TY - JOUR

T1 - Critical Percolation and the Minimal Spanning Tree in Slabs

AU - Newman, Charles

AU - Tassion, Vincent

AU - Wu, Wei

N1 - Funding Information:
Finally, to prove (4.3), we note that a map mi∆cmi∆cimimii∆mimi∆ci∆ B0\Bx\YA!B0\Bx\YA[B0\Bx\YA can be constructed in essentially the same way as above (in fact, one can skip Step 2 when constructing !0). Lemma 4.2 then implies mi∆cmi∆ci mimii mimi∆ci B0;Bx;YA B0;Bx;YA B0;Bx;YA for some C4 < 1. Together with (4.2) we obtain (4.3) with C3 D C4.1CC2/. □ Acknowledgment. We thank Artem Sapozhnikov for very helpful comments on earlier versions of the paper and Vladas Sidoravicius for useful comments. The research of C.M.N. and W.W. was supported in part by National Science Foundation grants DMS-1007524 and DMS-1507019. The research of V.T. was supported by the Swiss National Science Foundation.
Publisher Copyright:
© 2017 Wiley Periodicals, Inc.
Copyright:
Copyright 2017 Elsevier B.V., All rights reserved.

PY - 2017/11

Y1 - 2017/11

N2 - 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.

AB - 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.

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U2 - 10.1002/cpa.21714

DO - 10.1002/cpa.21714

M3 - Article

AN - SCOPUS:85029537471

SN - 0010-3640

VL - 70

SP - 2084

EP - 2120

JO - Communications on Pure and Applied Mathematics

JF - Communications on Pure and Applied Mathematics

IS - 11

ER -