Let X_N^rm ASEPsc(t) be the position of particle N in ASEPsc with flat initial condition, and use the functions J(beta,E) and Gamma(beta,E) defined by the model parameters. Define X^rm resc_N(t)=fracX_N^rm ASEPsc(t)-X_N^rm ASEPsc(0)-2J(beta,E)t-2 Gamma(beta,E)^1/3t^1/3. ASEPsc particle-position conjecture. For any N∈Z and every s∈R, lim_t→∞P(X^rm resc_N(t)≤ s)=F_rm GOE(2s), where F_rm GOE is the Tracy–Widom GOE distribution. This predicts the GOE fluctuation law for particle positions in ASEPsc under KPZ t^1/3 scaling; the source does not state that it has been proved.

ASEPsc particle-position Tracy–Widom GOE conjecture

The conjecture predicts that the rescaled position of a tagged particle in ASEPsc with flat initial data converges to the GOE Tracy–Widom law, $F_{\rm GOE}(2s)$ , under $t^{1/3}$ scaling. It is explicitly presented as a conjecture rather than a theorem.

December 2024 numerical evidence

A paper records the statement as Conjecture 2.9 and tests it numerically for $\beta=\log 4$ and $E=\infty$ , using $10^6$ trials at several times. The empirical distributions closely match $F_{\rm GOE}(2s)$ , with a decreasing finite-time shift, but no proof, counterexample, or independent verification is reported.

Current status (as of August 2026): The ASEPsc particle-position GOE conjecture is unproved; numerical evidence supports the predicted $F_{\rm GOE}(2s)$ limit, and the general case remains open.

ASEPsc particle-position Tracy–Widom GOE conjecture

Primary source

Progress summary

Solutions 0