Gibbs–non-Gibbs properties for n-vector lattice and mean-field models

Enter, Aernout van; Külske, Christof; Opoku, Alex Akwasi; Ruszel, Wioletta M.

doi:10.1214/09-bjps029

Cited by 33 publications

(34 citation statements)

References 30 publications

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“…Multiple histories were then shown to lead to jumps in conditional probabilities indicating non-Gibbsian behavior in the mean-field setting, see e.g. [5,7,14]. These special conditionings leading to multiple histories are the analogue of "bad configurations" (essential points of discontinuity of conditional probabilities of the measure at time t ) in the (lattice) Gibbs-non-Gibbs transition scenario.…”

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confidence: 99%

Gibbs-Non-Gibbs Transitions via Large Deviations: Computable Examples

Redig

Wang

2012

J Stat Phys

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We give new and explicitly computable examples of Gibbs-non-Gibbs transitions of mean-field type, using the large deviation approach introduced in (van Enter et al. in Mosc. Math. J. 10:687-711, 2010). These examples include Brownian motion with small variance and related diffusion processes, such as the Ornstein-Uhlenbeck process, as well as birth and death processes. We show for a large class of initial measures and diffusive dynamics both short-time conservation of Gibbsianness and dynamical Gibbs-non-Gibbs transitions.

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confidence: 99%

Gibbs-Non-Gibbs Transitions via Large Deviations: Computable Examples

Redig

Wang

2012

J Stat Phys

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“…The non-reversible time-evolutions we consider here suggest another set of questions, namely whether there are any non-Gibbsian pathologies along the trajectories depending on starting measures as found for reversible dynamics in [16,18,39,36,20,17,14,33,22]. Acknowledgement: This work is supported by the Sonderforschungsbereich SFB | TR12-Symmetries and Universality in Mesoscopic Systems.…”

Section: Ideas Of the Proofmentioning

confidence: 79%

A class of non-ergodic probabilistic cellular automata with unique invariant measure and quasi-periodic orbit

Jahnel

Külske

2015

Stochastic Processes and their Applications

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We provide an example of a discrete-time Markov process on the threedimensional infinite integer lattice with Z q -invariant Bernoulli-increments which has as local state space the cyclic group Z q . We show that the system has a unique invariant measure, but remarkably possesses an invariant set of measures on which the dynamics is conjugate to an irrational rotation on the continuous sphere S 1 . The update mechanism we construct is exponentially well localized on the lattice.

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“…The main point is the study of the dynamical Gibbs-non Gibbs transitions under rate-one symmetric independent spin-flip, keeping holes fixed, according to transition probabilities (11).…”

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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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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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Gibbs–non-Gibbs properties for n-vector lattice and mean-field models

Cited by 33 publications

References 30 publications

Gibbs-Non-Gibbs Transitions via Large Deviations: Computable Examples

Gibbs-Non-Gibbs Transitions via Large Deviations: Computable Examples

A class of non-ergodic probabilistic cellular automata with unique invariant measure and quasi-periodic orbit

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

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