Ground States and Gibbs Measures for the SOS Model with an External Field and Countable Set of Spin Values on a Cayley Tree

In the paper, we consider the solid-on-solid (SOS) model with nearest-neighbor interactions, where the spin takes values in the integers on the Cayley tree of order two. We study the translation-invariant, periodic and non-periodic ground states for the model. Moreover, we describe the ground states and verify the Peierls condition for the model. Using a contour argument, we show the existence of a countable distinct Gibbs measures.