Entropic Repulsion and Lack of the g-Measure Property for Dyson Models

Bissacot, Rodrigo; Endo, Eric O.; Enter, Aernout van; Ny, Arnaud Le

doi:10.1007/s00220-018-3233-6

Cited by 23 publications

(33 citation statements)

References 68 publications

(127 reference statements)

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“…In other words, controlling your borders and controlling the future are not the same things, except if you are a short-sighted Markovian. Although the result in [5] was proven under some restrictions on α, and the proof also requires the presence of a large enough nearest-neighbour interaction, in view of the results of [49,6] these conditions can presumably be removed. However, the situation for α = 2 remains unclear, as the main idea of our proof (mesoscopic localisation of the interface point) fails.…”

Section: Conclusion Final Remarks Higher Dimensionsmentioning

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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Section: Conclusion Final Remarks Higher Dimensionsmentioning

confidence: 99%

One-Sided Versus Two-Sided Stochastic Descriptions

Enter

Aernout²

2019

Statistical Mechanics of Classical and Disordered Systems

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“…We should emphasize that, although the use of contour arguments may look somewhat unwieldy in comparison with other approaches, it is much more robust. Indeed it has been used to analyze Dyson models in random [13,14] and periodic fields [25], for interface behaviour and phase separation [11,12], for entropic repulsion [8], and here for the model in decaying magnetic fields, all problems where alternative methods appear to break down.…”

Section: Introductionmentioning

confidence: 99%

Contour Methods for Long-Range Ising Models: Weakening Nearest-Neighbor Interactions and Adding Decaying Fields

Bissacot

Endo

Enter

et al. 2018

Ann. Henri Poincaré

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We consider ferromagnetic long-range Ising models which display phase transitions. They are one-dimensional Ising ferromagnets, in which the interaction is given by Jx,y = J(|x − y|) ≡ 1 |x−y| 2−α with α ∈ [0, 1), in particular, J(1) = 1. For this class of models one way in which one can prove the phase transition is via a kind of Peierls contour argument, using the adaptation of the Fröhlich-Spencer contours for α = 0, proposed by Cassandro, Ferrari, Merola and Presutti. As proved by Fröhlich and Spencer for α = 0 and conjectured by Cassandro et al for the region they could treat, α ∈ (0, α+) for α+ = log(3)/ log(2)−1, although in the literature dealing with contour methods for these models it is generally assumed that J(1) ≫ 1, we will show that this condition can be removed in the contour analysis. In addition, combining our theorem with a recent result of Littin and Picco we prove the persistence of the contour proof of the phase transition for any α ∈ [0, 1). Moreover, we show that when we add a magnetic field decaying to zero, given by hx = h * ·(1+|x|) −γ and γ > max{1 − α, 1 − α * } where α * ≈ 0.2714, the transition still persists.

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“…For long-range potentials, like Dyson models, nothing is known regarding concentration bounds. In that context, let us mention that g-measures can be different from Gibbs measures, see [2] and references therein.…”

Section: Introductionmentioning

confidence: 99%

On Concentration Inequalities and Their Applications for Gibbs Measures in Lattice Systems

2017

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We consider Gibbs measures on the configuration space S Z d , where mostly d ≥ 2 and S is a finite set. We start by a short review on concentration inequalities for Gibbs measures. In the Dobrushin uniqueness regime, we have a Gaussian concentration bound, whereas in the Ising model (and related models) at sufficiently low temperature, we control all moments and have a stretched-exponential concentration bound. We then give several applications of these inequalities whereby we obtain various new results. Amongst these applications, we get bounds on the speed of convergence of the empirical measure in the sense of Kantorovich distance, fluctuation bounds in the Shannon-McMillan-Breiman theorem, fluctuation bounds for the first occurrence of a pattern, as well as almost-sure central limit theorems.

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Entropic Repulsion and Lack of the g-Measure Property for Dyson Models

Cited by 23 publications

References 68 publications

One-Sided Versus Two-Sided Stochastic Descriptions

One-Sided Versus Two-Sided Stochastic Descriptions

Contour Methods for Long-Range Ising Models: Weakening Nearest-Neighbor Interactions and Adding Decaying Fields

On Concentration Inequalities and Their Applications for Gibbs Measures in Lattice Systems

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