DLR existence theorem

Statement

Let $\Phi$ be a translation-invariant interaction (potential) on $\mathbb{Z}^d$ with finite single-spin space $S$ (e.g. $S=\{\pm 1\}$) and assume $\Phi$ is uniformly absolutely summable (in particular, any finite-range interaction qualifies). Let $\gamma^\Phi$ be the associated Gibbs specification constructed from the lattice Hamiltonian .

Then the set of infinite-volume Gibbs measures consistent with $\gamma^\Phi$ is nonempty. Equivalently, there exists at least one probability measure $\mu$ on $\Omega=S^{\mathbb{Z}^d}$ such that $\mu$ satisfies the DLR equation for every finite $\Lambda\Subset\mathbb{Z}^d$.

Moreover, for any fixed boundary condition $\eta$ and any increasing sequence of finite volumes $\Lambda_n\uparrow\mathbb{Z}^d$, every weak limit point of the corresponding finite-volume Gibbs measures $\mu_{\Lambda_n}^{\eta}$ is an infinite-volume Gibbs measure for $\Phi$.

Key hypotheses

• Configuration space: $\Omega=S^{\mathbb{Z}^d}$ with $S$ finite (or more generally compact Polish).
• Interaction: $\Phi$ is translation-invariant and uniformly absolutely summable (e.g. finite range).
• Specification: $\gamma^\Phi$ is the Gibbs specification induced by $\Phi$ (quasilocal, consistent, proper).

(Probability-theoretic background: $\mu$ is a probability measure on $(\Omega,\mathcal{F})$.)

Key conclusions

• Existence: $\mathcal{G}(\gamma^\Phi)\neq\varnothing$, where $\mathcal{G}(\gamma^\Phi)$ denotes the set of $\mu$ satisfying the DLR equation .
• Compactness mechanism: the family $\{\mu_{\Lambda}^{\eta}\}_{\Lambda\Subset\mathbb{Z}^d}$ has weakly convergent subnet/subsequence along any exhaustion, and each limit point is in $\mathcal{G}(\gamma^\Phi)$.
• Equilibrium exists at all parameters: for every inverse temperature (absorbed into $\Phi$) and external parameters appearing in $\Phi$, at least one infinite-volume equilibrium state exists.

Proof idea / significance (sketch)

For finite spin spaces, $\Omega=S^{\mathbb{Z}^d}$ is compact in the product topology (Tychonoff), hence the space of probability measures on $\Omega$ is weak-* compact. Fix a boundary condition $\eta$ and consider $\mu_{\Lambda}^{\eta}$. Any sequence along $\Lambda_n\uparrow\mathbb{Z}^d$ admits weak limit points. Quasilocality/consistency of the specification implies that the DLR consistency relations pass to the limit for local test functions, yielding a measure $\mu$ that satisfies the DLR equations.

Conceptually: this theorem guarantees that the DLR formalism is not empty—there is always at least one infinite-volume Gibbs state to study for standard lattice interactions (e.g. the Ising model ).