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arxiv: 2602.18310 · v2 · submitted 2026-02-20 · 🧮 math.CO · cs.DM· cs.IT· math.IT· math.PR

Recoverable systems and the maximal hard-core model on the triangular lattice

Geyang Wang , Alexander Barg , Navin Kashyap This is my paper

Pith reviewed 2026-05-15 20:31 UTC · model grok-4.3

classification 🧮 math.CO cs.DMcs.ITmath.ITmath.PR
keywords recoverable systemsmaximal hard-core modeltriangular latticeGibbs measurescapacity boundsperiodic Gibbs measuresnon-uniqueness
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The pith

The maximal hard-core model on the triangular lattice has bounded recoverable capacity with non-unique Gibbs measures at high activity.

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

The paper extends the study of recoverable systems and the maximal hard-core model from the square lattice to the triangular lattice. It derives bounds on the capacity of the recoverable system, demonstrates non-uniqueness of Gibbs measures in the high-activity regime, and characterizes extremal periodic Gibbs measures at low activity. A reader would care because these results clarify phase transitions and information capacities in lattice-based constrained systems used in statistical physics and coding theory.

Core claim

The maximal hard-core model on the triangular lattice A yields bounds on the capacity of its associated recoverable system, non-uniqueness of Gibbs measures in the high-activity regime, and a characterization of extremal periodic Gibbs measures for low activity.

What carries the argument

The recoverable system tied to the maximal hard-core model on the triangular lattice, encoding admissible configurations and their recovery properties.

If this is right

  • Bounds on capacity give concrete limits on coding rates achievable with hard-core constraints on triangular lattices.
  • Non-uniqueness of Gibbs measures implies multiple possible equilibrium distributions at high activity parameters.
  • Characterization of periodic Gibbs measures allows precise identification of extremal states in the low-activity regime.
  • The extension confirms transferability of analytic methods between different lattice geometries.

Where Pith is reading between the lines

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

  • Similar non-uniqueness and capacity results may hold for other lattices such as the hexagonal lattice with appropriate adjustments.
  • These findings could inform simulations of particle packing on triangular grids in physical systems.
  • Computational verification of the bounds through Monte Carlo sampling of configurations would provide further support.

Load-bearing premise

Analytic techniques developed for the square lattice apply directly to the triangular lattice without additional geometric assumptions or obstructions.

What would settle it

An explicit example of a unique Gibbs measure at high activity or a computed capacity value outside the established bounds on the triangular lattice would falsify the claims.

Figures

Figures reproduced from arXiv: 2602.18310 by Alexander Barg, Geyang Wang, Navin Kashyap.

Figure 1
Figure 1. Figure 1: Maximal independent sets in Z 2 and A. It is easy to see that X(A) is equivalent to an SFT defined by forbidding the following patterns: 1 1 1 1 1 1 0 0 0 0 0 0 0 . An SFT is called strongly irreducible if there exists an r > 0 such that for all finite regions A, B ∈ L with ℓ1 distance ≥ r, for every pair of configurations ω, η ∈ X, there exists σ ∈ X such that σA = ωA and σB = ωB. As shown in [23], a stro… view at source ↗
Figure 3
Figure 3. Figure 3: A coloring of the vertices of A Let ω r , ω g , and ω b be the configurations supported on all red, green, and blue vertices, respectively. Recall that Λn = {1, . . . , n}× {1, . . . , n} ⊂ A is a rhombus and Λ c n = A\Λn is its complement. Consider the set of configurations ω ∈ X(A) such that all the vertices of some fixed color, say blue, in Λ c n are occupied; see Fig. 5a below. We say that these ω’s fo… view at source ↗
Figure 2
Figure 2. Figure 2: Connectivity and contours [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 4
Figure 4. Figure 4: Deep holes can be edge-connected or isolated. [PITH_FULL_IMAGE:figures/full_fig_p004_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Illustrating the contour erasing procedure. [PITH_FULL_IMAGE:figures/full_fig_p005_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Two possible cases with an empty triangle (filled in [PITH_FULL_IMAGE:figures/full_fig_p005_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Faces in γ at distance 1 in Gγ lie on a circuit. Let us prove that Gγ is 2-connected. We need to show that no v ∈ V (Gγ) forms a cut or that (by Menger’s theorem) for any pair of distinct vertices u and v, there is a circuit (i.e., a simple cycle) that contains both of them. We prove this by induction on dGγ (u, v), the length of the shortest path between u and v in Gγ. Assume that dGγ (u, v) = 1, which oc… view at source ↗
Figure 8
Figure 8. Figure 8: The induction step. While there are results on the growth rate of the number of 2-connected planar graphs [31], the estimates for the count of the graphs they give do not improve on our estimate. The same is true for other known results on the count of planar graphs of various kinds [32]–[34]. 4. PHASE COEXISTENCE: THE LOW-ACTIVITY REGIME At low activity, Gibbs distributions are determined by the ground st… view at source ↗
Figure 10
Figure 10. Figure 10: A sparse PGS: each unoccupied vertex is adjacent to [PITH_FULL_IMAGE:figures/full_fig_p009_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: A single step of contour construction in the proof of Theorem [PITH_FULL_IMAGE:figures/full_fig_p013_11.png] view at source ↗
read the original abstract

