{"id":"d03c1780-2895-47c0-a1f7-f4e3a410b25c","arxiv_id":"2412.12026","paper_version":2,"verdict":"ACCEPT","confidence":"HIGH","novelty_score":7.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"For the fan region of open ASEP, the stationary height profile satisfies an LDP with rate function depending only on the two boundary effective densities, equal to the open TASEP rate function.","lead":"This paper proves a large deviation principle for the large-scale shape of the stationary measure of the open asymmetric simple exclusion process (ASEP) in the fan region of its phase diagram. It is the first rigorous proof of a 2003 physics prediction, extending the 2024 TASEP result to general ASEP parameters.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Path-space LDP relies on Lemma 4.8 to pass from pointwise to sup-norm topology, but that lemma is false as stated and the needed sup-norm goodness of pI is not verified before use.","rationale":"The reader's ACCEPT is justified if the topological step is repaired. The main statistical comparison (Corollary 2.10) and the two-layer separation estimates are sound and well detailed. The single most load-bearing issue in the written proof is the unjustified topology enhancement: Lemma 4.8 is false as stated, and the paper does not show that pI is a good rate function in the sup-norm topology before applying the lemma. This is a proof gap rather than a counterexample to the central claim, so a conditional acceptance (requiring the missing verification) is appropriate. The reader's weakest assumption concerned external representations and the TASEP LDP; my concern is different and arises inside the paper's own argument, hence agreement is 'disagree'.","tokens_in":26165,"tokens_out":46297,"duration_ms":391382,"concrete_test":"Verify the missing hypothesis by proving that pI(f) = sup_{0<θ1<...<θ_{d+1}=1} I_θ(f(θ1),...,f(θ_{d+1})) is lower semicontinuous in the sup-norm topology on C0([0,1],R) and has compact level sets. Concretely, compute pI for a sequence of piecewise-linear profiles f_n converging uniformly to a given 1-Lipschitz profile f, using the alternative formula in Remark 1.4, and check that liminf_n pI(f_n) ≥ pI(f) and that {pI≤α} is compact; if either fails, Lemma 4.8 cannot be applied and the path-space LDP in Theorem 1.2 is not established by the given proof.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"In Step 2 of the proof of Theorem 1.2 (Section 4.3), the LDP is first established only for the pointwise topology on X, then transferred to C0([0,1],R) with the sup-norm topology by Lemma 4.8. Lemma 4.8 (credited to Corollary 4.2.6 of [DZ09]) is not true as stated: an LDP in a weaker topology plus exponential tightness in a stronger topology does not in general imply an LDP in the stronger topology with the same rate function (e.g., X_N = 1/N on R with the trivial versus usual topology). The missing hypothesis is that the rate function pI is already a good rate function in the sup-norm topology, or equivalently that the lower-semicontinuous envelope of pI for the sup norm coincides with pI on the support. The paper asserts pI = I^{(a,b)} only after this strengthening, via uniqueness, so this is not circular only if the strengthening step is independently justified. In the present setting the gap is likely repairable: the measures are supported on Y, the set of nondecreasing 1-Lipschitz functions, where the pointwise and uniform topologies coincide, and pI is sup-norm lower semicontinuous as a supremum of continuous functionals. But these facts are not stated or proved before Lemma 4.8 is invoked, so the written proof has a genuine missing step.","agreement_with_reader":"disagree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proves a large deviation principle (LDP) for the height profiles of the open asymmetric simple exclusion process (ASEP) on a finite interval, in the fan region ab<1 of the phase diagram, under the stationary measure. The rate function I^{(a,b)} is explicit, depends only on the effective boundary densities a and b, and coincides with the open TASEP rate function. The proof has three main ingredients: (i) a two-layer representation of the ASEP stationary measure derived from the Enaud–Derrida matrix product representation (Theorem 2.5); (ii) an estimate (Proposition 2.9) showing that, under the open TASEP two-layer measure, the two layers remain separated on all but a small fraction of sites with probability decaying slower than exponential; and (iii) a comparison of ASEP and TASEP two-layer weights (Corollary 2.10) showing that their logarithms differ by o(N) on the relevant finite-dimensional events. These yield finite-dimensional LDPs, which the Dawson–Gärtner theorem upgrades to a path-space LDP in the pointwise topology; exponential tightness on the compact set of nondecreasing 1-Lipschitz profiles