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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

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 

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.

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Copyright information

© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  1. 1.5030 Institut für Mathematische StrukturtheorieTechnische Universität GrazGrazAustria

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