Abstract.
In this paper, we consider an Ising model with three competing interactions on a triangular chandelier-lattice (TCL). We describe the existence, uniqueness, and non-uniqueness of translation-invariant Gibbs measures associated with the Ising model. We obtain an explicit formula for Gibbs measures with a memory of length 2 satisfying consistency conditions. It is proved rigorously that the model exhibits phase transitions only for given values of the coupling constants. As a consequence of our approach, the dichotomy between alternative solutions of Hamiltonian models on TCLs is solved. Finally, two numerical examples are given to illustrate the usefulness and effectiveness of the proposed theoretical results. Keywords: Chandelier lattices, Gibbs measures, Ising model, phase transition. PACS: 05.70.Fh; 05.70.Ce; 75.10.Hk.

Contents

1 Introduction

2 Preliminary

2.1 Triangular Chandelier Lattice

2.2 Kolmogorov consistency condition

3 Gibbs Measures

3.1 New Gibbs measures

3.2 Basic Equations

4 Translation-Invariant Gibbs measures (TIGMs) on a TCL

5 Phase translations

5.1 An example indicating a phase transition

6 Conclusions

1. Introduction

As is known, Cayley tree (or Bethe lattice), introduced by Hans Bethe in 1935, is a non-realistic lattice. Since other operations and the calculations on this lattice are easier to understand than the dݑ‘ditalic_d-dimensional ݐ™dsuperscriptݐ™ݑ‘\mathbfZ^dbold_Z start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT lattice, many of the topics in statistical physics have recently been taken into account on the Cayley tree [2]. Thus, the results obtained on the Cayley tree became a source of inspiration for the dݑ‘ditalic_d-dimensional ݐ™dsuperscriptݐ™ݑ‘\mathbfZ^dbold_Z start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT lattice. As a result, many researchers have employed the Ising and Potts models [7, 41] in conjunction with the Cayley tree [4, 7, 8, 9]. The Ising model has relevance to physical, chemical, and biological systems [5, 10, 11].

We were then able to identify a similar lattice, which we identified as triple, quadruple, quintuple, and so on. So we examined the dynamic behaviors of Ising models on these Cayley-like lattices. Up to now, some studies have been done [30, 31, 32, 35]. Although the results are similar to the results for the models on the Cayley tree, we think that we would get many different results in the future. We called this model a triangular, rectangular, pentagonal and similar ”Chandelier” model. Compared with ݐ™dsuperscriptݐ™ݑ‘\mathbfZ^dbold_Z start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT lattice, we think that the chandelier lattice is more realistic than the Cayley tree [30, 31, 32, 33, 35]. In this paper, we deal with a Cayley tree-like lattice [32] which we called a Triangular Chandelier Lattice (shortly, TCL) from the configuration model.

The theory of probability is one of the basic branches of mathematics lying at the base of the theory of statistical mechanics [10, 13, 14, 15, 17, 19, 22]. As is known, one of the fundamental problems of statistical mechanics is to specify the set of all Gibbs measures associated to the given Hamiltonian [18, 20, 21, 25, 34]. A Gibbs measure is a probability measure frequently used in many problems of probability theory and statistical mechanics. It is also known that such measures form a convex compact subset that is different from the void in the set of all probability measures. The number of translation-invariant splitting Gibbs measures associated with the Ising model on a Cayley tree can only be one or more than one, depending on temperature [38]. The pݑpitalic_p-adic counterpart of the Ising-Vanniminus model on the Cayley tree of order two was first studied in [26]. There, it was proposed a measure-theoretical approach to investigate the model in the pݑpitalic_p-adic setting.

In this present paper, we want to investigate translation-invariant Gibbs measures (TIGMs) corresponding to an Ising model on the TCL. It is well known that the understanding of phase transitions is one of the most interesting, perhaps the central, problems of equilibrium statistical mechanics [5]. By the phase transition we mean the existence of at least two distinct Gibbs measures associated with the given model [2, 3, 10, 17]. We will investigate the existence of translation invariant Gibbs measures on a wide class of the TCL, restricted only to the memory of length 2. We derive specific realizations for the ANNNI model on these structures. We derive the results within the Markov random field framework, making use of the Kolmogorov consistency conditions. We express the solutions of recurrence relations warranting consistency in terms of the fixed points of a function f(x)ݑ“ݑ¥f(x)italic_f ( italic_x ). We provide some diagrams of the behavior of the function f(x)ݑ“ݑ¥f(x)italic_f ( italic_x ) for different values of the model parameters. We present analytical developments allowing for the identification of Gibbs measures along the usual procedures.

