### Abstract

Let p ≠ 1/2 be the open-bond probability in Broadbent and Hammersley's percolation model on the square lattice. Let W_{x}be 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_{n}of sites inside γ(n) which are connected by open paths to γ(n) is approximately normal for large circuits γ(n).

Original language | English (US) |
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Pages (from-to) | 237-251 |

Number of pages | 15 |

Journal | Journal of Statistical Physics |

Volume | 25 |

Issue number | 2 |

DOIs | |

State | Published - Jun 1 1981 |

### Keywords

- Percolation
- asymptotic normality
- circuits
- semi-invariants

### ASJC Scopus subject areas

- Statistical and Nonlinear Physics
- Mathematical Physics

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## Cite this

Cox, J. T., & Grimmett, G. (1981). Central limit theorems for percolation models.

*Journal of Statistical Physics*,*25*(2), 237-251. https://doi.org/10.1007/BF01022185