{"id":"641bc4c1-c7db-42aa-a4ea-e686db5a634e","arxiv_id":"2412.11626","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":1,"one_line_summary":"For TASEP, the backward geodesic endpoint converges to the argmax of Airy_2 minus a parabola, and the same universal law is conjectured and numerically supported for ASEP and speed-changed ASEP.","lead":"This paper proves that for totally asymmetric exclusion processes starting from a flat density, the endpoint of the backward geodesic converges, after scaling, to the location where the Airy_2 process minus a parabola attains its maximum. It then defines quasi-geodesics for non-integrable exclusion processes, including ASEP and speed-changed ASEP, and presents numerical evidence that the same limit law holds there.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The non-integrable conjectures rest on an unproved characteristic-line centering; failure would break Conjectures 2.7/2.8 even if the t^{2/3} fluctuation scale is correct.","rationale":"The reader's weakest assumption identifies exactly the same load-bearing concern: the centering in Conjectures 2.7 and 2.8 is inherited from the unproved statement that the quasi-geodesic follows the characteristic line in the macroscopic limit. I agree that this is the main point on which the non-integrable claims could fail while the rest of the paper survives. Credit should be given where it is due: Theorem 2.6 is a genuine rigorous result for TASEP, the proof strategy via the identity with the argmax of the rescaled step-current process is coherent, and the numerical simulations are extensive and show decay of finite-time corrections at plausible rates. The concern is not that the conjectures are false, but that their derivation contains a heuristic step that is not proven and is not fully settled by the numerics, since the simulations cover a finite set of times and model parameters and cannot exclude a small nonzero linear drift. This warrants a conditional verdict rather than rejection: the central TASEP theorem stands, and the conjectures are credible and well supported, but the precise analytic centering for ASEP and ASEPsc is not rigorously established. No change to the reader's conditional verdict is needed.","tokens_in":26384,"tokens_out":12843,"duration_ms":130064,"concrete_test":"For ASEP at p = 3/4, simulate the suppression-based endpoint M_s(t) = N(t↓0) and, on the same clock realization, the variational endpoint M_v(t) = argmin_{M≤N}{X^{step,X,M}_{N−M}(t) + X_M(0)}, with M restricted to a window of width O(t^{2/3}) around the predicted characteristic label N − (p−q)t/4. Record the rescaled difference (M_v(t) − M_s(t)) / t^{2/3} and the means of both endpoints at t = 2000, 4000, 8000. If M_v and M_s do not converge to the same t^{2/3} fluctuation, or if either endpoint's raw mean drifts linearly away from (p−q)t/4, then the characteristic-line centering in Conjecture 2.7 is not the correct macroscopic velocity of the quasi-geodesic.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The load-bearing gap is in Section 3.2.2, where the centering for the quasi-geodesic endpoint is derived from the assertion that 'the backwards geodesic mimics the characteristic line,' giving macroscopic velocity J(ρ)/ρ − J′(ρ). For TASEP this follows from the variational identity (2.9). For ASEP, however, (2.9) is replaced by the inequality (2.12), and the discrepancy D_N(t) in (2.13) is strictly positive; the suppression-based 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)) by a nonzero constant c, then the rescaled variable in Conjectures 2.7 and 2.8 contains an extra −c t term and diverges proportionally to c t^{1/3}; the proposed Airy_2−u^2 limit would fail exactly through the centering, even if the t^{2/3} fluctuation order and the GOE-type shape were correct. The numerical evidence in Section 4.1.1 shows the empirical mean of the rescaled endpoint decaying like t^{−1/3}, which is consistent with the conjectured centering and is strong evidence, but it is not a proof and covers only p = 3/4 for ASEP and one parameter choice for ASEPsc. Conjecture 2.11, that D_N(t) converges to a finite random variable, does not by itself control this drift, since D_N(t) is defined relative to the step process and does not fix the label drift of the quasi-geodesic.