The structure of the present article is as follows: in section 2, we give the necessary definition and preliminaries about Ising model with three competing interactions on a TCL. In section 3, we establish the Gibbs measure associated with the model. In section 4, we describe the existence, uniqueness and non-uniqueness of translation-invariant Gibbs measures associated with the Ising model on a TCL. In section 5, it is proved rigorously that the model exhibits phase transitions only for given values of the coupling constants. As a consequence of our approach, the dichotomy between alternative solutions of Hamiltonian models on TCLs is solved. Finally, in section 6, the relevance of the results obtained for systems on the TCL is discussed and the results are compared to ones on the Cayley tree.

2. Preliminary

2.1. Triangular Chandelier Lattice

Chandelier lattices are simple connected undirected graphs G=(V,E)ݐºݑ‰ݐ¸G=(V,E)italic_G = ( italic_V , italic_E ) (Vݑ‰Vitalic_V set of vertices, Eݐ¸Eitalic_E set of edges). V as the set of vertices. The set of edges. It is clear that the root vertex x(0)superscriptݑ¥0x^(0)italic_x start_POSTSUPERSCRIPT ( 0 ) end_POSTSUPERSCRIPT has kݑ˜kitalic_k nearest neighbors. The notation iݑ–iitalic_i represents the incidence function corresponding to each edge e∈Eݑ’ݐ¸e\in Eitalic_e ∈ italic_E, with end points x1,x2∈Vsubscriptݑ¥1subscriptݑ¥2ݑ‰x_1,x_2\in Vitalic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_x start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ∈ italic_V. There is a distance d(x,y)ݑ‘ݑ¥ݑ¦d(x,y)italic_d ( italic_x , italic_y ) on Vݑ‰Vitalic_V the length of the minimal point from xݑ¥xitalic_x to yݑ¦yitalic_y, with the assumed length of 1 for any edge (see Figure 1). Let’s consider a chandelier with 3 lamps. Hanging on the ceiling. Suppose that the same three quadrants hung on each lamp of the first chandelier were added. In this case, we get a weave that resembles a semi-infinite Cayley tree. We assume here that each lamp is connected to the lamps in the nearest neighbors. Thus, we can have the possibility to investigate the titles examined in statistical physics by calculating the internal, external and full energies corresponding to a Hamiltonian on the chandelier lattice that we have defined.

The distance d(x,y),x,y∈Vݑ‘ݑ¥ݑ¦ݑ¥ݑ¦ݑ‰d(x,y),x,y\in Vitalic_d ( italic_x , italic_y ) , italic_x , italic_y ∈ italic_V, on the chandelier lattice Cksuperscriptݐ¶ݑ˜C^kitalic_C start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT (k>2ݑ˜2k>2italic_k >2), is the number of edges in the shortest path from xݑ¥xitalic_x to yݑ¦yitalic_y. The fixed vertex x(0)superscriptݑ¥0x^(0)italic_x start_POSTSUPERSCRIPT ( 0 ) end_POSTSUPERSCRIPT is called the 00-th level and the vertices in WnsubscriptݑŠݑ›W_nitalic_W start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT are called the nݑ›nitalic_n-th level. For the sake of simplicity we put |x|=d(x,x(0))ݑ¥ݑ‘ݑ¥superscriptݑ¥0|x|=d(x,x^(0))| italic_x | = italic_d ( italic_x , italic_x start_POSTSUPERSCRIPT ( 0 ) end_POSTSUPERSCRIPT ), x∈Vݑ¥ݑ‰x\in Vitalic_x ∈ italic_V. We denote the sphere of radius nݑ›nitalic_n on Vݑ‰Vitalic_V by

Wn(P)=x∈V:d(x,x(0))=nsuperscriptsubscriptݑŠݑ›ݑƒconditional-setݑ¥ݑ‰ݑ‘ݑ¥superscriptݑ¥0ݑ›W_n^(P)=\x\in V:d(x,x^(0))=n\italic_W start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_P ) end_POSTSUPERSCRIPT = italic_x ∈ italic_V : italic_d ( italic_x , italic_x start_POSTSUPERSCRIPT ( 0 ) end_POSTSUPERSCRIPT ) = italic_n
and the ball of radius nݑ›nitalic_n by

Vn(P)=x∈V:d(x,x(0))≤n,superscriptsubscriptݑ‰ݑ›ݑƒconditional-setݑ¥ݑ‰ݑ‘ݑ¥superscriptݑ¥0ݑ›V_n^(P)=\x\in V:d(x,x^(0))\leq n\,italic_V start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_P ) end_POSTSUPERSCRIPT = italic_x ∈ italic_V : italic_d ( italic_x , italic_x start_POSTSUPERSCRIPT ( 0 ) end_POSTSUPERSCRIPT ) ≤ italic_n ,
where vertexes xݑ¥xitalic_x is prolonged downwards relative to x(0)superscriptݑ¥0x^(0)italic_x start_POSTSUPERSCRIPT ( 0 ) end_POSTSUPERSCRIPT.

