{"id":"a7ea4225-5744-4063-94b0-97ddbdc01811","arxiv_id":"2603.26319","paper_version":3,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Unbounded spin systems with super-Gaussian single-site distributions are regular on arbitrary graphs, with a sharp boundary-growth threshold that is double-exponential for nearest-neighbour interactions.","lead":"This paper proves a general regularity estimate for unbounded spin systems on arbitrary graphs, allowing boundary conditions to grow much faster than earlier results permitted. The result makes it possible to construct infinite-volume 'plus' Gibbs measures for models such as φ⁴ on general graphs.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Plus measure is maximal only among a-regular Gibbs measures; unqualified 'extremal' in abstract is unsupported, though Theorem 1.1 is sound.","rationale":"The reader's conditional verdict is essentially right, and the 'maximal, hence extremal' overclaim is the concrete place where the paper's advertised conclusion exceeds what the proof delivers. The reader's formal weakest_assumption (C2) is a legitimate hypothesis but not a flaw: C2 is precisely what makes the branching-process comparison in Lemma 3.4 work, and the paper does not claim to cover interactions outside it. By contrast, the extremality statement is an internal overclaim relative to the standard definition of extremal Gibbs measure, and it appears in the abstract and Section 5.3. The core regularity theorem appears internally consistent: the exploration cluster thresholds, use of Lemmas 3.1–3.2, the subcritical branching comparison in Lemma 3.4, and the A-function estimates are all coherent. I therefore do not see a reason to move the verdict; the paper needs a clarification/qualification rather than a change to the main theorem.","tokens_in":31474,"tokens_out":37142,"duration_ms":350183,"concrete_test":"Analytic check: remove the a-regularity assumption from the proof of Proposition 5.8 and attempt to derive ν[F_Λ^c]→0 for an arbitrary Gibbs measure using only the DLR equation and Theorem 4.1. If the derivation fails, construct or identify a non-regular Gibbs measure for a super-Gaussian single-site measure with admissible nearest-neighbour interactions (e.g., a limit of finite-volume measures with boundary conditions growing faster than any admissible A) and verify it violates Definition 1.4; existence of such a measure shows the 'hence extremal' sentence must be weakened.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The load-bearing gap is in the advertised extremality of the plus measure, not in Theorem 1.1. Proposition 5.8 proves ν⪯ν+ only for a-regular Gibbs measures ν. The proof of ν[F_Λ^c]→0 uses the regularity bound explicitly: ν[F_Λ^c] ≤ B ρ_a[e^{a'φ^2}]/ρ_a(R) Σ_{x∉Λ} |B_{d(o,x)}(o)|^{-a'}. If ν is merely a Gibbs measure, the DLR equation and Theorem 4.1 give no uniform control of Σ A(x,Λ,ξ)^2 on the support of ν, so this tail estimate can fail. Consequently the abstract's claim of constructing an 'extremal measure' is not established in the standard sense (extreme point of all Gibbs measures). The main regularity theorem is unaffected; the conclusion should be qualified as extremal/maximal among a-regular Gibbs measures unless all Gibbs measures are shown regular.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper develops regularity estimates for finite-volume Gibbs measures of unbounded spin systems on arbitrary countable graphs with pair interactions satisfying an admissibility condition (C2). The main theorem (Theorem 1.1) bounds the marginal density of the finite-volume measure with boundary conditions ξ by a product of non-interacting single-site measures with super-Gaussian tails, with an exponent controlled by a function A(x,Λ,ξ,C). The proof constructs an exploration cluster of large spins and dominates its size by a subcritical branching process. Applications include tightness for boundary conditions with double-exponential growth in nearest-neighbour models, an optimality result for P(φ) models, a construction of plus/minus infinite-volume Gibbs measures that are regular and maximal among a-regular Gibbs measures, and an alternative finite-volume construction of the plus measure in the nearest-neighbour case via random boundary conditions or vertex-dependent single-site measures.","tokens_in":31687,"tokens_out":38641,"duration_ms":347301,"significance":"If correct, this is a substantial advance over earlier regularity results of Lebowitz–Presutti and Ruelle, which were limited to Z^d and logarithmic boundary growth; the proof is self-contained and works on arbitrary graphs, with explicit control of constants. The exploration/branching argument is a genuinely new tool for this class of models. The paper also gives clean tightness criteria and new constructions of infinite-volume measures. The main regularity theorem is not affected by the overclaim discussed below, but the advertised extremality of the plus measure is presently not proven.","major_comments":[{"comment":"The claim that the plus measure is 'extremal' is not supported by the proof. Proposition 5.8 establishes ν^- ⪯ ν ⪯ ν^+ only for a-regular Gibbs measures ν. Maximality in the stochastic order among a-regular Gibbs measures does not imply extremality in the convex set of all Gibbs measures, and the sentence 'maximal, hence extremal' is therefore unjustified. Since the abstract advertises the construction of an 'extremal measure', this is a load-bearing overclaim, though it does not affect Theorem 1.1. Please qualify the abstract (e.g., 'maximal among a-regular Gibbs measures') or add an