{"id":"aa1e0578-567c-4531-8cdb-f3db964e61fa","arxiv_id":"2607.11608","paper_version":1,"verdict":"ACCEPT","confidence":"HIGH","novelty_score":5.5,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"At the L2-critical power, frequency-truncated focusing Gibbs measures with harmonic potential converge to the free Gaussian (or cut-off Gaussian) precisely when the coupling is weaker than (KN + log N)^{-2/d}, and diverge otherwise.","lead":"The paper finds a sharp critical threshold for the coupling strength of focusing Gibbs measures of the harmonic NLS at the L2-critical power. Below the threshold the truncated measures converge to the free Gaussian field; above it they diverge. This settles a Brydges-Slade-type question for the harmonic oscillator setting.","discovery_kind":"extension","skeptic_critique":{"model":"grok-4.5","headline":"No significant objection identified","rationale":"The reader correctly isolates the radial assumption as the principal modelling limitation, yet that assumption is explicit in the statement of Theorem 1.3 and is the setting in which the eigenbasis, spectral projectors and Wick renormalisation are controlled. No internal inconsistency, missing estimate, or unjustified passage from free-energy divergence to non-existence of a TV limit was found. The variational programme, the GNS scaling that produces the precise power -2/d, and the dyadic Wiener-chaos control of the L^2-norm of the drift are all self-contained and match the pattern of the earlier log-correlated and harmonic-potential works. Consequently the ACCEPT verdict and low correctness-risk assessment stand.","tokens_in":23429,"tokens_out":575,"duration_ms":63292,"concrete_test":"Independently recompute the multi-index sum bound |S_{d,n}|≲n^{-(d+3)/4} (display (2.19)) for d=2,3 and n=10^3–10^4 via Stirling; if the constant fails to be uniform the coefficient decay of Lemma 2.3 (and therefore the whole non-normalizability construction) collapses.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim (existence of a sharp threshold λ∗≥λ∗>0 separating TV-convergence of ρ_N to the free Gaussian/cut-off Gaussian from divergence of the partition function) is proved under the stated hypotheses, including the radial restriction for d≥2. The new analytic inputs—decay of Hermite/Laguerre coefficients of the rescaled profile f_M (Lemma 2.3), the resulting approximation P_N f_M\to f_M in H^1 and L^p when M^eta∼N with eta>eta∗ (Corollary 2.4), the cut-off probability lower bound under the drifted measures (3.7)/(3.16), and the absorption of the critical GNS term by the Dirichlet energy when λ_N(K_N+log N)^{-2/d} is small (Proposition 4.1)—close without circularity or hidden gaps. The inference “Z_N=\to∞ ⇒ no TV-convergent subsequence” is the standard mass-escape argument of the literature and is justified by the free-energy lower bound constructed via the same drifts.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.5","summary":"The paper constructs and analyzes frequency-truncated focusing Gibbs measures for the nonlinear Schrödinger equation with harmonic potential on R^d, at the L^2-critical power p^*=2+4/d, when the coupling λ_N tends to zero. Under a radial assumption for d≥2, Theorem 1.3 establishes a phase transition: if λ_N ≤ λ^*(K_N+log N)^{-2/d} then the truncated measures ρ_N converge in total variation to the base Gaussian free field (or the same field with a renormalized L^2 cut-off), while if λ_N ≥ λ_*(K_N+log N)^{-2/d} the partition function diverges, so no total-variation limit exists even along subsequences. The proofs rely on the Boué–Dupuis variational formula, new decay estimates for Hermite/Laguerre coefficients of rescaled profiles (Lemma 2.3, Corollary 2.4), explicit drifts for the supercritical regime, and a dyadic decomposition controlling the critical GNS term in the subcritical regime (Proposition 4.1).","tokens_in":23660,"tokens_out":724,"duration_ms":15420,"significance":"The work answers, in the harmonic-potential setting, the Brydges–Slade question on weakly interacting focusing measures and complements the fixed-coupling normalizability theory of Robert–Seong–Tolomeo–Wang as well as the log-correlated analysis of Greco–Oh–Tao–Tolomeo. The new analytic inputs (coefficient decay, approximation of blow-up profiles by spectral projections, and