Rafael de la Llave, Enrico Valdinoci
Ground states and critical points for Aubry-Mather theory in
statistical mechanics
(526K, PostScript)
ABSTRACT. We consider statistical mechanics systems
that are defined on a set with some symmetry properties.
Each of the sites has a real order parameter.
We assume that the interaction is ferromagnetic
as well as symmetric
with respect to the action of a
group which is assumed to be finitely generated and
residually finite.
Given any cocycle of the symmetry group,
we prove that
there are ground states
which satisfy an order condition (known as
Birkhoff property) and that are at a bounded distance
to the cocycle.
Such ground states are organized into a (possibly
singular) lamination.
Furthermore, we show that, given any completely irrational
cocycle, either
the above lamination consists of a foliation made of
a continuous one parameter family of ground states, or,
inside any gap of the lamination,
there is a
well-ordered critical point
which is not a ground state.