Self-avoiding walks on random fractal environments

Abstract

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 λ is defined and calculated analytically [λ measures the dependence of the variance in the partition function (or in the effective fugacity per step)v∼L ^λ on the end-to-end distance of the SAW,L]. For lattices which are stable against weak disorder (λ<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 λ are discussed.