  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 $\xi\equiv r/t^{1/4}\ll1$. It is shown to obey the scaling form
$P_d(r,t)=\rho(r) t^{-1/2} \xi^{-2} f_d(\xi)$, where $\rho(r)\sim r^{2-d}$ is
the density of the chain. Expanding $f_d(\xi)$ in powers of $\xi$, 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 $f_d(\xi)$. Namely, for
$d=2$, $f_2(\xi)\simeq a_2\xi^2\ln\xi+b_2\xi^2$; for $3\le d\le 5$,
$f_d(\xi)\simeq a_d\xi^2+b_d\xi^d$; for $d=6$, $f_6(\xi)\simeq
a_6\xi^2+b_6\xi^6\ln\xi$; for $7\le d\le 9$, $f_d(\xi)\simeq
a_d\xi^2+b_d\xi^6+c_d\xi^d$; for $d=10$, $f_{10}(\xi)\simeq
a_{10}\xi^2+b_{10}\xi^6+c_{10}\xi^{10}\ln\xi$, {\it etc.\/} In particular, for
$d=2$, this implies that the temporal dependence of the probability density of
being close to the origin $Q_2(r,t)\equiv P_2(r,t)/\rho(r)\simeq t^{-1/2}\ln
t$.
