Lattice Gases and Antiferromagnetic Potts Models — Question 2

Definitions and mathematical statement

The supplied excerpt does not define the partition function $Z$ or specify the lattice-gas model. Consequently, the following is a conditional formulation of the question's quantifiers, rather than a complete definition of its mathematical object.

Fix the parameter $\Delta$. Let $\mathcal G_\Delta$ stand for the source's intended class of graphs of maximum degree $\Delta$. For $G\in\mathcal G_\Delta$, write $V(G)$ for its vertex set and use $Z_G((w_x)_{x\in V(G)})$ as a placeholder for the unspecified partition function, with complex vertex parameters $w_x$. Its equal-parameter specialization is denoted by \[ Z_G^{\mathrm{eq}}(w)=Z_G((w)_{x\in V(G)}). \] Here a complex domain means a connected open subset of ${\mathbb{C}}$.

Equal-parameter version. Determine the largest domain $D\subseteq{\mathbb{C}}$, containing $0$, such that \[ \forall G\in\mathcal G_\Delta\;\forall w\in D, \qquad Z_G^{\mathrm{eq}}(w)\ne0. \] Largest means containing every other domain with this property; existence of such a largest domain is part of the intended question and is not asserted here.

Unequal-parameter version. Seek domains $D\subseteq{\mathbb{C}}$, containing $0$, such that \[ \forall G\in\mathcal G_\Delta\;\forall (w_x)_{x\in V(G)}\in D^{V(G)}, \qquad Z_G((w_x)_{x\in V(G)})\ne0. \] The source explicitly cautions that there need not be a unique maximal such domain. A domain is maximal here if no strictly larger domain containing it satisfies the same condition; this differs from being a largest domain.

A complete statement requires the source's definition of $Z$, its graph conventions, the allowed values of $\Delta$, and clarification of whether "maximum degree $\Delta$" means degree at most $\Delta$ or maximum degree exactly $\Delta$. No particular partition function or graph convention is supplied by this draft.

Short English statement

Sokal's Question 2 asks for the largest complex domain containing zero in which the partition function stays nonzero uniformly over the specified graphs. It distinguishes equal vertex parameters from independently varying vertex parameters; in the latter version, a unique maximal domain need not exist. The supplied excerpt leaves the partition function and graph conventions undefined, so the displayed formulation remains conditional.

Sources

• Local UnsolvedMath registry, permanent ID $800571$, source problem code catalogue entry, supplied title: "Lattice Gases and Antiferromagnetic Potts Models — Question 2." The registry attributes the problem to Sokal, proposed in 2000, and labels it open; this is a supplied status label, not an independently verified assessment.
• Sokal, A Personal List of Unsolved Problems Concerning Lattice Gases and Antiferromagnetic Potts Models (2000), Question 2. Supplied source URL: https://arxiv.org/abs/cond-mat/0004231. The supplied provenance records access on 2026-10-16 and extraction by arxiv_latex; these are supplied provenance claims, not a claim of independent access or verification.
• Supplied upstream revision: d371ee917ac9647fbb06cd6fc529deec6f3dda82.