Gibbs–non-Gibbs transitions in the fuzzy Potts model with a Kac-type interaction: Closing the Ising gap

Henning, Florian; Kraaij, Richard C.; Kuelske, Christof

doi:10.3150/18-bej1045

Cited by 9 publications

(9 citation statements)

References 17 publications

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“…As in the mean-field models, there is again a single-site limiting kernel, however the limiting empirical distribution ν which appeared as a conditioning in the mean-field model is replaced by a whole profile of spin densities on the unit torus. For details of these definitions and results, see [18,16].…”

Section: Sequential Gibbsianness For Mean-field (And Kac-models On Tomentioning

confidence: 99%

“…For mean-field systems and Kac-systems there are also path-large-deviation principles available which lead to fixed-end-point variational problems for trajectories of empirical measures. While in an abstract sense this is a solution, the analytical understanding of the structure of minimizers of such problems can be quite hard (see however [16]). It is an open challenge to fully develop the analogous theory on the lattice, with ideas as suggested in [9].…”

Section: Relation To Disordered Systemsmentioning

confidence: 99%

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Gibbs-Non Gibbs Transitions in Different Geometries: The Widom-Rowlinson Model Under Stochastic Spin-Flip Dynamics

Külske

2019

Statistical Mechanics of Classical and Disordered Systems

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The Widom-Rowlinson model is an equilibrium model for point particles in Euclidean space. It has a repulsive interaction between particles of different colors, and shows a phase transition at high intensity. Natural versions of the model can moreover be formulated in different geometries: in particular as a lattice system or a mean-field system. We will discuss recent results on dynamical Gibbs-non Gibbs transitions in this context. Main issues will be the possibility or impossibility of an immediate loss of the Gibbs property, and of full-measure discontinuities of the time-evolved models.(Collaborations with Benedikt Jahnel, Sascha Kissel, Utkir Rozikov) space, on the lattice, as a mean-field model, and on a regular tree. Our aim here is to provide an overview; for detailed statements and proofs we refer to the original articles.2 Gibbs on lattice, sequentially Gibbs, marked Gibbs point processes, and the Widom-Rowlinson modelWe start by recalling the notion of an infinite-volume Gibbs measure for lattice systems. For the purpose of the discussion of the Widom-Rowlinson model and all measures appearing under timeevolution defined below from it, it is sufficient to restrict to the local state-space {−1, 0, 1} for particles carrying spins plus or minus, and holes. Our site space is the lattice Z d . The space of infinite-volume configurations is Ω = {−1, 0, 1} Z d . Specifications and Gibbs measures on the latticeThe central object in Gibbsian theory on a countable site space which defines the model is a specification. This covers both cases of infinite lattices and trees. It is a candidate system for conditional probabilities of an infinite-volume Gibbs measure µ (probability measure on Ω) to be defined by DLR equations µ(γ Λ (f |·)) = µ(f ). A specification γ is by definition a family of probability kernels γ = (γ Λ ) Λ Z d , indexed by finite subvolumes Λ, where γ Λ (dω|η) is a probability measure on Ω, for each fixed configuration η. It must have the following properties. The first is the consistency which means that

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Section: Sequential Gibbsianness For Mean-field (And Kac-models On Tomentioning

confidence: 99%

Section: Relation To Disordered Systemsmentioning

confidence: 99%

Gibbs-Non Gibbs Transitions in Different Geometries: The Widom-Rowlinson Model Under Stochastic Spin-Flip Dynamics

Külske

2019

Statistical Mechanics of Classical and Disordered Systems

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“…As a general consequence, if a mean-field model µ N is sequentially Gibbs, the resulting specification kernel α f → γ(•|α f ) is continuous as a self-map on the simplex M 1 ({−1, 0, 1}) (cf. [32], [14]). This makes clear that the sequential Gibbs property provides us with continuous dependence of conditional probabilities (here: in the limit), which is an essential requirement for Gibbsian theory on the lattice ( [31], [11]).…”

Section: Definition 22 a Sequence Of Exchangeable Measures µmentioning

confidence: 99%

“…The notion of sequential Gibbsianness is to be used for Kac-models on the torus, too, for which spin configurations have a spatial structure, where it relates to hydrodynamic scaling, cf. [8], [16], [14]. In the time-evolved Curie-Weiss Ising model non-Gibbsian behavior at low temperatures appears with symmetry-breaking in the set of bad magnetizations for an intermediate time-interval, and this happens already under independent spin-flip.…”

Section: Introductionmentioning

confidence: 99%

Dynamical Gibbs-non-Gibbs transitions in Curie-Weiss Widom-Rowlinson models

Kissel,

Kuelske

2018

Preprint

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We consider the Curie-Weiss Widom-Rowlinson model for particles with spins and holes, with a repulsion strength β > 0 between particles of opposite spins. We provide a closed solution of the model, and investigate dynamical Gibbs-non-Gibbs transitions for the time-evolved model under independent stochastic symmetric spin-flip dynamics. We show that, for sufficiently large β after a transition time, continuously many bad empirical measures appear. These lie on (unions of) curves on the simplex whose time-evolution we describe.

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“…In the present paper we are aiming to contribute to the understanding of Gibbs-non-Gibbs transformations for mean-field models, in the sense of the sequential Gibbs property [6,[9][10][11]14,17,18,21]. Usually there is a somewhat incomplete picture for lattice models, due to the difficulty to find sharp critical parameters.…”

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confidence: 99%

Dynamical Gibbs–non-Gibbs Transitions in the Curie–Weiss Potts Model in the Regime$$\beta <3$$

Kuelske

Meißner

2021

J Stat Phys

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We consider the Curie–Weiss Potts model in zero external field under independent symmetric spin-flip dynamics. We investigate dynamical Gibbs–non-Gibbs transitions for a range of initial inverse temperatures $$\beta <3$$ β < 3 , which covers the phase transition point $$\beta = 4\log 2$$ β = 4 log 2 (Ellis and Wang in Stoch Process Appl 35(1):59–79, 1990). We show that finitely many types of trajectories of bad empirical measures appear, depending on the parameter $$\beta $$ β , with a possibility of re-entrance into the Gibbsian regime, of which we provide a full description.

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Gibbs–non-Gibbs transitions in the fuzzy Potts model with a Kac-type interaction: Closing the Ising gap

Cited by 9 publications

References 17 publications

Gibbs-Non Gibbs Transitions in Different Geometries: The Widom-Rowlinson Model Under Stochastic Spin-Flip Dynamics

Gibbs-Non Gibbs Transitions in Different Geometries: The Widom-Rowlinson Model Under Stochastic Spin-Flip Dynamics

Dynamical Gibbs-non-Gibbs transitions in Curie-Weiss Widom-Rowlinson models

Dynamical Gibbs–non-Gibbs Transitions in the Curie–Weiss Potts Model in the Regime$$\beta <3$$

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