Ln=x,y∈Vn.fragmentssubscriptݐ¿ݑ›fragmentslx,yL.L_n=\x,y\in V_n\.italic_L start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT = italic_x , italic_y ∈ italic_V start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT .
For example, W2(P)=zv(u):u,v=1,2,3superscriptsubscriptݑŠ2ݑƒconditional-setsubscriptsuperscriptݑ§ݑ¢ݑ£formulae-sequenceݑ¢ݑ£123W_2^(P)=\z^(u)_v:u,v=1,2,3\italic_W start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_P ) end_POSTSUPERSCRIPT = italic_z start_POSTSUPERSCRIPT ( italic_u ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_v end_POSTSUBSCRIPT : italic_u , italic_v = 1 , 2 , 3 (see Figure 1).

The set of direct prolonged successors of any vertex x∈Wnݑ¥subscriptݑŠݑ›x\in W_nitalic_x ∈ italic_W start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT is denoted by

Sk(P)(x)=y∈Wn+1:d(x,y)=1.superscriptsubscriptݑ†ݑ˜ݑƒݑ¥conditional-setݑ¦subscriptݑŠݑ›1ݑ‘ݑ¥ݑ¦1S_k^(P)(x)=\y\in W_n+1:d(x,y)=1\.italic_S start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_P ) end_POSTSUPERSCRIPT ( italic_x ) = italic_y ∈ italic_W start_POSTSUBSCRIPT italic_n + 1 end_POSTSUBSCRIPT : italic_d ( italic_x , italic_y ) = 1 .
The set of same-level neighborhoods of any vertex x∈Wnݑ¥subscriptݑŠݑ›x\in W_nitalic_x ∈ italic_W start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT will be denoted by

SLk(x)=y∈Wn:d(x,y)=1.ݑ†subscriptݐ¿ݑ˜ݑ¥conditional-setݑ¦subscriptݑŠݑ›ݑ‘ݑ¥ݑ¦1SL_k(x)=\y\in W_n:d(x,y)=1\.italic_S italic_L start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ( italic_x ) = italic_y ∈ italic_W start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT : italic_d ( italic_x , italic_y ) = 1 .
It is clear that |SLk(x)|=2ݑ†subscriptݐ¿ݑ˜ݑ¥2|SL_k(x)|=2| italic_S italic_L start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ( italic_x ) | = 2, for all vertexes x∈Wnݑ¥subscriptݑŠݑ›x\in W_nitalic_x ∈ italic_W start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT.

Definition 2.1.

Hereafter, we will use the following definitions for neighborhoods.

(1)
Two vertices xݑ¥xitalic_x and yݑ¦yitalic_y, x,y∈Vݑ¥ݑ¦ݑ‰x,y\in Vitalic_x , italic_y ∈ italic_V are called nearest-neighbors (NN) if there exists an edge e∈Eݑ’ݐ¸e\in Eitalic_e ∈ italic_E connecting them, which is denoted by e=fragmentsex,ye=italic_e = .

(2)
The nearest-neighbor vertices x,y∈Vݑ¥ݑ¦ݑ‰x,y\in Vitalic_x , italic_y ∈ italic_V that are not prolonged are called same-level nearest-neighbors (SLNN) if |x|=|y|ݑ¥ݑ¦|x|=|y|| italic_x | = | italic_y | and are denoted by >x,y<~~fragmentsx,y\widetilde>x,yitalic_x , italic_y
(3)
Two vertices x,y∈Vݑ¥ݑ¦ݑ‰x,y\in Vitalic_x , italic_y ∈ italic_V are called the next-nearest-neighbors (NNN) if there exists a vertex z∈Vݑ§ݑ‰z\in Vitalic_z ∈ italic_V such that x,zݑ¥ݑ§x,zitalic_x , italic_z and y,zݑ¦ݑ§y,zitalic_y , italic_z are NN, that is if d(x,y)=2ݑ‘ݑ¥ݑ¦2d(x,y)=2italic_d ( italic_x , italic_y ) = 2.

(4)
The next-nearest-neighbor vertices x∈Wnݑ¥subscriptݑŠݑ›x\in W_nitalic_x ∈ italic_W start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT and y∈Wn+2ݑ¦subscriptݑŠݑ›2y\in W_n+2italic_y ∈ italic_W start_POSTSUBSCRIPT italic_n + 2 end_POSTSUBSCRIPT are called prolonged next-nearest-neighbors (PNNN) if |x|≠|y|ݑ¥ݑ¦|x|\neq|y|| italic_x | ≠| italic_y | and is denoted by >x,yx,y<> italic_x , italic_y <(see Figure 1).

