Exact dynamics of a class of aggregation models

The dynamics of a class of aggregation models proposed by Takayasu and co-workers are solved exactly in one dimension and in the mean field limit. These models describe the aggregation of positive and negative charges. In one dimension, we find the dynamical cluster-size exponents z=3/2 and ${\mathit{z}}_{\mathit{c}}$=3/4 when the average flux of injected charges is nonzero and zero, respectively. We also find the crossover exponent near the transition to be φ=4/3. Within mean field theory, we find these exponents to be z=2, ${\mathit{z}}_{\mathit{c}}$=1, and φ=1. Assuming dynamic scaling, we show that in any dimension, these exponents are related to one single static exponent.