{"id":"9fbc8482-4a83-4a1e-b9de-2abe842e0bfe","arxiv_id":"2412.16714","paper_version":1,"verdict":"REJECT","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"high","formal_verification":"none","parameter_count":2,"one_line_summary":"A new consistency-stability method yields the first quantitative hydrodynamic limit rates for the genuinely nonlinear symmetric zero-range process in d=1 and d=2.","lead":"The authors introduce a consistency-stability framework for quantitative hydrodynamic limits and apply it to the symmetric zero-range process on the torus. They derive explicit convergence rates in Monge-Kantorovich distance and relative entropy for dimensions one and two, uniform in time.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The consistency rates (5.20)–(5.21) rely on the random-walk estimate (5.16), which assumes a uniform adjacent-difference lower bound not present in Theorem 3.1 and treats the occupancy environment as fixed, so Proposition 5.1 and Theorem 3.1 are not established.","rationale":"The abstract Theorem 2.1 is sound if (H1)-(H4) hold, and the microscopic and macroscopic stability sections are plausible. The decisive point is the Monge-Kantorovich consistency estimate, where the only quantitative control of the Dirichlet form for general test functions is Method 2 in Section 5.7. Its key large-deviation estimate (5.16) is not a consequence of the stated hypotheses: the theorem's coarse growth condition is a blockwise increment condition, not a per-edge lower bound, and the environment is not frozen. The reader's verdict identifies exactly this gap, and I agree. Since the paper's main quantitative rates depend on this estimate, the central claim is not established as written, so no change to the REJECT verdict is needed.","tokens_in":29302,"tokens_out":9804,"duration_ms":91622,"concrete_test":"Attempt to derive (5.16) from the coarse condition alone. A minimal discriminating example is g(k)=delta*k + c_g*floor(k/n0) (with delta>0 arbitrarily small), which satisfies every hypothesis of Theorem 3.1 but has per-step differences equal to delta on flat stretches; for delta small the waiting-time lower bound used in (5.18) degrades, and for the limiting flat case the gamma convolution bound is invalid. Run the basic coupling on a moderately large torus, initializing eta and zeta one particle apart at sites in a flat stretch, and record the number of jumps of the defect S(t) up to macroscopic time t. If the probability of fewer than beta*t*N^2 jumps is not exponentially small in N^2, or if the derivation requires an additional uniform lower bound not stated in the theorem, then (5.20)-(5.21) are unsupported.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Section 5.7 (Method 2) is the only route to (5.20)–(5.21) for general Lipschitz test functions in d=1,2. The proof of (5.16) is explicitly based on '0 < g_*1 <= |g(k)-g(k+1)| <= g_*2 < infinity', but Theorem 3.1 hypothesizes only g(n')-g(n) >= c_g for n' >= n+n0, together with monotonicity and a global Lipschitz bound. The coarse growth condition permits flat stretches of length n0-1 on which adjacent differences vanish; hence a positive per-step lower bound is not a consequence of the hypotheses. The estimate (5.16) also models x_-(t), x_+(t) as a symmetric random walk in a fixed environment, while the jump rates (g(eta_{x_-})-g(eta_{x_-}-1))_+ and (g(zeta_{x_+}+1)-g(zeta_{x_+}))_+ evolve with the process. No argument is given that controls the waiting times or the return probabilities under this dynamic environment using only the coarse condition. Since (5.16) directly feeds the 'too few jumps' event and the non-return estimates used to obtain (5.20) and (5.21), the consistency Proposition 5.1, and hence the quantitative hydrodynamic limit Theorem 3.1, are not proven as stated.