absorption of the critical interaction by Dirichlet energy) are cleanly isolated and close without circularity. The result is a solid, self-contained contribution to the constructive theory of focusing Gibbs measures and invariant measures for dispersive PDEs.","major_comments":[],"minor_comments":[{"comment":"Several typographical slips should be corrected: “sequence sequence” (Theorem 1.3 statement), “main main idea” (p. 17), “te reader” (p. 19), “daydic” for “dyadic” (p. 24), and occasional missing spaces or duplicated words.","section":null},{"comment":"In the statement of Theorem 1.3 the relation λ^* ≥ λ_* > 0 is written with the same symbol family; a brief remark that the method yields only a possibly non-sharp gap between the two constants would help the reader.","section":null},{"comment":"The definition of β^* in (2.6) and the subsequent restriction M^β ∼ N appear only in the strong-coupling section; a forward reference in the introduction or in Corollary 2.4 would improve readability.","section":null},{"comment":"Notation for the Wick product and the spectral projector is consistent, but the shorthand Θ_N = P_N I(θ)(1) is introduced twice (after (3.1) and again in §4); a single global definition would avoid minor confusion.","section":null}],"recommendation":"minor_revision","confidential_remarks":"The manuscript is a natural and technically competent sequel to the authors’ earlier works and to the Greco–Oh–Tao–Tolomeo paper on log-correlated measures. Scope and novelty are appropriate for a strong probability/analysis journal; the radial restriction is clearly flagged and does not undermine the main claim."},"author_rebuttal":null,"desk_editor":{"model":"grok-4.5","letter":"This paper settles the Brydges-Slade-type question for the focusing NLS with harmonic potential at the mass-critical power p*=2+4/d. The punchline is clean: there is a critical coupling of order (KN + log N)^{-2/d} that separates total-variation convergence of the frequency-truncated measures to the free Gaussian (or free Gaussian with Wick L2 cut-off) from divergence of the partition function (hence no TV limit even along subsequences).\n\nWhat is new is the precise threshold itself and the analytic work needed to get it. The strategy is the now-standard Boué-Dupuis variational program used for log-correlated fields, but the harmonic oscillator forces genuine new estimates: decay of Hermite/Laguerre coefficients of the rescaled profile f_M (Lemma 2.3) and the resulting approximation PN f_M \to f_M in H1 and Lp when M^eta ~ N with eta large enough (Corollary 2.4). Those close the gap that would otherwise prevent the drifts from working. The weak-coupling side (Proposition 4.1) uses a dyadic decomposition plus hypercontractivity to absorb the critical GNS term; the strong-coupling side constructs explicit drifts (stochastic when KN is comparable to log N, deterministic when KN grows faster) that produce a free-energy lower bound blowing up. Both regimes look carefully written and free of circularity.\n\nThe only real modelling limitation is the radial restriction for d≥2 (and the extra range restriction on p for d≥3). It is standard in this literature and clearly flagged, but it means the result does not yet cover the full non-radial Gaussian free field. Everything else—Wick renormalization, spectral projectors, cut-off probabilities—is controlled by the same estimates that appear in the earlier normalizability theory of Robert-Seong-Tolomeo-Wang.\n\nThis is for people who work on constructive QFT or invariant measures for dispersive PDEs. The math is solid, the citations are appropriate, and the paper deserves a serious referee. I would accept it for peer review and expect it to appear after the usual polishing.","headline":"Solid, sharp phase-transition threshold for focusing harmonic Gibbs measures at the L2-critical power; the new Hermite/Laguerre estimates make the adaptation work.","tokens_in":24278,"tokens_out":547,"would_cite":true,"duration_ms":5539,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["60H30","81T08","35Q53","35Q55","35L71"],"pacs":[],"model":"grok-4.5","headline":"A critical threshold for the coupling strength decides whether weakly focusing critical Gibbs measures with harmonic potential collapse to the free Gaussian field or fail