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A Cluster Expansion and the Decay of Correlations of the 1D Long-Range Ising Model at Low Temperatures

T0 review · 2 major / 4 minor · reviewed 2026-08-02 · deepseek-v4-flash

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Pith's one-line read At low temperature, the 1D long-range Ising model's two-point correlation decays at the exact rate |x-y|^{-α} for all α∈(1,2].

desk verdict Full-α cluster expansion and exact α decay rate for the 1D long-range Ising model, built on the authors' own contour estimates—serious work, referee it. read the letter →

arxiv 2602.12447 v3 pith:MFRXJ6IE submitted 2026-02-12 math-ph cond-mat.stat-mechmath.MPphysics.class-ph

Rodrigo Bissacot , Henrique Corsini This is my paper · ORCID
classification math-phcond-mat.stat-mechmath.MPphysics.class-ph MSC 82B2082B0582B26 PACS 05.50.+q
keywords long-rangeIsingmodelclusterexpansionM-contourspolymergaspower-lawdecaylow-temperaturecorrelationsone-dimensionallatticemodels
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper proves that the one-dimensional long-range ferromagnetic Ising model with interaction J(r)=r^{-α}, for every α in (1,2], has a convergent low-temperature cluster expansion in terms of a hard-core gas of polymers built from the model's M-contours. The convergence is absolute and the expansion is analytic in a polydisk of external fields. As an application, the paper shows that the two-point truncated correlation decays as |x-y|^{-α}; combined with a lower bound known since the 1970s, the decay rate is exactly α. It also gives explicit tree-graph upper bounds for N-point truncated correlations. The whole construction is carried out without imposing an artificially large nearest-neighbor coupling, covering the full region where the 1D model is known to order.

What carries the argument

The central objects are M-contours: finite collections of spin flips that cannot be split into two pieces spaced farther apart than M times the smaller diameter to the 3/2 power; they play the role of local excitations with controlled energy and count. The paper packages compatible M-contours into polymers via a positive-compatibility relation and rewrites the partition function as a hard-core polymer gas. Each polymer's activity is bounded by z̄_β(Γ)=∏_{γ∈Γ} e^{-βH(γ)/2} Σ_{T∈T(Γ)} ∏_{edges} Φ(edge), with Φ the interaction between minus-interiors. The convergence argument is carried by a hierarchy of tree-sum estimates—one-vertex, two-vertex, and many-vertex—that reduce sums over trees of p

What would settle it

Find, for some α∈(1,2], a contour or collection of contours violating Hypothesis 1, e.g., H(Γ∖γ)+(7/8)H(γ)>H(Γ), or violating Hypothesis 3, e.g., a contour with anomalously low energy relative to its size; such an example would falsify Lemma 3.8's activity bound and, with it, Theorems 1.1–1.3. A direct numerical check of these hypotheses at α close to 1 for large β would be a concrete way to look for it.

Watch

Extended reading notes

Core claim

The claim is that at sufficiently low temperature the partition function of the 1D long-range Ising model with J(r)=r^{-α}, 1<α≤2, admits a genuine convergent cluster expansion: log Z_{Λ,β} is an absolutely convergent series over polymers built from M-contours, and the same series is analytic in a polydisk of external fields. From this expansion the paper derives concrete correlation bounds: the two-point truncated correlation is at most 2e^{-c₃β}|x-y|^{-α}, which, together with a lower bound known since the 1970s, makes the decay rate exactly α. For N points, the correlation is bounded by a sum over trees of products of |a-b|^{-α}.

Load-bearing premise

The load-bearing premise is that the three contour estimates of §2.4 (quasi-additivity, a counting bound, and exponential smallness of activity) hold for every α∈(1,2]; the paper states them as hypotheses and cites a companion paper for their proof, and if any of them fails the cluster expansion and the correlation theorems collapse.

