Strict convexity of the free energy of the canonical ensemble under decay of correlations

TL;DR

This paper proves that the free energy of a one-dimensional lattice system with unbounded spins and strong interactions is uniformly strictly convex in large systems, using decay of correlations and convergence to grand canonical ensembles.

Contribution

It establishes uniform convergence and strict convexity of the free energy in the canonical ensemble for systems with strong, attractive nearest-neighbor interactions.

Findings

Uniform convergence of canonical to grand canonical free energy in C^2 norm

Strict convexity of the free energy for large systems

Quantitative local Cramér theorem for the coarse-grained Hamiltonian

Abstract

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 converges uniformly in $C^2$ to the free energy of the grand canonical ensembles. The error estimates are quantitative. A direct consequence is that the free energy of the canonical ensemble is uniformly strictly convex for large systems. Another consequence is a quantitative local Cram\'er theorem which yields the strict convexity of the coarse-grained Hamiltonian. With small adaptations, the argument could be generalized to systems with finite-range interactions on a graph, as long as the degree of the graph is uniformly bounded and the associated grand canonical ensemble has uniform decay of correlations.