Dobrushin uniqueness theorem

A quantitative small-dependence condition on single-site conditional distributions implies uniqueness of the infinite-volume Gibbs measure and strong mixing.

Dobrushin uniqueness theorem

Statement

Let $\gamma$ be a Gibbs specification on $\Omega=S^{\mathbb{Z}^d}$ with finite single-spin space $S$ . For each site $i\in\mathbb{Z}^d$ , write $\gamma_i(\cdot\mid\omega)$ for the single-site conditional distribution at $i$ given the external configuration $\omega$ .

Define the Dobrushin influence coefficients

C_{ij} := \sup_{\substack{\omega,\omega'\in\Omega\\ \omega_k=\omega'_k\ \forall k\neq j}} \big\|\gamma_i(\cdot\mid\omega)-\gamma_i(\cdot\mid\omega')\big\|_{\mathrm{TV}},

where $\|\cdot\|_{\mathrm{TV}}$ denotes total variation distance.

If the Dobrushin constant

\alpha := \sup_{i\in\mathbb{Z}^d}\ \sum_{j\neq i} C_{ij}

satisfies $\alpha<1$ , then there exists a unique infinite-volume Gibbs measure $\mu$ consistent with $\gamma$ (equivalently, there is a unique $\mu$ satisfying the DLR equation ).

In addition, the unique Gibbs state enjoys strong mixing properties; in particular, boundary-condition influence decays quantitatively with distance, yielding (as a consequence) exponential correlation decay as formalized in exponential decay of correlations in the uniqueness regime .

Key hypotheses

Finite spin space: $S$ finite (so single-site conditionals are uniformly well-behaved).
Quasilocal specification: $\gamma$ is a genuine Gibbs specification arising from some lattice Hamiltonian (or at least satisfies the usual measurability/properness/consistency conditions).
Small interdependence: $\alpha<1$ for the influence matrix $(C_{ij})$ .

Key conclusions

Uniqueness: there is exactly one $\mu$ consistent with $\gamma$ .
High-temperature regime: for many concrete models (e.g. Ising with small $\beta J$ ), one can bound $C_{ij}$ explicitly and verify $\alpha<1$ .
Quantitative mixing: local expectations are stable under far-away boundary perturbations, with explicit bounds in terms of $(C_{ij})$ .

Proof idea / significance (sketch)

One constructs a “single-site update” operator on the space of probability measures (or on the space of boundary conditions), and shows it is a contraction in an appropriate metric when $\alpha<1$ . The coefficients $C_{ij}$ quantify how much flipping the spin at $j$ can change the conditional law at $i$ ; the condition $\sup_i\sum_j C_{ij}<1$ forces influence to die out under iteration.

This theorem is a standard, robust criterion for the uniqueness of equilibrium at high temperature and is often the entry point to analyticity and exponential mixing results.