{"id":"7306accc-757f-4c94-9ede-48de6308e6ee","arxiv_id":"2608.11603","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":1,"one_line_summary":"Rescaled point-to-point partition functions of continuous-time polymers with correlated Brownian noise converge in law to the critical Stochastic Heat Flow with an explicit strength parameter.","lead":"The paper proves that rescaled partition functions of a class of two-dimensional continuous-time random polymers in a correlated Brownian environment converge to the critical Stochastic Heat Flow. The proof goes through convergence of all moments and then upgrades to convergence in law on path space using a moment-based characterization.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The proof of Lemma 3.1(d) rests on an inequality labeled 'purely heuristic' in §3.3.1; without a rigorous proof, the diagonal resolvent convergence and Theorem 1.1 are incomplete.","rationale":"The reader's weakest assumption points to Tsai's uniqueness theorem; that is a genuine external dependency, but it is a published result and not the paper's own gap. The most load-bearing concern in the manuscript itself is the admitted 'purely heuristic' inequality in §3.3.1, because it sits inside the proof of Lemma 3.1(d), the technical core of Theorem 1.1. Without a rigorous proof of that inequality, the convergence of the diagonal resolvent is not established, and the moment convergence used to verify the delta-Bose moments of subsequential limits is unsupported. The passage explicitly says 'a rigorous proof of this bound is possible,' which is an admission that the proof is incomplete as written. A referee can reasonably ask for that proof before acceptance. This does not change the reader's CONDITIONAL verdict; it reinforces it. I agree partially with the reader: the external uniqueness theorem is a valid concern, but the internal heuristic step is more immediately load-bearing because it affects the paper's own derivation, not just the cited characterization. The concrete check—proving the inequality with explicit constants and re-deriving (3.39)—would settle whether the concern actually lands.","tokens_in":60353,"tokens_out":5677,"duration_ms":53531,"concrete_test":"Prove the inequality h(x)=x tan^{-1}(1/x)−C log(1+x) ≤ 0 for all x>0 with an explicit constant C (e.g., C=2), or otherwise rigorously derive the bound leading to (3.39); then verify that the resulting logarithmic kernel is square-integrable against κ(y_r)κ(y'_r) using assumption (1.3). If the inequality cannot be established, Lemma 3.1(d) is unproven and Theorem 1.1 collapses.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Section 3.3.1 derives the convergence of the prelimiting diagonal resolvent K^{θ}_{εα}(λ) to K^{θ*}_α(λ), which is indispensable for Theorem 1.1 and hence for the moment convergence used in Proposition 1.2. In estimating D_{2,1}(ε, η_{c∪[n]\\α}), the text introduces h(x)=x tan^{-1}(1/x)−C log(1+x), claims h(x)≤0 for large x, and states: 'We stress that this is purely heuristic and a rigorous proof of this bound is possible.' This bound is then used to obtain the logarithmic factor leading to (3.39), whose integrability against κ(y_r)κ(y'_r) under assumption (1.3) is essential for the dominated convergence argument. Since the paper explicitly labels a load-bearing estimate as heuristic, the proof of diagonal operator convergence is not complete as written. The later skipped proofs (§3.2 off-diagonal error terms; §4 Kolmogorov inequality 'same as [46]') compound the issue, but the heuristic inequality is the sharpest single gap because it is internal and admitted.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies a class of two-dimensional continuous-time directed polymers in a correlated Brownian environment. It defines the diffusively rescaled point-to-point partition functions and shows, via a resolvent argument adapted from Gu–Quastel–Tsai, that all moment semigroups converge to the delta-Bose semigroup with an explicitly computed parameter θ*. It then verifies tightness and the axioms of Tsai's moment-based characterization of the critical Stochastic Heat Flow, concluding that the partition functions converge in law to SHF(θ*) in C(R²_<, M_+(R²×R²)). The main results are Theorem 1.1 (semigroup/resolvent convergence with explicit θ*) and Proposition 1.2 (path-space convergence to the SHF).","tokens_in":60521,"tokens_out":8293,"duration_ms":77988,"significance":"If the proof is completed, this is