Anomalous diffusion on regular and random models for diffusion-limited aggregation

Abstract

Families of regular fractal models for Witten-Sander (DLA) clusters in d dimensions are proposed resembling the computer generated aggregates. The authors evaluate random walk dimensionalities exactly using resistance scaling ideas. The applicability of the conflicting conjectures of Alexander-Orbach (1982) (AO), dw=3df/2, and Aharony-Stauffer (1984) (AS), dw=df+1, to DLA is then examined. One finds that the dendritic nature of the fractals presented ensures that AS is satisfied exactly in all dimensions. An RSRG fugacity transformation method is applied to the simplest of the regular fractals in d=2. The result for dw is compared with the exact resistance scaling result to obtain an estimate of the error introduced by the RSRG. The RG method is then generalised to treat the random DLA problem. The resulting estimate for dw is in fortuitously good agreement with AS, the corresponding Monte Carlo data and the results above for the regular fractal models.