Abstract
The minimal spanning forest on ℤd 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 ℤ2 × {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
- General Mathematics
- Applied Mathematics