Critical Percolation and the Minimal Spanning Tree in Slabs

Charles Newman, Vincent Tassion, Wei Wu

Research output: Contribution to journalArticlepeer-review


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.

Original languageEnglish (US)
Pages (from-to)2084-2120
Number of pages37
JournalCommunications on Pure and Applied Mathematics
Issue number11
StatePublished - Nov 2017

ASJC Scopus subject areas

  • General Mathematics
  • Applied Mathematics


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