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes an abstract consistency-stability framework for quantitative hydrodynamic limits. Theorem 2.1 bounds the dual-Lipschitz distance (and relative entropy) between the microscopic law and a local Gibbs measure in terms of the corresponding initial discrepancy and a consistency error, assuming microscopic stability, macroscopic stability, and consistency conditions (H1)-(H4). The framework is then applied to the symmetric zero-range process on the d-dimensional torus with d=1,2 under parabolic scaling. Theorem 3.1 claims explicit rates N^{-1/6+epsilon} in d=1 and (ln N)^{-1/8+epsilon} in d=2 for both the Monge-Kantorovich distance and the relative entropy, uniformly in time. The proof of the application verifies the abstract assumptions for the ZRP; the main technical work is the Monge-Kantorovich consistency estimate in Section 5, whose core is a random-walk recurrence estimate in Section 5.7.","tokens_in":29693,"tokens_out":7669,"duration_ms":66832,"significance":"If fully established, the result would provide a quantitative hydrodynamic limit for a genuinely nonlinear zero-range process without block estimates, with rates in a weak distance and in relative entropy, uniform in time. The abstract framework of Section 2 is clean and the microscopic stability and entropic-consistency parts of the ZRP verification are careful and potentially reusable. The paper also identifies an interesting direction for degenerate limit equations. However, as detailed in the major comments, the central consistency estimate relies on an unproved and, under the stated hypotheses, unjustified random-walk bound. The main theorem is therefore not established as stated, and the paper is not suitable for publication in its current form.","major_comments":[{"comment":"The proof of the 'too few jumps' estimate (5.16) explicitly assumes 0 < g_*1 <= |g(k+1)-g(k)| <= g_*2 < infinity for every k. This assumption is not part of Theorem 3.1, whose hypotheses are monotonicity, global Lipschitz regularity, g(0)=0, g(n)>0, and the coarse growth condition g(n')-g(n) >= c_g for n' >= n+n0. The coarse condition permits flat stretches of length n0-1 on which adjacent increments vanish, so a uniform positive per-step lower bound is not a consequence of the assumptions. Since (5.16) controls the J_slow term in the decomposition of J_N^t and is used to derive (5.20)-(5.21), Proposition 5.1 and hence Theorem 3.1 are not proved for general Lipschitz test functions under the stated hypotheses.","section":"Section 5.7, Eq. (5.16)"},{"comment":"The random-walk estimate treats x_-(t) and x_+(t) as a symmetric random walk in a fixed random environment, with jump rates (g(eta_{x_-}) - g(eta_{x_-}-1))_+ and (g(zeta_{x_+}+1) - g(zeta_{x_+}))_+. The manuscript's own phrase 'provided we consider eta as given' acknowledges that the environment is frozen in this reasoning, but no argument is provided to transfer the fixed-environment waiting-time and return-probability estimates to the actual dynamic environment, where the occupancies eta and zeta evolve and the jump rates change in time. Under the theorem's hypotheses, a site's g-increment may vanish on long stretches, so the lower bound on the number of jumps of the coupled difference is not justified for the actual process.","section":"Section 5.7, random-walk argument"},{"comment":"The return-probability estimates quoted from [LL10, Prop 4.2.4] are applied to the difference process x_+(t)-x_-(t), but it is not verified that this process satisfies the hypotheses of the cited result (in particular, symmetry or aperiodicity are not established from the assumptions on p). Moreover, the control of the unbounded factor g(eta_{y1}) is left to the reader ('We leave this easy calculation to the reader'), yet it is part of the quantitative estimate that determines the final rate in Theorem 3.1. These omissions are substantial because the resulting bounds (5.20)-(5.21) directly feed into the consistency error in Proposition 5.1.","section":"Section 5.7, Eqs. (5.20)-(5.21)"}],"minor_comments":[{"comment":"The statement contains the duplicated phrase 'jump rate jump rate g'; it should read 'the jump rate g'.","section":"Section 3, Theorem 3.1"},{"comment":"There are typographical errors: 'threrein' should be 'therein', 'mesure' should be 'measure', and 'opened' should be 'open'.","section":"Introduction, Section 1.1"},{"comment":"The reference [DMOW] is missing publication details (year and journal or arXiv number); please complete it.","section":"References"},{"comment":"The notation E_{vartheta^{ell,m}} and E_{vartheta^ell_m} is confusing, because the difference E_{vartheta^{ell,m}}[g(eta_{y0})] - E_{vartheta^ell_m}[g(eta_{y0})] appears