{"id":"5aeda0de-7770-4f8d-abb3-34ae77909b9b","arxiv_id":"2606.17859","paper_version":1,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":5.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"Derives full Γ-expansion of level-two LDP rate functional I_ε for non-reversible 1D diffusions on torus as ε→0: I_ε = (1/ε)J^(-1) + J^(0) + sum (1/θ_ε^(p)) J^(p).","lead":"The paper derives a full Gamma-expansion for the level-two large deviation rate functional of a non-reversible one-dimensional diffusion on the torus as temperature epsilon approaches zero, breaking it into terms with different time scales for metastable behavior. A smart generalist might read it to see how advanced asymptotic tools in probability theory handle rare events in stochastic systems with applications in physics and biology.","discovery_kind":"new_method","skeptic_critique":{"model":"grok-4.3","headline":"No significant objection identified","rationale":"Reader already flagged the same minimal assumptions and correctly set UNVERDICTED for lack of text. With the claim now visible, the derivation is presented as the paper's contribution; no load-bearing gap appears in the statement of what must hold for the expansion to be valid. Verdict therefore remains UNVERDICTED pending inspection of the actual proof.","tokens_in":1899,"tokens_out":333,"duration_ms":20215,"concrete_test":"Confirm that the diffusion SDE with the given a,b satisfies the standard conditions (uniform ellipticity, C² coefficients, periodic boundary) that guarantee existence of a unique strong solution and the level-2 LDP; then verify that the first two terms of the claimed expansion recover the known small-noise quasipotential and its first correction on the circle.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim is a derivation, under the stated C² regularity and auxiliary conditions on a,b, of an explicit multi-scale Γ-expansion for the level-2 rate functional I_ε of the empirical occupation measure. The abstract and claim structure indicate that the authors identify the leading quasipotential term (1/ε)J^(-1), the order-1 correction J^(0), and a finite number of sub-exponential corrections scaled by the metastable time scales θ_ε^(p). No internal inconsistency, missing hypothesis, or unjustified passage is visible in the claim itself; the existence of I_ε and the Γ-structure are taken as given once the diffusion is well-defined on the torus.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.3","summary":"The paper considers the non-reversible diffusion dX_ε = b(X_ε)dt + √(2ε a(X_ε)) dW on the one-dimensional torus T=[0,1) with C² coefficients a,b satisfying auxiliary conditions ensuring well-posedness. It derives the full Γ-expansion I_ε = (1/ε) J^{-1} + J^0 + ∑_{p=1}^{q̂} (1/θ_ε^p) J^p as ε→0, where the J functionals are ε-independent rate functionals on P(T) and the θ_ε^p are the metastable time scales of the process.","tokens_in":1993,"tokens_out":352,"duration_ms":20546,"significance":"If the derivation is complete, the explicit multi-scale Γ-expansion supplies a precise asymptotic description of the level-two rate functional that separates the quasipotential, the order-1 correction, and sub-exponential metastable corrections; this is a concrete advance for the analysis of large deviations and metastability for non-reversible one-dimensional diffusions on the circle.","major_comments":[],"minor_comments":[{"comment":"The abstract states the expansion but supplies neither the explicit forms of the J functionals nor the definitions of the time scales θ_ε^p; the full manuscript should include these expressions in the main theorem statement.","section":null},{"comment":"Notation: the symbol \bb T for the torus and \bb P(T) for the space of probabilities should be introduced once and used consistently; the current abstract mixes \bb T and \bb P(T) with \bb T appearing only in the SDE.","section":null}],"recommendation":"uncertain","confidential_remarks":null},"author_rebuttal":{"model":"grok-4.3","summary":"We thank the referee for their summary of our manuscript and for recognizing the potential significance of the full multi-scale Γ-expansion for the level-two rate functional of non-reversible one-dimensional diffusions on the torus. We note that the referee's recommendation is listed as 'uncertain' but no specific major comments or points of concern were provided in the report. Accordingly, we have no point-by-point responses to address at this stage.","responses":[],"tokens_in":1418,"tokens_out":104,"duration_ms":12449,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"The central result is a derivation of the full Γ-expansion I_ε = (1/ε) J^{-1} + J^{(0)} + sum (1/θ_ε^p) J^{(p)} for the level-two rate functional of the empirical occupation measure of this non-reversible diffusion on the torus.\n\nThis is new because it moves beyond the reversible setting where the drift is a gradient. The authors handle a general C² drift b, which can break reversibility, and still get the multi-scale structure with the metastable times θ_ε^p. The work does well by making the expansion explicit in terms of ε-independent functionals J and tying the corrections directly to the time scales where metastability occurs. The setup with periodic boundary conditions on [0,1) is standard and keeps the state space compact.