{"id":"83a4ab72-8250-4f0d-9f75-f64dde3bbd71","arxiv_id":"2607.09841","paper_version":1,"verdict":"ACCEPT","confidence":"HIGH","novelty_score":7.5,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"Systematic scan for the mean-field q-state ferromagnetic Potts model exhibits cutoff at mixing time c(β,q) log n + Θ(1) for every β < β_s.","lead":"The paper proves that systematic scan dynamics for the mean-field ferromagnetic Potts model mixes in c(β,q) log n + Θ(1) full scans with cutoff, for all β below the metastability threshold β_s. This supplies the first general cutoff theorem for systematic scan on spin systems, a global non-reversible chain class where cutoff is poorly understood.","discovery_kind":"extension","skeptic_critique":{"model":"grok-4.5","headline":"No significant objection identified","rationale":"The reader correctly isolates Lemma 1.1 as the technically heaviest step, yet that step is self-contained and uses only elementary martingale and fixed-point tools already standard for the mean-field Potts model. The remainder of the multiphase coupling (matching geometric rates, relative-entropy coalescence) follows the same pattern as earlier cutoff proofs for Glauber and Swendsen–Wang dynamics and contains no additional soft spots. Because the load-bearing pieces check out on inspection, the ACCEPT verdict with high confidence is unchanged.","tokens_in":38092,"tokens_out":508,"duration_ms":4299,"concrete_test":"Independently re-derive the supermartingale property of M_t (Lemma 2.2) from the definition of g^{i\to j}_eta and the mean-value bound of Lemma 2.6 without invoking the global drift function G_eta; if the inequality E[M_{t+1}-M_t|F_t]≤0 fails for any configuration with a coordinate outside [1/e^eta+(q-1),1], the burn-in probability bound of Lemma 1.1 would require an extra truncation argument.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim (Theorem 1) rests on a three-phase coupling whose most delicate step is the burn-in of Lemma 1.1. That lemma is proved by a standard drift-plus-fluctuation decomposition: the one-dimensional spin-count process is controlled by the known contraction of G_eta,n toward 1/q for eta<eta_s (Lemma 2.4, citing the same fixed-point analysis used for Glauber dynamics) together with a custom Maximal-Azuma bound (Lemma 2.3) on the supermartingale of fluctuations. Both ingredients are written out completely, the supermartingale property follows from a short mean-value comparison (Lemma 2.6), and the subsequent geometric contraction rates for Hamming distance and proportions (Sections 3–5) match exactly, determining the constant c(eta,q). No hidden circularity, missing case, or unjustified interchange of limits appears on close reading. The lower bound (Section 7) uses the same contraction rate and standard concentration of \nu, so the cutoff window is tight. The argument therefore holds under the stated hypotheses.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.5","summary":"The paper proves that the heat-bath systematic scan dynamics for the q-state ferromagnetic mean-field Potts model, for every q≥2 and every inverse temperature β below the spinodal threshold β_s, has mixing time T_mix = c(β,q) log n + Θ(1) (measured in full scans) and therefore exhibits cutoff with an O(1) window. The constant c(β,q) is identified explicitly as 1/(2b) where b>0 solves β(1-e^{-b})/(q b)=e^{-b}. The argument is a three-phase coupling: an O(1)-scan burn-in that places both chains in a fine-grained set Σ_\rho^n of well-spread nearly-equiproportional configurations (Lemma 1.1), a subsequent c(β,q) log n phase that contracts Hamming distance to O(√n) while the proportions vectors contract to O(1/√n) of the equiproportion vector (Lemmas 1.2, 3.4, 5.1), and a final one-scan relative-entropy coupling that finishes coalescence (Lemma 1.3). A matching lower bound is obtained from the same contraction rate together with concentration of the Gibbs measure about equiproportion.","tokens_in":38371,"tokens_out":932,"duration_ms":7637,"significance":"The result supplies the first general cutoff theorem for systematic scan dynamics of a spin system. Systematic scan is both global and non-reversible, two regimes in which cutoff remains poorly understood; the paper therefore contributes both to the concrete analysis of Potts models and to the broader theory of non-reversible Markov chains. The leading constant is shown to be at least a factor of two smaller than the corresponding Glauber constant, confirming a long-standing folklore speed-up. All proofs are self-contained (Sections 2–7), the constant c(β,q) is obtained from an explicit transcendental equation rather than fitted, and the argument is tight both in the cutoff window and in the temperature range β<β_s.","major_comments":[],"minor_comments":[{"comment":"Figure 1 caption and surrounding text: the comparison of c(β,q) with the Glauber constant čc(β,q) would be clearer if the asymptotic relation γ_n=1-b/n+O(n^{-2}) (Lemma 4.1) were referenced explicitly in