to compare a conditional measure with itself unless the two symbols are intended to denote different measures; please distinguish them clearly.","section":"Appendix A, Theorem A.1"},{"comment":"The flattening error is ultimately bounded after choosing M = 2theta' ln(N ell^{-d-1}), but this optimization is not shown in the text; please state explicitly how the two terms of (5.8) are balanced.","section":"Section 5.5, Eq. (5.8)"}],"recommendation":"reject","confidential_remarks":"The abstract framework and the stability arguments are promising, but the gap in Section 5.7 is load-bearing for the main theorem. The error cannot be repaired by a local edit within the stated scope because the hypotheses of Theorem 3.1 would need to be strengthened to include a uniform per-step non-degeneracy condition, which would materially change the main result. If the authors can prove the consistency estimate under the coarse growth condition, or if they wish to restate the theorem under the stronger condition and provide a complete proof of the random-walk estimate, the paper might be reconsidered."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear colleague,\n\nThe abstract consistency-stability framework is genuinely nice. Theorem 2.1 is short and transparent, and it really does avoid block estimates. The microscopic stability via the basic coupling, the support-contraction lemma (Lemma 4.1), and the quantitative local limit theorem in the appendix are careful and useful. The entropic consistency section is also clean.\n\nThe problem is the application. Theorem 3.1 rests on the consistency rates in Proposition 5.1, and for general Lipschitz test functions in d=1,2 those rates come from Method 2 in Section 5.7. That section assumes '0 < g*_1 ≤ |g(k+1)-g(k)| ≤ g*_2 < ∞' and uses it to prove the large-deviation bound (5.16) on the number of jumps. Theorem 3.1 only assumes g(n')-g(n) ≥ c_g for n' ≥ n+n0, monotonicity, and a global Lipschitz bound. The coarse condition permits flat stretches of length n0-1, so a uniform positive lower bound on adjacent differences is not implied. The environment is also treated as fixed: the jump rates of x_± depend on η, which evolves, and no argument controls waiting times or return probabilities under that dynamic environment. Since (5.16) feeds directly into the too-few-jumps event and the non-return estimates that give (5.20)-(5.21), Proposition 5.1 and Theorem 3.1 are not established as stated.\n\nMethod 1 is fine but limited to d=1, nearest-neighbour jumps, and test functions of the form Φ0 = N^{-1}Ση_x φ_x. The general theorem falls back on Method 2, so this is a load-bearing gap, not a cosmetic one. The fix is not obvious: either the hypotheses on g need to be strengthened to a uniform increment bound, or (5.16) needs a proof under the actual coarse condition with a dynamic environment. The abstract framework looks salvageable, and the first-quantitative-nonlinear-ZRP claim is plausible, but the current paper does not deliver it.\n\nI would send this to a serious referee, expecting major revision. The gap is central but localized, and the rest of the paper is honest and careful. Not publishable as is.","headline":"A clean abstract framework and an unproven central theorem: Section 5.7's random-walk estimate assumes a uniform increment bound that Theorem 3.1 never states.","tokens_in":30120,"tokens_out":3766,"would_cite":false,"duration_ms":31042,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["60K35","82C22","60F05","35K55"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves explicit, time-uniform convergence rates for the hydrodynamic limit of the symmetric zero-range process in dimensions 1 and 2, in a Monge-Kantorovich distance and in relative entropy.","keywords":["zero-range process","hydrodynamic limit","quantitative error estimates","Monge-Kantorovich distance","relative entropy","parabolic scaling","local Gibbs measure","consistency-stability method"],"falsifier":"One concrete check: take the admissible jump rate $g(k)=\\lfloor(k+1)/2\\rfloor$, which satisfies the theorem's stated growth condition but has flat steps where $g(k+1)=g(k)$, and compute the probability bound (5.16) for the coupled difference process; at flat sites the holding rate vanishes, so