A sharp transition for the two-dimensional Ising percolation

Higuchi, Yasunari

doi:10.1007/bf01192961

Cited by 24 publications

(24 citation statements)

References 18 publications

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“…This follows by standard methods from the exponential tail decay of the size of Ising spin clusters on Z 2 at high temperatures [22,34]. Theorem 1 concerns the joint measure P β,r , but as anticipated, it has implications for the critical value of Bernoulli percolation on the realisations of the random graphs obtained from the FK clusters as explained in Sect.…”

Section: Theorem 1 For Allmentioning

confidence: 82%

“…It is interesting to compare our result to a result of Higuchi, who has extensively studied the percolation properties of the two-dimensional Ising model (see [18][19][20][21][22]). If one considers the Ising model on the square lattice (which we simply denote by Z 2 ) at inverse temperature β with an external field h, then there exists a critical value h c (β) such that for all h > h c (β), there is an infinite cluster of +1 spins, whereas there is no such infinite cluster for h < h c (β).…”

mentioning

confidence: 77%

“…A careful reading of the proof of this proposition shows that the same method works on the triangular lattice T. Moreover, we can take the parallelogram S n,6n instead of a square, consider a horizontal crossing R ∈ R n,4n , condition on the spins of vertices in D(R) instead of L(R) ∪ R, require that the (+)-path from a neighbour of v ∈ R to the top of S n,6n be in A(R) ∩ S n,6n , and still conclude that, under the assumption that (+) and (−)-crossings in S 3n,n have probabilities bounded away from 0, the expected number of special vertices (which here are cut points of R in S n,6n ) goes to infinity as n → ∞. In fact, using Higuchi's notation in [21], we see that since all the cut points considered in the proof are found inside annuli A j which are at distance at least 5 2 · 4 j from one another (where only integers j satisfying √ n ≤ 2 · 4 j are consideredsee (5.21) in [22]), all cut points considered are automatically at distance at least 5 4 · √ n from one another. Therefore, if E β denotes the expected value w.r.t.…”

Section: Cut Pointsmentioning

confidence: 91%

“…We shall use (a slightly modified version of) a result of Higuchi from [21] (see also Proposition 4.2 in [22]) about the Ising model. In order to state the theorem, we need a few definitions.…”

Section: Cut Pointsmentioning

confidence: 99%

“…Similarly, one can define h * c (β) as the critical value for percolation of +1 spins in the matching lattice of Z 2 which is obtained by adding diagonal edges inside the faces of Z 2 . Higuchi showed in [22] that for all subcritical β, the duality relation h c (β) + h * c (β) = 0 holds. (Note that since it is easier to percolate on the matching lattice than on Z 2 , it is not surprising that h * c (β) is negative.)…”

mentioning

confidence: 99%

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The High Temperature Ising Model on the Triangular Lattice is a Critical Bernoulli Percolation Model

2010

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We define a new percolation model by generalising the FK representation of the Ising model, and show that on the triangular lattice and at high temperatures, the critical point in the new model corresponds to the Ising model. Since the new model can be viewed as Bernoulli percolation on a random graph, our result makes an explicit connection between Ising percolation and critical Bernoulli percolation, and gives a new justification of the conjecture that the high temperature Ising model on the triangular lattice is in the same universality class as Bernoulli percolation.Keywords Ising model · Random-cluster measures · Dependent percolation · DaC models · Sharp phase transition · Duality · p c = 1/2 Motivation, Background and SynopsisIf one considers the percolation properties of spin clusters, the high temperature (β < β c ) Ising model on the triangular lattice T with no external field shows critical behaviour in the sense that the mean spin cluster size is divergent (see, e.g., [3]) and the probability that two spins are in the same spin cluster has a power law decay (see [17]). The relevant universality class is conjectured to be that of two-dimensional Bernoulli (independent) percolation. This conjecture includes convergence of certain interfaces to SLE 6 in the scaling limit, and

