  Phase separation is a fairly common physical phenomenon with examples
including the formation of water droplets from humid air (fog, rain), the
separation of a crystalline structure from an isotropic material such as a
liquid or even the formation of the sizzling gas bubbles when a soda can is
opened. It was recognized long ago (at least on a phenomenological level) that
systems exhibiting several phases in equilibrium can be described with an
appropriate variational principle: the phases arrange themselves in such a way
that the energy associated with the phase boundaries is minimal. Typically this
leads to an almost deterministic behavior and the phase boundaries are fairly
regular. However, when looked at from a microscopic point of view, the system
consists of a bunch of erratically moving molecules with relatively strong
short-range interaction and the simplicity of the above macroscopic description
looks more than miraculous. Indeed, when starting from the molecular level,
there are many more questions to be asked and understood: which are the phases
which we will see? why do only those occur? why are the phase boundaries sharp?
how should we find (define) the energy associated with the interfaces? Only
then can we ask the question: why does the system minimize this energy? It is
only in the last decade that a mathematically satisfactory understanding of
this phenomenon has been achieved. The main goal of the talk is to present the
current state of affairs focusing thereby on results obtained in joint works
with Raphael Cerf. The connection to fields of mathematics other than
probability theory or statistical mechanics will be highlighted; namely, to
geometric measure theory and to the calculus of variations.
