Local-Energy Classes of Summable Lattice-Gas Interactions

Definitions and mathematical statement

Let $d$ denote the lattice dimension and let $X=\mathbb{N}^{\mathbb{Z}^d}$ be the configuration space. The source does not explicitly specify the range of $d$ or the convention for $\mathbb{N}$. Write $\Phi=(\Phi_\Lambda)_\Lambda$ for a translation-invariant lattice-gas interaction on $X$. Under the conventional interpretation that the interaction indices $\Lambda$ are finite subsets of $\mathbb{Z}^d$, the supplied summability class is \[ S=\left\{\Phi:\sum_{\substack{\Lambda\subseteq\mathbb{Z}^d\text{ finite}\\0\in\Lambda}}\lVert\Phi_\Lambda\rVert_\infty<\infty\right\}, \] where $\lVert\Phi_\Lambda\rVert_\infty$ denotes the supremum of the absolute value of the interaction term over its configurations. The finite-subset indexing convention is implicit in the terminology and requires confirmation against the report.

Let $\mathcal{A}_S$ denote the class of local energy functions generated by interactions in $S$, as in the supplied record. For the two exponents needed below, write \[ \mathcal{R}_k=\left\{f:\sum_{n=1}^{\infty}n^k\delta_n(f)<\infty\right\},\qquad k\in\{d-1,d\}. \] Here $\delta_n(f)$ is the source's unspecified modulus-of-continuity quantity. The record does not define it, the common ambient class of functions $f$, or the rule assigning a local energy function to an interaction. In particular, it does not specify the neighborhoods used to measure continuity, the normalization or allocation of interaction terms, or whether energy functions are identified under any equivalence relation. These conventions must be supplied before the classes above determine a fully precise mathematical problem.

Question, conditional on those conventions. Taking the inclusions \[ \mathcal{R}_d\subseteq\mathcal{A}_S\subseteq\mathcal{R}_{d-1} \] as supplied premises, determine whether each inclusion is an equality or is strict. If both inclusions are strict, give a necessary and sufficient characterization of membership in $\mathcal{A}_S$ in terms of the modulus of continuity of the local energy function. The source does not specify the intended form of that characterization. The displayed inclusions are reported by the supplied record and are not independently established here.

Short English statement

The question asks which continuity conditions characterize local energies arising from absolutely summable, translation-invariant lattice-gas interactions on $\mathbb{N}^{\mathbb{Z}^d}$. It asks whether either of two supplied inclusions is an equality and, if both are strict, requests an exact continuity criterion. Definitions of the continuity quantities and the interaction-to-energy correspondence are missing, so the formulation remains conditional.

Sources

• Local UnsolvedMath problem registry, permanent ID $30001961$, source problem code catalogue entry, title: Local-Energy Classes of Summable Lattice-Gas Interactions. The supplied record lists proposed year 2012 and names no proposer. Its status is supplied as open; its statement status is supplied as corrected_verified, with a claimed manual semantic, typographic, and LaTeX review. These labels are source metadata, not independent verification.
• Supplied report citation: Mini-Workshop: Thermodynamic Formalism, Geometry and Stochastics (2012), Oberwolfach Reports, pp. 77--129, DOI 10.4171/owr/2012/03. The record describes Oberwolfach Reports as documenting mathematical workshops and their open-problem discussions. Report URL and supplied literature-source link labeled "original Oberwolfach report": https://doi.org/10.4171/owr/2012/03. No problem-specific page or named contributor is supplied.
• Supplied literature-source link labeled "real analyticity for exponential maps": https://www.cambridge.org/core/journals/ergodic-theory-and-dynamical-systems/article/real-analyticity-of-hausdorff-dimension-of-finer-julia-sets-of-exponential-family/40CE052F557F04507E762ABE161CF857.
• Literature assessment date: 2026-08-21. This is supplied dated curation, not independently verified. The supplied assessment asserts real analyticity of radial Julia dimension in hyperbolic parameters and says that supremum and monotonicity questions remain open. Those assertions concern a different apparent subject from the lattice-gas question and do not substantiate its status. The background explicitly requests independent verification of time-sensitive status and cited claims.