Bond percolation on honeycomb and triangular lattices

1981 ◽  
Vol 13 (2) ◽  
pp. 298-313 ◽  
Author(s):  
John C. Wierman
Keyword(s):  
Lower Bounds ◽  
Triangular Lattice ◽  
Cluster Size ◽  
Honeycomb Lattice ◽  
Bond Percolation ◽  
Critical Probability ◽  
Lattice Graph ◽  
The Mean ◽  
Triangular Lattices ◽  
Crossing Probabilities

The two common critical probabilities for a lattice graph L are the cluster size critical probability pH(L) and the mean cluster size critical probability pT(L). The values for the honeycomb lattice H and the triangular lattice T are proved to be pH(H) = pT(H) = 1–2 sin (π/18) and PH(T) = pT(T) = 2 sin (π/18). The proof uses the duality relationship and the star-triangle relationship between the two lattices, to find lower bounds for sponge-crossing probabilities.

Bond percolation on honeycomb and triangular lattices

1981 ◽  
Vol 13 (02) ◽  
pp. 298-313 ◽  
Author(s):  
John C. Wierman
Keyword(s):  
Lower Bounds ◽  
Triangular Lattice ◽  
Cluster Size ◽  
Honeycomb Lattice ◽  
Bond Percolation ◽  
Critical Probability ◽  
Lattice Graph ◽  
The Mean ◽  
Triangular Lattices ◽  
Crossing Probabilities

The two common critical probabilities for a lattice graphLare the cluster size critical probabilitypH(L) and the mean cluster size critical probabilitypT(L). The values for the honeycomb latticeHand the triangular latticeTare proved to bepH(H) =pT(H) = 1–2 sin (π/18) andPH(T) =pT(T) = 2 sin (π/18). The proof uses the duality relationship and the star-triangle relationship between the two lattices, to find lower bounds for sponge-crossing probabilities.

Inequalities with applications to percolation and reliability

1985 ◽  
Vol 22 (3) ◽  
pp. 556-569 ◽  
Author(s):  
J. Van Den Berg ◽  
H. Kesten
Keyword(s):  
Lower Bound ◽  
Probability Measure ◽  
Cluster Size ◽  
Discrete Analog ◽  
Reliability Theory ◽  
Probability Measures ◽  
Bond Percolation ◽  
Critical Probability ◽  
New Better Than Used ◽  
Better Than

A probability measure μ on ℝn+ is defined to be strongly new better than used (SNBU) if for all increasing subsets . For n = 1 this is equivalent to being new better than used (NBU distributions play an important role in reliability theory). We derive an inequality concerning products of NBU probability measures, which has as a consequence that if μ1, μ2, ···, μn are NBU probability measures on ℝ+, then the product-measure μ = μ × μ2 × ··· × μn on ℝn+ is SNBU. A discrete analog (i.e., with N instead of ℝ+) also holds.Applications are given to reliability and percolation. The latter are based on a new inequality for Bernoulli sequences, going in the opposite direction to the FKG–Harris inequality. The main application (3.15) gives a lower bound for the tail of the cluster size distribution for bond-percolation at the critical probability. Further applications are simplified proofs of some known results in percolation. A more general inequality (which contains the above as well as the FKG-Harris inequality) is conjectured, and connections with an inequality of Hammersley [12] and others ([17], [19] and [7]) are indicated.

Inequalities with applications to percolation and reliability

1985 ◽  
Vol 22 (03) ◽  
pp. 556-569 ◽  
Author(s):  
J. Van Den Berg ◽  
H. Kesten
Keyword(s):  
Lower Bound ◽  
Probability Measure ◽  
Cluster Size ◽  
Discrete Analog ◽  
Reliability Theory ◽  
Probability Measures ◽  
Bond Percolation ◽  
Critical Probability ◽  
New Better Than Used ◽  
Better Than

A probability measure μ on ℝn + is defined to be strongly new better than used (SNBU) if for all increasing subsets . For n = 1 this is equivalent to being new better than used (NBU distributions play an important role in reliability theory). We derive an inequality concerning products of NBU probability measures, which has as a consequence that if μ 1, μ 2, ···, μn are NBU probability measures on ℝ+, then the product-measure μ = μ × μ 2 × ··· × μn on ℝn + is SNBU. A discrete analog (i.e., with N instead of ℝ+) also holds. Applications are given to reliability and percolation. The latter are based on a new inequality for Bernoulli sequences, going in the opposite direction to the FKG–Harris inequality. The main application (3.15) gives a lower bound for the tail of the cluster size distribution for bond-percolation at the critical probability. Further applications are simplified proofs of some known results in percolation. A more general inequality (which contains the above as well as the FKG-Harris inequality) is conjectured, and connections with an inequality of Hammersley [12] and others ([17], [19] and [7]) are indicated.

