Shearer's measure and stochastic domination of product measures

TL;DR

This paper explores the conditions under which Shearer's measure exists on a graph and how it relates to stochastic domination of Bernoulli fields, extending previous homogeneous results and applying lattice gas models.

Contribution

It establishes a new equivalence between Shearer's measure existence and stochastic domination, generalizes prior homogeneous results, and connects these concepts with lattice gas models.

Findings

Shearer's measure exists iff certain stochastic domination conditions are met.

Derived a uniform lower bound for the parameters of dominated Bernoulli fields.

Connected Shearer's measure with lattice gas models to apply cluster expansion bounds.

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.