Mathematical Physics Seminar, November 24, 2008
Convexity of the surface tension for 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-Lowener 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. 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 of the microscopic subsystems (particles)?
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 Mueller.