Scaling theory for the anisotropic behavior of generalized diffusion-limited aggregation clusters in two dimensions

A scaling description of the crossover from isotropic to anisotropic cluster growth for ordinary diffusion-limited aggregation (DLA) in two dimensions developed recently by Family and Hentschel is extended to the generalized DLA or $\eta$ model. The dependence of various exponents necessary to characterize the anisotropic growth on the local-growth probability exponent $\eta$ of the generalized DLA is obtained explicitly. The $\eta$ dependence of the exponent $\beta$ describing the variation of the crossover mass $N_c$ on the degree of symmetry $m$, $N_c\sim m^{\beta }$, is derived. The results indicate that the anisotropic star-shaped clusters can be easily observed for $\eta >1\mathrm{}$, while their appearance is much more difficult for $\eta <1\mathrm{}$. All our results are consistent with those of computer simulations reported so far.