{"id":"4694a5cf-4963-4c1a-bf3b-98658c50d47e","arxiv_id":"2607.21443","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The averaging process has KPZ critical scale λ=7/8 rather than the moment-criterion value 3/4, with tilted fields converging to the multiplicative SHE of noise 1/√2.","lead":"The averaging process—where neighboring piles of mass repeatedly split the difference—has random KPZ-type fluctuations at a different scale than the standard moment rule predicts. This paper proves the true critical scale is 7/8 (not 3/4) and that the rescaled field converges to the multiplicative stochastic heat equation with noise coefficient 1/√2.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Central theorem depends on unproved continuous-time analogues and an imported moment characterization; if either fails, Theorem 1.1 collapses.","rationale":"The paper’s main computation is detailed and internally coherent: the two-point discrepancy, Green function, Dobrushin scaling, and the resulting moment limits are carefully derived. The central weakness is exactly the dependency flagged by the reader: the continuous-time analogues of [DP25] are stated without proof, and the final identification of the limit uses the imported [Par25] characterization. These are not cosmetic; Lemma 3.4 supplies the uniform exponential bounds needed for convergence of moments, and Lemma 3.5 supplies the local-time limits for the nonzero-mean Φ2 term. In particular, for k≥3 the tilted k-point motion is not reversible and pair gaps are not autonomous, so Lemma 3.5 is not a routine translation of the discrete-time reversible case. I also noted a secondary issue in Theorem 1.3’s supercritical proof: inequality (7.1) is not obviously valid pointwise on paths that visit ±1 without 0, but Theorem 1.1 does not depend on that part, so I do not base the verdict on it. Overall, the concern does not move the reader’s CONDITIONAL verdict; it confirms it.","tokens_in":44268,"tokens_out":53118,"duration_ms":469730,"concrete_test":"Independently re-derive Lemma 3.5 for k=3, β_N∼N^{-1/8}, h=1_{0}, using the explicit generator in Proposition 2.9 and the V_{β,k} formula (4.13). Verify that N^{-1/2}∫_0^{tN} h(R^1_s−R^2_s) ds, under P^{(β_N,3)}_{x_N}, converges in law to L^{W^1−W^2}_t jointly with N^{-1/2}(R^1_{tN}−R^2_{tN}), and that the error from configurations where a third coordinate is within distance 1 is o_P(1). If this invariance principle fails for this basic h, the moment limits in Proposition 4.4 break.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Section 3.2 imports Lemmas 3.4–3.5 as “continuous-time analogues” of [DP25] with no proof. Lemma 3.5 asserts a pair-gap invariance principle for the tilted k-point motion, but for k≥3 the tilted k-point motion is nonreversible and the pair gap R^i−R^j is not Markovian; the claimed coefficient γ2(h)=Σh and the joint convergence to (W,γ2(h)L^W) require a nontrivial estimate on time spent near third coordinates, exactly the kind of estimate the paper later develops only for heat kernels. Lemma 3.4’s exponential occupation-time bound is used to justify uniform integrability in Proposition 4.4 and to discard triple collisions; without it, the moment asymptotics (4.11) are not justified. These two lemmas, together with the imported moment-based characterization Proposition 5.1 from [Par25], are the load-bearing bridge from k-point motion estimates to the SHE limit: if any one fails, Proposition 5.2 and hence Theorem 1.1 fail.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the one-dimensional averaging process viewed as a random walk in a space-time random environment, and considers tilted quenched density fields Y_{λ,N} at spatial scale N^λ with diffusive window N^{1/2}. The main theorem (Theorem 1.1) states that at λ=7/8 the field converges in law in D([0,∞); M_f(R)) to the Itô–Walsh solution of ∂_t Y = (1/2)∂_z^2 Y + (1/√2)Y ξ with initial condition δ_0. The proof is built on a moment representation via tilted k-point motions, a Dobrushin-type local-time invariance principle for mean-zero additive functionals, refined heat-kernel and collision estimates, and an imported moment-based characterization of multiplicative SHE propagators. The paper also states Theorem 1.3: for λ<7/8 the variance of the tilted field vanishes, while for λ>7/8 it diverges.","tokens_in":44578,"tokens_out":5998,"duration_ms":62610,"significance":"If Theorem 1.1 is correct, the paper gives the first rigorous example where the true KPZ critical exponent λ_c=7/8 exceeds the moment-criterion value λ_{c,1}=3/4, and it exhibits a new mechanism in which a mean-zero Dobrushin fluctuation