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

Using Monte Carlo simulations, we report the behaviour of the total number of clusters, the cluster size diversity and the lattice animals (LA) divers ity on randomly occupied square lattices. The critical probability associat ed with the maximum of these variables is determined in comparison with the percolation probability p(c). Our simulations indicate that p(c) and the c ritical 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 T sang I J 1997 J. Phys. A: Math. Gen. 30 L239), the probability for the maxi mum 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 d ifferent LA and determined the critical probability for the maximum LA dive rsity, p(c)(D-f max) = 0.45 +/- 0.02. We derive the exponents characterizin g the relation between D-s max, D-f max, and N-max and several scaling rela tions between the variables measured, the lattice size, and the probability of occupation p. In addition, we show the scaling behaviour of LA diversit y versus cluster size diversity for each value of p.