## Abstract

Renormalization arguments are developed and applied to independent nearest-neighbor percolation on various subsets ℕ of ℤ^{d}, d≧2, yielding:Equality of the critical densities, p_{c}(ℕ), for ℕ a half-space, quarter-space, etc., and (for d>2) equality with the limit of slab critical densities. Continuity of the phase transition for the half-space, quarter-space, etc.; i.e., vanishing of the percolation probability, θ_{ℕ}(p), at p=p_{c}(ℕ). Corollaries of these results include uniqueness of the infinite cluster for such ℕ's and sufficiency of the following for proving continuity of the full-space phase transition: showing that percolation in the full-space at density p implies percolation in the half-space at the same density.

Original language | English (US) |
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Pages (from-to) | 111-148 |

Number of pages | 38 |

Journal | Probability Theory and Related Fields |

Volume | 90 |

Issue number | 1 |

DOIs | |

State | Published - Mar 1991 |

## ASJC Scopus subject areas

- Analysis
- Statistics and Probability
- Statistics, Probability and Uncertainty