An improvement of extremality regions for Gibbs measures of the Potts model on a Cayley tree

We give a condition of extemelity for translation-invariant Gibbs measures of q—state Potts model on a Cayley tree. We'll improve the regions of extremality for some measures considered in [14]. Moreover, some results in [14] are generalized.


Introduction
The Potts model is a generalization of the Ising model. In [3], [4] the q-state Potts model on a Cayley tree of order k ≥ 2 was studied, and it has been known for a long time that at sufficiently low temperatures, there are at least q + 1 translation-invariant Gibbs measures. This measures can be considered as tree-indexed Markov chains. Such translation-invariant tree-indexed measures are equivalently called translation-invariant splitting Gibbs measures (TISGMs).
In [6] the uniqueness of the translation-invariant Gibbs measure of the antiferromagnetic Potts model with an external field is proved. In [7] the Potts model with a countable number of states and nonzero external field on a Cayley tree was considered. In that paper, it was established that the model has a unique translation-invariant Gibbs measure.
In [13] all TISGMs (tree-indexed Markov chains) for the Potts model are found on the Cayley tree of order k ≥ 2, and it is shown that at sufficiently low temperatures their number is 2 q − 1. In the case k = 2 the explicit formulae for the critical temperatures and all TISGMs are given. Further, in [14] by means of methods and results of [10], [21], [15] it has been found some regions for the temperature parameter ensuring that a given TISGM is (non-)extreme in the set of all Gibbs measures. In particular, it was shown the existence of a temperature interval for which there are at least 2 q−1 + q extreme TISGMs. In case of the order of the tree is two, it was given an explicit formulae and some numerical values of the critical temperature. Note that other properties of the Potts model on a Cayley tree were studied in [1,5,8,9,11,12], [17]- [20].
In this paper, we consider the q−state Potts model on the Cayley tree of order two. Some results of [14] will be improved. Moreover, we will extend Theorem 6 of [14].

Definitions and known facts
A Cayley tree ℑ k of order k ≥ 1 is an infinite tree, i.e. a graph without cycles, such that exactly k + 1 edges originate from each vertex. Let ℑ k = (V, L, i), where V is the set of vertices ℑ k , L the set of edges and i is the incidence function setting each edge l ∈ L into the correspondence with its endpoints x, y ∈ V . If i(l) = {x, y}, then the vertices x and y are called the nearest neighbors, denoted by l = ⟨x, y⟩. The distance d(x, y), x, y ∈ V on a Cayley tree is defined by Namely, S(x) is the set of direct successors of x. We consider the model in which the spin variables take values in the set Φ = {1, 2, . . . , q}, (q ≥ 2) and which are located at the tree vertices. For A ⊂ V a configuration σ A on A is an arbitrary function σ A : A → Φ. Note that Ω A = Φ A is the set of all configurations. We denote that Ω = Ω V and σ = σ V .
A Hamiltonian of the Potts model is defined as where J ∈ R and δ ij is the Kronecker symbol.
In this paper, we restrict ourselves to the case of ferromagnetic interaction J > 0. Define a finite-dimensional distribution of a probability measure µ in the volume V n by where β = 1/T , T > 0-temperature, Z −1 n is the normalizing factor, {h x = (h 1,x , . . . , h q,x ) ∈ R q , x ∈ V } is a collection of vectors, and is the restriction of Hamiltonian to V n .
The probability distributions (2.2) are called compatible if for all n ≥ 1 and here σ n−1 ∨ ω n is the concatenation of configurations. In this case, by the well-known Kolmogorov's extension theorem, there exists a unique measure µ on Φ V such that, for all n and σ n ∈ Φ Vn µ({σ ∈ Ω : σ| Vn = σ n }) = µ n (σ n ).
This measure µ is called a splitting Gibbs measure corresponding to the Hamiltonian (2.1) and vector-valued function h x , x ∈ V .
The following statement describes conditions onh x guaranteeing compatibility of {µ n }.

the last equation can be written as follows
From [13] the following facts are known: 1. By solving (2.6) the full set of TISGMs is described. It is shown that any TISGM of the Potts model corresponds to a solution of the following equation . If θ < θ 1 then there exists a unique TISGM for k ≥ 2, J > 0. Moreover, each θ m is a critical value for the change of the number of TISGMs.
From [14] it is known that for each fixed m, the equation (2.7) has up to three solutions: [13, Step 1 of the proof of where Let us first give some necessary definitions from [15]. Considering finite complete subtrees T that are initial points of Cayley tree Γ k , i.e. share the same root; if T has depth d (i.e. the vertices of T are within distance ≤ d from the root) then it has (k d+1 − 1)/(k − 1) vertices, and its boundary ∂T consists the neighbors (in Γ k \ T ) of its vertices, i.e., |∂T | = k d+1 . We identify subgraphs of T with their vertex sets and write E(A) for the edges within a subset A and ∂A for the boundary of A, i.e., the neighbors of A in (T ∪ ∂T ) \ A.
For a given subtree T of Γ k and a vertex x ∈ T , we write T x for the (maximal) subtree of T rooted at x. When x is not the root of T , let µ s Tx denote the (finite-volume) Gibbs measure in which the parent of x has its spin fixed to s and the configuration on the bottom boundary of T x (i.e., on ∂T x \ {parent of x}) is specified by τ .
For two measures µ 1 and µ 2 on Ω, ∥µ 1 − µ 2 ∥ x denotes the variation distance between the projections of µ 1 and µ 2 onto the spin at x, i.e., Let η x,s be the configuration η with the spin at x set to s. Following [15] we define where the maximum is taken over all boundary conditions η, all sites y ∈ ∂A, all neighbors x ∈ A of y, and all spins s, s ′ ∈ {1, . . . , q}. We apply [15,Theorem 9.3], which says that for an arbitrary channel P = (P ij ) q i,j=1 on a tree reconstruction of the corresponding tree-indexed Markov chain (splitting Gibbs measure) is impossible if kκγ < 1.
Note that κ has the particularly simple form (see [15]) and γ is a constant which does not have a clean general formula, but can be estimated in specific models (as Ising, Hard-Core etc. Using (3.3) and (3.2) for i ̸ = j we get (see [14]) where a and b are defined by Clearly, (3.5) We consider the case z ̸ = 1 (where z = x 2 and x is a solution to (2.7)) and fix the solution of (2.6), which has the form (z, z, . . . , z m , 1, . . . , 1) and the corresponding matrix is P.
The following Proposition improves the part 2) of Proposition 1.
Proof. We shall find maximum values of functions K 2 (p 1 , p 2 , u) and K 4 (p 1 , p 2 , u). We consider Hence, it is sufficient to find the maximum of function K 2 (p 1 , p 2 , u) for α = 1.