A model for interfaces and its mesoscopic limit

We study a system of N layers with a Kac horizontal interaction of parameter γ > 0 and a Kac vertical interaction of parameter γ 1/2. We shall prove that the limit free energy functional is the rate function of the large deviations of the Gibbs measure (of a canonical constrained magnetization). The limit free energy functional is achieved as a Γ-limit for γ →0 for magnetizations with fixed average. Among all such magnetizations there exists a quasiconstant magnetization that minimizes the energy