Spatial Log Periodic Oscillations of First-Passage Observables in   Fractals
  For transport processes in geometrically restricted domains, the mean first-passage time (MFPT) admits a general scaling dependence on space parameters for diffusion, anomalous diffusion, and diffusion in disordered or fractal media. For transport in self-similar fractal structures, we obtain a new expression for the source-target distance dependence of the MFPT that exhibits both the leading power law behavior, depending on the Hausdorff and spectral dimension of the fractal, as well as small log periodic oscillations that are a clear and definitive signal of the underlying fractal structure. We also present refined numerical results for the Sierpinski gasket that confirm this oscillatory behavior. 
Eric Akkermans, Olivier Benichou, Gerald Dunne, Alexander Teplyaev and   Raphael Voituriez
cond-mat.stat-mech cond-mat.dis-nn