Editorial extensions

If this is right

  • For every α∈(1,2] and sufficiently large β, the two-point truncated correlation has the exact low-temperature decay |x-y|^{-α}; the upper bound supplied here complements a lower bound known since the 1970s.
  • The pressure series converges absolutely and is analytic in an external-field polydisk, so thermodynamic derivatives such as magnetization, susceptibilities, and n-point correlations may be computed term by term.
  • N-point truncated correlations obey a tree-graph bound with weights |a-b|^{-α} on each tree edge—the long-range analogue of the usual exponential decay bound for short-range models.
  • The cluster expansion covers the entire phase-transition interval 1<α≤2 without imposing an artificially large nearest-neighbor coupling, removing an assumption used in earlier contour proofs.
  • The same expansion provides a rigorous starting point for further low-temperature quantities, such as phase-separation points and surface tensions in the long-range 1D model.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The theorems inherit their validity from the three contour estimates stated in §2.4 and proved in a companion paper; the present text does not reproduce those proofs, so the reader should check them for the full interval (1,2] before relying on Theorem 1.1.
  • The site-estimate lemmas, such as the product-to-tree inequality for lattice points, are stated quite generally and look reusable for nearby problems—random-field perturbations, decaying fields, or phase separation in 1D long-range models—without repeating the contour analysis.
  • At α=2 the upper bound matches the critical decay; a known intermediate phase with a slower algebraic exponent exists elsewhere in the phase diagram, so the exact-α statement should be read as a low-temperature statement, not a claim about all β.
  • The argument uses α>1 through summability of 1/r^α and cannot extend to α≤1, where no long-range order occurs; the behavior right at α=1 is therefore untouched by this method.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 4 minor

Summary. The paper develops a convergent low-temperature cluster expansion for the one-dimensional ferromagnetic Ising model with interaction J(r)=r^{-α}, α∈(1,2]. The main results are: absolute convergence of the cluster expansion for log Z, analyticity in a magnetic-field polydisk (Theorem 1.1); an upper bound |<σ_x;σ_y>| ≤ 2 e^{-c β}|x−y|^{-α} (Theorem 1.2); and analogous tree-summation bounds for N-point truncated correlations (Theorem 1.3). Together with the known lower bound of Iagolnitzer–Souillard [26], Theorem 1.2 is claimed to give the exact algebraic decay rate α. The proof proceeds by constructing a polymer gas of positive Fröhlich–Spencer contours, bounding the polymer activity in Lemma 3.8, developing an abstract tree-summing formalism in Section 4, and then translating contour estimates into lattice-site estimates in Sections 6–7. The decisive input is the set of three 'Crucial estimates' (Hypotheses 1–3 in Section 2.4), which are not proved in this manuscript but are imported from the same-group preprint [2].

Significance. If the imported Hypotheses 1–3 are valid for all α∈(1,2], the paper gives a substantial extension of Imbrie's α=2 cluster expansion to the full phase-transition region of the one-dimensional long-range Ising model, without the artificial large nearest-neighbor interaction used in earlier contour/cluster works. The abstract weighted-tree lemmas (Section 4) are stated with explicit constants and are potentially reusable. The correlation bounds are falsifiable and quantitative. The manuscript is not machine-checked and contains no code, but it does provide a coherent chain of explicit estimates. Its significance, however, is conditional: the geometric energy-entropy core is outsourced to an unpublished preprint from the same group, so the paper cannot stand alone as a proof of the stated all-α theorem.

