THE FRACTAL BEHAVIOR OF THE SHORTEST-PATH AGGREGATION - SIMULATIONS AND RG CALCULATIONS

The shortest-path aggregation (SPA) is an irreversible process in which particles are attached to the cluster on the closer perimeter sites to the release site. Randomness is introduced via the choice of sites from which the particles are sequentially released. This model generates random fractals which have dendritic structures. Simulations and kinetic real-space renormalization group calculations on a two-dimensional square lattice are presented. Numerically, we find D(f) = 1.20 +/- 0.01 for the fractal dimension of the clusters. The universal behavior of the model against the way of releasing particles is observed. We find a transition from a weak correlation region to a strong correlation region. In particular, we find the fractal dimension D = 1.20 +/- 0.01 when the released particles are weakly correlated, while a one-dimensional object is obtained when the correlation is strong. Analytically, 2 x 2 and 3 x 3 cell real-space renormalization group calculations yield D(f) = 1.19 and 1.21, respectively, in good agreement with the simulations. A set of hierarchical dimensions D(q) are also computed for the SPA cluster on the square lattice