## Abstract

Consider a one-dimensional independent bond percolation model with p_{j} denoting the probability of an occupied bond between integer sites i and i±j, j≧1. If p_{j} is fixed for j≧2 and {Mathematical expression}j^{2}p_{j}>1, then (unoriented) percolation occurs for p_{1} sufficiently close to 1. This result, analogous to the existence of spontaneous magnetization in long range one-dimensional Ising models, is proved by an inductive series of bounds based on a renormalization group approach using blocks of variable size. Oriented percolation is shown to occur for p_{1} close to 1 if {Mathematical expression}j^{s}p_{j}>0 for some s<2. Analogous results are valid for one-dimensional site-bond percolation models.

Original language | English (US) |
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Pages (from-to) | 547-571 |

Number of pages | 25 |

Journal | Communications In Mathematical Physics |

Volume | 104 |

Issue number | 4 |

DOIs | |

State | Published - Dec 1986 |

## ASJC Scopus subject areas

- Statistical and Nonlinear Physics
- Mathematical Physics