major comments (2)
  1. [§2.4, Hypotheses 1–3; used in §3.3, §5.2, §5.3] The three crucial estimates — quasi-additivity of contour energies, the contour-counting bound, and exponential smallness of the contour activity — are load-bearing for every main theorem, but they are not proved here. They are used in Lemma 3.8 (activity bound), Lemma 5.3 and Corollaries 5.4/5.8 (tree bounds), and Lemma 5.11 (closing the polymer-tree sum). If any of them fails for some α∈(1,2], the convergence proof of Theorem 1.1 and the correlation bounds in Theorems 1.2–1.3 break down. Since [2] is a same-group arXiv preprint, the manuscript's central claim 'α∈(1,2]' is not self-contained. Please either include proofs of Hypotheses 1–3 in an appendix or explicitly state Theorems 1.1–1.3 as conditional on these hypotheses.
  2. [§6–§7, Lemma 5.7 and its descendants] The passage from contour sums to site sums relies on Lemma 5.7, whose proof is terse at the point where the summation over B is replaced by factors f_k(B). Because Corollaries 5.8–5.10 and Lemmas 7.1/7.5 all depend on this inequality, the proof should spell out exactly how |B| and diam(B) compensate for the dropped b-sums and how the summation over the last A_i is counted. If this step is not correct, the site-translation estimates in §7 would not follow. I am not asserting a definite error, but this is a central step that needs fuller justification.
minor comments (4)
  1. [§1.2 / statement of Theorem 1.2] The claim 'decay rate exactly α' depends on the external lower bound [26]. Please state the precise form of that bound, including its temperature/constant assumptions, so the combination of Theorem 1.2 and [26] is unambiguous.
  2. [§8, Concluding Remarks] The phrase 'αPp1,3´3 log 2s' appears to contain a typo; the text elsewhere uses 3−log_2 3. Please correct the formula.
  3. [Lemma 5.7 proof, around Eq. (58)] The notation 'E_φ' and the ordering condition are not fully explained; in particular, the displayed inequality after 'Finally, since dist(...)' skips the bookkeeping of the b-summation. Expanding this two-line argument would greatly improve readability.
  4. [Notation, §2.3–2.4] A few notational inconsistencies (e.g. 'CΛ' vs 'C_Λ', 'H˚h,γ' subscripts, and the use of 'Sym(k)' vs 'Sympkq') should be cleaned up. They do not affect the mathematics.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the cluster expansion and decay estimates are derived from explicit contour-energy hypotheses imported from [2] and an external lower bound [26]; no prediction is equivalent to its input by construction.

full rationale

The main theorems are proved by reducing the polymer-gas activity to contour estimates. The paper states in §2.4: 'For the region α∈(1,2], Hypotheses 1, 2 and 3 were shown to hold in [2]', and these hypotheses are then used in Lemma 3.8, Corollary 5.8, Lemma 5.11 and the proof of Theorem 1.1. This is an import of prior results rather than a derivation of the target from itself: Hypotheses 1–3 concern quasi-additivity, counting and exponential smallness of contour energies, not the correlation decay that is being proved. The two-point upper bound is obtained from the convergent cluster expansion and lattice-site estimates (Theorem 1.2), and the matching lower bound is quoted from the external paper [26]; neither quantity is fitted from the correlations being predicted. No equation in the paper is used as both input and output, and no fitted parameter is renamed as a prediction. The same-group citation [2] is load-bearing in the sense that Theorem 1.1 collapses if those hypotheses fail, but by the review standards this is an external, parameter-free result with stated assumptions that do not include the target; it is a dependency/completeness concern, not circularity.

Assumptions & free parameters 1 free parameters · 4 assumptions · 0 invented entities

The paper is an exact theorem proof, not a data fit. It introduces no new physical entities; contours and polymers are mathematical constructs from prior work. The main upstream inputs are contour estimates and lower-bound results from prior papers, while M is a proof parameter rather than an empirical free parameter.

free parameters (1)
  • M = M=M(α)>1, not numerically specified
    Definition 2.4 fixes a linear parameter M in the contour compatibility distance; Hypotheses 1–3 require M sufficiently large depending on α. It is an internal proof scale, not an empirical fitted value.
assumptions (4)
  • domain assumption Hypotheses 1–3 hold for all α∈(1,2] for the Fröhlich–Spencer contours with M=M(α).
    Invoked in §2.4 and used in Lemma 3.8, Lemma 5.3, Corollary 5.8; proved in Affonso et al. [2], not in this paper.
  • domain assumption The Iagolnitzer–Souillard lower bound [26] applies to the plus-boundary (or limiting equilibrium) truncated two-point function and gives algebraic decay with rate at least α.
    Used after Theorem 1.2 to upgrade the upper bound to 'exactly α'; hypotheses of [26] are not restated.
  • standard math Standard graph-tree bound |φ_n(Γ)| ≤ Σ_{T∈T_n} 1_{E(T)⊂E(Γ)} and the standard cluster expansion identity for log Z.
    Used in §3.2 and §5.1; standard results from Penrose [34] and Fernández–Procacci [22].
  • standard math Mayer identity (Equation 17) and analytic continuation of log partition function with respect to the field h.
    Used in Lemma 3.6 and in the proof of Theorem 1.1.

