Uniform Coupling under Gibbs Uniqueness

Definitions and mathematical statement

Let $\gamma=(\gamma_\Lambda)_{\Lambda\Subset{\mathbb{Z}}^d}$ be a specification: a family of consistent probability kernels prescribing conditional distributions in finite sets $\Lambda$, with the exterior configuration fixed. Denote its underlying configuration space by $\Omega\subseteq S^{{\mathbb{Z}}^d}$, where the single-site space $S$, its measurable structure, and any restrictions defining $\Omega$ are not specified in the supplied record. A Gibbs state for $\gamma$ is a probability measure $\mu$ on $\Omega$ satisfying the Dobrushin--Lanford--Ruelle equations \[ \mu\gamma_\Lambda=\mu\qquad\text{for every finite }\Lambda\subset{\mathbb{Z}}^d. \] Assume that exactly one such Gibbs state exists.

Write $\Lambda_n$ for the source's $n$-box. For an exterior boundary condition $\eta$, let $\gamma_{\Lambda_n}^{\eta}$ denote the induced probability law on configurations in $\Lambda_n$. A coupling of $\gamma_{\Lambda_n}^{\eta}$ and $\gamma_{\Lambda_n}^{\xi}$ is a joint probability law $\pi$ of configurations $X,Y$ in $\Lambda_n$ having these respective marginals. Define the total number of disagreements by \[ D_{\Lambda_n}(X,Y)=\sum_{v\in\Lambda_n}\mathbf{1}_{\{X_v\ne Y_v\}}. \] This definition presupposes that the disagreement events are measurable.

Question. Does uniqueness of the Gibbs state imply \[ \begin{gathered} \forall\varepsilon>0\ \exists N\in{\mathbb{N}}\ \forall n\in{\mathbb{N}},\ n\ge N,\quad \forall\eta,\xi\text{ boundary conditions outside }\Lambda_n,\\ \exists\pi_{n,\eta,\xi}\in \operatorname{Cpl}\left(\gamma_{\Lambda_n}^{\eta},\gamma_{\Lambda_n}^{\xi}\right) \quad\text{such that}\quad \int D_{\Lambda_n}(X,Y)\,\pi_{n,\eta,\xi}({\,\mathrm{d}} X,{\,\mathrm{d}} Y) \le\varepsilon? \end{gathered} \] Here $\operatorname{Cpl}(\alpha,\beta)$ denotes the set of probability measures with marginals $\alpha$ and $\beta$. The threshold $N$ may depend on the specification and on $\varepsilon$, but must be uniform over both boundary conditions and all $n\ge N$. The disagreement count is unnormalized and covers the entire box.

Unresolved scope. The source does not specify the dimension range, the state space, the exact boxes, regularity or interaction assumptions on the specification, or which exterior configurations count as boundary conditions. The displayed quantification can be read only where the finite-volume laws are defined; whether all exterior configurations or only admissible ones are intended requires clarification. No additional hypotheses resolving these omissions are imposed here.

Short English statement

Does a unique Gibbs state guarantee that, uniformly over every pair of exterior boundary conditions, the resulting laws on a sufficiently large box admit a coupling with expected total number of disagreements at most any prescribed $\varepsilon>0$? The supplied record leaves the state space, specification hypotheses, box convention, and boundary-condition domain unresolved.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30004434$, title Uniform Coupling under Gibbs Uniqueness, source problem code catalogue entry, in the Oberwolfach Reports Open Problems collection. No individual proposer is identified; the proposed year is recorded as 2020.
• Supplied report citation: Mini-Workshop: One-sided and Two-sided Stochastic Descriptions (2021), Oberwolfach Reports, pp. 601--637, DOI 10.4171/owr/2020/11. Supplied link: https://doi.org/10.4171/owr/2020/11. The background attributes the problem to workshop open-problem discussions documented in Oberwolfach Reports.
• Supplied literature source, labeled OWR-2020-11: https://ems.press/content/serial-article-files/46847.
• Literature assessment dated exactly 2026-08-22: the supplied curation labels the problem open and says that the implication is explicitly posed as a question, with the checked literature neither establishing it in the stated generality nor providing a demonstrably decisive counterexample. This is supplied dated curation derived from the OWR material, not an independently verified assessment; time-sensitive status and cited claims require independent verification.
• The registry labels the statement corrected_verified and reports manual semantic, typographic, and LaTeX review as confirming preservation of the mathematical target. These are supplied editorial assertions, not independent verification of completeness or mathematical correctness.