Bond percolation critical probability bounds for three Archimedean lattices

Wierman, John C.

doi:10.1002/rsa.10029

Cited by 16 publications

(17 citation statements)

References 27 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…. , and Wierman [16] established rigorous bounds of (0.738598, 0.744900) using the substitution method. Using the symmetry reduction technique, we tightened this bound to (0.739399, 0.741757) in May and Wierman [9].…”

Section: Results For the (3 12 2 ) Bond Modelmentioning

confidence: 99%

See 1 more Smart Citation

The Application of Non-Crossing Partitions to Improving Percolation Threshold Bounds

May¹,

Wierman²

2006

Combinator. Probab. Comp.

Self Cite

View full text Add to dashboard Cite

We describe how non-crossing partitions arise in substitution method calculations. By using efficient algorithms for computing non-crossing partitions we are able to substantially reduce the computational effort, which enables us to compute improved bounds on the percolation thresholds for three percolation models. For the Kagomé bond model we improve bounds from 0.5182 p c 0.5335 to 0.522197 p c 0.526873, improving the range from 0.0153 to 0.004676. For the (3, 12 2 ) bond model we improve bounds from 0.7393 p c 0.7418 to 0.739773 p c 0.741125, improving the range from 0.0025 to 0.001352. We also improve the upper bound for the hexagonal site model, from 0.794717 to 0.743359.

show abstract

Section: Results For the (3 12 2 ) Bond Modelmentioning

confidence: 99%

“…We define an up-set of n to be a subset U of n such that, if π 1 ∈ U and π 1 < π 2 , then π 2 ∈ U . If P G p (π ) and P H p c (π ) are probability measures on n , then P G p is stochastically no greater than P H p c (denoted [16,17].…”

Section: The Substitution Methodsmentioning

confidence: 99%

The Application of Non-Crossing Partitions to Improving Percolation Threshold Bounds

May¹,

Wierman²

2006

Combinator. Probab. Comp.

Self Cite

View full text Add to dashboard Cite

show abstract

“…The remarkable fact that allows exact bond percolation threshold values to be obtained is that it is possible to choose the parameters p and q so that the two probability measures are exactly equal. (Note that in cases with more boundary vertices, where the probability measures cannot be made equal, the concept of stochastic ordering of probability measures may be used to determine mathematically rigorous bounds for percolation thresholds, using the substitution method [12,13,22,24,25,26,27].) By the duality relationship between G and G * , we have that for each configuration of open and closed edges, the following five statements hold: While these statements are intuitively clear by drawing diagrams, the proofs of these statements rely on duality.…”

Section: Reduction To a Single Equationmentioning

confidence: 99%

Self-dual Planar Hypergraphs and Exact Bond Percolation Thresholds

Wierman¹,

Ziff²

2011

Electron. J. Combin.

Self Cite

View full text Add to dashboard Cite

A generalized star-triangle transformation and a concept of triangle-duality have been introduced recently in the physics literature to predict exact percolation threshold values of several lattices. We investigate the mathematical conditions for the solution of bond percolation models, and identify an infinite class of lattice graphs for which exact bond percolation thresholds may be rigorously determined as the solution of a polynomial equation. This class is naturally described in terms of hypergraphs, leading to definitions of planar hypergraphs and self-dual planar hypergraphs. We show that there exist infinitely many self-dual planar 3-uniform hypergraphs, and, as a consequence, that there exist infinitely many real numbers a ∈ [0, 1] for which there are infinitely many lattices that have bond percolation threshold equal to a.

show abstract

“…Suding and Ziff presented precise thresholds for site percolation on eight Archimedean lattices determined by the hullwalk gradient-percolation simulation method [26]. Rigorous bounds for the bond percolation critical probability are determined for three Archimedean lattices by Wierman [27]. In addition, Scullard and Ziff showed that the exact determination of the bond percolation threshold for the Martini lattice can be used to provide approximations to the Kagome and (3, 12 2 ) lattices [28].…”

Section: Introductionmentioning

confidence: 99%

Universality of the Ising and theS=1model on Archimedean lattices: A Monte Carlo determination

et al. 2012

View full text Add to dashboard Cite

The Ising model S = 1/2 and the S = 1 model are studied by efficient Monte Carlo schemes on the (3,4,6,4) and the (3,3,3,3,6) Archimedean lattices. The algorithms used, a hybrid Metropolis-Wolff algorithm and a parallel tempering protocol, are briefly described and compared with the simple Metropolis algorithm. Accurate Monte Carlo data are produced at the exact critical temperatures of the Ising model for these lattices. Their finite-size analysis provide, with high accuracy, all critical exponents which, as expected, are the same with the well known 2d Ising model exact values. A detailed finite-size scaling analysis of our Monte Carlo data for the S = 1 model on the same lattices provides very clear evidence that this model obeys, also very well, the 2d Ising model critical exponents. As a result, we find that recent Monte Carlo simulations and attempts to define effective dimensionality for the S = 1 model on these lattices are misleading. Accurate estimates are obtained for the critical amplitudes of the logarithmic expansions of the specific heat for both models on the two Archimedean lattices.

show abstract

Bond percolation critical probability bounds for three Archimedean lattices

Cited by 16 publications

References 27 publications

The Application of Non-Crossing Partitions to Improving Percolation Threshold Bounds

The Application of Non-Crossing Partitions to Improving Percolation Threshold Bounds

Self-dual Planar Hypergraphs and Exact Bond Percolation Thresholds

Universality of the Ising and theS=1model on Archimedean lattices: A Monte Carlo determination

Contact Info

Product

Resources

About