On the Ornstein-Zernike behaviour for the supercritical Random-Cluster model on $\mathbb{Z}^{d},d\geq3.$

Abstract

We prove Ornstein-Zernike behaviour in every direction for finite connection functions of the random cluster model on $\mathbb{Z}^{d},d\geq3,$ for $q\geq1,$ when occupation probabilities of the bonds are close to $1.$ Moreover, we prove that equi-decay surfaces are locally analytic, strictly convex, with positive Gaussian curvature.