{"id":"7a5b5d3d-345b-4e24-86ed-8c6a992ea179","arxiv_id":"2504.18292","paper_version":3,"verdict":"CONDITIONAL","confidence":"HIGH","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Exact large-time cumulants and upper-tail large deviation rate for the open KPZ equation on an interval are derived from replica Bethe ansatz and from a scaling limit of open ASEP.","lead":"The paper computes the large-time growth rates of all cumulants of the height of the KPZ equation on a finite interval with general boundary parameters. It gives explicit formulas for these cumulants and for the upper-tail large deviation rate, with a scaling limit toward the KPZ fixed point on an interval.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Higher cumulants c_k for k≥3 depend on an unvalidated exchange of limits: the ASEP-to-KPZ scaling limit is applied to the cumulant generating function, and the independent replica check stops at c2.","rationale":"The reader correctly identified the replica analytic continuation and the ASEP-to-KPZ cumulant exchange as the principal weaknesses, and assigned CONDITIONAL with high confidence. My reading agrees that the central computational claim is not fully rigorous, but I locate the most load-bearing gap more narrowly: the functional equation (1.16) is derived in Section 2 through a scaling limit of the ASEP cumulant generating function, and the paper's own caveat in Section 1.4 acknowledges that weak convergence of height processes does not imply convergence of large-deviation observables. The matching of c1 and c2 against the replica method in Section 5 is real independent support, but that support stops at second order. For c3 and beyond, the ASEP-scaling derivation is the only route, so the correctness of the headline claim for all cumulants depends on an assumption that is not tested anywhere in the paper. I do not see an internal inconsistency or a contradiction with known results; the issue is an unverified exchange of limits. The proposed test—extending the replica computation to third order—would directly settle whether the two independent methods continue to agree beyond c2. If they do agree at c3, the case for the higher cumulants would be substantially stronger; if they do not, the paper's main claim would need to be restricted to c1 and c2. Because the reader's verdict is already CONDITIONAL and my concern is consistent with that assessment, I recommend leaving the verdict unchanged.","tokens_in":39580,"tokens_out":3523,"duration_ms":40651,"concrete_test":"Extend Section 5 to order n^2: solve the m_2(z) term in M(z)=Σ n^k m_k(z) from the integral equation (5.28) and the functional Bethe equation (5.29), compute c3=-∂_n^3 E0(n,L)|_{n=0}, and compare with the Section 3.3 result (3.52). If the two expressions agree, the exchange-of-limits assumption is supported at the first unverified order; if they differ, the central claim for k≥3 fails. If extending the replica computation is infeasible, a direct numerical simulation of the open KPZ equation at fixed small L, measuring κ3(h(0,t))/t at large t, would also distinguish the prediction.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper's central claim—that the coefficients c_k(L,u,v) for all k are given by the functional-equation system (1.14)–(1.16)—rests, for k≥3, on a single unvalidated exchange of limits. Section 2 derives (1.16) by taking the weakly-asymmetric scaling limit ε→0 of the ASEP cumulant generating function E(µ), using the known height-process convergence of [38,39]. That convergence controls typical fluctuations, not the exponential tilts that define E(µ); the paper explicitly flags this issue in Section 1.4. The only independent check is the replica Bethe-ansatz computation of Section 5, but that computation is carried out explicitly only for c1 and c2, in (5.49) and (5.66), and is never extended to c3. Consequently c3 and higher cumulants have no cross-check: they inherit the unproved assumption that large-time cumulants commute with the ASEP-to-KPZ limit. A failure of this exchange at order s^3 would change the predicted c3 while leaving the verified c1 and c2 untouched.