{"id":"0e0fa4cf-44f0-472a-bdc5-bf61b283cef5","arxiv_id":"2507.23452","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The zero-range process in the whole space satisfies a full large deviation principle in every dimension under mild jump-rate hypotheses.","lead":"This paper proves matching large deviation upper and lower bounds for the hydrodynamic scaling of the zero-range process on the infinite lattice in every dimension. It settles a problem left open in Kipnis and Landim's book, using a new averaging estimate and a broader theory of degenerate parabolic equations.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 6.5 does not justify removing the spatial and velocity cutoffs for non-L1 initial data; the claimed global L^1 uniqueness is unsupported, and Theorem 8.3 inherits the gap.","rationale":"The reader's conditional verdict is appropriate. The paper's central claim is a matching large deviation principle whose upper and lower bounds are only reconciled through the analytic equality of rate functions in Theorem 8.3. That equality depends on uniqueness of renormalised kinetic solutions in the whole space, Theorem 6.5. The proof's most delicate step is the removal of spatial and velocity cutoffs for initial data with finite relative entropy but no global L^1 mass. I agree with the reader that this cutoff limit is the weakest assumption, and I sharpen it: the final line of Theorem 6.5 asserts a global L^1 uniqueness that is not even defined for the data class, and the preceding estimates do not supply a uniform bound or integrable majorant for the cutoff-dependent quantities. This is a genuine gap in the written argument, not merely a disagreement with consensus. No formal verification or reproducible code is provided, and several steps are delegated as 'small adaptations', which increases the risk. The proposed concrete test would settle whether the limit interchange can be made rigorous; until then, the conditional verdict is the right one.","tokens_in":97,"tokens_out":18401,"duration_ms":468104,"concrete_test":"Write out the missing limit interchange in Theorem 6.5: for fixed M, prove that sup_t lim_{R→∞} ∫_{B_R}∫ |χ1-χ2|^2 ζ_M(ξ) dξ dx is finite and equals the full-space integral, using only Proposition 6.4 and the local L^1 integrability of ρ_i; then pass M→∞ with a dominated-convergence argument and explicit constants C_M(∫Ψ(ρ0)+∥g∥^2_{L^2}). If the R-limit cannot be bounded this way, replace the 'monotone convergence' step in the conclusion of Theorem 6.5 with a rigorous localisation argument; otherwise Theorem 8.3 remains unsupported.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The matching upper and lower bounds in Theorem 1.2 are reduced by Theorem 8.3 to equality of the rate functions, which rests on the uniqueness theorem 6.5 for renormalised kinetic solutions with initial data in Ent_{Φ,γ}. In the proof of Theorem 6.5, the quantity I^{ε,δ,M,R}_t of (6.6) is shown to have a nonpositive time derivative after ε,δ→0 up to the terms (6.19)-(6.23), and then the cutoffs R→∞ and M→∞ are removed by 'monotone convergence'. This is not justified for the intended data class: ρ1,ρ2 need not lie in L^1(R^d), so the final conclusion max_t ∥ρ1(·,t)-ρ2(·,t)∥_{L^1(R^d)}=0 is not even meaningful. Moreover, the localised difference ∫|χ1-χ2|^2 ζ_M φ_R need not be bounded uniformly in R for fixed M: under finite relative entropy, the set of x where both densities lie in [M^{-1},2M] can have infinite volume. The derivative estimates only show that cutoff errors vanish asymptotically; no uniform-in-R (and later in-M) bound or integrable majorant is supplied that would allow passage to the limit in the time-integrated inequality. Thus uniqueness—and hence the equality I=I^{lo} in Theorem 8.3—is not established as written. At best a separate local uniqueness statement would need to be proven, and the rate-function identification does not follow from the present text.