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":26779,"tokens_out":13876,"duration_ms":123646,"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":[{"comment":"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.","section":"Section 3.2.2 / Conjectures 2.7 and 2.8"}],"minor_comments":[{"comment":"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.","section":"Section 3.1.1, Lemma 3.2"},{"comment":"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.","section":"Section 4.1.2, Conjecture 2.11"},{"comment":"There is a typo: 'KPT universality class' should be 'KPZ universality class'.","section":"Remark 4.1"},{"comment":"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.","section":"References"},{"comment":"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.","section":"Section 3.2.1, Equation (3.39)"}],"recommendation":"major_revision","confidential_remarks":"The TASEP theorem appears sound and is a genuine contribution. The main risk in the paper is the status of the centering in the non-integrable conjectures: it is derived from a heuristic 'backwards geodesic mimics the characteristic line' that is not proved and is not a consequence of the inequality (2.12). Because the abstract claims numerical verification of the universal limit, this needs to be addressed either by an explicit caveat or by stronger direct numerical tests of the label drift. I would not recommend rejection, since the conjectures are clearly labelled and the theorem is rigorous, but the revision should make the conditional nature of the centering prominent."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"This paper delivers a rigorous theorem and a set of clearly-labeled conjectures. The theorem — for TASEP with flat initial condition, the endpoint of the backwards geodesic, scaled by t^{2/3}, converges to the argmax of the Airy_2 process minus a parabola — is new and the proof looks correct. The strategy is sensible: map the endpoint to the argmax of a rescaled step-initial-condition process, show tightness via large deviation bounds, and rely on known uniqueness of the argmax. The extension of the backwards geodesic to non-integrable models (ASEP and speed-changed ASEP) as 'quasi-geodesics' is a genuine contribution, and the scaling constants are computed analytically through KPZ theory, so the main conjectures have no free parameters. The numerical work is extensive (10^6 samples) and shows power-law convergence consistent with the conjectures.\n\nThe main soft spot is the centering in Conjectures 2.7 and 2.8. Section 3.2.2 assumes the backwards geodesic mimics the characteristic line, giving a macroscopic velocity J(ρ)/ρ − J′(ρ). For TASEP this follows from a variational identity; for ASEP, the identity is only an inequality, and the quasi-geodesic is not known to solve any variational principle. If the label drift differs from the conjectured velocity by a nonzero constant, the rescaled variable in those conjectures would pick up a term growing like t^{1/3} and the Airy_2–parabola limit would fail. The numerical evidence shows the empirical mean decaying like t^{−1/3}, which is consistent with the conjectured centering and is strong evidence, but it is not a proof and covers only p = 3/4 for ASEP and one parameter set for ASEPsc. This is a real gap, but the authors are transparent that these are conjectures.\n\nMinor concerns: the discrepancy conjecture (2.11) is supported by statistics that drift slowly, so the evidence is suggestive rather than conclusive. Also, Remark 4.1 fits a shift of 0.385 in the GOE check; this is auxiliary and disclosed, but it is a fitted constant.\n\nThe citation pattern is appropriate; the proof of the TASEP theorem is careful and self-contained enough. The paper honestly separates theorem from conjecture.\n\nWho it is for: researchers in KPZ universality, exclusion processes, and geodesics. The rigorous result alone is citable; the conjectures give a concrete target for future work.\n\nRecommendation: send it to peer review. A referee can push on the centering and ask for more justification or at least a sharper heuristic, but the paper deserves referee time.","headline":"Rigorous TASEP endpoint theorem plus clearly-labeled conjectures for non-integrable models; the unproved characteristic-line centering is the main gap.","tokens_in":27254,"tokens_out":3053,"would_cite":true,"duration_ms":25387,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["60K35","82C22"],"pacs":[],"model":"deepseek-v4-flash","headline":"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.","keywords":["TASEP","ASEP","speed-changed ASEP","backwards geodesic","quasi-geodesic","Airy_2 process","KPZ universality","flat initial condition"],"falsifier":"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.","tokens_in":26213,"feed_emoji":"🎲","tokens_out":9541,"duration_ms":77350,"temperature":0.7,"pith_summary":"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.","feed_headline":"Backwards geodesic endpoints obey one universal law","feed_subtitle":"The same Airy-2 argmax law governs TASEP and, by simulation, non-integrable exclusion processes.","key_machinery":"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 $\\Gamma(\\rho)=-A(\\rho)^2J''(\\rho)$.