In a previous paper (arXiv:2510.19746), we have studied the maximal hard-code model on the square lattice ${\mathbb Z}^2$ from the perspective of recoverable systems. Here we extend this study to the case of the triangular lattice ${\mathbb A}$. The following results are obtained: (1) We derive bounds on the capacity of the associated recoverable system on ${\mathbb A}$; (2) We show non-uniqueness of Gibbs measures in the high-activity regime; (3) We characterize extremal periodic Gibbs measures for sufficiently low values of activity.

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 paper extends the recoverable-systems framework from the square lattice to the triangular lattice A. It claims three main results: explicit bounds on the capacity of the associated recoverable system, non-uniqueness of Gibbs measures in the high-activity regime, and a characterization of extremal periodic Gibbs measures for sufficiently small activity.

Significance. If the central claims hold, the work supplies the first recoverable-system treatment of the maximal hard-core model on a non-bipartite lattice of degree 6. The capacity bounds and the low-activity characterization would furnish concrete, lattice-specific information that can be compared with existing results on Z^2; the high-activity non-uniqueness statement would confirm that the phase-transition phenomenon persists under the changed geometry.

major comments (2)
  1. [§4] §4 (high-activity non-uniqueness): the contour or cluster-expansion argument is stated to carry over from the square-lattice case, yet the surface-to-volume ratio and the minimal contour length change with coordination number 6. The manuscript must supply the revised exponential-decay constant or the adjusted activity threshold that guarantees disagreement percolation; without this explicit re-derivation the non-uniqueness claim rests on an unverified transfer.
  2. [§3] §3 (capacity bounds): the upper and lower bounds on capacity are obtained by direct substitution of the triangular-lattice independence number into the formulas of the earlier Z^2 paper. Because the maximal independent-set density on A is 1/3 rather than 1/2, the resulting numerical bounds should be recomputed and compared with the square-lattice values; the current presentation leaves the dependence on the new geometry implicit.
minor comments (2)
  1. [Abstract] The abstract lists the three results but does not indicate the methods (Peierls contour, cluster expansion, or transfer-matrix) used for each; a single sentence identifying the principal technique for each claim would improve readability.
  2. [§2] Notation for the triangular lattice (denoted A) and the activity parameter should be introduced once in §2 and used consistently thereafter; occasional reversion to Z^2 symbols appears in the high-activity section.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our extension of recoverable systems to the triangular lattice. We address each major comment below and will revise the manuscript to incorporate the requested clarifications and explicit calculations.

read point-by-point responses

  1. Referee: [§4] §4 (high-activity non-uniqueness): the contour or cluster-expansion argument is stated to carry over from the square-lattice case, yet the surface-to-volume ratio and the minimal contour length change with coordination number 6. The manuscript must supply the revised exponential-decay constant or the adjusted activity threshold that guarantees disagreement percolation; without this explicit re-derivation the non-uniqueness claim rests on an unverified transfer.

    Authors: We agree that the specific constants require explicit recalculation for the triangular lattice. In the revised version we will supply the full re-derivation of the exponential-decay constant and the adjusted activity threshold, accounting for the changed surface-to-volume ratio and minimal contour length at coordination number 6. This will render the disagreement-percolation argument self-contained and verifiable. revision: yes

  2. Referee: [§3] §3 (capacity bounds): the upper and lower bounds on capacity are obtained by direct substitution of the triangular-lattice independence number into the formulas of the earlier Z^2 paper. Because the maximal independent-set density on A is 1/3 rather than 1/2, the resulting numerical bounds should be recomputed and compared with the square-lattice values; the current presentation leaves the dependence on the new geometry implicit.