and an ordinary local-time contribution jointly produce the SHE noise coefficient. The moment computations are explicit and the noise coefficient 1/√2 is derived from σ^2(Φ_1)=2 and γ_2(Φ_2)=1, not fitted. The proof contains substantial nontrivial material: the renewal-equation estimates for the gradient field, the heat-kernel bootstrap in Lemma 4.1, and the tightness argument via Aldous criteria. These are genuine strengths. However, the paper’s central limit identification depends on several results that are either imported without proof or only sketched, and one step in the supercritical variance divergence appears incorrect as written.","major_comments":[{"comment":"Lemmas 3.4 and 3.5 are stated as continuous-time analogues of results from [Par26, DP25] but no proof is given. These lemmas are load-bearing: Lemma 3.4 supplies the uniform integrability used in Proposition 4.4 and the exponential occupation-time bound used throughout, and Lemma 3.5 is used in Proposition 3.7 and Theorem 3.6 to identify γ2(h)=Σh and the joint local-time limits. For k≥3 the tilted k-point motion is non-reversible and the pair gap R^i−R^j is not Markovian; the asserted scalings require control of time spent near third coordinates. The paper later develops exactly such estimates in Corollary 4.2, but those are proved after Lemma 3.5 is invoked, and the relevant k≥3 statement is not literally contained in the cited discrete-time/reversible results. The authors should either provide full proofs of Lemmas 3.4 and 3.5 in the continuous-time, tilted, k≥3 setting, or state preci","section":"§5, Proposition 5.1"},{"comment":"The proof of the supercritical half of Theorem 1.3 (λ>7/8) uses the claimed almost-sure inequality E_{βλ,N,2}(T) ≥ (E_{β7/8,N,2}(T))^{N^ε}. This inequality is not valid because Φ_1 takes negative values: for a path that spends most of its time at gap ±1, the exponent on the left contains −(1/2)(c_{βλ}−1)T, while the right-hand exponent is approximately −(1/2)N^ε(c_{β7/8}−1)T. Since c_{βλ}−1 ≈ N^{2ε}(c_{β7/8}−1), the left side is smaller, not larger, for large N. Thus the divergence of Var(Y_{λ,N}(t,1)) for λ>7/8 is not established by this argument. The supercritical part of Theorem 1.3 needs a different proof (or should be removed from the main claims if it is not needed for Theorem 1.1).","section":"§7, Eq. (7.1)"},{"comment":"The proof of Theorem 4.3 begins: 'By the general results of [DP25], the first marginals satisfy...' and then uses Lemma 3.4/3.5-type bounds for the non-reversible tilted process. Since Theorem 4.3 is the key step for all higher moments k≥3, the paper should isolate the precise external theorem being applied and verify its hypotheses, especially the short-range interaction and uniform mixing conditions for the β-tilted process with |β|≤C N^{-1/8}. If the continuous-time extension is not already in [DP25], this is a substantial missing proof rather than a routine adaptation.","section":"§4.2, Theorem 4.3"}],"minor_comments":[{"comment":"The title as printed reads 'A VERAGING' with an extra space/letter; presumably should be 'AVERAGING'.","section":"Title"},{"comment":"The paragraph after Theorem 1.3 states that the subcritical fluctuation regime is 'well understood' via [DP25], but no precise theorem or verification of hypotheses is supplied. Since these statements are not used in the proof of Theorem 1.1, they could be compressed or made conditional.","section":"§1.4.2"},{"comment":"In the compact-containment proof, the family X^α_N(t) is called a martingale with mean one; this is correct for each fixed α, but the argument that the exponential prefactor is uniformly bounded for t∈[0,T] should spell out the cancellation of the linear term more explicitly, since that cancellation is essential.","section":"§6.4"},{"comment":"In Definition 2.2, the displayed line 'p_t(y−x)=p_t(x−y)=p^{(1)}_t(x,y)' has a probability kernel on the left and a transition probability on the right; this is understandable but slightly abusive notation. Also, the phrase 'in Z^k' after 'p^{(1)}_t' is a typo for 'on Z'.","section":"§2.1"}],"recommendation":"major_revision","confidential_remarks":"The main theorem is likely correct and the paper contains substantial new ideas, but the current version relies on two unproved imported lemmas that are load-bearing for Theorem 1.1, and the supercritical half of Theorem 1.3 contains an apparently invalid inequality. I recommend major revision: the authors should supply or precisely reference the missing continuous-time estimates, and repair or restrict the supercritical claim. I would be willing to review a revised version."