Symmetries of Open-Boundary Asymmetric Exclusion Processes

Definitions and mathematical statement

An asymmetric simple exclusion process (ASEP) is a continuous-time particle process with at most one particle at each lattice site: particles attempt directionally biased jumps to neighbouring sites, and jumps to occupied sites are suppressed. Open boundaries allow particle exchange with external reservoirs. The supplied record does not specify the lattice, its size, the jump rates, or the boundary exchange rules and rates.

For a finite configuration space $\Omega$, write $c(\eta,\xi)\geq 0$ for the transition rate from $\eta$ to $\xi$, for distinct configurations. Using the convention that the intensity matrix acts on functions $f:\Omega\to{\mathbb{R}}$, its action is \[ (Qf)(\eta)=\sum_{\substack{\xi\in\Omega\\\xi\ne\eta}}c(\eta,\xi)\left(f(\xi)-f(\eta)\right). \] This gives a finite-state framework only; the supplied record does not explicitly restrict the intended process to finite volume. In a linear-operator interpretation, a symmetry operator $S$ commutes with the intensity matrix, $SQ=QS$. A non-Abelian family of such symmetries contains operators $S,T$ satisfying $ST\ne TS$. The source does not identify the intended symmetry operators or algebra, so this interpretation does not establish which symmetries the report means.

The supplied research question is: What role do the non-Abelian symmetries of the intensity matrix play for ASEP with open boundaries?

The retained report passage reads: "An intriguing open problem is the role of such symmetries for the ASEP with open boundaries." The antecedent of "such symmetries" is absent from the supplied excerpt; its identification as non-Abelian symmetries of the intensity matrix comes from the supplied curated statement. Neither a particular consequence to establish nor parameter assumptions and quantifiers are supplied. Consequently, this is an unresolved research question requiring further source context, rather than a uniquely specified conjecture or theorem statement.

Short English statement

The question asks how non-Abelian symmetries of the transition-rate matrix affect ASEP when particles can enter or leave at the boundaries. The intended symmetries, boundary parameters, and mathematical consequences are unspecified. The registry labels the problem open; this draft preserves that supplied label without independently verifying it.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30006287$, title: Symmetries of Open-Boundary Asymmetric Exclusion Processes. Exact source problem code: catalogue entry. Collection: Oberwolfach Reports Open Problems. The supplied proposed year is 2025; no individual proposer is identified.
• Supplied report citation: Statistical Physics Out of Equilibrium: Quantitative Results and Universality (2025), Oberwolfach Reports, pp. 1333--1388, DOI 10.4171/owr/2025/26. Supplied source URL and literature-source link labelled "original OWR report": https://doi.org/10.4171/owr/2025/26. No page within this report range is identified for the question.
• Literature assessment dated exactly 2026-08-22: the supplied curation labels the problem open and states that symmetry and reverse-duality results are known for several ASEP-related systems, while the role of these symmetries for open-boundary ASEP remains explicitly posed as an open problem. This is supplied dated curation, not an independently verified literature assessment. Its listed checked source is the original OWR report linked above.
• The registry assigns statement status corrected_verified and reports that the official report was checked, the question isolated from surrounding exposition, and its notation repaired for self-contained publication. These are supplied provenance claims; this draft does not claim to have visited or independently checked the report.