{"id":"9aac0372-a2b4-4d2d-b5eb-aab65b4d3e92","arxiv_id":"2507.11537","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Open ASEP height functions with general local boundary asymmetries converge to the open KPZ equation without Liggett's condition or explicit invariant measures.","lead":"A rigorous proof shows that a family of particle systems, open ASEP with boundary reservoir speeds that depend on nearby particles, converges to the open KPZ equation. This removes a restrictive technical condition used in all previous derivations, answering a question posed by Corwin and Himwich.","discovery_kind":"first_principles","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Boundary sign conventions in Lemmas 3.4–3.5 and Eq. (2.8) break the O(N^{3/2}) cancellation required for Theorem 2.1.","rationale":"The reader's concern about the homogenization estimates in Lemma 5.2 and Proposition 5.6 is reasonable: those estimates are the technical heart of the boundary error control. However, before those estimates are even used, the boundary evolution equations must be internally consistent. As typeset, the paper contains a more immediate, load-bearing issue: the sign of the boundary height increment and the corresponding jump factors in Lemmas 3.4-3.5 appear to prevent the O(N^{3/2}) terms from canceling, and (2.8) as printed conflicts with the mean-zero condition for f_left used to apply Proposition 5.6. These are internal inconsistencies rather than disagreement with prior consensus. If a direct re-derivation shows the printed signs are simply transcription errors, the main argument may still be valid after correction. But as submitted, the boundary parameter formulas and the martingale identification are not yet secure. Hence the verdict remains conditional, matching the reader's overall assessment, though for an additional and more elementary reason.","tokens_in":38774,"tokens_out":38086,"duration_ms":414583,"concrete_test":"Recompute Lemma 3.4 directly: apply the generator (2.1)-(2.4) to the explicit function Z^N_{t,0} in (2.5)-(2.6), using the stated h^N_{t,0} convention, and collect the coefficient of N^{3/2} in dZ^N_{t,0}. If the coefficient is not zero, (3.6) is missing a term and the formulas (2.8)-(2.9) for A,B must be revised; if it is zero only after changing the sign of the flip factor or of h^N_{t,0}, then the model in Section 2 and the boundary evolution equations are inconsistent as printed.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"Under the definition in Section 2, h^N_{t,0}=2N^{-1/2}(#removed - #created), so a flip at site 1 from -1 to +1 decreases h^N_{t,0}; hence Z^N_{t,0} should jump by exp(+2 lambda N^{-1/2}). Expanding the boundary terms in (3.10) gives an O(N^{3/2}) contribution -(lambda/2) eta_{t,1} Z^N_{t,0}, the same sign as the O(N^{3/2}) contribution from the Robin Laplacian in (3.9); the two do not cancel. The proof nevertheless writes (3.6) with only O(N) drift, so either (3.6) is missing an O(N^{3/2}) term or a sign in Section 2/3 is wrong. The same issue appears at the right boundary: creation at N increases h^N_{t,N} by 2N^{-1/2}, so the jump factor in Lemma 3.5 should be exp(-2 lambda N^{-1/2}), not the stated Exp(+2 lambda N^{-1/2}). In addition, Theorem 2.1's (2.8) is typeset with E0{eta_1 (alpha-gamma)} while Lemma 3.4's f_left and the surrounding discussion require E0{eta_1 (alpha+gamma)}; if (2.8) is literal, f_left is not mean-zero under P0 and Proposition 5.6 cannot be applied to R^N_{1,t}. Since the boundary parameters A,B and the martingale identification depend on these signs, the central convergence claim is not established as written, though a sign correction may repair it.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies open ASEP on {1,...,N} and on the half-line, with reservoir rates whose N^{3/2}-order coefficients are arbitrary local functions α, β, γ, δ of the configuration near the boundary. The main result (Theorem 2.1) asserts that the Gärtner-transformed height function converges in law to the solution of the open stochastic heat equation on [0,1] with Robin boundary parameters A, B given by (2.8)-(2.9), equivalently that the height function converges to the open KPZ equation. The proof proceeds by deriving an evolution equation for Z^N (Section 3), proving tightness (Lemma 4.1), reducing the limit identification to vanishing of boundary error terms (Proposition 4.4), and estimating those terms via entropy/Dirichlet-form estimates and a Kipnis-Varadhan inequality with respect to the non-invariant