Mirror Symmetry of Height-Periodic Gradient Gibbs Measures of an SOS Model on Cayley Trees

For the solid-on-solid model with spin values from the set of all integers on a Cayley tree we give gradient Gibbs measures (GGMs). Such a measure corresponds to a boundary law (which is an infinite-dimensional vector-valued function defined on vertices of the Cayley tree) satisfying an infinite system of functional equations. We give several concrete GGMs of boundary laws which are independent from vertices of the Cayley tree and (as an infinite-dimensional vector) have periodic, (non-)mirror-symmetric coordinates. Namely, the particular class of height-periodic boundary laws of period q ≤ 5 is studied, where solutions are classified by their period and (two-)mirror-symmetry.