Critical Percolation and the Minimal Spanning Tree in Slabs

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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 arXiv:1401.7130. 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 arXiv:1401.7130, where the absence of an infinite cluster at criticality was first established.