The critical manifolds of inhomogeneous bond percolation on bow-tie and checkerboard lattices

@article{Ziff2012TheCM,
  title={The critical manifolds of inhomogeneous bond percolation on bow-tie and checkerboard lattices},
  author={Robert M. Ziff and Christian R Scullard and John C. Wierman and M. Sedlock},
  journal={Journal of Physics A: Mathematical and Theoretical},
  year={2012},
  volume={45}
}
We give a conditional derivation of the inhomogeneous critical percolation manifold of the bow-tie lattice with five different probabilities, a problem that does not appear at first to fall into any known solvable class. Although our argument is mathematically rigorous only on a region of the manifold, we conjecture that the formula is correct over its entire domain, and we provide a non-rigorous argument for this that employs the negative probability regime of the triangular lattice critical… 

Potts critical frontiers of inhomogeneous and asymmetric bow-tie lattices

We study the critical frontiers of the Potts model on two-dimensional bow-tie lattices with fully inhomogeneous coupling constants. Generally, for the Potts critical frontier to be found exactly, the

Exactly solvable percolation problems.

We propose a simple percolation criterion for arbitrary percolation problems. The basic idea is to decompose the system of interest into a hierarchy of neighborhoods, such that the percolation

Duality with real-space renormalization and its application to bond percolation.

  • Masayuki Ohzeki
  • Physics, Mathematics
    Physical review. E, Statistical, nonlinear, and soft matter physics
  • 2013
The resultant generic formulas from the derivation can give several estimations for the bond-percolation thresholds on other lattices rather than the square lattice.

Percolation on hypergraphs with four-edges

We study percolation on self-dual hypergraphs that contain hyperedges with four bounding vertices, or ‘four-edges’, using three different generators, each containing bonds or sites with three

How Inhomogeneous Site Percolation Works on Bethe Lattices: Theory and Application

Using the above theory, the diffusion behaviour of an infectious disease (SARS) is discussed in detail and specific disease-control strategies in consideration of groups with different infection probabilities are presented.

Recent advances in percolation theory and its applications

Erratum: Percolation on hypergraphs with four-edges (2015 J. Phys. A: Math. Theor. 48 405004)

We study percolation on self-dual hypergraphs that contain hyperedges with four bounding vertices, or ‘ four-edges ’ , using three different generators, each containing bonds or sites with three

Criticality, universality, and isoradiality

Critical points and singularities are encountered in the study of critical phenomena in proba- bility and physics. We present recent results concerning the values of such critical points and the

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

The upper bound for the bond percolation threshold of the cubic lattice is reduced from 0.447 792 to 0.347 297 by a growth process approach which views the open cluster of a bond perColation model as a dynamic process.

Accepted for publication in Communications in Mathematical Physics

We prove the existence of long-range order at sufficiently low temperatures, including zero temperature, for the three-state Potts antiferromagnet on a class of quasi-transitive plane

References

SHOWING 1-10 OF 26 REFERENCES

Percolation on Self-Dual Polygon Configurations

Recently, Scullard and Ziff noticed that a broad class of planar percolation models are self-dual under a simple condition that, in a parametrized version of such a model, reduces to a single

Exact bond percolation thresholds in two dimensions

Recent work in percolation has led to exact solutions for the site and bond critical thresholds of many new lattices. Here we show how these results can be extended to other classes of graphs,

Potts and percolation models on bowtie lattices.

The exact critical frontier of the Potts model on bowtie lattices is given, and the critical frontier yields the thresholds of bond percolation on these lattices, which are exactly consistent with the results given by Ziff et al.

Bond percolation on isoradial graphs: criticality and universality

In an investigation of percolation on isoradial graphs, we prove the criticality of canonical bond percolation on isoradial embeddings of planar graphs, thus extending celebrated earlier results for

Critical frontier of the Potts and percolation models on triangular-type and kagome-type lattices. II. Numerical analysis.

Numerical determination of critical properties such as conformal anomaly and magnetic correlation length verifies that the universality principle holds and infers that the homogeneity assumption determines critical frontiers with an accuracy of 5 decimal places or higher.

Universality for bond percolation in two dimensions

All (in)homogeneous bond percolation models on the square, triangular, and hexagonal lattices belong to the same universality class, in the sense that they have identical critical exponents at the

Exact site percolation thresholds using a site-to-bond transformation and the star-triangle transformation.

  • C. Scullard
  • Mathematics
    Physical review. E, Statistical, nonlinear, and soft matter physics
  • 2006
A correlated bond problem on the hexagonal lattice is solved by use of the star-triangle transformation and the site problem is solved, by a particular choice of correlations derived from a site-to-bond transformation, on the martini lattice.

Percolation transitions in two dimensions.

The amplitude of the power-law correction associated with X_{t2}=4 is found to be dependent on the orientation of the lattice with respect to the cylindrical geometry of the finite systems.

Random Cluster Models on the Triangular Lattice

We study percolation and the random cluster model on the triangular lattice with 3-body interactions. Starting with percolation, we generalize the star–triangle transformation: We introduce a new

Analytic results for the percolation transitions of the enhanced binary tree.

  • P. MinnhagenS. Baek
  • Physics
    Physical review. E, Statistical, nonlinear, and soft matter physics
  • 2010
Percolation for a planar lattice has a single percolation threshold, whereas percolation for a negatively curved lattice displays two separate thresholds. The enhanced binary tree (EBT) can be viewed