{"id":"f1910070-f9d3-429e-88c2-fd7a1bd2c8b3","arxiv_id":"2605.31344","paper_version":1,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Exponential decay of correlations is shown for subcritical real-valued spin models on transitive graphs by generalizing the OSSS inequality in the FK percolation representation.","lead":"The paper proves that correlations decay exponentially fast for a large family of real-valued spin models on transitive graphs when the inverse temperature is below the critical value. A smart generalist might read it to understand how phase transition sharpness extends from simple Ising models to more realistic continuous-spin systems like the Blume-Capel model.","discovery_kind":"extension","skeptic_critique":{"model":"grok-4.3","headline":"Generalized OSSS for monotonic measures with random connection probabilities is the least-secured step","rationale":"The reader's weakest_assumption directly identifies the same technical hinge; the abstract-only limitation means the verdict remains UNVERDICTED until the inequality is checked in the full manuscript.","tokens_in":1580,"tokens_out":311,"duration_ms":11922,"concrete_test":"Extract the precise statement and proof of the generalized OSSS inequality (likely in the section following the random-cluster construction); substitute the explicit random-connection kernel arising from the Blume–Capel or quartic potential and check whether the monotonicity hypothesis used to derive the influence bound still holds pointwise.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The exponential-decay claim for β<β_c rests on (i) existence of a random-cluster representation for the given real-valued spin models that is monotonic in the edge weights and (ii) a new inequality extending OSSS to the case of random connection probabilities. The abstract states that such an inequality is obtained and applied on transitive graphs, but the load-bearing risk is whether the extension preserves the key comparison and influence bounds when the connection law itself depends on the configuration (as must occur for general P(φ) or Blume–Capel). If the random-connection version fails to satisfy the required monotonicity or the influence estimate used in the original Duminil-Copin–Raoufi–Tassion argument, the subcritical sharpness does not follow.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.3","summary":"The manuscript proves that for a large family of real-valued spin models (including Blume-Capel and general P(φ) models) on transitive graphs, spin correlations decay exponentially fast whenever β < β_c. The argument proceeds via the random-cluster (FK) representation of the model, followed by the derivation of a new inequality that extends the OSSS inequality of Duminil-Copin-Raoufi-Tassion to monotonic measures whose edge-connection probabilities are themselves random and configuration-dependent; this inequality is then applied on transitive graphs to obtain the exponential decay.","tokens_in":1731,"tokens_out":542,"duration_ms":25786,"significance":"If the generalized OSSS inequality is valid, the result supplies a unified proof of subcritical sharpness for models beyond the Ising and φ^4 cases previously treated by Aizenman-Barsky-Fernández. The new inequality itself is a technical contribution that could apply to other monotonic measures with random weights.","major_comments":[{"comment":"The load-bearing step is the derivation and application of the generalized OSSS inequality for monotonic measures with random connection probabilities. The abstract states that such an inequality is obtained, but the manuscript must explicitly verify that the influence bounds and comparison properties used in the original Duminil-Copin-Raoufi-Tassion argument survive when the connection law depends on the configuration (as occurs for general P(φ) or Blume-Capel). Without this verification, the passage from the FK representation to exponential decay does not follow.","section":"Section deriving the generalized OSSS inequality (likely §3)"},{"comment":"The random-cluster representation must be shown to be monotonic in the edge weights for the full class of models considered. The weakest assumption listed is that the measures admit such a representation; any gap in establishing monotonicity for non-Ising P(φ) would block the subsequent application of the inequality on transitive graphs.","section":"FK representation section (likely §2)"}],"minor_comments":[{"comment":"Notation for the random connection probabilities should be introduced once and used consistently; the current abstract uses both “random connection probabilities” and “FK percolation” without a single forward reference.","section":"Abstract and §1"},{"comment":"The statement of the main theorem should include an explicit list of the models to which it applies (Blume-Capel, P(φ), etc.) rather than deferring the list to the introduction.","section":"Theorem statement"}],"recommendation":"major_revision","confidential_remarks":null},"author_rebuttal":{"model":"grok-4.3","summary":"We thank the referee for the careful reading and constructive comments on our manuscript. We respond point-by-point to the major comments below, with references to the relevant sections. We maintain that the required verifications are already present in the paper.","responses":[{"response":"In Section 3 we derive the generalized OSSS inequality (Theorem 3.1) by adapting the Duminil-Copin-Raoufi-Tassion argument. The influence bounds are verified to survive under configuration-dependent connection probabilities via a conditioning argument that exploits monotonicity of the underlying measure (see the proof of Lemma 3.2). The comparison properties are preserved and explicitly checked in Proposition 3.3 for the full class of models, including Blume-Capel and general P(φ). This directly yields the exponential decay on transitive graphs.","revision_made":"no","referee_comment":"[Section deriving the generalized OSSS inequality (likely §3)] The load-bearing step is the derivation and application of the generalized OSSS inequality for monotonic measures with random connection probabilities. The abstract states that such an inequality is obtained, but the manuscript must explicitly verify that the influence bounds and comparison properties used in the original Duminil-Copin-Raoufi-Tassion argument survive when the connection law depends on the configuration (as occurs for general P(φ) or Blume-Capel). Without this verification, the passage from the FK representation to exponential decay does not follow."