Shearer's measure and stochastic domination of product measures

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 Lovasz Local Lemma, with marginal distribution determined by \vec{p} exists on G iff every Bernoulli random field with the same marginals and dependency graph G dominates stochastically a non-trivial Bernoulli product field. Additionaly we derive a lower non-trivial uniform bound for the parameter vector of the dominated Bernoulli product field. This generalizes previous results by Liggett, Schonmann & Stacey in the homogeneous case, in particular on the k-fuzz of Z. Using the connection between Shearer's measure and lattice gases with hardcore interaction established by Scott & Sokal, we apply bounds derived from cluster expansions of lattice gas partition functions to the stochastic domination problem.