Noteworthy fractal features and transport properties of Cantor tartans

This Letter is focused on the impact of fractal topology on the transport processes governed by different kinds of random walks on Cantor tartans. We establish that the spectral dimension of the infinitely ramified Cantor tartan d(s) is equal to its fractal (self-similarity) dimension D. Consequently, the random walk on the Cantor tartan leads to a normal diffusion. On the other hand, the fractal geometry of Cantor tartans allows for a natural definition of power-law distributions of the waiting times and step lengths of random walkers. These distributions are Levy stable if D > 1.5. Accordingly, we found that the random walk with rests leads to sub-diffusion, whereas the Levy walk leads to ballistic diffusion. The Levy walk with rests leads to super-diffusion, if D > or sub-diffusion, if 1.5 < D < root 3. (C) 2018 Elsevier B.V. All rights reserved.