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,

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

Statement

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

1. $\mu$ is extremal in $\mathcal{G}$ (it cannot be written as a nontrivial convex combination of two distinct measures in $\mathcal{G}$).

2. 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).

Proof idea / significance

If the tail $\sigma$-algebra is nontrivial, one can condition $\mu$ on a tail event $A\in\mathcal{T}$ with $0<\mu(A)<1$ and obtain two distinct conditional measures that are still DLR-consistent; this yields a nontrivial convex decomposition of $\mu$, so $\mu$ is not extremal. Conversely, if $\mu$ admits a nontrivial convex decomposition, the component label can be realized as a tail-measurable random variable, producing a nontrivial tail event. This links convex geometry (extreme points) to probabilistic ergodic/tail behavior.