LAUSR

We consider a 1D solid-on-solid (SOS) model with external field in which the single-spin space is the set of all integers. Then, we construct a Gibbs specification for the model and get a functional equation such that every positive solution defines an infinite volume Gibbs measure. We show that there exist infinitely many Gibbs measures for the 1D SOS model with external field for some values of parameters θ and θ 1 (the last one is responsible to the external field). Moreover, by the main theorem, we conclude that a phase transition occurs for the 1D SOS model with external field and there is no Gibbs measure for the 1D SOS model without an external field.