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) Gamma is hard-core if for some lambda :Gamma --> [0, infinity), the probability p(M) of M is an element of M is proportional to Pi(A is an element of M) lambda(A). We Show that s uch distributions enjoy substantial approximate stochastic independenc e properties. This is based on showing that, with M chosen according t o the hard-core distribution p, MP(Gamma) the matching polytope of Gam ma, and delta > 0, if the vector of marginals, (Pr(A is an element of M):A an edge of Gamma), is in (1 - delta)MP(Gamma), then the weights l ambda(A) are bounded by some Lambda(delta). This eventually implies, f or example, that under the same assumption, with delta fixed, Pr(A, B is an element of M)/Pr(A is an element of M)Pr(B is an element of m) - -> 1 as the distance between A, B is an element of Gamma tends to infi nity. Thought to be of independent interest, our results have already been applied in the resolutions of several questions involving asympto tic behaviour of graphs and hypergraphs (see [14, 16], [11]-[13]).