# Extremality of Disordered Phase of \(\lambda\)-Model on Cayley Trees
## 3 Splitting Gibbs Measures  
In this section, we recall the definition of splitting Gibbs measures. Now, let us define a finite-dimensional distribution of probability measure \(\mu^{(n)}\) on \(\Omega_{n}:=\Phi^{V_{n}}\) as follows  
\[\mu^{(n)}(\sigma_{n})=Z_{n}^{-1}\exp\{-\beta H_{n}(\sigma_{n})+\sum_{x\in W_{n}}h _{\sigma(x),x}\},\;\sigma_{n}\in\Omega_{n} \tag{2}\]  
where \(\beta=1/T\), \(T>0\) is the temperature, and  
\[Z_{n}^{-1}=\sum_{\sigma\in\Phi^{V_{n}}}\exp\{-\beta H_{n}(\sigma_{n})+\sum_{x \in W_{n}}h_{\sigma(x),x}\}\]  
is the normalizing factor. Moreover, in (2), \(\{h_{x}=(h_{1,x},\dots,h_{q,x})\in\mathbb{R}^{q},x\in V\}\) is the set of vectors and  
\[H_{n}(\sigma_{n})=\sum_{\langle x,y\rangle\in L_{n}}\lambda(\sigma(x),\sigma( y)).\]  
We say that a sequence of a probability distribution \(\{\mu^{(n)}\}\) is _consistent_ if, for all \(n\geq 1\) and \(\sigma_{n-1}\in\Omega_{n-1}\), one has  
\[\sum_{\omega_{n}\in\Phi^{V_{n}}}\mu^{(n)}(\sigma_{n-1}\vee\omega_{n})=\mu^{(n -1)}(\sigma_{n-1}). \tag{3}\]  
Here, \(\sigma_{n-1}\vee\omega_{n}\) is the concatenation of the configurations. In this setting, there is a unique measure \(\mu\) on \(\Omega\), such that for all \(n\) and \(\sigma_{n}\in\Omega_{n}\),  
\[\mu(\{\sigma|V_{n}=\sigma_{n}\})=\mu^{(n)}(\sigma_{n}).\]  
Such a measure is called a _splitting Gibbs measure (SGM)_, corresponding to the Hamiltonian (1) and a vector-valued function \(h_{x},x\in V\) (for more information, see [27]). The next statement describes the condition on \(h_{x}\) ensuring that the sequence \(\{\mu^{(n)}\}\) is consistent. Theorem 1 ([19; 24]): _The measures \(\mu^{(n)},n=1,2,\dots\) satisfy the consistency condition if and only if for any \(x\in V\) the following equation holds:_  
\[u_{k,x}=\prod_{y\in S(x)}\frac{\sum_{j=1}^{q-1}\exp\{\beta\lambda(k,j)\}u_{j, y}+\exp\{\beta\lambda(k,q)\}}{\sum_{j=1}^{q-1}\exp\{\beta\lambda(q,j)\}u_{j, y}+\exp\{\beta\lambda(q,q)\}} \tag{4}\]  
_where \(u_{k,x}=\exp\{h_{k,x}-h_{q,x}\},k=\overline{1,q-1}\)._  
In what follows, we restrict ourself to the case \(\Phi=\{1,2,3\}\) (i.e., \(q=3\)), and we consider the following function:  
\[\lambda(i,j)=\left\{ \begin{array}{ll}\overline{a},&\text{if}\quad|i-j|=2,\\ \overline{b},&\text{if}\quad|i-j|=1,\\ \overline{c}_{1},&\text{if}\quad i=j=1,\\ \overline{c}_{2},&\text{if}\quad i=j=2,\\ \overline{c}_{3},&\text{if}\quad i=j=3,\end{array} \right. \tag{5}\]  
where \(\overline{a},\overline{b},\overline{c}_{1},\overline{c}_{2},\overline{c}_{3} \in\mathbb{R}\) for some given numbers. Remark 1: _We point out that the considered model is more general than the well known Potts model [12], since if \(\overline{a}=\overline{b}\) and \(\overline{a}\
eq\overline{c}_{1}=\overline{c}_{2}=\overline{c}_{3}\), then this model reduces to the mentioned one. Namely, in this setting, the function \(\lambda\) is defined by \(\lambda(i,j)=\overline{a}+(\overline{c}_{1}-\overline{a})\delta_{ij}\). The case of \(\overline{c}_{1}=\overline{c}_{2}=\overline{c}_{3}\) has been investigated in [24; 25]. If \(\lambda(i,j)=\left|i-j\right|\), then the model reduces to the Solid-on-SOlid (SOS) model [28]. If \(\lambda(i,j)=J_{p}\delta_{ij}+J|i-j|\), then the model becomes the Potts-SOS model [23]._Now, we are going to find the disordered phase of the model which is related to the translation-invariant splitting Gibbs measure (for short, SGM). Here, through the use of the the translation-invariant SGM, it is an SGM that corresponds to a solution \(\mathbf{u}_{x}\) of (4), which satisfies \(\mathbf{u}_{x}=\mathbf{u}_{y}\) for all \(x,y\in V\). This means that \(\mathbf{u}_{x}=\mathbf{u}\), where \(\mathbf{u}=(u_{1},u_{2}),u_{1},u_{2}>0\). Due to Theorem 1, \(u_{1}\) and \(u_{2}\) must satisfy the following system equations:  
\[\left\{ \begin{array}{l}u_{1}=\left(\frac{C_{1}u_{1}+Bu_{2}+A}{Au_{1}+Bu_{2}+ C_{3}}\right)^{k},\\ \\ u_{2}=\left(\frac{Bu_{1}+C_{2}u_{2}+B}{Au_{1}+Bu_{2}+C_{3}}\right)^{k},\end{array} \right.