Uniqueness implies analyticity (no phase transition in the uniqueness region)

Statement (absence of nonanalyticity under uniqueness)

Let a finite-range lattice spin system (e.g. the Ising model ) be specified by a Gibbs specification and associated finite-volume Gibbs measures with partition functions $Z_\Lambda(\beta,\mathbf{h})$.

Assume the system lies in a uniqueness region, for instance one covered by the Dobrushin uniqueness theorem (or any hypothesis implying a unique infinite-volume Gibbs measure and exponential mixing).

Then:

1. There is a unique infinite-volume Gibbs measure $\mu_{\beta,\mathbf{h}}$ satisfying the DLR equation .
2. The infinite-volume pressure (free energy density) $p(\beta,\mathbf{h})=\lim_{\Lambda\uparrow \mathbb{Z}^d}\frac{1}{|\Lambda|}\log Z_\Lambda(\beta,\mathbf{h})$ exists and is analytic in parameters throughout that uniqueness region.
3. All finite-volume expectations of local observables converge to $\mu_{\beta,\mathbf{h}}$ uniformly in boundary conditions, and the resulting infinite-volume correlation functions are analytic in parameters.

Key hypotheses

• Finite-range (or sufficiently fast decaying) interaction; well-defined specification.
• A criterion guaranteeing uniqueness and strong mixing, e.g. Dobrushin uniqueness or convergent cluster expansion in the parameter regime.
• Existence of the thermodynamic limit of the pressure (often proved independently, e.g. via subadditivity).

Conclusions

• No phase transition (in the thermodynamic sense of nonanalyticity of $p$) can occur within the uniqueness region.
• In particular, nonuniqueness of Gibbs states (phase coexistence) cannot happen there: phase transition as Gibbs nonuniqueness is excluded.

Cross-links (definitions and supporting results)

• infinite-volume Gibbs measure , extremal Gibbs measure
• lattice pressure , lattice partition function
• existence of thermodynamic limit of the pressure
• Dobrushin uniqueness theorem , high-temperature exponential decay

Proof idea / significance

Uniqueness plus strong mixing yields uniform control of boundary-condition influence. In regimes covered by a convergent expansion (Dobrushin method or cluster expansion), one obtains absolutely convergent series for $\log Z_\Lambda$ and for expectations of local observables, uniform in $\Lambda$. Passing to the limit gives analyticity of $p(\beta,\mathbf{h})$ and of correlation functions. This formalizes the slogan: nonanalyticity requires nonuniqueness.