Anomalous Diffusion on the Hanoi Networks

Diffusion is modeled on the recently proposed Hanoi networks by studying the mean- square displacement of random walks with time, <r^2>~t^{2/d_w}. 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-log_2(\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.