    Authors: We accept that the numerical bounds must be recomputed and displayed explicitly. In the revision we will calculate the concrete upper and lower bounds using the independence number 1/3, present the resulting numerical values, and add a direct comparison with the square-lattice bounds to make the geometric dependence transparent. revision: yes

Circularity Check

0 steps flagged

Minor self-citation to prior square-lattice work; central bounds and non-uniqueness claims remain independently derived for the triangular lattice

full rationale

The paper explicitly frames its contributions as an extension of the recoverable-system framework from arXiv:2510.19746, adapting capacity bounds, Gibbs-measure non-uniqueness arguments, and periodic-measure characterizations to the triangular lattice A. No equations or claims reduce by construction to fitted parameters or self-defined quantities within this manuscript; the lattice-specific geometry (degree 6) is handled through new derivations rather than by renaming or importing unverified uniqueness theorems from the authors' prior work. The self-citation supplies the general setup but does not bear the load of the specific results, which are presented as freshly obtained for A.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review supplies no explicit free parameters, axioms, or invented entities; the work relies on standard definitions of hard-core models and Gibbs measures from prior literature.

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Reference graph

Works this paper leans on

37 extracted references · 37 canonical work pages

  1. [1]

    Codes for distributed storage,

    V . Ramkumar, S. Balaji, B. Sasidharan, M. Vajha, M. N. Krishnan, and P. V . Kumar, “Codes for distributed storage,”Foundations and Trends in Communications and Information Theory, vol. 19, pp. 547–813, 2022

    work page 2022

  2. [2]

    Recoverable systems,

    O. Elishco and A. Barg, “Recoverable systems,”IEEE Trans. Inform. Theory, vol. 68, no. 6, pp. 3681–3699, Jun. 2022

    work page 2022

  3. [3]

    Recoverable systems as interaction models: A study by example,

    G. Wang, A. Barg, and N. Kashyap, “Recoverable systems as interaction models: A study by example,” in2025 IEEE International Symposium on Information Theory (ISIT). IEEE, 2025, pp. 1–6

    work page 2025

  4. [4]

    The maximal hard-core model as a recoverable system: Gibbs measures and phase coexistence,

    ——, “The maximal hard-core model as a recoverable system: Gibbs measures and phase coexistence,”arXiv preprint arXiv:2510.19746, 2025

    work page arXiv 2025

  5. [5]

    Counting independent sets up to the tree threshold,

    D. Weitz, “Counting independent sets up to the tree threshold,” in STOC’06: Proceedings of the 38th Annual ACM Symposium on Theory of Computing. ACM, New York, 2006, pp. 140–149

    work page 2006

  6. [6]

    Improved mixing condition on the grid for counting and sampling independent sets,

    R. Restrepo, J. Shin, P. Tetali, E. Vigoda, and L. Yang, “Improved mixing condition on the grid for counting and sampling independent sets,”Probab. Theory Related Fields, vol. 156, no. 1-2, pp. 75–99, 2013

    work page 2013

  7. [7]

    Phase coexistence for the hard-core model onZ 2,

    A. Blanca, Y . Chen, D. Galvin, D. Randall, and P. Tetali, “Phase coexistence for the hard-core model onZ 2,”Combin. Probab. Comput., vol. 28, no. 1, pp. 1–22, 2019

    work page 2019

  8. [8]

    Hard hexagons: exact solution,

    R. J. Baxter, “Hard hexagons: exact solution,”J. Phys. A, vol. 13, no. 3, pp. L61–L70, 1980

    work page 1980

  9. [9]

    High-density hard-core model on triangular and hexagonal lattices,

    A. Mazel, I. Stuhl, and Y . Suhov, “High-density hard-core model on triangular and hexagonal lattices,”Annales Henri Poincaré, vol. 26, pp. 3321–3381, 2025

    work page 2025

  10. [10]

    Bit-patterned magnetic recording: Theory, media fabrication, and recording performance,

