We consider an SOS (solid-on-solid) model, with spin values from the set of all integers, on a Cayley tree of order $k\geq 2$ and are interested in tree-automorphism invariant gradient Gibbs measures (GGMs) of the model. Such a measure corresponds to a boundary law (a function defined on vertices of the Cayley tree) satisfying a functional equation. In the ferromagnetic SOS case on the binary tree we find up to five solutions to a class of period-$4$ height-periodic boundary law equations (in particular, some period-2 height-periodic ones). We show that these boundary laws define up to four distinct GGMs. Moreover, we construct some period-$3$ height-periodic boundary laws on the Cayley tree of arbitrary order $k\geq 2$, which define GGMs different from the $4$-periodic ones.
"Gradient Gibbs measures for the SOS model with countable values on a Cayley tree." Electron. J. Probab. 24 1 - 23, 2019. https://doi.org/10.1214/19-EJP364