Parameter Dependence of Directed-Polymer Endpoint Limits

Definitions and mathematical statement

The supplied question concerns a directed polymer in $(1+1)$ dimensions, meaning one temporal and one spatial dimension. The symbols $\beta$ and $\alpha$ denote parameters whose definitions and admissible ranges are not supplied.

Source question. In the $(1+1)$-dimensional directed-polymer convergence diagram, does the putative scaling limit at the right-hand end depend on the parameters $\beta$ and $\alpha$?

Unresolved qualifications. The referenced convergence diagram is absent from the supplied record. The polymer model, environment law, roles of $\beta$ and $\alpha$, limiting regime, rescaling, observable, and mode of convergence are unspecified. The phrase "right-hand end" refers to a position in that missing diagram; it does not by itself identify a polymer endpoint observable. The record also does not specify whether dependence means dependence of a limiting law, dependence up to deterministic rescaling, or another equivalence criterion. Consequently, a complete quantified statement cannot be recovered from the supplied evidence. The existence of the putative limit is not asserted.

Short English statement

The question asks whether an unspecified directed-polymer scaling limit depends on $\beta$ and $\alpha$. Its precise meaning requires the missing convergence diagram and definitions of the model, parameters, scaling, and limiting object. This is a statement-only draft requiring human mathematical review.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30002188$, title "Parameter Dependence of Directed-Polymer Endpoint Limits," exact source problem code catalogue entry, in the Oberwolfach Reports Open Problems collection. The supplied proposed year is 2012; no individual proposer is identified.
• Supplied report citation: Scaling Limits in Models of Statistical Mechanics (2013), Oberwolfach Reports, pp. 2657--2745, DOI 10.4171/owr/2012/44. https://doi.org/10.4171/owr/2012/44. This same link is supplied under the literature-source label "Oberwolfach Report 44, polymer diagram." The collection describes these reports as records of mathematical workshops and their open-problem discussions.
• Supplied literature source: Alberts--Khanin--Quastel, "disorder regimes" (the supplied descriptive label). https://doi.org/10.1103/PhysRevLett.105.090603. No additional bibliographic details are supplied.
• Supplied literature assessment dated 2026-08-21: the record labels the problem open and describes it as a diagrammatic question about dependence of an unresolved polymer scaling limit on $\beta$ and $\alpha$. It states that known endpoint regimes do not determine the intermediate crossover requested here. This is supplied dated curation, not an independently verified literature assessment; the record itself calls for independent verification of time-sensitive status and cited claims.
• The supplied statement status is corrected_verified. Its verification note reports that the mathematical content was checked against the official Oberwolfach report and that notation, context, and question form were normalized without changing the source-level claim. That verification is reported provenance and has not been independently repeated for this draft.