COMBINATORICA Bolyai Society - Springer-Verlag COMBINATORICA 17 (3) (1997) 369--391 ON THE STOCHASTIC INDEPENDENCE PROPERTIES OF HARD-CORE DISTRIBUTIONS JEFF KAHN* and P. MARK KAYLLt Received June 4, 1996 A probability measure p on the set ~ of matchings in a graph (or, more generMly 2-bounded hypergraph) F is hard-core if for some A : F -+ [0,cx~), the probability p(M) of M CJA is proportional to YIAEM A(A). We show that such distributions enjoy substantial approximate stochastic independence properties. This is based oi1 showing that, with M chosen according to the hard-core distribution p, MP (F) the matching polytope of F, and 5 > 0, if the vector of marginals, (Pr (A E M): A an edge of F), is in (1 - 5)MP (F), then the weights A(A) are bounded by some A(5). This eventually implies, for example, that under the same assumption, with 5 fixed, Pr(A,BEM) 1 Pr(A6M)Pr(BEM) -'* 1 as the distance between A, B E F tends to infinity. Thought to be of independent interest, our results have already been applied in the resolu- tions of several questions im(olving asymptotic behaviour of graphs and hypergraphs (see [14, 16], [11]-[i31). 1. Introduction This paper is concerned with "hard-core" probability distributions p on the set J~ of matchings in a multigraph or, more generally, in a 2-bounded hypergraph F, and in particular with showing that such distributions exhibit a significant amount of approximate stochastic independence. (See below for omitted notation and terminology.) This phenomenon, mainly as developed in the present work, has proved to be of considerable utility in several applications of the probabilistic method; see [13, 16, 14, 11, 12] for a roughly chronological sequence of such applications. Consequently, the authors believe hard-core distributions have the potential to play a major r61e in the further development of this ubiquitous method (see, e.g., [1, 27]). For p a probability distribution on the set Jb~ of matchings of F, the marginals ofp are the numbers pA=Pr(AEM), for AcF. We sometimes write fp, or simply Mathematics Subject Classification (1991): 05C70, 05C65, 60C05; 52B12, 82B20. * Supported in part by NSF. t This work forms part of the author's doctoral dissertation [16]; see also [17]. The author gratefully acknowledges NSERC for partial support in the form of a 1967 Science and Engineering Scholarship. 0209-9683/97/$6.00 (~)1997 JAnos Bolyai Mathematical Society