Asymptotic structure of diffusion-limited aggregation clusters in two dimensions

Abstract

Owing to the symmetry of the underlying lattice or anisotropy in the growth mechanism, large-scale diffusion-limited aggregation (DLA) clusters exhibit a crossover from isotropic to anisotropic growth. A scaling description of this crossover behaviour is presented and various exponents, including those describing the divergences of the lengths along and perpendicular to the direction of anisotropy, are defined. Scaling relations among the exponents and the fractal dimension are derived. An idealized model, in which the asymptotic shape of an m-fold symmetric DLA is approximated by a star-shaped object with m symmetric spokes, is introduced. Using the analogy between DLA and an equivalent electrostatic problem, the m-spoke model is solved exactly via conformal mapping and the exponents are calculated exactly. The critical mass, N_c, above which DLA clusters exhibit an anisotropic structure is calculated and is found to grow as m^β, where β is an exponent that depends on the fractal dimension.