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper studies the KPZ equation on an interval [0,L] with Neumann boundary parameters u,v. The main claim is that, as t tends to infinity, the k-th cumulant of the height is κ_k(h(0,t)) = c_k(u,v,L)t + o(t), and that the cumulant generating function E(s) is given parametrically by (1.14)–(1.16): s = ∫ U, E = -s/24 + (1/2)∫ w^2 U, where U solves U(w) = -(1/2)log(1 - 2ζΨ(w)e^{kU}(w)), with Ψ(w) given by (1.17) and the operator k by (1.18)–(1.19). The authors give the first few cumulants explicitly, analyze the large-L scaling limit u = ũ/√L, v = ṽ/√L leading to a two-parameter KPZ fixed point, and use a Legendre transform to write an upper-tail rate function. Two methods are used: a weakly asymmetric scaling limit of open ASEP (Section 2) and a replica Bethe ansatz adapted from Brunet and Derrida (Section 5). The paper is explicitly heuristic: Section 1.4 lists the unproved steps, including the replica analytic continuation in the number of replicas, the exchange of large-time and scaling limits, and the selection of the ground-state branch among solutions of the functional Bethe equation.","tokens_in":39795,"tokens_out":17006,"duration_ms":156670,"significance":"If the formulas are correct, this is an important advance: it gives the first exact large-time cumulants for the open KPZ equation with general boundary parameters, including higher cumulants and a conjectural KPZ fixed point rate function. The agreement of the two methods for c1 and c2, the recovery of the periodic Brunet–Derrida results and of open-TASEP formulas in the large-L limit, and the explicit checks in special cases are significant strengths. The manuscript is also unusually candid about its lack of mathematical proof. However, the independent verification stops at order two in the replica expansion, so the higher cumulants, which are the paper's main new content, remain conditional on the heuristic steps.","major_comments":[{"comment":"The formulas for c_k with k ≥ 3 are not independently supported. Section 2.4 obtains E(s) by taking the weak-asymmetry limit ε → 0 in the ASEP cumulant generating function, but the convergence theorem [38,39] controls typical fluctuations rather than the exponential tilts entering E(µ); Section 1.4 explicitly acknowledges this. The replica method provides an independent route, but it is carried out only to order n^1 in the expansion of M(z): c1 and c2 are derived in (5.49) and (5.66), and no replica computation of c3 is given. Since c3 is the first order at which the two methods could disagree if the exchange of limits fails, the central claim that all c_k are given by (1.14)–(1.16) is not yet established for k ≥ 3. A concrete remedy is to extend the replica computation to order n^2 and compare with (3.52).","section":"§2.4 and §5.4"},{"comment":"There appears to be a factor-of-two inconsistency in the periodic-case reduction. If Uper = (1/2)U|_{Ψ=e^{L/2w^2}} as stated after (6.1), then substituting into (1.16) gives Uper = -(1/4)log(1 - 2ζΨ e^{2kUper}), not (6.1); conversely, the solution of (6.1) does not satisfy (1.16) under this identification. Moreover, combining (6.2)–(6.3) with U1 = Ψ' gives c1per = -1/24 - 1/(4L) (since ∫ν w^2 = -1/(2L)), whereas (6.6) states -1/24 - 1/(2L). Either (6.1), the identification Uper = U/2, or the relation (6.4) needs correction, and the claimed agreement with [16, Eq. (49)] should be rechecked.","section":"§6, Eqs. (6.1)–(6.4)"},{"comment":"The upper-tail large deviation statement (1.10)–(1.13) presupposes that the parametric representation (1.14)–(1.16) defines E(s) for all real s and that E(s) + s/24 is convex. The paper computes E(s) only as a power series around s = 0 (Section 1.2.3) and does not establish convergence or a global definition; the Legendre transform (1.13) therefore does not yet yield a rate function on the whole upper tail. A statement of the domain of s on which (1.14)–(1.16) is valid, or a direct large-deviation argument, is needed.","section":"§1.2.3 and §1.3.1"}],"minor_comments":[{"comment":"In the abstract, the sentence 'u = ũ/√L, u = ṽ/√L' should read 'v = ṽ/√L'.","section":"Abstract"},{"comment":"Equation (3.42) has unbalanced parentheses in the denominator of the last integral; the intended normalization is unclear as printed.","section":"Eq. (3.42)"},{"comment":"In Section 3.4.2, 'we already now that f(u,-u) = u^2' should be 'we already