  We consider a one-dimensional lattice system of unbounded, real-valued spins
with arbitrary strong, quadratic, finite-range interaction. We show the
equivalence of correlations of the grand canonical (gce) and the canonical
ensemble (ce). As a corollary we obtain that the correlations of the ce decay
exponentially plus a volume correction term. Then, we use the decay of
correlation to verify a conjecture that the infinite-volume Gibbs measure of
the ce is unique on a one-dimensional lattice. For the equivalence of
correlations, we modify a method that was recently used by the authors to show
the equivalence of the ce and the gce on the level of thermodynamic functions.
In this article we also show that the equivalence of the ce and the gce holds
on the level of observables. One should be able to extend the methods and
results to graphs with bounded degree as long as the gce has a sufficient
strong decay of correlations.
