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arxiv: 2606.29404 · v1 · pith:Q6AHUF2Ynew · submitted 2026-06-28 · 🧮 math-ph · math.MP

A Multi-Body Dobrushin-Sokal Criterion -- Part II

Jan Philipp Neumann This is my paper

Pith reviewed 2026-06-30 01:53 UTC · model grok-4.3

classification 🧮 math-ph math.MP
keywords Mayer cluster expansionlattice gasmulti-body interactionDobrushin-Sokal criterionhypergraph partitionpolymer expansionGruber-Kunz conditionindependence polynomial
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The pith

A partition scheme on spanning hypergraphs supplies a sufficient condition for absolute convergence of Mayer cluster expansions in lattice gases with complex multi-body interactions.

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

The paper establishes a sufficient condition that guarantees absolute convergence of Mayer cluster expansions for both log-partition functions and correlation functions. The condition applies to lattice gases whose interactions may be complex-valued and multi-body in nature. It recovers several classical convergence criteria as special cases while introducing a partition scheme for spanning hypergraphs that accommodates stronger interactions, such as higher-order hard-core repulsion arising in hypergraph independence polynomials. The same scheme combines readily with the Gruber-Kunz condition, yielding extended convergence domains for the associated polymer expansions.

Core claim

We prove a sufficient condition for the absolute convergence of Mayer cluster expansions of log-partition and correlation functions applicable to lattice gases with possibly complex-valued multi-body interactions. Not only are several classical results subsumed but a partition scheme for spanning hypergraphs also makes our methods well-suited for treating stronger multi-body interactions, including higher-order hard-core repulsion in the context of hypergraph independence polynomials. Furthermore, our approach is easily combined with the Gruber--Kunz condition to produce extended convergence results for the polymer expansion of lattice gases, rivalling those obtained not too long ago by Nguy

What carries the argument

The multi-body Dobrushin-Sokal criterion, which relies on a partition of spanning subhypergraphs to produce the contraction or bound required for absolute convergence.

If this is right

  • Absolute convergence of the Mayer expansions holds for a larger class of possibly complex multi-body interactions than covered by earlier criteria.
  • The criterion directly applies to the independence polynomials of hypergraphs that encode higher-order hard-core repulsions.
  • Combining the criterion with the Gruber-Kunz condition produces polymer-expansion convergence domains that match or exceed recent results of Nguyen and Fernández.
  • Several classical Dobrushin-type and Sokal-type bounds are recovered as special cases of the hypergraph partition scheme.

Where Pith is reading between the lines

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

  • The hypergraph formulation may allow direct transfer of convergence results between combinatorial enumeration problems and statistical-mechanical models.
  • Models with genuine three- or four-body forces, previously excluded by pairwise-interaction assumptions, become amenable to rigorous expansion techniques.
  • The partition scheme could be tested numerically on small hypergraphs to locate the precise boundary between convergent and divergent regimes.

Load-bearing premise

The multi-body interactions admit a hypergraph representation in which the spanning subhypergraphs can be partitioned so as to deliver the needed contraction bound.

What would settle it

A concrete lattice-gas model whose interactions satisfy the stated hypergraph-partition condition yet whose Mayer series diverges for some complex activity values would falsify the criterion.

read the original abstract

We prove a sufficient condition for the absolute convergence of Mayer cluster expansions of log-partition and correlation functions applicable to lattice gases with possibly complex-valued multi-body interactions. Not only are several classical results subsumed but a partition scheme for spanning hypergraphs also makes our methods well-suited for treating stronger multi-body interactions, including higher-order hard-core repulsion in the context of hypergraph independence polynomials. Furthermore, our approach is easily combined with the Gruber--Kunz condition to produce extended convergence results for the polymer expansion of lattice gases, rivalling those obtained not too long ago by Nguyen and Fern\'andez (2024).

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

1 major / 0 minor

Summary. The manuscript proves a sufficient condition for the absolute convergence of Mayer cluster expansions of the log-partition function and correlation functions for lattice gases with possibly complex-valued multi-body interactions. The proof employs a partition scheme for spanning hypergraphs to obtain the required contraction estimates. This subsumes several classical results, extends applicability to stronger interactions including higher-order hard-core repulsion in hypergraph independence polynomials, and combines with the Gruber-Kunz condition to yield extended convergence results for polymer expansions comparable to those of Nguyen and Fernández (2024).

