Anomalous diffusion on the Hanoi networks

Abstract

Abstract. -Diffusion is modeled on the recently proposed Hanoi networks by studying the meansquare displacement of random walks with time,˙r 2¸∼ t 2/dw . It is found that diffusionthe quintessential mode of transport throughout Nature -proceeds faster than ordinary, in one case with an exact, anomalous exponent dw = 2 − log 2 (φ) = 1.30576 . . .. It is an instance of a physical exponent containing the "golden ratio" φ =`1 + √ 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 φ 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 φ in this context remains elusive. These results provide an accurate test of recently proposed universal scaling forms for first passage times.

Renormalization Group for Critical Phenomena in Complex Networks

“…where $\varphi =\left(\sqrt{5}+1\right)∕2=1.6180339887\dots$ is the “golden ratio” (Boettcher and Goncalves, 2008). For bare couplings at this temperature, marked by a blue dot in the ( y  = 1)-plot of Figure 5, the RG flow marginally reaches the strong-coupling limit.…”

Applying Renormalization Group to Quantum Walks

“…Our walks are governed by the discrete-time "master" equation [3] for the state of the system, is known for DSG [1] and HN3 [13], or easily derived for MK [9]. The values for d QW w were first conjectured [8,9], then analytically verified (except HN3) [10] 0 (z) (in blue, with • for k = 2 and × for k = 3) and for the hopping parameters a k (z) (in red, with for k = 2 and + for k = 3) for quantum walk on the 1d-line and DSG with a Grover coin.…”

Renormalization of the unitary evolution equation for coined quantum walks

“…However, anomalous diffusion with d w = 2 may arise in many of those transport processes [18,[25][26][27]. This scaling affects many important observables, for instance, the mean-square displacement, x 2 t ∼ t 2/dw , or first-passage times [19,28,29].…”