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(±j|0)${\mathrm{〉}}_{\mathit{w}}$=${\mathit{T}}_0$ 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=${\lim }_{\mathit{j}\to \mathrm{\infty }}$ ${\mathit{j}}^2$〈1/${\mathit{T}}_0$ ${\mathrm{〉}}_{\mathit{c}}$, where 〈 ${\mathrm{〉}}_{\mathit{c}}$ stands for the configuration average over the realizations of the random comb. The diffusion constant in the mean-field approximation is given by ${\mathit{K}}_{\mathrm{MF}}$=${\lim }_{\mathit{j}\to \mathrm{\infty }}$ ${\mathit{j}}^2$/〈${\mathit{T}}_0$ ${\mathrm{〉}}_{\mathit{c}}$. We compute ${\mathit{T}}_0$ exactly for an arbitrary realization of the comb and then show rigorously that, owing to the suppression of the relative fluctuations in ${\mathit{T}}_0$ in the ‘‘thermodynamic limit’’ j→∞, we have ${\mathit{K}}_{\mathrm{MF}}$=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 ${\mathit{K}}_{\mathrm{MF}}$, although the transport remains diffusive, as opposed to those in which anomalous diffusion occurs. © 1996 The American Physical Society.