Diffusion on complex cellular automata patterns

We studied random walks on two-dimensional patterns formed by the sequence of configurations of complex elementary cellular automata (CA) with random initial configurations. The walkers are allowed to jump between nearest neighbours or next nearest neighbours 1 sites. On patterns of rules 22, 54, 90, 122, 126, 150 and 182, the diffusion is normal (ν=1/2). On patterns 18 and 146 the diffusion is anomalous, with ν=0.415±0.005, and the spectral dimension is D_s=1.25±0.1. From the analysis of the diffusion in the horizontal and vertical directions (space and time directions of the CA problem, respectively) of patterns 18 and 146, we obtained ν_x≈0.22 and ν_y≈0.42, because they have branches of 1 sites which are long in the vertical direction but narrow in the horizontal direction. Due to the anisotropic diffusion, the results do not satisfy the relation D_s=2D_F/D_w, with D_w=1/ν and fractal dimension D_F=2. Considering that the fractal dimension of the region visited by the walker is equal to the dimension of the substrate (D_F), we suggest the new scaling relation D_s=2(ν_x+ν_y). This relation is supported by our numerical results and may be generalized to other structures with anisotropic diffusion.