Raphael Cerf, Agoston Pisztora
Phase coexistence in Ising, Potts and percolation models
(715K, uuencoded gzipped Postscript file)
ABSTRACT. We study phase separation and phase coexistence phenomena
in Ising, Potts and random cluster models in
dimensions $d\ge 3$. The simultaneous occurrence of several phases
is typical for systems with appropriately arranged (mixed) boundary
conditions and for systems conditioned on certain large
deviation events. These phases define a partition of the space.
Our main results are large
deviations principles for (the shape of) the {\it empirical phase
partition} emerging in these models. More specifically, we establish a
general large deviation principle for the partition induced by large
(macroscopic) clusters in the
Fortuin--Kasteleyn model and transfer it to
the Ising--Potts model where we obtain a large deviation principle
for the natural partition
corresponding to the various phases. The rate function turns out to be the total
surface energy (associated with the surface tension of the model
and with boundary conditions) which can be
naturally assigned to each reasonable partition.
These LDP-s imply a weak law of large numbers:
the law of the phase partition is asymptotically determined by an
appropriate variational problem. More precisely, the phase partition
will be close to some partition which is compatible with the
constraints imposed on the system and which minimizes the total surface energy.
A general compactness argument
guarantees the existence of at least one such minimizing partition.
Our results are valid for temperatures $T$
below a limit of slab-thresholds $\tchat$ conjectured to
agree with the critical point $T_c$. Moreover, $T$ should be such that
there exists only one translation invariant infinite volume state in the
corresponding Fortuin--Kasteleyn model; a property
which can fail for at most countably
many values and which is conjectured to be true for every~$T$.