The random cluster representation for the infinite-spin Ising model: application to QCD pure gauge theory

Blanchard, Philippe; Chayes, L.; Gandolfo, Daniel

doi:10.1016/s0550-3213(00)00459-4

Cited by 9 publications

(6 citation statements)

References 27 publications

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“…It is evident that the percolation behaviour coincides fully with the thermal critical behaviour. This conclusion holds in general for all admissable distribution functions, as it was shown in [11].…”

Section: Model A: Spin Distribution Functionsupporting

confidence: 74%

“…Subsequently, we provide a suitable cluster definition in order to interpret the thermal phase transition of the models as a geometrical percolation transition. Concerning the second issue, it was recently proven rigorously that the result of [10] can be extended to the models A, B and C [11]. We believe, however, that it is useful to illustrate it by exploiting the power of the finite-size scaling analysis.…”

Section: Introductionmentioning

confidence: 96%

“…However, our result is valid only in the strong coupling limit of SU (2), where the partition function can be well approximated [9] by that of a continuous spin Ising model. In this model, the correspondence between percolation and thermal variables can be rigorously established [10,11]. For a general study of SU(2) gauge theory, the weak coupling limit must be taken into account, and there the approximations used in [9] are no longer valid.…”

Section: Introductionmentioning

confidence: 99%

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Percolation and magnetization for generalized continuous spin models

Fortunato¹,

Satz²

2001

Nuclear Physics B

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For the Ising model, the spin magnetization transition is equivalent to the percolation transition of Fortuin-Kasteleyn clusters; this result remains valid also for the conventional continuous spin Ising model. The investigation of more general continuous spin models may help to obtain a percolation formulation for the critical behaviour in SU(2) gauge theory. We therefore study a broad class of theories, introducing spin distribution functions, longer range interactions and self-interaction terms. The thermal behaviour of each model turns out to be in the Ising universality class. The corresponding percolation formulations are then obtained by extending the Fortuin-Kasteleyn cluster definition; in several cases they illustrate recent rigorous results.

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Section: Model A: Spin Distribution Functionsupporting

confidence: 74%

Section: Introductionmentioning

confidence: 96%

Section: Introductionmentioning

confidence: 99%

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Percolation and magnetization for generalized continuous spin models

Fortunato¹,

Satz²

2001

Nuclear Physics B

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“…We shall develop a graphical representation for the GI-system akin to the FK representation for the Potts model [29] that, for all intents and purposes, is the same as the one used in [10], where only the case of bounded fields is explicitly analyzed. Let us then consider the GI-Hamiltonian in finite volume with all notation pertaining to boundary conditions temporarily suppressed.…”

Section: Proof Of a High-temperature Phasementioning

confidence: 99%

“…Foremost, it is noted that the presentation in Eqs. (10) and (11) are, for all intents and purposes, the same as would have been obtained from the mean-field version of the so-called BEG model [12]. As such, some aspects of the current problem have been treated in [27].…”

Section: The Mean-field Equationsmentioning

confidence: 99%

Territorial Developments Based on Graffiti: a Statistical Mechanics Approach

Barbaro¹,

Chayes²,

D’Orsogna³

2011

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We study the well-known sociological phenomenon of gang aggregation and territory formation through an interacting agent system defined on a lattice. We introduce a two-gang Hamiltonian model where agents have red or blue affiliation but are otherwise indistinguishable. In this model, all interactions are indirect and occur only via graffiti markings, on-site as well as on nearest neighbor locations. We also allow for gang proliferation and graffiti suppression. Within the context of this model, we show that gang clustering and territory formation may arise under specific parameter choices and that a phase transition may occur between well-mixed, possibly dilute configurations and well separated, clustered ones. Using methods from statistical mechanics, we study the phase transition between these two qualitatively different scenarios. In the mean-fields rendition of this model, we identify parameter regimes where the transition is first or second order. In all cases, we have found that the transitions are a consequence solely of the gang to graffiti couplings, implying that direct gang to gang interactions are not strictly necessary for gang territory formation; in particular, graffiti may be the sole driving force behind gang clustering. We further discuss possible sociological -as well as ecological -ramifications of our results.

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Order by Disorder, without Order, in a Two-Dimensional Spin System with O(2) Symmetry

Biskup

Chayes

Kivelson

2004

Ann. Henri Poincaré

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Abstract:We present a rigorous proof of an ordering transition for a two-component two-dimensional antiferromagnet with nearest and next-nearest neighbor interactions. The low-temperature phase contains two states distinguished by local order among columns or, respectively, rows. Overall, there is no magnetic order in accord with the classic Mermin-Wagner theorem. The method of proof employs a rigorous version of "order by disorder," whereby a high degeneracy among the ground states is lifted according to the differences in their associated spin-wave spectra.

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The random cluster representation for the infinite-spin Ising model: application to QCD pure gauge theory

Cited by 9 publications

References 27 publications

Percolation and magnetization for generalized continuous spin models

Percolation and magnetization for generalized continuous spin models

Territorial Developments Based on Graffiti: a Statistical Mechanics Approach

Order by Disorder, without Order, in a Two-Dimensional Spin System with O(2) Symmetry

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