P. Butta', I. Merola, E. Presutti
On the validity of the van der Waals theory in Ising systems with
long range interactions.
(95K, Plain TeX)
ABSTRACT. We consider an Ising system in $d \ge 2$ dimensions with ferromagnetic
spin-spin interactions $-J_\gamma(x,y)\sigma(x)\sigma(y)$, $x$,
$y \in \Bbb Z^d$, where $J_\gamma(x,y)$ scales like a Kac potential.
We prove that when the temperature is below the mean field critical value,
for any $\gamma$ small enough (i.e. when the range of the interaction
is long but finite), there are only two pure homogeneous phases,
as stated by the van der Waals theory. The proof follows that in
[12] (G. Gallavotti, S. Miracle-Sole', 1972) on the translationally
invariant states at low temperatures for nearest neighbor interactions,
supplemented by a ``relative
uniqueness criterion for Gibbs fields" which yields
uniqueness in a restricted ensemble of measures, in a
context where there is a phase transition. This
criterion is derived by introducing special couplings as in
[2] (J. van den Berg, C. Maes, 1994) which reduce the proof of
relative uniqueness to the absence of percolation of
``bad events".