A hierarchical model for scaling structure in generalised diffusion-limited aggregations

Abstract

Deterministic fractal models are presented to be exactly solvable for the growth probability distribution on the surface of the cluster in a hierarchical lattice. The recursion relations of the electric field on the growth bond are obtained for diffusion-limited aggregation and the dielectric breakdown models. A hierarchy of generalised dimensions D(q) is calculated to describe the growth probability, by using the recursion relations. The partition of (q-1)D(q) into a density of singularities f(q) with singularity strength alpha (q) is made and the alpha -f spectra are studied for different dielectric breakdown models. The scaling of the highest growth probability pmax on the growth bond is analytically derived and the dependence of the fractal dimensions is found on the parameter eta describing the different dielectric breakdown models.