Estimation of bond percolation thresholds on the Archimedean lattices

Parviainen, Robert

doi:10.1088/1751-8113/40/31/005

Cited by 37 publications

(71 citation statements)

References 9 publications

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“…Noting that even this conservative estimate is in significant disagreement with the numerical result of Parviainen [54] (see Table 22) we therefore turn to a different way on constructing the snub hexagonal lattice. The second construction is based on the four-terminal representation shown in Figure 21.…”

Section: Snub Hexagonal Lattice (3 4 6)mentioning

confidence: 94%

High-precision percolation thresholds and Potts-model critical manifolds from graph polynomials

Jacobsen

2014

J. Phys. A: Math. Theor.

105

209

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The critical curves of the q-state Potts model can be determined exactly for regular two-dimensional lattices G that are of the three-terminal type. This comprises the square, triangular, hexagonal and bowtie lattices. Jacobsen and Scullard have defined a graph polynomial P B (q, v) that gives access to the critical manifold for general lattices. It depends on a finite repeating part of the lattice, called the basis B, and its real roots in the temperature variable v = e K − 1 provide increasingly accurate approximations to the critical manifolds upon increasing the size of B. Using transfer matrix techniques, these authors computed P B (q, v) for large bases (up to 243 edges), obtaining determinations of the ferromagnetic critical point v c > 0 for the (4, 8 2 ), kagome, and (3, 12 2 ) lattices to a precision (of the order 10 −8 ) slightly superior to that of the best available Monte Carlo simulations.In this paper we describe a more efficient transfer matrix approach to the computation of P B (q, v) that relies on a formulation within the periodic Temperley-Lieb algebra. This makes possible computations for substantially larger bases (up to 882 edges), and the precision on v c is hence taken to the range 10 −13 . We further show that a large variety of regular lattices can be cast in a form suitable for this approach. This includes all Archimedean lattices, their duals and their medials. For all these lattices we tabulate high-precision estimates of the bond percolation thresholds p c and Potts critical points v c . We also trace and discuss the full Potts critical manifold in the (q, v) plane, paying special attention to the antiferromagnetic region v < 0. Finally, we adapt the technique to site percolation as well, and compute the polynomials P B (p) for certain Archimedean and dual lattices (those having only cubic and quartic vertices), using very large bases (up to 243 vertices). This produces the site percolation thresholds p c to a precision of the order 10 −9 .

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Section: Snub Hexagonal Lattice (3 4 6)mentioning

confidence: 94%

High-precision percolation thresholds and Potts-model critical manifolds from graph polynomials

Jacobsen

2014

J. Phys. A: Math. Theor.

105

209

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“…Note that the cases (n, k) = (2, 0) and (2, 1) now produce different results. The bond threshold of this lattice has not been studied as thoroughly as that of the kagome lattice, and apparently the only high-precision result is Parviainen's [11], p c = 0.676 802 32(63). Our 4 × 4 results are within two standard deviations.…”

Section: (4 8 2 ) Latticementioning

confidence: 99%

Transfer matrix computation of generalized critical polynomials in percolation

Scullard¹,

Jacobsen²

2012

J. Phys. A: Math. Theor.

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Percolation thresholds have recently been studied by means of a graph polynomial P B (p), henceforth referred to as the critical polynomial, that may be defined on any periodic lattice. The polynomial depends on a finite subgraph B, called the basis, and the way in which the basis is tiled to form the lattice. The unique root of P B (p) in [0, 1] either gives the exact percolation threshold for the lattice, or provides an approximation that becomes more accurate with appropriately increasing size of B. Initially P B (p) was defined by a contraction-deletion identity, similar to that satisfied by the Tutte polynomial. Here, we give an alternative probabilistic definition of P B (p), which allows for much more efficient computations, by using the transfer matrix, than was previously possible with contraction-deletion.We present bond percolation polynomials for the (4, 8 2 ), kagome, and (3, 12 2 ) lattices for bases of up to respectively 96, 162, and 243 edges, much larger than the previous limit of 36 edges using contraction-deletion. We discuss in detail the role of the symmetries and the embedding of B. For the largest bases, we obtain the thresholds p c (4, 8 2 ) = 0.676 803 329 · · ·, p c (kagome) = 0.524 404 998 · · ·, p c (3, 12 2 ) = 0.740 420 798 · · ·, comparable to the best simulation results. We also show that the alternative definition of P B (p) can be applied to study site percolation problems.

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“…Note also that min-cut data for Et-OCS and Et-OCS(Me) fall along a similar trend (as with the bulk two-dimensional square lattice, the EMA provides only an approximation for similar networks. The square, (3,4,6,4), and (3,6,3,6) two-dimensional Archimedean lattices all have a coordination number of four but the bond percolation thresholds of these lattices are 0.5000, 0.5248, [ 30 ] and 0.5244, [ 31 ] respectively. The accuracy of the EMA in predicting the min-cut and fracture energy scaling for hybrid glasses means that remarkably, the fracture energy scaling with connectivity for amorphous, three-dimensional hybrid glasses can be well predicted to fi rst order by modeling the glass network as a square lattice of Si atoms.…”

Section: Cohesive Fracturementioning

confidence: 99%

Molecular Origins of the Mechanical Behavior of Hybrid Glasses

Oliver

Dubois

Sherwood

et al. 2010

Adv Funct Materials

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Hybrid organic-inorganic glasses exhibit unique electro-optical properties along with excellent thermal stability. Their inherently mechanically fragile nature, however, which derives from the oxide component of the hybrid glass network together with the presence of terminal groups that reduce network connectivity, remains a fundamental challenge for their integration in nanoscience and energy technologies. We report on a combined synthesis and computational strategy to elucidate the effect of molecular structure on mechanical properties of hybrid glass fi lms. We fi rst demonstrate the importance of rigidity percolation to elastic behavior. Secondly, using a novel application of graph theory, we reveal the complex 3-D fracture path at the molecular scale and show that fracture energy in brittle hybrid glasses is fundamentally governed by the bond percolation properties of the network. The computational tools and scaling laws presented provide a robust predictive capability for guiding precursor selection and molecular network design of advanced hybrid organic-inorganic materials.

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Estimation of bond percolation thresholds on the Archimedean lattices

Abstract: Abstract. We give accurate estimates for the bond percolation critical probabilities on seven Archimedean lattices, for which the critical probabilities are unknown, using an algorithm of Newman and Ziff.

Cited by 37 publications

References 9 publications

High-precision percolation thresholds and Potts-model critical manifolds from graph polynomials

High-precision percolation thresholds and Potts-model critical manifolds from graph polynomials

Transfer matrix computation of generalized critical polynomials in percolation

Molecular Origins of the Mechanical Behavior of Hybrid Glasses

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