Publisher's Note: Critical frontier of the Potts and percolation models on triangular-type and kagome-type lattices. II. Numerical analysis [Phys. Rev. E<b>81</b>, 061111 (2010)]

Ding, Chengxiang; Fu, Zhe; Guo, Wenan; Wu, F. Y.

doi:10.1103/physreve.81.069904

Cited by 15 publications

(60 citation statements)

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“…Our final values agree with-and are marginally more precise than-two sets of recent numerical results, obtained respectively from the crossings of effective critical exponents [27] and from the maximum of the effective central charge [1].…”

Section: Square Basessupporting

confidence: 87%

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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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Section: Square Basessupporting

confidence: 87%

“…It follows that both for q = 3 and q = 4, the accuracy of our final result improves on that of [27] by more than an order of magnitude.…”

Section: (3 12mentioning

confidence: 52%

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

Jacobsen¹,

Scullard²

2013

J. Phys. A: Math. Theor.

109

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“…7.8. Ruby lattice (3,4,6,4) For the ruby lattice, we used 2046 processors; it took about 5 hours to do the computation for a single prime, and we used 14 primes. Once again, only even n works for this problem.…”

Section: Snub Hexagonal Lattice (3 4 6)mentioning

confidence: 99%

“…The computation of critical thresholds in percolation and the Potts model has long been the domain of Monte Carlo methods [2,3,4] or transfer matrix techniques [5,6] of similar accuracy. Recently, we developed a radically different technique called the method of critical polynomials [7,8,9,10,11,12,13] which gives us the ability to compute percolation and Potts-model thresholds to precisions that are orders of magnitude greater than that obtained with traditional tools.…”

Section: Introductionmentioning

confidence: 99%

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Potts-model critical manifolds revisited

Scullard

Jacobsen

2016

J. Phys. A: Math. Theor.

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We compute the critical polymials for the q-state Potts model on all Archimedean lattices, using a parallel implementation of the algorithm of Ref.[1] that gives us access to larger sizes than previously possible. The exact polynomials are computed for bases of size 6×6 unit cells, and the root in the temperature variable v = e K −1 is determined numerically at q = 1 for bases of size 8 × 8. This leads to improved results for bond percolation thresholds, and for the Potts-model critical manifolds in the real (q, v) plane. In the two most favourable cases, we find now the kagome-lattice threshold to eleven digits and that of the (3, 12 2 ) lattice to thirteen. Our critical manifolds reveal many interesting features in the antiferromagnetic region of the Potts model, and determine accurately the extent of the Berker-Kadanoff phase for the lattices studied.

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Simultaneous analysis of three-dimensional percolation models

et al. 2013

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We simulate the bond and site percolation models on several three-dimensional lattices, including the diamond, body-centered cubic, and face-centered cubic lattices. As on the simple-cubic lattice [Phys. Rev. E, \textbf{87} 052107 (2013)], it is observed that in comparison with dimensionless ratios based on cluster-size distribution, certain wrapping probabilities exhibit weaker finite-size corrections and are more sensitive to the deviation from percolation threshold $p_c$, and thus provide a powerful means for determining $p_c$. We analyze the numerical data of the wrapping probabilities simultaneously such that universal parameters are shared by the aforementioned models, and thus significantly improved estimates of $p_c$ are obtained.Comment: 7 pages, 8 figure

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Publisher's Note: Critical frontier of the Potts and percolation models on triangular-type and kagome-type lattices. II. Numerical analysis [Phys. Rev. E81, 061111 (2010)]

Cited by 15 publications

References 0 publications

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

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

Potts-model critical manifolds revisited

Simultaneous analysis of three-dimensional percolation models

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