Some properties of a random walk on a comb structure

We analyze transport properties of a random walk on a comb structure, which serves as a model for a random walk on the backbone of a percolation cluster. It is shown that the random walk along the x axis, which is the analog of the backbone, exhibits anomalous diffusion in that 〈x²(n)〉 ∼ n ^{1 2}, and the expected number of x sites visited is proportional to n ^{1 4} for large n. The distribution function is found to be a two-dimensional Gaussian. If a field in the x direction, so that diffusion is asymmetric, the expected displacement is found to be asymptotically proportional to n ^{1 2}.