Morphological transitions in supercritical generalized percolation and moving interfaces in media with frozen randomness

We consider the growth of clusters in disordered media at zero temperature, as exemplified by supercritical generalized percolation and by the random field Ising model. We show that the morphology of such clusters and of their surfaces can be of different types: They can be standard compact clusters with rough or smooth surfaces, but there exists also a completely different “spongy” phase. Clusters in the spongy phase are compact' as far as the size-mass relation M ~ R^D is concerned (with D the space dimension), but have an outer surface (or hull’) whose fractal dimension is also D and which is indeed dense in the interior of the entire cluster. This behavior is found in all dimensions D >= 3. Slightly supercritical clusters can be of either type in $D=3$, while they are always spongy in D >= 4. Possible consequences for the applicability of KPZ (Kardar-Parisi-Zhang) scaling to interfaces in media with frozen randomness are studied in detail.