the claimed exponential bound on the number of jumps cannot hold, and a positive probability of too few jumps would show the stated rates need a stronger hypothesis.","tokens_in":29144,"feed_emoji":"🎲","tokens_out":13217,"duration_ms":109880,"temperature":0.7,"pith_summary":"The paper proves a quantitative hydrodynamic limit for the symmetric zero-range process — a lattice gas in which a particle jumps from a site at a rate depending only on the occupation of that site — in dimensions $1$ and $2$ under parabolic scaling. The claim is that the law of the process stays close, with an explicit algebraic or logarithmic rate, to the local Gibbs measure (product state matching the macroscopic density profile) built from the solution of the nonlinear diffusion equation. In dimension $1$ the rate is $N^{-1/6+\\varepsilon_0}$ and in dimension $2$ it is $(\\ln N)^{-1/8+\\varepsilon_0}$, uniformly in time. The distance is a Monge-Kantorovich (dual Lipschitz) distance whose macroscopic counterpart is the classical $L^1$ stability estimate for conservation laws, and the same rate is obtained for relative entropy. The method is a consistency-stability scheme that avoids block estimates and is presented as a general template for symmetric attractive particle systems.","feed_headline":"Explicit rates proven for zero-range hydrodynamic limit","feed_subtitle":"Convergence to the local Gibbs measure holds in a weak distance and in entropy, uniformly in time.","key_machinery":"The load-bearing object is the variation-of-constants formula for the difference between the density $F_t^N$ of the particle system and the density $G_t^N$ of the local Gibbs measure: $$F_t^N - G_t^N = $e^{{tL_N^*}}$(F_0^N - G_0^N) + \\int_0^t $e^{{(t-\\tau)L_N^*}}$\\bigl(L_N^* G_\\tau^N - L_\\infty^* G_\\tau^N\\bigr)\\,d\\tau.$$ The first term is bounded by the microscopic stability assumption, which says the basic coupling contracts the $\\ell^1$ distance between configurations; the second term is the Monge-Kantorovich consistency error, comparing the microscopic generator with the macroscopic evolution. That error is reduced through an intermediate box scale $\\ell$: local averages are projected onto constant-mass hyperplanes, a spectral-gap inequality on the box controls the non-equilibrium part, a quantitative local limit theorem (equivalence of ensembles) controls the projected part, and recurrence of the difference random walk gives the final decay in dimensions $1$ and $2$.","core_discovery":"On the torus in dimensions $1$ and $2$, for a centred symmetric zero-range process with a non-decreasing Lipschitz jump rate satisfying a coarse growth condition, and for initial profiles $f_0 \\in C^3(\\mathbb{T}^d)$ bounded below, the paper's Theorem 3.1 asserts that under parabolic scaling the law $\\mu_t^N$ and the local Gibbs measure $\\vartheta^N_{f_t}$ built from the solution $f_t$ of $\\partial_t f = A:\\nabla^2\\sigma(f)$ satisfy $$\\sup_{t\\in[0,T]}\\|\\mu_t^N - \\vartheta^N_{f_t}\\|_{\\mathrm{Lip}^*} \\leq \\|\\mu_0^N - \\vartheta^N_{f_0}\\|_{\\mathrm{Lip}^*} + C $N^{{-1/6+\\varepsilon_0}}$$$ in $d=1$, with $C(\\ln N)^{-1/8+\\varepsilon_0}$ in $d=2$, and the same bound holds for the rescaled relative entropy $H^N(\\mu_t^N|\\vartheta^N_{f_t})$. The estimate is uniform in time. The proof verifies four abstract assumptions for this model: microscopic stability, macroscopic stability, Monge-Kantorovich consistency, and entropic consistency. The consistency error is controlled by a site-by-site computation, an intermediate mesoscopic scale, a local spectral-gap inequality, a quantitative local limit theorem for the equivalence of ensembles, and, in dimensions $1$ and $2$, recurrence estimates for the difference of two configurations coupled by the basic coupling.","pith_inferences":["Not claimed by the paper: the argument's random-walk step could be repaired under the theorem's stated hypotheses, because the coarse growth condition on $g$ does not imply the uniform per-step lower bound used in Section 5.7; closing this gap is a concrete open problem.","Not claimed by the paper: the same consistency-stability layout may handle degenerate limit equations such as the porous medium equation, where quantitative relative entropy methods are currently unavailable; the