Scaling behavior of generalized diffusion-limited aggregation: The correct form of them-spoke model

The potential of the original m-spoke model, conceived by Family and Hentschel (FH) [Faraday Discuss. Chem. Soc. 83, 139 (1987)], does not satisfy the correct boundary conditions for diffusion-limited aggregation (DLA) or dielectric breakdown. Moreover, while the suggested scaling forms hold, if the growth probability exponent η of the dielectric breakdown model is larger than 2, they are not generally valid for η<2. The correct potential of an m-spoke structure is given, and the scaling behavior of this new model is calculated. Since the scaling dependence on the cluster size N is determined by the local-field behavior, the exponent ${\nu }_{?}$ found by FH is recovered with the new potential. For a careful evaluation of the m dependence, however, a nonlocal quantity must be considered, leading to novel results. We conclude that only if the fractal dimension is m dependent can the asymptotic shape of DLA clusters be m-spoke-like.