A coarse graining for the Fortuin–Kasteleyn measure in random media

By means of a multi-scale analysis we describe the typical geometrical structure of the clusters under the FK measure in random media. Our result holds in any dimension d ⩾ 2 provided that slab percolation occurs under the averaged measure, which should be the case for the whole supercritical phase. This work extends that of Pisztora [A. Pisztora, Surface order large deviations for Ising, Potts and percolation models, Probab. Theory Related Fields 104 (4) (1996) 427–466] and provides an essential tool for the analysis of the supercritical regime in disordered FK models and in the corresponding disordered Ising and Potts models.