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Pith. "Pith review of A Cluster Expansion and the Decay of Correlations of the 1D Long-Range Ising Model at Low Temperatures." pith.science (2026). https://pith.science/paper/MFRXJ6IE

@misc{pith2026260212447,
  author       = {Pith},
  title        = {Pith review of: A Cluster Expansion and the Decay of Correlations of the 1D Long-Range Ising Model at Low Temperatures},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/MFRXJ6IE}},
  note         = {Machine review of arXiv:2602.12447}
}
abstract

In this work, a convergent low-temperature cluster expansion of the one-dimensional long-range ferromagnetic Ising model with polynomial decay $\alpha\in (1,2]$ is developed; that is, $J(r)=r^{-\alpha}$. As an application, the $n$-point correlations are studied and the two-point correlation is shown to be algebraic with rate of decay exactly $\alpha$.

Figures

Figures reproduced from arXiv: 2602.12447 by the authors.

Figure 1
Figure 1. All three contours are compatible, but γ1 is not positively compatible with γ2. On the other hand, γ3 is positively compatible with both γ1 and γ2. Hence, Γ “ tγ1, γ2, γ3u is not a positive collection. Lemma 2.10. Let Γ1, Γ2 P P such that pγ1 P Γ1q ^ pγ2 P Γ2q ñ γ1 „△ γ2, then Γ1 \ Γ2 P P. In this case, we write Γ1 „△ Γ2. Proof. This follows directly from the fact that compatibility is checked in pairs. That is, Γ P… view at source ↗
Figure 2
Figure 2. Take Γ as in [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗
Figure 3
Figure 3. An example of the procedure whereof one obtains the coarsest de￾composition of Γ P E into compatible polymers. Let us explain the content of [PITH_FULL_IMAGE:figures/full_fig_p012_3.png] view at source ↗
Figures from the paper (13 more)
Figure 4
Figure 4. Figure 4: Another example of the procedure whereof one obtains the coarsest decomposition of Γ P E into compatible polymers. Once again, let us explain the content of Figure (4). For those who are colorblind, let us give an alternative description of the contours. Ordering spin …
Figure 5
Figure 5. Figure 5: A concrete example of Notation 4.5 where B “ t1, 2, 4u. Notation 4.5. Let V be a finite set. We denote the set of leaves of T P TpV q by LpTq :“ tv P V : δT pvq “ 1u. Let A Ă B Ť N, we define TpB, Aq :“ tT P TpBq : LpTq “ Au If B “ rns, we write TnpAq instead of Tprns,…
Figure 6
Figure 6. Figure 6: Visual description of the injection of vertices on a tree. The term pn!q ´1 comes from the labeling introduced in the set of trees, and without which there is no hope of computing sums of trees. In which case, fixing the value of specific vertices can be described by i…
Figure 7
