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, [Formula Presented](r,t), is analytically studied for the case ξ≡r/[Formula Presented]≪1. It is shown to obey the scaling form [Formula Presented](r,t)=ρ(r)[Formula Presented][Formula Presented][Formula Presented](ξ), where ρ(r)∼[Formula Presented] is the density of the chain. Expanding [Formula Presented](ξ) in powers of ξ, we find that there exists an infinite hierarchy of critical dimensions, [Formula Presented]=2,6,10,..., each one characterized by a logarithmic correction in [Formula Presented](ξ). Namely, for d=2, [Formula Presented](ξ)≃[Formula Presented][Formula Presented]lnξ+[Formula Presented][Formula Presented]; for 3⩽d⩽5, [Formula Presented](ξ)≃[Formula Presented][Formula Presented]+[Formula Presented][Formula Presented]; for d=6, [Formula Presented](ξ)≃[Formula Presented][Formula Presented]+[Formula Presented][Formula Presented]lnξ; for 7⩽d⩽9, [Formula Presented](ξ)≃[Formula Presented][Formula Presented]+[Formula Presented][Formula Presented]+[Formula Presented][Formula Presented]; for d=10, [Formula Presented](ξ)≃[Formula Presented][Formula Presented]+[Formula Presented][Formula Presented]+[Formula Presented][Formula Presented]lnj, etc. In particular, for d=2, this implies that the temporal dependence of the probability density of being close to the origin [Formula Presented](r,t)≡[Formula Presented](r,t)/ρ(r)≃[Formula Presented]lnt.