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, $P_d(r,t)$, is analytically studied for the case $ξ\equiv r/t^{1/4}\ll1$. It is shown to obey the scaling form $P_d(r,t)=ρ(r) t^{-1/2} ξ^{-2} f_d(ξ)$, where $ρ(r)\sim r^{2-d}$ is the density of the chain. Expanding $f_d(ξ)$ in powers of $ξ$, we find that there exists an infinite hierarchy of critical dimensions, $d_c=2,6,10,\ldots$, each one characterized by a logarithmic correction in $f_d(ξ)$. Namely, for $d=2$, $f_2(ξ)\simeq a_2ξ^2\lnξ+b_2ξ^2$; for $3\le d\le 5$, $f_d(ξ)\simeq a_dξ^2+b_dξ^d$; for $d=6$, $f_6(ξ)\simeq a_6ξ^2+b_6ξ^6\lnξ$; for $7\le d\le 9$, $f_d(ξ)\simeq a_dξ^2+b_dξ^6+c_dξ^d$; for $d=10$, $f_{10}(ξ)\simeq a_{10}ξ^2+b_{10}ξ^6+c_{10}ξ^{10}\lnξ$, {\it etc.\/} In particular, for $d=2$, this implies that the temporal dependence of the probability density of being close to the origin $Q_2(r,t)\equiv P_2(r,t)/ρ(r)\simeq t^{-1/2}\ln t$.