Three-state $p$-SOS models on binary Cayley trees

We consider a version of the solid-on-solid model on the Cayley tree of order two in which vertices carry spins of value $0,1$ or $2$ and the pairwise interaction of neighboring vertices is given by their spin difference to the power $p>0$. We exhibit all translation-invariant splitting Gibbs measures (TISGMs) of the model and demonstrate the existence of up to seven such measures, depending on the parameters. We further establish general conditions for extremality and non-extremality of TISGMs in the set of all Gibbs measures and use them to examine selected TISGMs for a small and a large $p$. Notably, our analysis reveals that extremality properties are similar for large $p$ compared to the case $p=1$, a case that has been explored already in previous work. However, for the small $p$, certain measures that were consistently non-extremal for $p=1$ do exhibit transitions between extremality and non-extremality.