2.2. Kolmogorov consistency condition

Kolmogorov’s extension theorem allows us for the construction of a variety of measures on infinite-dimensional spaces (see [36] for details).

Now let us explain this theorem for a one-dimensional situation. Let S=0,2,…,k-1ݑ†02…ݑ˜1S=\0,2,...,k-1\italic_S = 0 , 2 , … , italic_k - 1 be a finite state space. On the infinite product space Ω=SℤΩsuperscriptݑ†ℤ\Omega=S^\mathbbZroman_Ω = italic_S start_POSTSUPERSCRIPT blackboard_Z end_POSTSUPERSCRIPT, one can define the product σݜŽ\sigmaitalic_σ-algebra, which is generated by cylinder sets [i1,…,iN]m=x∈Sݐ™:xm=i0,…,xm+N-1=iNfragmentssubscriptfragments[subscriptݑ–1,…,subscriptݑ–ݑ]ݑšfragmentsxsuperscriptݑ†ݐ™:subscriptݑ¥ݑšsubscriptݑ–0,…,subscriptݑ¥ݑšݑ1subscriptݑ–ݑ{}_m[i_1,...,i_N]=\x\in S^\mathbfZ:x_m=i_0,\ldots,x_m+N-1=i_% N\start_FLOATSUBSCRIPT italic_m end_FLOATSUBSCRIPT [ italic_i start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , italic_i start_POSTSUBSCRIPT italic_N end_POSTSUBSCRIPT ] = italic_x ∈ italic_S start_POSTSUPERSCRIPT bold_Z end_POSTSUPERSCRIPT : italic_x start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT = italic_i start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , … , italic_x start_POSTSUBSCRIPT italic_m + italic_N - 1 end_POSTSUBSCRIPT = italic_i start_POSTSUBSCRIPT italic_N end_POSTSUBSCRIPT of length NݑNitalic_N based on the block (i1,…,iN)subscriptݑ–1…subscriptݑ–ݑ(i_1,...,i_N)( italic_i start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , italic_i start_POSTSUBSCRIPT italic_N end_POSTSUBSCRIPT ) at the place mݑšmitalic_m. Note that a cylinder set is a set of sequences where we fix which symbol can occur in a finite number of places. We denote by ݔ(Sℤ)ݔsuperscriptݑ†ℤ\mathfrakM(S^\mathbbZ)fraktur_M ( italic_S start_POSTSUPERSCRIPT blackboard_Z end_POSTSUPERSCRIPT ) the set of all measures on Sℤsuperscriptݑ†ℤS^\mathbbZitalic_S start_POSTSUPERSCRIPT blackboard_Z end_POSTSUPERSCRIPT. The set of all σݜŽ\sigmaitalic_σ-invariant measures in Sℤsuperscriptݑ†ℤS^\mathbbZitalic_S start_POSTSUPERSCRIPT blackboard_Z end_POSTSUPERSCRIPT is denoted by ݔσ(Sℤ)subscriptݔݜŽsuperscriptݑ†ℤ\mathfrakM_\sigma(S^\mathbbZ)fraktur_M start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT ( italic_S start_POSTSUPERSCRIPT blackboard_Z end_POSTSUPERSCRIPT ), where σݜŽ\sigmaitalic_σ is the shift transformation.

Proposition 2.2.

[1, (8.1) Proposition] For μ∈ݔσ(Sℤ)ݜ‡subscriptݔݜŽsuperscriptݑ†ℤ\mu\in\mathfrakM_\sigma(S^\mathbbZ)italic_μ ∈ fraktur_M start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT ( italic_S start_POSTSUPERSCRIPT blackboard_Z end_POSTSUPERSCRIPT ) the following properties are valid:

(1)
∑i∈Sμ(0[i])=1fragmentssubscriptݑ–ݑ†μfragmentssubscript(0fragments[i])1\sum\limits_i\in S\mu{(}_0[i])=1∑ start_POSTSUBSCRIPT italic_i ∈ italic_S end_POSTSUBSCRIPT italic_μ ( start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT [ italic_i ] ) = 1;

(2)
μ(n[i0,…,ik])≥0fragmentsμfragmentssubscript(ݑ›fragments[subscriptݑ–0,…,subscriptݑ–ݑ˜])0\mu(_n[i_0,...,i_k])\geq 0italic_μ ( start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT [ italic_i start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , … , italic_i start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ] ) ≥ 0 for any block (i0,i1,…,ik)∈Sk+1subscriptݑ–0subscriptݑ–1…subscriptݑ–ݑ˜superscriptݑ†ݑ˜1(i_0,i_1,...,i_k)\in S^k+1( italic_i start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_i start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , italic_i start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) ∈ italic_S start_POSTSUPERSCRIPT italic_k + 1 end_POSTSUPERSCRIPT and any n∈ℤݑ›ℤn\in\mathbbZitalic_n ∈ blackboard_Z;

