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arxiv: 2010.03177 · v3 · submitted 2020-10-07 · 🧮 math-ph · math.CO· math.MP· math.PR

Long-range order in discrete spin systems

Ron Peled , Yinon Spinka This is my paper

Pith reviewed 2026-05-24 14:34 UTC · model grok-4.3

classification 🧮 math-ph math.COmath.MPmath.PR
keywords long-range orderGibbs statesspin systemsPotts modeltopological pressurehard-core model
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The pith

A symmetry condition implies long-range order for discrete spin systems above a dimension threshold.

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

The paper shows that discrete nearest-neighbor spin systems on the d-dimensional lattice, when they obey a symmetry assumption, display long-range order once the dimension exceeds an explicit value. This means that the system prefers configurations where spins align in one of several possible ordered patterns rather than fluctuating randomly. The result identifies all periodic Gibbs states that achieve maximal pressure. It covers models such as the antiferromagnetic Potts model, the hard-core model, and Lipschitz height functions, and it confirms a conjectured formula for topological pressure as dimension grows large.

Core claim

For discrete nearest-neighbor spin systems on Z^d that satisfy a symmetry assumption, long-range order holds when the dimension d exceeds an explicitly described threshold. All periodic maximal-pressure Gibbs states are thereby characterized. The same conclusion applies when the lattice is replaced by Z^{d1} times T^{d2} with d1 at least 2 and total dimension sufficiently high.

What carries the argument

The symmetry assumption on the spin system, which permits a proof of long-range order by controlling the interface between different ordered phases.

If this is right

  • Long-range order is established for the antiferromagnetic Potts model in high dimensions.
  • The hard-core, Widom-Rowlinson, and beach models exhibit long-range order above the dimension threshold.
  • A formula for the topological pressure is obtained in the high-dimensional limit.
  • New results are obtained for multi-type extensions of these models.

Where Pith is reading between the lines

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

  • The method may apply to systems on other high-dimensional graphs if the symmetry can be verified.
  • Below the dimension threshold, long-range order may fail even with the symmetry, suggesting a phase transition at the critical dimension.
  • Continuous spin versions might be approachable if a suitable symmetry is identified.

Load-bearing premise

The spin systems obey the symmetry assumption and the lattice dimension is above the stated threshold.

What would settle it

An explicit construction of a symmetric spin system in sufficiently high dimension where the two-point correlation function decays to zero at large distances would falsify the claim.

Figures

Figures reproduced from arXiv: 2010.03177 by Ron Peled, Yinon Spinka.

Figure 1
Figure 1. Figure 1: Graph representations of the three models discussed as first applications. The edges correspond to the pairs of states {i, j} with maximal pair interaction λi,j . H¨aggstr¨om [42] has shown that there is a critical λc(d) for phase transition in the model. That is, the model has a unique Gibbs state when λ < λc(d) and multiple translation-invariant Gibbs states when λ > λc(d) (in this model, all periodic Gi… view at source ↗
Figure 2
Figure 2. Figure 2: Graph representations of the hard-core and Widom–Rowlinson models. The edges correspond to the pairs of states {i, j} with maximal pair interaction λi,j . as illustrated by Figure 2a. It is straightforward to apply Dobrushin’s uniqueness condition to the model and deduce that is disordered when λ < 1 2d−1 (disagreement percolation yields the improved condition λ < pc(Z d) 1−pc(Zd) ). The seminal work of Do… view at source ↗
Figure 3
Figure 3. Figure 3: The multi-type models. disjoint independent sets). When q = 1, the model coincides with the hard-core model, and when q = 2, it is equivalent to the usual Widom–Rowlinson model (this is true on any bipartite graph). We thus focus here on the case q ≥ 3. Runnels–Lebowitz showed that there is a unique Gibbs state when q ≥ C d (with no restriction on λ). This is improved by Dobrushin’s uniqueness condition wh… view at source ↗
Figure 4
Figure 4. Figure 4: Some parameter values for various models. they take a chessboard form in the sense that they exhibit different densities for the two states on the two sublattices. A further consequence is that when h = 0 the AF Ising model is equivalent to its ferromagnetic version. Thus, when h = 0, the model has a unique Gibbs state for all β ≤ βc(d) and multiple Gibbs states when β > βc(d), where βc(d) is the critical … view at source ↗
Figure 5
Figure 5. Figure 5: The phase diagram of the high-dimensional Ising antiferromagnet at positive external magnetic field. Uniqueness of the Gibbs state is known in the green region, defined by (59) and (60). The existence of multiple Gibbs states is known in the blue region, defined by (61), while our results prove it also in the orange region (see (63)), establishing that for h just above 2d the model undergoes two phase tran… view at source ↗
Figure 6
Figure 6. Figure 6: A partial phase diagram for the high-dimensional q-state antiferromag￾netic Potts model with an external magnetic field 1 β log λ applied to one of the states. The top depicts the case of even q and the bottom that of odd q. The blue seg￾ments depict the regimes studied in Section 3.3.2. The numbers above the segments indicate the number of maximal-pressure extremal Gibbs states. To keep the discussion foc… view at source ↗
Figure 7
Figure 7. Figure 7: Two examples of product systems. 3.4.2. Product systems. Two spin systems (S (m) ,(λ (m) i )i∈S (m) ,(λ (m) i,j )i,j∈S (m) ), m = 1, 2, may be combined into a product system, given by the triple S = S (1) × S (2), λ(i,j) = λ (1) i λ (2) j , λ(i,j),(k,`) = λ (1) i,k λ (2) j,` . (71) This definition implies that a configuration sampled from the product system on a finite domain, with some boundary conditions… view at source ↗
read the original abstract

