Extremal Gibbs measures are ergodic (pure phases)

Extremality in the convex set of Gibbs measures is equivalent to trivial tail σ-algebra; with translation invariance, this implies translation ergodicity.

Extremal Gibbs measures are ergodic (pure phases)

Fix a lattice interaction/specification and let $\mathcal{G}$ denote the convex set of all infinite-volume Gibbs measures consistent with it.

Let $\mathcal{T}$ be the tail $\sigma$ -algebra,

\mathcal{T}=\bigcap_{\Lambda\Subset\mathbb{Z}^d}\mathcal{F}_{\Lambda^c},

i.e. events that depend only on spins “arbitrarily far away”.

Statement

For $\mu\in\mathcal{G}$ , the following are equivalent:

$\mu$ is extremal in $\mathcal{G}$ (it cannot be written as a nontrivial convex combination of two distinct measures in $\mathcal{G}$ ).
The tail $\sigma$ -algebra is $\mu$ -trivial: for every $A\in\mathcal{T}$ , one has $\mu(A)\in\{0,1\}$ .

If, in addition, the interaction/specification is translation-covariant and $\mu$ is translation invariant, then these conditions imply (and in common settings are equivalent to) translation ergodicity: every translation-invariant event has probability $0$ or $1$ under $\mu$ .

Key hypotheses

$\mathcal{G}$ is the convex set of all DLR-consistent Gibbs measures for a fixed specification.
(For the translation-ergodic refinement) translation covariance of the specification and translation invariance of $\mu$ .

Conclusions

Extremal Gibbs measures are “pure phases”: they have no nontrivial decomposition into distinct Gibbs components.
Tail-triviality provides a practical criterion for purity, often used in phase-transition analysis (compare phase transition and coexistence phenomena).