a substantial contribution: it enlarges the class of discrete/semi-discrete polymer models known to converge to the critical Stochastic Heat Flow, gives an explicit formula for the limiting parameter θ*, and demonstrates the power of the moment-based axiomatic route. The paper is not circular: the delta-Bose semigroup and the uniqueness of the SHF are taken as external inputs from [33] and [46], and θ* is derived rather than fitted. The argument is detailed and the operator-norm framework is appropriate. However, several load-bearing estimates are either labeled heuristic or deferred to previous works, and the constant in Theorem 1.1 appears to contain a factor error; these issues need to be repaired before the claims can be accepted.","major_comments":[{"comment":"The convergence of the diagonal resolvent, and hence of the full resolvent in Theorem 1.1, rests on the bound h(x)=x tan^{-1}(1/x)-C log(1+x)≤0 for large x. The paper explicitly labels this bound 'purely heuristic' and uses it to obtain the logarithmic factor displayed in (3.39), whose square-integrability against κ(y_r)κ(y'_r) under assumption (1.3) is essential for the dominated convergence argument. As written, the proof of this key inequality is missing; please supply a rigorous proof or an alternative bound before the diagonal convergence can be accepted.","section":"§3.3.1, around Eq. (3.39)"},{"comment":"The printed formula for θ* appears inconsistent with the derivation in §3.3.2. There the author obtains θ⋆=2θ+4π(θ1+θ2+θ3+θ4)+2logπ, with θ4=log2/(2π)−G/π^2, so the constant term should be 2log2−4G/π rather than log2/(2π)−G/π^2 as written in the theorem. Unless the notation θ4 is being used differently in the two places, the theorem statement needs correction. Since θ* is the explicit parameter identifying the limit, this is load-bearing.","section":"Theorem 1.1"},{"comment":"The norm convergence of the off-diagonal resolvents (3.1c) is asserted with the sentence 'We skip the details.' This is one of the four convergence statements in Lemma 3.1 that feed directly into the resolvent convergence of Theorem 1.1, so the omission is load-bearing. Please provide the full argument, or a precise reduction showing exactly how each error term O_r(R,ε) is bounded using Lemma 2.2.","section":"§3.2"},{"comment":"The Kolmogorov-type estimate E|⟨h,(Zθ,ε_{s,t}-Zθ,ε_{s',t'})h'⟩|^{2m}≤c(...) is stated and its proof is deferred with 'the proof for the above inequality is the same as in [46]. We skip this part.' This estimate is the basis of tightness and hence of Proposition 1.2. If the proof is genuinely identical, a precise statement with the required parameter mapping should be given; otherwise this is a gap in the main convergence argument.","section":"§4, after Lemma 4.1"}],"minor_comments":[{"comment":"The same letter K is used for the integer cutoff in the summation and for the value of the double sum; please disambiguate the two uses.","section":"§1.1, Eq. (1.3)"},{"comment":"The displayed identity writes the last term as Zθ,ε_{s,u}(⌊x′⌋ε,⌊x′⌋ε), which appears to be a typo for Zθ,ε_{s,u}(⌊x⌋ε,⌊x′⌋ε).","section":"§4, Lemma 4.2 proof"},{"comment":"The definition of the simplex Σ(t) is written as {τ_k/2 ...}, but the notation is inconsistent with the subsequent use of τ_k as the integration variables; please clarify.","section":"§2, display after (2.2)"},{"comment":"Lemma 2.2 is quoted from [33, Lemma 5.1] without proof; since it is used repeatedly in the off-diagonal estimates, a short proof or a more precise statement of the cited lemma would improve self-containedness.","section":"§2, Lemma 2.2"},{"comment":"The operator D2 is introduced in several forms (D2(ε), D2(ε,η_{c∪[n]\\α}), D2,1, D2,2) without a uniform convention; please make the dependencies explicit throughout the subsection.","section":"§3.3.1"}],"recommendation":"major_revision","confidential_remarks":"This is a promising manuscript with a plausible central argument and an explicit, falsifiable limiting parameter. The main obstacles are the unproved heuristic inequality in §3.3.1, the skipped off-diagonal and tightness proofs, and the apparent factor error in the constant of Theorem 1.1. These appear fixable within the scope of the paper, so I recommend major revision rather than rejection. The overlap with the work of Drillick–Hou–Parekh is acknowledged in a remark, and the citation pattern is appropriate."