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Quasi-geodesics in integrable and non-integrable exclusion processes

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Pith's one-line read For flat TASEP, the backwards geodesic endpoint converges to the Airy-2 argmax; the same law is conjectured and numerically supported for quasi-geodesics in ASEP and speed-changed ASEP.

desk verdict Rigorous TASEP endpoint theorem plus clearly-labeled conjectures for non-integrable models; the unproved characteristic-line centering is the main gap. read the letter →

arxiv 2412.11626 v1 pith:MXTAZPMB submitted 2024-12-16 math.PR

Patrik L. Ferrari , Min Liu This is my paper · ORCID
classification math.PR MSC 60K3582C22
keywords TASEPASEPspeed-changedbackwardsgeodesicquasi-geodesicAiry_2processKPZuniversalityflatinitialcondition
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper establishes a fluctuation limit for the endpoint of the backwards geodesic in TASEP with flat initial condition: after subtracting the linear drift and rescaling by $t^{2/3}$, the endpoint converges in distribution to the maximizer of the Airy$_2$ process minus a parabola. The authors then define the same backwards-index construction for non-integrable exclusion processes — ASEP and speed-changed ASEP — calling the resulting trajectories quasi-geodesics, and derive precise scaling conjectures for their endpoints using KPZ scaling theory. All non-universal constants in those conjectures are computed analytically, so the predictions have no free parameters, and simulations at moderate times support them. The broader point is that the endpoint fluctuation is universal across integrable and non-integrable models in the KPZ class, and that the backwards geodesic provides a tractable observable to test that universality.

What carries the argument

The backwards geodesic and its index process: a trajectory built backwards in time from a chosen particle, switching labels whenever a suppressed jump of that particle is encountered, so that it tracks the space-time randomness that actually influences the particle's position. For TASEP this object satisfies the exact variational identity (2.9), $X_N(t)=\min_{M\le N}\{X^{\mathrm{step},X,M}_{N-M}(t)+X_M(0)\}$, which reduces the endpoint to the argmax of a rescaled height function $H_t$ that converges weakly to $A_2(u)-u^2$; tightness of the argmax then transfers the limit. For ASEP and speed-changed ASEP the same construction defines the quasi-geodesic, and KPZ scaling theory provides the centering and scaling constants through the stationary current $J(\rho)$, the integrated covariance $A(\rho)$, and $\Gamma(\rho)=-A(\rho)^2J''(\rho)$.

What would settle it

Run high-precision simulations of ASEP with $p$ close to $1/2$ (say $p=0.55$) and of speed-changed ASEP with different $\beta$ and $E$, at times where the $t^{-1/3}$ drift is resolved, and test whether the empirical endpoint distribution, after subtracting the paper's analytically computed centering, converges to $\hat{u}$ with the predicted decays of the first four moments; any $t^{2/3}$-scale residual drift or variance mismatch that persists at large $t$ would disconfirm Conjectures 2.7 and 2.8.

Watch

Extended reading notes

Core claim

In Theorem 2.6 the paper proves for TASEP with $X_n(0)=-2n$ that $(X_{N(t\downarrow 0)}(0)-X_N(0)-t/2)/(2^{1/3}t^{2/3})$ converges in distribution to $\hat{u}=\arg\max_{u\in\mathbb{R}}\{A_2(u)-u^2\}$, where $A_2$ is the Airy$_2$ process. It conjectures the identical limit for the quasi-geodesic endpoint in ASEP (Conjecture 2.7) and in speed-changed ASEP (Conjecture 2.8), with the centering $(p-q)t/2$ and the model-dependent but analytically known constants $J(\beta,E)$, $\Gamma(\beta,E)$, $A(\beta)$ respectively, and provides numerical evidence for both. It also conjectures and tests numerically that the one-point particle distribution in speed-changed ASEP is GOE Tracy-Widom, and that the ASEP discrepancy $D_N(t)$ — the difference between the quasi-geodesic reconstruction and the true particle position — converges to a non-degenerate random variable without rescaling.

Load-bearing premise

The conjectured centering assumes the quasi-geodesic tracks the characteristic line at the macroscopic speed of information propagation in the model, a property proved only for TASEP; if that heuristic fails for ASEP or speed-changed ASEP, the conjectured limiting law would be off-center even though its fluctuation shape might still be right.

