Anomalous diffusion on a random comblike structure

We have recently studied a random walk on a comblike structure as an analog of diffusion on a fractal structure. In our earlier work, the comb was assumed to have a deterministic structure, the comb having teeth of infinite length. In the present paper we study diffusion on a one-dimensional random comb, the length of whose teeth are random variables with an asymptotic stable law distribution {φ}(L){~}L^-(1+γ) where 0<{γ}{≤}1. Two mean-field methods are used for the analysis, one based on the continuous-time random walk, and the second a self-consistent scaling theory. Both lead to the same conclusions. We find that the diffusion exponent characterizing the mean-square displacement along the backbone of the comb is dw=4/(1+{γ}) for {γ}<1 and dw=2 for {γ}{≥}1. The probability of being at the origin at time t is P₀(t){~}t^-dₛ/2 for large t with dₛ=(3-{γ})/2 for {γ}<1 and dₛ=1 for {γ}>1. When a field is applied along the backbone of the comb the diffusion exponent is dw=2/(1+{γ}) for {γ}<1 and dw=1 for {γ}{≥}1. The theoretical results are confirmed using the exact enumeration method.