Scaling and critical probabilities for cluster size and LA diversity on randomly occupied square lattices

摘要

Using Monte Carlo simulations, we report the behaviour of the total number of clusters, the cluster size diversity and the lattice animals (LA) diversity on randomly occupied square lattices. The critical probability associated with the maximum of these variables is determined in comparison with the percolation probability p(c). Our simulations indicate that p(c) and the critical probability of the maximum cluster size diversity p(c) (D-s (max)), occur at the same point. As indicated in a previous paper (Tsang I R and Tsang I J 1997 J. Phys. A: Math. Gen. 30 L239), the probability for the maximum number of clusters is obtained at a lower value, p(c)(N-max) = 0.27 +/- 0.01. We describe the cluster identification algorithm used to count the different LA and determined the critical probability for the maximum LA diversity, p(c)(D-f max) = 0.45 +/- 0.02. We derive the exponents characterizing the relation between D-s max, D-f max, and N-max and several scaling relations between the variables measured, the lattice size, and the probability of occupation p. In addition, we show the scaling behaviour of LA diversity versus cluster size diversity for each value of p. [References: 30]