Description of Specifications by Means of Probability Distributions in Small Volumes Under Condition of Very Weak Positivity

Dachian, Sergueı̈; Nahapetian, Boris

doi:10.1023/b:joss.0000044069.91072.0b

Cited by 11 publications

(29 citation statements)

References 8 publications

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“…The construction in the proof of Theorem A.4 of [12] shows that the continuity of the single-site kernels extend to all finite-volume kernels. Alternatively the theorem can be proven using the conditions from [6]. 4.…”

Section: Definition 33mentioning

confidence: 99%

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On the relation between Gibbs and -measures

Berghout¹,

Fernández²,

Verbitskiy³

2018

Ergod. Th. Dynam. Sys.

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Thermodynamic formalism, the theory of equilibrium states, is studied both in dynamical systems and probability theory. Various closely related notions have been developed: e.g. Dobrushin–Lanford–Ruelle Gibbs, Bowen–Gibbs and $g$-measures. We discuss the relation between Gibbs and $g$-measures in a one-dimensional context. Often $g$-measures are also Gibbs, but recently an example to the contrary has been presented. In this paper we discuss exactly when a $g$-measure is Gibbs and how this relates to notions such as uniqueness and reversibility of $g$-measures.

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Section: Definition 33mentioning

confidence: 99%

“…we also have f σ 0 ,η 0 (ω) f η 0 ,σ 0 (ω) = 1; f σ 0 ,η 0 is bounded from above as a continuous function on the compact X , it is bounded below by 1/ f σ 0 ,η 0 ∞ , for all σ 0 , η 0 . Finally, equation (6) implies that…”

Section: Definition 33mentioning

confidence: 99%

On the relation between Gibbs and -measures

Berghout¹,

Fernández²,

Verbitskiy³

2018

Ergod. Th. Dynam. Sys.

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“…These consistency conditions can be equivalently rewritten in the following form (see the paper [4] of the authors): for all I ⊂ V ∈ W, x, u ∈ X V \I , y ∈ X I and x ∈ X Z d \V , it holds…”

Section: Preliminariesmentioning

confidence: 99%

“…It should be noted that this definition is a particular case of a more general (not requiring the strict positivity assumption) definition of 1-specification introduced in the paper [4] of the authors.…”

Section: Preliminariesmentioning

confidence: 99%

“…The argumentation used in this work follows the idea formulated by Sinai in [34] that for infinite volume systems, the Hamiltonian cannot be defined as a function on the space of all configurations; however, it makes sense to consider the "differences of the Hamiltonians" for configurations which differ in a finite number of points. At the same time, our considerations are in many ways related to Dobrushin's problem of description of specifications by means of systems of consistent one-point probability distributions with infinite boundary conditions, which was solved (by introducing the notion of 1-specification) in the works [2][3][4] of the authors (see also Fernández and Maillard [14,15]).…”

Section: Introductionmentioning

confidence: 99%

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On the relationship of energy and probability in models of classical statistical physics

Dachian,

Nahapetian

2018

Preprint

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In this paper we present a new point of view on the mathematical foundations of statistical physics of infinite volume systems. This viewpoint is based on the newly introduced notions of transition energy function, transition energy field and one-point transition energy field. The former of them, namely the transition energy function, is a generalization of the notion of relative Hamiltonian introduced by Pirogov and Sinai. However, unlike the (relative) Hamiltonian, our objects are defined axiomatically by their natural and physically well-founded intrinsic properties. The developed approach allowed us to give a proper mathematical definition of the Hamiltonian without involving the notion of potential, to propose a justification of the Gibbs formula for infinite systems and to answer the problem stated by D. Ruelle of how wide the class of specifications, which can be represented in Gibbsian form, is. Furthermore, this approach establishes a straightforward relationship between the probabilistic notion of (Gibbs) random field and the physical notion of (transition) energy, and so opens the possibility to directly apply probabilistic methods to the mathematical problems of statistical physics.

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Partially Ordered Models

Deveaux

Fernández

2010

J Stat Phys

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We provide a formal definition and study the basic properties of partially ordered chains (POC). These systems were proposed to model textures in image processing and to represent independence relations between random variables in statistics (in the later case they are known as Bayesian networks). Our chains are a generalization of probabilistic cellular automata (PCA) and their theory has features intermediate between that of discrete-time processes and the theory of statistical mechanical lattice fields. Its proper definition is based on the notion of partially ordered specification (POS), in close analogy to the theory of Gibbs measure. This paper contains two types of results. First, we present the basic elements of the general theory of POCs: basic geometrical issues, definition in terms of conditional probability kernels, extremal decomposition, extremality and triviality, reconstruction starting from single-site kernels, relations between POM and Gibbs fields. Second, we prove three uniqueness criteria that correspond to the criteria known as bounded uniformity, Dobrushin and disagreement percolation in the theory of Gibbs measures.

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Description of Specifications by Means of Probability Distributions in Small Volumes Under Condition of Very Weak Positivity

Cited by 11 publications

References 8 publications

On the relation between Gibbs and -measures

On the relation between Gibbs and -measures

On the relationship of energy and probability in models of classical statistical physics

Partially Ordered Models

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