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Critical percolation and the minimal spanning tree in slabs
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Newman, Charles, Tassion, Vincent and Wu, Wei (2017) Critical percolation and the minimal spanning tree in slabs. Communications on Pure and Applied Mathematics, 70 (11). pp. 2084-2120. doi:10.1002/cpa.21714
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WRAP-critical-percolation-minimal-spanning-tree-slabs-Wu-2020.pdf - Accepted Version - Requires a PDF viewer. Download (980Kb) | Preview |
Official URL: http://dx.doi.org/10.1002/cpa.21714
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.
| Item Type: | Journal Article | ||||||||||||
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| Subjects: | Q Science > QA Mathematics Q Science > QD Chemistry T Technology > TA Engineering (General). Civil engineering (General) |
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| Divisions: | Faculty of Science > Statistics | ||||||||||||
| Library of Congress Subject Headings (LCSH): | Percolation (Statistical physics), Trees (Graph theory), Slabs, Topological graph theory | ||||||||||||
| Journal or Publication Title: | Communications on Pure and Applied Mathematics | ||||||||||||
| Publisher: | John Wiley & Sons | ||||||||||||
| ISSN: | 0010-3640 | ||||||||||||
| Official Date: | November 2017 | ||||||||||||
| Dates: |
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| Volume: | 70 | ||||||||||||
| Number: | 11 | ||||||||||||
| Page Range: | pp. 2084-2120 | ||||||||||||
| DOI: | 10.1002/cpa.21714 | ||||||||||||
| Status: | Peer Reviewed | ||||||||||||
| Publication Status: | Published | ||||||||||||
| Publisher Statement: | This is the peer reviewed version of the following article: Newman, C., Tassion, V. and Wu, W. (2017), Critical Percolation and the Minimal Spanning Tree in Slabs. Comm. Pure Appl. Math., 70: 2084-2120. doi:10.1002/cpa.21714, which has been published in final form at http://dx.doi.org/10.1002/cpa.21714. This article may be used for non-commercial purposes in accordance with Wiley Terms and Conditions for Use of Self-Archived Versions. | ||||||||||||
| Access rights to Published version: | Restricted or Subscription Access | ||||||||||||
| RIOXX Funder/Project Grant: |
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