  The interface between the plus and minus phases in the low temperature 3D
Ising model has been intensely studied since Dobrushin's pioneering works in
the early 1970's established its rigidity. Advances in the last decade yielded
the tightness of the maximum of the interface of this Ising model on the
cylinder of side length $n$, around a mean that is asymptotically $c\log n$ for
an explicit $c$ (temperature dependent). In this work, we establish analogous
results for the 3D Potts and random cluster (FK) models. Compared to 3D Ising,
the Potts model and its lack of monotonicity form obstacles for existing
methods, calling for new proof ideas, while its interfaces (and associated
extrema) exhibit richer behavior. We show that the maxima and minima of the
interface bounding the blue component in the 3D Potts interface, and those of
the interface bounding the bottom component in the 3D FK model, are governed by
4 different large deviation rates, whence the corresponding global extrema
feature 4 distinct constants $c$ as above. Due to the above obstacles, our
methods are initially only applicable to 1 of these 4 interface extrema, and
additional ideas are needed to recover the other 3 rates given the behavior of
the first one.
