Predictions of bond percolation thresholds for the kagomé and Archimedean<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:mo>(</mml:mo><mml:mn>3</mml:mn><mml:mo>,</mml:mo><mml:msup><mml:mn>12</mml:mn><mml:mn>2</mml:mn></mml:msup><mml:mo>)</mml:mo></mml:mrow></mml:math>lattices

Scullard, Christian R.; Ziff, Robert M.

doi:10.1103/physreve.73.045102

Cited by 53 publications

(118 citation statements)

References 25 publications

(59 reference statements)

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“…Nevertheless, it is still believed to be false; see Ziff and Suding [46], for example. More recently, Scullard and Ziff [28] have predicted certain values for p b c for the Kagomé and (3, 12 2 ) lattices, using a heuristic version of the star-triangle transformation. Although they leave open the 'possibility' that one of these values might be exact, there seems no reason (to us, or, apparently, to them) to really believe this: the method is (as they admit) non-rigorous, and the value obtained in the same way for the Kagomé lattice (given earlier by Hori and Kitahara without derivation) is outside the error bounds of existing experimental results.…”

Section: Discussionmentioning

confidence: 99%

Rigorous confidence intervals for critical probabilities

Riordan

Walters

2007

Phys. Rev. E

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We use the method of Balister, Bollobás and Walters [3] to give rigorous 99.9999% confidence intervals for the critical probabilities for site and bond percolation on the 11 Archimedean lattices. In our computer calculations, the emphasis is on simplicity and ease of verification, rather than obtaining the best possible results. Nevertheless, we obtain intervals of width at most 0.0005 in all cases.

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Section: Discussionmentioning

confidence: 99%

Rigorous confidence intervals for critical probabilities

Riordan

Walters

2007

Phys. Rev. E

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“…The solution on the triangular lattice can be extended by decoration [8,9,10] and to the closely related bowtie lattices [11,12,13]. The critical manifold-which is the set of points in (q, v) space at which the model stands at a phase transition-is obviously of special interest.…”

Section: Introductionmentioning

confidence: 99%

Transfer matrix computation of critical polynomials for two-dimensional Potts models

Jacobsen¹,

Scullard²

2013

J. Phys. A: Math. Theor.

109

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Abstract.In our previous work [1] we have shown that critical manifolds of the q-state Potts model can be studied by means of a graph polynomial P B (q, v), henceforth referred to as the critical polynomial. This polynomial may be defined on any periodic twodimensional lattice. It depends on a finite subgraph B, called the basis, and the manner in which B is tiled to construct the lattice. The real roots v = e K − 1 of P B (q, v) either give the exact critical points for the lattice, or provide approximations that, in principle, can be made arbitrarily accurate by increasing the size of B in an appropriate way. In earlier work, P B (q, v) was defined by a contraction-deletion identity, similar to that satisfied by the Tutte polynomial. Here, we give a probabilistic definition of P B (q, v), which facilitates its computation, using the transfer matrix, on much larger B than was previously possible.We present results for the critical polynomial on the (4, 8 2 ), kagome, and (3, 12 2 ) lattices for bases of up to respectively 96, 162, and 243 edges, compared to the limit of 36 edges with contraction-deletion. We discuss in detail the role of the symmetries and the embedding of B. The critical temperatures v c obtained for ferromagnetic (v > 0) Potts models are at least as precise as the best available results from Monte Carlo simulations or series expansions. For instance, with q = 3 we obtain v c (4, 82 ) = 3.742 489 (4), v c (kagome) = 1.876 459 7 (2), and v c (3, 122 ) = 5.033 078 49 (4), the precision being comparable or superior to the best simulation results. More generally, we trace the critical manifolds in the real (q, v) plane and discuss the intricate structure of the phase diagram in the antiferromagnetic (v < 0) region.

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“…For example, choosing a simple triangle with uncorrelated bonds in Figure 2(a), gives the kagomé lattice. This lattice's bond threshold cannot be found by the following method and, in fact, the problem remains unsolved [18].…”

Section: Triangle-triangle Transformationmentioning

confidence: 99%

“…This results in a critical surface rather than a critical point and application of (7) implies that it is given by [17,18] Table 1. Bond percolation thresholds of the lattices in Fig.…”

Section: Martini and Related Latticesmentioning

confidence: 99%

Exact bond percolation thresholds in two dimensions

Ziff

Scullard

2006

J. Phys. A: Math. Gen.

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Abstract. Recent work in percolation has led to exact solutions for the site and bond critical thresholds of many new lattices. Here we show how these results can be extended to other classes of graphs, significantly increasing the number and variety of solved problems. Any graph that can be decomposed into a certain arrangement of triangles, which we call self-dual, gives a class of lattices whose percolation thresholds can be found exactly by a recently introduced triangle-triangle transformation. We use this method to generalize Wierman's solution of the bow-tie lattice to yield several new solutions. We also give another example of a self-dual arrangement of triangles that leads to a further class of solvable problems. There are certainly many more such classes.

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Predictions of bond percolation thresholds for the kagomé and Archimedean(3,122)lattices

Cited by 53 publications

References 25 publications

Rigorous confidence intervals for critical probabilities

Rigorous confidence intervals for critical probabilities

Transfer matrix computation of critical polynomials for two-dimensional Potts models

Exact bond percolation thresholds in two dimensions

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