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We consider a one-dimensional lattice system of unbounded, real-valued spinswith arbitrary strong, quadratic, finite-range interaction. We show theequivalence of the grand canonical ensemble (gce) and the canonical ensemble(ce), in the sense of observables and correlations. A direct consequence isthat the correlations of the ce decay exponentially plus a volume correctionterm. The volume correction term is uniform in the external field, the meanspin and scales optimally in the system size. This extends prior results ofCancrini & Martinelli for bounded discrete spins to unbounded continuous spins.The result is obtained by adapting Cancrini & Martinelli's method combined withauthors' recent approach on continuous real-valued spin systems.

Equivalence of the grand canonical ensemble and the canonical ensemble on 1d-lattice systems