ar X iv : m at h / 03 02 18 5 v 1 [ m at h . PR ] 1 7 Fe b 20 03 NEGATIVE ASSOCIATION IN UNIFORM FORESTS AND CONNECTED GRAPHS

We consider three probability measures on subsets of edges of a given finite graph G, namely those which govern, respectively, a uniform forest, a uniform spanning tree, and a uniform connected subgraph. A conjecture concerning the negative association of two edges is reviewed for a uniform forest, and a related conjecture is posed for a uniform connected subgraph. The former conjecture is verified numerically for all graphs G having eight or fewer vertices, or having nine vertices and no more than eighteen edges, using a certain computer algorithm which is summarised in this paper. Negative association is known already to be valid for a uniform spanning tree. The three cases of uniform forest, uniform spanning tree, and uniform connected subgraph are special cases of a more general conjecture arising from the random-cluster model of statistical mechanics. Throughout this paper, G = (V, E) denotes a finite labelled graph with vertex set V and edge set E. An edge e with endpoints x, y is written e = x, y. For reasons discussed at the end of Section 1, we shall assume that G has neither loops nor multiple edges. We shall consider three probability measures on the set of subsets of E, and shall discuss certain results and conjectures concerning these measures. Since each such measure is a uniform measure on a given subset of E, each of our conclusions and conjectures may be expressed as a purely combinatorial statement. The three measures are given as follows. Let F be the set of all subsets of E which contain no cycle, noting that the elements of F are exactly the subgraphs of G which are forests. Let C be the set of all subsets C of E such that (V, C) is connected, and let T = F ∩ C be the set of all spanning trees of G. We write F , C, T (respectively) for elements of F , C, T chosen uniformly at random, and we call them a uniform forest, a uniform connected subgraph, and a uniform spanning tree, respectively. The uniform spanning tree has been studied extensively, see [3, 18] for example. The uniform forests of this paper are different from those of [3] in that, in [3], the term uniform spanning forest denotes effectively the probability measure on infinite subsets of a given infinite graph G which is obtained as the infinite-volume weak limit …