know'.","section":"§3.4.2"},{"comment":"Remark 2.3 says 'for u = 0, v = 1/2 one has Ψ(u) = 4π e^{w^2L}'; the argument should be Ψ(w), not Ψ(u).","section":"Remark 2.3"},{"comment":"There are two spelling typos: 'Stritly speaking' in footnote 1 and 'Brownien motion' in Section 1.2.1.","section":"Footnote 1 and §1.2.1"}],"recommendation":"major_revision","confidential_remarks":"The manuscript is transparent about its heuristic status, and I do not think the lack of full mathematical proof is by itself disqualifying. The decisive issue is the missing c3 cross-check: the two methods agree only for c1 and c2, so the higher cumulants, which are the paper's main new content, remain unverified. I would encourage the editor to request that the authors either extend the replica computation to order n^2 or otherwise provide a check at order s^3, and to correct the factor inconsistency in the periodic-case reduction."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"What you should know: this is a serious, useful paper. The main new result is a functional-equation system, (1.14)-(1.16), that gives all large-time cumulants c_k(L,u,v) for the open KPZ equation on a finite interval with general Neumann boundary parameters. That is genuinely new; the periodic analogues and open-ASEP results are prior art, but the interval case with two-sided boundaries was open. The paper also gives the upper-tail large deviation function and a large-L scaling limit to the two-parameter KPZ fixed point, where the cumulants match earlier open-TASEP formulas. The two derivations, one from an ASEP scaling limit and one from a replica Bethe-ansatz adaptation of Brunet-Derrida, agree for c1 and c2, and special cases reproduce known periodic and half-line results. That is real evidence. The authors are also honest about the limits of what they have proved.\n\nThe weak spot is exactly where the stress-test note puts it. The ASEP-to-KPZ convergence proved in [38,39] controls typical fluctuations, not the exponential tilts that define the cumulant generating function. The authors flag this in Section 1.4, then use the limit anyway. Their independent check is the replica computation, but it is carried out only for c1 and c2; c3 and higher have no cross-check from that side. So the coefficients c_k for k at least 3 do rest on a single unvalidated exchange of limits. A failure at order s^3 would change c3 without disturbing c1 or c2. This is a load-bearing assumption, though not a hidden one: it is stated plainly, and the first two cumulants plus the structural consistency of the formulas give some confidence. The other issues, replica analytic continuation in n, Bethe-ansatz completeness, and branch selection, are real but secondary; they are the standard folkloric costs of the replica method, and the authors do not pretend otherwise.\n\nThe math presented is coherent, the special-case checks are meaningful, and the citation pattern looks clean. This is not a paper with a hidden circular parameter; nothing is fitted to the target cumulants. The novelty is high within the subfield, and the formulas are concrete enough that others can test them. I would bring it to a reading group and would likely cite the c1-c2 results. A serious referee should be sent to engage with the exchange-of-limits question in detail, and the authors should be pushed to extend the replica check even one order higher, or to explain why c3 should be trusted. Conditional accept with a request for a deeper discussion of the k>=3 step is the right verdict; desk rejection would be wrong.","headline":"First explicit formulas for all large-time cumulants of the open KPZ equation on an interval, cross-checked through c2 by two independent methods; c3 and above rest on a clearly flagged but unproved exchange of limits.","tokens_in":40347,"tokens_out":1626,"would_cite":true,"duration_ms":18960,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["60H15","82B23","82C22"],"pacs":[],"model":"deepseek-v4-flash","headline":"A single functional equation determines every large-time cumulant of the open KPZ height.","keywords":["open KPZ equation","large-time cumulants","functional equation","replica Bethe ansatz","open ASEP","KPZ fixed point on an interval","upper tail large deviations","cumulant