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear colleague,\n\nThe paper proves that rescaled partition functions of a broad class of continuous-time directed polymers in 2D, with correlated Brownian environments, converge in law to the critical Stochastic Heat Flow. The route is through moments: strong convergence of the delta-Bose semigroup via resolvents, tightness in C(R^2_<, M_+), and identification via Tsai's uniqueness theorem. The novelty is the resolvent comparison for non-compactly supported κ, with an explicit θ* that is derived rather than fitted. The paper is honest and mostly self-contained; it even states where it leans on [46] and [44].\n\nThe main soft spot is the one the stress-test note flags. In §3.3.1, the inequality x tan^{-1}(1/x) − C log(1+x) ≤ 0 for large x is introduced and explicitly labeled 'purely heuristic', then used as the key estimate to get the logarithmic bound (3.39). That bound is necessary for the dominated convergence argument that shows D_{2,1} vanishes against κ(y_r)κ(y'_r). Without a rigorous proof of this inequality, the convergence of the diagonal resolvent, and therefore Theorem 1.1, is incomplete as written. The inequality is likely true for a sufficiently large C, so this is a fixable gap rather than a fatal error, but a referee has to demand the proof.\n\nThe other gaps are milder. Section 3.2 skips the off-diagonal error estimates with 'we skip the details'; Section 4 takes the Kolmogorov tightness criterion from [46] verbatim. These are routine checks, but they should be written out to make the paper verifiable. There is no circularity: the identification step uses Tsai's characterization as an external theorem, and the self-citation [44] is to work the author actually uses.\n\nThe announced overlap with [26] reduces the novelty of finite-dimensional convergence, but this paper's moment-based path-space argument is a distinct contribution, and the result is worth having. My recommendation is to send it to peer review. The referee should focus first on §3.3.1 and require a proof of the heuristic inequality, then check the skipped estimates. After those are supplied, this will be a solid paper.\n\nBest.","headline":"A credible moment-based path-space convergence result for critical 2D polymers, but the proof of the diagonal resolvent bound contains an admitted heuristic inequality that must be repaired before the paper is complete.","tokens_in":61104,"tokens_out":4987,"would_cite":true,"duration_ms":49697,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["60K35","60H15","82B44","60F17"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves that diffusively rescaled partition functions of a broad class of two-dimensional continuous-time directed polymers converge in law to the Critical Stochastic Heat Flow SHF(θ*), with the parameter θ* given explicitly in…","keywords":["directed polymers","critical stochastic heat flow","two-dimensional stochastic heat equation","delta-Bose gas","resolvent method","moment convergence","tightness","critical window"],"falsifier":"Compute the two-point function of the rescaled polymer for a concrete choice of $\\kappa$ and $\\nu$ and compare it with the explicit two-particle delta-Bose semigroup with the paper's $\\theta^{*}$: the two-particle semigroup is explicit in center-of-mass and relative coordinates, so a mismatch beyond numerical error would falsify Theorem 1.1. A cheaper check is to verify the stated value of the constant $\\theta_4 = \\log 2/(2\\pi) - G/\\pi^{2}$ by direct numerical integration of $\\int_{\\hat{T}^{2}\\setminus B(0;\\pi)} \\|\\eta\\|^{-2}\\, d\\eta$.","tokens_in":60094,"feed_emoji":"🎲","tokens_out":10999,"duration_ms":101608,"temperature":0.7,"pith_summary":"The paper takes a class of two-dimensional continuous-time directed polymers in a correlated Brownian environment and shows that, in the critical window of couplings, their rescaled point-to-point partition functions converge in law to the Critical Stochastic Heat Flow (SHF), the conjectural solution object of the two-dimensional stochastic heat equation. The proof goes through moments: all n-point correlation functions of the polymer converge to the delta-Bose semigroup of the SHF, the family is tight on the space of continuous measure-valued processes, and the resulting limit points satisfy the four axioms of the flow's moment-based characterization. Because that characterization is unique, the whole family converges, not just along a subsequence. A sympathetic reader should care because this transfers a limit theorem previously