product measure P0 (Sections 5-6). Corollary 2.2 deduces convergence of stationary measures of the increment process, and Theorem 2.3 gives the half-space analog.","tokens_in":39113,"tokens_out":44355,"duration_ms":453734,"significance":"If the proof can be repaired, this is a substantial step: it removes Liggett's condition for open ASEP, removes product-invariance assumptions for speed-change boundary dynamics, answers the cited open questions of Corwin and Himwich, and introduces a useful technical framework (Kipnis-Varadhan estimates with respect to a non-invariant measure on fattened boundary intervals). The paper is largely self-contained in its main estimates and identifies A and B parameter-free through expectations under P0. However, the right-boundary evolution equation and the displayed formulas for A and B contain sign/type inconsistencies that currently break the proof at load-bearing points; the central convergence claim is therefore not established as written.","major_comments":[{"comment":"The jump factors at the right boundary are inconsistent with the height definition (2.5). Under (2.5), a flip at site N from η_N = -1 to +1 increases h^N_{t,N} by 2N^{-1/2}, so Z^N_{t,N} should multiply by Exp(-2λN^{-1/2}) - 1; the lemma instead uses Exp(+2λN^{-1/2}) - 1, and the two factors in (3.12)-(3.13) are reversed. With the stated factors, the O(N^{3/2}) part of the flip drift is -λ/2 η_N Z^N_{t,N}, while the discrete Robin Laplacian (3.1) at x=N contributes +λ/2 η_N Z^N_{t,N}; the difference contains an O(N^{3/2}) term that cannot be absorbed into the O(N) term λN f_right Z^N_{t,N}. Lemma 3.5 is therefore false as written, and the right-boundary error R^N_{2,t} in Proposition 4.4 is not controlled.","section":"Section 3, Lemma 3.5, Eqs. (3.11)-(3.13)"},{"comment":"The last expectation in (2.8) is printed as E0{η1 (α[η] - γ[η])}, but f_left in Lemma 3.4 is 3λ/4 + α - γ - η1(α + γ) - A/2. With the printed formula one obtains E0 f_left = -2E0[η1 γ], which is generically nonzero because γ may depend on η1 near the boundary. The mean-zero condition required by Proposition 5.6 for applying it to R^N_{1,t} therefore fails. The sentence immediately after (2.8) states the intended function is 2(α - γ) - 2η1(α + γ), so the last term in (2.8) should read -2E0{η1 (α + γ)}; this must be corrected in the statement of the main theorem.","section":"Theorem 2.1, Eq. (2.8)"},{"comment":"Even taking the manuscript's stated f_right = 3λ/4 + δ - β - η_N(δ + β) - B/2 and the value of B in (2.9), one gets E0 f_right = 3λ/2 - 2E0β, which is not zero in general. Hence R^N_{2,t} cannot be handled by Proposition 5.6 either. This is not an isolated typo: the right-boundary sign error in (3.12)-(3.13) changes the derivation of f_right, so (2.9) and the definition of f_right must be rederived consistently with the height convention (2.5).","section":"Eqs. (2.9) and Lemma 3.5"},{"comment":"The stress-test concern about the left boundary O(N^{3/2}) cancellation does not actually land: in (3.6) the term (1/2)N^2 Δ_{A,B} Z^N_{t,0} appears on the right-hand side, and both this term and the flip drift in (3.10) contain -λ/2 N^{3/2} η_{t,1} Z^N_{t,0}; their difference is O(N), which is exactly what the definition of f_left requires. However, the prose in the proof of Lemma 3.4 says that a flip from -1 to +1 makes h^N_{t,0} go up by 2λN^{-1/2}, whereas under (2.5) it goes down; the displayed jump factor in (3.7) is consistent with (2.5), but the sentence is misleading and should be corrected.","section":"Section 3, Lemma 3.4 proof"}],"minor_comments":[{"comment":"The function f_right is labeled f_left in the displayed definition; the name should be f_right.","section":"Lemma 3.5, bullet after (3.11)"},{"comment":"The exponent written as N^{-1/2 ρ} is ambiguous; if it means N^{-ρ/2}, please write it as such.","section":"Section 6, Eq. (6.7)"},{"comment":"The notation P0 is used both for the initial density and for the reference measure; switching to a different symbol for one of these would improve clarity.","section":"Lemma 5.4 proof"},{"comment":"The symbols w_left/w_right and m_left/m_right are used inconsistently for the same object; please unify the notation.","section":"Section 4, Eqs. (4.5)-(4.7)"}],"recommendation":"major_revision","confidential_remarks":"The sign errors are localized and arguably fixable, but they affect the statements of Theorem 2.1 and Lemma 3.5 as well as the boundary parameters A and B, so the author should be asked to rederive the boundary evolution equations and the formulas (2.8)-(2.9) from scratch. The half-space theorem inherits the (2.8) issue, and Corollary 2.2 depends on the corrected boundary parameters. I would not recommend acceptance until the right-boundary signs and the mean-zero identities for f_left and f_right are verified in the revised manuscript."