    T. R. Albrecht, H. Arora, V . Ayanoor-Vitikkate, J.-M. Beaujour, D. Be- dau, D. Berman, A. L. Bogdanov, Y .-A. Chapuis, J. Cushen, E. E. Dobisz et al., “Bit-patterned magnetic recording: Theory, media fabrication, and recording performance,”IEEE Transactions on Magnetics, vol. 51, no. 5, pp. 1–42, 2015

    work page 2015

  11. [11]

    Stabilization of mag- netic skyrmions on arrays of self-assembled hexagonal nanodomes for magnetic recording applications,

    F. Tejo, D. Toneto, S. Oyarzún, J. Hermosilla, C. S. Danna, J. L. Palma, R. B. da Silva, L. S. Dorneles, and J. C. Denardin, “Stabilization of mag- netic skyrmions on arrays of self-assembled hexagonal nanodomes for magnetic recording applications,”ACS Applied Materials & Interfaces, vol. 12, no. 47, pp. 53 454–53 461, 2020

    work page 2020

  12. [12]

    Friedli and Y

    S. Friedli and Y . Velenik,Statistical Mechanics of Lattice Systems: A Concrete Mathematical Introduction. Cambridge University Press, Cambridge, 2018

    work page 2018

  13. [13]

    Simon,Phase Transitions in the Theory of Lattice Gases

    B. Simon,Phase Transitions in the Theory of Lattice Gases. Cambridge University Press, 2025

    work page 2025

  14. [14]

    On runlength codes,

    E. Zehavi and J. K. Wolf, “On runlength codes,”IEEE Transactions on Information Theory, vol. 34, no. 1, pp. 45–54, 1988

    work page 1988

  15. [15]

    Asymptotics of input-constrained binary sym- metric channel capacity,

    G. Han and B. Marcus, “Asymptotics of input-constrained binary sym- metric channel capacity,”The Annals of Applied Probability, vol. 19, no. 3, pp. 1063 – 1091, 2009

    work page 2009

  16. [16]

    Generalized belief propagation for the noiseless capacity and information rates of run-length limited con- straints,

    G. Sabato and M. Molkaraie, “Generalized belief propagation for the noiseless capacity and information rates of run-length limited con- straints,”IEEE Transactions on Communications, vol. 60, no. 3, pp. 669–675, 2012

    work page 2012

  17. [17]

    Monte Carlo algorithms for the partition function and information rates of two-dimensional channels,

    M. Molkaraie and H.-A. Loeliger, “Monte Carlo algorithms for the partition function and information rates of two-dimensional channels,” IEEE Transactions on Information Theory, vol. 59, no. 1, pp. 495–503, 2013

    work page 2013

  18. [18]

    The feedback capacity of the binary erasure channel with a no-consecutive-ones input constraint,

    O. Sabag, H. H. Permuter, and N. Kashyap, “The feedback capacity of the binary erasure channel with a no-consecutive-ones input constraint,” IEEE Transactions on Information Theory, vol. 62, no. 1, pp. 8–22, 2016

    work page 2016

  19. [19]

    Asymptotics of input-constrained erasure channel capacity,

    Y . Li and G. Han, “Asymptotics of input-constrained erasure channel capacity,”IEEE Transactions on Information Theory, vol. 64, no. 1, pp. 148–162, 2018

    work page 2018

  20. [20]

    Feedback capacity and coding for the BIBO channel with a no-repeated-ones input constraint,

    O. Sabag, H. H. Permuter, and N. Kashyap, “Feedback capacity and coding for the BIBO channel with a no-repeated-ones input constraint,” IEEE Transactions on Information Theory, vol. 64, no. 7, pp. 4940– 4961, 2018

    work page 2018

  21. [21]

    High-density hard-core model onZ 2 and norm equations in ringZ[ 6√−1],

    A. Mazel, I. Stuhl, and Y . Suhov, “High-density hard-core model onZ 2 and norm equations in ringZ[ 6√−1],”arXiv preprint arXiv:1909.11648, 2019

    work page arXiv 1909

  22. [22]

    Statistical mechanics of maximal independent sets,

    L. Dall’Asta, P. Pin, and A. Ramezanpour, “Statistical mechanics of maximal independent sets,”Physical Review E, vol. 80, no. 6, p. 061136, Dec. 2009

    work page 2009

  23. [23]

    Non-uniqueness of measures of maximal entropy for subshifts of finite type,

    R. Burton and J. E. Steif, “Non-uniqueness of measures of maximal entropy for subshifts of finite type,”Ergodic Theory Dynam. Systems, vol. 14, no. 2, pp. 213–235, 1994

    work page 1994

  24. [24]

