### Abstract

In independent bond percolation on Z^{d} with parameter p, if one removes the vertices of the infinite cluster (and incident edges), for which values of p does the remaining graph contain an infinite connected component? Grimmett-Holroyd-Kozma used the triangle condition to show that for d≥ 19 , the set of such p contains values strictly larger than the percolation threshold p_{c}. With the work of Fitzner-van der Hofstad, this has been reduced to d≥ 11. We improve this result by showing that for d≥ 10 and some p> p_{c}, there are infinite paths consisting of “shielded” vertices—vertices all whose adjacent edges are closed—which must be in the complement of the infinite cluster. Using values of p_{c} obtained from computer simulations, this bound can be reduced to d≥ 7. Our methods are elementary and do not require the triangle condition.

Original language | English (US) |
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Pages (from-to) | 789-807 |

Number of pages | 19 |

Journal | Journal of Statistical Physics |

Volume | 179 |

Issue number | 3 |

DOIs | |

State | Published - May 1 2020 |

### Keywords

- High dimensions
- Percolation
- Shielded vertices
- Triangle condition

### ASJC Scopus subject areas

- Statistical and Nonlinear Physics
- Mathematical Physics

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## Cite this

*Journal of Statistical Physics*,

*179*(3), 789-807. https://doi.org/10.1007/s10955-020-02558-4