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Here is my read. The result is important if true: it removes Liggett's condition in both interval and half-space and gives a new formula for the boundary parameters. The technical engine is real and substantial: relative entropy with respect to a non-invariant product measure, the Kipnis-Varadhan bound for time-averaged mean-zero local functions on a fattened boundary interval, and the localization coupling are all nontrivial, and they are presented with enough detail for a specialist to follow. Credit where due.\n\nBut the stress-test's sign concern is correct, and it is load-bearing. With h^N_{t,0} defined as 2N^{-1/2}(#removed - #created), a flip at site 1 from -1 to +1 decreases h^N_{t,0}, so the jump factor in Lemma 3.4's Q_{t,0} is exp(+2λN^{-1/2}). Expanding the boundary drift in (3.10) then gives -λ/2 N^{3/2} η_{t,1} Z^N_{t,0}. The discrete Robin Laplacian in (3.9) contributes the same sign, -λ/2 N^{3/2} η_{t,1} Z^N_{t,0}. The two add instead of cancel, so (3.6) is missing an O(N^{3/2}) drift. The same issue appears at the right boundary: creation at N increases h^N_{t,N} by 2N^{-1/2}, so the jump factor in Lemma 3.5 should be exp(-2λN^{-1/2}), not the stated Exp(+2λN^{-1/2}). This is not cosmetic; the uncancelled term is too large to vanish after space-time averaging, and the identification of A and B collapses. The boundary parameter formula in (2.8) also has a typo: it should be E0{η_1(α+γ)} in the last term, otherwise f_left is not mean-zero under P0 and Proposition 5.6 cannot be applied. These errors are likely fixable by changing the sign convention for h^N_{t,0} or the boundary speeds, but as written the proof of Theorem 2.1 does not go through.\n\nThe half-space proof is explicitly a sketch, and Corollary 2.2 delegates a key step to [4]; those are secondary compared to the sign issue. If the signs are corrected and the cancellation restored, this is a strong paper. As it stands, the main theorem is not proven as written, and a referee should not pass it without a substantial revision.\n\nRecommendation: send to peer review, because the result and the technique matter, but insist on a careful correction of the signs and a re-derivation of (3.6) and the right-boundary analogue.","headline":"Important if true, but a verifiable sign error breaks the central cancellation; the main theorem is not proven as written.","tokens_in":39672,"tokens_out":13161,"would_cite":false,"duration_ms":128355,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["60K35","82C22","60H15"],"pacs":[],"model":"deepseek-v4-flash","headline":"Open ASEP with arbitrary local boundary speeds converges to the open KPZ equation, removing Liggett's condition.","keywords":["open KPZ equation","open ASEP","boundary speed changes","Gartner transform","Cole-Hopf transform","non-invariant measure","Kipnis-Varadhan estimate","stationary measure"],"falsifier":"Simulate the boundary-localized process on a fattened interval of length $N^\\kappa$ for parameters outside Liggett's condition, and measure the second moment of the time-average of $f_{\\mathrm{left}}$ over $\\tau = N^{-2+\\rho}$: Proposition 5.6 predicts decay $N^{-\\rho} + N^{-1/3}$. Alternatively, for a fixed non-Liggett choice of $\\alpha,\\beta,\\gamma,\\delta$, compute the stationary height increment measure of open ASEP numerically on large intervals and compare it with the open KPZ stationary measure with parameters (2.8)-(2.9); a mismatch at leading order would refute Corollary 2.2.","tokens_in":38541,"feed_emoji":"📈","tokens_out":9184,"duration_ms":94586,"temperature":0.7,"pith_summary":"This paper proves that the open ASEP particle system on a finite interval, with reservoir dynamics whose speeds are general local functions of the configuration, has a height function whose exponential (Gärtner transform) converges in law to the solution of the open stochastic heat equation; equivalently the height function itself converges to the open KPZ equation. The boundary parameters of the limiting SPDE are explicit expectations of the boundary speed functions under a product measure that is not invariant for the dynamics. This removes the longstanding Liggett condition (constant