Figure 7. Figure 7: We have φ˜p1q “ 3, φ˜p2q “ 4, φ˜p3q “ 2, φ˜p4q “ 1, φ˜p5q “ 5, φ˜p6q “ 6. Note that T ‰ F ‹ φ˜T. Since F ‹ φ˜ is a bijection for any φ˜ P Sympnq, introducing the variable change T 1 “ F ‹ φ˜T, we have (28) Wβ,npwq “ ÿ φPInjpm,nq 1 n! ÿ T1PTn ÿ v1PV n 1v1 |rms“wwβpT 1 ,…
Figure 8
Figure 8. Figure 8: Note that there might not exist n 1 P N such that Ψn,2pTq P Tn. We define the total edge function family E “ tEβ : β ą 1u, where Eβ : V 2 Ñ Rě0 is given by (35) EβpΩ1, Ω2q :“ ÿ8 n“2 ÿ vPV n 1ϑ1“Ω1 1ϑn“Ω2 nź´1 i“2 Wβpϑiq nź´1 j“1 epϑj , ϑj`1q. It is also useful to defin…
Figure 9
Figure 9. Figure 9: An example of the correspondences between T P T2 and pn, φq, and between T P Ψ ´1 n,2 pTq and pP, Tq. For each T P T2 , provided n ě N, the set Ψ ´1 n,2 pTq is in a one-to-one correspondence with (1) an ordered partition P “ pP1, ..., Pnq of rns such that i P rns ñ φpi…
Figure 10
Figure 10. Figure 10: A graphical description of the algorithm in case n “ 2. The number to the left is the current step (0 denotes the starting position). A dotted circle represents a weight of 1, a blank circle represents a weight of vβp¨q and a gray circle represents a weight of vβ{2 p¨…
Figure 11
Figure 11. Figure 11: The graphic description of the application Ψ ˚ m with an example where m “ 4. Fix m ě 3. By Hypothesis 4, Lemma (4.9) and Equation (39) (consequence of Lemma 4.14), if β ě β0 is sufficiently large, we may suppose that Wβpϑq ď rp8m ´ 12q!s ´1{mvβ{2 pϑq, Eβpϑ1, ϑ2q ď v …
Figure 12
Figure 12. Figure 12: Graphs arising after removing the black vertex. Squares indicate the vertices carrying the extra v 1 β p¨q term. Note that this graph operation preserves the set of leaves and the degree of every remaining vertex [PITH_FULL_IMAGE:figures/full_fig_p033_12.png]
Figure 13
Figure 13. Figure 13: In this case, we obtain 1Γ0ȷΓ1 1Γ1ȷΓ2 ď ÿ Υ 1A pγ0,1, γ1,0q1˜pγ1,0, γ1,2q1A pγ1,2, γ2,1q1γ0,1PΓ0 1γ1,0,γ1,2PΓ1 1γ2,1PΓ2 , where Υ “ pγ0,1, γ1,0, γ1,2, γ2,1q P C 4 and 1˜ : C 2 Ñ t0, 1u is given by (73) 1˜pγ1, γ2q :“ 1γ1“γ2 ` 1γ1„△ γ2 . This generalizes rather easily t…
Figure 13
Figure 13. Figure 13: Graphical description of the translation of polymer incompatibility in terms of contours. where Υ “ pγ0,1, γ1,2, ..., γn,n`1, γn`1,nq P C 2n`2 . Writing źn j“1 1˜pγj,j´1, γj,j`1q “ÿ JĂrns ź j1PJ 1γj1´1,j1„△ γj1,j1`1 ź j2PrnszJ 1γj2,j2´1“γj2,j2`1 “: ÿ JĂrns 1˜J pΥq, we…
Figure 14
Figure 14. Figure 14: Condensed notation employed in this proof. Condensing notation as in [PITH_FULL_IMAGE:figures/full_fig_p048_14.png]
Figure 15
Figure 15. Figure 15: Example of TP , where A “ r8s and P “ tt1, 5u, t2, 3, 6u, t4u, t7, 8uu. We are now in a position to tackle the case of a single set partition in Equation (68). Lemma 7.3. There exists β0 ą 0 such that if β ě β0 and A P PpZq, then ÿ γPC 1AĂI´pγqe ´βHpγq ď r2p1 ` diampA…

