Eulerian-circuit continuum limits and SLE

Definitions and mathematical statement

Write ${\mathbb{Z}}_N={\mathbb{Z}}/N{\mathbb{Z}}$. The source concerns uniform Eulerian circuits on the lattice torus ${\mathbb{Z}}_N^2$. An Eulerian circuit is a closed edge-traversing walk using every relevant edge exactly once. The exact graph, directed-versus-undirected convention, edge labels, and rooting/orientation equivalence used for the uniform law are not supplied.

In a usual geometric embedding, scaling lattice coordinates by $1/N$ produces curves on the unit flat torus. A continuum-limit statement would need interpolation, starting-point law, time normalization, curve topology, and a limit as $N\to\infty$. The source explicitly calls that limit conjectural and does not specify these choices in the extracted item.

The question is whether that proposed limit is related to space-filling $\operatorname{SLE}_\kappa$ for some $\kappa>8$. SLE denotes Schramm--Loewner evolution, driven in its standard planar formulation by Brownian motion with variance parameter $\kappa$. A precise relation would have to specify the domain or torus version, boundary/marking data, value or range of $\kappa$, parametrization, and whether the claim is equality in law, a local limit, a coupling, or another correspondence. No such relation is chosen in the record, and the endpoint $\kappa=8$ is not silently substituted for the strict range $\kappa>8$.

The research summary inconsistently identifies the uniform-spanning-tree Peano limit with both $\operatorname{SLE}_{16}$ and $\operatorname{SLE}_8$. Those are different parameter claims; its $16$ attribution is not adopted as a verified theorem or as a proposed parameter for this Eulerian-circuit question. Established facts about a different random curve would not by themselves determine the conjectural Eulerian limit.

Short English statement

Is the conjectural scaling limit of uniform Eulerian circuits on the square lattice torus related to space-filling SLE with parameter greater than eight? Both the limit construction and the meaning of relation need definition.

The supplied summary reports no Eulerian-circuit scaling-limit theorem or SLE identification, points to separate excursion questions, and mentions Le Jan's loop/free-field connections as distinct work. Its conflicting UST/SLE parameter statements are flagged. The OPEN-TRIAGE label and Aldous source are retained with current primary-literature audit required.

Sources

• Local UnsolvedMath registry, permanent ID 9700017, source code catalogue entry, "Eulerian-circuit continuum limits and SLE," proposed by David Aldous. Dataset curation and metadata: Ulam AI and UnsolvedMath Contributors, CC BY 4.0.
• David Aldous, Open problems, "Random Eulerian Circuits," final question, index https://www.stat.berkeley.edu/~aldous/Research/OP/index.html; HTML-text extraction with linked primary-PDF text, accessed 2026-07-29. No exact primary PDF citation is supplied.
• Supplied summary context: Lawler--Schramm--Werner, including a 2004 attribution, and Miller--Sheffield on space-filling SLE/imaginary geometry; complete references are absent and the UST parameter attribution is inconsistent.
• The summary cites arXiv:1405.2879 in the Le Jan loop/free-field context and cross-refers to registry source codes AMR-096-0013 and AMR-096-0014 for excursion questions; no permanent numeric IDs are inferred.
• Provenance records a maintained index of 23 active pages; current primary-literature status audit and redistribution terms NEEDS_REVIEW, separately from dataset attribution.