available for mollified equations to a genuine discrete polymer model with non-compactly supported spatial correlations.","feed_headline":"2D directed polymers converge to the critical stochastic heat flow","feed_subtitle":"Moment-by-moment limits match the delta-Bose semigroup, so no other limit object survives.","key_machinery":"Two mechanisms carry the argument. The first is the resolvent expansion of the delta-Bose gas: the $n$-th moment semigroup is written as the free heat semigroup plus a diagram sum over all sequences of collisions, with incoming, outgoing, off-diagonal, and diagonal operators, following [33]; the paper's discrete version uses $\\varepsilon$-scaled center-of-mass and relative coordinates and proves the resolvents converge in norm, using the uniform symbol lower bound of Lemma 3.2 to control the lattice Green's function. The second is the axiomatic characterization of the SHF [46]: continuity, Chapman–Kolmogorov, independent increments, and the delta-Bose moment identity for $n = 1,2,3,4$ determine the law uniquely, so moment convergence plus tightness replaces the traditional finite-dimensional distribution proof. The critical coupling $\\beta_{\\varepsilon} \\approx 2\\pi/|\\log \\varepsilon|$ exactly cancels the logarithmic divergence of the diagonal resolvent, and tracking that cancellation produces the explicit fine-tuning constant $\\theta^{*}$.","core_discovery":"The central claim, stated as Proposition 1.2, is that the rescaled partition functions $Z^{\\theta,\\varepsilon}_{s,t}$, extended piecewise constantly, converge in law in $C(R^{2}_{<}, M_{+}(R^{2} \\times R^{2}))$ to $\\mathrm{SHF}(\\theta^{*})$. The explicit constant $\\theta^{*}$ is assembled from the environment correlation $\\kappa$, the jump measure $\\nu$ of the underlying compound Poisson walk, and universal constants such as $\\log 2$ and the constant $G$. The proof identifies the limit without constructing it pathwise: Theorem 1.1 proves strong convergence of the semigroups $Q^{[n],\\theta}_{\\varepsilon}(t)$ to the delta-Bose semigroup $Q^{[n],\\theta^{*}}(t)$ for every $n$, Lemma 4.1 supplies weighted norm bounds that yield tightness, and Lemma 4.2 verifies the Chapman–Kolmogorov property. Then every subsequential limit satisfies Definition 1.1, so the uniqueness result [46] forces all limits to share the law of $\\mathrm{SHF}(\\theta^{*})$.","pith_inferences":["The same strategy should transfer to other two-dimensional polymer models whose first four moment functions match the delta-Bose semigroup: only the orders $1$ to $4$ are used for identification, while Theorem 1.1 supplies the higher moments for free.","The explicit dependence of $\\theta^{*}$ on $\\kappa$ and $\\nu$ suggests that the SHF parameter is not universal, and comparisons across different correlation functions could check whether two models with equal $\\theta^{*}$ are statistically indistinguishable at the critical scale.","Tracking the resolvent error terms, which are polynomial in $\\varepsilon R$, $R^{-1/2}$, and $\\sqrt{\\varepsilon}$, could upgrade the qualitative convergence to explicit rates at which finite-dimensional distributions approach the SHF; the paper does not pursue that refinement.","The lattice-correction estimates indicate that convergence should persist for any symmetric finite-variance jump measure with aperiodic support, so the standing assumption on $\\nu$ is likely flexible."],"forward_implications":["For every pair of test functions $h, h'$, the real-valued processes $\\langle h, Z^{\\theta,\\varepsilon}_{s,t} h' \\rangle$ converge in law on compact time intervals to the corresponding marginals of $\\mathrm{SHF}(\\theta^{*})$.","All moments of all orders converge: the $n$-point correlation functions converge to the delta-Bose semigroup for every $n$, not only for the orders needed by the characterization.","The class of models with SHF as scaling limit now includes continuous-time random-walk polymers with a spatially correlated, non-compactly supported noise, and not only mollified stochastic heat equations.","The limit is unique: tightness plus the axioms leave no room for a different subsequential limit, so convergence in law holds for the entire family as $\\varepsilon \\to 0$.","Any model in this class with the same $\\theta^{*}$ has the same asymptotic statistics, making $\\theta^{*}$ the only model-dependent information that survives scaling."],"supporting_citations":[{"why":"Supplies the uniqueness theorem: any two