Metastates in Finite-type Mean-field Models: Visibility, Invisibility, and Random Restoration of Symmetry

Iacobelli, Giulio; Külske, Christof

doi:10.1007/s10955-010-9979-7

Cited by 10 publications

(23 citation statements)

References 22 publications

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“…The models are some of the easiest disordered mean-field models and have been studied intensively over the last decades, see e. g. [23,2,1] and the references therein. For results on the dynamics or metastates of these models see e. g. [17,14,3,16]. It may be worthwhile noting that in the present paper, unlike many of the above authors, we will neither assume the random external fields to be symmetrically Bernoulli distributed nor to be bounded at all.…”

Section: Introductionmentioning

confidence: 94%

Moderate Deviations for Random Field Curie-Weiss Models

Löwe¹,

Meiners²

2012

J Stat Phys

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The random field Curie-Weiss model is derived from the classical Curie-Weiss model by replacing the deterministic global magnetic field by random local magnetic fields. This opens up a new and interestingly rich phase structure. In this setting, we derive moderate deviations principles for the random total magnetization S n , which is the partial sum of (dependent) spins. A typical result is that under appropriate assumptions on the distribution of the local external fields there exist a real number m, a positive real number λ, and a positive integer k such that (S n − nm)/n α satisfies a moderate deviations principle with speed n 1−2k(1−α) and rate function λx 2k /(2k)!, where 1 − 1/(2(2k − 1)) < α < 1.The Curie-Weiss model is a mean-field model of a ferromagnet. Due to its mean-field structure the spatial location of the spins is unimportant. The Hamiltonian of the Curie-Weiss model with external magnetic field h ∈ R can therefore be described

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Section: Introductionmentioning

confidence: 94%

Moderate Deviations for Random Field Curie-Weiss Models

Löwe¹,

Meiners²

2012

J Stat Phys

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“…Its intuitive meaning is that it describes the limiting empirical measure for the occurrence of Gibbs states for fixed environment ξ in a sufficiently sparsely chosen increasing volume sequence, see equation (4). Hence, it carries the additional information about the weight, or relevance, of a particular Gibbs measure, compare [IK10]. To show existence of a metastate, e.g., in situations with a compact local spin space, is always possible, see [AW90,NS97,Bov06] for Ising systems, and as we sketch in Section 2.2.…”

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confidence: 99%

Extremal decomposition for random Gibbs measures: from general metastates to metastates on extremal random Gibbs measures

Cotar¹,

Jahnel²,

Külske³

2018

Electron. Commun. Probab.

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The concept of metastate measures on the states of a random spin system was introduced to be able to treat the large-volume asymptotics for complex quenched random systems, like spin glasses, which may exhibit chaotic volume dependence in the strong-coupling regime. We consider the general issue of the extremal decomposition for Gibbsian specifications which depend measurably on a parameter that may describe a whole random environment in the infinite volume. Given a random Gibbs measure, as a measurable map from the environment space, we prove measurability of its decomposition measure on pure states at fixed environment, with respect to the environment. As a general corollary we obtain that, for any metastate, there is an associated decomposition metastate, which is supported on the extremes for almost all environments, and which has the same barycenter.In this note, we consider quenched random spin models in the infinite volume on lattices or more generally countably-infinite graphs. This includes examples like the random-field Ising model and the Edwards-Anderson nearest-neighbor spin glass, but also more generally continuous spin models. We prove an extended version of the known extremal decomposition (see [Föl75] and [Geo11, Theorem 7.26]) for the infinite-volume Gibbs measures in terms of pure states, i.e., the extremal Gibbs measures, which is measurable w.r.t. the random environment, see Theorem 2.1. We also present a connection between this result and the theory of metastates via the notion of the decomposition metastate, see Corollary 2.3.What are the difficulties? Indeed, measurability w.r.t. random environments in the infinite volume, which can be taken for granted in simple situations, becomes nontrivial for general systems. The difficulty comes from the following physical phenomenon that may happen for general random systems and in particular those with frustrated interactions like spin glasses. A system is frustrated, if it has competing interactions, in the sense that it is impossible to minimize the total finite-volume

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“…As a consequence, the model has been used as a playground to test new ideas. We refer to [APZ92] for the characterization of infinite volume Gibbs states; [KLN07] for Gibbs/non-Gibbs transitions; [Kül97,IK10,FKR12] for the study of metastates; [MP98, FMP00, BBI09] for the metastability analysis; and references therein. From a static viewpoint, the behavior of the fluctuations for this system is clear.…”

Section: Introductionmentioning

confidence: 99%

Path-space moderate deviation principles for the random field Curie-Weiss model

Collet¹,

Kraaij²

2018

Electron. J. Probab.

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We analyze the dynamics of moderate fluctuations for macroscopic observables of the random field Curie-Weiss model (i.e., standard Curie-Weiss model embedded in a site-dependent, i.i.d. random environment). We obtain pathspace moderate deviation principles via a general analytic approach based on convergence of non-linear generators and uniqueness of viscosity solutions for associated Hamilton-Jacobi equations. The moderate asymptotics depend crucially on the phase we consider and moreover, the space-time scale range for which fluctuations can be proven is restricted by the addition of the disorder.

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Metastates in Finite-type Mean-field Models: Visibility, Invisibility, and Random Restoration of Symmetry

Cited by 10 publications

References 22 publications

Moderate Deviations for Random Field Curie-Weiss Models

Moderate Deviations for Random Field Curie-Weiss Models

Extremal decomposition for random Gibbs measures: from general metastates to metastates on extremal random Gibbs measures

Path-space moderate deviation principles for the random field Curie-Weiss model

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