Correlations in random cluster model at $q=1$

Let $μ$ be a measure that samples a subset of a finite ground set, and let Aₑ be the event that element e is sampled. The measure $μ$ is negatively correlated if for any pair of elements $e, f$ one has μ(Aₑ ∩ Af) - μ(Aₑ) μ(Af) ≤ 0. A measure is positively correlated if the direction of the inequality is reversed. For the random cluster model on graphs positive correlation between edges is known for q ≥ 1 due to the FKG inequality, while the negative correlation is only conjectured for 0 ≤ q ≤ 1. The main result of this paper is to give a combinatorial formula for the difference in question at $q=1$. Previously, such a formula was known in the uniform spanning tree case, which is a limit of the random cluster model at $q=0$.