Splitting Gibbs Measures Research Articles

Three-state p-SOS models on binary Cayley trees

Abstract We consider a version of the solid-on-solid model on the Cayley tree of order two in which vertices carry spins of value 0 , 1 or 2 and the pairwise interaction of neighboring vertices is given by their spin difference to the power p > 0. We exhibit all translation-invariant splitting Gibbs measures (TISGMs) of the model and demonstrate the existence of up to seven such measures, depending on the parameters. We further establish general conditions for extremality and non-extremality of TISGMs in the set of all Gibbs measures and use them to examine selected TISGMs for a small and a large p. Notably, our analysis reveals that extremality properties are similar for large p compared to the case p = 1, a case that has been explored already in previous work. However, for the small p, certain measures that were consistently non-extremal for p = 1 do exhibit transitions between extremality and non-extremality.

The qualitative properties of 1D mixed-type Potts-SOS model with 1-spin and its dynamical behavior

In this investigation, we consider the one-dimensional (1D) mixed-type Potts-SOS model, where the spin is within the range of {−1, 0, 1}. We elaborate thermodynamic characteristics of 1D Potts-SOS model through the application of three distinct analytical approaches. We provide a brief overview of all translation-invariant splitting Gibbs measures (TISGMs) applicable to this model. For the model with a boundary field condition, we provide a comprehensive analysis of the uniqueness and non-uniqueness properties of the subset of fully homogeneous splitting Gibbs masures (SGMs). Our demonstration reveals that the model under consideration does not exhibit a phase transition phenomenon. We are also curious in the stability study of the suggested fixed points associated with the Gibbs measures. We show that the magnetization decreases to zero. By means of the transfer matrix method, we compute the free energy, entropy and internal energy of the model.

The extremality of disordered phases for the mixed spin-(1,1/2) Ising model on a Cayley tree of arbitrary order

This paper continues the exploration of translation-invariant splitting Gibbs measures (TISGMs) within the framework of the Ising model with mixed spin-(1,1/2) (abbreviated as (1,1/2)-MSIM) on a Cayley tree of arbitrary order. Building upon our prior work (Akin and Mukhamedov 2022 J. Stat. Mech. 053204), where we extensively elucidated TISGMs and investigated the extremality of disordered phases employing a Markov chain indexed by a tree on a semi-infinite Cayley tree of order two, our current research extends this analysis. In the present study, we begin by constructing TISGMs and tree-indexed Markov chains tailored to the (1,1/2)-MSIM. Notably, we extend our scope to encompass Cayley trees of varying orders, thereby affording a comprehensive examination of the extremality characteristics inherent to disordered phases. Utilizing the Kesten–Stigum condition, we delve into the non-extremality aspects of disordered phases by scrutinizing the eigenvalues of the stochastic matrix associated with the (1,1/2)-MSIM on Cayley trees with order k ( k⩾3 ). This approach allows for a nuanced exploration of the model’s behavior and stability under different conditions.

Phase transition for the Ising model with mixed spins on a Cayley tree

In the present paper, we consider the Ising model with mixed spin- (1, 1/2) on the second order Cayley tree. For this model, a construction of splitting Gibbs measures is given that allows us to establish the existence of the phase transition (non-uniqueness of Gibbs measures). We point out that, in the phase transition region, the considered model exhibits three translation-invariant Gibbs measures in the ferromagnetic and anti-ferromagnetic regimes, respectively, while the classical Ising model does not possess such Gibbs measures in the anti-ferromagnetic setting. It turns out, that like the classical Ising model, we can find a disordered Gibbs measure, therefore, its non-extremity and extremity are investigated by means of tree-indexed Markov chains.

Extremality of Disordered Phase of λ-Model on Cayley Trees

In this paper, we consider the λ-model for an arbitrary-order Cayley tree that has a disordered phase. Such a phase corresponds to a splitting Gibbs measure with free boundary conditions. In communication theory, such a measure appears naturally, and its extremality is related to the solvability of the non-reconstruction problem. In general, the disordered phase is not extreme; hence, it is natural to find a condition for their extremality. In the present paper, we present certain conditions for the extremality of the disordered phase of the λ-model.

Extremality of translation-invariant Gibbs measures for the Potts–SOS model on the Cayley tree

In this paper, we consider the Potts–SOS model where the spin takes values in the set {0, 1, 2} on the Cayley tree of order two. We describe all the translation-invariant splitting Gibbs measures (GMs) for this model in some conditions. Moreover, we investigate whether these GMs are extremal or non-extremal in the set of all GMs.

Uniqueness of Gibbs Measures for an Ising Model with Continuous Spin Values on a Cayley Tree

In this paper we consider an Ising model with nearest-neighbour interactions with spin space [0, 1] on a Cayley tree. We present a sufficient condition under which the Ising model has a unique splitting Gibbs measure.

