Exponential decay of correlations in the uniqueness regime

Statement

Assume the hypotheses of the Dobrushin uniqueness theorem for a Gibbs specification $\gamma$ on $\Omega=S^{\mathbb{Z}^d}$, and let $\mu$ be the (unique) infinite-volume Gibbs measure satisfying the DLR equation .

Then there exist constants $C<\infty$ and $c>0$ (depending on the Dobrushin matrix $(C_{ij})$) such that for any bounded local functions $f,g:\Omega\to\mathbb{R}$ with supports contained in finite sets $A,B\Subset\mathbb{Z}^d$,

In particular, for single-site observables (e.g. spins), the two-point correlation function decays exponentially in the distance between the sites.

Key hypotheses

• Uniqueness criterion: the Dobrushin constant $\alpha<1$ as in Dobrushin uniqueness .
• Local observables: $f$ and $g$ depend only on finitely many coordinates.
• Covariance: $\mathrm{Cov}_\mu(f,g)=\mathbb{E}_\mu[fg]-\mathbb{E}_\mu[f]\mathbb{E}_\mu[g]$, with expectations taken in the sense of expectation .

Key conclusions

• Exponential mixing: correlations between spatially separated regions decay exponentially with separation.
• Stability under boundary conditions: local expectations are exponentially insensitive to far-away boundary perturbations (a strong form of uniqueness).
• No phase coexistence in this regime: exponential decay is incompatible with symmetry-breaking coexistence typical of phase transitions .

Proof idea / significance (sketch)

Dobrushin’s comparison method bounds how changing a boundary condition at site $j$ influences the conditional distribution at site $i$, and then propagates this bound along paths in the lattice via the interdependence matrix $(C_{ij})$. When $\alpha<1$, influences contract and admit an expansion whose coefficients decay exponentially in graph distance. Converting “influence decay” into a covariance estimate yields the stated exponential bound.

This result formalizes the heuristic that in the high-temperature (unique-phase) regime, distant regions are nearly independent.