equal boundary creation and removal speeds) required in earlier derivations, and also removes the assumption of product invariant measures. The same result is proved for the half-space, and for the interval the stationary measure of the height increment process is shown to converge to that of the open KPZ increment process.","feed_headline":"Open ASEP hits open KPZ without Liggett's condition","feed_subtitle":"General local boundary speeds still converge to the same universal SPDE, with explicit boundary parameters.","key_machinery":"The central object is the Gärtner transform $Z^N_{t,x} = \\exp\\{-\\lambda h^N_{t,x} + (\\tfrac{\\lambda^2}{2}N + \\tfrac{\\lambda^4}{24})t\\}$, the discrete analogue of the Cole-Hopf map that turns the height equation into a multiplicative stochastic heat equation. The paper derives its infinitesimal evolution: the bulk ASEP generator plus boundary contributions containing fluctuating terms $f_{\\mathrm{left}}$ and $f_{\\mathrm{right}}$ that are mean-zero with respect to the non-invariant product measure $P_0$, together with bounded remainder terms. The main work is homogenizing these boundary fluctuations without an invariant measure; the engine is a Kipnis-Varadhan-type estimate (Proposition 5.6) for time-averaged mean-zero local functions on a fattened boundary interval, built on entropy-production bounds (Lemma 5.2), log-Sobolev and one-block estimates (Lemma 5.4), and a semigroup comparison to the symmetric generator that preserves $P_0$ (Lemma 5.7).","core_discovery":"The central claim is that, under the stated moment and Lipschitz bounds on the initial data, the linearly interpolated Gärtner transform $Z^N_{t,NX}$ converges in law in $D([0,1],C([0,1]))$ to the solution of the open stochastic heat equation with boundary parameters $A$ and $B$ given by the explicit expectations displayed in (2.8)-(2.9); equivalently, the height function converges to the open KPZ equation. This holds for general local functions $\\alpha,\\beta,\\gamma,\\delta$ governing the reservoir speeds, without Liggett's condition $\\alpha=\\gamma$, $\\delta=\\beta$ and without any product invariant measure for open ASEP. The half-space version with only a left reservoir converges to the half-space open KPZ equation with the same parameter $A$. The paper further proves that the stationary measure of the height increment process converges weakly to the unique stationary measure of the open KPZ increment process when $\\lambda=-1$.","pith_inferences":["If the theorem is right, the same homogenization mechanism should apply when the bulk asymmetry is of order $N$ rather than $N^{3/2}$, or when the boundary functions depend on longer but still mesoscopic windows, suggesting that the open KPZ universality class is stable under an open set of boundary-speed perturbations.","A testable extension is to compute finite-dimensional stationary correlations of the increment process and compare them with the known open KPZ stationary correlations for parameters outside Liggett's condition; Corollary 2.2 predicts they match at leading order.","The appearance of $P_0$, not the true invariant measure, in the boundary parameters suggests that for boundary-driven systems the continuum limit can be read off from the local equilibrium distribution even when global equilibrium is non-product; this principle may extend to other boundary-driven exclusion processes with speed changes."],"forward_implications":["Earlier open-ASEP-to-open-KPZ derivations required Liggett's condition (constant $\\alpha=\\gamma$ and $\\delta=\\beta$); this result removes it, answering the question posed in [3] and [14].","The limiting boundary parameters are computed from expectations under the non-invariant product measure $P_0$, so the open KPZ fixed point organizes a whole family of boundary dynamics, not only those with simple invariant measures.","The half-space open KPZ equation is also derived from half-space open ASEP with general local left-boundary speeds, without Liggett's condition.","The stationary measure of the height increment process converges weakly to the unique stationary measure of the open KPZ increment process (for $\\lambda=-1$), so equilibrium fluctuations of the particle system match those of the continuum SPDE.","The proof provides a template for boundary homogenization when invariant measures are unavailable, extending beyond product-measure and constant-boundary settings."],"supporting_citations":[{"why":"Establishes the open-ASEP weak-asymmetry convergence framework, the