Editorial extensions

If this is right

  • The endpoint of the backwards geodesic for flat TASEP has the same limit law as the endpoint of the point-to-line geodesic in exponential LPP, namely the argmax of $A_2$ minus a parabola.
  • If Conjectures 2.7 and 2.8 hold, then the endpoint fluctuation of quasi-geodesics in ASEP and speed-changed ASEP is universal and given by that same law, with no fitted constants.
  • Conjecture 2.9 implies that the one-point distribution of a tagged particle in speed-changed ASEP with flat initial condition is the GOE Tracy-Widom distribution $F_{\mathrm{GOE}}(2s)$.
  • Conjecture 2.11 states that for ASEP the difference between the quasi-geodesic value and the true particle position stays random but bounded as $t\to\infty$, meaning the minimization identity (1.2) fails only by an $O(1)$ error.
  • The numerical evidence suggests the convergence rates: for both ASEP and speed-changed ASEP, the empirical mean of the scaled endpoint approaches that of $\hat{u}$ at rate $t^{-1/3}$, while variance and skewness converge at rate $t^{-2/3}$.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The same KPZ scaling formula in Conjecture 3.9 should apply to other non-integrable exclusion-type models with known stationary measures; testing it on, say, the inclusion process or multi-species exclusion would separate the characteristic-line heuristic from the universal fluctuation law.
  • If Conjecture 2.11 is right, the ASEP height function is within $O(1)$ of a line-ensemble maximum, which suggests a concrete route to a rigorous ASEP analogue of the LPP variational formula.
  • The empirical shift of about $0.385$ in the speed-changed ASEP one-point function hints at a universal $O(1)$ correction to the KPZ scaling; a refined next-order theory could turn the numerical fits into a sharper test.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

1 major / 5 minor

Summary. The paper studies backwards geodesics (and their generalization, quasi-geodesics) in exclusion processes. For TASEP with flat initial condition X_n(0)=-2n, it proves Theorem 2.6: the endpoint of the backwards geodesic, after centering by t/2 and scaling by 2^{1/3}t^{2/3}, converges in distribution to argmax_{u}{A_2(u)-u^2}. The proof uses the variational identity (2.9), a distributional reduction to step initial conditions, weak convergence of the rescaled height function to A_2-u^2, and exponential tightness of the argmax. The paper then defines the same backwards index process for ASEP and speed-changed ASEP, derives (via KPZ scaling theory) explicit scaling conjectures for the endpoint (Conjectures 2.7 and 2.8), a GOE one-point conjecture for ASEPsc (Conjecture 2.9), and a conjecture that the ASEP discrepancy D_N(t) converges to a finite random variable (Conjecture 2.11). These are tested with 10^6-sample simulations.

Significance. If the conjectures are correct, the paper gives a clean and natural analogue for dynamical backwards geodesics of the point-to-line LPP geodesic endpoint, and it provides parameter-free KPZ predictions for non-integrable models. The rigorous TASEP theorem is a solid contribution, with tightness handled through large-deviation bounds rather than only by soft compactness arguments. The non-integrable part is also valuable: all non-universal coefficients are computed analytically from the known stationary measures, and the numerical evidence is extensive and power-law in time. The main weakness is that the centering of the endpoint conjectures rests on an unproved characteristic-line heuristic; the paper is honest in labelling the statements as conjectures, but the abstract's wording 'numerically verify' overstates the strength of the evidence for the centering.

