Analysis of multiscaling structure in diffusion-limited aggregation: A kinetic renormalization-group approach

A new kinetic real-space renormalization-group (KRG) approach to diffusion-limited aggregation (DLA) is presented. It is based on the combined length- and time-scale invariance of the clusters. The probabilities relating different local configurations under rescaling are determined from the growth process itself. Recursion relations between the masses of the cluster and of its interface on consecutive iterations are obtained. The common fractal dimension ($D_f$) of the cluster and of its interface is calculated. The approach is applied to DLA on square, cubic, and two regular fractal lattices. We find $D_f$=1.727 (2.494) for the square two-dimensional (cubic three-dimensional) lattices, in very good agreement with the most extensive numerical results. Corrections to scaling exponents are found from the smaller eigenvalues. In addition, the multifractal spectrum D(q) of the local-growth probability’s moments and the f-α curve are derived.