Directed aggregation on the Bethe lattice: Scaling, mappings, and universality

We study two types of directed aggregation on the Bethe lattice—directed diffusion-limited aggregation (DDLA) and ballistic aggregation (BA). In the DDLA problem on finite lattices, we construct an exact nonlinear recursion relation for the probability distribution of the density. The mean density tends to zero as the lattice size is taken to infinity. Using a mapping between the model with perfect adhesion on contact and another model with a particular value of the adhesion probability, we show that the adhesion probability is irrelevant over an interval of values. The aggregates in a problem on the infinite lattice display marginal scaling behavior: They are compact up to a logarithmic correction for any coordination number of the lattice. In the BA problem we calculate the mean level number and consider fluctuation effects. The implications of a mapping from the BA problem to the DDLA problem on finite lattices are discussed.