Issue
EPL
Volume 84, Number 3, November 2008
Article Number 30002
Number of page(s) 6
Section General
DOI http://dx.doi.org/10.1209/0295-5075/84/30002
Published online 14 October 2008
EPL, 84 (2008) 30002
DOI: 10.1209/0295-5075/84/30002

Anomalous diffusion on the Hanoi networks

S. Boettcher and B. Gonçalves

Department of Physics, Emory University - Atlanta, GA 30322, USA


received 20 February 2008; accepted in final form 15 September 2008; published November 2008
published online 14 October 2008

Abstract
Diffusion is modeled on the recently proposed Hanoi networks by studying the mean-square displacement of random walks with time, $\langle$ r2$\rangle$ ~ t 2/dw. It is found that diffusion —the quintessential mode of transport throughout Nature— proceeds faster than ordinary, in one case with an exact, anomalous exponent dw=2- log2($\phi$ )=1.30576... . It is an instance of a physical exponent containing the “golden ratio” $\phi$=(1+$\sqrt{5}$)/2 that is intimately related to Fibonacci sequences and since Euclid's time has been found to be fundamental throughout geometry, architecture, art, and Nature itself. It originates from a singular renormalization group fixed point with a subtle boundary layer, for whose resolution $\phi$ is the main protagonist. The origin of this rare singularity is easily understood in terms of the physics of the process. Yet, the connection between network geometry and the emergence of $\phi$ in this context remains elusive. These results provide an accurate test of recently proposed universal scaling forms for first passage times.

PACS
05.40.-a - Fluctuation phenomena, random processes, noise, and Brownian motion.
64.60.aq - Networks.
64.60.ae - Renormalization-group theory.

© EPLA 2008