Random Walks on a Fractal Solid

Abstract

It is established that the trapping of a random walker undergoing unbiased, nearest-neighbor displacements on a triangular lattice of Euclidean dimension d=2 is more efficient (i.e., the mean walklength 〈n〉 before trapping of the random walker is shorter) than on a fractal set, the Sierpinski tower, which has a Hausdorff dimension D exactly equal to the Euclidean dimension of the regular lattice. We also explore whether the self similarity in the geometrical structure of the Sierpinski lattice translates into a “self similarity” in diffusional flows, and find that expressions for 〈n〉 having a common analytic form can be obtained for sites that are the first- and second-nearest-neighbors to a vertex trap.