Random walks on fractal structures and percolation clusters

The notion of spectral dimensionality of a self-similar (fractal) structure is recalled, and its value for the family of Sierpinski gaskets derived via a scaling argument. Various random walk properties such as the probability of closed walks and the mean number of visited sites are shown to be governed by this spectral dimension. It is suggested that the number SN of distinct sites visited during an N-step random walk on an infinite cluster at percolation threshold varies asymptotically as : SN ∼ N2/3, in any dimension.