Real-space renormalization of diffusion-limited aggregation

We present a quantitative treatment of the interplay between underlying anisotropy and the inherent degree of randomness (noise) in growth processes. Using a renormalization procedure, we show how the noise and anisotropy of diffusion-limited aggregates evolve with cluster size. The fixed point associated with isotropically growing clusters is stable with respect to noise, but at a value 2 orders of magnitude below the ‘‘bare’’ value in simple diffusion-limited aggregation. This fixed point is unstable with respect to any anisotropy in the growing conditions amplifying itself in the growing cluster, causing flow to (in our case one of two) absolutely stable fixed points associated with strongly anisotropic growth. The location and exponents of the isotropic fixed point explain why clusters of order ${10}^4$ particles are required to see sensitivity to lattice anisotropy in the Witten-Sander model.