Title: Strict convexity of the surface tension for gradient fields
with non-convex potentials

Recently the study of gradients fields has attained a lot of attention
because they are space-time analogy of Brownian motions, and are
connected to the Schramm-Loewner evolution. The corresponding discrete
versions arise in equilibrium statistical mechanics, e.g., as
approximations of critical systems and as effective interface
models. The latter models - seen as gradient fields - enable one to
study effective descriptions of phase coexistence.  Gradient fields
have a continuous symmetry and coexistence of different phases breaks
this symmetry. In the probabilistic setting gradient fields involve
the study of strongly correlated random variables. As a result the
asymptotic behavior (free energy, measures) depends on the boundary
constraint (enforced tilt). Main challenge is the question of
uniqueness of Gibbs measures and the strict convexity of the free
energy (surface tension) for any non-convex interaction potential. We
present in the talk the first break through for low temperatures using
Gaussian measures and renormalization group techniques yielding an
analysis in terms of dynamical systems. Our main input is a finite
range decomposition for a family of Gaussian measures depending on non
isotropic tuning parameters. 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. Mueller.