### Abstract

The minimal spanning forest on ℤ^{d} is known to consist of a single tree for d ≤ 2 and is conjectured to consist of infinitely many trees for large d. In this paper, we prove that there is a single tree for quasi-planar graphs such as ℤ^{2} × {0,…,k}^{d−2}. Our method relies on generalizations of the “gluing lemma” of Duminil-Copin, Sidoravicius, and Tassion. A related result is that critical Bernoulli percolation on a slab satisfies the box-crossing property. Its proof is based on a new Russo-Seymour-Welsh-type theorem for quasi-planar graphs. Thus, at criticality, the probability of an open path from 0 of diameter n decays polynomially in n. This strengthens the result of Duminil-Copin et al., where the absence of an infinite cluster at criticality was first established.

Original language | English (US) |
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Pages (from-to) | 2084-2120 |

Number of pages | 37 |

Journal | Communications on Pure and Applied Mathematics |

Volume | 70 |

Issue number | 11 |

DOIs | |

State | Published - Nov 2017 |

### ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics

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## Cite this

*Communications on Pure and Applied Mathematics*,

*70*(11), 2084-2120. https://doi.org/10.1002/cpa.21714