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)∼$L^{\mathrm{-}(1+\gamma )}$ 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 $d_w$=4/(1+γ) for γ<1 and $d_w$=2 for γ≥1. The probability of being at the origin at time t is $P_0$(t)∼$t^{\mathrm{-}{d}_s/2\mathrm{}}$ for large t with $d_s$=(3-γ)/2 for γ<1 and $d_s$=1 for γ>1. When a field is applied along the backbone of the comb the diffusion exponent is $d_w$=2/(1+γ) for γ<1 and $d_w$=1 for γ≥1. The theoretical results are confirmed using the exact enumeration method.