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Section: Theorem 1 For Allmentioning

confidence: 82%

mentioning

confidence: 77%

Section: Cut Pointsmentioning

confidence: 91%

Section: Cut Pointsmentioning

confidence: 99%

mentioning

confidence: 99%

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The High Temperature Ising Model on the Triangular Lattice is a Critical Bernoulli Percolation Model

2010

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Constrained percolation, Ising model, and XOR Ising model on planar lattices

2020

Random Struct Algorithms

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We study site percolation models on planar lattices including the [m, 4, n, 4] lattice and the square tilings on the Euclidean plane (R 2 ) or the hyperbolic plane (H 2 ), satisfying certain local constraints on degree-4 faces. These models are closely related to Ising models and XOR Ising models (product of two i.i.d Ising models) on regular tilings of R 2 or H 2 . In particular, we obtain a description of the numbers of infinite "+" and "−" clusters of the ferromagnetic Ising model on a vertex-transitive triangular tiling of H 2 for different boundary conditions and coupling constants. Our results show the possibility that such an Ising configuration has infinitely many infinite "+" and "−" clusters, while its random cluster representation has no infinite open clusters. Percolation properties of corresponding XOR Ising models are also discussed.

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Percolation and disordered systems

Grimmett

1997

Lecture Notes in Mathematics

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PREFACEThis course aims to be a (nearly) self-contained account of part of the mathematical theory of percolation and related topics. The rst nine chapters summarise rigorous results in percolation theory, with special emphasis on results obtained since the publication of my book 156] entitled`Percolation', and sometimes referred to simply as G] in these notes. Following this core material are chapters on random walks in random labyrinths, and fractal percolation. The nal two chapters include material on Ising, Potts, and random-cluster models, and concentrate on a`percolative' approach to the associated phase transitions.The rst target of this course is to draw a picture of the mathematics of percolation, together with its immediate mathematical relations. Another target is to present and summarise recent progress. There is a considerable overlap between the rst nine chapters and the contents of the principal reference G]. On the other hand, the current notes are more concise than G], and include some important extensions, such as material concerning strict inequalities for critical probabilities, the uniqueness of the in nite cluster, the triangle condition and lace expansion in high dimensions, together with material concerning percolation in slabs, and conformal invariance in two dimensions. The present account di ers from that of G] in numerous minor ways also. It does not claim to be comprehensive. A second edition of G] is planned, containing further material based in part on the current notes.A special feature is the bibliography, which is a fairly full list of papers published in recent years on percolation and related mathematical phenomena. The compilation of the list was greatly facilitated by the kind responses of many individuals to my request for lists of publications.Many people have commented on versions of these notes, the bulk of which have been typed so superbly by Sarah Shea-Simonds. I thank all those who have contributed, and acknowledge particularly the suggestions of Ken Alexander, Carol Bezuidenhout, Philipp Hiemer, Anthony Quas, and Alan Stacey, some of whom are mentioned at appropriate points in the text. In addition, these notes have bene ted from the critical observations of various members of the audience at St Flour.Members of the 1996 summer school were treated to a guided tour of the library of the former seminary of St Flour. We were pleased to nd there a copy of the Encyclop edie, ou Dictionnaire Raisonn e des Sciences, des Arts et des M etiers, compiled by Diderot and D'Alembert, and published in Geneva around 1778. Of the many illuminating entries in this substantial work, the following de nition of a probabilist was not overlooked. 3 PROBABILISTE, s. m. (Gram. Th eol.) celui qui tient pour la doctrine abominable des opinions rendues probables par la d ecision d'un casuiste, & qui assure l'innocence de l'action faite en cons equence. Pascal a foudroy e ce systême, qui ouvroit la porte au crime en accordant a l'autorit e les pr erogatives de la certitude, a l'opinion & la s ec...

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A sharp transition for the two-dimensional Ising percolation

Cited by 24 publications

References 18 publications

The High Temperature Ising Model on the Triangular Lattice is a Critical Bernoulli Percolation Model

The High Temperature Ising Model on the Triangular Lattice is a Critical Bernoulli Percolation Model

Constrained percolation, Ising model, and XOR Ising model on planar lattices

Percolation and disordered systems

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