High-temperature exponential decay of correlations

Statement (exponential clustering in the uniqueness region)

Let $\mu$ be the unique infinite-volume Gibbs measure of a finite-range lattice system in a high-temperature/uniqueness regime (e.g. satisfying the Dobrushin uniqueness condition ).

Then there exist constants $C<\infty$ and $m>0$ such that for any two bounded local observables $F,G$ with finite supports $\mathrm{supp}(F)$ and $\mathrm{supp}(G)$,

where $\operatorname{Cov}_\mu(F,G)=\mu(FG)-\mu(F)\mu(G)$.

In particular, the two-point function (and connected two-point function) decays exponentially with separation.

Key hypotheses

• Finite-range (or sufficiently fast decaying) interaction on $\mathbb{Z}^d$.
• A parameter regime ensuring uniqueness and contractivity/mixing, e.g. Dobrushin uniqueness or cluster expansion convergence .
• Observables are local (depend on finitely many sites).

Conclusions

• The system has a finite correlation length $\xi = 1/m$ (up to constants).
• No long-range order compatible with persistent connected correlations can occur within this regime.
• Together with uniqueness, this supports analyticity of thermodynamic functions (see uniqueness implies analyticity ).

Cross-links (definitions and supporting results)

• two-point correlation function
• connected correlations as derivatives
• exponential decay from uniqueness theorem
• Dobrushin uniqueness theorem

Proof idea / significance

In Dobrushin’s approach, one shows that conditional distributions form a contraction in an appropriate metric; the influence of boundary conditions at distance $r$ decays like $e^{-mr}$. This converts directly into an exponential bound on covariances of local observables. In cluster expansion regimes, exponential decay follows from exponential summability of connected polymer contributions. Exponential decay is a quantitative expression of “high temperature = strong mixing.”