Significance. If the derivation of the contraction bound holds, the result meaningfully extends Dobrushin-Sokal-type criteria to the multi-body setting with complex weights. The hypergraph partition scheme provides a concrete tool for handling higher-arity terms that classical single-body criteria cannot address directly, and the combination with Gruber-Kunz enlarges the domain of guaranteed convergence for polymer expansions. These features would strengthen the toolkit for rigorous analysis of lattice gases and related combinatorial models.

major comments (1)
  1. [Partition scheme for spanning hypergraphs (abstract and §3)] The hypergraph partition scheme (central to the sufficient condition stated in the abstract): the manuscript must explicitly verify that every admissible partition of spanning subhypergraphs produces a uniform bound on the absolute values of the cluster coefficients that remains strictly smaller than the reciprocal of the maximum weighted degree (or its multi-body analogue) once complex phases are included. If overlapping higher-arity configurations are omitted or the bound fails to stay below 1, the absolute-convergence guarantee does not follow. This step is load-bearing for the main claim.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the positive assessment and for highlighting the load-bearing step in the hypergraph partition argument. We address the single major comment below.

read point-by-point responses

  1. Referee: [Partition scheme for spanning hypergraphs (abstract and §3)] The hypergraph partition scheme (central to the sufficient condition stated in the abstract): the manuscript must explicitly verify that every admissible partition of spanning subhypergraphs produces a uniform bound on the absolute values of the cluster coefficients that remains strictly smaller than the reciprocal of the maximum weighted degree (or its multi-body analogue) once complex phases are included. If overlapping higher-arity configurations are omitted or the bound fails to stay below 1, the absolute-convergence guarantee does not follow. This step is load-bearing for the main claim.

    Authors: We agree that an explicit verification for every admissible partition, including the effect of complex phases, is required. In the proof of the main contraction estimate (Theorem 3.2 and the surrounding lemmas in §3), we enumerate all admissible partitions of each spanning subhypergraph exactly as defined in Definition 3.1. For each such partition the absolute value of the corresponding term is bounded by replacing every complex weight w_e by |w_e| and applying the triangle inequality; the resulting sum is then majorized by a product over vertices that reproduces the classical Dobrushin-Sokal factor. Because the enumeration is exhaustive, no overlapping higher-arity configurations are omitted. The bound is shown to be strictly smaller than the reciprocal of the multi-body weighted degree whenever the hypothesis of the theorem holds, independently of the phases. We will insert one additional sentence immediately after the statement of the partition bound to make this uniform control explicit. revision: yes

Circularity Check

0 steps flagged

Direct mathematical proof with no load-bearing self-reference or definitional reduction

full rationale

The manuscript states a direct proof of a sufficient condition for absolute convergence of Mayer cluster expansions via a hypergraph partition scheme for spanning subhypergraphs. No equations or steps are presented that define a radius or bound in terms of a fitted parameter later called a prediction, nor does any central claim reduce to a self-citation whose content is itself unverified within the paper. The abstract explicitly frames the result as a proof that subsumes classical results and extends them, with the partition scheme introduced as a methodological tool rather than an ansatz smuggled from prior work by the same authors. The derivation chain is therefore self-contained against external benchmarks and receives the default non-circularity finding.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review provides no explicit list of free parameters, axioms, or invented entities; the central claim rests on standard mathematical assumptions of lattice gas models and hypergraph theory that are not detailed here.

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discussion (0)

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Reference graph

Works this paper leans on

29 extracted references · 24 canonical work pages

  1. [1]

    Random Struct

    Bencs, F., Buys, P.: Optimal zero-free regions for the independence polynomial of bounded degree hypergraphs. Random Struct. & Algorithms66(4), e70018 (2025). https://doi.org/10.1002/rsa.70018

    work page doi:10.1002/rsa.70018 2025

  2. [2]

    Bissacot, R., Fern´ andez, R., Procacci, A.: On the convergence of cluster expansions for polymer gases. J. Stat. Phys.139(4), 598–617 (2010).https://doi.org/10.1007/ s10955-010-9956-1

    2010

  3. [3]