\n\nOn the soft side, the paper assumes the existence of I_ε and then derives the expansion. That is reasonable but means the result is as strong as the underlying LDP. The actual expressions for the J functionals are not shown in the abstract, so it is difficult to assess how much new computation is involved versus applying known Gamma-convergence techniques. The regularity C² on a and b is given, but one wonders if weaker conditions would suffice.\n\nThe citation pattern looks standard for this area. This is aimed at researchers in stochastic processes who care about precise asymptotics for large deviations in non-equilibrium systems. Someone studying metastability or hydrodynamic limits might use this as a reference for the rate functional structure.\n\nIt deserves a serious referee because the claim is specific and the setting is clean. The math seems internally consistent from the description. I would recommend sending it to peer review.","headline":"Derives explicit multi-scale Γ-expansion for level-2 LDP rate functional of non-reversible 1D diffusion on the torus, extending reversible cases.","tokens_in":2487,"tokens_out":423,"would_cite":false,"duration_ms":23078,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"grok-4.3","headline":"The level two large deviation rate functional of non-reversible diffusions on the one-dimensional torus admits a full Gamma-expansion as the temperature vanishes.","keywords":["large deviations","Gamma-convergence","diffusions","metastability","non-reversible","periodic boundary conditions","rate functionals","one-dimensional"],"falsifier":"Computing the rate functional I_ε numerically for small ε and verifying whether it matches the proposed combination of the J functionals at the appropriate scales would test the claim; mismatch at any scale would falsify it.","tokens_in":2779,"feed_emoji":"","tokens_out":647,"duration_ms":37227,"temperature":0.7,"pith_summary":"The paper derives a full Gamma-expansion for the level-two large deviation rate functional I_ε of a one-dimensional diffusion process on the torus as ε approaches zero. The expansion writes I_ε as a sum of an ε-independent functional scaled by 1/ε, plus another ε-independent term, plus terms scaled by the inverses of metastable time scales. This structure reveals how the rate of rare events changes across different time scales in non-reversible systems. A reader would care because it gives a detailed asymptotic picture of metastability and large deviations without relying on reversibility assumptions.","feed_headline":"Diffusion rate functionals admit full Gamma expansion","feed_subtitle":"The expansion breaks the level-two rate into equilibrium and metastable scale contributions for non-reversible torus diffusions.","key_machinery":"The full Γ-expansion of the level-two large deviation rate functional I_ε into ε-independent components J weighted by 1/ε and the metastable time scales 1/θ_ε.","core_discovery":"Consider the diffusion dX_ε(t) = b(X_ε(t)) dt + sqrt(2ε a(X_ε(t))) dW_t on the torus. The level-two rate functional I_ε of this process satisfies I_ε = (1/ε) J^{-1} + J^{(0)} + sum (1/θ_ε^{(p)}) J^{(p)} as ε→0, where the J's are rate functionals independent of ε and the θ's are the metastable time scales of the process.","pith_inferences":["The explicit expansion may allow for more accurate approximations in Monte Carlo simulations of rare events.","It highlights the role of non-reversibility in determining the metastable time scales θ_ε."],"forward_implications":["The leading term (1/ε) J^{-1} dominates the large deviation behavior at the fastest scale.","Additional terms capture the contributions from metastable transitions at slower time scales.","The expansion is valid under periodic boundary conditions for non-reversible drifts with C² coefficients.","Explicit forms of the J functionals can be computed from the diffusion coefficients a and b."],"fun_headline_variants":["Gamma expansion for level-two rate functionals of torus diffusions","Full Gamma expansion of non-reversible diffusion rate functionals","Level-two rates of one-dimensional diffusions receive Gamma expansion","Torus diffusions yield Gamma-expanded level-two large deviation rates"],"cache_read_input_tokens":2112,"weakest_assumption_plain":"The diffusion coefficients a and b are of class C² on the torus and satisfy conditions making the process well-defined, while the level-two rate functional I_ε is assumed to exist and possess the given Gamma-expansion form.","fun_headline_variants_meta":{"raw":{"variants":["Gamma expansion for level-two rate functionals of torus diffusions","Full Gamma expansion of non-reversible diffusion rate functionals","Level-two rates of one-dimensional diffusions receive Gamma expansion","Torus diffusions yield Gamma-expanded level-two large deviation rates"]},"model":"grok-4.3","cost_usd":0.007579,"raw_usage":{"total_tokens":3550,"prompt_tokens":822,"num_sources_used":0,"completion_tokens":58,"cost_in_usd_ticks":75787000,"prompt_tokens_details":{"text_tokens":822,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":2670,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":822,"tokens_out":58,"duration_ms":23491,"temperature":1.0,"reasoning_tokens":2670,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-06-26T22:48:37.987970+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"Computing the rate functional I_ε numerically for small ε and verifying whether it matches the proposed combination of the J functionals at the appropriate scales would test the claim; mismatch at any scale would falsify it.","supporting_citations":[],"review_version":1}