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The main news is real: the averaging process has critical KPZ scale 7/8, not the 3/4 the moment criterion would predict, and the tilted density field converges to the multiplicative SHE with coefficient 1/√2. This is the first concrete example where the general moment criterion is not sharp, and the two-mechanism explanation (mean-zero Φ1 Dobrushin term plus nonzero Φ2 local-time term) is convincing. The moment asymptotics and tightness argument are carefully done; the proof of the k-moment limits via tilted k-point motions and the triple-collision estimates in Section 4 is the core technical contribution and looks solid.\n\nThe soft spots are real but manageable. Lemmas 3.4 and 3.5 are imported as continuous-time analogues of [DP25] without proof. Lemma 3.5 is effectively proved later in Theorem 4.3, including the k≥3 non-Markovian case, so that concern is less serious than it first appears. Lemma 3.4, the exponential occupation-time bound, remains unproved and is load-bearing for uniform integrability and for Lemma 4.1. It is a plausible technical estimate, but a serious referee would ask for a proof or a precise statement of the discrete-time result it is adapted from. The supercritical half of Theorem 1.3 is sketched: the proof for general test functions is left to the reader, and the argument for ϕ≡1 uses an inequality (7.1) that is sensible but not fully detailed. This is a secondary result, so the sketch is acceptable but should be filled in.\n\nThe imported moment characterization Proposition 5.1 from [Par25] is used as a black box. That is fine if the source is reliable; it is a published or at least widely circulated result. No circularity or fitting found. The citation pattern is honest.\n\nOverall: this is a significant paper for the probability community. It deserves a serious referee, not a desk reject. The referee should focus on the continuous-time analogues and ask for more details on Lemma 3.4 and the supercritical proof. I would cite it and bring it to reading group.","headline":"Genuinely new result — the averaging process has critical KPZ scale 7/8, not 3/4 — but the proof leans on unproved continuous-time analogues and a sketchy supercritical argument; still deserves a serious referee.","tokens_in":45018,"tokens_out":5455,"would_cite":true,"duration_ms":52784,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["60K35","82C22","35R60","60F17"],"pacs":[],"model":"deepseek-v4-flash","headline":"At scale 7/8, the averaging process converges to a multiplicative SHE with noise coefficient 1/√2, above the predicted 3/4.","keywords":["averaging process","KPZ equation","multiplicative stochastic heat equation","random walk in random environment","critical exponent","Dobrushin local time","tilted k-point motion","interacting particle system"],"falsifier":"Compute, by simulation or exact diagonalization, the second moment E[Y_{7/8,N}(t,1)²] for large N at λ = 7/8: if it does not approach E[ exp( L_t^{BM}/2 ) ] — equivalently, if the limiting noise coefficient is not 1/√2 — the central claim is false.","tokens_in":44191,"feed_emoji":"🧮","tokens_out":4110,"duration_ms":43023,"temperature":0.7,"pith_summary":"The paper's central claim is that the averaging process — the interacting particle system in which neighbouring masses are replaced by their arithmetic mean whenever a Poisson clock rings — has a KPZ critical exponent of 7/8, not the 3/4 that the standard moment criterion predicts. At that scale the tilted quenched density field converges in law to the multiplicative stochastic heat equation with noise coefficient 1/√2, starting from a Dirac mass. The reason the criterion fails is a degeneracy: the first random moment produces a mean-zero correction, so its leading local-time contribution vanishes; the actual critical scale is set by the interplay of that mean-zero fluctuation and the ordinary second-order contribution. The paper also pins down the surroundings of the critical scale: below 7/8 the field concentrates around the annealed profile, above it the second moment diverges, signalling intermittency.","feed_headline":"Averaging process hits KPZ at 7/8, not 3/4","feed_subtitle":"Two fluctuation mechanisms combine to make the true critical exponent 7/8, where the field converges to the multiplicative SHE.","key_machinery":"The central object is the gap process X = R¹ − R² of the tilted two-particle motion, a symmetric random walk whose tilted coordinates have opposite drifts but whose gap is unbiased. Its exponential functional decomposes into Φ₁(x) = 1_{x=0} − ½(1_{x=1} + 1_{x=−1}), a mean-zero function