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proves matching large-deviation upper and lower bounds for the hydrodynamic rescaling of the zero-range process on the whole space Z^d, in every dimension d≥1, under the canonical hypotheses (A1)-(A2) on the jump rate. The proof has two main parts. The first is probabilistic: a superexponential estimate (Theorem 3.3) adapted to infinite volume and d≥2, based on a new local-averaging restriction (Lemma 3.5) that replaces the unavailable global averaging used on the torus. The second is analytic: an extension of the skeleton equation theory of [FG23] to initial data with finite relative entropy with respect to a constant density, including well-posedness of renormalised kinetic solutions (Theorem 6.5), equivalence of weak and kinetic solutions under a novel 'defective concavity' estimate (Lemma 7.4), and equality of the variational rate functions I=I^lo=I^up (Theorem 8.3). Theorem 1.2 is then deduced from the partial large deviation principle of Theorem 3.1 and the rate-function identification.","tokens_in":40020,"tokens_out":9344,"duration_ms":121952,"significance":"If the central claims hold, this resolves a long-standing open problem from [KL99] by giving matching bounds in the whole space without global convexity/concavity assumptions on the nonlinearity. The probabilistic contribution is substantial: the two-block estimate in arbitrary dimension is genuinely new in infinite volume, and the local-typicality Lemma 3.5 is an elegant replacement for the torus averaging argument. The analytic contribution is also significant: the defective-concavity inequality (7.7) and the infinite-volume renormalised-solution theory extend [FG23] in a non-obvious way. The paper is detailed, self-contained in its main estimates, and free of fitted parameters. However, the proof of the central uniqueness theorem contains a gap in the passage to the global L^1 limit, and the statement of Theorem 3.3 contains a limit-direction typo. These issues are local and likely repairable, but they currently affect the identification of the rate functions.","major_comments":[{"comment":"The conclusion 'max_{t∈[0,T]} ∥ρ1(·,t)-ρ2(·,t)∥_{L^1(R^d)}=0' is not justified for initial data in Ent_{Φ,γ}(R^d), since such data need not lie in L^1(R^d). The quantity I^{ε,δ,M,R}_t defined in (6.6) is only localised by φ_R and ζ_M; for fixed M the integral ∫ |χ1-χ2|^2 ζ_M can be infinite after R→∞ because the level set where both densities lie in the support of ζ_M may have infinite Lebesgue measure under only a finite relative entropy condition. The argument removes the cutoffs ε,δ, then R, then M using 'monotone convergence', but monotone convergence does not apply to a limit that is infinite and no uniform-in-R bound or integrable majorant is supplied. The derivative estimates (6.19)-(6.23) only show that cutoff errors vanish asymptotically; they do not provide the integrated inequality I^{M,R}(t) ≤ I^{M,R}(0)+∫_0^t error(R,s) ds from which one could let R→∞ and obtain local uniqueness, e.g. ∫_{B_K}|ρ1-ρ2|=0 for each K. As written, uniqueness of renormalised kinetic solutions is therefore not established, and Theorem 8.3, which uses Theorem 6.5 to identify the weak limit in Step 3, inherits this gap. The proof should be repaired by proving a localised uniqueness statement and then using it in Theorem 8.3; for the rate-function equality, local L^1_loc uniqueness is sufficient.","section":"Section 6.2, proof of Theorem 6.5"},{"comment":"The limit in (3.10) is written as lim sup_{ϵ→∞}, but the averaging scale ϵ is macroscopic and should go to 0; this is consistent with (3.13), with the proof's discussion of ℓ,ε,N→∞, and with the use of ε→0 in Lemma 4.2. As printed, the statement is the wrong direction for the cutoff.","section":"Theorem 3.3, Eq. (3.10)"},{"comment":"The proof asserts that the minimising control g in the definition of I(ρ) is unique. This is not proved and does not obviously follow from convexity of the admissible set without a closedness argument. The proof actually only needs that the weak limit g̃ is a minimizer, not that it equals the original g, so the uniqueness assertion is unnecessary; it should be removed or replaced by a direct argument that ∥g̃∥_{L^2}=∥g∥_{L^2} makes g̃ admissible and hence a minimizer.","section":"Theorem 8.3, Step 3"}],"minor_comments":[{"comment":"The initial data ρ0,n,R are defined using a spatial cutoff φ_R, but the displayed formula writes 'ρ0,n' without the R dependence; the notation should be consistent throughout Step 1 and Step 2.","section":"Section 8, Step 1"},{"comment":"In the line 'I(ρn,R) = 1/2 ∫ Φ(ρn)|∇Hn,R|^2', the argument of Φ should be ρn,R, not ρn.","section":"Section 8, Step 3"},{"comment":"The