Deterministic Limits of Trace-Reinforced Ant Walks

Definitions and mathematical statement

Let $G=(V,E)$ be a finite graph equipped with the trace-reinforced ant model intended in the cited source. Write $W_e(n)$ for the weight of edge $e\in E$ after $n$ completed random walks, and let $\mathbb P$ denote the law of that process, including its prescribed initial configuration.

The supplied record does not define the process: it gives neither the transition probabilities nor the reinforcement rule, initial weights, nest and food configuration, or admissibility conditions on the graph. Accordingly, the notation here describes the intended convergence target, conditional on those definitions being supplied; it is not a complete definition of the model.

Deterministic-limit conjecture. For every finite graph covered by the intended model, does there exist a deterministic vector $(c_e)_{e\in E}\in{\mathbb{R}}^E$ such that \[ \mathbb P\!\left(\bigcap_{e\in E}\left\{\lim_{n\to\infty}\frac{W_e(n)}{n}=c_e\right\}\right)=1? \] The normalization is by the number of random walks, not by their lengths. The constants may depend on the graph and the prescribed model configuration, but are required to be nonrandom. The record does not specify the quantification over initial weights or nest and food configurations, so no universal quantifier over those choices is added here. The phrase "every finite graph" also requires clarification of connectivity, orientation, and any conditions ensuring that the walks terminate.

Several food sources. The record additionally asks for a meaningful extension of the model to several food sources. It supplies neither rules for this extension nor a mathematical criterion for "meaningful." This is therefore retained as an unresolved model-construction question; the record does not explicitly assert the deterministic-limit conjecture for such an extension.

The supplied report passage describes deterministic convergence in previously treated settings and conjectures its validity on general finite graphs. The identities and hypotheses of those previously treated settings are not included in the record.

Short English statement

The main question asks whether, on every finite graph allowed by the intended trace-reinforced ant model, each edge weight divided by the number of completed walks converges almost surely to a deterministic constant. A separate question asks how to extend the model to several food sources. The process rules and the requirements for that extension are missing, so this is a conditional statement-only draft requiring mathematical review.

Sources

• Local UnsolvedMath registry, supplied complete source record, permanent ID $30005449$, title Deterministic Limits of Trace-Reinforced Ant Walks; exact source problem code catalogue entry. Collection: Oberwolfach Reports Open Problems; proposed year recorded as 2023. No individual proposer is supplied.
• MATRIX-MFO Tandem Workshop: Stochastic Reinforcement Processes and Graphs (2023), Oberwolfach Reports, pp. 639--679, DOI 10.4171/owr/2023/12. Supplied source report and literature reference labeled "original Oberwolfach report": https://doi.org/10.4171/owr/2023/12. The supplied excerpt is the provenance for the general finite-graph conjecture and the several-food-sources question.
• Supplied literature reference attributed to Kious--Mailler--Schapira, labeled "trace-reinforced ants": https://math.univ-lyon1.fr/~schapira/articlespdf/antsJEP.pdf. The record supplies no further bibliographic details and does not identify these authors as the individual proposers of this extracted problem.
• Literature assessment dated exactly 2026-08-22: the supplied curation labels the problem open and reports that deterministic convergence remains conjectural for general finite graphs, with results only for several graph classes and examples. This is supplied dated curation, not an independently verified literature assessment.
• The registry labels the statement corrected_verified and says it was checked against the official report. That verification claim is supplied metadata; neither the report nor the linked paper has been independently consulted for this draft.