Multifractal structures of mass and growth probability distributions in diffusion-limited aggregation on hierarchical lattices

The scaling structures of the mass and growth probability distributions in the generalized diffusion-limited aggregation ({η} model) on the hierarchical lattices are derived by making use of the real-space renormalization-group method. In the dielectric breakdown language, the process is formulated as a random multiplicative process of current on the surface of the aggregate. The mass and growth probability distributions are related to the bulk and surface structures of the aggregate. The mass scaling of the clusters generated by the processes is found and compared with the scaling structure of the growth probability distribution. The dependences of the {α}-f spectra of the mass and growth probability distributions are shown on the parameter {η} describing the different dielectric breakdown models.