Fluctuation Limits for Exclusion Processes

Definitions and mathematical statement

In the standard continuous-time one-dimensional TASEP on ${\mathbb{Z}}$, a configuration is $\eta\in\{0,1\}^{{\mathbb{Z}}}$, where $\eta(i)=1$ indicates a particle. Each particle attempts to jump from $i$ to $i+1$ at rate one, and the jump occurs only when the destination is empty. Equivalently, on cylinder functions the generator is \[(\mathcal LF)(\eta)=\sum_{i\in{\mathbb{Z}}}\eta(i)(1-\eta(i+1))\left(F(\eta^{i,i+1})-F(\eta)\right),\] where $\eta^{i,i+1}$ interchanges the two indicated occupations. This is the usual model interpretation; the extract does not itself specify state space, time convention, or rates. Its question is: for the various classes of initial conditions, what are the limiting one-point fluctuation laws and the limiting fluctuation process as $t\to\infty$? To formulate a particular limit one must choose an observable $O_t$, observation locations $x_t(s)$, deterministic centering $a_t(s)$, and positive fluctuation scale $b_t(s)$, and ask for limits of \[Z_t(s)=\frac{O_t(x_t(s))-a_t(s)}{b_t(s)}.\] This notation does not select a current, height, or particle-position convention, or assert a common scaling across all initial conditions. A one-point limiting law concerns $Z_t(s)$ for a fixed parameter, whereas a process limit also requires joint distributions and an appropriate mode or topology of convergence. The supplied record does not enumerate the initial-condition classes, fix these normalizations, or specify that topology. The 2026-08-21 triage labels the record partially solved and reports proved Airy-process and Tracy--Widom scaling limits for step and several other initial conditions, while saying a complete long-time classification across initial conditions and models remains open. The displayed source question itself concerns TASEP; the broader mention of models is retained as a literature qualification rather than a new universal statement.

Short English statement

Classify the long-time one-point fluctuation laws and fluctuation processes of TASEP for different initial conditions. The chosen observable, centering, spatial and fluctuation scales, initial-condition classes, and convergence topology must be specified for each regime.

Sources

• Local UnsolvedMath registry, permanent ID 30000839, source problem catalogue entry, Fluctuation Limits for Exclusion Processes, proposed year 2007; no individual proposer supplied.
• Supplied citation: Large Scale Stochastic Dynamics (2008), Oberwolfach Reports, DOI 10.4171/owr/2007/42, pp. 2443--2530: https://doi.org/10.4171/owr/2007/42. Supplied literature label: original Oberwolfach problem.
• Supplied literature label: TASEP Airy limits, https://link.springer.com/article/10.1007/s00440-021-01074-0.
• Supplied literature label: Airy-one TASEP limit, https://arxiv.org/abs/math-ph/0611071.