Radius, perimeter, and density profile for percolation clusters and lattice animals

Monte Carlo simulation and “series expansion” shows the radius of gyration of large clusters withs sites each to vary assρ withρ≃0.56 in two andρ≃0.47 in three dimensions at the percolation threshold, and withρ(d=2)≃0.65 andρ(d=3)≃0.53 for random lattice animals (zero concentration). Clusters up tos=100 were used. The perimeter of random animals approaches 2.8s for larges on the simple cubic lattice. Monte Carlo simulation of the Eden process (“growing animals”) up tos=5,000 indicates a systematic variation of about ±0.05 for the effective exponentρ=ρ(s) and thus suggests that the true asymptotic exponents may be compatible with the predictions of “hyper”-scaling.

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