    In: Dobrushin, R.L., Minlos, R.A., Shubin, M.A., Vershik, A.M., (eds.): Topics in Statistical and Theoretical Physics – F

    Dobrushin, R.L.: Estimates of semi-invariants for the Ising model at low tempera- tures. In: Dobrushin, R.L., Minlos, R.A., Shubin, M.A., Vershik, A.M., (eds.): Topics in Statistical and Theoretical Physics – F. A. Berezin Memorial Volume. Am. Math. Soc. Transl., Ser. 2,177, 59–81 (1996).https://doi.org/10.1090/trans2/177/05

    work page doi:10.1090/trans2/177/05 1996

  4. [4]

    In: Bernard, P

    Dobrushin, R.L.: Perturbation methods of the theory of Gibbsian fields. In: Bernard, P. (ed.): Lectures on Probability Theory and Statistics. Lecture Notes in Mathematics, vol.1648, Ecole d’ ´Et´ e de Probabilit´ es de Saint-Flour XXIV - 1994, 1-66. Springer, Berlin, Heidelberg (1996).https://doi.org/10.1007/BFb0095674

    work page doi:10.1007/bfb0095674 1994

  5. [5]

    Faris, W.G.: A connected graph identity and convergence of cluster expansions. J. Math. Phys.49(11), 113302 (2008).https://doi.org/10.1063/1.2976217

    work page doi:10.1063/1.2976217 2008

  6. [6]

    Fern´ andez, R., Nguyen, T.X.: High-temperature cluster expansion for classical and quantum spin lattice systems with multi-body interactions. J. Stat. Phys.191(2), 13 (2024).https://doi.org/10.1007/s10955-024-03231-w

    work page doi:10.1007/s10955-024-03231-w 2024

  7. [7]

    New bounds from an old approach

    Fern´ andez, R., Procacci, A.: Cluster expansion for abstract polymer models. New bounds from an old approach. Commun. Math. Phys.274(1), 123–140 (2007).https: //doi.org/10.1007/s00220-007-0279-2

    work page doi:10.1007/s00220-007-0279-2 2007

  8. [8]

    Gallavotti, G., Miracle-Sol´ e, S.: Correlation functions of a lattice system. Commun. Math. Phys.7(4), 274–288 (1968).https://doi.org/10.1007/BF01646661

    work page doi:10.1007/bf01646661 1968

  9. [9]

    Gallavotti, G., Miracle-Sol´ e, S., Robinson, D.W.: Analyticity properties of a lattice gas. Phys. Lett. A,25(7), 493-494 (1967).https://doi.org/10.1016/0375-9601(67) 90004-7

    work page doi:10.1016/0375-9601(67 1967

  10. [10]

    Comb., Probab

    Galvin, D., McKinley, G., Perkins, W., Sarantis, M., Tetali, P.: On the zeroes of hypergraph independence polynomials. Comb., Probab. and Comput.33(1), 65-84 (2024).https://doi.org/10.1017/S0963548323000330

    work page doi:10.1017/s0963548323000330 2024

  11. [11]

    Gruber, C., Kunz, H.: General properties of polymer systems. Commun. Math. Phys. 22(2), 133–161 (1971).https://doi.org/10.1007/BF01651334

    work page doi:10.1007/bf01651334 1971

  12. [12]

    Preprint (2025).https://doi.org/10.48550/arXiv.2509.04287

    Helmuth, T., Pappik, M., Perkins, W.: Simple analyticity criteria for repulsive multi- body potentials. Preprint (2025).https://doi.org/10.48550/arXiv.2509.04287

    work page doi:10.48550/arxiv.2509.04287 2025

  13. [13]

    Jansen, S.: Cluster expansions for Gibbs point processes. Adv. Appl. Probab.51(4), 1129–1178 (2019).https://doi.org/10.1017/apr.2019.46

    work page doi:10.1017/apr.2019.46 2019

  14. [14]

    Jansen, S.: Thermodynamics of a hierarchical mixture of cubes. J. Stat. Phys.179(2), 309–340 (2020).https://doi.org/10.1007/s10955-020-02531-1

    work page doi:10.1007/s10955-020-02531-1 2020

  15. [15]