argument that all Gibbs measures are regular under the stated hypotheses.","section":"Abstract; Section 5.3, Proposition 5.8"}],"minor_comments":[{"comment":"The abstract announces 'an alternative construction ... regular up to the boundary' without noting that Section 5.4 is restricted to nearest-neighbour interactions on bounded-degree graphs. Please add the qualification.","section":"Abstract; Section 5.4"},{"comment":"The second stochastic domination ('hence ... ν^0_{Λ,β,˜ρ}') uses monotonicity in β for nonnegative single-site measures, which is not stated at this point. It follows from FKG together with H≥0 on the support of ˜ρ, but should be mentioned to avoid a gap.","section":"Corollary 1.2"},{"comment":"The measures ρ_a and ρ_{a/2} are not normalized, so ν^0_{Λ,0,ρ_a,0} is not a probability measure. The normalization convention should be stated explicitly where 'density' is used.","section":"Definition 1.4; Theorem 1.1"},{"comment":"Several displayed formulas lose superscripts in the arXiv text, e.g., 'CA(n−1)m' should read 'C A^{(n-1)^m}'. Please proofread the mathematical expressions in the final version.","section":"Section 2.2"},{"comment":"The definition of D_{i,j} is hard to parse; a display with fully parenthesized exponents would improve readability.","section":"Proof of Proposition 5.2"}],"recommendation":"major_revision","confidential_remarks":"The core regularity theorem appears sound and the paper is a strong contribution. The main issue is the overstatement of extremality of the plus measure; if the authors qualify that statement, the paper should be acceptable. I would not reject on this basis."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"What you should know: Theorem 1.1 is the real product. It gives a Radon-Nikodym bound for finite-volume measures with growing boundary conditions on any countable graph, for both short- and long-range interactions, with the threshold encoded in the A-function. The proof is self-contained (exploration cluster, branching process, FKG) and the constants are explicit. That is a clean advance over Lebowitz–Presutti and Ruelle on Z^d and over [7] for vertex-transitive polynomial-growth graphs. The nearest-neighbour applications are also sharp: double-exponential boundary growth for n>2, exponential for n=2, and Propositions 5.2 and 5.3 show the threshold is optimal for P(phi) models. The alternative construction of the plus measure in Section 5.4 (random boundary conditions, regular up to the boundary, stochastically monotone in volume) is a nice extra and likely useful for later work.\n\nSoft spots. The one that matters is the extremality claim. The abstract and Section 5.3 say the plus measure is extremal, but Proposition 5.8 only shows it is maximal among a-regular Gibbs measures. The proof that nu[F_Lambda^c] -> 0 for arbitrary Gibbs nu uses the regularity bound on nu; if nu is not regular, that step has no support. So 'maximal, hence extremal' is not justified in the standard sense of an extreme point of all Gibbs measures. The main theorem and the construction of nu+ are unaffected; the claim just needs to be qualified to 'maximal among a-regular Gibbs measures' unless the authors can show every Gibbs measure is regular. I don't see an easy fix in the present paper, but it is a presentation issue, not a hole in the core result.\n\nThe admissibility condition (C2) is the other thing to check before using the theorem. The f-function is doing real work: it controls the tails of the interactions so that Lemma 3.2 can absorb a neighbouring spin and Lemma 3.4 can keep the branching process subcritical. For long-range interactions with slow decay, you need to know such an f exists. The paper gives reasonable examples (log and power-law type f), so this is an assumption to verify in each application, not a hidden flaw.\n\nI found no sign of circularity or fitted parameters. The proof is long, and I did not machine-check every inequality, but the main line is clear and the lemmas line up. The paper is honest about its limits, mentioning k-body extensions only as expected.\n\nBottom line: this is a paper for specialists in statistical mechanics and Gibbs measures, and it should go to a serious referee. The extremality overclaim should be fixed, but it is fixable in revision and does not undermine the main regularity theorem.","headline":"Strong, self-contained regularity theorem for unbounded spins on arbitrary graphs, a real advance over the Z^d literature, but the advertised extremality of the plus measure is only proved among regular Gibbs measures.","tokens_in":32141,"tokens_out":4202,"would_cite":true,"duration_ms":38507,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["60K35","82B20"],"pacs":[],"model":"deepseek-v4-flash","headline":"For unbounded spin systems on arbitrary graphs, boundary conditions can grow double-exponentially without destroying regularity of the Gibbs measure.","keywords":["unbounded spin systems","Gibbs measures","regularity estimates","tightness","plus measure","super-Gaussian tails","branching process exploration","long-range interactions"],"falsifier":"Take V=Z with nearest-neighbour interactions, ρ(u)=e^{-a|u|^3}, and boundary conditions ξ_z = exp(exp(3|z|)). Proposition 5.2 predicts these finite-volume measures are not tight. Computing the law of φ_0 and testing whether it has a weak limit would settle the optimality claim.","tokens_in":31357,"feed_emoji":"🎲","tokens_out":7070,"duration_ms":77864,"temperature":0.7,"pith_summary":"This paper proves a regularity estimate for finite-volume Gibbs measures with unbounded real spins whose single-site