pith. sign in

arxiv: 2605.31344 · v1 · pith:HPQ5JQJ6new · submitted 2026-05-29 · 🧮 math.PR · math-ph· math.MP

Subcritical sharpness for real-valued spin models

Christoforos Panagiotis , William Veitch This is my paper

Pith reviewed 2026-06-28 21:04 UTC · model grok-4.3

classification 🧮 math.PR math-phmath.MP
keywords subcritical sharpnessrandom-cluster representationOSSS inequalityreal-valued spin modelsexponential decayBlume-Capel modelP(phi) modelstransitive graphs
0
0 comments X

The pith

Real-valued spin models on transitive graphs exhibit exponentially decaying correlations below the critical inverse temperature.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

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.

Core claim

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.

What carries the argument

Generalized OSSS inequality for monotonic measures with random connection probabilities, obtained from the random-cluster (FK percolation) representation on transitive graphs.

If this is right

  • 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.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • 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.

Load-bearing premise

The models admit a random-cluster representation whose measures are monotonic and allow the generalized OSSS inequality to be applied on transitive graphs.

What would settle it

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.

read the original abstract

In this paper, we consider a large family of real-valued spin models on general transitive graphs. We show that, in the subcritical regime $\beta<\beta_c$, the correlations of the model decay exponentially fast. To prove this result, we consider the random cluster (a.k.a. FK percolation) 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 $P(\varphi)$ models, going beyond the cases of the Ising and $\varphi^4$ models treated by Aizenman, Barsky, and Fern\'andez.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

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.

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 (2)
  1. [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.
  2. [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.
minor comments (2)
  1. [Abstract and §1] 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.
  2. [Theorem statement] 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.

Simulated Author's Rebuttal

2 responses · 0 unresolved

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.

read point-by-point responses

  1. Referee: [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.

    Authors: 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: no

  2. Referee: [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.

    Authors: 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: no

Circularity Check

0 steps flagged

No circularity: new generalized OSSS inequality derived from FK representation

full rationale

The paper's central claim of subcritical exponential decay is obtained by first establishing a random-cluster representation for the real-valued spin models, then deriving a new inequality that extends OSSS to monotonic measures with random connection probabilities, and finally applying this on transitive graphs. This derivation chain is independent of fitted parameters, self-citations, or renamings; the extension of Duminil-Copin-Raoufi-Tassion is presented as original work rather than presupposed. No load-bearing step reduces by construction to the target result or to prior author work.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper is a pure existence proof with no numerical fitting. The only non-standard assumption is monotonicity of the measures in the random-cluster representation.

axioms (1)
  • domain assumption The measures arising from the random-cluster representation are monotonic
    Invoked to extend the OSSS inequality to random connection probabilities.

pith-pipeline@v0.9.1-grok · 5655 in / 1073 out tokens · 22919 ms · 2026-06-28T21:04:32.971333+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

39 extracted references · 3 canonical work pages · 2 internal anchors

  1. [1]

    Aizenman and D

    M. Aizenman and D. J. Barsky. Sharpness of the phase transition in percolation models. Communications in Mathematical Physics, 108(3):489–526, 1987

    1987

  2. [2]

    Aizenman, D

    M. Aizenman, D. J. Barsky, and R. Fern´ andez. The phase transition in a general class of Ising-type models is sharp.Journal of Statistical Physics, 47:343–374, 1987

    1987

  3. [3]

    M. Blume. Theory of the first-order magnetic phase change in UO 2.Physical review, 141(2):517–524, 1966

    1966

  4. [4]

    Borgs and J

    C. Borgs and J. Z. Imbrie. A unified approach to phase diagrams in field theory and statistical mechanics.Communications in Mathematical Physics, 123(2):305–328, 1989

    1989

  5. [5]

    H. W. Capel. On the possibility of first-order phase transitions in Ising systems of triplet ions with zero-field splitting.Physica, 32(5):966–988, 1966

    1966

  6. [6]

    Lectures on the Ising and Potts models on the hypercubic lattice

    H. Duminil-Copin. Lectures on the Ising and Potts models on the hypercubic lattice. In PIMS-CRM Summer School in Probability, pages 35–161. Springer, 2017. Available at https://arxiv.org/abs/1707.00520

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  7. [7]

    Duminil-Copin, S

    H. Duminil-Copin, S. Goswami, A. Raoufi, F. Severo, and A. Yadin. Existence of phase transition for percolation using the Gaussian free field.Duke Mathematical Journal, 169(18):3539 – 3563, 2020

    2020

  8. [8]

    Duminil-Copin, A

    H. Duminil-Copin, A. Raoufi, and V. Tassion. Sharp phase transition for the random- cluster and Potts models via decision trees.Annals of Mathematics, 189(1):75–99, 2019

    2019

  9. [9]

    Duminil-Copin and V

    H. Duminil-Copin and V. Tassion. A new proof of the sharpness of the phase transition for Bernoulli percolation and the Ising model.Communications in Mathematical Physics, 343:725–745, 2016

    2016

  10. [10]

    P. Easo, F. Severo, and V. Tassion. Counting minimal cutsets andp c <1.Forum of Mathematics, Pi, 13, 2025. 28

    2025

  11. [11]

    R. G. Edwards and A. D. Sokal. Generalization of the Fortuin-Kasteleyn-Swendsen-Wang representation and Monte Carlo algorithm.Physical Review D, 38(6):2009–2012, 1988

    2009

  12. [12]

    Friedli and Y

    S. Friedli and Y. Velenik.Statistical Mechanics of Lattice Systems: A Concrete Mathemat- ical Introduction. Cambridge University Press, 2017

    2017

  13. [13]