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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 $t^{-2/3}$."],"supporting_citations":[{"why":"Introduces the backwards geodesic and index process for TASEP, the construction this paper extends to other exclusion processes.","marker":"[26]"},{"why":"Provides the weak convergence of the rescaled height function $H_t$ to $A_2(u)-u^2$ used in the proof of Theorem 2.6.","marker":"[15]"},{"why":"Gives the limit law of the argmax of $A_2$ minus a parabola, the target distribution $\\hat{u}$.","marker":"[39]"},{"why":"Proves uniqueness of the argmax of $A_2(u)-u^2$, needed to transfer weak convergence of processes to convergence of the argmax.","marker":"[21]"},{"why":"Defines the Airy$_2$ process, the limiting object in the endpoint distribution.","marker":"[47]"},{"why":"Supplies the KPZ scaling theory used to derive the parameter-free centering and scaling constants for the conjectures.","marker":"[54]"},{"why":"Provides the stationary measures and hydrodynamic equation for speed-changed ASEP, from which $J(\\rho)$ and $A(\\rho)$ are computed.","marker":"[40]"},{"why":"Establishes convergence of ASEP height functions to the KPZ fixed point, used to prove that the rescaled minimization discrepancy vanishes in Lemma 2.10.","marker":"[48]"}],"fun_headline_variants":["Backwards geodesic endpoints obey one universal law","Same Airy-2 argmax law for TASEP and quasi-geodesics","Universal endpoint law: from TASEP to non-integrable exclusion","Quasi-geodesics: one endpoint law for integrable and non-integrable","TASEP proof, ASEP simulation: one geodesic endpoint law"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Backwards geodesic endpoints obey one universal law","Same Airy-2 argmax law for TASEP and quasi-geodesics","Universal endpoint law: from TASEP to non-integrable exclusion","Quasi-geodesics: one endpoint law for integrable and non-integrable","TASEP proof, ASEP simulation: one geodesic endpoint law"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000815,"raw_usage":{"total_tokens":3532,"prompt_tokens":864,"completion_tokens":2668,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":480,"completion_tokens_details":{"reasoning_tokens":2569}},"tokens_in":480,"tokens_out":2668,"duration_ms":17591,"temperature":1.0,"reasoning_tokens":2569,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T14:45:04.148783+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"Ferrari, Finite GUE distribution with cut-off at a shock , J","cited_arxiv_id":null,"evidence_quote":"Introduces the backwards geodesic and index process for TASEP, the construction this paper extends to other exclusion processes."},{"cited_title":"Borodin, A","cited_arxiv_id":null,"evidence_quote":"Provides the weak convergence of the rescaled height function $H_t$ to $A_2(u)-u^2$ used in the proof of Theorem 2.6."},{"cited_title":"Johansson, Discrete polynuclear growth and determinantal processes , Com- mun","cited_arxiv_id":null,"evidence_quote":"Gives the limit law of the argmax of $A_2$ minus a parabola, the target distribution $\\hat{u}$."},{"cited_title":"Corwin and A","cited_arxiv_id":null,"evidence_quote":"Proves uniqueness of the argmax of $A_2(u)-u^2$, needed to transfer weak convergence of processes to convergence of the argmax."},{"cited_title":"Spohn, KPZ scaling theory and the semidiscrete directed polymer mo del, Random Matrix Theory, Interacting Particle Systems and Integra ble Systems 65 (2014), 483–493","cited_arxiv_id":null,"evidence_quote":"Supplies the KPZ scaling theory used to derive the parameter-free centering and scaling constants for the conjectures."},{"cited_title":"Krug and H","cited_arxiv_id":null,"evidence_quote":"Provides the stationary measures and hydrodynamic equation for speed-changed ASEP, from which $J(\\rho)$ and $A(\\rho)$ are computed."},{"cited_title":"Quastel and S","cited_arxiv_id":null,"evidence_quote":"Establishes convergence of ASEP height functions to the KPZ fixed point, used to prove that the rescaled minimization discrepancy vanishes in Lemma 2.10."}],"review_version":1}