Decay of correlations and uniqueness of the infinite-volume Gibbs measure of the canonical ensemble of 1d-lattice systems

TL;DR

This paper proves that in a one-dimensional lattice system with unbounded spins, correlations decay exponentially and the canonical ensemble's Gibbs measure is unique, extending understanding of phase behavior in such systems.

Contribution

The authors establish the equivalence of correlations between the grand canonical and canonical ensembles and prove the uniqueness of the infinite-volume Gibbs measure for 1D lattice systems.

Findings

Correlations in the canonical ensemble decay exponentially with volume correction.

The infinite-volume Gibbs measure of the canonical ensemble is unique in 1D.

Equivalence of ensembles holds at the observable level.

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 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.