New disordered phases of the (s,1/2)-mixed spin Ising model for arbitrary spin s

In this paper, we introduce an Ising model with mixed spin $(s,1/2)$ (abbreviated as $(s,1/2)$-MSIM) for any spin set $[-s,s]\cap Z$ on a semi-infinite second-order Cayley tree and construct translation-invariant splitting Gibbs measures (TISGMs) associated with the model. We prove that as the weight of the $s$-spin value increases, the repelling region of the fixed point ${\ell }_0^{\left(s\right)}$, corresponding to the TISGM, expands, leading to a broadening of the phase transition region. We also study tree-indexed Markov chains associated with the $(s,1/2)$-MSIM. Additionally, we clarify the extremality of the associated disordered phases by utilizing the method of Martinelli, Sinclair, and Weitz (Martinelli et al., 2007). By examining the non-extremality of the disordered phases related to the $(s,1/2)$-MSIM on the Cayley tree using the Kesten–Stigum condition, we extend previous research findings to encompass any set of spins in $[-s,s]\cap Z$. Furthermore, we prove that as the weight of the $s$-spin value increases, the region where the disordered phase corresponding to the $(s,1/2)$-MSIM is extreme narrows, while the region where it is non-extreme widens.