The extremality of disordered phases for the mixed spin-(1,1/2) Ising model on a Cayley tree of arbitrary order

Abstract This paper continues the exploration of translation-invariant splitting Gibbs measures (TISGMs) within the framework of the Ising model with mixed spin-(1,1/2) (abbreviated as (1,1/2)-MSIM) on a Cayley tree of arbitrary order. Building upon our prior work (Akin and Mukhamedov 2022 J. Stat. Mech. 053204), where we extensively elucidated TISGMs and investigated the extremality of disordered phases employing a Markov chain indexed by a tree on a semi-infinite Cayley tree of order two, our current research extends this analysis. In the present study, we begin by constructing TISGMs and tree-indexed Markov chains tailored to the (1,1/2)-MSIM. Notably, we extend our scope to encompass Cayley trees of varying orders, thereby affording a comprehensive examination of the extremality characteristics inherent to disordered phases. Utilizing the Kesten–Stigum condition, we delve into the non-extremality aspects of disordered phases by scrutinizing the eigenvalues of the stochastic matrix associated with the (1,1/2)-MSIM on Cayley trees with order k ( k ⩾ 3 ). This approach allows for a nuanced exploration of the model’s behavior and stability under different conditions.