Universality Classes for Interacting Particle Fluctuations

Definitions and mathematical statement

Terminology. A one-dimensional interacting particle system describes particles whose stochastic evolution depends on their configuration in one spatial dimension. In an exclusion system, each site can contain at most one particle. Finite-range exclusion permits jumps only over a bounded spatial distance; nearest-neighbour exclusion restricts jumps to adjacent sites. The source does not specify the underlying spatial domain, jump rates, or allowed interactions.

Integrated particle-density fluctuations refer to a spatially accumulated, centered particle-density field, with a reference point and a time-dependent current term where required by the chosen height convention. The centering, reference convention, and initial distribution are not specified in the supplied record. KPZ denotes the Kardar--Parisi--Zhang universality class, and EW denotes the Edwards--Wilkinson universality class. The record does not specify a mathematical definition of membership in either class.

Source research problem. Determine explicit hypotheses on one-dimensional interacting particle systems extending beyond nearest-neighbour simple exclusion, including finite-range exclusion systems, under which their integrated particle-density fluctuations rigorously exhibit the predicted KPZ or EW scaling behaviour. The desired results should specify which hypotheses lead to each class and establish the corresponding fluctuation limits.

Unresolved formulation. For a precise convergence assertion, one must specify the stochastic dynamics, spatial domain and boundary conditions, initial laws or stationary measures, density regime, centering and observation frame, space--time and fluctuation rescalings, limiting random object, and mode of convergence. None of these is fixed by the supplied statement. In particular, it does not distinguish convergence of a single observable, convergence of finite-dimensional distributions, and convergence of a random field or process. It also does not say whether KPZ or EW behaviour is intended to mean scaling exponents, limiting distributions, or a particular stochastic evolution. Thus the supplied record states a research programme seeking sufficient hypotheses, rather than a fully quantified conjecture for a specified family of systems; a unique precise limit theorem cannot be extracted without additional source information.

Short English statement

Find conditions ensuring that integrated density fluctuations in one-dimensional interacting particle systems beyond nearest-neighbour exclusion, including finite-range exclusion, have KPZ or Edwards--Wilkinson scaling limits. The supplied record leaves the models, hypotheses, rescalings, and convergence targets unspecified, so a precise theorem remains to be formulated.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30001618$, title: Universality Classes for Interacting Particle Fluctuations; exact source problem code: catalogue entry. Collection: Oberwolfach Reports Open Problems. No individual proposer is supplied; the proposed year is recorded as 2010.
• Supplied report citation: Large Scale Stochastic Dynamics (2011), Oberwolfach Reports, pp. 2877--2975, DOI 10.4171/owr/2010/50. Supplied source URL and literature-source link, labelled "original OWR report": https://doi.org/10.4171/owr/2010/50. The background describes Oberwolfach Reports as documenting mathematical workshops and their open-problem discussions. No pinpoint page or verbatim report passage is supplied for this problem.
• Literature assessment dated 2026-08-21: the supplied curation labels the problem open, attributes to the report an explicit identification of rigorous KPZ and Edwards--Wilkinson universality beyond nearest-neighbour exclusion as an important unresolved problem, and states that later results cover special finite-range and integrable models rather than the general extension. This is supplied dated curation, not independently verified here. The record itself requests independent verification of time-sensitive status and cited claims; it supplies no separate references for those later results.
• The supplied statement status is corrected_verified; its accompanying verification note states only that TeX-style punctuation was normalized without changing the mathematical content. This metadata does not establish mathematical correctness or verify the literature assessment.