Gibbsian versus Non-Gibbsian Measures: Some Results and Some Questions in Renormalization Group Theory and Stochastic Dynamics

Vanenter, Acd; Fernández, Roberto; Sokal, Ad

doi:10.1007/978-1-4615-2460-1_15

Cited by 5 publications

(13 citation statements)

References 18 publications

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“…This prediction is known to be correct for q = 2 [32,33,34] and is believed to be correct also for q = 4 [5,35,36]. On the other hand, for q = 3 the conjecture contradicts the rigorous result [37], based on Pirogov-Sinai theory, that there is a low-temperature phase with long-range order and small correlation length. 4 In any case, for q > 4 we expect that the triangular-lattice Potts model is noncritical even at zero temperature; this has recently been confirmed by Monte Carlo simulation of the models with q = 5, 6 [38].…”

Section: Introductionmentioning

confidence: 94%

Transfer Matrices and Partition-Function Zeros for Antiferromagnetic Potts Models

2006

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We study the chromatic polynomial P G (q) for m × n square-and triangular-lattice strips of widths 2 ≤ m ≤ 8 with cyclic boundary conditions. This polynomial gives the zero-temperature limit of the partition function for the antiferromagnetic q-state Potts model defined on the lattice G. We show how to construct the transfer matrix in the Fortuin-Kasteleyn representation for such lattices and obtain the accumulation sets of chromatic zeros in the complex q-plane in the limit n → ∞. We find that the different phases that appear in this model can be characterized by a topological parameter. We also compute the bulk and surface free energies and the central charge.

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Section: Introductionmentioning

confidence: 94%

Transfer Matrices and Partition-Function Zeros for Antiferromagnetic Potts Models

2006

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“…This prediction of a zero-temperature critical point is known to be correct for q = 2 [57,58,59] and is believed to be correct also for q = 4 [60,61,62]. On the other hand, for q = 3 this prediction contradicts the rigorous result [63], based on Pirogov-Sinai theory, that there is a low-temperature phase with long-range order and small correlation length. 2 For the model (1.5), Baxter [54] computed three different expressions λ i (q) [i = 1, 2, 3] that he argued correspond to the dominant eigenvalues of the transfer matrix in different regions D i of the complex q-plane; in a second paper [55] he provided corrected estimates for the precise locations of D 1 , D 2 , D 3 .…”

Section: Introductionmentioning

confidence: 95%

Transfer Matrices and Partition-Function Zeros for Antiferromagnetic Potts Models VI. Square Lattice with Extra-Vertex Boundary Conditions

Salas

Sokal

2011

J Stat Phys

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We study, using transfer-matrix methods, the partition-function zeros of the square-lattice q-state Potts antiferromagnet at zero temperature (= square-lattice chromatic polynomial) for the boundary conditions that are obtained from an m × n grid with free boundary conditions by adjoining one new vertex adjacent to all the sites in the leftmost column and a second new vertex adjacent to all the sites in the rightmost column. We provide numerical evidence that the partition-function zeros are becoming dense everywhere in the complex q-plane outside the limiting curve B ∞ (sq) for this model with ordinary (e.g. free or cylindrical) boundary conditions. Despite this, the infinite-volume free energy is perfectly analytic in this region.

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“…This prediction of a zero-temperature critical point is known to be correct for q = 2 [8], and there is heuristic analytical evidence that it is correct also for q = 4 [18,25]. On the other hand, for q = 3 this prediction contradicts the rigorous result [26], based on Pirogov-Sinai theory, that there is a low-temperature phase with long-range order and small correlation length. Indeed, a recent Monte Carlo study of the q = 3 model has found strong evidence for a first-order transition (to an ordered phase) at β ≈ 1.594 [27].…”

Section: Introductionmentioning

confidence: 98%

Absence of phase transition for antiferromagnetic Potts models via the Dobrushin uniqueness theorem

1997

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We prove that the q-state Potts antiferromagnet on a lattice of maximum coordination number r exhibits exponential decay of correlations uniformly at all temperatures (including zero temperature) whenever q > 2r. We also prove slightly better bounds for several two-dimensional lattices: square lattice (exponential decay for q ≥ 7), triangular lattice (q ≥ 11), hexagonal lattice (q ≥ 4), and Kagomé lattice (q ≥ 6). The proofs are based on the Dobrushin uniqueness theorem.

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Gibbsian versus Non-Gibbsian Measures: Some Results and Some Questions in Renormalization Group Theory and Stochastic Dynamics

Cited by 5 publications

References 18 publications

Transfer Matrices and Partition-Function Zeros for Antiferromagnetic Potts Models

Transfer Matrices and Partition-Function Zeros for Antiferromagnetic Potts Models

Transfer Matrices and Partition-Function Zeros for Antiferromagnetic Potts Models VI. Square Lattice with Extra-Vertex Boundary Conditions

Absence of phase transition for antiferromagnetic Potts models via the Dobrushin uniqueness theorem

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