martingale problem characterization, and the tightness strategy that this paper adapts.","marker":"[5]"},{"why":"Supplies the heat kernel estimates for the discrete Robin Laplacian and the moment/tightness bounds used in Lemma 4.1.","marker":"[24]"},{"why":"Previous derivation of a boundary KPZ-type equation assuming product invariant measures; this paper removes that assumption.","marker":"[12]"},{"why":"Provides the non-invariant homogenization technique for speed-change exclusion processes that the author adapts to the boundary setting.","marker":"[27]"},{"why":"Provides the coupling construction used to prove convergence of stationary measures of the increment process in Corollary 2.2.","marker":"[4]"},{"why":"Supplies the standard entropy inequalities, log-Sobolev bounds, and Kipnis-Varadhan inequalities that underlie the stochastic estimates in Section 5.","marker":"[16]"}],"fun_headline_variants":["Open ASEP without Liggett's condition reaches open KPZ","General reservoir speeds in open ASEP still yield KPZ","No Liggett's condition: open ASEP converges to KPZ","Open ASEP with general boundary asymmetry flows to KPZ","Liggett-free open ASEP limit is open KPZ equation"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof depends on the assumption that near the boundary the dynamics mixes fast enough relative to the non-invariant product measure $P_0$: specifically, that the time-integrated discrepancy from $P_0$ is $O(N^{-1/2})$ and that time-averaged mean-zero local boundary functions decay at the stated rates; if either decay is slower, the boundary error terms survive and the limiting boundary parameters would change.","fun_headline_variants_meta":{"raw":{"variants":["Open ASEP without Liggett's condition reaches open KPZ","General reservoir speeds in open ASEP still yield KPZ","No Liggett's condition: open ASEP converges to KPZ","Open ASEP with general boundary asymmetry flows to KPZ","Liggett-free open ASEP limit is open KPZ equation"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000184,"raw_usage":{"total_tokens":1272,"prompt_tokens":852,"completion_tokens":420,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":468,"completion_tokens_details":{"reasoning_tokens":334}},"tokens_in":468,"tokens_out":420,"duration_ms":4624,"temperature":1.0,"reasoning_tokens":334,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T17:06:47.014798+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Simulate the boundary-localized process on a fattened interval of length $N^\\kappa$ for parameters outside Liggett's condition, and measure the second moment of the time-average of $f_{\\mathrm{left}}$ over $\\tau = N^{-2+\\rho}$: Proposition 5.6 predicts decay $N^{-\\rho} + N^{-1/3}$. Alternatively, for a fixed non-Liggett choice of $\\alpha,\\beta,\\gamma,\\delta$, compute the stationary height increment measure of open ASEP numerically on large intervals and compare it with the open KPZ stationary measure with parameters (2.8)-(2.9); a mismatch at leading order would refute Corollary 2.2.","supporting_citations":[{"cited_title":"Open ASEP in the weakly asymmetric regime","cited_arxiv_id":null,"evidence_quote":"Establishes the open-ASEP weak-asymmetry convergence framework, the martingale problem characterization, and the tightness strategy that this paper adapts."},{"cited_title":"The KPZ Limit of ASEP with Boundary","cited_arxiv_id":null,"evidence_quote":"Supplies the heat kernel estimates for the discrete Robin Laplacian and the moment/tightness bounds used in Lemma 4.1."},{"cited_title":"Derivation of the stochastic Burgers equation with Dirichlet boundary conditions from the W ASEP","cited_arxiv_id":null,"evidence_quote":"Previous derivation of a boundary KPZ-type equation assuming product invariant measures; this paper removes that assumption."},{"cited_title":"KPZ equation from ASEP plus general speed-change drift","cited_arxiv_id":"2409.10513","evidence_quote":"Provides the non-invariant homogenization technique for speed-change exclusion processes that the author adapts to the boundary setting."},{"cited_title":"Stationary measure for the open KPZ equation","cited_arxiv_id":null,"evidence_quote":"Provides the coupling construction used to prove convergence of stationary measures of the increment process in Corollary 2.2."},{"cited_title":"Kipnis, C","cited_arxiv_id":null,"evidence_quote":"Supplies the standard entropy inequalities, log-Sobolev bounds, and Kipnis-Varadhan inequalities that underlie the stochastic estimates in Section 5."}],"review_version":1}