Critical Percolation and the Minimal Spanning Tree in Slabs

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. © 2017 Wiley Periodicals, Inc.