Fractal geometry and anomalous diffusion in the backbone of percolation clusters

The backbone of the infinite percolation cluster is shown by Monte Carlo studies to be a fractal object, on short length scales. Its measured fractal dimensionality d B , at two dimensions, is found: d B =1.68+or-0.02. Numerical calculations for the anomalous diffusion on the backbone are also performed on the same samples. The value of the spectral dimensionality d B , extracted from the power law behaviour of the average number of distinct visited sites, is obtained: d B /2=0.625+or-0.005. Both results show clearly fractal-to-Euclidean cross-over above the percolation threshold. A general relation between fractal and spectral dimensionalities pertaining to the infinite percolation cluster and to the backbone is also given and checked numerically.