Equivalence of the Potts model or Whitney polynomial with an ice-type model

Baxter, R. J.; Kelland, S B; Wu, Friedrich

doi:10.1088/0305-4470/9/3/009

Cited by 242 publications

(300 citation statements)

References 11 publications

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“…In the above expression, a power of x is associated to each bond in the graph. Taking into account the summation over colours one arrives to the expansion [19] 5) where N b is the total number of bonds in the graph G and N c is the number of connected components (clusters) in G (each isolated site is also counted as a cluster). In terms of the partition function (2.5) the q-state Potts model is well defined even for noninteger values of q.…”

Section: Scattering Theory Of the Scaling Potts Modelmentioning

confidence: 99%

Universal amplitude ratios in the two-dimensional q-state Potts model and percolation from quantum field theory

1998

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We consider the scaling limit of the two-dimensional q-state Potts model for q ≤ 4. We use the exact scattering theory proposed by Chim and Zamolodchikov to determine the one and two-kink form factors of the energy, order and disorder operators in the model. Correlation functions and universal combinations of critical amplitudes are then computed within the two-kink approximation in the form factor approach. Very good agreement is found whenever comparison with exact results is possible. We finally consider the limit q → 1 which is related to the isotropic percolation problem. Although this case presents a serious technical difficulty, we predict a value close to 74 for the ratio of the mean cluster size amplitudes above and below the percolation threshold. Previous estimates for this quantity range from 14 to 220.

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Section: Scattering Theory Of the Scaling Potts Modelmentioning

confidence: 99%

Universal amplitude ratios in the two-dimensional q-state Potts model and percolation from quantum field theory

1998

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“…Taking a conformal gauge g µν (x) =ĝ µν e φ(x) and introducing DDK arguments, we can rewrite the first and second terms of the eq. (2.25 ) as 26) where the term e α −1 φ is introduced to keep the Weyl invariance of the second line of eq. (2.26) in accordance with the arguments of eq.…”

Section: Fractal Dimension From Diffusion Equation Of Random Walkmentioning

confidence: 99%

“…Thus there are Q N terms in the summation. It has been shown [20,26] that the partition function (3.3) can be expressed as a dichromatic polynomial [27]. In order to see this, let us expand the partition function as a product of terms associated with nearest-neighbor vertices.…”

Section: Generalized Potts Model ≡ Weighted Percolation Cluster Modelmentioning

confidence: 99%

“…Baxter, Kelland and Wu (BKW) [26] have later found a very elegant derivation of the result of Temperley and Lieb. They use a construction known as the BKW construction which makes many exact results obvious, including the critical temperature, self-duality and energy at criticality of the Potts model.…”

Section: Generalized Potts Model ≡ Weighted Percolation Cluster Modelmentioning

confidence: 99%

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Numerical study for the c-dependence of fractal dimension in two-dimensional quantum gravity

Kawamoto

Yotsuji

2002

Nuclear Physics B

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We numerically investigate the fractal structure of two-dimensional quantum gravity coupled to matter central charge c for −2 ≤ c ≤ 1. We reformulate Q-state Potts model into the model which can be identified as a weighted percolation cluster model and can make continuous change of Q, which relates c, on the dynamically triangulated lattice. The c-dependence of the critical coupling is measured from the percolation probability and susceptibility. The c-dependence of the string susceptibility of the quantum surface is evaluated and has very good agreement with the theoretical predictions. The c-dependence of the fractal dimension based on the finite size scaling hypothesis is measured and has excellent agreement with one of the theoretical predictions previously proposed except for the region near c ≈ 1.

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“…We shall call this variant of the n-vector model the Nienhuis model, or Loop-gas model (as it is a 'hard-core lattice gas' in which the elementary objects are self-avoiding loops). Also the critical behaviour of the Potts model can be analyzed exactly, thanks to the mapping onto an ice-type model, which has been constructed both algebrically [11] and combinatorially [12]. This is once more mapped to a solid-on-solid model [13].…”

Section: Introductionmentioning

confidence: 99%

Spanning Forests on Random Planar Lattices

Caracciolo

Sportiello

2009

J Stat Phys

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The generating function for spanning forests on a lattice is related to the q-state Potts model in a certain q → 0 limit, and extends the analogous notion for spanning trees, or dense self-avoiding branched polymers. Recent works have found a combinatorial perturbative equivalence also with the (quadratic action) O(n) model in the limit n → −1, the expansion parameter t counting the number of components in the forest. We give a random-matrix formulation of this model on the ensemble of degree-k random planar lattices. For k = 3, a correspondence is found with the Kostov solution of the loop-gas problem, which arise as a reformulation of the (logarithmic action) O(n) model, at n = −2. Then, we show how to perform an expansion around the t = 0 theory. In the thermodynamic limit, at any order in t we have a finite sum of finite-dimensional Cauchy integrals. The leading contribution comes from a peculiar class of terms, for which a resummation can be performed exactly.

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Equivalence of the Potts model or Whitney polynomial with an ice-type model

Cited by 242 publications

References 11 publications

Universal amplitude ratios in the two-dimensional q-state Potts model and percolation from quantum field theory

Universal amplitude ratios in the two-dimensional q-state Potts model and percolation from quantum field theory

Numerical study for the c-dependence of fractal dimension in two-dimensional quantum gravity

Spanning Forests on Random Planar Lattices

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