(3)
μ(n[i0,…,ik])=∑ik+1∈Sμ(n[i0,…,ik,ik+1])fragmentsμfragmentssubscript(ݑ›fragments[subscriptݑ–0,…,subscriptݑ–ݑ˜])subscriptsubscriptݑ–ݑ˜1ݑ†μfragmentssubscript(ݑ›fragments[subscriptݑ–0,…,subscriptݑ–ݑ˜,subscriptݑ–ݑ˜1])\mu(_n[i_0,...,i_k])=\sum\limits_i_k+1\in S\mu{(}_n% [i_0,...,i_k,i_k+1])italic_μ ( start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT [ italic_i start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , … , italic_i start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ] ) = ∑ start_POSTSUBSCRIPT italic_i start_POSTSUBSCRIPT italic_k + 1 end_POSTSUBSCRIPT ∈ italic_S end_POSTSUBSCRIPT italic_μ ( start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT [ italic_i start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , … , italic_i start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT , italic_i start_POSTSUBSCRIPT italic_k + 1 end_POSTSUBSCRIPT ] );

(4)
μ(n[i0,…,ik])=∑i-1∈Sμ(n[i-1,i0,…,ik])fragmentsμfragmentssubscript(ݑ›fragments[subscriptݑ–0,…,subscriptݑ–ݑ˜])subscriptsubscriptݑ–1ݑ†μfragmentssubscript(ݑ›fragments[subscriptݑ–1,subscriptݑ–0,…,subscriptݑ–ݑ˜])\mu(_n[i_0,...,i_k])=\sum\limits_i_-1\in S\mu{(}_n[% i_-1,i_0,...,i_k])italic_μ ( start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT [ italic_i start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , … , italic_i start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ] ) = ∑ start_POSTSUBSCRIPT italic_i start_POSTSUBSCRIPT - 1 end_POSTSUBSCRIPT ∈ italic_S end_POSTSUBSCRIPT italic_μ ( start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT [ italic_i start_POSTSUBSCRIPT - 1 end_POSTSUBSCRIPT , italic_i start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , … , italic_i start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ] ).

By a special case of Kolmogoroff’s consistency theorem (see [1]), these properties are sufficient to define a measure. It is well known that a Gibbs measure is a generalization of a Markov measure to any graph, therefore any Gibbs measure should satisfy the conditions in the Proposition 2.2.

We shall give two examples satisfied the conditions in Proposition 2.2 and to illustrate the consistency conditions.

Example 2.1.

Let π=(pi)i∈Sݜ‹subscriptsubscriptݑݑ–ݑ–ݑ†\pi=(p_i)_i\in Sitalic_π = ( italic_p start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) start_POSTSUBSCRIPT italic_i ∈ italic_S end_POSTSUBSCRIPT be any probability vector on the state set Sݑ†Sitalic_S. For each n≥0ݑ›0n\geq 0italic_n ≥ 0, define

(2.1) μπ(m[i0,…,in])=pn(i0,i1,…,in)=pi0pi1…pin,fragmentssubscriptݜ‡ݜ‹fragmentssubscript(ݑšfragments[subscriptݑ–0,…,subscriptݑ–ݑ›])subscriptݑݑ›fragments(subscriptݑ–0,subscriptݑ–1,…,subscriptݑ–ݑ›)subscriptݑsubscriptݑ–0subscriptݑsubscriptݑ–1…subscriptݑsubscriptݑ–ݑ›,\mu_\pi(_m[i_0,...,i_n])=p_n(i_0,i_1,...,i_n)=p_i_0% p_i_1\ldots p_i_n,italic_μ start_POSTSUBSCRIPT italic_π end_POSTSUBSCRIPT ( start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT [ italic_i start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , … , italic_i start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ] ) = italic_p start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_i start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_i start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , italic_i start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ) = italic_p start_POSTSUBSCRIPT italic_i start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_p start_POSTSUBSCRIPT italic_i start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT … italic_p start_POSTSUBSCRIPT italic_i start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT end_POSTSUBSCRIPT ,
where i0,i1,…,in∈Ssubscriptݑ–0subscriptݑ–1…subscriptݑ–ݑ›ݑ†i_0,i_1,...,i_n\in Sitalic_i start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_i start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , italic_i start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ∈ italic_S. It is clear that pnn≥0subscriptsubscriptݑݑ›ݑ›0\p_n\_n\geq 0 italic_p start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUBSCRIPT italic_n ≥ 0 end_POSTSUBSCRIPT satisfies the consistency conditions (1)-(4) in Proposition 2.2 (see [1]). Such a measure μπsubscriptݜ‡ݜ‹\mu_\piitalic_μ start_POSTSUBSCRIPT italic_π end_POSTSUBSCRIPT is called a Bernoulli measure. One of motivation examples is the Bernoulli measure, which also satisfies the compatible property.