text says 'as in the proof of Theorem 6.5 we first pass to the limit R→∞' inside the proof of Theorem 6.5 itself; this self-reference should be corrected.","section":"Section 6.2, Eq. (6.22)"},{"comment":"The kinetic measure p is defined on R^d × (0,∞) × [0,T], but the equation in item (3) integrates p over ξ∈R; the domain of ξ-integration should be stated consistently, with p extended by zero or restricted to positive ξ.","section":"Definition 6.1"}],"recommendation":"major_revision","confidential_remarks":"The paper is a strong contribution and the overall strategy is credible, but the uniqueness gap in Theorem 6.5 is load-bearing for the rate-function identification and must be fixed before the main theorem can be accepted. I would suggest the authors add a localised uniqueness statement for renormalised kinetic solutions in Ent_{Φ,γ}, proved via an integrated cutoff inequality, and adjust Theorem 8.3 and its Step 3 accordingly. The theorem statement typo in Theorem 3.3 should also be corrected."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"This paper makes a serious run at a longstanding open problem and introduces tools that are worth keeping. The probabilistic side is the most convincing part: Lemma 3.5 is a clever replacement for the lost torus averaging in the two-block estimate, and Proposition 3.4 nicely explains why the standard path argument must fail in d≥2. The defective concavity lemma (Lemma 7.4) is also new and plausibly removes the convexity/concavity assumptions that blocked earlier work. If the analytic machinery holds up, Theorem 1.2 would be a major result.\n\nThe soft spot is Theorem 6.5. The proof tracks a localized difference with cutoffs in space and in the kinetic variable, and it shows, after letting ε,δ→0, that the time derivative of this localized quantity is bounded by terms that vanish as the cutoffs are removed. What is missing is a legitimate passage to the limit R→∞, M→∞. The initial data live in Ent_{Φ,γ}, so they need not be globally L1, and the final line claiming max_t ∥ρ1−ρ2∥_{L1(Rd)}=0 is not even meaningful for that class. More concretely, the derivative bounds do not give a uniform-in-R (or later in-M) control that would justify integrating the differential inequality and then taking the cutoff limits. The phrase \"monotone convergence\" is doing more work than the estimates support. That is a load-bearing gap because Theorem 8.3 inherits it: without genuine uniqueness of renormalised kinetic solutions, the rate-function equality does not follow.\n\nI do not think this is a cosmetic error. But it may be repairable. A local L1 uniqueness statement, or a weighted-L1 contraction, would likely be enough for the applications in Section 8, and the existing estimates (Proposition 5.5, local entropy bounds) could be reorganized to yield it. There are also small typos—Theorem 3.3 sends ϵ→∞ instead of ϵ→0—that should be swept up.\n\nBottom line: the paper deserves serious refereeing, not desk rejection. The probabilistic contributions and the PDE framework are substantial enough that even a conditional acceptance with a request for a repaired uniqueness proof is reasonable. I would want to see the gap closed before relying on Theorem 1.2.","headline":"Strong paper with a genuinely new probabilistic technique and a real gap in the infinite-volume uniqueness proof; the main theorem is conditional on a fix.","tokens_in":40497,"tokens_out":7450,"would_cite":false,"duration_ms":89808,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["60F10","60K35","82C22","35K55"],"pacs":[],"model":"deepseek-v4-flash","headline":"The zero-range process on Z^d satisfies a full large deviation principle in the whole space for every dimension d≥1, with rate function given by relative entropy plus an optimal L2 control, under only the canonical jump-rate hypotheses…","keywords":["zero-range process","large deviations","hydrodynamic limit","skeleton equation","renormalised kinetic solutions","superexponential estimate","infinite volume","entropy dissipation"],"falsifier":"Compute, for a jump rate satisfying (A1)–(A2), two renormalised kinetic solutions of ∂tρ = Δφ(ρ) − ∇·($φ^{{1/2}}$(ρ)g) on R^d with the same initial datum ρ0 ∈ Ent_{φ,γ} and the same control g, and show their L1 distance on a bounded ball is positive at some time; Theorem 6.5 