Self-avoiding walks on random fractal environments

Self-avoiding random walks (SAWs) are studied on several hierarchical lattices in a randomly disordered environment. An analytical method to determine whether their fractal dimensionD_saw is affected by disorder is introduced. Using this method, it is found that for some lattices,D_saw is unaffected by weak disorder; while for othersD_saw changes even for infinitestimal disorder. A weak disorder exponent is defined and calculated analytically [ measures the dependence of the variance in the partition function (or in the effective fugacity per step)vL on the end-to-end distance of the SAW,L]. For lattices which are stable against weak disorder (<0) a phase transition exists at a critical valuev=v^* which separates weak- and strong-disorder phases. The geometrical properties which contribute to the value of are discussed.