    Jansen, S., Kolesnikov, L.: Cluster expansions: Necessary and sufficient conver- gence conditions. J. Stat. Phys.189(3), 33 (2022).https://doi.org/10.1007/ s10955-022-02992-6

    2022

  16. [16]

    Jansen, S., Neumann, J.P.: Hierarchical cubes: Gibbs measures and decay of correlations. J. Stat. Phys.191(12), 161 (2024).https://doi.org/10.1007/ s10955-024-03375-9 A MULTI-BODY DOBRUSHIN-SOKAL CRITERION – PART II 29

    2024

  17. [17]

    Jansen, S., Tsagkarogiannis, D.: Cluster expansions with renormalized activities and applications to colloids. Ann. Henri Poincar´ e21(1), 45–79 (2020).https://doi.org/ 10.1007/s00023-019-00868-2

    work page doi:10.1007/s00023-019-00868-2 2020

  18. [18]

    Koteck´ y, R., Preiss, D.: Cluster expansion for abstract polymer models. Commun. Math. Phys.103(3), 491–498 (1986).https://doi.org/10.1007/BF01211762

    work page doi:10.1007/bf01211762 1986

  19. [19]

    Moraal, H.: The Kirkwood–Salsburg equation and the virial expansion for many- body potentials. Phys. Lett. A59(1), 9-10 (1976).https://doi.org/10.1016/ 0375-9601(76)90334-0

    1976

  20. [20]

    Preprint (2025)

    Neumann, J.P.: A multi-body Dobrushin–Sokal criterion – part i. Preprint (2025). https://doi.org/10.48550/arXiv.2508.12078

    work page doi:10.48550/arxiv.2508.12078 2025

  21. [21]

    Poghosyan, S., Ueltschi, D.: Abstract cluster expansion with applications to statisti- cal mechanical systems. J. Math. Phys.50(5), 053509 (2009).https://doi.org/10. 1063/1.3124770

    2009

  22. [22]

    Procacci, A., Scoppola, B.: Polymer gas approach ton-body lattice systems. J. Stat. Phys.96(1-2), 49–68 (1999).https://doi.org/10.1023/A:1004564214528

    work page doi:10.1023/a:1004564214528 1999

  23. [23]

    Procacci, A., Scoppola, B.: The gas phase of continuous systems of hard spheres interacting vian-body potential. Comm. Math. Phys.211(2), 487–496 (2000).https: //doi.org/10.1007/s002200050823

    work page doi:10.1007/s002200050823 2000

  24. [24]

    Procacci, A., Yuhjtman, S.A.: Convergence of Mayer and virial expansions and the Penrose tree-graph identity. Lett. Math. Phys.107(1), 31–46 (2017).https://doi. org/10.1007/s11005-016-0918-7

    work page doi:10.1007/s11005-016-0918-7 2017

  25. [25]

    World Scientific, (1999).https: //doi.org/10.1142/4090

    Ruelle, D.: Statistical Mechanics: Rigorous Results. World Scientific, (1999).https: //doi.org/10.1142/4090

    work page doi:10.1142/4090 1999

  26. [26]

    Scott, A.D., Sokal, A.D.: The repulsive lattice gas, the independent-set polynomial, and the Lov´ asz local lemma. J. Stat. Phys.118(5-6), 1151–1261 (2005).https:// doi.org/10.1007/s10955-004-2055-4

    work page doi:10.1007/s10955-004-2055-4 2005

  27. [27]

    Skrypnyk, V.I.: Solutions of the Kirkwood–Salsburg equation for particles with finite- range nonpairwise repulsion. Ukr. Math. J.60(8), 1329–1334 (2008).https://doi. org/10.1007/s11253-009-0122-3

    work page doi:10.1007/s11253-009-0122-3 2008

  28. [28]

    Sokal, A.D.: Bounds on the complex zeros of (di)chromatic polynomials and Potts- model partition functions. Comb. Probab. Comput.10(1), 41-77 (2001).https:// doi.org/10.1017/S0963548300004612

    work page doi:10.1017/s0963548300004612 2001

  29. [29]

    Ueltschi, D.: Cluster expansions and correlation functions. Mosc. Math. J.4(2), 511- 522 (2004).https://doi.org/10.17323/1609-4514-2004-4-2-511-522

    work page doi:10.17323/1609-4514-2004-4-2-511-522 2004