Correlated Matroids

Semple and Welsh [5] introduced the concept of correlated matroids, which relate to conjectures by Grimmett and Winkler [2], and Pemantle [4], respectively, that the uniformly random forest and the uniformly random connected subgraph of a finite graph have the edge-negative-association property. In this paper, we extend results of Semple and Welsh, and show that the Grimmett and Winkler, and Pemantle conjectures are equivalent to statements about correlated graphic matroids. We also answer some open questions raised in [5] regarding correlated matroids, and in particular show that the 2-sum of correlated matroids is correlated.