Anomalous diffusion on fractal lattices with site disorder

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Abstract
Like random walks on Euclidean lattices, random walks on fractal lattices are modified by a waiting time (site) disorder when the first moment of the waiting time distribution diverges. It is shown that for lattices which support recurrent walks (spectral dimension smaller than two) the inverse of the diffusion exponent in the presence of disorder is increased by the difference between the waiting time fractal dimension and the usual fractal dimension. This hyperscaling relation is derived for Sierpinski gaskets in arbitrary dimension with scale-dependent waiting times. This provides qualitative insight into this problem and the exponent relation derived should also hold for statistically self-similar structure such as percolation clusters. For lattices whose usual spectral dimension is larger than two, a mean-field result holds.

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