Nucleation for One-Dimensional Long-Range Ising Models

Enter, Aernout van; Kimura, Bruno; Ruszel, Wioletta M.; Spitoni, Cristian

doi:10.1007/s10955-019-02238-y

Cited by 6 publications

(7 citation statements)

References 29 publications

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“…|Δ − e (u −u ) cap(B L ) Δ | ≤ 3(u − u) 1 + cap(B L ) 2 e (u −u ) cap(B L ) . (55) This completes the proof of ( 25) and hence of Lemma 2. With this last ingredient the proof of Proposition 1 is now complete.…”

Section: Conceivably Misleading)supporting

confidence: 63%

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The Roles of Random Boundary Conditions in Spin Systems

Enter

Endo

2020

Progress in Probability

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Section: Conceivably Misleading)supporting

confidence: 63%

“…The proof proceeds as in Case 1, γ > 1 above, by writing gλ (u) = R 2 bλ (u, t)dt with A as in (55),…”

Section: Case 1 and γmentioning

confidence: 99%

The Roles of Random Boundary Conditions in Spin Systems

Enter

Endo

2020

Progress in Probability

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“…[18] and references therein). For some recent results for these long-range Ising models with polynomially decaying interactions, see [79,34,10,21,33,23,69].…”

Section: Remark 5 the Borderline Casementioning

confidence: 99%

Markov and almost Markov properties in one, two or more directions

van Enter,

Ny,

Paccaut

2020

Preprint

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In this review-type paper written at the occasion of the Oberwolfach workshop Onesided vs. Two-sided stochastic processes (february 22-29, 2020), we discuss and compare Markov properties and generalisations thereof in more directions, as well as weaker forms of conditional dependence, again either in one or more directions. In particular, we discuss in both contexts various extensions of Markov chains and Markov fields and their properties, such as g-measures, Variable Length Markov Chains, Variable Neighbourhood Markov Fields, Variable Neighbourhood (Parsimonious) Random Fields, and Generalized Gibbs Measures.

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“…[10] and references therein). For some recent results for these long-range Dyson models with polynomially decaying interactions, see [49,21,6,12,20,14].…”

Section: Dyson Modelsmentioning

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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Nucleation for One-Dimensional Long-Range Ising Models

Cited by 6 publications

References 29 publications

The Roles of Random Boundary Conditions in Spin Systems

The Roles of Random Boundary Conditions in Spin Systems

Markov and almost Markov properties in one, two or more directions

One-Sided Versus Two-Sided Stochastic Descriptions

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