asserts this distance is zero. A more targeted check is whether the terms in (6.22) involving Δφ_R and |∇φ_R| fail to vanish as R→∞ for some admissible φ; the equality of rate functions in Theorem 8.3 depends on that vanishing.","tokens_in":39428,"feed_emoji":"🎲","tokens_out":6705,"duration_ms":68090,"temperature":0.7,"pith_summary":"The paper aims to close the long-standing gap between the upper and lower large-deviation bounds for the hydrodynamic rescaling of the zero-range process on Z^d, for every d≥1 and in the whole space. The authors claim that, under the canonical jump-rate hypotheses (A1)–(A2), the large deviation rate function can be taken to be the relative entropy of the initial state plus half the squared L2 norm of the optimal control driving the skeleton equation. To get there, they prove that the superexponential estimate holds in arbitrary dimension and infinite volume, and that the skeleton equation has a well-posed theory on the whole space for initial data with finite relative entropy but infinite mass. A sympathetic reader would care because matching bounds are exactly what turns a variational candidate into a genuine large deviation principle, and the proof removes the convexity/concavity restrictions that limited earlier treatments.","feed_headline":"Matching large deviation bounds for zero-range process on Z^d","feed_subtitle":"Upper and lower bounds agree in the whole space for every dimension, closing a problem opened in 1999.","key_machinery":"The load-bearing object is the skeleton equation ∂tρ = Δφ(ρ) − ∇·($φ^{{1/2}}$(ρ)g), together with its kinetic formulation, in which the kinetic function χ(ρ,ξ)=1_{0<ξ<ρ} satisfies a transport equation with a nonnegative parabolic defect measure p satisfying δ0(ξ−ρ)φ′(ρ)|∇ρ|² ≤ p. Uniqueness for this equation is proved in the class of renormalised kinetic solutions by a relative-entropy/kinetic comparison that cuts off large velocity and distant space, using interpolation and Sobolev estimates to control the spatial cutoffs at infinity. The probabilistic half of the argument is carried by a new restriction lemma (Lemma 3.5) that limits the superexponential variational problem to densities whose Dirichlet-form contributions are uniformly typical, dx,y(f) ≤ $zN^{{-2}}$, with an error vanishing as z→∞; this substitutes for the translation averaging that previously forced compact geometry. A 'defective concavity' estimate, ($φ^{{1/2}}$(u))^ε ≤ ϑ $φ^{{1/2}}$(u^ε), replaces the global convexity or concavity of $φ^{{1/2}}$ and makes the weak-to-kinetic passage work for zero-range nonlinearities.","core_discovery":"On the paper's own terms, the central discovery is Theorem 1.2: if the local jump rate λ satisfies (A1)–(A2) and the zero-range process starts from the invariant measure ν_γ, then for every open U, liminf $N^{{-d}}$ log P(π^N ∈ U) ≥ -inf_{π∈U} I(π), and for every closed A, limsup $N^{{-d}}$ log P(π^N ∈ A) ≤ -inf_{π∈A} I(π). The rate function I assigns to a trajectory the relative entropy H_Φ(π0|γ) plus 1/2 of the infimum of ||g||²_{L²} over controls for which ρ solves the skeleton equation ∂tρ = Δφ(ρ) − ∇·($φ^{{1/2}}$(ρ)g). This equality of upper and lower rate functions is obtained by proving that the weak and renormalised kinetic solution theories of the skeleton equation coincide on the whole space, so that the lower bound can be extended from smooth fluctuations to all trajectories, and by removing the global convexity/concavity assumptions on $φ^{{1/2}}$ through a defective-concavity estimate.","pith_inferences":["Editorial inference: the defective-concavity estimate is not tied to zero-range structure and could plausibly be carried over to other parabolic-hyperbolic equations whose diffusivity is only locally elliptic.","Editorial inference: Lemma 3.5's typicality cutoff suggests a general recipe for infinite-volume interacting particle systems: replace global averaging over a transitive group by local averaging across the scale of the test function; one could test this on symmetric exclusion or Kac models.","Editorial inference: because the rate-function equality is proved through uniqueness of kinetic solutions, any future model where weak and kinetic solutions diverge would automatically exhibit a mismatch between upper and lower large-deviation bounds, along the lines of earlier