Validity of the mean-field approximation for diffusion on a random comb

We consider unbiased diffusion on a random comb structure (an infinitely long backbone with loopless branches of arbitrary length emanating from it). If 〈t(±j0)[Formula Presented]=[Formula Presented] is the mean time (averaged over all random walks) for first passage from an arbitrary origin 0 on the backbone to either of the sites +j or -j on it in a given realization of the structure, the exact diffusion constant for the problem is defined as K=[Formula Presented][Formula Presented]〈1/[Formula Presented][Formula Presented], where 〈 [Formula Presented] stands for the configuration average over the realizations of the random comb. The diffusion constant in the mean-field approximation is given by [Formula Presented]=[Formula Presented][Formula Presented]/〈[Formula Presented][Formula Presented]. We compute [Formula Presented] exactly for an arbitrary realization of the comb and then show rigorously that, owing to the suppression of the relative fluctuations in [Formula Presented] in the “thermodynamic limit” j→∞, we have [Formula Presented]=K whenever the moments of certain random variables Γ(L,α,β) are finite; here the site-dependent random variables L, α, and β are, respectively, the branch length, stay probability at the tip of a branch, and the backbone-to-branch jump probability. Finally, we discuss different situations in which K will not be equal to [Formula Presented], although the transport remains diffusive, as opposed to those in which anomalous diffusion occurs. © 1996 The American Physical Society.