A Disproof of Tsallis' Bond Percolation Threshold Conjecture for the Kagome Lattice

Wierman, John C.; Yu, Gaoran; Huang, Theodore

doi:10.37236/5117

Cited by 6 publications

(10 citation statements)

References 43 publications

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“…Despite the problem's apparent simplicity, the value of p c is unknown for most lattices, with exact solutions available only on a restricted class [4][5][6][7][8][9]. There have been great advances in the understanding of the continuum limit using conformal invariance and stochastic Loewner evolution [10][11][12][13], in which the details of the underlying lattice are irrelevant, but progress on unsolved lattice-dependent quantities, such as the critical probabilities of most of the Archimedean lattices, has been limited to the derivation of rigorous bounds [14][15][16][17][18][19][20] (though these are ever-tightening) and numerical studies [21][22][23]. For example, the critical probability for bond percolation on the kagome lattice is known rigorously to satisfy [20] 0.522551 < p c < 0.526490,…”

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confidence: 99%

Bond percolation thresholds on Archimedean lattices from critical polynomial roots

Scullard

Jacobsen

2020

Phys. Rev. Research

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We present percolation thresholds calculated numerically with the eigenvalue formulation of the method of critical polynomials; developed in the last few years, it has already proven to be orders of magnitude more accurate than traditional techniques. Here we report the result of large parallel calculations to produce what we believe may become the reference values of bond percolation thresholds on the Archimedean lattices for years to come. For example, for the kagome lattice we find pc = 0.524 404 999 167 448 20(1), whereas the best estimate using standard techniques is pc = 0.524 404 99(2). We further provide strong evidence that there are two classes of lattices: one for which the first three scaling exponents characterizing the finite-size corrections to pc are ∆ = 6, 7, 8, and another for which ∆ = 4, 6, 8. We discuss the open questions related to the method, such as the full scaling law, as well as its potential for determining critical points of other models.

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confidence: 99%

Bond percolation thresholds on Archimedean lattices from critical polynomial roots

Scullard

Jacobsen

2020

Phys. Rev. Research

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“…The substitution method provides most of the current best mathematicallyrigorous percolation threshold bounds for two-and three-dimensional lattice models. Most notably, it produced bounds that determined the first two digits of the Kagome lattice bond percolation threshold [38] and the first three digits of the (3, 12 2 ) lattice bond percolation threshold [39], disproving conjectured exact values in [40]. It also establishes equality of the critical exponents (assuming they exist) of a dual pair of planar bond percolation models [41,42].…”

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confidence: 88%

New bounds for the site percolation threshold of the hexagonal lattice

Wierman¹

2022

J. Phys. A: Math. Theor.

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The site percolation threshold of the hexagonal lattice satisfies $0.656246 < p_c < 0.739695$. For comparison, the largest previous lower bound of 0.652703... was established in 1981, and the smallest previous upper bound of 0.743359 was derived in 2007. The bound is obtained by using the substitution method to compare the hexagonal lattice site model to an exactly-solved two-parameter site percolation model on the martini lattice. Computational reductions involving graph-welding, symmetry, non-crossing partitions, and network flow computations overcome challenges to establishing stochastic ordering between the models.

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“…Besides being a mathematically challenging problem, research on bounds potentially develops techniques which may eventually lead to exact solutions, as it did for the square lattice bond percolation model. Mathematically rigorous bounds have become increasingly accurate, recently providing three-digit accuracy for the (3, 12 2 ) lattice [48] and two-digit accuracy for the kagome lattice [50], disproving long-standing conjectured exact values [37]. Percolation threshold bounds are also applied in the study of related models.…”

Section: Introductionmentioning

confidence: 99%

An upper bound for the bond percolation threshold of the cubic lattice by a growth process approach

Wierman

2021

J. Appl. Probab.

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We reduce the upper bound for the bond percolation threshold of the cubic lattice from 0.447 792 to 0.347 297. The bound is obtained by a growth process approach which views the open cluster of a bond percolation model as a dynamic process. A three-dimensional dynamic process on the cubic lattice is constructed and then projected onto a carefully chosen plane to obtain a two-dimensional dynamic process on a triangular lattice. We compare the bond percolation models on the cubic lattice and their projections, and demonstrate that the bond percolation threshold of the cubic lattice is no greater than that of the triangular lattice. Applying the approach to the body-centered cubic lattice yields an upper bound of 0.292 893 for its bond percolation threshold.

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A Disproof of Tsallis' Bond Percolation Threshold Conjecture for the Kagome Lattice

Cited by 6 publications

References 43 publications

Bond percolation thresholds on Archimedean lattices from critical polynomial roots

Bond percolation thresholds on Archimedean lattices from critical polynomial roots

New bounds for the site percolation threshold of the hexagonal lattice

An upper bound for the bond percolation threshold of the cubic lattice by a growth process approach

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