Equivalence of the Grand Canonical Ensemble and the Canonical Ensemble of 1d-Lattice Systems

#### Y. Kwon, J. Lee, G. Menz

2021, v.27, Issue 1, 63-110

ABSTRACT

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

Keywords: canonical ensemble, decay of correlations, infinite-volume Gibbs measure, phase transition, one-dimensional lattice, equivalence of ensembles