## Abstract

We consider critical site percolation on the triangular lattice in the upper half-plane. Let

^{u1},^{u2}be two sites on the boundary and w a site in the interior. It was predicted by Simmons et al. (2007) that the ratio P(^{nu1虠nu2虠nw)2}/P(n^{u1}虠n^{u2}).P(n^{u1}虠nw).P(n^{u2}虠nw) converges to^{KF}as n→8, where x虠y denotes that x and y are in the same cluster, and^{KF}is a constant. Beliaev and Izyurov (2012) proved an analog of this in the scaling limit. We prove, using their result and a generalized coupling argument, the earlier mentioned prediction. Furthermore we prove a factorization formula for P(n^{u2}虠[n^{u1},n^{u1}+s];nw虠[n^{u1},n^{u1}+s]), where s>0.Original language | English |
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Pages (from-to) | 4102-4116 |

Journal | Stochastic Processes and Their Applications |

Volume | 125 |

Issue number | 11 |

DOIs | |

Publication status | Published - 2015 |