Examples of DLR States Which are Not Weak Limits of Finite Volume Gibbs Measures with Deterministic Boundary Conditions

Coquille, Loren

doi:10.1007/s10955-015-1211-3

Cited by 13 publications

(12 citation statements)

References 33 publications

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“…This is the case in four or more spatial dimensions, where bounded transverse fluctuations of an interface allow non-translationally invariant extremal equilibrium states (satisfying cluster decomposition) to exist. 8 Such states may be viewed as the limit of finite volume equilibrium states in which one fixes degrees of freedom on the boundary of the volume in a non-uniform manner which pins the interface location on the boundary and selects the desired volume fraction x [29]. But in three or fewer spatial dimensions, which the following discussion assumes, the transverse fluctuations in interface position diverge as V → ∞.…”

Section: )mentioning

confidence: 99%

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Large N phase transitions and the fate of small Schwarzschild-AdS black holes

Yaffe

2018

Phys. Rev. D

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Sufficiently small Schwarzschild-AdS black holes in asymptotically global AdS 5 × S 5 spacetime are known to become dynamically unstable toward deformation of the internal S 5 geometry. The resulting evolution of such an unstable black hole is related, via holography, to the dynamics of supercooled plasma which has reached the limit of metastability in maximally supersymmetric large-N Yang-Mills theory on R × S 3 . Puzzles related to the resulting dynamical evolution are discussed, with a key issue involving differences between the large-N limit in the dual field theory and typical large volume thermodynamic limits.

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Section: )mentioning

confidence: 99%

“…This use of "pure states" in statistical physics, as a synonym of "extremal", is distinct from pure states in quantum mechanics 8. For lattice theories in three space dimensions, non-translationally invariant equilibrium states exist below the interface roughening temperature[27][28][29]. The lower limit of four spatial dimensions…”

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confidence: 99%

Large N phase transitions and the fate of small Schwarzschild-AdS black holes

Yaffe

2018

Phys. Rev. D

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“…the question whether equality of the two sets holds is discussed in [6]. A Gibbs state µ ∈ G(β) is said to be translation invariant if for any local function f : Ω → R, and any translation θ :…”

Section: Definition Of the Modelsmentioning

confidence: 99%

Absence of Dobrushin States for 2d Long-Range Ising Models

Coquille

Enter

et al. 2018

J Stat Phys

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We consider the two-dimensional Ising model with long-range pair interactions of the form Jxy ∼ |x − y| −α with α > 2, mostly when Jxy ≥ 0. We show that Dobrushin states (i.e. extremal non-translation-invariant Gibbs states selected by mixed ±-boundary conditions) do not exist. We discuss possible extensions of this result in the direction of the Aizenman-Higuchi theorem, or concerning fluctuations of interfaces. We also mention the existence of rigid interfaces in two long-range anisotropic contexts. IntroductionWe are interested in the possible existence of so-called interface states (or Dobrushin states) for long-range Ising models in dimension d = 2. Dobrushin states are extremal infinite-volume Gibbs measures selected by mixed ±-boundary conditions originally described in [8] for the standard (nearest-neighbour) Ising model in dimension three.Depending on the question one asks, a long-range interaction can behave similarly or not to nearest-neighbour (n.n.) models. It is well-known that for n.n. interactions, interface states do not exist, see e.g. [1,16,29,24,7,9]. On the other hand there exist such extremal and non-translation-invariant Gibbs states in d ≥ 3 [2,8].In this note we consider three different examples of long-range models, with interactions which are either isotropic, long-range horizontally and n.n. vertically or bi-axial long-range (i.e. long-range in both horizontal and vertical directions with possibly different decays). By long-range interaction we mean pair interactions with coupling constants given byfor all x, y ∈ Z 2 , where | · | denotes the Euclidean norm.We first prove that for all α > 2 there are no Dobrushin states in the isotropic case, see Theorem 1. This result is similar to what happens for short-range models. However, the precise statements we can prove are weaker than what is known for n.n. models, in particular regarding the full picture of the convex set of Gibbs measures. The Aizenman-Higuchi theorem indeed states [1,24] that all infinite-volume Gibbs measures of the 2d ferromagnetic n.n. Ising model are convex combinations of the pure + and − phases. As we shall see, we have either a statement for a subset of boundary conditions, or at low enough temperatures, but we provide directions to study these problems in more generality. Note that the case of fast decays α > 3 falls within the framework of the Gertzik-Pirogov-Sinai theory. Indeed, in this decay range, a first-moment condition on the interaction applies, and allows the proof of the existence of a spontaneous magnetisation for Ising models with n.n interaction plus a long-range perturbation [21,20]. Moreover, in such a 2010 Mathematics Subject Classification. 82B05, 82B20, 82B26.

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“…In [1] it was shown that for non-extremal Gibbs measures on Z d a Gibbs measure need not be a limiting Gibbs measure (see [1] and [2] for more details).…”

Section: Definitionsmentioning

confidence: 99%

Boundary Conditions for Translation-Invariant Gibbs Measures of the Potts Model on Cayley Trees

Gandolfo

Рахматуллаев²,

Rozikov³

2017

J Stat Phys

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Abstract. We consider translation-invariant splitting Gibbs measures (TISGMs) for the qstate Potts model on a Cayley tree of order two. Recently a full description of the TISGMs was obtained, and it was shown in particular that at sufficiently low temperatures their number is 2 q − 1. In this paper for each TISGM µ we explicitly give the set of boundary conditions such that limiting Gibbs measures with respect to these boundary conditions coincide with µ.Mathematics Subject Classifications (2010). 82B26; 60K35.

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Examples of DLR States Which are Not Weak Limits of Finite Volume Gibbs Measures with Deterministic Boundary Conditions

Cited by 13 publications

References 33 publications

Large N phase transitions and the fate of small Schwarzschild-AdS black holes

Large N phase transitions and the fate of small Schwarzschild-AdS black holes

Absence of Dobrushin States for 2d Long-Range Ising Models

Boundary Conditions for Translation-Invariant Gibbs Measures of the Potts Model on Cayley Trees

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