distribution has super-Gaussian tails, on an arbitrary countable graph with short- or long-range interactions. The estimate controls the Radon–Nikodym derivative of the interacting measure against a non-interacting product measure by an exponential factor built from a function A(x,Λ,ξ,C) that measures the influence of boundary conditions at each site. In the nearest-neighbour case this permits boundary conditions growing double-exponentially in the distance to the boundary, and the paper shows this growth rate is optimal. The result yields infinite-volume 'plus' and 'minus' Gibbs measures that are regular with respect to a product measure, and provides a construction of the plus measure that avoids growing boundary conditions altogether.","feed_headline":"Spins can grow double-exponentially and still be regular","feed_subtitle":"A Radon-Nikodym bound gives non-Gaussian Cameron-Martin control on any graph.","key_machinery":"The argument rests on an exploration process that grows a cluster C of sites whose spins exceed thresholds calibrated by A(x,Λ,ξ,C), with thresholds rising as the exploration moves away from the target region. Lemmas 3.1 and 3.2 use Young-type inequalities to absorb the interaction energy of such large spins into the single-site potential, at a cost controlled by the tail exponent n. The cluster size is then dominated by the total progeny of a subcritical branching process whose offspring distribution is given by tail probabilities of the single-site measure; Lemma 3.4 uses this comparison to sum the exploration contributions and close the bound. The admissibility condition (C2) is what keep","core_discovery":"The central claim is Theorem 1.1: under the admissibility condition (C2) — the interactions are symmetric and there is a function f with f(t) ≥ log(1/|t|)^{1/n} near zero such that ∑_y |J_{xy}| f(J_{xy}) is bounded uniformly in x — the density of the finite-volume Gibbs measure restricted to Λ′ is bounded by ∏_{x∈Λ′} e^{C̃ A(x,Λ,ξ,C)^n} dν^0_{Λ′,0,ρ_{a/2},0}. Here A(x,Λ,ξ,C) is the smallest value that keeps boundary contributions along every walk from x within a prescribed decay envelope; it behaves like a non-Gaussian analogue of the harmonic extension of the boundary condition. When A stays bounded in the bulk, the finite-volume measures are tight, and the paper constructs the extremal plu","pith_inferences":["Because the proof never uses translation invariance or amenability, it is plausible that the same scheme extends to graphs of unbounded degree and to non-geometric or random graphs, as long as the uniform summability condition (C2) holds.","The paper notes but does not prove that the argument should adapt to k-body interactions when the tail exponent exceeds k; a natural test is to replace the pairwise Young bounds with a k-variable inequality.","The bound is one-sided. An exact non-Gaussian Cameron–Martin identity would require identifying interactions and boundary conditions for which the inequality becomes an equality, which the paper leaves open."],"forward_implications":["For nearest-neighbour interactions with n>2, boundary conditions growing at most like K^{(n-1)^{d(o,x)}} are admissible, and this double-exponential threshold is shown to be optimal for non-negative boundary conditions.","Long-range interactions are handled with the same proof; the allowed boundary growth is governed by the decay of the interaction kernel and the admissibility function f.","The plus and minus measures are constructed as limits of finite-volume measures, are regular Gibbs measures, and dominate all regular Gibbs measures in stochastic order.","A second construction of the plus measure uses random boundary conditions from a non-interacting product measure, giving finite-volume measures that are regular up to the boundary and stochastically decreasing in the volume."],"fun_headline_variants":["Non-Gaussian Cameron-Martin bound on any graph","Super-Gaussian tails give regular Gibbs measures on general graphs","Double-exponential boundary growth still yields regular measures","Radon-Nikodym bound for unbounded spins beyond log growth"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The load-bearing assumption is the admissibility condition (C2): the interaction weights must admit one function f, growing at least like log(1/|t|)^{1/n} near zero, such that ∑_y |J_{xy}| f(J_{xy}) is bounded uniformly in the vertex x. If the interaction graph has long-range edges that make this sum infinite for every such f, the regularity estimate is not established.","fun_headline_variants_meta":{"raw":{"variants":["Non-Gaussian Cameron-Martin bound on any graph","Super-Gaussian tails give regular Gibbs measures on general graphs","Double-exponential boundary growth still yields regular measures","Radon-Nikodym bound for unbounded spins beyond log growth"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001665,"raw_usage":{"total_tokens":6518,"prompt_tokens":889,"completion_tokens":5629,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":633,"completion_tokens_details":{"reasoning_tokens":5563}},"tokens_in":633,"tokens_out":5629,"duration_ms":41176,"temperature":1.0,"reasoning_tokens":5563,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-02T17:16:34.788772+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take V=Z with nearest-neighbour interactions, ρ(u)=e^{-a|u|^3}, and boundary conditions ξ_z = exp(exp(3|z|)). Proposition 5.2 predicts these finite-volume measures are not tight. Computing the law of φ_0 and testing whether it has a weak limit would settle the optimality claim.","supporting_citations":[],"review_version":2}