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, Pd(r, t), is analytically studied for the case ξ ≡ r/t1/4 ≪ 1. It is shown to obey the scaling form Pd(r, t) = ρ(r)tξfd(ξ), where ρ(r) ∼ r 2−d is the density of the chain. Expanding fd(ξ) in powers of ξ, we find that there exists an infinite hierarchy of critical dimensions, dc = 2, 6, 10, . . ., each one characterized by a logarithmic correction in fd(ξ). Namely, for d = 2, f2(ξ) ≃ a2ξ 2 ln ξ + b2ξ 2; for 3 ≤ d ≤ 5, fd(ξ) ≃ adξ 2 + bdξ d; for d = 6, f6(ξ) ≃ a6ξ 2 + b6ξ 6 ln ξ; for 7 ≤ d ≤ 9, fd(ξ) ≃ adξ 2 + bdξ 6 + cdξ d; for d = 10, f10(ξ) ≃ a10ξ 2 + b10ξ 6 + c10ξ 10 ln ξ, etc. In particular, for d = 2, this implies that the temporal dependence of the probability density of being close to the origin Q2(r, t) ≡ P2(r, t)/ρ(r) ≃ t −1/2 ln t. Pacs: 5.40.+j, 05.60.+w, 66.30.-h