Example 2.2.

Let π=(pi)i∈Sݜ‹subscriptsubscriptݑݑ–ݑ–ݑ†\pi=(p_i)_i\in Sitalic_π = ( italic_p start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) start_POSTSUBSCRIPT italic_i ∈ italic_S end_POSTSUBSCRIPT be any probability vector on the state set Sݑ†Sitalic_S and let P=(pij)i,j∈Sݑƒsubscriptsubscriptݑݑ–ݑ—ݑ–ݑ—ݑ†P=(p_ij)_i,j\in Sitalic_P = ( italic_p start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT ) start_POSTSUBSCRIPT italic_i , italic_j ∈ italic_S end_POSTSUBSCRIPT be any stochastic matrix, i.e., 0≤pij≤10subscriptݑݑ–ݑ—10\leq p_ij\leq 10 ≤ italic_p start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT ≤ 1 end ∑k∈Spik=1subscriptݑ˜ݑ†subscriptݑݑ–ݑ˜1\sum_k\in Sp_ik=1∑ start_POSTSUBSCRIPT italic_k ∈ italic_S end_POSTSUBSCRIPT italic_p start_POSTSUBSCRIPT italic_i italic_k end_POSTSUBSCRIPT = 1 for each i,j∈S.ݑ–ݑ—ݑ†i,j\in S.italic_i , italic_j ∈ italic_S . Thus, πݜ‹\piitalic_π is defined as a probability vector such that πP=πݜ‹ݑƒݜ‹\pi P=\piitalic_π italic_P = italic_π. If PݑƒPitalic_P is irreducible, πݜ‹\piitalic_π is uniquely defined.

For each n≥0ݑ›0n\geq 0italic_n ≥ 0, the function defined by

μπP(m[i0,…,in])=pn(i0,i1,…,in)=pi0pi0i1…pin-1in,wherei0,i1,…,in∈Sfragmentssubscriptݜ‡ݜ‹ݑƒfragmentssubscript(ݑšfragments[subscriptݑ–0,…,subscriptݑ–ݑ›])subscriptݑݑ›fragments(subscriptݑ–0,subscriptݑ–1,…,subscriptݑ–ݑ›)subscriptݑsubscriptݑ–0subscriptݑsubscriptݑ–0subscriptݑ–1…subscriptݑsubscriptݑ–ݑ›1subscriptݑ–ݑ›,wheresubscriptݑ–0,subscriptݑ–1,…,subscriptݑ–ݑ›S\mu_\pi P(_m[i_0,...,i_n])=p_n(i_0,i_1,...,i_n)=p_i_0p_i_% 0i_1\ldots p_i_n-1i_n,\ \ \ \ where\ i_0,i_1,...,i_n\in Sitalic_μ start_POSTSUBSCRIPT italic_π italic_P end_POSTSUBSCRIPT ( start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT [ italic_i start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , … , italic_i start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ] ) = italic_p start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_i start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_i start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , italic_i start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ) = italic_p start_POSTSUBSCRIPT italic_i start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_p start_POSTSUBSCRIPT italic_i start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT italic_i start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT … italic_p start_POSTSUBSCRIPT italic_i start_POSTSUBSCRIPT italic_n - 1 end_POSTSUBSCRIPT italic_i start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT end_POSTSUBSCRIPT , italic_w italic_h italic_e italic_r italic_e italic_i start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_i start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , italic_i start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ∈ italic_S
satisfies the consistency conditions (1)-(4) in Proposition 2.2 (see [1]). Such a measure μπPsubscriptݜ‡ݜ‹ݑƒ\mu_\pi Pitalic_μ start_POSTSUBSCRIPT italic_π italic_P end_POSTSUBSCRIPT is called a Markov measure.

