ABSTRACT:
In this talk I will discuss recent results (joint work with R. Cerf)
about phase separation and coexistence in certain lattice models.
There are two basic questions in this context. First: why do we observe
tipically specific distinct phases separated by sharp phase
boundaries? Second: what is the law determining the configuration and
shape of the emerging phases?
To analyze these questions we study Ising and Potts models below the
critical temperature in dimensions $\ge 2$ (although we focus on $d\ge
3$). These models are probably the simplest possible
ones to exhibit several phases simultaneously. In order to obtain
more than one phase we have to impose certain constraints, such as
boundary conditions or restrictions on the total
magnetization, etc. One well known example is the Wulff problem. In this
context we can pose it as follows: Consider the Ising model
in a large box with plus boundary conditions and condition the system
to have an excess amount of negative spins so that the empirical
magnetization
is smaller than the spontaneous magnetization $m^*$. What is the
typical picture we will see?
It is expected that the excess amount of minus spins will create
a region filled with the minus phase (surrounded by the plus phase)
and this region has an asymptotically definite shape, called the Wulff
crystall (droplet) which minimizes the surface energy associated
with the phase boundary.
One of our results is a proof of the validity of the
picture described above. In the talk I will try to give an outline
of our approach which involves the use of the Fortuin-Kasteleyn
represenation, large deviation theory, coarse-graining and geometric
measure theory.