P-adic boundary laws and Markov chains on trees

In this paper, we consider a potential on general infinite trees with q spin values and nearest-neighbor p-adic interactions given by a stochastic matrix. We show the uniqueness of the associated Markov chain (splitting Gibbs measures) under some sufficient conditions on the stochastic matrix. Moreover, we find a family of stochastic matrices for which there are at least two p-adic Markov chains on an infinite tree (in particular, on a Cayley tree). When the p-adic norm of q is greater (resp. less) than the norm of any element of the stochastic matrix then it is proved that the p-adic Markov chain is bounded (resp. is not bounded). Our method uses a classical boundary law argument carefully adapted from the real case to the p-adic case, by a systematic use of some nice peculiarities of the ultrametric (p-adic) norms.

On the uniqueness of Gibbs measure in the Potts model on a Cayley tree with external field**To the memory of H-O Georgii.

The paper concerns the q-state Potts model (i.e. with spin values in ) on a Cayley tree of degree (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 versus 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 2q − 1, and we make a conjecture (supported by computer calculations) that this bound is valid for all and .

Hard-core and soft-core Widom–Rowlinson models on Cayley trees

We consider both hard-core and soft-core Widom–Rowlinson models with spin values on a Cayley tree of order and we are interested in the Gibbs measures of the models. The models depend on three parameters: the order k of the tree, describing the strength of the (ferromagnetic or antiferromagnetic) interaction, and describing the intensity for particles. The hard-core Widom–Rowlinson model corresponds to the case .For the binary tree k = 2, and for k = 3 we prove that the ferromagnetic model has either one or three splitting Gibbs measures (tree-automorphism invariant Gibbs measures (TISGMs) which are tree-indexed Markov chains). We also give the exact form of the corresponding critical curves in parameter space. For higher values of k we give an explicit sufficient bound ensuring non-uniqueness which we conjecture to be the exact curve. Moreover, for the antiferromagnetic model we explicitly give two critical curves , , and prove that on these curves there are exactly two TISGMs; between these curves there are exactly three TISGMs; otherwise there exists a unique TISGM. Some periodic and non-periodic SGMs are also constructed in the ferromagnetic model.

Phase transitions for a model with uncountable spin space on the Cayley tree: the general case

In this paper we complete the analysis of a statistical mechanics model on Cayley trees of any degree, started in Botirov (Positivity 21(3):955–961, 2017), Eshkabilov et al. (J Stat Phys 147(4):779–794, 2012), Eshkabilov and Rozikov (Math Phys Anal Geom 13:275–286, 2010), Botirov et al. (Lobachevskii J Math 34(3):256–263 2013) and Jahnel et al. (Math Phys Anal Geom 17:323–331 2014). The potential is of nearest-neighbor type and the local state space is compact but uncountable. Based on the system parameters we prove existence of a critical value $$\theta _{\mathrm{c}}$$ such that for $$\theta \le \theta _{\mathrm{c}}$$ there is a unique translation-invariant splitting Gibbs measure. For $$\theta _{\mathrm{c}}<\theta $$ there is a phase transition with exactly three translation-invariant splitting Gibbs measures. The proof rests on an analysis of fixed points of an associated non-linear Hammerstein integral operator for the boundary laws.

Nontranslation invariant Gibbs measures for models with uncountable set of spin values on a Cayley tree

We consider models with nearest-neighbour interactions and with the set [0, 1] of spin values, on a Cayley tree of order k ≥ 1. It is known that the “splitting Gibbs measures” of the model can be described by solutions of a nonlinear integral equation. Recently, by solving this integral equation some periodic (in particular translation invariant) splitting Gibbs measures were found. In this paper we give three constructions of new sets of nontranslation invariant splitting Gibbs measures. Our constructions are based on known solutions of the integral equation (1.5) .

Four competing interactions for models with an uncountable set of spin values on a Cayley tree

We consider models with four competing interactions (external field, nearest neighbor, second neighbor, and three neighbors) and an uncountable set [0, 1] of spin values on the Cayley tree of order two. We reduce the problem of describing the splitting Gibbs measures of the model to the problem of analyzing solutions of a nonlinear integral equation and study some particular cases for Ising and Potts models. We also show that periodic Gibbs measures for the given models either are translation invariant or have the period two. We present examples where periodic Gibbs measures with the period two are not unique.

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

We consider translation-invariant splitting Gibbs measures (TISGMs) for the q-state 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 $$\mu $$ we explicitly give the set of boundary conditions such that limiting Gibbs measures with respect to these boundary conditions coincide with $$\mu $$ .

