arXiv subjects

Armin Bunde

Publications and source records attributed to Armin Bunde.

39 records · Page 3Linked to original sources

arXiv1999

Distribution of Dangling Ends on the Incipient Percolation Cluster

We study numerically and by scaling arguments the probability P(M)dM that a given dangling end of the incipient percolation cluster has a mass between M and M + dM. We find by scaling arguments that P(M) decays with a power law, P(M)~M^(-(1+k)), with an exponent k=dBf/df, where df and dBf are the fractal dimensions of the cluster and its backbone, respectively. Our numerical results yield k=0.83 in d=2 and k=0.74 in d=3 in very good agreement with theory.

cond-mat.stat-mech↗

arXiv1998

Extended Fractons and Localized Phonons on Percolation Clusters

We investigate the localization behavior of vibrational modes of infinite percolation clusters above the critical concentration in two and three dimensions using the method of level statistics. We find that all eigenstates are localized in d=2, including the low frequency phonon states. In d=3 we obtain evidence for a localization-delocalization transition. But contrary to the common view this transition occurs for frequencies above the phonon fracton crossover giving rise to a new regime of extended fracton states.

cond-mat.dis-nn↗

arXiv1996

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 $Q_2(r,t)\equiv P_2(r,t)/ρ(r)\simeq t^{-1/2}\ln t$.

cond-mat↗