to converge.","keywords":["Gibbs measure","focusing NLS","harmonic potential","phase transition","weak coupling","Wick renormalization","variational method","critical threshold"],"falsifier":"Construct an explicit sequence of coupling constants sitting exactly on the critical curve λ_N = c(K_N+log N)^{-2/d} and check whether the partition function remains bounded or diverges for that particular constant c; any finite nonzero limit would contradict the claimed sharp dichotomy.","tokens_in":24329,"feed_emoji":"⚖️","tokens_out":719,"duration_ms":6495,"temperature":0.7,"pith_summary":"The paper asks what happens to frequency-truncated focusing Gibbs measures for the nonlinear Schrödinger equation with harmonic potential when the interaction strength λ_N is allowed to tend to zero while the L^{2} cut-off K_N may grow. At the mass-critical power p*=2+4/d the authors identify a sharp threshold of order (K_N+log N)^{-2/d}. Below the threshold the truncated measures converge in total variation to the free Gaussian free field (or to the same field restricted by a renormalized L^{2} cut-off). Above the threshold the partition function diverges, so no total-variation limit exists even along subsequences. The result answers, in the harmonic setting, the critical-coupling question first raised by Brydges–Slade for the two-dimensional Φ^{4} model, and shows that the critical measures essentially trivialise.","feed_headline":"Critical coupling decides if focusing Gibbs measures collapse","feed_subtitle":"Below a sharp threshold they converge to free Gaussian noise; above it they diverge.","key_machinery":"The Boué–Dupuis variational formula applied to carefully chosen drifts that approximate –Y_N plus a rescaled blow-up profile f_M; the same formula yields both the uniform bound on the partition function in the subcritical regime and its divergence in the supercritical regime.","core_discovery":"There exist positive constants λ*≥λ_*>0 such that, for the L^{2}-critical nonlinearity p*=2+4/d, the frequency-truncated focusing Gibbs measures ρ_N converge in total variation to the free Gaussian measure (possibly with a renormalized L^{2} cut-off) whenever λ_N ≤ λ*(K_N+log N)^{-2/d}, while the partition function diverges (hence no total-variation limit exists even along subsequences) whenever λ_N ≥ λ_*(K_N+log N)^{-2/d}.","pith_inferences":[],"forward_implications":[],"fun_headline_variants":["Weak coupling threshold sets focusing Gibbs measure convergence","Focusing Gibbs measures converge below critical λ, diverge above","Sharp phase transition for weakly interacting focusing Gibbs measures","Critical λ splits free Gaussian limit from focusing measure collapse","Harmonic focusing Gibbs: converge only under weak-coupling threshold"],"cache_read_input_tokens":16512,"weakest_assumption_plain":"The argument for dimensions greater than one requires the fields to be radial, so that the eigenfunctions of the harmonic oscillator form a complete basis of the radial L^{2} space and the spectral projectors stay under control.","fun_headline_variants_meta":{"raw":{"variants":["Weak coupling threshold sets focusing Gibbs measure convergence","Focusing Gibbs measures converge below critical λ, diverge above","Sharp phase transition for weakly interacting focusing Gibbs measures","Critical λ splits free Gaussian limit from focusing measure collapse","Harmonic focusing Gibbs: converge only under weak-coupling threshold"]},"model":"grok-4.5","effort":"low","cost_usd":0.0068,"raw_usage":{"total_tokens":1666,"prompt_tokens":703,"num_sources_used":0,"completion_tokens":78,"cost_in_usd_ticks":68000000,"prompt_tokens_details":{"text_tokens":703,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":885,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":703,"tokens_out":78,"duration_ms":8129,"temperature":1.0,"reasoning_tokens":885,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-14T04:25:00.192233+00:00","model_set":{"reader":"grok-4.5"},"falsifier":"Construct an explicit sequence of coupling constants sitting exactly on the critical curve λ_N = c(K_N+log N)^{-2/d} and check whether the partition function remains bounded or diverges for that particular constant c; any finite nonzero limit would contradict the claimed sharp dichotomy.","supporting_citations":[],"review_version":1}