Central limit theorems for percolation models

Let p ≠ 1/2 be the open-bond probability in Broadbent and Hammersley's percolation model on the square lattice. Let W_xbe the cluster of sites connected to x by open paths, and let γ(n) be any sequence of circuits with interiors {Mathematical expression}. It is shown that for certain sequences of functions {f_n}, {Mathematical expression} converges in distribution to the standard normal law when properly normalized. This result answers a problem posed by Kunz and Souillard, proving that the number S_nof sites inside γ(n) which are connected by open paths to γ(n) is approximately normal for large circuits γ(n).