Absence of Infinite Cluster for Critical Bernoulli Percolation on Slabs

Duminil, Hugo; Sidoravičius, Vladas; Tassion, Vincent

doi:10.1002/cpa.21641

Cited by 20 publications

(44 citation statements)

References 22 publications

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“…Item 3 implies in particular that critical percolation on the slab S k does not have an infinite cluster. It strengthens the previous result of [11]. Moreover, our proof leads to the bound ı C k for some C < 1.…”

Section: Existence Of Blocking Surfaces With Positive Probabilitysupporting

confidence: 89%

“…The main purpose of this paper is to make progress toward a proof that at least for some low dimensions above d D 2, there is a single tree by showing that this is the case for the approximation of, say, Z 3 by a thick slab Z 2 f0; : : : ; kg. In the process, we also obtain a new result for critical percolation on such slabs, where it was only recently proved that there is no infinite cluster [11]; the new result is inverse power law decay for the probability of large-diameter finite clusters at criticality (see Corollary 3.3 in Section 3).…”

Section: Introductionmentioning

confidence: 74%

“…This follows from the box-crossing property for critical Bernoulli percolation on S k , which we will prove in Section 3 (see Theorem 3.1), and a result (Theorem 3.12) that allows one to create open circuits in annuli with positive probability. The proof uses a new Russo-Seymour-Welsh-type theorem, based on gluing lemmas given in Section 3 below (like those first established in [11]).…”

Section: Single-tree Resultsmentioning

confidence: 99%

“…Now given ! 0 in the image ofˆ, by the same argument as in [11],´is the only site in the new minimal path .! .´/ / (from A to B) that is connected to C without using any edge in .!…”

Section: Gluing Lemmasmentioning

confidence: 87%

“…The construction ofˆis similar to that in [11], as we now describe. For any´2 U.!/ that is not one of the eight corners of S , we will construct a new configuration !…”

Section: Gluing Lemmasmentioning

confidence: 96%

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Critical Percolation and the Minimal Spanning Tree in Slabs

Newman

Tassion

2017

Comm Pure Appl Math

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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. © 2017 Wiley Periodicals, Inc.

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Section: Existence Of Blocking Surfaces With Positive Probabilitysupporting

confidence: 89%

Section: Introductionmentioning

confidence: 74%

Section: Single-tree Resultsmentioning

confidence: 99%

“…Now given ! 0 in the image ofˆ, by the same argument as in [11],´is the only site in the new minimal path .! .´/ / (from A to B) that is connected to C without using any edge in .!…”

Section: Gluing Lemmasmentioning

confidence: 87%

“…The construction ofˆis similar to that in [11], as we now describe. For any´2 U.!/ that is not one of the eight corners of S , we will construct a new configuration !…”

Section: Gluing Lemmasmentioning

confidence: 96%

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Critical Percolation and the Minimal Spanning Tree in Slabs

Newman

Tassion

2017

Comm Pure Appl Math

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Lectures on the Ising and Potts Models on the Hypercubic Lattice

Duminil-Copin

2019

Springer Proceedings in Mathematics &Amp; Statistics

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Phase transitions are a central theme of statistical mechanics, and of probability more generally. Lattice spin models represent a general paradigm for phase transitions in finite dimensions, describing ferromagnets and even some fluids (lattice gases). It has been understood since the 1980s that random geometric representations, such as the random walk and random current representations, are powerful tools to understand spin models. In addition to techniques intrinsic to spin models, such representations provide access to rich ideas from percolation theory. In recent years, for two-dimensional spin models, these ideas have been further combined with ideas from discrete complex analysis. Spectacular results obtained through these connections include the proofs that interfaces of the two-dimensional Ising model have conformally invariant scaling limits given by SLE curves, that the connective constant of the self-avoiding walk on the hexagonal lattice is given by 2 + √ 2. In higher dimensions, the understanding also progresses with the proof that the phase transition of Potts models is sharp, and that the magnetization of the three-dimensional Ising model vanishes at the critical point. These notes are largely inspired by [39,41,42].

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