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

Abstract

We consider unbiased diffusion on a random comb structure (an infinitely long backbone with loopless branches of arbitrary length emanating from it). If 〈t(\ifmmode\pm\else\textpm\fi{}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=${\mathrm{lim}}_{\mathit{j}\ensuremath{\rightarrow}\mathrm{\ensuremath{\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 ${\mathit{K}}_{\mathrm{MF}}$=${\mathrm{lim}}_{\mathit{j}\ensuremath{\rightarrow}\mathrm{\ensuremath{\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\ensuremath{\rightarrow}\ensuremath{\infty}, we have ${\mathit{K}}_{\mathrm{MF}}$=K whenever the moments of certain random variables \ensuremath{\Gamma}(L,\ensuremath{\alpha},\ensuremath{\beta}) are finite; here the site-dependent random variables L, \ensuremath{\alpha}, and \ensuremath{\beta} 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. \textcopyright{} 1996 The American Physical Society.