Universal Fluctuation Limits for Exclusion Processes

Definitions and mathematical statement

For the totally asymmetric simple exclusion process (TASEP), classify the initial conditions yielding a common scaling-limit distribution and a common limit process for particle-position fluctuations.

In the standard continuous-time one-dimensional model on ${\mathbb{Z}}$, a configuration is $\eta\in\{0,1\}^{{\mathbb{Z}}}$; a particle at $x$ attempts to jump to $x+1$ at rate one, and the jump occurs only if $x+1$ is empty. This explains the usual name, but the source does not specify lattice, time convention, jump rate, or boundary conditions. These model choices must be confirmed.

With labeled particle positions $X_j(t)$, a fluctuation process requires a chosen observation regime, deterministic centering, and positive scaling. One may write the general form \[ Z_t(u)=\frac{X_{j(t,u)}(t)-m_t(u)}{s_t(u)}, \] but the source supplies none of $j(t,u)$, $m_t(u)$, $s_t(u)$, or the parameter domain. This notation identifies required data, not a selected scaling or fluctuation exponent.

The classification must specify which initial configurations or laws lead, after the intended normalization, to the same one-point limiting distribution and to the same limiting process. The record does not say whether a particular common limit is prescribed or whether all equivalence classes of initial data sharing limits are to be determined. Deterministic versus random initial conditions, restrictions on densities or large-scale profiles, and averaging over initial randomness are also unspecified.

Convergence of a one-point law is distinct from convergence of finite-dimensional distributions or convergence in a function-space topology. The source requests both a distribution and a process, so the process target is not silently reduced to one-point universality. No named limit law, process, centering, or scaling exponent is inserted. The supplied assessment reports identified limits for several initial conditions but no full classification; it gives no list or theorem hypotheses.

Short English statement

The question asks which TASEP initial data share normalized particle-fluctuation limits, both as individual distributions and as processes. The observation scale, normalization, initial-data class, and common limiting object are not specified.

Sources

• Local UnsolvedMath registry, permanent ID 30000586, code catalogue entry, "Universal Fluctuation Limits for Exclusion Processes." Proposed year 2006; proposer unspecified.
• Spatial Random Processes and Statistical Mechanics (2007), Oberwolfach Reports, pp. 2447--2504, DOI 10.4171/owr/2006/41; https://doi.org/10.4171/owr/2006/41. The displayed report year 2007 differs from the DOI and proposed year 2006.
• Statement status exact; the supplied verification calls the statement clear and internally complete and says it was retained byte-for-byte. The omitted scaling and initial-data conventions still require source review.
• Literature triage checked 2026-08-21: partially solved. It says several TASEP initial conditions have identified scaling limits but no complete common-distribution/process classification is known. No individual cases or further references are supplied. Created 2026-08-25; updated 2026-09-16. Current status and source claims require independent verification.
• Registry curation by Ulam AI / UnsolvedMath Contributors, CC BY 4.0; the underlying report retains its supplied provenance. Unreviewed statement-only draft.