Strict Convexity of the Free Energy of the Canonical Ensemble Under Decay of Correlations

We consider a one-dimensional lattice system of unbounded, real-valued spins. We allow arbitrary strong, attractive, nearest-neighbor interaction. We show that the free energy of the canonical ensemble (ce) converges uniformly in $C^2$ to the free energy of the grand ce (gce). The error estimates are quantitative. A direct consequence is that the free energy of the ce is uniformly strictly convex for large systems. Another consequence is a quantitative local Cramér theorem which yields the strict convexity of the coarse-grained Hamiltonian. With small adaptations, the argument could be generalized to systems with finite-range interaction on a graph, as long as the degree of the graph is uniformly bounded and the associated gce has uniform decay of correlations.