Journal of Statistical Physics

, Volume 157, Issue 6, pp 1225–1254 | Cite as

Sufficient Conditions for Uniform Bounds in Abstract Polymer Systems and Explorative Partition Schemes

  • Christoph Temmel


We present several new sufficient conditions for uniform boundedness of the reduced correlations and free energy of an abstract polymer system in a complex multidisc around zero fugacity. They resolve a discrepancy between two incomparable and previously known extensions of Dobrushin’s classic condition. All conditions arise from an extension of the tree-operator approach introduced by Fernández and Procacci combined with a novel family of partition schemes of the spanning subgraph complex of a cluster. The key technique is the increased transfer of structural information from the partition scheme to a tree-operator on an enhanced space.


Cluster expansion Abstract polymer system Partition scheme Tree-operator Hardcore gas 



I am grateful to Roberto Fernández for the time he took to explain me his work and encourage my attempts at extending it. The visits at the University of Utrecht have been supported by the RGLIS short visit Grants 4076 and 4446 from the European Science Foundation (ESF). This work has been also supported by the Austrian Science Fund (FWF), Project W1230-N13. I also want to thank Maria Eichlseder for help with the graphics. I am thankful of the comments of my referees, pushing me towards an improved exposition of the motivation and high-level overviews.


  1. 1.
    Bissacot, R., Fernández, R., Procacci, A.: On the convergence of cluster expansions for polymer gases. J. Stat. Phys. 139(4), 598–617 (2010)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  2. 2.
    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)CrossRefzbMATHGoogle Scholar
  3. 3.
    Dijkstra, E.W.: A note on two problems in connexion with graphs. Numer. Math. 1, 269–271 (1959)CrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    Dobrushin, R. L.: Estimates of semi-invariants for the Ising model at low temperatures. In: Topics in statistical and theoretical physics, vol. 177 of Am. Math. Soc. Transl. Ser. 2, pp. 59–81. Am. Math. Soc., Providence, RI, 1996Google Scholar
  5. 5.
    Erdős, P., Lovász, L.: Problems and results on \(3\)-chromatic hypergraphs and some related questions. In: Infinite and finite sets (Colloq., Keszthely, 1973; dedicated to P. Erdős on his 60th birthday), vol. II, pp. 609–627. Colloquia Mathematica Societatis János Bolyai, vol. 10. North-Holland, Amsterdam, 1975Google Scholar
  6. 6.
    Faris, W.G.: Combinatorics and cluster expansions. Probab. Surv. 7, 157–206 (2010)CrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    Fernández, R., Procacci, A.: Cluster expansion for abstract polymer models. New bounds from an old approach. Commun. Math. Phys. 274(1), 123–140 (2007)ADSCrossRefzbMATHGoogle Scholar
  8. 8.
    Gruber, C., Kunz, H.: General properties of polymer systems. Commun. Math. Phys. 22, 133–161 (1971)ADSCrossRefMathSciNetGoogle Scholar
  9. 9.
    Korte, B., Vygen, J.: Combinatorial optimization: theory and algorithms. In: Algorithms and Combinatorics, vol. 21, 3rd edn. Springer, Berlin (2006)Google Scholar
  10. 10.
    Kotecký, R., Preiss, D.: Cluster expansion for abstract polymer models. Commun. Math. Phys. 103(3), 491–498 (1986)ADSCrossRefzbMATHGoogle Scholar
  11. 11.
    Miracle-Solé, S.: On the theory of cluster expansions. Markov Process. Relat. Fields 16(2), 287–294 (2010)zbMATHGoogle Scholar
  12. 12.
    Penrose, O.: Convergence of fugacity expansions for classical systems. In Bak T.A., (eds.), Statistical Mechanics: Foundations and Applications, p. 101 (1967)Google Scholar
  13. 13.
    Scott, A.D., 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)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  14. 14.
    Shearer, J.B.: On a problem of Spencer. Combinatorica 5(3), 241–245 (1985)CrossRefzbMATHMathSciNetGoogle Scholar
  15. 15.
    Temmel, C.: Properties and applications of Bernoulli random fields with strong dependency graphs. PhD thesis, Institut für Mathematische Strukturtheorie, TU Graz, 2012Google Scholar
  16. 16.
    Ursell, H.D.: The evaluation of Gibbs’ phase-integral for imperfect gases. Math. Proc. Camb. Philos. Soc. 23(06), 685–697 (1927)ADSCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.Department of Mathematics, Faculty of SciencesVU University AmsterdamAmsterdamThe Netherlands

Personalised recommendations