On the random-cluster model

Fortuin, C.M.; Kasteleyn, P.W.

doi:10.1016/0031-8914(72)90045-6

Cited by 1,539 publications

(971 citation statements)

References 15 publications

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“…The factor Q N C arises because a given cluster can be in any of the Q possible spin states. Equation (4) is the celebrated Fortuin-Kasteleyn (FK) representation of the Potts model [8]. It gives an equivalent representation of that spin model in terms of FK clusters obtained from the naive geometrical clusters of nearest neighbor spins in the same spin state, discussed in the Introduction, by putting bonds with a probability p = 1 − e −β between nearest neighbors.…”

Section: A Fortuin-kasteleyn Representationmentioning

confidence: 99%

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Two-dimensional critical Potts and its tricritical shadow

Janke

Schakel

2006

Braz. J. Phys.

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These notes give examples of how suitably defined geometrical objects encode in their fractal structure thermal critical behavior. The emphasis is on the two-dimensional Potts model for which two types of spin clusters can be defined. Whereas the Fortuin-Kasteleyn clusters describe the standard critical behavior, the geometrical clusters describe the tricritical behavior that arises when including vacant sites in the pure Potts model. Other phase transitions that allow for a geometrical description discussed in these notes include the superfluid phase transition and Bose-Einstein condensation.

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Section: A Fortuin-kasteleyn Representationmentioning

confidence: 99%

“…(5). From the asymptotic behavior (8), the divergence of the correlation length, and the vanishing (6) of the parameter θ as T c is approached, the relation…”

Section: B Fk Clustersmentioning

confidence: 99%

Two-dimensional critical Potts and its tricritical shadow

Janke

Schakel

2006

Braz. J. Phys.

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“…The reasons for this extension of the pure Potts model are twofold. First, in the limit s → 1, the s-state Potts model describes (bond) percolation [2], which is naturally formulated in terms of clusters. The extension thus allows for the investigation of cluster properties of the original model when the limit s → 1 is taken.…”

Section: Diluted Q-state Potts Modelmentioning

confidence: 99%

“…The two-dimensional q-state Potts models [1] can be equivalently formulated in terms of Fortuin-Kasteleyn (FK) clusters of like spins [2]. These FK clusters are obtained from the geometrical spin clusters, which consist of nearest neighbor sites with their spin variables in the same state, by laying bonds with a certain probability between the nearest neighbors.…”

Section: Introductionmentioning

confidence: 99%

Geometrical vs. Fortuin–Kasteleyn clusters in the two-dimensional q-state Potts model

Janke

Schakel

2004

Nuclear Physics B

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The tricritical behavior of the two-dimensional q-state Potts model with vacancies for 0 ≤ q ≤ 4 is argued to be encoded in the fractal structure of the geometrical spin clusters of the pure model. The known close connection between the critical properties of the pure model and the tricritical properties of the diluted model is shown to be reflected in an intimate relation between Fortuin-Kasteleyn and geometrical clusters: The same transformation mapping the two critical regimes onto each other also maps the two cluster types onto each other. The map conserves the central charge, so that both cluster types are in the same universality class. The geometrical picture is supported by a Monte Carlo simulation of the high-temperature representation of the Ising model (q = 2) in which closed graph configurations are generated by means of a Metropolis update algorithm involving single plaquettes.

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“…For these two cases a more general cluster construction is possible which does not automatically link all neighbors with equal center elements, but puts them in the same cluster only after some cutoff is applied [3], a step which corresponds to the construction of Fortuin-Kasteleyn clusters that are known to percolate at the same temperature where Potts models with continuous transitions demagnetize [11]. The construction allows one to use the cutoff parameter of the clusters to establish a continuum limit for the percolation description.…”

Section: Discussion Of the Resultsmentioning

confidence: 99%

No coincidence of center percolation and deconfinement in SU(4) lattice gauge theory

Dirnberger¹,

Gattringer²,

Maas³

2012

Physics Letters B

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We study the behavior of center sectors in pure SU(4) lattice gauge theory at finite temperature. The center sectors are defined as spatial clusters of neighboring sites with values of their local Polyakov loops near the same center elements. We study the connectedness and percolation properties of the center clusters across the deconfinement transition. We show that for SU(4) gauge theory deconfinement cannot be described as a percolation transition of center clusters, a finding which is different from pure SU(2) or pure SU(3) Yang Mills theory, where the percolation description even allows for a continuum limit.

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On the random-cluster model

Cited by 1,539 publications

References 15 publications

Two-dimensional critical Potts and its tricritical shadow

Two-dimensional critical Potts and its tricritical shadow

Geometrical vs. Fortuin–Kasteleyn clusters in the two-dimensional q-state Potts model

No coincidence of center percolation and deconfinement in SU(4) lattice gauge theory

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