CARDY'S FORMULA FOR CERTAIN MODELS OF THE BOND-TRIANGULAR TYPE

2007 ◽  
Vol 19 (05) ◽  
pp. 511-565 ◽  
Author(s):  
L. CHAYES ◽  
H. K. LEI
Keyword(s):  
Critical Behavior ◽  
Triangular Lattice ◽  
Continuum Limit ◽  
Percolation Model ◽  
Bond Percolation ◽  
Local Correlations ◽  
Triangular Type ◽  
Crossing Probabilities ◽  
The Continuum

We introduce and study a family of 2D percolation systems which are based on the bond percolation model of the triangular lattice. The system under study has local correlations, however, bonds separated by a few lattice spacings act independently of one another. By avoiding explicit use of microscopic paths, it is first established that the model possesses the typical attributes which are indicative of critical behavior in 2D percolation problems. Subsequently, the so-called Cardy–Carleson functions are demonstrated to satisfy, in the continuum limit, Cardy's formula for crossing probabilities. This extends the results of S. Smirnov to a non-trivial class of critical 2D percolation systems.

A simple method to determine the mean cluster size in a molecular beam

1991 ◽  
Vol 21 (3) ◽  
pp. 265-269 ◽  
Author(s):  
J. Cuvellier ◽  
P. Meynadier ◽  
P. Pujo ◽  
O. Sublemontier ◽  
J-P Visticot ◽  
...  
Keyword(s):  
Cluster Size ◽  
Molecular Beam ◽  
Simple Method ◽  
The Mean

Distribution of the number of alleles in multigene families

1992 ◽  
Vol 29 (04) ◽  
pp. 759-769
Author(s):  
R. C. Griffiths
Keyword(s):  
Stochastic Model ◽  
Gene Conversion ◽  
Lower Bounds ◽  
Multigene Families ◽  
Expected Number ◽  
Allele Type ◽  
The Mean

The distribution of the number of alleles in samples from r chromosomes is studied. The stochastic model used includes gene conversion within chromosomes and mutation at loci on the chromosomes. A method is described for simulating the distribution of alleles and an algorithm given for computing lower bounds for the mean number of alleles. A formula is derived for the expected number of samples from r chromosomes which contain the allele type of a locus chosen at random.

Kinetic Monte Carlo Simulation of Impurity Effects on Nucleation and Growth of SiC Clusters on Si(100)

2013 ◽  
Vol 740-742 ◽  
pp. 393-396
Author(s):  
Maxim N. Lubov ◽  
Jörg Pezoldt ◽  
Yuri V. Trushin
Keyword(s):  
Monte Carlo Simulation ◽  
Monte Carlo ◽  
Cluster Size ◽  
Kinetic Monte Carlo ◽  
Nucleation And Growth ◽  
Nucleation Density ◽  
Nucleation Process ◽  
Impurity Effects ◽  
Kinetic Monte Carlo Simulations ◽  
The Mean

The influence of attractive and repulsive impurities on the nucleation process of the SiC clusters on Si(100) surface was investigated. Kinetic Monte Carlo simulations of the SiC clusters growth show that that increase of the impurity concentration (both attractive and repulsive) leads to decrease of the mean cluster size and rise of the nucleation density of the clusters.

Lower bounds for the critical probability in percolation models with oriented bonds

1980 ◽  
Vol 17 (04) ◽  
pp. 979-986 ◽  
Author(s):  
Lawrence Gray ◽  
John C. Wierman ◽  
R. T. Smythe
Keyword(s):  
Lower Bounds ◽  
Oriented Percolation ◽  
Critical Probability ◽  
Critical Percolation ◽  
Simple Method

In completely or partially oriented percolation models, a conceptually simple method, using barriers to enclose all open paths from the origin, improves the best previous lower bounds for the critical percolation probabilities.

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