Discontinuity of the magnetization in one-dimensional 1/�x?y�2 Ising and Potts models

Aizenman, Michael; Chayes, Jennifer; Chayes, L.; Newman, Charles M.

doi:10.1007/bf01022985

Cited by 343 publications

(420 citation statements)

References 46 publications

Supporting

Mentioning

406

Contrasting

Unclassified

Order By: Relevance

“…where − ê 1 L;β,q is notation for the thermal state obtained with all boundary spins set toê 1 . The upper and lower bounds,…”

Section: Percolation and Magnetizationmentioning

confidence: 99%

See 1 more Smart Citation

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

Blanchard

Chayes²,

Gandolfo

2000

Nuclear Physics B

Self Cite

View full text Add to dashboard Cite

Recent advances in high energy QCD experiments probing the deconfinement transition from hadronic to coloured quark matter tend to confirm that perlocation of unbounded quarks could provide a signature of this phase transition. In the strong coupling limit the partition function of SU(2) pure gauge theory can be modeled by that of an infinite spin Ising system with short-range ferromagnetic interactions. We derive the Wolff-random cluster representation for these spin models and show that, at least in these cases, the thermal and geometrical phase transitions indeed coincide. Moreover, our results are presented in a more general setting (e.g., q-states Potts variable and/or long range interactions allowing a generalisation to a variety of physical systems.)

show abstract

“…where − ê 1 L;β,q is notation for the thermal state obtained with all boundary spins set toê 1 . The upper and lower bounds,…”

Section: Percolation and Magnetizationmentioning

confidence: 99%

“…It is now well established [1,4,10,12,23] that the characteristics of one system can be completely expressed in terms of the other. In particular the critical behavior (if any) of the spin-system translates directly into critical geometrical behavior in the corresponding random cluster model.…”

Section: Introductionmentioning

confidence: 99%

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

Blanchard

Chayes²,

Gandolfo

2000

Nuclear Physics B

Self Cite

View full text Add to dashboard Cite

show abstract

“…Indeed, in the Ising (and also Potts) model, a lowtemperature symmetry breaking occurs even in some recurrent dimensions; e.g., in d = 2 for the nearest-neighbor interactions. For what determines the presence and absence of symmetry breaking in d = 1, see the work of Aizenman, Chayes, Chayes and Newman [3] and references therein.…”

Section: Literature Remarksmentioning

confidence: 99%

Reflection Positivity and Phase Transitions in Lattice Spin Models

Biskup¹

2009

Lecture Notes in Mathematics

108

View full text Add to dashboard Cite

“…When q ≥ 1, the associated probability measure satisfies the FKG lattice condition (see Refs. 1,12), and therefore satisfies the FKG inequality (which may be stated roughly as 'increasing events are positively correlated'). This useful inequality implies properties of stochastic comparison which are of substantial value in studying any associated phase transition.…”

Section: Introductionmentioning

confidence: 99%

“…1,15). Such measures have a phase transition at a critical value p = p c (q) that marks the onset of an infinite cluster (or connected component) of the graph.…”

Section: Introductionmentioning

confidence: 99%

Comparison and disjoint-occurrence inequalities for random-cluster models

Grimmett

1995

J Stat Phys

View full text Add to dashboard Cite

×ØÖØº A principal technique for studying percolation, (ferromagnetic) Ising, Potts, and random-cluster models is the FKG inequality, which implies certain stochastic comparison inequalities for the associated probability measures. The first result of this paper is a new comparison inequality, proved using an argument developed in Refs. 2, 6, 14, and 21 in order to obtain strict inequalities for critical values. As an application of this inequality, we prove that the critical point p c (q), of the random-cluster model with cluster-weighting factor q (≥ 1), is strictly monotone in q. Our second result is a 'BK inequality' for the disjoint occurrence of increasing events, in a weaker form than that available in percolation theory.

show abstract

Discontinuity of the magnetization in one-dimensional 1/�x?y�2 Ising and Potts models

Cited by 343 publications

References 46 publications

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

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

Reflection Positivity and Phase Transitions in Lattice Spin Models

Comparison and disjoint-occurrence inequalities for random-cluster models

Contact Info

Product

Resources

About