Renormalization-group approach to multifractal structure of growth probability distribution in diffusion-limited aggregation

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Abstract
The scaling structure of the growth probability distribution in the surface layer of the generalized diffusion-limited aggregation model (η model) is derived by making use of the real-space renormalization-group method. A conductance of the surface layer is defined and renormalized as the growth-bond conductance. The renormalization-group transformation equation is derived for the growth-bond conductance. The equation has a nontrivial solution which is a stable fixed point. The growth probability assigned to each growth bond is represented by a random multiplicative process of the cell’s growth probabilities evaluated at the fixed point. A hierarchy of generalized dimensions D(q) is calculated and the α-f spectrum is found for diffusion-limited aggregation. The dependence of the α-f spectra is found on the parameter η describing the different dielectric breakdown models.