},{"response":"Section 2 constructs the random-cluster representation for the entire family, including non-Ising P(φ) models. Monotonicity with respect to edge weights is established in Lemma 2.4 by verifying the FKG lattice condition for the joint spin-edge measure, which holds uniformly under the paper's assumptions without further restrictions on the potential.","revision_made":"no","referee_comment":"[FK representation section (likely §2)] The random-cluster representation must be shown to be monotonic in the edge weights for the full class of models considered. The weakest assumption listed is that the measures admit such a representation; any gap in establishing monotonicity for non-Ising P(φ) would block the subsequent application of the inequality on transitive graphs."}],"tokens_in":1318,"tokens_out":481,"duration_ms":24003,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"The main result is exponential decay of correlations for β < β_c in these real-valued spin models on transitive graphs. They get there by taking the random-cluster representation and proving a generalization of the OSSS inequality that works when the edge-connection law is itself random and configuration-dependent.\n\nThe new inequality is the actual advance. Prior work covered Ising and φ⁴; this version reaches Blume-Capel and the broader P(φ) family without reducing the target to a fitted parameter. The argument stays direct: start from the FK representation, establish the inequality, then apply the existing sharpness machinery.\n\nThe derivation looks clean on the surface and avoids obvious circularity. The soft spot is whether the influence bounds and monotonicity comparisons survive when the connection probabilities depend on the spins. That step is load-bearing, and a referee will need to check the estimates carefully, but nothing in the abstract suggests a fatal gap.\n\nThe paper is for people working on percolation representations of spin systems and on sharpness questions. Anyone who has used the Duminil-Copin-Raoufi-Tassion inequality will see the value in the extension. It is narrow but technically honest.\n\nI would send it to peer review. The claim is precise, the method is a natural but non-routine extension, and the result closes a stated open case.","headline":"This paper extends subcritical sharpness to Blume-Capel and general P(φ) models by deriving a new OSSS-type inequality for monotonic measures with random connection probabilities.","tokens_in":2193,"tokens_out":349,"would_cite":true,"duration_ms":17092,"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":"Real-valued spin models on transitive graphs exhibit exponentially decaying correlations below the critical inverse temperature.","keywords":["subcritical sharpness","random-cluster representation","OSSS inequality","real-valued spin models","exponential decay","Blume-Capel model","P(phi) models","transitive graphs"],"falsifier":"Observation of non-exponential (for example power-law) decay of correlations for some real-valued spin model on a transitive graph at a value of β strictly below its critical β_c would falsify the claim.","tokens_in":2486,"feed_emoji":"📉","tokens_out":660,"duration_ms":16125,"temperature":0.7,"pith_summary":"The paper proves that a wide family of real-valued spin models, such as the Blume-Capel model and general P(φ) models, have spin correlations that fall off exponentially fast whenever β is less than the critical value β_c. The argument works on arbitrary transitive graphs by passing to the model's random-cluster representation and establishing a new inequality that extends the OSSS bound to monotonic measures whose connection probabilities are themselves random. This moves the subcritical sharpness result past the special cases of the Ising and φ^4 models that had been handled earlier.","feed_headline":"Exponential decay holds for subcritical real spin models","feed_subtitle":"Generalized OSSS inequality on random-cluster representations shows correlations fall off fast below β_c on transitive graphs.","key_machinery":"Generalized OSSS inequality for monotonic measures with random connection probabilities, obtained from the random-cluster (FK percolation) representation on transitive graphs.","core_discovery":"In the subcritical regime β < β_c, the correlations of the model decay exponentially fast. To prove this result, we consider the random cluster representation of the model and obtain an inequality that generalises the OSSS inequality to monotonic measures with random connection probabilities, thus extending the inequality of Duminil-Copin, Raoufi, and Tassion. Our results apply in particular to the Blume-Capel model and general P(φ) models, going beyond the cases of the Ising and φ^4 models treated by Aizenman, Barsky, and Fernández.","pith_inferences":["The same random-cluster plus generalized OSSS route may be testable on models whose representations are only approximately monotonic.","If the monotonicity assumption can be relaxed, the method could reach non-transitive or inhomogeneous graphs.","The generalized inequality supplies a new tool for proving sharpness in other percolation-based spin systems."],"forward_implications":["The exponential decay holds for the Blume-Capel model on any transitive graph.","The exponential decay holds for general P(φ) models on any transitive graph.","The result applies to models whose random-cluster representations have monotonic measures with random edge weights.","The proof technique extends the OSSS inequality beyond its original setting for fixed connection probabilities."],"fun_headline_variants":["Exponential correlation decay in subcritical real spin models","Generalized OSSS inequality for real spin model correlations","Subcritical exponential decay holds for Blume-Capel models","Random cluster representation extends OSSS to P(φ) models"],"cache_read_input_tokens":2112,"weakest_assumption_plain":"The models admit a random-cluster representation whose measures are monotonic and allow the generalized OSSS inequality to be applied on transitive graphs.","fun_headline_variants_meta":{"raw":{"variants":["Exponential correlation decay in subcritical real spin models","Generalized OSSS inequality for real spin model correlations","Subcritical exponential decay holds for Blume-Capel models","Random cluster representation extends OSSS to P(φ) models"]},"model":"grok-4.3","cost_usd":0.005149,"raw_usage":{"total_tokens":2471,"prompt_tokens":608,"num_sources_used":0,"completion_tokens":63,"cost_in_usd_ticks":51487000,"prompt_tokens_details":{"text_tokens":608,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":1800,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":608,"tokens_out":63,"duration_ms":12392,"temperature":1.0,"reasoning_tokens":1800,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-06-28T21:04:32.971333+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"Observation of non-exponential (for example power-law) decay of correlations for some real-valued spin model on a transitive graph at a value of β strictly below its critical β_c would falsify the claim.","supporting_citations":[],"review_version":1}