then transfers the LDP to the uniform topology. The authors also prove a convex-envelope identity (Lemma 4.3) used to simplify the lower bound for open sets.","tokens_in":26479,"tokens_out":39791,"duration_ms":340826,"significance":"The result is significant: it provides the first rigorous proof of the Derrida–Lebowitz–Speer large deviation functional for the height profile of open ASEP in the fan region, a prediction from 2002–2003. The rate function is independent of the asymmetry q and of the reservoir parameters beyond the effective densities, a striking universality feature. The proof is largely self-contained, and the novel comparison argument between ASEP and TASEP two-layer measures, built on separation estimates for non-intersecting random walks, is a useful methodological contribution that may extend to the shock region. The paper relies appropriately on cited results (matrix product ansatz, Enaud–Derrida representation, USW normalization asymptotics, and the Bryc–Zatitskii TASEP LDP) and gives precise citations. The stress-test objection to Lemma 4.8 is not borne out: the lemma's hypotheses include Hausdorffness of the weaker topology, which holds for the pointwise topology on C0; the trivial-topology counterexample violates that hypothesis.","major_comments":[],"minor_comments":[{"comment":"The application of Lemma 4.7 with E = C0 is not directly justified because C0, the set of continuous functions, is not a Borel subset of the pointwise product space X. This is repairable by taking E = Y (which is closed and contains the support), obtaining the LDP on Y with the pointwise topology, and extending the rate function by infinity to C0; alternatively, the LDP on C0 follows directly from the LDP on X because the profiles take values in Y. Please adjust the argument accordingly.","section":"Section 4.3, Step 2"},{"comment":"The identification pI = I^{(a,b)} after Lemma 4.8 is terse; it would help to spell out that the TASEP finite-dimensional marginals also yield the pointwise LDP with rate function pI by the Dawson–Gärtner theorem, so uniqueness of rate functions on the regular space X gives equality with the known TASEP rate function.","section":"Section 4.3, Step 2"},{"comment":"In the displayed formula for the sum over λ2, the product index is i while the indicators use j; please make the index consistent.","section":"Theorem 2.5, proof, equation (2.7)"},{"comment":"The phrase 'by conditioning on the values λ1(⌊θjN⌋) and λ1(⌊θjN⌋)' should read 'λ1(⌊θjN⌋) and λ2(⌊θjN⌋)'.","section":"Proposition 3.9, proof, Step 2"},{"comment":"The statement is a correct citation of Dembo–Zeitouni Corollary 4.2.6 when the weaker topology is Hausdorff; a parenthetical noting that the pointwise topology on C0 is Hausdorff and that exponential tightness holds on the compact set Y would help the reader apply the lemma without concern.","section":"Lemma 4.8"}],"recommendation":"minor_revision","confidential_remarks":"The paper is mathematically sound and the main theorem is significant. The only technical gap is the application of Lemma 4.7 to a non-Borel set, which is easily repaired by working with the closed support set Y. The stress-test concern about Lemma 4.8 is not valid because the lemma's hypotheses include Hausdorffness of the weaker topology, and the pointwise topology on C0 satisfies this. I recommend minor revision."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Here's the short version: this paper proves a long-standing physics prediction, the Derrida–Lebowitz–Speer large deviation rate for the height profile of open ASEP, in the fan region ab<1, for the full range of asymmetry q and boundary parameters c,d. It does so by reducing the problem to the recently proved TASEP LDP of Bryc and Zatitskii via a two-layer representation. That reduction is the real new work, and it is mostly convincing.\n\nThe genuinely new ingredients are the ordered two-layer representation with a free start for the second layer, and the separation estimate (Proposition 2.9) which says that, under the TASEP two-layer measure, the two layers stay apart on all but an ε-fraction of sites with probability e^{-o(N)}. This is then used to show that the ASEP weights differ from the TASEP weights only by subexponential factors (Corollary 2.10). The rate function is explicit and parameter-free, and the comparison argument is elegant. The finite-dimensional part of the proof is careful, including the handling of the cone boundary by the perturbation argument in Lemma 4.3.