# Extremality of Disordered Phase of \(\lambda\)-Model on Cayley Trees
## 4 Three-Indexed Markov Chains of the Disordered Phase
### Conditions for Extremality  
From the previous subsection, we infer the non-extremality conditions for the disordered phase. However, in the present subsection, we are going to find sufficient conditions for its extremality (or non-reconstructability in information-theoretic language [8; 14; 15; 30]). If, from the tree \(\Gamma^{k}\), one removes an arbitrary edge \(\langle x^{0},x^{1}\rangle=l\in L\), then it is divided into two components \(\Gamma^{k}_{x^{0}}\) and \(\Gamma^{k}_{x^{1}}\), each of which is called a _semi-infinite Cayley tree_ or _Cayley subtree_. Let us first give some necessary definitions from [30]. We consider finite complete subtrees \(\mathcal{T}\) that are initial points of the Cayley tree \(\Gamma^{k}_{x^{0}}\). The boundary \(\partial\mathcal{T}\) of a subtree \(\mathcal{T}\) consists of the neighbors which are on \(\Gamma^{k}_{x^{0}}\setminus\mathcal{T}\). We identify subgraphs of \(\mathcal{T}\) with their vertex sets and write \(E(A)\) for the edges within a subset \(A\) and \(\partial A\). Following [30], there are two key ingredients which are the quantities \(\kappa\) and \(\gamma\). Both of them are properties of the collection of Gibbs measures \(\{\mu^{\Gamma}_{\mathcal{T}}\}\), where the boundary condition \(\tau\) is fixed and \(\mathcal{T}\) ranges over all initial finite complete subtrees of \(\Gamma^{k}_{x^{0}}\). For a given subtree \(\mathcal{T}\) of \(\Gamma^{k}_{x^{0}}\) and a vertex \(x\in\mathcal{T}\), we write \(\mathcal{T}_{x}\) for the (maximal) subtree of \(\mathcal{T}\) rooted at \(x\). When \(x\) is not the root of \(\mathcal{T}\), let \(\mu^{\Gamma}_{\mathcal{T}_{x}}\) 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 \(\mathcal{T}_{x}\) (i.e., on \(\partial\mathcal{T}_{x}\setminus\{\text{parent of }x\}\)) is specified by \(\tau\). For two measures \(\mu_{1}\) and \(\mu_{2}\) on \(\Omega\), \(\|\mu_{1}-\mu_{2}\|_{x}\) denotes the variation distance between the projections of \(\mu_{1}\) and \(\mu_{2}\) onto the spin at \(x\), i.e.,  
\[\parallel\mu_{1}-\mu_{2}\parallel_{x}=\frac{1}{2}\sum_{i\in\{1,2,3\}} \mid\mu_{1}(\sigma(x)=i)-\mu_{2}(\sigma(x)=i)\mid.\]  
Let \(\eta^{x,s}\) be the configuration \(\eta\) with the spin at \(x\) set to \(s\). Following [30], the quantities \(\kappa(\mu)\) and \(\kappa(\mu)\) are defined by  
\[\kappa(\mu)=\sup_{x\in\Gamma^{k}}\max_{x,s,s^{\prime}}\parallel\mu^{\Gamma}_{ \mathcal{T}_{x}}-\mu^{s^{\prime}}_{\mathcal{T}_{x}}\parallel_{x},\]  
\[\gamma(\mu)=\sup_{A\subset\Gamma^{k}}\max\parallel\mu^{\eta^{y,s}}_{A}-\mu^{ \eta^{y,s^{\prime}}}_{A}\parallel_{x},\]  
where the maximum is taken over all boundary conditions \(\eta\), all sites \(y\in\partial A\), all neighbors \(x\in A\) of \(y\) and all spins \(s,s^{\prime}\in\{1,2,3\}\). It is known that a sufficient condition for the extremality of the translation-invariant Gibbs measure is \(k\kappa(\mu)\gamma(\mu)<1\)[30] (Theorem 9.3). In what follows, the measure \(\mu\) is taken as the disordered phase; therefore, \(\kappa(\mu)\) and \(\gamma(\mu)\) are denoted by \(\kappa\) and \(\gamma\). Note that \(\kappa\) has the particularly simple form  
\[\kappa(\mu)=\frac{1}{2}\max\sum_{u\in\{1,2,3\}}\mid P_{iu}-P_{ju}\mid.\]  
Hence, from (16), one has  
\[\kappa=\frac{1}{2(2B+C_{2})}\max  
where  
\[\|\mu_{A}^{\eta^{y,i}}-\mu_{A}^{\eta^{y,i}}\|_{x} = \frac{1}{2}\sum_{s\in\{1,2,3\}}|\mu_{A}^{\eta^{y,1}}(\sigma(x)=s)- \mu_{A}^{\eta^{y,2}}(\sigma(x)=s)|\] \[= \frac{1}{2}\big{(}|P_{i,1}-P_{j,1}|+|P_{i,2}-P_{j,2}|+|P_{i,3}-P_{ j,3}|\big{)}\] \[= \frac{1}{2(2B+C_{2})}(|C_{1}-B|+|C_{2}-B|+|A-B|)\]  
Hence, \(\gamma=\kappa\). Hence, the extremality condition reduces to \(k\kappa^{2}<1\). By (18), we have the following result. Theorem 4.: _Assume that, for the \(\lambda\)-model given by (5), the condition (7) is satisfied.