Statistical mechanics, structure of matter Rigorous equilibrium theory Theorem

Dobrushin uniqueness criterion

Builds on Gibbs measure Total variation distance between probability measures Probability measure

Statement

Let $S$ be a finite set and $\Omega = S^{\Z^d}$ . Let $\Phi = (\Phi_A)$ be an interaction with $\sup_{x} \sum_{A \ni x} \norm{\Phi_A}_\infty < \infty$ , let $\beta > 0$ , and let $\mathcal{G}$ be the set of Gibbs measures for $\beta \Phi$ . For a site $i$ and $\omega \in \Omega$ , let $\gamma_i(\cdot \mid \omega)$ be the single-site Gibbs kernel: the probability measure on $S$ proportional to $s \mapsto e^{-\beta H_{\set{i}}^\omega(s)}$ . For sites $i \ne j$ , define the Dobrushin coefficient $C_{ij} = \sup \set{ \norm{\gamma_i(\cdot \mid \omega) - \gamma_i(\cdot \mid \omega')}_{TV} : \omega_k = \omega'_k \text{ for all } k \ne j },$ and $C_{ii} = 0$ . Here $\norm{\cdot}_{TV}$ is the total variation distance, $\norm{p - q}_{TV} = \max_{B \subseteq S} \abs{p(B) - q(B)}$ . If $\sup_{i} \sum_{j} C_{ij} < 1$ , then $\mathcal{G}$ has exactly one element.

Remarks

Existence holds for any such interaction, so the content is uniqueness. Proof idea: for two Gibbs measures, bound the total variation distance of their marginal at site $i$ by $\sum_j C_{ij}$ times the distances at the sites $j$ . Iterating this contraction drives the distances to $0$ . A simple sufficient condition is $4 \beta \sup_i \sum_{A \ni i} (\abs{A} - 1) \norm{\Phi_A}_\infty < 1$ , so the criterion applies at high temperature, that is, for small $\beta$ . Example: take the nearest-neighbour Ising model on $\Z^d$ with coupling $J > 0$ and no field. Then $C_{ij} = \frac{1}{2} \tanh(2 \beta J)$ for neighbours $i, j$ and $0$ otherwise, so uniqueness holds when $d \tanh(2 \beta J) < 1$ . The criterion is not necessary: in dimension one the Ising model has a unique Gibbs measure at every $\beta$ .

Going deeper

For a student

The criterion is a contraction argument, in the spirit of the Banach fixed-point theorem, on the space of measures. Besides uniqueness, it gives decay of correlations for the unique Gibbs measure. It is a high-temperature result; at low temperature, uniqueness can fail, as for the Ising model in dimension $d \ge 2$ . It also gives fast mixing of the Glauber dynamics, and it has a close analogue, the path coupling method, in the analysis of Markov chains.

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Gibbs measure concept
Probability measure concept
Total variation distance between probability measures concept
Dobrushin uniqueness criterion statement

Tag	`8JH9AJ4P2A`
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MSC	82B05, 82B20, 60K35
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About	Gibbs measure
Uses	Gibbs measure, Total variation distance between probability measures, Probability measure
Changed	3 November 2026