Asymptotics of even-even correlations in the Ising model

We consider finite-range ferromagnetic Ising models on ${\mathbb{Z}}^d$ in the regime $\beta <{\beta }_c$. We analyze the behavior of the prefactor to the exponential decay of $\mathrm{C}\mathrm{o}\mathrm{v}({\sigma }_A,{\sigma }_B)$, for arbitrary finite sets $A$ and $B$ of even cardinality, as the distance between $A$ and $B$ diverges.