that produces fluctuations at the finer N^{−1/4} scale through a classical local-time invariance principle for mean-zero additive functionals, and Φ₂(x) = 1_{x=0}, a non-zero-mean function producing the standard N^{−1/2} local time. Both terms reach order one exactly at λ = 7/8 and their combination yields the noise coefficient 1/√2. For higher moments the argument extends to tilted k-point motions, with heat-","core_discovery":"Theorem 1.1 establishes that for λ = 7/8 the normalized tilted density field Y_{λ,N} of the averaging process converges in law in D([0,∞); M_f(R)) to the unique Itô–Walsh solution of ∂_t Y = ½ ∂_z² Y + (1/√2) Y ξ, with initial condition δ_0. Because 7/8 exceeds the moment-criterion value λ_{c,1} = 3/4, this is the first example in the RWRE setting where the actual critical KPZ scale is strictly larger than the one predicted by the general moment criterion. The critical contribution arises from two simultaneous fluctuation mechanisms, one driven by a mean-zero additive functional and one by a non-zero-mean functional, which become critical at the same scale 7/8.","pith_inferences":["The same two-contribution mechanism should appear in any averaging-type model where the update rule forces quenched gradients to be locally smooth; a testable prediction is that block-averaging variants (averaging over intervals of size k ≥ 2) also exhibit a critical exponent larger than the moment-criterion value.","If the cancellation of the first-order local-time term is robust, then the phenomenon extends to higher-order hierarchies only when a mean-zero p-th level and a non-zero-mass 2p-th level resonate; the paper's formal reasoning suggests such resonances may be impossible under space-time i.i.d. environments for p ≥ 2.","A direct simulation or exact computation of the second moment of Y_{7/8,N} — checking that it approaches the SHE prediction exp(L_t^{BM}/2) averaged over Brownian motion — would give a clean numerical test of the whole critical-scale picture."],"forward_implications":["The averaging process has critical KPZ exponent λ_c = 7/8, not the 3/4 given by the general moment criterion, providing the first counterexample to the identification λ_c = λ_{c,1}.","At λ < 7/8 the tilted field concentrates around the local Gaussian profile, while at λ > 7/8 the second moment diverges, sharply separating subcritical from supercritical behaviour.","The limiting noise coefficient in the multiplicative SHE is the sum of a mean-zero Dobrushin contribution and an ordinary local-time contribution; each component alone would predict a different scale.","Subcritical fluctuations of the tilted field, for λ ∈ (1/2, 7/8) and λ ≤ 1/2, are described by additive SHE limits with explicit noise coefficients, as adaptations of earlier methods would show."],"fun_headline_variants":["Averaging process's KPZ scale is 7/8, not 3/4","Moment criterion fails: averaging process KPZ scale is 7/8","Why averaging process KPZ scale is 7/8, not 3/4","Two fluctuation mechanisms set averaging process KPZ scale to 7/8","Averaging process: KPZ limit appears at 7/8, beyond predicted 3/4"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The proof depends on two imported estimate packages: the continuous-time analogues of the heat-kernel and occupation-time bounds for tilted multi-particle motions, and the moment-based uniqueness characterization of multiplicative SHE propagators; if either fails at the required N^{−1/4} scale, the finite-dimensional convergence and hence the main theorem collapse.","fun_headline_variants_meta":{"raw":{"variants":["Averaging process's KPZ scale is 7/8, not 3/4","Moment criterion fails: averaging process KPZ scale is 7/8","Why averaging process KPZ scale is 7/8, not 3/4","Two fluctuation mechanisms set averaging process KPZ scale to 7/8","Averaging process: KPZ limit appears at 7/8, beyond predicted 3/4"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001222,"raw_usage":{"total_tokens":4868,"prompt_tokens":754,"completion_tokens":4114,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":498,"completion_tokens_details":{"reasoning_tokens":4002}},"tokens_in":498,"tokens_out":4114,"duration_ms":26128,"temperature":1.0,"reasoning_tokens":4002,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-01T07:24:40.208042+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute, by simulation or exact diagonalization, the second moment E[Y_{7/8,N}(t,1)²] for large N at λ = 7/8: if it does not approach E[ exp( L_t^{BM}/2 ) ] — equivalently, if the limiting noise coefficient is not 1/√2 — the central claim is false.","supporting_citations":[],"review_version":1}