Cardinality of Spin-Glass Ground-State Sets

Definitions and mathematical statement

Let $\Gamma$ denote a graph, let $J$ denote the spin-glass data referred to in the supplied record, and let $G(J)$ denote its set of ground states. The record does not define the spin-glass model, the data $J$, the ground-state condition, or the phrase "distribution of linear growth." These are unresolved qualifications, so the following is a schematic mathematical question rather than a complete formal statement.

The requested cardinality assertion is: for every graph $\Gamma$ and every distribution satisfying the source's unspecified "linear growth" condition, does the corresponding ground-state set satisfy \[ \lvert G(J)\rvert=2 \quad\text{or}\quad G(J)\text{ is infinite}? \] Here "infinite" does not specify an infinite cardinality. If $J$ is random under the distribution in question, the source does not specify whether the assertion concerns every realization, almost every realization, or another probabilistic qualification.

The graph class, coupling assumptions, spin space, boundary conditions, and whether ground states are counted individually or modulo a symmetry are not specified. In particular, this draft does not impose finiteness, countability, connectedness, or local finiteness of the graph.

The record also asks whether there is a connection between random walks and spin-glass ground states on graphs beyond trees. It does not specify the random walk, the ground-state property to be related to it, or a proposed equivalence or implication. This broader question therefore remains a research direction without a determinate mathematical assertion.

Short English statement

The supplied question asks whether every graph and every distribution of "linear growth" yield exactly two or infinitely many spin-glass ground states, and whether random walks help describe ground states beyond trees. Essential definitions and the probabilistic quantifier are missing, so a precise unconditional formulation cannot be recovered from this record alone.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30004436$, titled "Cardinality of Spin-Glass Ground-State Sets," in the Oberwolfach Reports Open Problems collection. Exact source problem code: catalogue entry. No individual proposer is supplied; 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 describes Oberwolfach Reports as records of mathematical workshops and their open-problem discussions. The citation's 2021 date and the record's 2020 proposed-year metadata are retained without reconciliation.
• Supplied literature source, labelled 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 reports that the cardinality assertion and the random-walk connection beyond trees are explicitly listed as open problems, with no general resolution found. This is supplied dated curation, not independently verified literature status or source verification. The background expressly calls for independent verification of time-sensitive status and cited claims.
• The registry labels the statement corrected_verified and reports a manual semantic, typographic, and LaTeX review asserting that the clean statement is self-contained and preserves the target. These are supplied review claims; they do not resolve the missing definitions identified in this draft. No cited external source has been visited for this preparation.