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arXiv · 1512.09107

Critical Percolation and the Minimal Spanning Tree in Slabs

Charles M. Newman·Vincent Tassion·Wei Wu
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Abstract

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

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Charles M. Newman, Vincent Tassion, Wei Wu. 2015-12-30. Critical Percolation and the Minimal Spanning Tree in Slabs. https://arxiv.org/abs/1512.09107

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