    Fr¨ ohlich and B

    J. Fr¨ ohlich and B. Simon. Pure states for generalP(φ) 2 theories: construction, regularity and variational equality.Annals of Mathematics, pages 493–526, 1977

    1977

  14. [14]

    Fr¨ ohlich, B

    J. Fr¨ ohlich, B. Simon, and T Spencer. Infrared bounds, phase transitions and continuous symmetry breaking.Communications in Mathematical Physics, 50(1):79–95, 1976

    1976

  15. [15]

    Glimm and A

    J. Glimm and A. Jaffe. Positivity of theφ 4 3 hamiltonian.Fortschritte der Physik, 21(7):327– 376, 1973

    1973

  16. [16]

    Glimm and A

    J. Glimm and A. Jaffe.Quantum Physics : A Functional Integral Point of View. Springer New York, 2nd edition, 1987

    1987

  17. [17]

    Glimm, A

    J. Glimm, A. Jaffe, and T. Spencer. Phase transitions forφ 4 2 quantum fields.Communi- cations in Mathematical Physics, 45:203–216, 1975

    1975

  18. [18]

    B. T. Graham and G. R. Grimmett. Random-cluster representation of the Blume–Capel model.Journal of Statistical Physics, 125(2):283–316, 2006

    2006

  19. [19]

    R. B. Griffiths and B. Simon. The (φ 4)2 field theory as a classical Ising model.Commu- nications in Mathematical Physics, 33(2):145–164, 1973

    1973

  20. [20]

    G. R. Grimmett.The Random-Cluster Model, volume 333 ofGrundlehren der mathema- tischen Wissenschaften. Springer-Verlag, 2006

    2006

  21. [21]

    T. S. Gunaratnam, D. Krachun, and C. Panagiotis. Existence of a tricritical point for the Blume–Capel model onZ d.Probability and Mathematical Physics, 5(3):785–845, 2024

    2024

  22. [22]

    T. S. Gunaratnam, C. Panagiotis, R. Panis, and F. Severo. Random tangled currents for φ4: Translation invariant Gibbs measures and continuity of the phase transition.Journal of the European Mathematical Society (to appear), 2025

    2025

  23. [23]

    T. S. Gunaratnam, C. Panagiotis, R. Panis, and F. Severo. The supercritical phase of the φ4 model is well behaved.arXiv:2501.05353, 2025

    work page arXiv 2025

  24. [24]

    Hutchcroft

    T. Hutchcroft. New critical exponent inequalities for percolation and the random cluster model.Probability and Mathematical Physics, 1(1):147–165, 2020

    2020

  25. [25]

    Ising.Beitrag zur theorie des ferro-und paramagnetismus

    E. Ising.Beitrag zur theorie des ferro-und paramagnetismus. PhD thesis, Grefe & Tiede- mann Hamburg, Germany, 1924

    1924

  26. [26]

    Lammers and S

    P. Lammers and S. Ott. Delocalisation and absolute-value-FKG in the solid-on-solid model. Probability Theory and Related Fields, 188:63–87, 2024

    2024

  27. [27]

    J. L. Lebowitz. Coexistence of phases in Ising ferromagnets.Journal of Statistical Physics, 16(6):463–476, 1977

    1977

  28. [28]

    J. L. Lebowitz and E. Presutti. Statistical mechanics of systems of unbounded spins. Communications in Mathematical Physics, 50(3):195–218, 1976

    1976

  29. [29]

    W. Lenz. Beitrag zum Verst¨ andnis der magnetischen Erscheinungen in festen K¨ orpern.Z. Phys., 21:613–615, 1920. 29

    1920

  30. [30]

    T. M. Liggett, R. H. Schonmann, and A. M. Stacey. Domination by product measures. The Annals of Probability, 25(1):71–95, 1997

    1997

  31. [31]

    M. V. Mensikov. Coincidence of critical points in percolation problems. InSoviet Mathe- matics Doklady, volume 33, pages 856–859, 1986

    1986

  32. [32]

    E. Nelson. A quartic interaction in two dimensions. InMathematical Theory of Elementary Particles, (Dedham, Mass., 1965), pages 69–73, 1966

    1965

  33. [33]

    O’Donnell, M

    R. O’Donnell, M. Saks, O. Schramm, and R. A. Servedio. Every decision tree has an influential variable. In46th Annual IEEE Symposium on Foundations of Computer Science (FOCS’05), pages 31–39, 2005

    2005

  34. [34]

    Regularity of Gibbs measures for unbounded spin systems on general graphs

    C. Panagiotis and W. Veitch. Regularity of Gibbs measures for unbounded spin systems on general graphs.arXiv:2603.26319, 2026

    work page internal anchor Pith review Pith/arXiv arXiv 2026

  35. [35]

    R. Peierls. On Ising’s model of ferromagnetism.Mathematical Proceedings of the Cambridge Philosophical Society, 32(3):477–481, 1936

    1936

  36. [36]

    D. Ruelle. Superstable interactions in classical statistical mechanics.Communications in Mathematical Physics, 18(2):127–159, 1970

    1970

  37. [37]

    D. Ruelle. Probability estimates for continuous spin systems.Communications in Mathe- matical Physics, 50(3):189–194, 1976

    1976

  38. [38]

    Vanneuville

    H. Vanneuville. Exponential decay of the volume for Bernoulli percolation: a proof via stochastic comparison.Annales Henri Lebesgue, 8:101–112, 2025

    2025

  39. [39]

    Werner and E

    W. Werner and E. Powell.Lecture Notes on the Gaussian Free Field, volume 28 ofCours Sp´ ecialis´ es. Soci´ et´ e Math´ ematique de France, Paris, 2021. 30

    2021