major comments (1)
  1. [Section 3.2.2 / Conjectures 2.7 and 2.8] The centering in Conjectures 2.7 and 2.8 is derived from the assertion that 'the backwards geodesic mimics the characteristic line', giving the macroscopic velocity J(ρ)/ρ − J′(ρ) (Section 3.2.2, around equations (3.43)–(3.46)). For TASEP this follows from the equality (2.9), but for ASEP the equality is replaced by the inequality (2.12), and the index process of Definition 2.1 is not known to solve any variational principle whose implied velocity is the characteristic velocity. If the actual label drift of N(t↓0) differs from (J(ρ)/ρ − J′(ρ))t by a constant c, then the numerator in Conjectures 2.7 and 2.8 contains an extra −ct term and the rescaled variable diverges like c t^{1/3}; the conjectured limit would fail exactly through the centering, even if the t^{2/3} fluctuation scale and the Airy_2−u^2 shape were correct. The numerical evidence in Section 4.1.1 (empirical mean of B_t^{ASEP} decaying like t^{−1/3}) is consistent with c=0 for p=3/4, and Figure 13 gives similar evidence for one parameter choice of ASEPsc, but this does not prove the centering and covers only a single point in parameter space. I recommend that the paper explicitly state that the characteristic-line centering is an unproved additional assumption, and, if feasible, add a direct check of the macroscopic label drift (for example, regressing X_{N(t↓0)}(0)−X_N(0) on t) for more than one asymmetry or density.
minor comments (5)
  1. [Section 3.1.1, Lemma 3.2] The proof says 'we only need to show the case N=t/4', but Theorem 2.6 is stated for a fixed N; please add a sentence explaining that the scaled label difference is asymptotically independent of the fixed offset N, since the relevant backward label drifts by about t/4.
  2. [Section 4.1.2, Conjecture 2.11] The evidence for convergence of the unscaled discrepancy D_N(t) is suggestive but not conclusive: Table 1 shows the empirical mean increasing from 4.34 at t=400 to 4.95 at t=2000 and the variance increasing from 12.96 to 14.07, with no error bars or extrapolation. Please soften the wording and state more clearly that the data are consistent with, but do not establish, a finite limit.
  3. [Remark 4.1] There is a typo: 'KPT universality class' should be 'KPZ universality class'.
  4. [References] In reference [32], 'Eletron. J. Probab.' should be 'Electron. J. Probab.'; also, the BonnData repository reference [29] should include a stable URL or DOI to be useful.
  5. [Section 3.2.1, Equation (3.39)] The phrase 'there exist universal constants c1 and c2' followed by 'The constants are c1=2 and c2=1' is slightly redundant; consider clarifying that these constants are fixed by matching the TASEP case.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the TASEP theorem is proved from external convergence and tightness results, and the ASEP/ASEPsc statements are explicitly labeled conjectures with analytically computed coefficients rather than fitted restatements of the target.

full rationale

The paper's central rigorous result, Theorem 2.6, is not circular. The endpoint of the TASEP backwards geodesic is identified with the argmax of a TASEP height-function process via the distributional identity (3.5), proved in Lemma 3.2; the limiting weak convergence of that process to A_2(u)-u^2 is imported from the independent result of [15], and tightness is proved in Lemma 3.3. The limit law therefore comes from established external inputs, not from assuming the answer. For ASEP and ASEPsc, the paper formulates Conjectures 2.7 and 2.8 by applying the KPZ scaling theory: universal constants c1, c2 are calibrated once from the TASEP result (using [16]), while the model-dependent coefficients J, A, Gamma are computed analytically from the stationary measures (Lemma 3.10). The endpoint prediction is not used to fit any of these parameters, so it is not a fitted input called a prediction. The characteristic-line centering in Section 3.2.2 is explicitly a heuristic assumption ("the backwards geodesic mimics the characteristic line"), and the paper flags the unsolved discrepancy for ASEP; an unproved assumption is a correctness risk, not circularity. Numerical checks compare with the known density of argmax{A_2-u^2} obtained from the independent formula of [35]; no parameter is fitted for the geodesic endpoint. The only empirically adjusted constant, 0.385 in Remark 4.1, is an explicitly finite-time shift in an auxiliary GOE one-point check and is not used to derive or verify Conjectures 2.7/2.8. Self-citations to prior work of the authors (e.g., [26], [28], [15], [16], [20]) supply established tools and estimates, not the paper's own conclusions, and none of those citations smuggles in the target result. Accordingly, no circular step can be exhibited, and the appropriate score is 0.

Assumptions & free parameters 1 free parameters · 5 assumptions · 0 invented entities

The central theorem relies on standard TASEP/LPP estimates and the uniqueness of the Airy_2 argmax. The conjectures for non-integrable models additionally assume the KPZ scaling theory and a characteristic-line heuristic for the quasi-geodesic; these are the main extra postulates.

free parameters (1)
  • mean shift constant = 0.385
    In Remark 4.1, Eq. (4.6), this constant is estimated from the empirical mean offset of the rescaled speed-changed ASEP particle position and subtracted to improve convergence of the GOE check. It is not used for the endpoint conjectures.
assumptions (5)
  • domain assumption Assumption 3.7: spatially ergodic and time stationary measures are labeled by average density rho, with |rho| <= 1.
    Used in Section 3.2 to define J(rho), A(rho), Gamma(rho) and to apply the KPZ scaling theory to ASEP and ASEPsc.
  • domain assumption The KPZ scaling theory universality (Conjectures 3.8 and 3.9): rescaled height functions and backward geodesic endpoints converge to universal objects with constants c1=2, c2=1 determined from TASEP.
    Invoked in Section 3.2 to derive the explicit centering and scaling for Conjectures 2.7, 2.8, and 2.9.
  • ad hoc to paper The backwards geodesic follows the characteristic line with macroscopic velocity J'(rho).
    Section 3.2.2, around Eq. (3.44), this heuristic gives the law of large numbers term (J(rho)/rho - J'(rho))t used in Conjecture 3.9; it is not proved for non-integrable models.
  • standard math Large deviation and tightness estimates for TASEP and LPP from [2] and [28].
    Used in the proofs of Lemma 3.3 and Lemma 3.6 to bound the probability of the geodesic endpoint falling outside [-L,L]; these are external published results.
  • standard math Uniqueness of the maximizer of A_2(u) - u^2.
    Used in Proposition 3.5 to convert weak convergence of the process to convergence of the argmax; established in [21].

