Topological properties of diffusion limited aggregation and cluster-cluster aggregation

The detailed topological or 'connectivity' properties of the clusters formed in diffusion limited aggregation (DLA) and cluster-cluster aggregation (CCA) are considered for spatial dimensions d=2, 3 and 4. Specifically, for both aggregation phenomena the authors calculate the fractal dimension d _min= nu ^-1 defined by l approximately R^d(min) where l is the shortest path between two points separated by a Pythagorean distance R. For CCA, they find that d_min increases monotonically with d, presumably tending toward a limiting value d_min=2 at the upper critical dimensionality d_c as found previously for lattice animals and percolation. For DLA, on the other hand, they find that d_min=1 within the accuracy of the calculations for d=2, 3 and 4, suggesting the absence of an upper critical dimension. They also discuss some of the subtle features encountered in calculating d_min for DLA.