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 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.