Percolation, fractals, and anomalous diffusion

Both the infinite cluster and its backbone are self-similar at the percolation threshold, p_c. This self-similarity also holds at concentrations p near p_c, for length scales L which are smaller than the percolation connectedness length, ξ. For L<ξ, the number of bonds on the infinite cluster scales as L^D, where the fractal dimensionality D is equal to (d-β/v). Geometrical fractal models, which imitate the backbone and on which physical models are exactly solvable, are presented. Above six dimensions, one has D=4 and an additional scaling length must be included. The effects of the geometrical structure of the backbone on magnetic spin correlations and on diffusion at percolation are also discussed.