On the stochastic independence properties of hard-core distributions
A probability measure \(p\) on the set \({\mathcal M}\) of matchings in a graph (or, more generally 2-bounded hypergraph) \(\Gamma\) is hard-core if for some \(\lambda :\Gamma\mapsto [0,\infty)\), the probability \(p(M)\) of \(M\in{\mathcal M}\) is proportional to \(\prod _{A\in M}\lambda (A)\). It is shown in this paper that such distributions have substantial approximate stochastic independence properties, e.g. the main theorem is: For every \(\delta \in (0,1)\), \(\eta >0\) and \(\ell\geq 1\), there exist \(\varepsilon >0\) and an integer \(D\) such that the following is true. Let \(\Gamma\) be a multigraph and \({\mathcal M}\) the set of matchings in \(\Gamma\). Let \(p\) be a hard-core distribution on \({\mathcal M}\) with marginals \(f_p\in (1-\delta)\text{MP}(\Gamma)\) (here MP\((\Gamma)\) is the matching polytope of \(\Gamma\)), and let \(\Gamma ' =\{A\in\Gamma :p_A>\varepsilon\}\). If \(F_1,\dots F_\ell\in\Gamma\) are pairwise at distance at least \(D\) in \(\Gamma '\), then Pr\((F_1,\dots F_\ell\in M)=\eta\prod_{i=1}^\ell p_{F_i}\). This result is obtained through a sequence of lemmata which are interesting at their own, too. The obtained results are very useful in the resolution of several questions involving the asymptotic behaviour of graphs and hypergraphs.
scientific article; zbMATH DE number 1226325