Geometry of contours and Peierls estimates in d=1 Ising models with long range interactions

Cassandro, M.; Ferrari, Pablo A.; Merola, Immacolata; Presutti, Errico

doi:10.1063/1.1897644

Cited by 35 publications

(136 citation statements)

References 24 publications

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“…Our approach is somewhat akin to the bulk of work on the so-called Kac limit of lattice [14][15][16][17] as well as continuum [30,36,37] systems. Here one considers finite-range interactions of unit total strength which are smeared out over a region of scale 1 / γ .…”

Section: Motivationmentioning

confidence: 99%

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Mean-Field Driven First-Order Phase Transitions in Systems with Long-Range Interactions

Biskup¹,

Chayes²,

Crawford³

2006

J Stat Phys

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We consider a class of spin systems on Z d with vector valued spins (S x ) that interact via the pair-potentials J x,y S x · S y . The interactions are generally spreadout in the sense that the J x,y 's exhibit either exponential or power-law fall-off. Under the technical condition of reflection positivity and for sufficiently spread out interactions, we prove that the model exhibits a first-order phase transition whenever the associated mean-field theory signals such a transition. As a consequence, e.g., in dimensions d ≥ 3, we can finally provide examples of the 3-state Potts model with spread-out, exponentially decaying interactions, which undergoes a first-order phase transition as the temperature varies. Similar transitions are established in dimensions d = 1, 2 for power-law decaying interactions and in high dimensions for next-nearest neighbor couplings. In addition, we also investigate the limit of infinitely spread-out interactions. Specifically, we show that once the mean-field theory is in a unique "state," then in any sequence of translation-invariant Gibbs states various observables converge to their meanfield values and the states themselves converge to a product measure.

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Section: Motivationmentioning

confidence: 99%

“…[14][15][16][17] are based on coarse-graining arguments. As a consequence, we have no difficulty treating models with complicated single-spin spaces-even those exhibiting continuous internal symmetries or leading to power-law decay of correlations-or nearestneighbor systems in large dimensions.…”

Section: Motivationmentioning

confidence: 99%

Mean-Field Driven First-Order Phase Transitions in Systems with Long-Range Interactions

Biskup¹,

Chayes²,

Crawford³

2006

J Stat Phys

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show abstract

“…Dyson proved a part of the Kac-Thompson conjecture, namely that for long-range models with interactions of the form J(n) = n −α with α ∈ (1, 2), there is a phase transition at low temperatures. Later different proofs were found, [32,42,10,1] and also the case α = 2 was shown to have a transition [33].…”

Section: Dyson Modelsmentioning

confidence: 99%

“…Here we will make use of the approach of [10], which has been extended to a number of other situations (Dyson models in random fields [13], interfaces [11], phase separation [12], inhomogeneous decaying fields [6], etc). The disadvantage of this approach is that it works only at very low temperatures, as it is perturbative, and it works only for a reduced set of α-values, α * < α < 2, with α * = 3 − ln 3 ln 2 .…”

Section: Remarkmentioning

confidence: 99%

One-Sided Versus Two-Sided Stochastic Descriptions

Enter

Aernout²

2019

Statistical Mechanics of Classical and Disordered Systems

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It is well-known that discrete-time finite-state Markov Chains, which are described by onesided conditional probabilities which describe a dependence on the past as only dependent on the present, can also be described as one-dimensional Markov Fields, that is, nearestneighbor Gibbs measures for finite-spin models, which are described by two-sided conditional probabilities. In such Markov Fields the time interpretation of past and future is being replaced by the space interpretation of an interior volume, surrounded by an exterior to the left and to the right. If we relax the Markov requirement to weak dependence, that is, continuous dependence, either on the past (generalising the Markov-Chain description) or on the external configuration (generalising the Markov-Field description), it turns out this equivalence breaks down, and neither class contains the other. In one direction this result has been known for a few years, in the opposite direction a counterexample was found recently. Our counterexample is based on the phenomenon of entropic repulsion in long-range Ising (or "Dyson") models.

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“…An alternative approach to the investigation of ferromagnetic systems based on detailed investigation of the geometry of spin configurations is given in Ref. [13] (for 1.5 ≤ γ ≤ 2).…”

Section: Introductionmentioning

confidence: 99%

The one-dimensional long-range ferromagnetic Ising model with a periodic external field

Kerimov

2012

Physica A: Statistical Mechanics and its Applications

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Geometry of contours and Peierls estimates in d=1 Ising models with long range interactions

Cited by 35 publications

References 24 publications

Mean-Field Driven First-Order Phase Transitions in Systems with Long-Range Interactions

Mean-Field Driven First-Order Phase Transitions in Systems with Long-Range Interactions

One-Sided Versus Two-Sided Stochastic Descriptions

The one-dimensional long-range ferromagnetic Ising model with a periodic external field

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