Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: 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 $\xi\equiv r/t^{1/4}\ll1$. 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 $Q_2(r,t)\equiv P_2(r,t)/\rho(r)\simeq t^{-1/2}\ln
t$.

23
The categories of this paper are: solv-int
20
The categories of this paper are: acc-phys
36
The categories of this paper are: cmp-lg
29
The categories of this paper are: funct-an
8
The categories of this paper are: hep-ex
7
The categories of this paper are: hep-th
2
The categories of this paper are: math
18
The categories of this paper are: econ
27
The categories of this paper are: comp-gas
33
The categories of this paper are: plasm-ph
15
The categories of this paper are: nucl-ex
25
The categories of this paper are: patt-sol
17
The categories of this paper are: q-fin
37
The categories of this paper are: supr-con
9
The categories of this paper are: nlin
6
The categories of this paper are: astro-ph