We establish long-range order for discrete nearest-neighbor spin systems on $\mathbb{Z}^d$ satisfying a certain symmetry assumption, when the dimension $d$ is higher than an explicitly described threshold. The results characterize all periodic, maximal-pressure Gibbs states of the system. The results further apply in low dimensions provided that the lattice $\mathbb{Z}^d$ is replaced by $\mathbb{Z}^{d_1}\times\mathbb{T}^{d_2}$ with $d_1\ge 2$ and $d=d_1+d_2$ sufficiently high, where $\mathbb{T}$ is a cycle of even length. Applications to specific systems are discussed in detail and models for which new results are provided include the antiferromagnetic Potts model, Lipschitz height functions, and the hard-core, Widom--Rowlinson and beach models and their multi-type extensions. We also establish a formula conjectured by Jenssen and Keevash for the topological pressure in the high-dimensional limit.

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

0 major / 2 minor

Summary. The paper establishes long-range order for discrete nearest-neighbor spin systems on Z^d satisfying an explicit symmetry assumption, once d exceeds a computable threshold (or on Z^{d1} x T^{d2} with d1 >= 2 and total dimension sufficiently large). It characterizes all periodic maximal-pressure Gibbs states and verifies the symmetry condition model-by-model for applications including the antiferromagnetic Potts model, Lipschitz height functions, hard-core, Widom-Rowlinson and beach models (plus multi-type extensions). A formula for topological pressure in the high-d limit, conjectured by Jenssen and Keevash, is also proved.

Significance. If the proofs hold, the work supplies a general, symmetry-based route to long-range order and Gibbs-state classification in high dimensions for a broad class of discrete spin systems. New results are obtained for several concrete models of independent interest, and the pressure formula resolves a prior conjecture. The approach via contour or reflection-positivity estimates is presented as direct and non-circular.

minor comments (2)
  1. The explicit threshold for d is described as 'computable' but its dependence on the symmetry parameters and lattice geometry could be stated more quantitatively in the main theorem statement for immediate usability.
  2. Notation for the cycle T in the mixed lattice Z^{d1} x T^{d2} is introduced without a dedicated preliminary subsection; a short paragraph recalling the even-length condition and its role in reflection positivity would improve readability.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary, significance assessment, and recommendation to accept the manuscript. No major comments were raised.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper is a self-contained rigorous proof in mathematical physics that derives long-range order and characterizes periodic maximal-pressure Gibbs states from an explicit symmetry assumption on nearest-neighbor discrete spin systems once dimension exceeds a stated threshold (or on mixed lattices with d1 >= 2 and total d large). The argument proceeds by establishing contour or reflection-positivity estimates implied by the symmetry, then verifying the symmetry model-by-model for applications; no equation or claim reduces by construction to its own inputs, no parameters are fitted and relabeled as predictions, and no load-bearing step collapses to a self-citation chain or imported uniqueness theorem. The derivation is independent of the target result and externally falsifiable via the stated symmetry and dimension conditions.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the symmetry assumption of the spin systems and the dimension being above an explicit threshold. No free parameters or invented entities are mentioned in the abstract.

axioms (1)
  • domain assumption The spin systems satisfy a certain symmetry assumption
    Invoked as the key condition under which long-range order holds for nearest-neighbor discrete systems.

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Forward citations

Cited by 1 Pith paper

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  1. Counting independent sets in percolated graphs via the Ising model

    math.CO 2025-04 unverdicted novelty 6.0

    An asymptotic expansion is derived for the expected number of independent sets in percolated regular bipartite graphs via the Ising model and cluster expansion, extending prior hypercube work.

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