Upper and Lower Bounds for the Kagomé Lattice Bond Percolation Critical Probability

Wierman, John C.

doi:10.1017/s0963548302005370

Cited by 14 publications

(21 citation statements)

References 33 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…This result is within the error bars of [8] and falls within the rigorous bounds of [19], which raises the possibility that the result is exact. Clearly, more precise numerical work for both lattices is called for.…”

supporting

confidence: 82%

See 1 more Smart Citation

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

2006

View full text Add to dashboard Cite

Here we show how the recent exact determination of the bond percolation threshold for the martini lattice can be used to provide approximations to the unsolved kagomé and (3, 122 ) lattices. We present two different methods, one of which provides an approximation to the inhomogeneous kagomé and (3, 122 ) bond problems, and the other gives estimates of pc for the homogeneous kagomé (0.5244088...) and (3,12 2 ) (0.7404212...) problems that respectively agree with numerical results to five and six significant figures.

show abstract

supporting

confidence: 82%

“…This work is largely carried out by Wierman and co-workers [19,20], using a technique called substitution. The method is such that continual refinements are possible and the most current rigorous bounds are [21]:…”

mentioning

confidence: 99%

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

2006

View full text Add to dashboard Cite

show abstract

“…While this is much more stringent than Wierman's bounds [31], 0.5209 < p c < 0.5291 (30) his result is completely rigourous while ours is only a guess based on the observed monotonicity in the estimates with n. In fact, as we will soon see, the (3, 12 2 ) lattice violates this monotonicity for the n = 3 hexagonal basis, making (29) even less certain. Nevertheless, the kagome n = 2 and 3 (i.e., 72 and 162 edges) predictions appear to be converged to at least seven digits, and agree with the transfer matrix result p c = 0.524 404 99(2) of Feng, Deng, and Blöte [33] to eight decimal places (the limit of their accuracy).…”

Section: Hexagonal Basesmentioning

confidence: 47%

Transfer matrix computation of generalized critical polynomials in percolation

Scullard¹,

Jacobsen²

2012

J. Phys. A: Math. Theor.

View full text Add to dashboard Cite

Percolation thresholds have recently been studied by means of a graph polynomial P B (p), henceforth referred to as the critical polynomial, that may be defined on any periodic lattice. The polynomial depends on a finite subgraph B, called the basis, and the way in which the basis is tiled to form the lattice. The unique root of P B (p) in [0, 1] either gives the exact percolation threshold for the lattice, or provides an approximation that becomes more accurate with appropriately increasing size of B. Initially P B (p) was defined by a contraction-deletion identity, similar to that satisfied by the Tutte polynomial. Here, we give an alternative probabilistic definition of P B (p), which allows for much more efficient computations, by using the transfer matrix, than was previously possible with contraction-deletion.We present bond percolation polynomials for the (4, 8 2 ), kagome, and (3, 12 2 ) lattices for bases of up to respectively 96, 162, and 243 edges, much larger than the previous limit of 36 edges using contraction-deletion. We discuss in detail the role of the symmetries and the embedding of B. For the largest bases, we obtain the thresholds p c (4, 8 2 ) = 0.676 803 329 · · ·, p c (kagome) = 0.524 404 998 · · ·, p c (3, 12 2 ) = 0.740 420 798 · · ·, comparable to the best simulation results. We also show that the alternative definition of P B (p) can be applied to study site percolation problems.

show abstract

“…However, the Kagomé lattice bond model is not exactly solved, although very accurate bounds, .5182 Ͻ p c ͑Kagomé bond͒ Ͻ .5335, have been proved [27], and recently [29] improved to .5209 Ͻ p c ͑Kagomé bond͒ Ͻ .5291.…”

Section: Bounds For the (4 6 12) And (4 8 2 ) Latticesmentioning

confidence: 99%

Bond percolation critical probability bounds for three Archimedean lattices

Wierman

2002

Random Struct Algorithms

Self Cite

View full text Add to dashboard Cite

ABSTRACT:Rigorous bounds for the bond percolation critical probability are determined for three Archimedean lattices: .7385 Ͻ p c ((3, 12 2 ) bond) Ͻ .7449, .6430 Ͻ p c ((4, 6, 12) bond) Ͻ .7376, .6281 Ͻ p c ((4, 8 2 ) bond) Ͻ .7201. Consequently, the bond percolation critical probability of the (3, 12 2 ) lattice is strictly larger than those of the other ten Archimedean lattices. Thus, the (3, 12 2 ) bond percolation critical probability is possibly the largest of any vertex-transitive graph with bond percolation critical probability that is strictly less than one.

show abstract

Upper and Lower Bounds for the Kagomé Lattice Bond Percolation Critical Probability

Cited by 14 publications

References 33 publications

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

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

Transfer matrix computation of generalized critical polynomials in percolation

Bond percolation critical probability bounds for three Archimedean lattices

Contact Info

Product

Resources

About