Single random walker on disordered lattices

Abstract

Random walks on square lattice percolating clusters were followed for up to 2×105 steps. The mean number of distinct sites visited 〈 (S N ⊃> gives a spectral dimension of d s = 1.30±0.03 consistent with superuniversality ( d s =4J3) but closer to the alternative d s = 182/139, based on the low dimensionality correction. Simulations are also given for walkers on an energetically disordered lattice, with a jump probability that depends on the local energy mismatch and the temperature. An apparent fractal behavior is observed for a low enough reduced temperature. Above this temperature, the walker exhibits a “crossover” from fractal-to-Euclidean behavior. Walks on two- and three-dimensional lattices are similar, except that those in three dimensions are more efficient.