Critical dimensions for random walks on random-walk chains

The probability distribution of random walks on linear structures generated by random walks in d-dimensional space, ${\mathit{P}}_{\mathit{d}}$(r,t), is analytically studied for the case ξ≡r/${\mathit{t}}^{1/4}$≪1. It is shown to obey the scaling form ${\mathit{P}}_{\mathit{d}}$(r,t)=ρ(r)${\mathit{t}}^{\mathrm{-}1/2}$ ${\xi }^{\mathrm{-}2}$ ${\mathit{f}}_{\mathit{d}}$(ξ), where ρ(r)∼${\mathit{r}}^{2\mathrm{-}\mathit{d}}$ is the density of the chain. Expanding ${\mathit{f}}_{\mathit{d}}$(ξ) in powers of ξ, we find that there exists an infinite hierarchy of critical dimensions, ${\mathit{d}}_{\mathit{c}}$=2,6,10,..., each one characterized by a logarithmic correction in ${\mathit{f}}_{\mathit{d}}$(ξ). Namely, for d=2, ${\mathit{f}}_2$(ξ)≃${\mathit{a}}_2$ ${\xi }^2$lnξ+${\mathit{b}}_2$ ${\xi }^2$; for 3⩽d⩽5, ${\mathit{f}}_{\mathit{d}}$(ξ)≃${\mathit{a}}_{\mathit{d}}$ ${\xi }^2$+${\mathit{b}}_{\mathit{d}}$ ${\xi }^{\mathit{d}}$; for d=6, ${\mathit{f}}_6$(ξ)≃${\mathit{a}}_6$ ${\xi }^2$+${\mathit{b}}_6$ ${\xi }^6$lnξ; for 7⩽d⩽9, ${\mathit{f}}_{\mathit{d}}$(ξ)≃${\mathit{a}}_{\mathit{d}}$ ${\xi }^2$+${\mathit{b}}_{\mathit{d}}$ ${\xi }^6$+${\mathit{c}}_{\mathit{d}}$ ${\xi }^{\mathit{d}}$; for d=10, ${\mathit{f}}_{10}$(ξ)≃${\mathit{a}}_{10}$ ${\xi }^2$+${\mathit{b}}_{10}$ ${\xi }^6$+${\mathit{c}}_{10}$ ${\xi }^{10}$lnj, etc. In particular, for d=2, this implies that the temporal dependence of the probability density of being close to the origin ${\mathit{Q}}_2$(r,t)≡${\mathit{P}}_2$(r,t)/ρ(r)≃${\mathit{t}}^{\mathrm{-}1/2}$lnt. © 1996 The American Physical Society.