# Universal condition for critical percolation thresholds of kagomé-like lattices.

@article{Ziff2008UniversalCF, title={Universal condition for critical percolation thresholds of kagom{\'e}-like lattices.}, author={Robert M. Ziff and Hang Gu}, journal={Physical review. E, Statistical, nonlinear, and soft matter physics}, year={2008}, volume={79 2 Pt 1}, pages={ 020102 } }

Lattices that can be represented in a kagomé-like form are shown to satisfy a universal percolation criticality condition, expressed as a relation between P3 , the probability that all three vertices in the triangle connect, and P0 , the probability that none connect. A linear approximation for P3(P0) is derived and appears to provide a rigorous upper bound for critical thresholds. A numerically determined relation for P3(P0) gives thresholds for the kagomé, site-bond honeycomb, (3-12;{2…

## 27 Citations

### A Disproof of Tsallis' Bond Percolation Threshold Conjecture for the Kagome Lattice

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- 2015

The substitution method is used, which is based on stochastic ordering, to compare the probability distribution of connections in the homogeneous bond percolation model on the kagome lattice to those of an exactly-solved inhomogeneous bondPercolation models on the martini lattice.

### Transfer matrix computation of generalized critical polynomials in percolation

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Percolation thresholds have recently been studied by means of a graph polynomial PB(p), henceforth referred to as the critical polynomial, that may be defined on any periodic lattice. The polynomial…

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- PhysicsJournal of Physics A: Mathematical and Theoretical
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We study the percolation critical surface of the kagome lattice in which each triangle is allowed an arbitrary connectivity. Using the method of critical polynomials, we find points along this…

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Improved bounds are proved for bond percolation thresholds for certain Archimedean lattices using the substitution method with new comparisons between models and more efficient computational…

### New bounds for the site percolation threshold of the hexagonal lattice

- Computer ScienceJournal of Physics A: Mathematical and Theoretical
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The site percolation threshold of the hexagonal lattice satisfies 0.656 246 < p c < 0.739 695, and this bound is obtained by using the substitution method to compare the hexagon lattice site model to an exactly-solved two-parameter site perColation model on the martini lattice.

### Critical manifold of the kagome-lattice Potts model

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Any two-dimensional infinite regular lattice G can be produced by tiling the plane with a finite subgraph B⊆G; we call B a basis of G. We introduce a two-parameter graph polynomial PB(q, v) that…

### Tight bounds for the bond percolation threshold of the (3, 122) lattice

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Improved mathematically rigorous upper and lower bounds for the bond percolation threshold are established using the substitution method, in which stochastic ordering is checked using a network flow algorithm.

### Polynomial sequences for bond percolation critical thresholds

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- 2011

In this paper, I compute the inhomogeneous (multi-probability) bond critical surfaces for the (4, 6, 12) and (34, 6) lattices using the linearity approximation described in Scullard and Ziff (2010 J.…

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