SELF-AVOIDING RANDOM-WALKS ON A FAMILY OF DIAMOND-TYPE HIERARCHICAL LATTICES

We have used the exact renormalization-group method proposed by Dhar to study self-avoiding random walks (SAW's) on a family of hierarchical lattices. The generator of the lattices is made of l branches and each branch has m bonds. We expect that since the lattices are infinitely ramified, the critical exponents of SAW's should be different from that on finitely ramified lattices and belong to a new universal class. We calculated the critical exponents alpha, nu, and gamma under the condition l < m, and, with D(f) the fractal dimension, obtained the scaling law D(f) nu = 2-alpha, which agrees with other authors's conclusions. When l greater-than-or-equal-to m, we cannot work out the problem, and some discussion is given.