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Witness Trees in the Moser–Tardos Algorithmic Lovász Local Lemma and Penrose Trees in the Hard-Core Lattice Gas

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Abstract

We point out a close connection between the Moser–Tardos algorithmic version of the Lovász local lemma, a central tool in probabilistic combinatorics, and the cluster expansion of the hard-core lattice gas in statistical mechanics. We show that the notion of witness trees given by Moser and Tardos is essentially coincident with that of Penrose trees in the Cluster expansion scheme of the hard-core gas. Such an identification implies that the Moser–Tardos algorithm is successful in a polynomial time if the cluster expansion converges.

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Notes

  1. Be careful! Here ’maximal’ means maximal in \(W_i\), so that \(\tau ^{(i)}(s)\) can have vertices with depth greater than those in \(\tilde{W}_i\) as long as the labels in these vertices are all compatible with \(C(i-1)\)

References

  1. Alon, N.: A parallel algorithmic version of the local lemma. Random Struct. Algorithms 2(4), 367–378 (1991)

    Article  MATH  Google Scholar 

  2. Alon, N., Spencer, J.: The Probabilistic Method, 3rd edn. Wiley-Interscience, New York (2008)

    Book  MATH  Google Scholar 

  3. Beck, J.: An algorithmic approach to the Lovász local lemma. Random Struct. Algorithms 2(4), 343–365 (1991)

    Article  MATH  Google Scholar 

  4. Böttcher, J., Kohayakawa, Y., Procacci, A.: Properly coloured copies and rainbow copies of large graphs with small maximum degree. Random Struct. Algorithms 40(4), 425–436 (2012)

    Article  MATH  Google Scholar 

  5. Bissacot, R., Fernández, R., Procacci, A., Scoppola, B.: An improvement of the Lovász local lemma via cluster expansion. Comb. Probab. Comput. 20(5), 709–719 (2011)

    Article  MATH  Google Scholar 

  6. Cammarota, C.: Decay of correlations for infinite range interactions in unbounded spin systems. Comm. Math. Phys. 85, 517–28 (1982)

    Article  ADS  MathSciNet  Google Scholar 

  7. 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, pp. 1–66. Springer-Verlag, Berlin (1996)

  8. Erdös, P.; Lovász, L.: Problems and results on 3-chromatic hypergraphs and some related questions. In Infinite and Finite Sets, vol. 10, pp. 609–627. Colloq. Math. Soc. Janos Bolyai, Amsterdam (1975)

  9. Esperet, L., Parreau, A.: Acyclic edge-coloring using entropy compression. Eur. J. Comb. 34(6), 1019–1027 (2013)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  ADS  MATH  Google Scholar 

  11. Gruber, C.: General properties of polymer systems. Commun. Math. Phys. 22, 133–161 (1971)

    Article  ADS  Google Scholar 

  12. Grytczuk, J., Kozik, K., Micek, P.: New approach to nonrepetitive sequences. Random Struct. Algorithms 42(2), 214–225 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  13. 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. J. Comb. Theory, Series B 103, 21–45 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  14. Kolipaka, K.B.R., Szegedy, M.: Moser and Tardos meet Lovász. In: Proceedings of the 43rd annual ACM symposium on Theory of computing, pp. 235–244. ACM New York, NY (2011)

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

    Article  ADS  MATH  MathSciNet  Google Scholar 

  16. Moser, R.A.: A constructive proof of the Lovász local lemma. In: Proceedings of the 41st Annual ACM Symposium on the Theory of Computing (STOC). ACM, New York (2009)

  17. Moser, R., Tardos, G.: A constructive proof of the general Lovász local Lemma. JACM 57, 11 (2010)

    Article  MathSciNet  Google Scholar 

  18. Ndreca, S., Procacci, A., Scoppola, B.: Improved bounds on coloring of graphs. Eur. J. Comb. 33(4), 592–609 (2012)

    Article  MATH  MathSciNet  Google Scholar 

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

    Google Scholar 

  20. Pegden, W.: An extension of the Moser–Tardos algorithmic local lemma, SIAM J. Discrete Math. (2013). To appear http://arxiv.org/abs/1102.2853

  21. Procacci, A., de Lima, B.N.B., Scoppola, B.: A remark on high temperature polymer expansion for lattice systems with infinite range pair interactions. Lett. Math. Phys. 45, 303–322 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  22. Procacci, A., Scoppola, B.: Polymer gas approach to \(N\)-body lattice systems. J. Statist. Phys. 96, 49–68 (1999)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  23. Ruelle, D.: Statistical Mechanics: Rigorous Results. W. A. Benjamin Inc, New York-Amsterdam (1969)

    MATH  Google Scholar 

  24. Scott, A., Sokal, A.D.: The repulsive lattice gas, the independent-set polynomial, and the Lovász local lemma. J. Stat. Phys. 118(5–6), 1151–1261 (2005)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  25. Scott, A., Sokal, A.D.: On dependency graphs and the lattice gas. Comb. Probab. Comput. 15, 253–279 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  26. Seiler, E.: Gauge Theories as a Problem of Constructive Quantum Field Theory and Statistical Mechanics. Lecture Notes in Physics, vol. 159. Springer-Verlag, Berlin (1982)

  27. Shearer, J.B.: On a problem of spencer. Combinatorica 5, 241–245 (1985)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    MATH  MathSciNet  Google Scholar 

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Acknowledgments

Aldo Procacci has been partially supported by the Brazilian agencies Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq) and Fundação de Amparo à Pesquisa do estado de Minas Gerais (FAPEMIG—Programa de Pesquisador Mineiro).

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Authors and Affiliations

  1. Departamento de Matemática-ICEB, UFOP, Ouro Preto, MG, 35400-000, Brazil

    Rogério Gomes Alves

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

    Aldo Procacci

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  1. Rogério Gomes Alves
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  2. Aldo Procacci
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Correspondence to Aldo Procacci.

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Alves, R.G., Procacci, A. Witness Trees in the Moser–Tardos Algorithmic Lovász Local Lemma and Penrose Trees in the Hard-Core Lattice Gas. J Stat Phys 156, 877–895 (2014). https://doi.org/10.1007/s10955-014-1054-3

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  • DOI: https://doi.org/10.1007/s10955-014-1054-3

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