture that the universality class of the self-avoiding walk on graphs is not determined only by the fractal dimension.