Analytic Theory of Fractal Growth Patterns in 2 Dimensions

Abstract

Diffusion Limited Aggregation (DLA) is a model of fractal growth that was introduced in 1981 and had since attained a paradigmatic status due to its simplicity and its underlying role for a variety of pattern forming processes. Despite tremendous efforts there is no theory to compute the fractal dimension of DLA from first principles. In this Letter we offer such a theory for fractal growth patterns in two dimensions, including DLA as a particular case. In this theory the fractal dimension of the asymptotic cluster manifests iteself as a renormalization exponent observable already at very early growth stages. Using early stage dynamics we compute 1.6896<D<1.7135, and explain why traditional numerical estimates converged so slowly. We discuss similar computations for other fractal growth processes in 2-dimensions.