Exact Relation between the Solid-on-Solid Model and the $\mathrm{XY}$ Model

An exact relation is established between the solid-on-solid model describing the growth of crystals and the classical $\mathrm{XY}$ model. Application of this relation to a special case leads to an exact correspondence between the $\mathrm{XY}$ model and the Coulomb gas which matches with the relation obtained by Kosterlitz in the strong-coupling limit.