On the stochastic independence properties of hard-core distributions

A probability measure p on the set M of matchings in a graph (or, more generally 2-bounded hypergraph) Γ is hard-core if for some λ:Γ → [0, ∞), the probability p(M) of M ∈ M is proportional to Π_A∈M λ(A). We show that such distributions enjoy substantial approximate stochastic independence properties. This is based on showing that, with M chosen according to the hard-core distribution p, MP (Γ) the matching polytope of Γ, and δ > 0, if the vector of marginals, (Pr(A ∈ M): A an edge of Γ), is in (1 - δ)MP (Γ), then the weights λ(A) are bounded by some A(δ). This eventually implies, for example, that under the same assumption, with δ fixed, Pr(A,B∈M)/Pr(A∈M)Pr(B∈M) → 1 as the distance between A, B ∈ Γ tends to infinity. Thought to be of independent interest, our results have already been applied in the resolutions of several questions involving asymptotic behaviour of graphs and hypergraphs (see [14, 16], [11]-[13]).