Extremal Gibbs measure

Fix a Gibbs specification $\gamma$ and let $\mathcal{G}(\gamma)$ denote the set of infinite-volume Gibbs measures consistent with $\gamma$.

A Gibbs measure $\mu \in \mathcal{G}(\gamma)$ is extremal if it cannot be written as a nontrivial convex combination of other Gibbs measures: whenever

it follows that $\mu_1=\mu_2=\mu$.

A standard equivalent characterization is tail triviality: if $\mathcal{T}$ is the tail σ-algebra (events depending only on spins arbitrarily far away),

then $\mu(A)\in\{0,1\}$ for every $A\in\mathcal{T}$.

Key properties

• Pure-state building blocks: Extremal Gibbs measures are the “indecomposable” equilibrium states. Every Gibbs measure can be represented as a mixture of extremal ones.

• No hidden randomization at infinity: Tail triviality means there is no residual macroscopic randomness left after conditioning on arbitrarily large scales. Intuitively, an extremal state does not randomly select among competing phases.

• Ergodic behavior under symmetries (when applicable): For translation-invariant interactions , many physically relevant extremal Gibbs measures are translation-ergodic (though extremality is defined relative to the Gibbs set, not relative to translations).

• Characterization of coexistence: When multiple distinct extremal Gibbs measures exist at the same parameters, they correspond to distinct macroscopic phases (see phase transition and pure phase ).

Physical interpretation

An extremal Gibbs measure corresponds to a single homogeneous equilibrium phase of an infinite system, with no macroscopic phase coexistence encoded by random mixing. In models with symmetry breaking (e.g. the Ising model at low temperature), the “plus” and “minus” magnetized states are paradigmatic examples of distinct extremal Gibbs measures, while a symmetry-invariant state may be their non-extremal mixture.