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

Gandolfo, Daniel; Рахматуллаев, Музаффар Мухаммаджанович; Rozikov, U. A.

doi:10.1007/s10955-017-1771-5

Cited by 8 publications

(12 citation statements)

References 10 publications

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“…To the contrary, suppose first that v m < 1 for some θ > θ m . Then, according to Lemma 6.6, we have 19) and furthermore, Assume now that v m = 1 for some θ > θ m . Then…”

Section: 2mentioning

confidence: 99%

“…Remark 2.8. In the existing studies of Gibbs measures on trees (see, e.g., [19,30]), it is common to use the term "translation invariant" (and the abbreviation TISGM) having in mind just completely homogeneous SGM. We prefer to keep the terminological distinction between "single-coloured" completely homogeneous GBC ȟ(x) ≡ ȟ0 and "multi-coloured" translation-invariant GBC as characterized by Proposition 2.6.…”

Section: Translation Invariancementioning

confidence: 99%

“…Now, take the conditional expectation E(• | F n ) on both sides of (4. 19), noting that the factor in front of the sum is a random variable measurable with respect to the sigma-algebra F n , so it can be pulled out of the expectation (see [56, Property K*, page 216]). This yields…”

Section: And Hence Minmentioning

confidence: 99%

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On the uniqueness of Gibbs measure in the Potts model on a Cayley tree with external field

Bogachev

Rozikov²

2019

J. Stat. Mech.

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The paper concerns the q-state Potts model (i.e., with spin values in {1, . . . , q}) on a Cayley tree T k of degree k ≥ 2 (i.e., with k + 1 edges emanating from each vertex) in an external (possibly random) field. We construct the so-called splitting Gibbs measures (SGM) using generalized boundary conditions on a sequence of expanding balls, subject to a suitable compatibility criterion. Hence, the problem of existence/uniqueness of SGM is reduced to solvability of the corresponding functional equation on the tree. In particular, we introduce the notion of translation-invariant SGMs and prove a novel criterion of translation invariance. Assuming a ferromagnetic nearest-neighbour spin-spin interaction, we obtain various sufficient conditions for uniqueness. For a model with constant external field, we provide in-depth analysis of uniqueness vs. non-uniqueness in the subclass of completely homogeneous SGMs by identifying the phase diagrams on the "temperature-field" plane for different values of the parameters q and k. In a few particular cases (e.g., q = 2 or k = 2), the maximal number of completely homogeneous SGMs in this model is shown to be 2 q − 1, and we make a conjecture (supported by computer calculations) that this bound is valid for all q ≥ 2 and k ≥ 2. Contents2010 Mathematics Subject Classification. Primary 82B26; Secondary 60K35. 4 L. V. BOGACHEV AND U. A. ROZIKOV1.2. Set-up. We start by summarizing the basic concepts for Gibbs measures on a Cayley tree, and also fix some notation.1.2.1. Cayley tree. Let T k be a (homogeneous) Cayley tree of degree k ≥ 2, that is, an infinite connected cycle-free (undirected) regular graph with each vertex incident to k + 1 edges. 3 For example, T 1 = Z. Denote by V = {x} the set of the vertices of the tree and by E = { x, y } the set of its (non-oriented) edges connecting pairs of neighbouring vertices. The natural distance d(x, y) on T k is defined as the number of edges on the unique path connecting vertices x, y ∈ V . In particular, x, y ∈ E whenever d(x, y) = 1. A (non-empty) set Λ ⊂ V is called connected if for any x, y ∈ Λ the path connecting x and y lies in Λ. We denote the complement of Λ by Λ c := V \ Λ and its boundary by ∂Λ := {x ∈ Λ c : ∃y ∈ Λ, d(x, y) = 1}, and we writeΛ = Λ ∪ ∂Λ. The subset of edges in Λ is denotedFix a vertex x • ∈ V , interpreted as the root of the tree. We say that y ∈ V is a direct successor of x ∈ V if x is the penultimate vertex on the unique path leading from the root x • to the vertex y; that is, d(x • , y) = d(x • , x) + 1 and d(x, y) = 1. The set of all direct successors of x ∈ V is denoted S(x). It is convenient to work with the family of the radial subsets centred at x • , defined for n ∈ N 0 := {0} ∪ N byinterpreted as the "ball" and "sphere", respectively, of radius n centred at the root x • . Clearly, ∂V n = W n+1 . Note that if x ∈ W n then S(x) = {y ∈ W n+1 : d(x, y) = 1}. In the special case x = x • we have S(x • ) = W 1 . For short, we set E n := E Vn .Remark 1.1. Note that the sequence of balls (V n ) (n ∈ N 0 ) is ...

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“…To the contrary, suppose first that v m < 1 for some θ > θ m . Then, according to Lemma 6.6, we have 19) and furthermore, Assume now that v m = 1 for some θ > θ m . Then…”

Section: 2mentioning

confidence: 99%

Section: Translation Invariancementioning

confidence: 99%

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On the uniqueness of Gibbs measure in the Potts model on a Cayley tree with external field

Bogachev

Rozikov²

2019

J. Stat. Mech.

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“…For this reason, we are considering positive elements h x ∈ B x,+ as the boundary condition. For more information about the boundary conditions related to classical models we refer [29,50].…”

Section: Construction Of Quantum Markov Chains On Cayley Treementioning

confidence: 99%

Phase Transitions for Quantum Markov Chains Associated with Ising Type Models on a Cayley Tree

2016

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The main aim of the present paper is to prove the existence of a phase transition in quantum Markov chain (QMC) scheme for the Ising type models on a Cayley tree. Note that this kind of models do not have one-dimensional analogous, i.e. the considered model persists only on trees. In this paper, we provide a more general construction of forward QMC. In that construction, a QMC is defined as a weak limit of finite volume states with boundary conditions, i.e. QMC depends on the boundary conditions. Our main result states the existence of a phase transition for the Ising model with competing interactions on a Cayley tree of order two. By the phase transition we mean the existence of two distinct QMC which are not quasi-equivalent and their supports do not overlap. We also study some algebraic property of the disordered phase of the model, which is a new phenomena even in a classical setting.Mathematics Subject Classification: 46L53, 60J99, 46L60, 60G50, 82B10, 81Q10, 94A17.

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“…In [29] we have obtained all measures µ i by changing boundary conditions (configurations). To give the main result of [29] we need the following result of [50]. Denote…”

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confidence: 99%

Gibbs measures of Potts model on Cayley trees: a survey and applications

Rozikov

2021

Preprint

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In this paper we give a systematic review of the theory of Gibbs measures of Potts model on Cayley trees (developed since 2013) and discuss many applications of the Potts model to real world situations: mainly biology, physics, and some examples of alloy behavior, cell sorting, financial engineering, flocking birds, flowing foams, image segmentation, medicine, sociology etc. 2020 Mathematics Subject Classification. 82B20 (82B26).

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

Cited by 8 publications

References 10 publications

On the uniqueness of Gibbs measure in the Potts model on a Cayley tree with external field

On the uniqueness of Gibbs measure in the Potts model on a Cayley tree with external field

Phase Transitions for Quantum Markov Chains Associated with Ising Type Models on a Cayley Tree

Gibbs measures of Potts model on Cayley trees: a survey and applications

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