The proof of the Proposition 2.2 can clearly be checked for both the Bernoulli and the Markov measures on σݜŽ\sigmaitalic_σ-algebra [1]. For any cylinder set [i0,…,in]m=x∈Sݐ™:xm=i0,…,xm+n-1=infragmentssubscriptfragments[subscriptݑ–0,…,subscriptݑ–ݑ›]ݑšfragmentsxsuperscriptݑ†ݐ™:subscriptݑ¥ݑšsubscriptݑ–0,…,subscriptݑ¥ݑšݑ›1subscriptݑ–ݑ›{}_m[i_0,...,i_n]=\x\in S^\mathbfZ:x_m=i_0,\ldots,x_m+n-1=i_% n\start_FLOATSUBSCRIPT italic_m end_FLOATSUBSCRIPT [ italic_i start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , … , italic_i start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ] = italic_x ∈ italic_S start_POSTSUPERSCRIPT bold_Z end_POSTSUPERSCRIPT : italic_x start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT = italic_i start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , … , italic_x start_POSTSUBSCRIPT italic_m + italic_n - 1 end_POSTSUBSCRIPT = italic_i start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT and any k≥1,ݑ˜1k\geq 1,italic_k ≥ 1 , we have

[i0,…,in]m=⋃in+1∈S…⋃in+k∈S(m[i0,…,in,in+1,…,in+k])fragmentssubscriptfragments[subscriptݑ–0,…,subscriptݑ–ݑ›]ݑšsubscriptsubscriptݑ–ݑ›1ݑ†…subscriptsubscriptݑ–ݑ›ݑ˜ݑ†fragmentssubscript(ݑšfragments[subscriptݑ–0,…,subscriptݑ–ݑ›,subscriptݑ–ݑ›1,…,subscriptݑ–ݑ›ݑ˜]){}_m[i_0,...,i_n]=\bigcup_i_n+1\in S...\bigcup_i_n+k\in S(_m[i% _0,...,i_n,i_n+1,...,i_n+k])start_FLOATSUBSCRIPT italic_m end_FLOATSUBSCRIPT [ italic_i start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , … , italic_i start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ] = ⋃ start_POSTSUBSCRIPT italic_i start_POSTSUBSCRIPT italic_n + 1 end_POSTSUBSCRIPT ∈ italic_S end_POSTSUBSCRIPT … ⋃ start_POSTSUBSCRIPT italic_i start_POSTSUBSCRIPT italic_n + italic_k end_POSTSUBSCRIPT ∈ italic_S end_POSTSUBSCRIPT ( start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT [ italic_i start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , … , italic_i start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT , italic_i start_POSTSUBSCRIPT italic_n + 1 end_POSTSUBSCRIPT , … , italic_i start_POSTSUBSCRIPT italic_n + italic_k end_POSTSUBSCRIPT ] )
and

pn(m[i0,…,in])=∑in+1∈S…∑in+k∈Spn+k(m[i0,…,in,in+1,…,in+k]).fragmentssubscriptݑݑ›fragmentssubscript(ݑšfragments[subscriptݑ–0,…,subscriptݑ–ݑ›])subscriptsubscriptݑ–ݑ›1ݑ†…subscriptsubscriptݑ–ݑ›ݑ˜ݑ†subscriptݑݑ›ݑ˜fragmentssubscript(ݑšfragments[subscriptݑ–0,…,subscriptݑ–ݑ›,subscriptݑ–ݑ›1,…,subscriptݑ–ݑ›ݑ˜]).p_n(_m[i_0,...,i_n])=\sum_i_n+1\in S\text{...}\sum_i_n+k\in Sp% _n+k(_m[i_0,\text{...},i_n,i_n+1,\text{...},i_n+k]).italic_p start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT [ italic_i start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , … , italic_i start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ] ) = ∑ start_POSTSUBSCRIPT italic_i start_POSTSUBSCRIPT italic_n + 1 end_POSTSUBSCRIPT ∈ italic_S end_POSTSUBSCRIPT … ∑ start_POSTSUBSCRIPT italic_i start_POSTSUBSCRIPT italic_n + italic_k end_POSTSUBSCRIPT ∈ italic_S end_POSTSUBSCRIPT italic_p start_POSTSUBSCRIPT italic_n + italic_k end_POSTSUBSCRIPT ( start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT [ italic_i start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , … , italic_i start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT , italic_i start_POSTSUBSCRIPT italic_n + 1 end_POSTSUBSCRIPT , … , italic_i start_POSTSUBSCRIPT italic_n + italic_k end_POSTSUBSCRIPT ] ) .
The following is the Kolmogorov extension theorem.

Theorem 2.3.