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Works this paper leans on

40 extracted references · 8 linked inside Pith

  1. [2]

    Affonso, R

    L. Affonso, R. Bissacot, H. Corsini, and K. Welsch. Phase Transitions on 1d Long-Range Ising Models with Decaying Fields: A Direct Proof via Contours, 2024, ArXiv:2412.07098

    arXiv 2024

  2. [26]

    Iagolnitzer and B

    D. Iagolnitzer and B. Souillard. Decay of correlations for slowly decreasing potentials.Physical Review A, 16:1700–4, 1977

    1977

  3. [6]

    Affonso, R

    L. Affonso, R. Bissacot, J. Maia, J. Rodrigues, and K. Welsch. Cluster Expansion and Decay of Correlations for Multidimensional Long-Range Ising Models, 2025, ArXiv:2508.15666

    arXiv 2025

  4. [1]

    L. Affonso. Multidimensional Contours à la Fröhlich-Spencer and Boundary Conditions for Quantum Spin Systems. Ph.D. Thesis - University of São Paulo, 2023, ArXiv:2310.07946

    arXiv 2023

  5. [3]

    Affonso, R

    L. Affonso, R. Bissacot, E. O. Endo, and S. Handa. Long-range Ising models: Contours, phase transitions and decaying fields.Journal of the European Mathematical Society, 27(4):1679–1714, 2025

    2025

  6. [4]

    Affonso, R

    L. Affonso, R. Bissacot, G. Faria, and K. Welsch. Phase Transition in Long-Range Ferromagneticq´state Models: Contours, Random and Decaying Fields, 2024, ArXiv:2410.01234

    arXiv 2024

  7. [5]

    Affonso, R

    L. Affonso, R. Bissacot, and J. Maia. Phase Transitions in Multidimensional Long-Range Random Field Ising Models, 2024, ArXiv:2307.14150

    arXiv 2024

  8. [7]

    Aizenman, J

    M. Aizenman, J. T. Chayes, L. Chayes, and C. M. Newman. Discontinuity of the magnetization in one- dimensional1{|x´y| 2 Ising and Potts models.Journal of Statistical Physics, 50(1-2):1–40, 1988

    1988

Show all 40 references
  1. [8]

    Bissacot, E

    R. Bissacot, E. O. Endo, A. C. D. van Enter, B. Kimura, and W. M. Ruszel. Contour Methods for Long- Range Ising Models: Weakening Nearest-Neighbor Interactions and Adding Decaying Fields.Annales Henri Poincaré, 19(8):2557–2574, 2018

    2018

  2. [9]

    Bissacot, R

    R. Bissacot, R. Fernández, and A. Procacci. On the Convergence of Cluster Expansions for Polymer Gases. Journal of Statistical Physics, 139:598–617, 2010

    2010

  3. [10]

    Bricmont, J

    J. Bricmont, J. Lebowitz, and C.-E. Pfister. On the surface tension of lattice systems.Annals of the New York Academy of Sciences, 337:214, 1980

    1980

  4. [11]

    Burkov and Y

    S. Burkov and Y. G. Sinai. Phase diagrams of one-dimensional lattice models with long-range anti- ferromagnetic interaction.Russian Mathematical Surveys, 38, 1983

    1983

  5. [12]

    Cassandro, P

    M. Cassandro, P. A. Ferrari, I. Merola, and E. Presutti. Geometry of contours and Peierls estimates ind“1 Ising models with long range interactions.Journal of Mathematical Physics, 46:053305, 2005

    2005

  6. [13]

    Cassandro, I

    M. Cassandro, I. Merola, and P. Picco. Phase separation for the long range one-dimensional Ising model. Journal of Statistical Physics, 167:351–382, 2017

    2017

  7. [14]

    M.Cassandro, I.Merola, P.Picco, andU.Rozikov.One-dimensionalIsingmodelswithlongrangeinteractions: cluster expansion, phase-separating point.Communications in Mathematical Physics, 327:951–991, 2014

    2014

  8. [15]

    Cassandro, E

    M. Cassandro, E. Orlandi, and P. Picco. Phase Transition in the 1d Random Field Ising Model with long range interaction.Communications in Mathematical Physics, 288(2):731–744, 2009

    2009

  9. [16]

    J. Ding, F. Huang, and J. Maia. Phase transitions in low-dimensional long-range random field Ising models, 2024, ArXiv:2412.19281

    2024 arXiv

  10. [17]

    R. L. Dobrushin.Estimates of semi-invariants for the Ising model at low temperatures, volume 177 ofAMS Translations (2), pages 59–82. American Mathematical Society, 1996

    1996

  11. [18]

    R. L. Dobrushin.Perturbation methods of the theory of Gibbsian fields, volume 1648 ofLecture Notes in Mathematics, pages 1–66. Springer-Verlag, 1996

    1996

  12. [19]

    F. J. Dyson. Existence of a phase-transition in a one-dimensional Ising ferromagnet.Communications in Mathematical Physics, 12:91–107, 1969

    1969

  13. [20]

    F. J. Dyson. Non-existence of spontaneous magnetization in a one-dimensional Ising ferromagnet.Commu- nications in Mathematical Physics, 12:212–215, 1969

    1969

  14. [21]

