Anomalous Diffusion on the Hanoi Networks

Abstract

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(ϕ) = 1.30576 . . .. It is an instance of a physical exponent containing the "golden ratio" ϕ=(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 ϕis 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 ϕin this context remains elusive. These results provide an accurate test of recently proposed universal scaling forms for first passage times.

6 pages, 6 figures incl.; for related work, see http://www.physics.emory.edu/faculty/boettcher/, some new material, as to appear in Europhysics Letters

References in corpus (4)

• First-passage times in complex scale-invariant media
• Hierarchical, Regular Small-World Networks
• Ultrametricity and Memory in a Solvable Model of Self-Organized Criticality
• Geometry and Dynamics for Hierarchical Regular Networks

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• Spatial Search Algorithms on Hanoi Networks
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• Renormalization of the Unitary Evolution Equation for Coined Quantum Walks
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• Evaluation of Spectral Zeta-Functions with the Renormalization Group
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• Optimal Vertex Cover for the Small-World Hanoi Networks
• Renormalization Group Solution of the Chutes&Ladder Model
• Modified Hanoi Towers Groups and Limit Spaces