Negative edge correlation for weighted spanning forests of complete graphs

Let $\mathbb{P}_t$ be the arboreal gas on the complete graph $K_n$: the probability measure on acyclic edge sets $F$ proportional to $t^{|F|}$. We prove that for every $n$ and every $t>0$ any two distinct edges $e,f$ satisfy $\mathbb{P}_t(e,f\in F)<\mathbb{P}_t(e\in F)\,\mathbb{P}_t(f\in F)$. The case $t=1$ is due to Gupta. For two edges sharing a vertex the inequality follows from the real-rootedness of the degree polynomial of a vertex, which is the characteristic polynomial of a rank-one perturbation of a Hermite Jacobi matrix. For disjoint edges the edge marginal satisfies an explicit Riccati equation, and the covariance is controlled by a barrier argument along it.