Dimension of fractal growth patterns as a dynamical exponent

Abstract

We consider a conformal theory of fractal growth patterns in two dimensions, including diffusion limited aggregation (DLA) as a particular case. In this theory the fractal dimension of the asymptotic cluster manifests itself as a dynamical exponent observable already at very early growth stages. Using a renormalization relation we show from early stage dynamics that the dimension D of DLA can be estimated, 1.69<D<1.72. We explain why traditional numerical estimates converged so slowly. We discuss similar computations for other fractal growth processes in two dimensions.