\n\nThe soft spot is at the very end. The paper transfers the LDP from the pointwise topology on C0([0,1]) to the sup-norm topology using Lemma 4.8, which is false as stated: an LDP in a weaker topology plus exponential tightness does not imply an LDP in the stronger topology unless the rate function is also a good rate function for the stronger topology. In this setting the missing hypothesis is satisfied—the measures are supported on the compact set Y of 1-Lipschitz, nondecreasing functions, where the two topologies coincide, and pI is sup-norm lower semicontinuous—but the paper does not state or prove these facts before invoking Lemma 4.8. So the written proof has a genuine gap in Step 2 of Theorem 1.2, not in the finite-dimensional LDP. It looks repairable without new ideas, but a referee should insist on a correct statement of the transfer theorem and the verification of sup-norm goodness of pI.\n\nThe citation pattern is healthy; the central external inputs (BZ24, ED04, USW04) are appropriate and the reduction to BZ24 is explicit. I don't see a circularity problem.\n\nBottom line: this is a significant result that deserves a serious referee. With a fix of the last step, it should be accepted. I'd bring it to a reading group on exactly solvable models or large deviations.","headline":"Solid proof of the DLS03 LDP for open ASEP in the fan region, with a patchable gap in the final path-space step.","tokens_in":27005,"tokens_out":3990,"would_cite":true,"duration_ms":34304,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["60F10","60K35","82C22"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper proves that, in the fan region $ab<1$, the stationary height profiles of the open asymmetric simple exclusion process satisfy a large deviation principle with an explicit rate function $I^{(a,b)}$ that depends only on the…","keywords":["large deviations","asymmetric simple exclusion process","open boundaries","stationary measure","fan region","two-layer representation","matrix product ansatz","height profile"],"falsifier":"For a fixed pair $(a,b)$ with $ab<1$, compute via the matrix product ansatz the stationary probability that $h_N(1/2) > 0.75$ for $q=0$ and $q=1/2$ (with $c=d=0$ so $a,b$ are the same). The theorem implies the difference of the two log-probabilities, divided by $N$, goes to $0$; observing a nonzero limit would falsify it.","tokens_in":25982,"feed_emoji":"📈","tokens_out":6638,"duration_ms":57901,"temperature":0.7,"pith_summary":"The paper proves that, in the fan region $ab<1$, the stationary height profiles of the open asymmetric simple exclusion process (ASEP) satisfy a large deviation principle with an explicit rate function $I^{(a,b)}$. The rate function depends only on the effective boundary densities $a$ and $b$, and coincides with the rate function for the totally asymmetric (TASEP) case, so the asymmetry parameter $q$ and the auxiliary parameters $c,d$ drop out entirely. This settles, for the fan region, a prediction from the physics literature obtained by the additivity principle and matrix product ansatz. The proof works by rewriting the stationary measure as the marginal of a two-layer random-walk measure, then showing that, on the large-deviation scale, the weights can be compared to those of open TASEP.","feed_headline":"Explicit large-deviation law proved for open ASEP height profiles","feed_subtitle":"Rate function is explicit and independent of the jump asymmetry, matching the predicted TASEP form.","key_machinery":"The central object is a two-layer representation: the stationary measure of open ASEP is realized as the marginal of a probability measure on pairs of non-crossing Bernoulli paths $(\\lambda_1,\\lambda_2)$. The weight of a pair factors into boundary rewards $a^{\\lambda_1(0)-\\lambda_2(0)}b^{\\lambda_1(N)-\\lambda_2(N)}$, a q-Pochhammer endpoint factor, and local transition weights $W^{(q,c,d)}$. This is a marginal of the Enaud-Derrida matrix product ansatz representation. The argument then compares these weights to the $q=0$ TASEP two-layer weights: whenever the two layers are separated by at least $r$, each local factor is bounded above and below by constants depending only on $q,r,c,d$, so the ratio of the weights is exponentially flat. A separation estimate (Proposition 2.9) shows that, conditional on the height-profile event, the two TASEP layers stay separated on all but an $\\varepsilon$-fraction of sites with probability at least $e^{-N^{4/5}}$, which is enough to make the large-deviation comparison.","core_discovery":"Under the stationary measure $\\mu_N$ of open ASEP with parameters satisfying $ab<1$, the sequence of height profiles $h_N$ satisfies the LDP on $C_0([0,1],\\mathbb{R})$ with good rate function $I^{(a,b)}$. This function is finite only for absolutely continuous, nondecreasing, 1-Lipschitz profiles, and is given by an integral of the entropy $H$ plus a convex-envelope correction and a boundary constant $-\\log J(a,b)$. The key discovery is that this rate function is independent of $q,c,d$; it matches the TASEP rate function established previously. The authors establish the LDP by a two-layer representation of the stationary measure, derived from the Enaud-Derrida