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Universal backward-geodesic endpoint argmax conjecture

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Consider an exclusion process with generator specified by the paper and initial condition Xn(0)=−⌊n/ρ⌋X_n(0)=-\lfloor n/\rho\rfloor for some ρ∈(0,1)\rho\in(0,1). Assume its translation-invariant stationary measure μρ\mu_\rho satisfies the paper's Assumption~. Let J(ρ)J(\rho) be the stationary current and let A(ρ)A(\rho) and Γ(ρ)\Gamma(\rho) be the associated KPZ parameters. Let N(t↓0)N(t\downarrow0) denote the backward-geodesic endpoint index, and define

u^=argmax⁡u∈R{A2(u)−u2}.\hat u=\operatorname*{argmax}_{u\in\mathbb{R}}\left\{{\cal A}_2(u)-u^2\right\}.

Universal backward-geodesic endpoint conjecture. There exists a universal constant c2c_2 such that

lim⁡t→∞XN(t↓0)(0)−XN(0)−(J(ρ)/ρ−J′(ρ))t21/3Γ(ρ)2/3A(ρ)−1t2/3=du^.\lim_{t\to\infty}\frac{X_{N(t\downarrow0)}(0)-X_N(0)-(J(\rho)/\rho-J'(\rho))t}{2^{1/3}\Gamma(\rho)^{2/3}A(\rho)^{-1}t^{2/3}}\stackrel{d}{=}\hat u.

This predicts the universal argmax distribution of the Airy-two process minus a parabola for backward-geodesic endpoints. The source motivates the scale using KPZ theory and the known TASEP result, but gives no general proof.

References

Primary source

Patrik L. Ferrari and Min Liu, “Quasi-geodesics in integrable and non-integrable exclusion processes”, arXiv:2412.11626 (2024).

air-y-processesexclusion-processeskpz-universality
Posted

Progress summary

Refreshed
Open

The conjecture remains unproved in general, with only a special case established for one standard exclusion process.

A 2024 preprint formulates the universal endpoint-distribution claim as Conjecture 3.93.9, motivated by KPZ scaling and the known TASEP case. It predicts convergence to the argmax law of the Airy-two process minus a parabola.

Known results

  • For flat-initial-condition TASEP\mathrm{TASEP}, Theorem 2.62.6 proves convergence of the suitably rescaled backward-geodesic endpoint to the stated argmax distribution.
  • The corresponding assertions for ASEP and speed-changed ASEP are presented as conjectures, not theorems.

2024 general formulation

The preprint extends the conjecture to exclusion processes whose stationary measures satisfy its Assumption 3.73.7, but supplies no general proof. The retrieved searches found no counterexample, claimed proof, verification, or subsequent correction through August 20262026.

Current status (as of August 2026): The conjecture is proved for flat-initial-condition TASEP\mathrm{TASEP} but remains open for the stated general class of exclusion processes.

Sources

Solutions 0

No solutions have been posted yet.