the caption.","section":null},{"comment":"Section 2.1, definition of G_{β,n}: the additive β/n correction is introduced without a one-line justification; a brief remark that it absorbs the mean-value error of Lemma 2.6 would help the reader.","section":null},{"comment":"Lemma 2.3 (Maximal-Azuma): the constant 32 in the exponent is slightly looser than the classical Azuma constant; a parenthetical note that the factor arises from the Doob decomposition and the two-sided bound would remove any appearance of arbitrariness.","section":null},{"comment":"Section 6, Fact 6.1: the O(∥s-s̃∥_2^{2}) relative-entropy bound is standard but the appendix proof is terse; expanding the Hessian remainder by one line would make the argument fully self-contained.","section":null},{"comment":"Throughout: the notation Σ_\rho^n for the fine-grained set is overloaded with the ordinary configuration space Σ_n; a slightly more distinctive symbol (e.g., Σ_\rho^{spread}) would reduce cognitive load.","section":null}],"recommendation":"accept","confidential_remarks":"The manuscript is unusually clean for a first submission on a technically demanding topic. The only potential editorial concern is length: the full proofs occupy roughly 35 pages, but every step is used and the paper is already tightly written. I see no citation or novelty issues."},"author_rebuttal":null,"desk_editor":{"model":"grok-4.5","letter":"This paper settles the mixing time and cutoff for systematic scan on the mean-field ferromagnetic Potts model for every q≥2 and β<β_s. The mixing time is c(β,q)log n+Θ(1) scans, where c is the reciprocal of twice the positive root of an explicit transcendental equation; the same constant governs the geometric contraction of both the proportions vector and the Hamming distance. That is new: prior systematic-scan bounds were only O(n^{2}log n) or required stronger assumptions, and no previous cutoff result existed for systematic scan on any spin system.\n\nThe argument is a three-phase coupling. Phase 1 (burn-in) drives an arbitrary configuration into a fine-grained set of well-spread nearly-equiproportional configurations after O(1) independent scans; the proof is a standard drift-plus-fluctuation decomposition using the known contraction of G_eta,n toward 1/q together with a short custom Maximal-Azuma inequality for the supermartingale of spin-count fluctuations. Phase 2 couples optimally site-by-site and shows matching geometric decay of Hamming distance and ℓ2-distance to equiproportion, which pins down the leading constant. Phase 3 finishes with a one-scan relative-entropy bound and Pinsker. The lower bound re-uses the same contraction rate plus concentration of μ, so the Θ(1) window is tight. Everything is written out in Sections 2–7; the supermartingale property follows from a mean-value comparison and there is no circularity.\n\nThe only soft spot is that the burn-in relies on a custom concentration inequality and a slightly stronger notion of “good spread” than the usual ℓ∞ neighborhood used for Glauber; both are fully proved and the rest of the argument is insensitive to the precise form of the inequality. The result is tight in temperature (exponential mixing for β>β_s follows from known comparison) and of genuine interest for non-reversible and global chains, where cutoff is still poorly understood.\n\nThis is for people who work on mixing times of spin systems or non-reversible Markov chains. The math is solid, the citations are appropriate, and a serious referee should see it. I would accept it for peer review and would cite the constant and the cutoff statement.","headline":"First clean cutoff theorem for systematic scan on a classical spin system, with matching constant and tight window; the three-phase coupling holds up.","tokens_in":38977,"tokens_out":610,"would_cite":true,"duration_ms":7456,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["60J10","82B20","68W20"],"pacs":[],"model":"grok-4.5","headline":"Systematic scan for the mean-field Potts model mixes in c(β,q) log n scans and exhibits cutoff for all β below the metastability threshold.","keywords":["mixing time","cutoff","systematic scan","mean-field Potts model","ferromagnetic Potts","non-reversible Markov chains","Glauber dynamics"],"falsifier":"Compute or rigorously bound the total-variation distance of the systematic-scan chain after c(β,q) log n - κ scans for large fixed κ and moderate n (say n=10^4); if the distance remains bounded away from 1, or if the distance after c(β,q) log n + κ scans remains bounded away from 0, the claimed cutoff location is false.","tokens_in":38997,"feed_emoji":"⏱","tokens_out":1175,"duration_ms":10250,"temperature":0.7,"pith_summary":"The paper establishes that the systematic-scan Markov chain for the q-state ferromagnetic Potts model on the complete graph mixes in a sharp number of full scans once the inverse temperature is below the known metastability point β_s. Systematic scan updates every site in a fixed order, rather than choosing a random site