Central limit theorems for percolation models

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