paper states this direction is under investigation.","Not claimed by the paper: if the jump-distance contraction of Section 5.7 were established in dimension $2$, the logarithmic rate would likely become algebraic, since the random-walk recurrence input would no longer be the bottleneck."],"forward_implications":["The empirical density field converges quantitatively to the solution of the nonlinear diffusion equation: the paper's Remark 1 transfers the Lip* bound into an explicit probability estimate for the empirical measure against test functions.","The same rate is obtained in relative entropy without any logarithmic Sobolev inequality, so the entropic route does not depend on that strong functional inequality.","Because the abstract theorem only needs the four consistency-stability assumptions, the method applies beyond the zero-range process to other symmetric attractive models; the paper notes that simple exclusion fits the same assumptions.","The paper's Remark 4 notes that in dimension 1, for test functions of the form $N^{-1}\\sum_x \\eta_x\\phi_x$ with $\\phi\\in C^1$, the jump-distance method gives a better consistency rate, and higher-order local limit expansions are expected to bring the general rate close to the optimal $N^{-1/2}$.","The uniformity in time means the quantitative closeness does not degrade as the macroscopic profile relaxes to its equilibrium."],"supporting_citations":[{"why":"Supplies the equilibrium structure, the macroscopic regularity of $\\sigma$, and the local limit theorem tools used throughout.","marker":"[KL99]"},{"why":"Provides the standard coupling and basic generator formalism used for the microscopic stability estimate.","marker":"[Lig85]"},{"why":"Gives the microscopic stability idea, adapted here to the quantitative consistency argument.","marker":"[Rez91]"},{"why":"Introduces the relative entropy method that the entropic consistency part makes quantitative.","marker":"[Yau91]"},{"why":"Establishes the qualitative hydrodynamic limit that the paper's rates refine.","marker":"[GPV88]"},{"why":"Provides the spectral-gap inequality on the box with constant proportional to $\\ell^2$.","marker":"[LSV96]"},{"why":"Source of the quantitative local limit theorem used for the equivalence of ensembles in the box.","marker":"[Pet75]"},{"why":"Gives the random-walk recurrence and return probabilities used to control the non-return event in dimensions 1 and 2.","marker":"[LL10]"}],"fun_headline_variants":["Uniform-in-time rates for zero-range hydrodynamic limit","Explicit convergence rates for zero-range process","Consistency-stability yields explicit ZRP rates","Zero-range limit: explicit uniform rates"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The argument's weakest premise is that every allowed jump changes the jump rate by at least a fixed positive amount; the stated hypotheses only guarantee a coarser growth condition, and the key probability estimate (5.16) that controls the consistency rates is asserted from that stronger premise without proof.","fun_headline_variants_meta":{"raw":{"variants":["Uniform-in-time rates for zero-range hydrodynamic limit","Explicit convergence rates for zero-range process","Consistency-stability yields explicit ZRP rates","Zero-range limit: explicit uniform rates"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000811,"raw_usage":{"total_tokens":3566,"prompt_tokens":962,"completion_tokens":2604,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":578,"completion_tokens_details":{"reasoning_tokens":2548}},"tokens_in":578,"tokens_out":2604,"duration_ms":15845,"temperature":1.0,"reasoning_tokens":2548,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T10:21:00.540273+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"One concrete check: take the admissible jump rate $g(k)=\\lfloor(k+1)/2\\rfloor$, which satisfies the theorem's stated growth condition but has flat steps where $g(k+1)=g(k)$, and compute the probability bound (5.16) for the coupled difference process; at flat sites the holding rate vanishes, so the claimed exponential bound on the number of jumps cannot hold, and a positive probability of too few jumps would show the stated rates need a stronger hypothesis.","supporting_citations":[],"review_version":1}