Negative correlation for the random-cluster model below one on high-degree regular graphs

The random-cluster model with cluster parameter \(0<Q<1\) is conjectured to exhibit negative dependence, but even pairwise negative correlation between distinct edges remains open on general finite graphs. We first show that restricting the problem to regular graphs of diverging degree does not essentially weaken the pairwise conjecture: validity for all such graph sequences, even when restricted to edge pairs at distance \(o(d/\log d)\), is equivalent to validity on arbitrary finite graphs. We then prove strict pairwise negative correlation for a broad class of high-degree regular graph sequences satisfying a two-scale edge-isoperimetric condition. More precisely, for every fixed \(p\in(0,1)\), all sufficiently large graphs in the sequence have strictly negative covariance between any two distinct edge indicators at distance \(o(d/\log d)\), uniformly in \(Q\in[0,1)\) and in the choice of the two edges. In particular, the result includes the \(Q=0\) endpoint, corresponding to Bernoulli bond percolation conditioned to be connected. The isoperimetric hypothesis is satisfied by high-degree expander families and, with probability tending to one, by uniformly random regular graphs of diverging degree; its two-scale form also allows product geometries such as hypercubes and fixed-side high-dimensional discrete tori. The proof is based on a polymer representation and cluster expansion, together with a geometric identification of the leading contribution to the two-edge correlation.