Superlocalization, Anomalous Diffusion and Self Avoiding Walks on Fractals

Abstract

Many physical systems exhibit fractal geometry, at least over some range of length scales.¹ Much of the recent experimental and theoretical interest in such systems concentrated on alloys near the percolation threshold, P_c.^2,3 At p_c, such alloys are self-similar, and their properties exhibit an anomalous power law dependence on their linear size. Diferent physical properties turn out to depend on different subsets of sites (or bonds) on the percolating cluster, and hence require knowledge of a plenitude of fractal dimensionalities.⁴ For example, the average total mass of the spanning cluster between two points a distance L apart scales as M∼L^D, and the fractal dimensionality D equals 91/48,∼2.5,∼3.2 and 4 for percolation clusters in dimensions d=2,3,4 and ≥ 6. The mass of the corresponding backbone (without the “dangling” bonds) has fractal dimensionalities D_B∼1.62, 1.83, 1.94 and 2, and that of the minimal path ⁵ (along bonds on the backbone) has d_min ≃ 1.13, 1.36, 1.62 and 2. Similar fractal properties apply to other physical structures, e.g. lattice animals,⁶ where d_min=D_B ≃ 1.19, 1.33, 1.47 and 2 in d= 2,3,4 and ≥ 8.