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Cite this review

Pith. "Pith review of Quasi-geodesics in integrable and non-integrable exclusion processes." pith.science (2026). https://pith.science/paper/MXTAZPMB

@misc{pith2026241211626,
  author       = {Pith},
  title        = {Pith review of: Quasi-geodesics in integrable and non-integrable exclusion processes},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/MXTAZPMB}},
  note         = {Machine review of arXiv:2412.11626}
}
abstract

Backwards geodesics for TASEP were introduced in [Fer18]. We consider flat initial conditions and show that under proper scaling its end-point converges to maximizer argument of the Airy$_2$ process minus a parabola. We generalize its definition to generic non-integrable models including ASEP and speed changed ASEP (call it quasi-geodesics). We numerically verify that its end-point is universal, where the scaling coefficients are analytically computed through the KPZ scaling theory.

Figures

Figures reproduced from arXiv: 2412.11626 by the authors.

Figure 1
Figure 1. Illustration for jump rate in ASEP. In the setting of ASEP, [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Illustration for jump rate of process ASEPsc. The left fig [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Illustration for backwards geodesic in ASEP setting for th [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figures from the paper (10 more)
Figure 4
Figure 4. Figure 4: Illustration for Lemma 3.1. On the left panel: from left to right, the black solid lines represent the trajectory of particles X2, X1, X0, X−1, and X−2 respectively. The green line is the backwards trajectory of particle X1 from time t. The end-point of this trajectory…
Figure 5
Figure 5. Figure 5: The solid lines represent particle trajectories, with the blu [PITH_FULL_IMAGE:figures/full_fig_p019_5.png]
Figure 6
Figure 6. Figure 6: Comparison between the density function of [PITH_FULL_IMAGE:figures/full_fig_p024_6.png]
Figure 7
Figure 7. Figure 7: Log-log plot for the statistics of BASEP t . The square, triangle, and circle markers represent the points corresponding to (ti , Eb(BASEP t )), (ti , Var( d BASEP t )), and (ti , Skew( [ BASEP t )), respectively, where ti ∈ {1000, 1500, . . ., 5000}. The blue, red, an…
Figure 8
Figure 8. Figure 8: Probability density function fDn(t) in ASEP with p = 3/4. Left panel: Ori￾ginal density function (x, fDn(t)(x)). Right panel: Log-linear plot (x, log fDn(t)(x)). The green, red, and blue points in the figure correspond to t = 200, t = 1000, and t = 2000, respectively. …
Figure 9
Figure 9. Figure 9: Numerical verification for Conjecture 2.9. The red, green, and blue dots represent the density function of Xresc n (t) with t ∈ {200, 600, 1000}. The black solid line is given by (s, 2F ′ GOE(2s)). The number of trials is 106 . We use the same notation as in (4.3), tha…
Figure 10
Figure 10. Figure 10: Log-log plot for the absolute difference between the sta [PITH_FULL_IMAGE:figures/full_fig_p027_10.png]
Figure 11
Figure 11. Figure 11: The red, green, and blue dots represent the density fu [PITH_FULL_IMAGE:figures/full_fig_p027_11.png]
Figure 12
Figure 12. Figure 12: Numerics for Conjecture 2.8. Left panel: The green, red, and blue dots represent the density function of BASEPsc t with t ∈ {200, 600, 1000}. Right panel: The green, red, and blue dots represent the density function of BˆASEPsc t with t ∈ {200, 600, 1000}. The black s…
Figure 13
Figure 13. Figure 13: , we plot the log-log plots of the first four statistics as compared with the ones of ˆu. We see that the expectation of BASEPsc t converges as t −1/3 , while the other three statistics converges faster, namely as t −2/3 . 400 600 800 1000 0.01 0.02 0.05 0.10 0.20 0.5…

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Reviewed August 11, 2026 · model on record in the stance chip above.