Strict convexity of the surface tension of Gradient models with non-convex interactions

Effective interface models - seen as gradient fields - enable one to study effective phase coexistence. In the probabilistic setting gradient fields involve the study of strongly correlated random variables. One major problem has been open for several decades. What can be proved for the free energy and the Gibbs states for non-convex interactions given a non-vanishing tilt at the boundary? We present in the talk the first break through for low temperature using Gaussian measures and renormalisation group techniques yielding an analysis in terms of dynamical systems. We outline also the connection to the Cauchy-Born rule which states that the deformation on the atomistic level is locally given by an affine deformation at the boundary. Work in cooperation with R. Kotecky and S Müller.