    Fannes, P

    M. Fannes, P. Vanheuverzwijn, and A. Verbeure. Energy-entropy inequalities for classical lattice systems. Journal of Statistical Physics, 29:547–560, 1982

    1982

  15. [22]

    Fernández and A

    R. Fernández and A. Procacci. Cluster Expansion for Abstract Polymer Models. New Bounds from an Old Approach.Communications in Mathematical Physics, 274(1):123–140, 2007

    2007

  16. [23]

    Fröhlich and T

    J. Fröhlich and T. Spencer. The Kosterlitz-Thouless transition in two-dimensional Abelian spin systems and the Coulomb gas.Communications in Mathematical Physics, 81:527–602, 1981

    1981

  17. [24]

    Fröhlich and T

    J. Fröhlich and T. Spencer. The Phase Transition in the one-dimensional Ising model with1{r2 interaction energy.Communications in Mathematical Physics, 84:87–101, 1982

    1982

  18. [25]

    Gallavotti and S

    G. Gallavotti and S. Miracle-Sole. Statistical Mechanics of Lattice Systems.Communications in Mathemtat- ical Physics, 5:317–323, 1967. 58 BISSACOT AND CORSINI

    1967

  19. [27]

    J. Z. Imbrie. Decay of correlations in the one dimensional Ising model withJij“|i´j| ´2.Communications in Mathematical Physics, 85:491–515, 1982

    1982

  20. [28]

    J. Z. Imbrie and C. M. Newman. An intermediate phase with slow decay of correlations in one dimensional 1{|x´y| 2 percolation, Ising and Potts models.Communications in Mathematical Physics, 118:303–336, 1988

    1988

  21. [29]

    Jansen and L

    S. Jansen and L. Kolesnikov. Cluster Expansions: Necessary and Sufficient Convergence Conditions.Journal of Statistical Physics, 189(3), 2022

    2022

  22. [30]

    Kotecký and D

    R. Kotecký and D. Preiss. Cluster expansion for abstract polymer models.Communications in Mathematical Physics, 103(3):491–498, 1986

    1986

  23. [31]

    Littin and P

    J. Littin and P. Picco. Quasi-additive estimates on the Hamiltonian for the one-dimensional long range Ising model.Journal of Mathematical Physics, 58, 2017

    2017

  24. [32]

    J. Maia. Phase Transitions in Ising models: the Semi-infinite with decaying field and the Random Field Long-range. Ph.D. Thesis - University of São Paulo, 2024, ArXiv:2403.04921

    2024 arXiv

  25. [33]

    V. A. Malyshev and R. A. Minlos.Gibbs random fields. Mathematics and its Applications. Springer, Dor- drecht, Netherlands, 1991 edition, 1991

    1991

  26. [34]

    O. Penrose. Convergence of Fugacity Expansions for Fluids and Lattice Gases.Journal of Mathematical Physics, 4(10):1312–1320, 1963

    1963

  27. [35]

    Procacci

    A. Procacci. Cluster expansion methods in rigorous statistical mechanics, 2023, ArXiv:2308.06380

    2023 arXiv

  28. [36]

    J. B. Rogers and C. J. Thompson. Absence of long-range order in one-dimensional spin systems.Journal of Statistical Physics, 25(4):669–678, 1981

    1981

  29. [37]

    D. Ruelle. Statistical mechanics of a one-dimensional lattice gas.Communications in Mathematical Physics, 9:267–278, 1968

    1968

  30. [38]

    A. D. Scott and A. D. Sokal. The Repulsive Lattice Gas, the Independent-Set Polynomial, and the Lovász Local Lemma.Journal of Statistical Physics, 118(5-6):1151–1261, 2005

    2005

  31. [39]

    C. Temmel. Properties and applications of Bernoulli random fields with strong dependency graph. Ph.D Thesis - Aix-Marseille Université and Graz University of Technology, 2012

    2012

  32. [40]

    C. Temmel. Sufficient Conditions for Uniform Bounds in Abstract Polymer Systems and Explorative Partition Schemes.Journal of Statistical Physics, 157(6):1225–1254, 2014

    2014

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Reviewed August 2, 2026 · model on record in the stance chip above.