  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 greater or equal to 2 provided that slab percolation
occurs under the averaged measure, which should be the case in the whole
supercritical phase. This work extends the one of Pisztora 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.