each step; it is therefore global and non-reversible, two regimes in which cutoff has been hard to prove. The authors show that after a constant-length burn-in the chain enters a set of well-spread nearly-equiproportional configurations, after which the Hamming distance between two coupled copies contracts geometrically at a precise rate that yields the constant c(β,q). The resulting mixing time is therefore c(β,q) log n + Θ(1), which forces the total-variation distance to drop from nearly 1 to nearly 0 inside a window of constant width. The same threshold is tight: above β_s the chain is exponentially slow. The result supplies the first general cutoff theorem for systematic scan on a spin system and shows that the scan can be at least twice as fast as Glauber dynamics on the same model.","feed_headline":"Systematic scan mixes in c log n steps with cutoff","feed_subtitle":"First sharp cutoff for a global non-reversible spin-system chain, twice as fast as Glauber","key_machinery":"A three-phase coupling that first drives both chains into the fine-grained set Σ_\rho^n of well-spread nearly-equiproportional configurations (via a one-dimensional drift-plus-supermartingale analysis controlled by a Maximal-Azuma inequality), then contracts Hamming distance and ℓ_{2} distance to equiproportion at the matching geometric rate γ_n determined by the characteristic equation of the linearised update map, and finally coalesces the chains in a single scan by a relative-entropy bound.","core_discovery":"For every q ≥ 2 and every inverse temperature β < β_s there exists a positive constant c(β,q) such that the systematic-scan dynamics of the mean-field ferromagnetic Potts model has mixing time exactly c(β,q) log n + Θ(1) and therefore exhibits cutoff; the constant is 1/(2b) where b solves β(1-e^{-b})/(q b)=e^{-b}.","pith_inferences":["The same drift-plus-contraction analysis should extend to other mean-field models whose update map linearises to a contraction below a known metastability point.","If the well-spread condition can be verified under weaker spatial-mixing hypotheses, cutoff for systematic scan may hold on sparse graphs as well.","The factor-of-two speed-up relative to Glauber is consistent with the folklore that deterministic scanning halves the coupon-collector overhead; the paper supplies the first sharp confirmation for a non-trivial interacting system."],"forward_implications":["Systematic scan mixes at least twice as fast as Glauber dynamics on the same model for every β < β_s.","Cutoff can hold for a global non-reversible Markov chain on a classical spin system.","The leading constant c(β,q) is completely determined by the root of a simple transcendental equation and can be plotted explicitly against β/q.","At the critical value β=β_s the same methods suggest polynomial mixing of order n^c for some c<1 without cutoff."],"fun_headline_variants":["Systematic scan for mean-field Potts cuts off at c log n","Cutoff at c(β,q) log n for systematic scan of ferromagnetic Potts","Mean-field Potts systematic scan mixes in c log n + Θ(1)","First cutoff for global non-reversible spin-system systematic scan","Systematic scan of q-state Potts shows Θ(log n) mixing with cutoff"],"cache_read_input_tokens":32896,"weakest_assumption_plain":"After a fixed number of independent scans the chain must enter, and then stay inside, the set of configurations that are both nearly equiproportional and well-spread along the scan order; this rests on a custom concentration inequality for the spin-count supermartingale and on the drift function contracting toward 1/q precisely when β is below the metastability threshold.","fun_headline_variants_meta":{"raw":{"variants":["Systematic scan for mean-field Potts cuts off at c log n","Cutoff at c(β,q) log n for systematic scan of ferromagnetic Potts","Mean-field Potts systematic scan mixes in c log n + Θ(1)","First cutoff for global non-reversible spin-system systematic scan","Systematic scan of q-state Potts shows Θ(log n) mixing with cutoff"]},"model":"grok-4.5","effort":"low","cost_usd":0.00783,"raw_usage":{"total_tokens":1945,"prompt_tokens":912,"num_sources_used":0,"completion_tokens":105,"cost_in_usd_ticks":78300000,"prompt_tokens_details":{"text_tokens":912,"audio_tokens":0,"image_tokens":0,"cached_tokens":128},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":928,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":912,"tokens_out":105,"duration_ms":9818,"temperature":1.0,"reasoning_tokens":928,"cache_read_input_tokens":128,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-14T15:06:59.526673+00:00","model_set":{"reader":"grok-4.5"},"falsifier":"Compute or rigorously bound the total-variation distance of the systematic-scan chain after c(β,q) log n - κ scans for large fixed κ and moderate n (say n=10^4); if the distance remains bounded away from 1, or if the distance after c(β,q) log n + κ scans remains bounded away from 0, the claimed cutoff location is false.","supporting_citations":[],"review_version":1}