generating function"],"falsifier":"Simulate the stochastic heat equation on $[0,1]$ with $u=v=1$, estimate $\\kappa_3(h(0,t))/t$ for large $t$, and compare with the closed-form $c_3$ from (3.52); a persistent disagreement beyond numerical error would invalidate the analytic continuation at $k=3$.","tokens_in":39325,"feed_emoji":"📈","tokens_out":10034,"duration_ms":89939,"temperature":0.7,"pith_summary":"The paper claims that for the Kardar-Parisi-Zhang equation on an interval with Neumann boundary parameters $u,v$, every height cumulant (connected moment) has a large-time limit $\\kappa_k(h(x,t)) \\sim c_k(L,u,v)\\, t$, and that all coefficients $c_k(L,u,v)$ are computable from one parametric functional equation. In that equation, the cumulant generating function $E(s)$ is fixed by $s = \\int_{i\\mathbb{R}} \\frac{dw}{2i\\pi} U(w)$ and $E(s) = -s/24 + \\frac12\\int_{i\\mathbb{R}} \\frac{dw}{2i\\pi} w^2 U(w)$, where $U(w)$ solves $U(w) = -\\frac12\\log\\bigl(1 - 2\\zeta \\Psi(w) e^{kU}(w)\\bigr)$; eliminating the free parameter $\\zeta$ between the two integrals yields $E(s)$, whose Taylor coefficients are the $c_k$. The authors reach this system by two independent routes, a scaling limit of open-ASEP cumulants and a replica Bethe ansatz adapted from Brunet and Derrida's periodic computation, and they check agreement for the first two cumulants. If correct, this yields the first exact computation of all large-time cumulants, of the upper-tail large deviation rate function, and of the corresponding fixed-point cumulants in the large-$L$ scaling $u=\\tilde u/\\sqrt{L}$, $v=\\tilde v/\\sqrt{L}$.","feed_headline":"Functional equation computes every KPZ interval cumulant","feed_subtitle":"All k-th height cumulants of the open KPZ equation follow from one parametric equation; the first two match the open-ASEP limit.","key_machinery":"The load-bearing object is the pair $(\\Psi, k)$: $\\Psi(w)$ is the meromorphic function that already normalizes the stationary measure of the open KPZ equation, and $k$ is an integral operator with kernel $-2(\\psi(1+w-w')+\\psi(1+w'-w))$, built from the digamma function (the logarithmic derivative of the Gamma function). The functional equation $U = -\\frac12\\log(1-2\\zeta\\Psi e^{kU})$ is the central mechanism; expanding it in $\\zeta$ produces a tree expansion in which every coefficient $U_n$ is a sum of terms $\\Psi^c \\prod_j k(F_j)$, and the same expansion, after reversion of the series $s(\\zeta)$, yields every cumulant $c_k$. The simplification in the large-$L$ limit is that $k$ becomes subleading, so the equation becomes local in $w$ and the fixed-point cumulants are determined by the single profile $\\phi(y)$.","core_discovery":"The central discovery is that the asymptotic cumulant generating function of the open KPZ height is carried by the functional equation $U(w) = -\\frac12\\log(1 - 2\\zeta \\Psi(w) e^{kU}(w))$, with $\\Psi(w) = \\frac{\\Gamma(u+w)\\Gamma(u-w)\\Gamma(v+w)\\Gamma(v-w) e^{w^2L}}{\\Gamma(2w)\\Gamma(-2w)}$ and $k$ the convolution operator whose kernel is $-2(\\psi(1+w-w')+\\psi(1+w'-w))$. Once $U$ is expanded in powers of $\\zeta$, the integrals $s=\\int_{i\\mathbb{R}} \\frac{dw}{2i\\pi} U(w)$ and $E=-s/24+\\frac12\\int_{i\\mathbb{R}} \\frac{dw}{2i\\pi}w^2 U(w)$ give $s$ and $E$ as power series in $\\zeta$; eliminating $\\zeta$ and re-expanding in $s$ produces the cumulants through the triangular recurrence (1.38). The same framework gives the first cumulant as $c_1=-1/24+\\frac12\\partial_L\\log Z_{u,v}(L)$ and the second as an integral against the stationary-measure normalizing density $\\nu(dw)$, and in the large-$L$ limit it reduces to a local equation whose coefficients are elementary integrals of $\\phi(y)=\\frac{4y^2 e^{-y^2}}{(\\tilde u^2+y^2)(\\tilde v^2+y^2)}$, characterizing the two-parameter KPZ fixed point on an interval.","pith_inferences":["The same functional equation may control the crossover regime $t \\sim L^{3/2}$, not only the limits $t\\gg L^{3/2}$ and $L\\to\\infty$; the paper does not derive finite-time formulas, so this is an extrapolation.","Because $c_2$ is expressed as an $L$-derivative of a kernel average, the higher $c_k$ might also be total $L$-derivatives of tree functionals of the stationary measure, which would make them