Random field of gradients.
Stefan Adams,
(Warwick Mathematics Institute)
jueves 24 de junio de 2010
Abstract: Random fields of gradients are a class of model systems arising in the studies of
random interfaces, random geometry, field theory (mass-less fields), and elasticity theory.
These random objects pose challenging problems for probabilists as even an a priori distribution
involves strong correlations. Gradient fields are likely to be an universal class of models
combining probability, analysis and physics in the study of critical phenomena. They emerge
in the following three areas, effective models for random interfaces, Gaussian Free Fields
(scaling limits), and mathematical models for the Cauchy-Born rule of materials. The latter
class of models requires that interaction energies are non-convex functions of the gradients.
Open problems over the last decades include unicity of Gibbs measures and strict convexity
of the free energy. We present in the talk the first break through for the free energy at
low temperatures using Gaussian measures and rigorous renormalization group techniques
yielding an analysis in terms of dynamical systems. The key ingredient is a finite range
decomposition for parameter dependent families of Gaussian measures. We outline the connection
to the Cauchy-Born rule which states that the deformation at the atomistic level is locally
given by an affine deformation at the boundary. Finally we discuss the behavior under scaling
limits.