representation of the matrix product ansatz, and by a comparison argument showing that ASEP two-layer weights and TASEP two-layer weights have the same exponential asymptotics.","pith_inferences":["Beyond the paper: this comparison strategy suggests that, once a two-layer representation is available in the shock region, the rate function there would also be independent of $q$ and equal to the TASEP rate function.","Beyond the paper: the separation estimate only needs polynomial lower bounds on probabilities, so the method may tolerate much weaker separation events and could apply to other integrable stochastic models with two-layer representations.","Beyond the paper: the explicit rate function, with its convex-envelope term, resembles the rate function for a single random walk conditioned to stay in an interval, hinting that the two-layer representation could yield finer fluctuation results, such as the order of pre-exponential factors.","Beyond the paper: if the universality in $q$ extends to the weakly asymmetric limit $q\\to 1$, the same $I^{(a,b)}$ could describe large deviations of the stationary measure for the KPZ fixed point with boundaries."],"forward_implications":["Every macroscopic height profile in the fan region has an explicit exponential cost: $-\\log \\mathbb{P}(h_N\\approx f) \\sim N I^{(a,b)}(f)$.","The rate function is universal in $q$: two open ASEPs with the same $a,b$ but different asymmetry or different $c,d$ have the same large-deviation speed and rate.","The LDP passes by contraction to any continuous observable of the height profile, including the total particle number and other linear statistics.","The normalization asymptotics $\\frac{1}{N}\\log Z_N \\to -\\log J(a,b)$ identify the free energy per site in the fan region, matching the three-phase formula for $J$.","The paper reports that an analogous argument, using a two-layer representation currently being developed, is expected to prove the LDP in the shock region."],"supporting_citations":[{"why":"Proves the LDP for open TASEP and supplies the alternative formula for the rate function used as the base case.","marker":"[BZ24]"},{"why":"Constructs the Enaud-Derrida matrix representation whose two-layer marginal yields the stationary measure.","marker":"[ED04]"},{"why":"Computes the asymptotic normalization constant needed for the rate function's boundary term.","marker":"[USW04]"},{"why":"Predicts the rate function and defines the fan/shock regions that frame the theorem.","marker":"[DLS03]"},{"why":"Establishes the matrix product ansatz formula for the stationary weights used at the start of the proof.","marker":"[DEHP93]"},{"why":"Provides the Dawson-Gärtner theorem and contraction principle that lift finite-dimensional LDPs to path space.","marker":"[DZ09]"},{"why":"Supplies the monotone coupling for non-intersecting Bernoulli bridges used in the separation estimate.","marker":"[CEP00]"},{"why":"Gives the supremum fluctuation bound for Bernoulli bridges used to keep the two layers separated.","marker":"[ACH24]"}],"fun_headline_variants":["Open ASEP height profiles obey explicit large-deviation law","LDP proved for open ASEP: rate function matches TASEP","Height-profile LDP for open ASEP, explicit and TASEP-like","First proof: large deviations for open ASEP stationary measures","Open ASEP large deviations: explicit rate function, TASEP form"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof's load-bearing premise is that the two-layer representation of the stationary measure, imported from the Enaud-Derrida matrix ansatz, is valid across the entire fan region and that the TASEP rate function it compares to is exactly $I^{(a,b)}$.","fun_headline_variants_meta":{"raw":{"variants":["Open ASEP height profiles obey explicit large-deviation law","LDP proved for open ASEP: rate function matches TASEP","Height-profile LDP for open ASEP, explicit and TASEP-like","First proof: large deviations for open ASEP stationary measures","Open ASEP large deviations: explicit rate function, TASEP form"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000214,"raw_usage":{"total_tokens":1395,"prompt_tokens":883,"completion_tokens":512,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":499,"completion_tokens_details":{"reasoning_tokens":421}},"tokens_in":499,"tokens_out":512,"duration_ms":4560,"temperature":1.0,"reasoning_tokens":421,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T14:23:59.070561+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"For a fixed pair $(a,b)$ with $ab<1$, compute via the matrix product ansatz the stationary probability that $h_N(1/2) > 0.75$ for $q=0$ and $q=1/2$ (with $c=d=0$ so $a,b$ are the same). The theorem implies the difference of the two log-probabilities, divided by $N$, goes to $0$; observing a nonzero limit would falsify it.","supporting_citations":[],"review_version":1}