Pinned gradient measures of SOS model associated with $H_A$-boundary laws on Cayley trees

This paper investigates pinned gradient measures for SOS (Solid-On-Solid) models associated with $H_A$-boundary laws of period two, a class that encompasses all 2-height periodic gradient Gibbs measures corresponding to a spatially homogeneous boundary law. While previous research has predominantly focused on a spatially homogeneous boundary law and corresponding GGMs on Cayley trees, this study extends the analysis by providing a comprehensive characterization of such measures. Specifically, it demonstrates the existence of pinned gradient measures on a set of $G$-admissible configurations and precisely quantifies their number under certain temperature conditions.