Critical Percolation and the Minimal Spanning Tree in Slabs

Abstract

The minimal spanning forest on ${\mathbb{Z}}^d$ is known to consist of a single tree for $d\le 2$ 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 ${\mathbb{Z}}^2\times {\{0,\dots ,k\}}^{d-2}$. 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.

References in corpus (1)

• Discontinuity of the phase transition for the planar random-cluster and Potts models with $q>4$

Cited by in corpus (1)

• The phase transitions of the random-cluster and Potts models on slabs with $q\ge 1$ are sharp