Theory of branched growth

We present a theory of branched growth processes, notably diffusion-limited aggregation (DLA). Using a simple model of the dynamics of screening of competing branches, we compute statistics of the growth probability distribution. These statistics are multifractal only for stochastic models. Applying this general approach to diffusion-limited aggregation, we obtain results for the dimension D of a DLA cluster that are extremely close to the Muthukumar formula, D=(${\mathit{d}}^2$+1)/(d+1), in spatial dimensionality d [Phys. Rev. Lett. 50, 839 (1983)]. This formula is believed to be accurate for DLA clusters in dimensions d>2. The maximum growth probability of the cluster scales as ${\mathit{p}}_{\max }$∼${\mathit{r}}^{1\mathrm{-}\mathit{D}}$, with r the cluster radius, as predicted by Turkevich and Scher [Phys. Rev. Lett. 55, 1026 (1985); Phys. Rev. A 33, 786 (1986)]. We also discuss the scaling of the minimum growth probability, the behavior of nonstochastic models, and possible approaches to direct computation of the screening dynamics. Our results are in good qualitative, and in some cases good quantitative, agreement with numerical studies of the screening dynamics.