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

We continue our study of the full set of translation-invariant splitting Gibbs measures TISGMs, translation-invariant tree-indexed Markov chains for the q-state Potts model on a Cayley tree. In our previous work Kulske et al., J Stat Phys 156 2014, 189-200 we gave a full description of the TISGMs, and showed in particular that at sufficiently low temperatures their number is 2q-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 2q-1+q extremal TISGMs. For the Cayley tree of order two we give explicit formulae and some numerical values. © 2016 Wiley Periodicals, Inc. Random Struct. Alg., 50, 636-678, 2017

Gibbs measures for the fertile three-state hard-core models on a Cayley tree

We study translation-invariant splitting Gibbs measures (TISGMs, tree-indexed Markov chains) for the fertile three-state hard-core models with activity $$\lambda >0$$?>0 on the Cayley tree of order $$k\ge 1$$k?1. There are four such models: wrench, wand, hinge, and pipe. These models arise as simple examples of loss networks with nearest-neighbor exclusion. It is known that (i) for the wrench and pipe cases $$\forall \lambda >0$$??>0 and $$k\ge 1$$k?1, there exists a unique TISGM; (ii) for hinge (resp. wand) case at $$k=2$$k=2 if $$\lambda 9/4$$?>9/4 (resp. $$\lambda >1$$?>1), there exist three TISGMs. In this paper we generalize the result (ii) for any $$k\ge 2$$k?2, i.e., for hinge and wand cases we find the exact critical value $$\lambda _\mathrm{cr}(k)$$?cr(k) with properties mentioned in (ii). Moreover, we find some regions for the $$\lambda $$? parameter ensuring that a given TISGM is extreme or non-extreme in the set of all Gibbs measures. For the Cayley tree of order two, we give explicit formulae and some numerical values.

Extremality of Translation-Invariant Phases for a Three-State SOS-Model on the Binary Tree

We consider the solid-on-solid model, with spin values $$0,1,2$$ , on the Cayley tree of order two (binary tree). We treat both ferromagnetic and antiferromagnetic coupling, with interactions which are proportional to the absolute value of the spin differences. We present a classification of all translation-invariant phases (splitting Gibbs measures) of the model. These measures are labeled by solutions to a nonlinear vector-valued functional equation. We show uniqueness in the case of antiferromagnetic interactions, and existence of up to seven phases in the case of ferromagnetic interactions, where the number of phases depends on the interaction strength. Next we investigate whether these states are extremal or non-extremal in the set of all Gibbs measures, when the coupling strength is varied, whenever they exist. We show that two states are always extremal, two states are always non-extremal, while three of the seven states make transitions between extremality and non-extremality. We provide explicit bounds on those transition values, making use of algebraic properties of the models, and an adaptation of the method of Martinelli, Sinclair, Weitz.

Description of the Translation-Invariant Splitting Gibbs Measures for the Potts Model on a Cayley Tree

For the $q$-state Potts model on a Cayley tree of order $k\geq 2$ it is well-known that at sufficiently low temperatures there are at least $q+1$ translation-invariant Gibbs measures which are also tree-indexed Markov chains. Such measures are called translation-invariant splitting Gibbs measures (TISGMs). In this paper we find all TISGMs, and show in particular that at sufficiently low temperatures their number is $2^{q}-1$. We prove that there are $[q/2]$ (where $[a]$ is the integer part of $a$) critical temperatures at which the number of TISGMs changes and give the exact number of TISGMs for each intermediate temperature. For the binary tree we give explicit formulae for the critical temperatures and the possible TISGMs. While we show that these measures are never convex combinations of each other, the question which of these measures are extremals in the set of all Gibbs measures will be treated in future work.

Uniqueness of Gibbs Measure for Models with Uncountable Set of Spin Values on a Cayley Tree

We consider models with nearest-neighbor interactions and with the set $[0,1]$ of spin values, on a Cayley tree of order $k\geq 1$. It is known that the "splitting Gibbs measures" of the model can be described by solutions of a nonlinear integral equation. For arbitrary $k\geq 2$ we find a sufficient condition under which the integral equation has unique solution, hence under the condition the corresponding model has unique splitting Gibbs measure.

On Models with Uncountable Set of Spin Values on a Cayley Tree: Integral Equations

We consider models with nearest-neighbor interactions and with the set [0, 1] of spin values, on a Cayley tree of order k ⩾ 1. We reduce the problem of describing the “splitting Gibbs measures” of the model to the description of the solutions of some nonlinear integral equation. For k = 1 we show that the integral equation has a unique solution. In case k ⩾ 2 some models (with the set [0, 1] of spin values) which have a unique splitting Gibbs measure are constructed. Also for the Potts model with uncountable set of spin values it is proven that there is unique splitting Gibbs measure.