    Katok and B

    A. Katok and B. Hasselblatt,Introduction to the modern theory of dynamical systems. Cambridge University Press, 1999. 12

    work page 1999

  25. [25]

    On phase transitions for subshifts of finite type,

    O. Häggström, “On phase transitions for subshifts of finite type,”Israel Journal of Mathematics, vol. 94, no. 1, pp. 319–352, 1996

    work page 1996

  26. [26]

    Bounds on the capacity of constrained two-dimensional codes,

    S. Forchhammer and J. Justesen, “Bounds on the capacity of constrained two-dimensional codes,”IEEE Trans. Inform. Theory, vol. 46, no. 7, pp. 2659–2666, 2000

    work page 2000

  27. [27]

    Efficient coding schemes for the hard-square model,

    R. M. Roth, P. H. Siegel, and J. K. Wolf, “Efficient coding schemes for the hard-square model,”IEEE Transactions on Information Theory, vol. 47, no. 3, pp. 1166–1176, 2002

    work page 2002

  28. [28]

    Two-dimensional constrained coding based on tiling,

    A. Sharov and R. M. Roth, “Two-dimensional constrained coding based on tiling,”IEEE transactions on information theory, vol. 56, no. 4, pp. 1800–1807, 2010

    work page 2010

  29. [29]

    The problem of uniqueness of a Gibbsian random field and the problem of phase transitions,

    R. L. Dobrushin, “The problem of uniqueness of a Gibbsian random field and the problem of phase transitions,”Functional analysis and its applications, vol. 2, no. 4, pp. 302–312, 1968

    work page 1968

  30. [30]

    Slow mixing of Markov chains using fault lines and fat contours,

    S. Greenberg and D. Randall, “Slow mixing of Markov chains using fault lines and fat contours,”Algorithmica, vol. 58, no. 4, pp. 911–927, 2010

    work page 2010

  31. [31]

    The number of labeled 2-connected planar graphs,

    E. A. Bender, Z. Gao, and N. C. Wormald, “The number of labeled 2-connected planar graphs,”The Electronic Journal of Combinatorics, pp. R43–R43, 2002

    work page 2002

  32. [32]

    Asymptotic enumeration and limit laws of planar graphs,

    O. Giménez and M. Noy, “Asymptotic enumeration and limit laws of planar graphs,”Journal of the American Mathematical Society, vol. 22, no. 2, pp. 309–329, 2009

    work page 2009

  33. [33]

    Enumeration of labelled 4- regular planar graphs ii: Asymptotics,

    M. Noy, V . Ravelomananana, and J. Rué, “Enumeration of labelled 4- regular planar graphs ii: Asymptotics,”Journal of Combinatorial Theory, Series B, vol. 154, pp. 273–316, 2022

    work page 2022

  34. [34]

    Cubic graphs and related triangulations on orientable surfaces,

    M. Bodirsky, M. Kang, M. Löffler, and C. McDiarmid, “Cubic graphs and related triangulations on orientable surfaces,”The Electronic Jour- nal of Combinatorics, vol. 25, no. 1, p. P1.30, 2018, arXiv preprint 1612.01358 (2016)

    work page arXiv 2018

  35. [35]

    A short course on the Pirogov-Sinai theory,

    M. Zahradník, “A short course on the Pirogov-Sinai theory,”Rendiconti di Matematica, Serie VII, vol. 18, pp. 411–486, 1998

    work page 1998

  36. [36]

    An alternate version of Pirogov-Sinai theory,

    ——, “An alternate version of Pirogov-Sinai theory,”Commun. Math. Phys., vol. 93, pp. 559–581, 1984

    work page 1984

  37. [37]

    The problem of translation invariance of Gibbs states at low temperatures,

    R. Dobrushin and S. Shlosman, “The problem of translation invariance of Gibbs states at low temperatures,”Sov. Sci. Rev. C Math. Phys., vol. 5, pp. 53–196, 1985. 13 (a) 4 neighbors, vertex-connected (b) 4 neighbors, edge-connected (c) 3 neighbors, vertex-connected (d) 3 neighbors, edge-connected (e) 2 neighbors, vertex-connected (f) 2 neighbors, edge-conn...

    work page 1985