processes satisfying the four SHF axioms with the same θ* are equal in law, which converts moment convergence and tightness into full convergence.","marker":"[46]"},{"why":"Provides the resolvent expansion and diagram formulas for the delta-Bose semigroup whose discrete analog is proven to converge in Theorem 1.1.","marker":"[33]"},{"why":"Constructs the Critical Stochastic Heat Flow as the target object and as the scaling limit of critical two-dimensional polymers.","marker":"[8]"},{"why":"Extends the delta-Bose semigroup to weighted L2 spaces including exponential weights; its bounds underpin the tightness estimates in Lemma 4.1.","marker":"[44]"},{"why":"Gives the original resolvent expansion for the two-dimensional delta-Bose gas that the moment semigroup formulas are built on.","marker":"[42]"},{"why":"Establishes the multi-particle point-interaction resolvent expansion in the plane that the diagram series relies on.","marker":"[25]"},{"why":"Supplies the tightness criterion for measure-valued processes used to lift tightness of scalar marginals to tightness in path space.","marker":"[41]"},{"why":"Defines the heat-kernel regularized product used in the Chapman–Kolmogorov axiom that the limit points must satisfy.","marker":"[19]"},{"why":"Provides the special-function identities and the evaluation of the constant G used in the explicit constant θ4 and the fine-tuning constant θ*.","marker":"[31]"}],"fun_headline_variants":["Moment method proves polymer limit to critical heat flow","Polymers converge to critical SHF via moment axioms","Critical SHF realized as polymer scaling limit","Moment-by-moment proof pins down polymer limit","2D polymer laws converge to critical SHF"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The identification of the limit depends on the uniqueness characterization of [46]: if two processes satisfying the four axioms with the same $\\theta^{*}$ could differ in law, or if a subsequential limit turned out to satisfy the moment identities of orders one through four but fail continuity or Chapman–Kolmogorov, the argument would not force convergence to $\\mathrm{SHF}(\\theta^{*})$.","fun_headline_variants_meta":{"raw":{"variants":["Moment method proves polymer limit to critical heat flow","Polymers converge to critical SHF via moment axioms","Critical SHF realized as polymer scaling limit","Moment-by-moment proof pins down polymer limit","2D polymer laws converge to critical SHF"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000184,"raw_usage":{"total_tokens":1304,"prompt_tokens":915,"completion_tokens":389,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":531,"completion_tokens_details":{"reasoning_tokens":316}},"tokens_in":531,"tokens_out":389,"duration_ms":4070,"temperature":1.0,"reasoning_tokens":316,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-16T00:33:52.874304+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the two-point function of the rescaled polymer for a concrete choice of $\\kappa$ and $\\nu$ and compare it with the explicit two-particle delta-Bose semigroup with the paper's $\\theta^{*}$: the two-particle semigroup is explicit in center-of-mass and relative coordinates, so a mismatch beyond numerical error would falsify Theorem 1.1. A cheaper check is to verify the stated value of the constant $\\theta_4 = \\log 2/(2\\pi) - G/\\pi^{2}$ by direct numerical integration of $\\int_{\\hat{T}^{2}\\setminus B(0;\\pi)} \\|\\eta\\|^{-2}\\, d\\eta$.","supporting_citations":[{"cited_title":"In:Probab","cited_arxiv_id":null,"evidence_quote":"Provides the resolvent expansion and diagram formulas for the delta-Bose semigroup whose discrete analog is proven to converge in Theorem 1.1."},{"cited_title":"In:Electron","cited_arxiv_id":null,"evidence_quote":"Extends the delta-Bose semigroup to weighted L2 spaces including exponential weights; its bounds underpin the tightness estimates in Lemma 4.1."},{"cited_title":"Dimock and S","cited_arxiv_id":null,"evidence_quote":"Establishes the multi-particle point-interaction resolvent expansion in the plane that the diagram series relies on."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the tightness criterion for measure-valued processes used to lift tightness of scalar marginals to tightness in path space."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the special-function identities and the evaluation of the constant G used in the explicit constant θ4 and the fine-tuning constant θ*."}],"review_version":1}