directly accessible to Monte Carlo sampling of the measure (1.5).","If the assumed commutation of limits holds, the same ASEP-to-KPZ scaling should yield the interval cumulants of other integrable stochastic models in the KPZ class, such as the stochastic six-vertex model, a route the paper mentions but does not take."],"forward_implications":["Every $c_k(L,u,v)$ can be computed iteratively from the coefficients $U_n$; the paper displays explicit $c_1,c_2,c_3$, and the recurrence (1.38) gives the rest.","The upper-tail probability obeys $P(h(0,t)/t + 1/24 > H) \\sim e^{-t\\Phi(H)}$, with $\\Phi$ obtained by Legendre transform of $E(s)+s/24$.","In the scaling $u=\\tilde u/\\sqrt{L}$, $v=\\tilde v/\\sqrt{L}$, the cumulants scale as $c_k \\sim L^{(k-3)/2}\\tilde c_k$, and the $\\tilde c_k$ are closed-form integrals in terms of $\\phi(y)$; this is the large-time characterization of the two-parameter KPZ fixed point on an interval.","The periodic KPZ equation is recovered by replacing $\\Psi(w)$ with $e^{Lw^2/2}$, and the paper verifies that the second cumulant agrees with the Brunet-Derrida result.","For general boundary parameters with $u,v>0$, $c_1$ is tied to the stationary-measure normalization through $c_1=-1/24+\\frac12\\partial_L\\log Z_{u,v}(L)$, and analytic continuation covers $u\\leq 0$ or $v\\leq 0$."],"supporting_citations":[{"why":"Introduced the replica Bethe ansatz computation of periodic KPZ large-time cumulants that this paper adapts to the interval.","marker":"[16,17]"},{"why":"Gave the functional-equation and tree structure for TASEP current cumulants that underlies the ASEP-limit method.","marker":"[18]"},{"why":"Provide the deformed matrix product ansatz expression for open ASEP cumulant generating function that is scaled to the KPZ equation.","marker":"[20,21]"},{"why":"Prove weak convergence of open ASEP to the KPZ/SHE, the scaling limit whose commutation with cumulants is assumed.","marker":"[38,39]"},{"why":"Supply the stationary measure of the open KPZ equation and the integral representation of its normalization featuring Psi, used for c1 and the measure nu.","marker":"[30,31,32,37]"},{"why":"Offers an exact functional Bethe ansatz derivation of the open ASEP top eigenvalue, cited as a potential way to justify the perturbative computation.","marker":"[23]"}],"fun_headline_variants":["All KPZ interval cumulants from one functional equation","KPZ interval cumulants via a single functional equation","One equation gives all KPZ interval height cumulants","Replica Bethe ansatz solves KPZ interval cumulants"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the particle-model-to-PDE convergence can be exchanged with the large-time cumulant limit (equivalently, that replica energies known for positive integers continue analytically to real parameters), a step verified only for the first two cumulants.","fun_headline_variants_meta":{"raw":{"variants":["All KPZ interval cumulants from one functional equation","KPZ interval cumulants via a single functional equation","One equation gives all KPZ interval height cumulants","Replica Bethe ansatz solves KPZ interval cumulants"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000609,"raw_usage":{"total_tokens":2909,"prompt_tokens":1091,"completion_tokens":1818,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":707,"completion_tokens_details":{"reasoning_tokens":1753}},"tokens_in":707,"tokens_out":1818,"duration_ms":11941,"temperature":1.0,"reasoning_tokens":1753,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-16T10:19:50.934287+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Simulate the stochastic heat equation on $[0,1]$ with $u=v=1$, estimate $\\kappa_3(h(0,t))/t$ for large $t$, and compare with the closed-form $c_3$ from (3.52); a persistent disagreement beyond numerical error would invalidate the analytic continuation at $k=3$.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Gave the functional-equation and tree structure for TASEP current cumulants that underlies the ASEP-limit method."},{"cited_title":"Lazarescu and V","cited_arxiv_id":null,"evidence_quote":"Offers an exact functional Bethe ansatz derivation of the open ASEP top eigenvalue, cited as a potential way to justify the perturbative computation."}],"review_version":1}