Scaling Limits Across the Directed-Polymer Crossover

Definitions and mathematical statement

The supplied record concerns directed polymers in $(1+1)$ dimensions, meaning one time coordinate and one spatial coordinate. It refers to a disorder environment $\omega_n$, a polymer path $S_n$, a parameter $\beta$, and a convergence diagram. The underlying environment law, reference path law, polymer measure, boundary conditions, rescalings, and convergence diagram are not supplied. Consequently, the following records the source questions schematically, rather than specifying a uniquely determined convergence problem.

1. Does the scaling limit represented by the $\beta=\infty$ end of the cited convergence diagram exist? The record does not specify the random object being scaled, the normalization, the convergence topology, or the order of the limits involving $n$ and $\beta$.
2. What are the properties of the intermediate-disorder crossover connecting the regimes in that diagram? The expression "crossover bridge" is not defined in the record; in particular, it does not specify a family of probability laws, its parameter range, its endpoint limits, or the properties to be established.
3. For environment scalings described as slower than $n^{-1/4}$, does the joint environment--path pair have a distributional limit, schematically \[ (\omega_n,S_n)\xrightarrow[n\to\infty]{\mathrm{law}}(\omega,S)? \] If such a limit exists, is its law the same as the corresponding last-passage-percolation limit? Here $(\omega,S)$ denotes only a prospective limiting pair. Neither its state space nor the last-passage-percolation object and identification used for comparison are specified.

The phrase "environment scalings slower than $n^{-1/4}$" does not identify the scaled quantity or the allowed sequences, so it does not determine a precise quantified rate condition. No particular environment distribution, endpoint constraint, centering, normalization, or topology is imposed in this draft. These missing specifications must be recovered from the cited report before the questions can become precise mathematical assertions.

The supplied dated literature assessment labels the slower-than-$n^{-1/4}$ regime and its joint environment--path limit as open in the cited research line. It describes the $n^{-1/4}$ intermediate-disorder limit and the faster-scaling Brownian regime as known, while describing the KPZ-side crossover as not fully established. These are supplied curation claims, not independently verified mathematical conclusions.

Short English statement

The source asks whether directed polymers have a limit at the $\beta=\infty$ endpoint, how the intermediate-disorder crossover behaves, and whether more slowly scaled disorder and the polymer path converge jointly to the corresponding last-passage-percolation limit. The model, normalizations, boundary conditions, convergence spaces, and comparison object are missing, so a fully precise formulation requires source review.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30002187$, source problem code catalogue entry, title "Scaling Limits Across the Directed-Polymer Crossover," in the Oberwolfach Reports Open Problems collection. The project identifies data/unsolvedmath/problems.json and the supplemental catalog/problems.json as local registry resources. This draft uses the supplied complete record; it does not claim an independent inspection of those files.
• Supplied source citation: Scaling Limits in Models of Statistical Mechanics (2013), Oberwolfach Reports, DOI 10.4171/owr/2012/44, pp. 2657--2745. https://doi.org/10.4171/owr/2012/44. The literature-source list labels this reference "Oberwolfach Report 44, convergence diagram." The record gives proposed year 2012 and names no proposer.
• Supplied literature reference: Alberts--Khanin--Quastel, "intermediate disorder." https://doi.org/10.1103/PhysRevLett.105.090603.
• Supplied literature reference: Alberts--Khanin--Quastel, "continuum polymer." https://arxiv.org/abs/1202.4403.
• Literature assessment date: 2026-08-21. The supplied assessment labels the problem open and makes the regime-specific claims recorded above. This is supplied dated curation, derived from the supplied Oberwolfach curation, and has not been independently verified; its time-sensitive status and cited claims require independent checking.
• 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. This is provenance supplied by the registry, not verification performed for this draft.