Critical Percolation and the Minimal Spanning Tree in Slabs

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 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.