Causality bound on the density of aggregates

The irreversible accretion of diffusing particles onto a large cluster results in a tenuous aggregate characterized by a fractal dimension $D_0$ smaller than that of space. The rate of aggregation onto the fastest-growing sites in such a process must not increase indefinitely as the cluster grows. This fact sets a lower limit on the fractal dimension, viz., the dimension $d$ of space minus 1. For aggregation of ballistically moving particles, this bound implies that the aggregate must be compact: The fractal dimension must equal that of space. In general, if the aggregating particles follow trajectories of fractal dimension $D_1$, the bound implies $D_0>~d-D_1+1\mathrm{}$.