[36, 4.18 Theorem] Let S=0,1,…,r-1ݑ†01normal-…ݑŸ1S=\0,1,...,r-1\italic_S = 0 , 1 , … , italic_r - 1 , for some r≥2.ݑŸ2r\geq 2.italic_r ≥ 2 . Let pnn≥0subscriptsubscriptݑݑ›ݑ›0\p_n\_n\geq 0 italic_p start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUBSCRIPT italic_n ≥ 0 end_POSTSUBSCRIPT be a sequence of functions satisfying the consistency conditions, where pnsubscriptݑݑ›p_nitalic_p start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT has domain Sn+1superscriptݑ†ݑ›1S^n+1italic_S start_POSTSUPERSCRIPT italic_n + 1 end_POSTSUPERSCRIPT. Then there exists a unique probability measure μݜ‡\muitalic_μ on the measurable space (Ω,B(Ω))normal-Ωݐµnormal-Ω(\Omega,B(\Omega))( roman_Ω , italic_B ( roman_Ω ) ) such that

μ(m[i0,…,in])=pn(i0,i1,…,in)fragmentsμfragmentssubscript(ݑšfragments[subscriptݑ–0,…,subscriptݑ–ݑ›])subscriptݑݑ›fragments(subscriptݑ–0,subscriptݑ–1,…,subscriptݑ–ݑ›)\mu(_m[i_0,...,i_n])=p_n(i_0,i_1,...,i_n)italic_μ ( start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT [ italic_i start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , … , italic_i start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ] ) = italic_p start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_i start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_i start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , italic_i start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT )
for all i0,i1,…,in∈Ssubscriptݑ–0subscriptݑ–1normal-…subscriptݑ–ݑ›ݑ†i_0,i_1,...,i_n\in Sitalic_i start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_i start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , italic_i start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ∈ italic_S and all n≥0ݑ›0n\geq 0italic_n ≥ 0.

3. Gibbs Measures

Let us consider the Ising model with competing nearest-neighbors interactions defined by the Hamiltonian

(3.1) H(σ)=-J∑∈Lnσ(x)σ(y),ݐ»ݜŽݐ½subscriptabsentݑ¥ݑ¦absentsubscriptݐ¿ݑ›ݜŽݑ¥ݜŽݑ¦H(\sigma)=-J\sum_\in L_n\sigma(x)\sigma(y),italic_H ( italic_σ ) = - italic_J ∑ start_POSTSUBSCRIPT ∈ italic_L start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_σ ( italic_x ) italic_σ ( italic_y ) ,
where the sum runs over nearest-neighbor vertices fragmentsx,y< italic_x , italic_y >and the spins σ(x)ݜŽݑ¥\sigma(x)italic_σ ( italic_x ) and σ(y)ݜŽݑ¦\sigma(y)italic_σ ( italic_y ) take values in the set Φ=-1,+1Φ11\Phi=\-1,+1\roman_Φ = - 1 , + 1 .

Let hxsubscriptℎݑ¥h_xitalic_h start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT be a real-valued function of x∈Vݑ¥ݑ‰x\in Vitalic_x ∈ italic_V. A finite-dimensional Gibbs distributions on ΦVnsuperscriptΦsubscriptݑ‰ݑ›\Phi^V_nroman_Φ start_POSTSUPERSCRIPT italic_V start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT end_POSTSUPERSCRIPT are defined by formula

(3.2) μn(σn)=1Znexp[-1THn(σn)+∑x∈Wnσ(x)hx]subscriptݜ‡ݑ›subscriptݜŽݑ›1subscriptݑݑ›1ݑ‡subscriptݐ»ݑ›subscriptݜŽݑ›subscriptݑ¥subscriptݑŠݑ›ݜŽݑ¥subscriptℎݑ¥\mu_n(\sigma_n)=\frac1Z_n\exp[-\frac1TH_n(\sigma_n)+\sum_x% \in W_n\sigma(x)h_x]italic_μ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_σ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ) = divide start_ARG 1 end_ARG start_ARG italic_Z start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT end_ARG roman_exp [ - divide start_ARG 1 end_ARG start_ARG italic_T end_ARG italic_H start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_σ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ) + ∑ start_POSTSUBSCRIPT italic_x ∈ italic_W start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_σ ( italic_x ) italic_h start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT ]
with the associated partition function defined as

Zn=∑σn∈ΦVnexp[-1THn(σ)+∑x∈Wnσ(x)hx],subscriptݑݑ›subscriptsubscriptݜŽݑ›superscriptΦsubscriptݑ‰ݑ›1ݑ‡subscriptݐ»ݑ›ݜŽsubscriptݑ¥subscriptݑŠݑ›ݜŽݑ¥subscriptℎݑ¥Z_n=\sum_\sigma_n\in\Phi^V_n\exp[-\frac1TH_n(\sigma)+\sum_x% \in W_n\sigma(x)h_x],italic_Z start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT = ∑ start_POSTSUBSCRIPT italic_σ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ∈ roman_Φ start_POSTSUPERSCRIPT italic_V start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT end_POSTSUPERSCRIPT end_POSTSUBSCRIPT roman_exp [ - divide start_ARG 1 end_ARG start_ARG italic_T end_ARG italic_H start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_σ ) + ∑ start_POSTSUBSCRIPT italic_x ∈ italic_W start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_σ ( italic_x ) italic_h start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT ]

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