Probability distribution and sizes of spanning clusters at the percolation thresholds

Abstract

For random percolation at p_c, the probability distribution P(n) of the number of spanning clusters (n) has been studied in large scale simulations. The results are compatible with $P(n) \sim \exp(-an^2)$ for all dimensions. We also study the variation of the average size (mass) of the spanning clusters when there are more than one spanning cluster. While the average size of the spanning clusters scales as usual with $L^D$ where $D = d- \beta/\nu$ for any number of clusters, it shows a smooth decrease as the number of spanning clusters increases.