scholarly journals Precise bond percolation thresholds on several four-dimensional lattices

2020 ◽  
Vol 2 (1) ◽  
Author(s):  
Zhipeng Xun ◽  
Robert M. Ziff
Keyword(s):  
Bond Percolation ◽  
Percolation Thresholds

scholarly journals Calculation of Percolation Thresholds in High Dimensions for FCC, BCC and Diamond Lattices

1998 ◽  
Vol 09 (04) ◽  
pp. 529-540 ◽  
Author(s):  
Steven C. van der Marck
Keyword(s):  
Bond Percolation ◽  
Face Centered Cubic ◽  
Body Centered Cubic ◽  
High Dimensions ◽  
The Face ◽  
Percolation Thresholds ◽  
Six Dimensions ◽  
Face Centered

Site and bond percolation thresholds are calculated for the face centered cubic, body centered cubic and diamond lattices in four, five and six dimensions. The results are used to study the behavior of percolation thresholds as a functions of dimension. It is shown that the predictions from a recently proposed invariant for percolation thresholds are not satisfactory for these lattices.

scholarly journals AN INVESTIGATION OF SITE-BOND PERCOLATION ON MANY LATTICES

1999 ◽  
Vol 10 (07) ◽  
pp. 1193-1204 ◽  
Author(s):  
YURIY YU. TARASEVICH ◽  
STEVEN C. VAN DER MARCK
Keyword(s):  
Bond Percolation ◽  
Threshold Values ◽  
Five Dimensions ◽  
Percolation Thresholds ◽  
Approximate Equations

A calculation of site-bond percolation thresholds in many lattices in two to five dimensions is presented. The line of threshold values has been parametrized in the literature, but we show here that there are strong deviations from the known approximate equations. We propose an alternative parametrization that lies much closer to the numerical values.

scholarly journals Aperiodic Non-Isomorphic Lattices with Equivalent Percolation and Random-Cluster Models

10.37236/320 ◽  
2010 ◽  
Vol 17 (1) ◽  
Author(s):  
Klas Markström ◽  
John C. Wierman
Keyword(s):  
Percolation Threshold ◽  
Cluster Model ◽  
Large Scale ◽  
Threshold Value ◽  
Bond Percolation ◽  
Random Cluster Model ◽  
Random Cluster ◽  
Random Cluster Models ◽  
Percolation Thresholds ◽  
Scale Properties

We explicitly construct an uncountable class of infinite aperiodic plane graphs which have equal, and explicitly computable, bond percolation thresholds. Furthermore for both bond percolation and the random-cluster model all large scale properties, such as the values of the percolation threshold and the critical exponents, of the graphs are equal. This equivalence holds for all values of $p$ and all $q\in[0,\infty]$ for the random-cluster model. The graphs are constructed by placing a copy of a rotor gadget graph or its reflection in each hyperedge of a connected self-dual 3-uniform plane hypergraph lattice. The exact bond percolation threshold may be explicitly determined as the root of a polynomial by using a generalised star-triangle transformation. Related randomly oriented models share the same bond percolation threshold value.

scholarly journals Percolation on anisotropic media, the Bethe lattice revisited. Application to fracture networks

1997 ◽  
Vol 4 (1) ◽  
pp. 11-18 ◽  
Author(s):  
M. Canals ◽  
M. Ayt Ougoudal
Keyword(s):  
Good Accuracy ◽  
Linear Equations ◽  
Bethe Lattice ◽  
Anisotropic Media ◽  
Percolation Model ◽  
Bond Percolation ◽  
Fracture Networks ◽  
Dead End ◽  
Percolation Thresholds ◽  
Non Linear Equations

Abstract. A bond-percolation model based on the Bethe Lattice is presented. This model handles anisotropic and multiscale situations where, typically, the bond probability is non unique and depends on the sites it connects. The model is governed by a set of non-linear equations which are solved numerically. As a result, the structure of the network is obtained: strengths of the backbone, dead-end roads and finite clusters. Percolation thresholds and cluster sizes are also obtained. Application to fissured media is presented and random simulations of 3D distributions of fractures show the good accuracy of the model.

scholarly journals A Boltzmann Approach to Percolation on Random Triangulations

2019 ◽  
Vol 71 (1) ◽  
pp. 1-43 ◽  
Author(s):  
Olivier Bernardi ◽  
Nicolas Curien ◽  
Grégory Miermont
Keyword(s):  
Percolation Model ◽  
Bond Percolation ◽  
Critical Percolation ◽  
Site Percolation ◽  
Percolation Clusters ◽  
Percolation Parameter ◽  
Stable Map ◽  
Percolation Thresholds ◽  
Boltzmann Model ◽  
Independent Derivation

AbstractWe study the percolation model on Boltzmann triangulations using a generating function approach. More precisely, we consider a Boltzmann model on the set of finite planar triangulations, together with a percolation configuration (either site-percolation or bond-percolation) on this triangulation. By enumerating triangulations with boundaries according to both the boundary length and the number of vertices/edges on the boundary, we are able to identify a phase transition for the geometry of the origin cluster. For instance, we show that the probability that a percolation interface has length$n$decays exponentially with$n$except at a particular value$p_{c}$of the percolation parameter$p$for which the decay is polynomial (of order$n^{-10/3}$). Moreover, the probability that the origin cluster has size$n$decays exponentially if$p<p_{c}$and polynomially if$p\geqslant p_{c}$.The critical percolation value is$p_{c}=1/2$for site percolation, and$p_{c}=(2\sqrt{3}-1)/11$for bond percolation. These values coincide with critical percolation thresholds for infinite triangulations identified by Angel for site-percolation, and by Angel and Curien for bond-percolation, and we give an independent derivation of these percolation thresholds.Lastly, we revisit the criticality conditions for random Boltzmann maps, and argue that at$p_{c}$, the percolation clusters conditioned to have size$n$should converge toward the stable map of parameter$\frac{7}{6}$introduced by Le Gall and Miermont. This enables us to derive heuristically some new critical exponents.

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