Growth histories and overlap distributions of diffusion-limited-aggregation clusters

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Abstract
The large variability in the ramified structure of diffusion-limited-aggregation clusters suggests an analogy between their branches and the phase-space valleys of spin glasses. We define the overlap q of two growth sites, and study numerically the overlap function q(x) of two- and three-dimensional aggregates. For isotropic aggregates we find that the average overlap decreases as a power law when their mass N increases, indicating that the number of branches increases with N, and that the overlap function and its fluctuations obey a scaling law. Analytical results are presented for the infinite-dimensional limit on the Cayley tree. For anisotropic aggregates with b-fold symmetry (b=2, 3, and 4), the average overlap obtained numerically has a finite limit for large N, and the overlap function is in very good agreement with analytical calculations on a simplified model.