Year: 1996
Type: article
Abstract: The probability distribution of random walks on linear structures generated by random walks in d-dimensional space, ${\mathit{P}}_{\mathit{d}}$(r,t), is analytically studied for the case \ensuremath{\xi}\ensuremath{\equiv}r/${\mathit{t}}^{1/4}$\ensuremath{\ll}1. It is shown to obey the scaling form ${\mathit{P}}_{\mathit{d}}$(r,t)=\ensuremath{\rho}(r)${\mathit{t}}^{\mathrm{\ensuremath{-}}1/2}$${\ensuremath{\xi}}^{\mathrm{\ensuremath{-}}2}$${\mathit{f}}_{\mathit{d}}$(\ensuremath{\xi}), where \ensuremath{\rho}(r)\ensuremath{\sim}${\mathit{r}}^{2\mathrm{\ensuremath{-}}\mathit{d}}$ is the density of the chain. Expanding ${\mathit{f}}_{\mathit{d}}$(\ensuremath{\xi}) in powers of \ensuremath{\xi}, we find that there exists an infinite hierarchy of critical dimensions, ${\mathit{d}}_{\mathit{c}}$=2,6,10,..., each one characterized by a logarithmic correction in ${\mathit{f}}_{\mathit{d}}$(\ensuremath{\xi}). Namely, for d=2, ${\mathit{f}}_{2}$(\ensuremath{\xi})\ensuremath{\simeq}${\mathit{a}}_{2}$${\ensuremath{\xi}}^{2}$ln\ensuremath{\xi}+${\mathit{b}}_{2}$${\ensuremath{\xi}}^{2}$; for 3\ensuremath{\leqslant}d\ensuremath{\leqslant}5, ${\mathit{f}}_{\mathit{d}}$(\ensuremath{\xi})\ensuremath{\simeq}${\mathit{a}}_{\mathit{d}}$${\ensuremath{\xi}}^{2}$+${\mathit{b}}_{\mathit{d}}$${\ensuremath{\xi}}^{\mathit{d}}$; for d=6, ${\mathit{f}}_{6}$(\ensuremath{\xi})\ensuremath{\simeq}${\mathit{a}}_{6}$${\ensuremath{\xi}}^{2}$+${\mathit{b}}_{6}$${\ensuremath{\xi}}^{6}$ln\ensuremath{\xi}; for 7\ensuremath{\leqslant}d\ensuremath{\leqslant}9, ${\mathit{f}}_{\mathit{d}}$(\ensuremath{\xi})\ensuremath{\simeq}${\mathit{a}}_{\mathit{d}}$${\ensuremath{\xi}}^{2}$+${\mathit{b}}_{\mathit{d}}$${\ensuremath{\xi}}^{6}$+${\mathit{c}}_{\mathit{d}}$${\ensuremath{\xi}}^{\mathit{d}}$; for d=10, ${\mathit{f}}_{10}$(\ensuremath{\xi})\ensuremath{\simeq}${\mathit{a}}_{10}$${\ensuremath{\xi}}^{2}$+${\mathit{b}}_{10}$${\ensuremath{\xi}}^{6}$+${\mathit{c}}_{10}$${\ensuremath{\xi}}^{10}$lnj, etc. In particular, for d=2, this implies that the temporal dependence of the probability density of being close to the origin ${\mathit{Q}}_{2}$(r,t)\ensuremath{\equiv}${\mathit{P}}_{2}$(r,t)/\ensuremath{\rho}(r)\ensuremath{\simeq}${\mathit{t}}^{\mathrm{\ensuremath{-}}1/2}$lnt. \textcopyright{} 1996 The American Physical Society.
Institutions: Bar-Ilan University, Justus-Liebig-Universität Gießen
Language: English
Subfield: Condensed Matter Physics
Field: Physics and Astronomy
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W2073087690