Self-avoiding walks on random fractal environments
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Self-avoiding random walks (SAWs) are studied on several hierarchical lattices in a randomly disordered environment. An analytical method to determine whether their fractal dimensionDsaw is affected by disorder is introduced. Using this method, it is found that for some lattices,Dsaw is unaffected by weak disorder; while for othersDsaw 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∼Lλ 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.
Key WordsSelf-avoiding walks disordered environment hierarchical lattices fractals renormalization
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