Fractal Dimensions and Corrections to Scaling for Critical Potts Clusters

Aharony, Amnon; Asikainen, J.

doi:10.1142/s0218348x03001665

Cited by 15 publications

(30 citation statements)

References 10 publications

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“…It turns out that also a few years ago, Aharony and Asikainen [20,21] proposed the same two leading correction exponents ω = 3/2 and 2 for the correction-toscaling for fractal properties of the complete hulls of percolation clusters, relating them to earlier results of den Nijs [41]. Evidently, these same corrections to apply to the cluster statistics.…”

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confidence: 61%

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Correction-to-scaling exponent for two-dimensional percolation

Ziff

2011

Phys. Rev. E

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We show that the correction-to-scaling exponents in two-dimensional percolation are bounded by Ω ≤ 72/91, ω = DΩ ≤ 3/2, and ∆1 = νω ≤ 2, based upon Cardy's result for the critical crossing probability on an annulus. The upper bounds are consistent with many previous measurements of site percolation on square and triangular lattices, and new measurements for bond percolation presented here, suggesting this result is exact. A scaling form evidently applicable to site percolation is also found.

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mentioning

confidence: 61%

“…Evidently, these same corrections to apply to the cluster statistics. Note that the results of [20] have not been verified numerically.…”

mentioning

confidence: 93%

Correction-to-scaling exponent for two-dimensional percolation

Ziff

2011

Phys. Rev. E

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“…The triangular lattice has the advantage of allowing for a unambiguous interface definition, i. e. with no need to define a "tiebreak" rule to deal with four-lines junctions as is the case for the square lattice; on the other hand, it seems that this geometry somewhat enhances the finite-size effects, requiring simulating very large systems to achieve a stable value for D (as opposed, for instance, to the rather stable results from moderate sizes for the square lattice [20]). Moreover, even though we could not apply the predictions in [27] for the subleading terms in S(L) (because they are formulated in a different setting than ours), there are some indications that fitting numerical data simply to the leading-order behaviour does not give reliable fractal dimensions. Another potential concern is our choice -widely common in literature -of working with square aspect ratios ℓ = 1: as shown by the test of jagged vs. non-jagged boundary conditions, a proper determination of D should reasonably be conducted on systems fulfilling to some extent the requirement L x ≫ L y .…”

Section: Discussionmentioning

confidence: 89%

“…We also tried, at the Potts point, other functional forms inspired by the RG arguments in [27], but they had to be discarded in favour of the above.…”

Section: Fractal Dimension From the Interface Lengthmentioning

confidence: 99%

Critical domain walls in the Ashkin–Teller model

Caselle¹,

Lottini²,

Rajabpour³

2011

J. Stat. Mech.

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Abstract. We study the fractal properties of interfaces in the 2d Ashkin-Teller model. The fractal dimension of the symmetric interfaces is calculated along the critical line of the model in the interval between the Ising and the four-states Potts models. Using Schramm's formula for crossing probabilities we show that such interfaces can not be related to the simple SLE κ , except for the Ising point. The same calculation on non-symmetric interfaces is performed at the four-states Potts model: the fractal dimension is compatible with the result coming from Schramm's formula, and we expect a simple SLE κ in this case.

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“…9. Here we show an alternate way of analyzing the data, where we plot s τ −2 P ≥s vs. s −Ω where Ω = 72/91 [21,22] …”

Section: Generator II On Hypergraph Amentioning

confidence: 99%

Percolation on hypergraphs with four-edges

Damavandi

Ziff

2015

J. Phys. A: Math. Theor.

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Abstract. 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 distinct probabilities p, r, and t connecting the four vertices. We find explicit values of these probabilities that satisfy the selfduality conditions discussed by Bollobás and Riordan. This demonstrates that explicit solutions of the self-duality conditions can be found using generators containing bonds and sites with independent probabilities. These solutions also provide new examples of lattices where exact percolation critical points are known. One of the generators exhibits three distinct criticality solutions (p, r, t). We carry out Monte-Carlo simulations of two of the generators on two different hypergraphs to confirm the critical values. For the case of the hypergraph and uniform generator studied by Wierman et al., we also determine the threshold p = 0.441374 ± 0.000001, which falls within the tight bounds that they derived. Furthermore, we consider a generator in which all or none of the vertices can connect, and find a soluble inhomogeneous percolation system that interpolates between site percolation on the union-jack lattice and bond percolation on the square lattice.arXiv:1506.06125v2 [cond-mat.dis-nn]

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Fractal Dimensions and Corrections to Scaling for Critical Potts Clusters

Cited by 15 publications

References 10 publications

Correction-to-scaling exponent for two-dimensional percolation

Correction-to-scaling exponent for two-dimensional percolation

Critical domain walls in the Ashkin–Teller model

Percolation on hypergraphs with four-edges

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