GRADIENT MODELS WITH NON-CONVEX INTERACTIONS

We outline recent results in collaboration with R.Kotecky and S. Muller in [1] and [2] about gradient models on an integer lattice with non-convex interactions. These Gibbsian models (continuous Ising models) emerge in various branches of physics and mathematics, with a particular frequency in quantum field theory. Our attention is however mostly devoted to interfaces, of which a massless field is an effective modelisation, however the motivation stems considering vector valued fields as displacements for atoms of crystal structures and the study of the Cauchy-Born rule for these models. For the interface case we prove the strict convexity of the surface tension (free energy) for low enough temperatures and small enough tilts using multi-scale (renormalisation group analysis) techniques. This is the complementary study of the high temperature regime in [3] and it is an extension of Funaki and Spohn's result [4].