counterexamples.","Editorial inference: in dimensions d≥3 the oscillating-profile example in Proposition 3.4 indicates that the two-block estimate genuinely fails without extra regularity; a quantitative version of that failure could inform whether higher-order corrections appear at the level of prefactors."],"forward_implications":["The full large deviation principle for the zero-range process in the whole space holds in every dimension d≥1, with the explicit rate function I of Definition 1.1.","The superexponential estimate is now available in arbitrary dimension and infinite volume, so the two-block estimate no longer requires the one-dimensional Sobolev embedding or a compact torus.","The skeleton equation admits unique renormalised kinetic solutions for initial data with finite relative entropy but infinite total mass, a class forced by the infinite-volume equilibrium.","Weak solutions and renormalised kinetic solutions coincide under hypotheses (A1)–(A2), so the upper-bound rate function, the lower-bound rate function, and the variational rate function are all equal.","No separate global convexity or concavity condition on φ^{1/2} is needed; the mild spectral-gap and Lipschitz conditions on the jump rate suffice."],"supporting_citations":[{"why":"Supplies the original finite-volume/d=1 large-deviation bounds and the one-block/two-block framework that the paper generalises.","marker":"[BKL95a]"},{"why":"Provides the Feynman-Kac superexponential-estimate strategy that the paper adapts to infinite volume.","marker":"[KOV89]"},{"why":"States the problem of matching bounds and gives the torus treatment the paper aims to extend to Z^d.","marker":"[KL99]"},{"why":"Treats general dimensions on the torus via averaging, the step that fails in the whole space.","marker":"[QR V99]"},{"why":"Establishes weak-solution well-posedness of the skeleton equation in finite volume under convexity/concavity, used as the analytic template here.","marker":"[FG23]"},{"why":"Gives examples where a candidate rate function fails to capture the LDP, motivating the need for matching bounds.","marker":"[Hey23]"},{"why":"Supplies the Schauder and elliptic regularity estimates used to construct the smooth approximating fluctuations in Theorem 8.3.","marker":"[LSU67]"},{"why":"Supplies the kinetic formulation of degenerate parabolic-hyperbolic equations on which the renormalised solution theory is based.","marker":"[CP03]"}],"fun_headline_variants":["Zero-range process: matching large deviations in whole space","Whole-space zero-range process gets matching large deviation bounds","Zero-range process: large deviation bounds match on Z^d, any dimension","Matching large deviation bounds for zero-range process on Z^d, any d"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the spatial cutoffs at infinity in the uniqueness proof of the skeleton equation vanish using only local entropy dissipation, the L2 control g, and interpolation, even though the initial data have infinite total mass.","fun_headline_variants_meta":{"raw":{"variants":["Zero-range process: matching large deviations in whole space","Whole-space zero-range process gets matching large deviation bounds","Zero-range process: large deviation bounds match on Z^d, any dimension","Matching large deviation bounds for zero-range process on Z^d, any d"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000782,"raw_usage":{"total_tokens":3430,"prompt_tokens":899,"completion_tokens":2531,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":515,"completion_tokens_details":{"reasoning_tokens":2457}},"tokens_in":515,"tokens_out":2531,"duration_ms":20776,"temperature":1.0,"reasoning_tokens":2457,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T10:43:40.201725+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute, for a jump rate satisfying (A1)–(A2), two renormalised kinetic solutions of ∂tρ = Δφ(ρ) − ∇·($φ^{{1/2}}$(ρ)g) on R^d with the same initial datum ρ0 ∈ Ent_{φ,γ} and the same control g, and show their L1 distance on a bounded ball is positive at some time; Theorem 6.5 asserts this distance is zero. A more targeted check is whether the terms in (6.22) involving Δφ_R and |∇φ_R| fail to vanish as R→∞ for some admissible φ; the equality of rate functions in Theorem 8.3 depends on that vanishing.","supporting_citations":[],"review_version":1}