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Journal of Theoretical Probability

, Volume 27, Issue 1, pp 22–40 | Cite as

Shearer’s Measure and Stochastic Domination of Product Measures

  • Christoph Temmel
Article
First Online:

Abstract

Let G=(V,E) be a locally finite graph. Let \(\vec{p}\in[0,1]^{V}\). We show that Shearer’s measure, introduced in the context of the Lovász Local Lemma, with marginal distribution determined by \(\vec{p}\), exists on G if and only if every Bernoulli random field with the same marginals and dependency graph G dominates stochastically a non-trivial Bernoulli product field. Additionally, we derive a non-trivial uniform lower bound for the parameter vector of the dominated Bernoulli product field. This generalises previous results by Liggett, Schonmann, and Stacey in the homogeneous case, in particular on the k-fuzz of ℤ. Using the connection between Shearer’s measure and a hardcore lattice gas established by Scott and Sokal, we transfer bounds derived from cluster expansions of lattice gas partition functions to the stochastic domination problem.

Keywords

Stochastic domination Lovász Local Lemma Product measure Bernoulli random field Stochastic order Hardcore lattice gas 

Mathematics Subject Classification

60E15 60G60 82B20 05D40 
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Notes

Acknowledgements

I want to thank Yuval Peres and Rick Durrett for pointing out [15] to me and Pierre Mathieu for listening patiently to my numerous attempts at understanding and solving this problem. This work has been partly done during a series of stays at the LATP, Aix-Marseille Université, financially supported by grants A3-16.M-93/2009-1 and A3-16.M-93/2009-2 from the Land Steiermark and by the Austrian Science Fund (FWF), project W1230-N13. I am also indebted to the anonymous referees for their constructive comments.

References

  1. 1.
    Andjel, E.D.: Characteristic exponents for two-dimensional bootstrap percolation. Ann. Probab. 21(2), 926–935 (1993) CrossRefzbMATHMathSciNetGoogle Scholar
  2. 2.
    Antal, P., Pisztora, A.: On the chemical distance for supercritical Bernoulli percolation. Ann. Probab. 24(2), 1036–1048 (1996) CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    Billingsley, P.: Probability and Measure, 3rd edn. Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics. Wiley, New York (1995) zbMATHGoogle Scholar
  4. 4.
    Bissacot, R., Fernández, R., Procacci, A.: On the convergence of cluster expansions for polymer gases. J. Stat. Phys. 139(4), 598–617 (2010) CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    Bissacot, R., Fernández, R., Procacci, A., Scoppola, B.: An improvement of the Lovász Local Lemma via cluster expansion. Comb. Probab. Comput. (2011). doi: 10.1017/S0963548311000253 zbMATHGoogle Scholar
  6. 6.
    Bollobás, B., Riordan, O.: Percolation. Cambridge University Press, Cambridge (2006) CrossRefzbMATHGoogle Scholar
  7. 7.
    Dobrushin, R.L.: Perturbation methods of the theory of Gibbsian fields. In: Lectures on Probability Theory and Statistics, Saint-Flour, 1994. Lecture Notes in Math., vol. 1648, pp. 1–66. Springer, Berlin (1996) CrossRefGoogle Scholar
  8. 8.
    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. Colloquia Mathematica Societatis János Bolyai, vol. 10, pp. 609–627. North-Holland, Amsterdam (1975) Google Scholar
  9. 9.
    Fernández, R., Procacci, A.: Cluster expansion for abstract polymer models. New bounds from an old approach. Commun. Math. Phys. 274(1), 123–140 (2007) CrossRefzbMATHGoogle Scholar
  10. 10.
    Fisher, D.C., Solow, A.E.: Dependence polynomials. Discrete Math. 82(3), 251–258 (1990) CrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    Grimmett, G.: Percolation, 2nd edn. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 321. Springer, Berlin (1999) CrossRefzbMATHGoogle Scholar
  12. 12.
    Gruber, C., Kunz, H.: General properties of polymer systems. Commun. Math. Phys. 22, 133–161 (1971) CrossRefMathSciNetGoogle Scholar
  13. 13.
    Hoede, C., Li, X.L.: Clique polynomials and independent set polynomials of graphs. Discrete Math. 125(1–3), 219–228 (1994). 13th British Combinatorial Conference (Guildford, 1991) CrossRefzbMATHMathSciNetGoogle Scholar
  14. 14.
    Liggett, T.M.: Interacting Particle Systems. Classics in Mathematics. Springer, Berlin (2005). Reprint of the 1985 original zbMATHGoogle Scholar
  15. 15.
    Liggett, T.M., Schonmann, R.H., Stacey, A.M.: Domination by product measures. Ann. Probab. 25(1), 71–95 (1997) CrossRefzbMATHMathSciNetGoogle Scholar
  16. 16.
    Mathieu, P., Temmel, C.: K-independent percolation on trees. Stoch. Process. Appl. (2012). doi: 10.1016/j.spa.2011.10.014
  17. 17.
    Russo, L.: An approximate zero–one law. Z. Wahrscheinlichkeitstheor. Verw. Geb. 61(1), 129–139 (1982) CrossRefzbMATHGoogle Scholar
  18. 18.
    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) CrossRefzbMATHMathSciNetGoogle Scholar
  19. 19.
    Shearer, J.B.: On a problem of Spencer. Combinatorica 5(3), 241–245 (1985) CrossRefzbMATHMathSciNetGoogle Scholar
  20. 20.
    Strassen, V.: The existence of probability measures with given marginals. Ann. Math. Stat. 36, 423–439 (1965) CrossRefzbMATHMathSciNetGoogle Scholar

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© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  1. 1.5030 Institut für Mathematische StrukturtheorieTechnische Universität GrazGrazAustria

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