Configuration space (lattice)

Fix a spin space $(S,\mathcal{S})$ and consider the integer lattice (see integer lattice ). The (infinite-volume) configuration space is

the set of all assignments $\sigma = (\sigma_i)_{i\in\mathbb{Z}^d}$ with $\sigma_i \in S$.

The natural measurable structure on $\Omega$ is the product sigma-algebra

generated by cylinder events (events depending only on finitely many sites). This is the standard setting for defining Gibbs measures as probability measures on $(\Omega,\mathcal{F})$ (see probability space ).

For a finite region $\Lambda$ (e.g. finite box ), the finite-volume configuration space is

with sigma-algebra $\mathcal{F}_\Lambda := \bigotimes_{i\in\Lambda}\mathcal{S}$.

Key properties

• Cylinder sigma-algebra and locality: Any local observable depending on finitely many spins is measurable with respect to $\mathcal{F}_\Lambda$ for some finite $\Lambda$. This matches the locality of typical interactions and Hamiltonians .
• Product measures and a priori structure: If a single-site a priori measure $\rho$ is chosen on $S$, one obtains a product measure on $\Omega$ (see product measure ). Finite-volume Gibbs measures are often absolutely continuous with respect to the corresponding product measure on $\Omega_\Lambda$.
• Shifts (translations): Translations of the lattice act on $\Omega$ by shifting coordinates. Translation invariance of interactions (see translation-invariant interaction ) often leads to translation-invariant Gibbs measures.
• Topology (when used): If $S$ is finite or compact and metrizable, $\Omega$ can be given the product topology, and then $\Omega$ is compact by Tychonoff’s theorem. This is convenient for existence/compactness arguments in equilibrium theory.

Physical interpretation

$\Omega$ is the phase space (more precisely, the configuration or sample space) of a lattice spin system: each $\sigma\in\Omega$ is a full microscopic state. Equilibrium statistical mechanics selects probability measures on $\Omega$—finite-volume (see finite-volume Gibbs measures ) and infinite-volume (see infinite-volume Gibbs measures )—that encode typical configurations and fluctuations, as formalized by the Gibbs specification and DLR equation .