# Extremality of Disordered Phase of \(\lambda\)-Model on Cayley Trees
## 4 Three-Indexed Markov Chains of the Disordered Phase
### Conditions for Extremality
Then, the following statements hold:_  
_(i)_ _If_ \(|C_{1}-B|+|C_{2}-B|+|A-B|>2|C_{1}-A|\) _and_  
\[\frac{|C_{1}-B|+|C_{2}-B|+|A-B|}{2B+C_{2}}<\frac{2}{\sqrt{k}},\] _then the disordered phase is extreme._ _(ii)_ _If_ \(|C_{1}-B|+|C_{2}-B|+|A-B|\leq 2|C_{1}-A|\) _and_  
\[\frac{|C_{1}-A|}{2B+C_{2}}<\frac{1}{\sqrt{k}},\] _then the disordered phase is extreme._  
Remark 4.: _We point out that if \(A=B\), then by (7) we infer that \(C_{1}=C_{2}\), which by Remark 1 the model reduces to the Potts model. The conditions of the extremality of the disordered phase of this model has been investigated in [8; 14; 29; 31]. Such a condition from Theorem 4 is given as follows_  
\[\frac{|C_{1}-A|}{2A+C_{1}}<\frac{1}{\sqrt{k}}.\]  
Due to Theorem 2, if  
\[\kappa(\mu)=\sup_{x\in\Gamma^{k}}\max_{x,s,s^{\prime}}\parallel\mu^{\Gamma}_{ \mathcal{T}_{x}}-\mu^{s^{\prime}}_{\mathcal{T}_{x}}\parallel_{x},\]0  
then the disordered phase is unique, and so it is extreme. Hence, we are interested in its extremality in the case of  
\[\kappa(\mu)=\sup_{x\in\Gamma^{k}}\max_{x,s,s^{\prime}}\parallel\mu^{\Gamma}_{ \mathcal{T}_{x}}-\mu^{s^{\prime}}_{\mathcal{T}_{x}}\parallel_{x},\]1  
where the phase transition exists. The last condition results in \(C_{2}>B\) (if \(k\geq 2\)). Now, let us consider several other possibilities for the values of \(A,B,C_{1},C_{2}\). **Case I. ** Assume that \(B>A\); then, due to (7), one has \(C_{1}-C_{2}=B-A\), which implies \(C_{1}>C_{2}>B>A\). Then, \((B-A)(C_{1}-B)>0\), which, from Theorem 3, means that if  
\[\kappa(\mu)=\sup_{x\in\Gamma^{k}}\max_{x,s,s^{\prime}}\parallel\mu^{\Gamma}_{ \mathcal{T}_{x}}-\mu^{s^{\prime}}_{\mathcal{T}_{x}}\parallel_{x},\]2  
the disordered phase is not extreme. On the other hand, we have  
\[\kappa(\mu)=\sup_{x\in\Gamma^{k}}\max_{x,s,s^{\prime}}\parallel\mu^{\Gamma}_{ \mathcal{T}_{x}}-\mu^{s^{\prime}}_{\mathcal{T}_{x}}\parallel_{x},\]3 \[\kappa(\mu)=\sup_{x\in\Gamma^{k}}\max_{x,s,s^{\prime}}\parallel\mu^{\Gamma}_{ \mathcal{T}_{x}}-\mu^{s^{\prime}}_{\mathcal{T}_{x}}\parallel_{x},\]4 \[\kappa(\mu)=\sup_{x\in\Gamma^{k}}\max_{x,s,s^{\prime}}\parallel\mu^{\Gamma}_{ \mathcal{T}_{x}}-\mu^{s^{\prime}}_{\mathcal{T}_{x}}\parallel_{x},\]5  
which, together with  
\[\kappa(\mu)=\sup_{x\in\Gamma^{k}}\max_{x,s,s^{\prime}}\parallel\mu^{\Gamma}_{ \mathcal{T}_{x}}-\mu^{s^{\prime}}_{\mathcal{T}_{x}}\parallel_{x},\]6  
and (ii) Theorem 4, implies that the disordered phase is extreme if  
\[\kappa(\mu)=\sup_{x\in\Gamma^{k}}\max_{x,s,s^{\prime}}\parallel\mu^{\Gamma}_{ \mathcal{T}_{x}}-\mu^{s^{\prime}}_{\mathcal{T}_{x}}\parallel_{x},\]7  
The last condition can be rewritten as follows by using (7)  
\[\kappa(\mu)=\sup_{x\in\Gamma^{k}}\max_{x,s,s^{\prime}}\parallel\mu^{\Gamma}_{ \mathcal{T}_{x}}-\mu^{s^{\prime}}_{\mathcal{T}_{x}}\parallel_{x},\]8 \[\kappa(\mu)=\sup_{x\in\Gamma^{k}}\max_{x,s,s^{\prime}}\parallel\mu^{\Gamma}_{ \mathcal{T}_{x}}-\mu^{s^{\prime}}_{\mathcal{T}_{x}}\parallel_{x},\]9 \[\gamma(\mu)=\sup_{A\subset\Gamma^{k}}\max\parallel\mu^{\eta^{y,s}}_{A}-\mu^{ \eta^{y,s^{\prime}}}_{A}\parallel_{x},\]0  
where \(\tau=\frac{A}{B}\). Hence, we obtain  
\[\gamma(\mu)=\sup_{A\subset\Gamma^{k}}\max\parallel\mu^{\eta^{y,s}}_{A}-\mu^{ \eta^{y,s^{\prime}}}_{A}\parallel_{x},\]1  
Therefore, one has the following result. **Theorem 5**. : _Assume that for the \(\lambda\)-model given by (5), the condition (7) is satisfied and \(C_{2}>B>A\). Then, the disordered phase is extreme if and only if_  
\[\gamma(\mu)=\sup_{A\subset\Gamma^{k}}\max\parallel\mu^{\eta^{y,s}}_{A}-\mu^{ \eta^{y,s^{\prime}}}_{A}\parallel_{x},\]2  
_where \(\tau=\frac{A}{B}\)._  
**Case II. ** Assume that \(A>B\); then, we have \(C_{1}<C_{2}\). However, this case has five subcases. We are going to consider all of them one by one. **Subcase (a). ** Let us suppose that \(C_{1}<B<C_{2}<A\). Then, \((B-A)(C_{1}-B)>0\), which from Theorem 3 means that if  
\[\gamma(\mu)=\sup_{A\subset\Gamma^{k}}\max\parallel\mu^{\eta^{y,s}}_{A}-\mu^{ \eta^{y,s^{\prime}}}_{A}\parallel_{x},\]3  
the disordered phase is not extreme.