Abstract
Letp ≠ 1/2 be the open-bond probability in Broadbent and Hammersley's percolation model on the square lattice. LetW x be the cluster of sites connected tox by open paths, and letγ(n) be any sequence of circuits with interiors\(|\mathop \gamma \limits^ \circ (n)| \to \infty \). It is shown that for certain sequences of functions {f n },\(S_n = \sum _{x \in \mathop \gamma \limits^ \circ (n)} f_n (W_x )\) 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 n of sites insideγ(n) which are connected by open paths toγ(n) is approximately normal for large circuitsγ(n).
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Cox, J.T., Grimmett, G. Central limit theorems for percolation models. J Stat Phys 25, 237–251 (1981). https://doi.org/10.1007/BF01022185
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DOI: https://doi.org/10.1007/BF01022185