Gibbs–non-Gibbs properties for evolving Ising models on trees

Enter, Aernout van; Ermolaev, Victor; Iacobelli, Giulio; Külske, Christof

doi:10.1214/11-aihp421

Cited by 25 publications

(30 citation statements)

References 20 publications

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“…Here one should take the boundary law z x,y (i) = exp(h i,x ), where y is the unique neighbor of x that lies closer to 3 The equality (2.6) means that the restriction of µn to the set Φ V n−1 coincides with µn−1. 4 To see that a SGM satisfies the DLR equation, we consider any finite volume D and note that for any finite n which is sufficiently large we have 8) which follows from the compatibility property of the finite-volume Gibbs measures.…”

Section: Preliminariesmentioning

confidence: 99%

“…Here the 3 The equality (2.6) means that the restriction of μ n to the set V n−1 coincides with μ n−1 . 4 To see that a SGM satisfies the DLR equation, we consider any finite volume D and note that for any finite n which is sufficiently large we have…”

Section: )mentioning

confidence: 99%

“…Furthermore, [4] provides an explicit counterexample for such a theorem to hold in full generality, by showing that the Gibbs property of a time-evolved Ising tree measure does depend on the phase of the Ising model, in certain parameter regimes. It would be very interesting nevertheless to understand better to what extent the phenomenon of independence of the renormalized Hamiltonian of the phase for measures on trees we find here might generalize beyond the case we find in our example.…”

Section: Introductionmentioning

confidence: 99%

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Fuzzy transformations and extremality of Gibbs measures for the potts model on a Cayley tree

Külske

Rozikov²

2016

Random Struct. Alg.

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Abstract. We continue our study of the full set of translation-invariant splitting Gibbs measures (TISGMs, translation-invariant tree-indexed Markov chains) for the qstate Potts model on a Cayley tree. In our previous work [14] we gave a full description of the TISGMs, and showed in particular that at sufficiently low temperatures their number is 2 q − 1. In this paper we find some regions for the temperature parameter ensuring that a given TISGM is (non-)extreme in the set of all Gibbs measures. In particular we show the existence of a temperature interval for which there are at least 2 q−1 + q extremal TISGMs. For the Cayley tree of order two we give explicit formulae and some numerical values.Mathematics Subject Classifications (2010). 82B26 (primary); 60K35 (secondary)

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Section: Preliminariesmentioning

confidence: 99%

Section: )mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Fuzzy transformations and extremality of Gibbs measures for the potts model on a Cayley tree

Külske

Rozikov²

2016

Random Struct. Alg.

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“…Since ρ = min{β,log λ} 2d+1 goes to infinity for β, λ → ∞ the right hand side of the inequality goes to 0 and we can define a(β, λ) := e −ρ (2d) 2 (…”

Section: It Remains To Show Thatmentioning

confidence: 99%

Dynamical Gibbs–Non-Gibbs Transitions in Lattice Widom–Rowlinson Models with Hard-Core and Soft-Core Interactions

Kissel

Kuelske

2020

J Stat Phys

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We consider the Widom-Rowlinson model on the lattice Z d in two versions, comparing the cases of a hard-core repulsion and of a soft-core repulsion between particles carrying opposite signs. For both versions we investigate their dynamical Gibbs-non-Gibbs transitions under an independent stochastic symmetric spin-flip dynamics. While both models have a similar phase transition in the high-intensity regime in equilibrium, we show that they behave differently under time-evolution: The time-evolved soft-core model is Gibbs for small times and loses the Gibbs property for large enough times. By contrast, the time-evolved hard-core model loses the Gibbs property immediately, and for asymmetric intensities, shows a transition back to the Gibbsian regime at a sharp transition time.

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“…The large N -behavior of such a sequence defines our model. The model is called sequentially Gibbs iff the volume-limit of the single-spin probabilities in the finite-volume measures lim N ↑∞ µ N (dω 1 |ω [2,N ] ) = γ(dω 1 |ν) (7) exists whenever the empirical distributions of a configuration (ω i ) i=2,3,4,... converge,…”

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

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 evolving Ising models on trees

Cited by 25 publications

References 20 publications

Fuzzy transformations and extremality of Gibbs measures for the potts model on a Cayley tree

Fuzzy transformations and extremality of Gibbs measures for the potts model on a Cayley tree

Dynamical Gibbs–Non-Gibbs Transitions in Lattice Widom–Rowlinson Models with Hard-Core and Soft-Core Interactions

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

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