Extremal inhomogeneous Gibbs states for SOS-models and finite-spin models on trees

We consider $\mathbb Z$-valued $p$-SOS-models with nearest neighbor interactions of the form $|\omega_v-\omega_w|^p$, and finite-spin ferromagnetic models on regular trees. This includes the classical SOS-model, the discrete Gaussian model and the Potts model. We exhibit a family of extremal inhomogeneous (i.e. tree automorphism non-invariant) Gibbs measures arising as low temperature perturbations of ground states (local energy minimizers), which have a sparse enough set of broken bonds together with uniformly bounded increments along them. These low temperature states in general do not possess any symmetries of the tree. This generalises the results of Gandolfo, Ruiz and Shlosman \cite{GRS12} about the Ising model, and shows that the latter behaviour is robust. We treat three different types of extensions: non-compact state space gradient models, models without spin-symmetry, and models in small random fields. We give a detailed construction and full proofs of the extremality of the low-temperature states in the set of all Gibbs measures, analysing excess energies relative to the ground states, convergence of low-temperature expansions, and properties of cutsets.