A Martingale Approach in the Study of Percolation Clusters on the Z ^d Lattice

Consider percolation on Z ^d with parameter p. Let K n be the number of occupied clusters in [–n, n] ^d . Here we use a martingale method to show that if p<>0, 1, K n satisfies the CLT for all d>1. Furthermore, we investigate the large deviations and concentration property for K n . Besides K n , we also consider the distribution of the number (lambda)n of such vertices connected by the infinite occupied cluster in a large box [–n, n] ^d . We show that (lambda)n satisfies the CLT and investigate the concentration property for (lambda)n , by using the martingale method in the supercritical phase.