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On the Convergence of Cluster Expansions for Polymer Gases

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Abstract

We compare the different convergence criteria available for cluster expansions of polymer gases subjected to hard-core exclusions, with emphasis on polymers defined as finite subsets of a countable set (e.g. contour expansions and more generally high- and low-temperature expansions). In order of increasing strength, these criteria are: (i) Dobrushin criterion, obtained by a simple inductive argument; (ii) Gruber-Kunz criterion obtained through the use of Kirkwood-Salzburg equations, and (iii) a criterion obtained by two of us via a direct combinatorial handling of the terms of the expansion. We show that for subset polymers our sharper criterion can be proven both by a suitable adaptation of Dobrushin inductive argument and by an alternative—in fact, more elementary—handling of the Kirkwood-Salzburg equations. In addition we show that for general abstract polymers this alternative treatment leads to the same convergence region as the inductive Dobrushin argument and, furthermore, to a systematic way to improve bounds on correlations.

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References

  1. Bissacot, R.: Técnicas para convergência da Expansão do Gás de Polímeros e uma aplicação ao Método Probabilístico. Thesis. Universidade Federal de Minas Gerais (2009)

  2. Bissacot, R., Fernández, R., Procacci, A., Scoppola, B.: An improvement of the Lovász local lemma via cluster expansion. Preprint, arXiv:0910.1824 [math.CO]

  3. Bovier, A., Zahradnik, M.: A simple inductive approach to the problem of convergence of cluster expansions of polymer models. J. Stat. Phys. 100, 765–778 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  4. Brydges, D.C.: A short cluster in cluster expansions. In: Osterwalder, K., Stora, R. (eds.) Critical Phenomena, Random Systems, Gauge Theories, pp. 129–183. Elsevier, Amsterdam (1984)

    Google Scholar 

  5. Brydges, D.C., Martin, P.A.: Coulomb systems at low density: a review. J. Stat. Phys. 96, 1163–1330 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  6. Cammarota, C.: Decay of correlations for infinite range interactions in unbounded spin systems. Commun. Math. Phys. 85, 517–528 (1982)

    Article  MathSciNet  ADS  Google Scholar 

  7. Dobrushin, R.L.: Estimates of semiinvariants for the Ising model at low temperatures. Top. Stat. Theor. Phys., Am. Math. Soc. Transl. 177(2), 59–81 (1996)

    MathSciNet  Google Scholar 

  8. Dobrushin, R.L.: Perturbation methods of the theory of Gibbsian fields. In: Ecole d’Eté de Probabilités de Saint-Flour XXIV—1994. Lecture Notes in Mathematics, vol. 1648, pp. 1–66. Springer, Berlin/Heidelberg/New York (1996)

    Google Scholar 

  9. Fernandez, R., Procacci, A.: Cluster expansion for abstract polymer models. New bounds from an old approach. Commun. Math. Phys. 274(1), 123–140 (2007)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  10. Fernandez, R., Procacci, A.: Regions without complex zeros for chromatic polynomials on graphs with bounded degree. Comb. Probab. Comput. 17, 225–238 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  11. Fernandez, R., Procacci, A., Scoppola, B.: The analyticity region of the hard sphere gas. Improved bounds. J. Stat. Phys. 128(5), 1139–1143 (2007)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  12. Jackson, B., Procacci, A., Sokal, A.D.: Complex zero-free regions at large |q| for multivariate Tutte polynomials (alias Potts-model partition functions) with general complex edge weights. Preprint arXiv:0810.4703v2 [math.CO]

  13. Gruber, C., Kunz, H.: General properties of polymer systems. Commun. Math. Phys. 22, 133–61 (1971)

    Article  MathSciNet  ADS  Google Scholar 

  14. Kotecký, R., Preiss, D.: Cluster expansion for abstract polymer models. Commun. Math. Phys. 103, 491–498 (1986)

    Article  MATH  ADS  Google Scholar 

  15. Miracle-Solé, S.: On the convergence of cluster expansions. Physica A 279, 244–249 (2000)

    Article  MathSciNet  ADS  Google Scholar 

  16. Nardi, F.R., Olivieri, E., Zahradnik, M.: On the Ising model with strongly anisotropic external field. J. Stat. Phys. 97, 87–144 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  17. Penrose, O.: Convergence of fugacity expansions for classical systems. In: Bak, A. (ed.) Statistical Mechanics: Foundations and Applications. Benjamin, New York (1967)

    Google Scholar 

  18. Pfister, C.-E.: Large deviation and phase separation in the two-dimensional Ising model. Helv. Phys. Acta 64, 953–1054 (1991)

    MathSciNet  Google Scholar 

  19. Ruelle, D.: Statistical Mechanics: Rigorous Results. Benjamin, New York/Amsterdam (1969)

    MATH  Google Scholar 

  20. Scott, A., Sokal, A.D.: The repulsive lattice gas, the independent-set polynomial, and the Lovász local lemma. J. Stat. Phys. 118, 1151–261 (2005)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  21. Sinai, Y.G.: Theory of Phase Transitions: Rigorous Results. Pergamon, Oxford (1982)

    MATH  Google Scholar 

  22. Sokal, A.D.: Bounds on the complex zeros of (di)chromatic polynomials and Potts-model partition functions. Comb. Probab. Comput. 10, 41–77 (2001)

    MATH  MathSciNet  Google Scholar 

  23. Ueltschi, D.: Cluster expansions and correlation functions. Mosc. Math. J. 4, 511–522 (2004) and p. 536

    MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Departamento de Matemática-ICEx, UFMG, CP 702, Belo Horizonte, MG, 30161-970, Brazil

    Rodrigo Bissacot & Aldo Procacci

  2. Laboratoire de Maths Raphael Salem, Université de Rouen, Rouen, 76801, France

    Rodrigo Bissacot & Roberto Fernández

  3. Department of Mathematics, Utrecht University, P.O. Box 80010, 3508 TA, Utrecht, The Netherlands

    Roberto Fernández

Authors
  1. Rodrigo Bissacot
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  2. Roberto Fernández
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  3. Aldo Procacci
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Correspondence to Aldo Procacci.

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Bissacot, R., Fernández, R. & Procacci, A. On the Convergence of Cluster Expansions for Polymer Gases. J Stat Phys 139, 598–617 (2010). https://doi.org/10.1007/s10955-010-9956-1

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